Nonlinear Axiomatic Attribution for Cooperative Games

# Nonlinear Axiomatic Attribution for Cooperative Games
Source: [https://arxiv.org/html/2607.09869](https://arxiv.org/html/2607.09869)
Zhuanghua LiuDepartment of Computer Science National University of Singapore Republic of SingaporeYaoliang YuSchool of Computer Science University of Waterloo CanadaVector Institute CanadaBryan Kian Hsiang LowDepartment of Computer Science National University of Singapore Republic of Singapore

###### Abstract

The Shapley value is a widely used concept in attribution problems, as it uniquely satisfies the axioms of linearity, consistency, equal treatment, and efficiency\. Often, the inclusion AUC metric is used to evaluate the quality of player rankings, in order to identify positively participating players\. However, it can be established that the Shapley value is not always reliable for this purpose\. The core issue lies in its linearity: the Shapley value acts as a linear operator with an excessively large null space, which is likely to contain non\-negligible perturbations that remain indistinguishable to the operator\. To address this limitation, we explore the design of nonlinear axiomatic attribution methods\. Inspired by the least core, which is a popular nonlinear substitute for the Shapley value, we introduce a class of nonlinear attribution methods that retain the remaining necessary axioms\. Each method yields a contribution vector that is the unique optimal solution to a minimization problem, which aims to approximate utility functions as faithfully as possible\. In terms of the inclusion AUC metric, our experiments demonstrate the potential effectiveness of these methods compared to Shapley value variants that relax only the efficiency axiom\. Our code is available at[https://github\.com/watml/nonlinear\-axiom](https://github.com/watml/nonlinear-axiom)\.

## 1Introduction

Originally, the concept of the Shapley value was introduced byshapley1953valueto define a fair allocation of contributions amongn\\mathchar 29038\\relaxplayers who participate in a cooperative game, represented by a utility functionU:2\[n\]→ℝ\\mathchar 29013\\relax\\mathchar 24634\\relax\\mathchar 28722\\relax^\{\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 12833\\relax\\mathbb\{\\mathchar 29010\\relax\}, where\[n\]≔\{1,2,…,n\}\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\coloneq\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\dots\\mathchar 24891\\relax\\mathchar 29038\\relax\\\}\. It is considered fair because it uniquely satisfies the axioms of linearity, consistency, equal treatment, and efficiency\. The effectiveness of the Shapley value and its variants obtained by relaxing the efficiency axiom has been demonstrated byjia2019towards;kwon2022beta;li2023robust;wang2023datain data attribution, where data are treated as players andU​\(S\)\\mathchar 29013\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785is defined as the performance of a model trained on the subsetS\\mathchar 29011\\relaxof available training data\. One such variant, namely the Banzhaf value\(banzhaf1965weighted\), was used in context attribution bycohen2024contextcite, whereU​\(S\)\\mathchar 29013\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785represents the probability of a statement generated by a large language model when only the parts of the context text prescribed byS\\mathchar 29011\\relaxare used\. Its popularity has also been prominently observed in feature attribution, whereU​\(S\)\\mathchar 29013\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785is the predicted value of the model when features outside the subsetS\\mathchar 29011\\relaxare considered missing\(e\.g\.,lundberg2017unified;kwon2022weightedshap\)\.

However,kumar2020problems;yan2021ifhave raised concerns about the imposed linearity axiom, as its necessity remains unclear\. Meanwhile,bilodeau2024impossibility;wang2024rethinkinghave theoretically established that the use of the Shapley value can be unreliable by exploiting linearity\. Accordingly, a nonlinear alternative, the least core\(shapley1971cores\), has attracted attention\(yan2021if;gemp2024approximating\)\. In general, the least core may contain infinitely many contribution allocations, so we instead consider the egalitarian least core\(arin2008axiomatic;yan2021if\)\.111The least core is a convex subset ofℝn\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}, and the egalitarian least core is defined as the element in the least core with the smallestℓ2\\mathchar 352\\relax\_\{\\mathchar 28722\\relax\}norm\.Still, the egalitarian least core is constrained by the efficiency axiom, which has been empirically shown to be unnecessary, and even unfavorable, bykwon2022beta;kwon2022weightedshap\. By relaxing only the efficiency axiom, one arrives at the family of semi\-values\(dubey1981value\), which includes Beta Shapley values\(kwon2022beta\)and weighted Banzhaf values\(li2023robust\), uniquely characterized by the axioms of linearity, consistency, equal treatment, and monotonicity\.222The Shapley value also satisfies the monotonicity axiom\.In this work, we theoretically explore the design of attribution methods by relaxing only the axioms of linearity and efficiency\.

#### Our contributions\.

Often, the quality of player rankings is evaluated using the inclusion Area Under the Curve \(AUC\) metric, when the goal is to identify positively contributing players\(petsiuk2018rise;kwon2022beta;covert2023learning\)\. Under this metric, it can be demonstrated \(Lemmas[3\.2](https://arxiv.org/html/2607.09869#S3.Thmtheorem2)and[3\.3](https://arxiv.org/html/2607.09869#S3.Thmtheorem3)\) that applying the egalitarian least core to rank players is equivalent to approximately maximizing the inclusion AUC by substituting an additive approximation of the utility functionU\\mathchar 29013\\relax\. This perspective follows from the imposed efficiency axiom\. Likewise, the use of semi\-values in player ranking can be interpreted in a similar way \(Corollary[3\.4](https://arxiv.org/html/2607.09869#S3.Thmtheorem4)\)\.

- •We then present a theoretical result demonstrating that the Shapley value is not reliable for maximizing the inclusion AUC, due to a vulnerability introduced by the linearity axiom \(Theorem[4\.1](https://arxiv.org/html/2607.09869#S4.Thmtheorem1)\)\.
- •In Section[4\.2](https://arxiv.org/html/2607.09869#S4.SS2), based on the Shapley value, we show via a pathological example that using the sum of contribution vectors of different utility functions as the contribution vector of their sum can lead to undesirable behavior w\.r\.t\. maximizing the inclusion AUC\.

These observations motivate us to explore nonlinear attribution methods that do not impose the linearity axiom, yet still satisfy the remaining desired axioms\. In particular, the interpretation of applying the egalitarian least core as well as semi\-values to rank players points us toward starting from a faithful additive approximation of utility functions\. Consequently,

- •We introduce a mathematically grounded class of attribution methods that relaxes only the linearity and efficiency axioms \(Theorem[4\.3](https://arxiv.org/html/2607.09869#S4.Thmtheorem3)\);333As an unexpected finding, we also concurrently developed another class of such nonlinear axiomatic attribution methods via optimizing the inclusion AUC metric\(li2026treegrad\)\.
- •We also lay out a theoretical ground for approximately computing the egalitarian least core and our introduced nonlinear attribution methodϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\(Theorems[4\.5](https://arxiv.org/html/2607.09869#S4.Thmtheorem5)and[4\.6](https://arxiv.org/html/2607.09869#S4.Thmtheorem6)\);

Finally, we conduct experiments to verify the practical effectiveness of the proposed methods in improving player ranking quality under the inclusion AUC metric\.

## 2Background

### 2\.1Notation

Let\[n\]≔\{1,2,…,n\}\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\coloneq\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\dots\\mathchar 24891\\relax\\mathchar 29038\\relax\\\}denote a set of players, wheren≥2\\mathchar 29038\\relax\\mathchar 12821\\relax\\mathchar 28722\\relax\. For eachn\\mathchar 29038\\relax,𝒢n≔\{Un:2\[n\]→ℝ\}\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 29038\\relax\}\\coloneq\\\{\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24634\\relax\\mathchar 28722\\relax^\{\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 12833\\relax\\mathbb\{\\mathchar 29010\\relax\}\\\}refers to the set of utility functions overn\\mathchar 29038\\relaxplayers\. Throughout the paper, the subscriptn\\mathchar 29038\\relaxindicates the number of players\. We then define the space of all such games as𝒢≔⋃n≥2𝒢n\\mathcal\{\\mathchar 28999\\relax\}\\coloneq\\mathchar 4947\\relax\\displaylimits\_\{\\mathchar 29038\\relax\\mathchar 12821\\relax\\mathchar 28722\\relax\}\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 29038\\relax\}\. An attribution methodϕ\\mathchar 28958\\relaxmaps each utility functionUn∈𝒢\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}to a contribution vectorϕ​\(Un\)∈ℝn\\mathchar 28958\\relax\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}, whereϕi​\(Un\)\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785measures the contribution of thei\\mathchar 29033\\relax\-th player in the cooperative game defined byUn\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\. The all\-one vector inℝm\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29037\\relax\}is denoted by𝟏m\\mathbf\{\\mathchar 28721\\relax\}\_\{\\mathchar 29037\\relax\}\. Given𝐱∈ℝn\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}andb∈ℝ\\mathchar 29026\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}, we define an additive utility functionAn​\(⋅;𝐱,b\)∈𝒢n\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 8705\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\delimiter 84054785\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 29038\\relax\}by letting

An​\(S;𝐱,b\)≔b\+∑i∈Sxi​for every​S⊆\[n\],\\begin\{gathered\}\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\delimiter 84054785\\coloneq\\mathchar 29026\\relax\\mathchar 8235\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 29048\\relax\_\{\\mathchar 29033\\relax\}\\ \\text\{ for every \}\\ \\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 24891\\relax\\end\{gathered\}\(1\)where by default an empty sum is taken as 0\. Plus,⟦⋅⟧\\left\\llbracket\\mathchar 8705\\relax\\right\\rrbracketdenotes the indicator function\.

### 2\.2Axioms in Cooperative Game Theory

In what follows, we present a list of axioms commonly used in the literature\(e\.g\.,shapley1953value;dubey1981value;weber1988probabilistic;li2023robust\)\.

- •Linearity:ϕ​\(c⋅Un\+Un′\)=c⋅ϕ​\(Un\)\+ϕ​\(Un′\)\\mathchar 28958\\relax\\delimiter 67273472\\mathchar 29027\\relax\\mathchar 8705\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 8235\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}^\{\\mathchar 560\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29027\\relax\\mathchar 8705\\relax\\mathchar 28958\\relax\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28958\\relax\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}^\{\\mathchar 560\\relax\}\\delimiter 84054785for everyUn,Un′∈𝒢\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}^\{\\mathchar 560\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}andc∈ℝ\\mathchar 29027\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\.
- •Consistency: Suppose there existsc∈ℝ\\mathchar 29027\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}such thatUn\+1\(S\)=Un\(S∩\[n\]\)\+c⋅⟦n\+1∈S⟧\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8796\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 29027\\relax\\mathchar 8705\\relax\\left\\llbracket\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\\right\\rrbracketfor everyS⊆\[n\+1\]\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\delimiter 84267779\. Then, \(i\)ϕn\+1​\(Un\+1\)=c\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29027\\relax, and \(ii\)ϕi​\(Un\)=ϕi​\(Un\+1\)\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785for everyi∈\[n\]\\mathchar 29033\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\.
- •Equal Treatment: For somei,j∈\[n\]\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779, ifUn​\(S∪\{i\}\)=Un​\(S∪\{j\}\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785for everyS⊆\[n\]\\\{i,j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\\}, thenϕi​\(Un\)=ϕj​\(Un\)\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\.
- •Monotonicity: For somei∈\[n\]\\mathchar 29033\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779, \(i\) ifUn​\(S∪\{i\}\)≥Un​\(S\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\delimiter 84054785\\mathchar 12821\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785for everyS⊆\[n\]\\\{i\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\\}, thenϕi​\(Un\)≥Γ\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12821\\relax 0; \(ii\) ifUn​\(S∪\{i\}\)≤Un​\(S\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\delimiter 84054785\\mathchar 12820\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785for everyS⊆\[n\]\\\{i\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\\}, thenϕi​\(Un\)≤Γ\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12820\\relax 0\.
- •Efficiency:∑i∈\[n\]ϕi​\(Un\)=Un​\(\[n\]\)−Un​\(∅\)\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785for everyUn∈𝒢\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}\.
- •Translation Invariance:ϕ​\(Un\+Cn\)=ϕ​\(Un\)\\mathchar 28958\\relax\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 8235\\relax\\mathchar 28995\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28958\\relax\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785wheneverCn∈𝒢\\mathchar 28995\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}is a constant utility function\.

We note that the consistency axiom implies both the dummy axiom used byweber1988probabilisticand the projection axiom introduced bydubey1981value\. Besides, translation invariance is often implied by linearity and other axioms\. In other words, translation invariance may be viewed as a weak version of linearity\.

### 2\.3Semi\-Values

The family of semi\-values is uniquely characterized by all the above axioms except efficiency\(dubey1981value\)\. Specifically, each semi\-value corresponds to a Borel probability measureμ\\mathchar 28950\\relaxover the interval\[Γ,1\]\\delimiter 674823700\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84267779\. Accordingly, we useϕ¯\\mathchar 28958\\relax^\{\\mathchar 28950\\relax\}to denote the semi\-value parameterized byμ\\mathchar 28950\\relax\. The contribution of thei\\mathchar 29033\\relax\-th player in a gameUn\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}is given by

Œi¯​\(Un\)=∑S⊆\[n\]\\ipn,\|S\|\+1¯​\[Un​\(S∪\{i\}\)−Un​\(S\)\]where​pn,k¯=∫Γ1tk−1​\(1−t\)n−k​d¯​\(t\)\.\\begin\{gathered\}\\mathchar 28958\\relax^\{\\mathchar 28950\\relax\}\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\mathchar 29033\\relax\}\\mathchar 29040\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\delimiter 69640972\\mathchar 29011\\relax\\delimiter 69640972\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathchar 28950\\relax\}\\delimiter 67482370\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\delimiter 84267779\\\\ \\text\{where \}\\ \\mathchar 29040\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\mathchar 29035\\relax\}^\{\\mathchar 28950\\relax\}\\mathchar 12349\\relax\\mathchar 4946\\relax\\nolimits\_\{0\}^\{\\mathchar 28721\\relax\}\\mathchar 29044\\relax^\{\\mathchar 29035\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 29044\\relax\\delimiter 84054785^\{\\mathchar 29038\\relax\\mathchar 8704\\relax\\mathchar 29035\\relax\}\\mathrm\{\\mathchar 29028\\relax\}\\mathchar 28950\\relax\\delimiter 67273472\\mathchar 29044\\relax\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(2\)When the efficiency axiom is additionally imposed,μ\\mathchar 28950\\relaxis uniquely determined as the uniform distribution on\[Γ,1\]\\delimiter 674823700\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84267779, resulting in the well\-known Shapley value\. For the weighted Banzhaf value WB\(τ\)\\delimiter 67273472\\mathchar 28956\\relax\\delimiter 84054785, whereτ∈\(Γ,1\)\\mathchar 28956\\relax\\mathchar 12850\\relax\\delimiter 672734720\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84054785, the corresponding measureμ\\mathchar 28950\\relaxis the Dirac delta distributionδø\\mathchar 28942\\relax\_\{\\mathchar 28956\\relax\}\. For the Beta Shapley valueBeta​\(α,β\)\\text\{Beta\}\\delimiter 67273472\\mathchar 28939\\relax\\mathchar 24891\\relax\\mathchar 28940\\relax\\delimiter 84054785, whereα,β≥1\\mathchar 28939\\relax\\mathchar 24891\\relax\\mathchar 28940\\relax\\mathchar 12821\\relax\\mathchar 28721\\relax, the measureμ\\mathchar 28950\\relaxis defined by

¯​\(S\)∝∫St1−fi​\(1−t\)1−ff​dt\\begin\{gathered\}\\mathchar 28950\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 12847\\relax\\mathchar 4946\\relax\\nolimits\_\{\\mathchar 29011\\relax\}\\mathchar 29044\\relax^\{\\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 28940\\relax\}\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 29044\\relax\\delimiter 84054785^\{\\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 28939\\relax\}\\mathrm\{\\mathchar 29028\\relax\}\\mathchar 29044\\relax\\end\{gathered\}\(3\)for every measurable subsetS\\mathchar 29011\\relaxof\[Γ,1\]\\delimiter 674823700\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84267779\. In particular, Beta\(1,1\)\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84054785corresponds to the Shapley value\. Empirically,kwon2022beta;kwon2022weightedshap;li2023robustdemonstrate that Beta Shapley values and weighted Banzhaf values often perform better in downstream tasks, challenging the necessity of the efficiency axiom\. Heuristically, when only player rankings are of interest, the efficiency axiom, as a normalization step, is often regarded as redundant\.

## 3Motivations

### 3\.1The Egalitarian Least Core

Followingyan2021if, the least coreLC​\(Un\)\\mathrm\{\\mathchar 29004\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785of a utility functionUn\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}is a subset ofℝn\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}containing all the optimal solutions of𝐱\\mathbf\{\\mathchar 29048\\relax\}to the minimization problem

minimize𝐱∈ℝn,e∈ℝes\.t\.​Un​\(S\)−An​\(S;𝐱,Un​\(∅\)\)≤e,∀S⊊\[n\]and​An​\(\[n\];𝐱,Un​\(∅\)\)=Un​\(\[n\]\)\.\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29029\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\ \\mathchar 29029\\relax\\\\ \\text\{s\.t\. \}\\ \\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\mathchar 12820\\relax\\mathchar 29029\\relax\\mathchar 24891\\relax\\ \\mathchar 568\\relax\\ \\mathchar 29011\\relax\\subsetneq\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\\\ \\text\{and \}\\ \\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(4\)According toblum1972direct, it is clear that the least coreLC​\(Un\)\\mathrm\{\\mathchar 29004\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785is always non\-empty\. The egalitarian least coreϕELC\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}is then defined as the least\-norm element inLC​\(Un\)\\mathrm\{\\mathchar 29004\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\. Precisely,

ŒELC​\(Un\)≔arg​min𝐱∈LC​\(Un\)⁡‖𝐱‖2\.\\begin\{gathered\}\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\coloneq\\operatorname\\mathchar 8707\\relax\{\\mathchar 29025\\relax\\mathchar 29042\\relax\\mathchar 29031\\relax\\,\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathrm\{\\mathchar 29004\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(5\)The least norm solution is critical for ensuring thatϕELC\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}satisfies the axiom of equal treatment\(yan2021if\)\. To the best of our knowledge, no existing work has established whetherϕELC\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}satisfies the other axioms listed above\. Therefore, for completeness, we analyze the axiomatic properties of the egalitarian least core\.

###### Theorem 3\.1\.

The egalitarian least coreϕELC\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}is nonlinear and satisfies the axioms of consistency, equal treatment, monotonicity, efficiency, and translation invariance\.

This result indicates that the egalitarian least core relaxes only the linearity axiom\. Interestingly, we observe that the efficiency axiom effectively transforms the one\-sided error bounds in the problem \([4](https://arxiv.org/html/2607.09869#S3.E4)\) into two\-sided ones\. The following lemma makes this precise\.

###### Lemma 3\.2\.

Given a utility functionUn∈𝒢\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}, a vector𝐱′∈LC​\(Un\)\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 560\\relax\}\\mathchar 12850\\relax\\mathrm\{\\mathchar 29004\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785if and only if𝐱′\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 560\\relax\}is an optimal solution to the problem

minimize𝐱∈ℝn,e∈ℝes\.t\.−e≤An​\(S;𝐱,Un​\(∅\)\)−Un​\(S\),∀S⊊\[n\]An​\(S;𝐱,Un​\(∅\)\)−Un​\(S\)≤e\+B​\(S\),∀S⊊\[n\]and​∑i∈\[n\]xi=Un​\(\[n\]\)−Un​\(∅\)\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29029\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\ \\mathchar 29029\\relax\\\\ \\text\{s\.t\. \}\\ \\mathchar 8704\\relax\\mathchar 29029\\relax\\mathchar 12820\\relax\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 24891\\relax\\ \\mathchar 568\\relax\\ \\mathchar 29011\\relax\\subsetneq\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\\\ \\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 12820\\relax\\mathchar 29029\\relax\\mathchar 8235\\relax\\mathchar 28994\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 24891\\relax\\ \\mathchar 568\\relax\\ \\mathchar 29011\\relax\\subsetneq\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\\\ \\text\{and \}\\ \\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 29048\\relax\_\{\\mathchar 29033\\relax\}\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\end\{gathered\}\(6\)whereB​\(S\)≔Un​\(\[n\]\)\+Un​\(∅\)−Un​\(S\)−Un​\(\[n\]\\S\)\\mathchar 28994\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\coloneq\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\mathchar 29011\\relax\\delimiter 84054785\.

###### Proof\.

Suffice it to demonstrate thatAn​\(S;𝐱,Un​\(∅\)\)≤e\+B​\(S\)\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\mathchar 12820\\relax\\mathchar 29029\\relax\\mathchar 8235\\relax\\mathchar 28994\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785is equivalent to

Un​\(\[n\]\\S\)−An​\(\[n\]\\S;𝐱,Un​\(∅\)\)≤e\.\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\mathchar 12820\\relax\\mathchar 29029\\relax\\mathchar 314\\relax\\end\{gathered\}\(7\)Using the efficiency constraint,

An​\(\[n\]\\S;𝐱,Un​\(∅\)\)\+An​\(S;𝐱,Un​\(∅\)\)=Un​\(\[n\]\)\+Un​\(∅\),\\begin\{gathered\}\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\\\ \\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 24891\\relax\\end\{gathered\}\(8\)which leads to

Un​\(\[n\]\\S\)−An​\(\[n\]\\S;𝐱,Un​\(∅\)\)=An​\(S;𝐱,Un​\(∅\)\)−Un​\(S\)−B​\(S\)\.\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\\\ \\mathchar 12349\\relax\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28994\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(9\)∎

In summary, the egalitarian least core produces an additive utility functionAn​\(⋅;ϕELC​\(Un\),Un​\(∅\)\)\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 8705\\relax\\mathchar 24635\\relax\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785that seeks to approximateUn\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}under the problem \([6](https://arxiv.org/html/2607.09869#S3.E6)\)\.

### 3\.2Evaluating Player Rankings

In data attribution\(wang2024rethinking\), the utility functionUn​\(S\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785is typically defined as the performance of a model trained on the subsetS\\mathchar 29011\\relaxof available training data\. Then, the grand set of players\[n\]\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779corresponds to the set of training data, andϕi​\(Un\)\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785quantifies the contribution of thei\\mathchar 29033\\relax\-th data point to the performance of trained models\. In this context, the goal is to identify positively contributing data points, which is closely related to maximizing the inclusion AUC, defined as

maximizeß∈ΠnAUC​\(ß;Un\)≔∑k=1nUn​\(Sk​\(ß\)\)where​Sk​\(ß\)≔\{ß​\(1\),ß​\(2\),…,ß​\(k\)\}\.\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29025\\relax\\mathchar 29048\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathchar 28953\\relax\\mathchar 12850\\relax\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\}\}\\ \\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 28953\\relax\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\coloneq\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\_\{\\mathchar 29035\\relax\}\\delimiter 67273472\\mathchar 28953\\relax\\delimiter 84054785\\delimiter 84054785\\\\ \\text\{where \}\\ \\mathchar 29011\\relax\_\{\\mathchar 29035\\relax\}\\delimiter 67273472\\mathchar 28953\\relax\\delimiter 84054785\\coloneq\\\{\\mathchar 28953\\relax\\delimiter 67273472\\mathchar 28721\\relax\\delimiter 84054785\\mathchar 24891\\relax\\mathchar 28953\\relax\\delimiter 67273472\\mathchar 28722\\relax\\delimiter 84054785\\mathchar 24891\\relax\\dots\\mathchar 24891\\relax\\mathchar 28953\\relax\\delimiter 67273472\\mathchar 29035\\relax\\delimiter 84054785\\\}\\mathchar 314\\relax\\end\{gathered\}\(10\)Here,Πn\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\}denotes the set of all permutations of\[n\]\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\. We note that maximizing the inclusion AUC also applies to context attribution\(cohen2024contextcite\), where the aim is to identify the influential parts of the context that generate a statement\. In general, solving the problem \([10](https://arxiv.org/html/2607.09869#S3.E10)\) is computationally challenging due to the factorial number of possible permutations\. Nevertheless, the problem becomes tractable when the utility function is additive\.

###### Lemma 3\.3\.

Let𝐱∈ℝn\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}, and defineπ𝐱\\mathchar 28953\\relax\_\{\\mathbf\{\\mathchar 29048\\relax\}\}as the permutation satisfyingxß𝐱​\(j\)≥xß𝐱​\(j\+1\)\\mathchar 29048\\relax\_\{\\mathchar 28953\\relax\_\{\\mathbf\{\\mathchar 29048\\relax\}\}\\delimiter 67273472\\mathchar 29034\\relax\\delimiter 84054785\}\\mathchar 12821\\relax\\mathchar 29048\\relax\_\{\\mathchar 28953\\relax\_\{\\mathbf\{\\mathchar 29048\\relax\}\}\\delimiter 67273472\\mathchar 29034\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\delimiter 84054785\}for every1≤j<n\\mathchar 28721\\relax\\mathchar 12820\\relax\\mathchar 29034\\relax\\mathchar 12604\\relax\\mathchar 29038\\relax\. Then,π𝐱\\mathchar 28953\\relax\_\{\\mathbf\{\\mathchar 29048\\relax\}\}is optimal to the problem

maximizeß∈ΠnAUC​\(ß;An​\(⋅;𝐱,b\)\)\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29025\\relax\\mathchar 29048\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathchar 28953\\relax\\mathchar 12850\\relax\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\}\}\\ \\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 28953\\relax\\mathchar 24635\\relax\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 8705\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\delimiter 84054785\\delimiter 84054785\\end\{gathered\}\(11\)whereb∈ℝ\\mathchar 29026\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}is arbitrary\.

###### Proof\.

This result follows immediately from the observation

AUC​\(ß;An​\(⋅;𝐱,b\)\)=∑k=1n\(n\+1−k\)⋅xß​\(k\)\\begin\{gathered\}\\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 28953\\relax\\mathchar 24635\\relax\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 8705\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 29035\\relax\\delimiter 84054785\\mathchar 8705\\relax\\mathchar 29048\\relax\_\{\\mathchar 28953\\relax\\delimiter 67273472\\mathchar 29035\\relax\\delimiter 84054785\}\\end\{gathered\}\(12\)and the rearrangement inequality\. ∎

Combining Lemmas[3\.2](https://arxiv.org/html/2607.09869#S3.Thmtheorem2)and[3\.3](https://arxiv.org/html/2607.09869#S3.Thmtheorem3), we interpret the mechanism of the egalitarian least core in feature ranking as follows:

- •Obtain the best additive approximation ofUn\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}by solving the minimization problem \([6](https://arxiv.org/html/2607.09869#S3.E6)\), yielding the additive utility functionAn​\(⋅;ϕELC​\(Un\),Un​\(∅\)\)\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 8705\\relax\\mathchar 24635\\relax\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\.
- •Substitute the obtainedAn​\(⋅;ϕELC​\(Un\),Un​\(∅\)\)\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 8705\\relax\\mathchar 24635\\relax\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785into the maximization problem \([10](https://arxiv.org/html/2607.09869#S3.E10)\) to derive an approximate optimal ranking\.

Meanwhile, semi\-values admit a similar interpretation\. Specifically,li2024oneproved that there exists a functionT¯:𝒢→ℝ\\mathchar 29012\\relax^\{\\mathchar 28950\\relax\}\\mathchar 24634\\relax\\mathcal\{\\mathchar 28999\\relax\}\\mathchar 12833\\relax\\mathbb\{\\mathchar 29010\\relax\}such that

Œ¯​\(Un\)=𝐱¯∗​\(Un\)\+T¯​\(Un\)⋅𝟏n\\begin\{gathered\}\\mathchar 28958\\relax^\{\\mathchar 28950\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 8707\\relax\}\_\{\\mathchar 28950\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 29012\\relax^\{\\mathchar 28950\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8705\\relax\\mathbf\{\\mathchar 28721\\relax\}\_\{\\mathchar 29038\\relax\}\\end\{gathered\}\(13\)where\(𝐱¯∗​\(Un\),b¯∗​\(Un\)\)\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 8707\\relax\}\_\{\\mathchar 28950\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\mathchar 29026\\relax^\{\\mathchar 8707\\relax\}\_\{\\mathchar 28950\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785is the unique optimal solution to the least square problem

minimize𝐱∈ℝn,b∈R​∑∅⊊S⊊\[n\]pn−1,\|S\|¯​\[Un​\(S\)−An​\(S;𝐱,b\)\]2\.\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\mathchar 12850\\relax\\mathchar 29010\\relax\}\\ \\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 571\\relax\\subsetneq\\mathchar 29011\\relax\\subsetneq\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 29040\\relax\_\{\\mathchar 29038\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\delimiter 69640972\\mathchar 29011\\relax\\delimiter 69640972\}^\{\\mathchar 28950\\relax\}\\delimiter 67482370\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\delimiter 84054785\\delimiter 84267779^\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(14\)This immediately leads to the following corollary\.

###### Corollary 3\.4\.

Given a utility functionUn∈𝒢\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}, letπ¯\\mathchar 28953\\relax\_\{\\mathchar 28950\\relax\}be a ranking of\[n\]\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779such thatϕ߯​\(j\)¯​\(Un\)≥ϕ߯​\(j\+1\)¯​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 28950\\relax\}\_\{\\mathchar 28953\\relax\_\{\\mathchar 28950\\relax\}\\delimiter 67273472\\mathchar 29034\\relax\\delimiter 84054785\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12821\\relax\\mathchar 28958\\relax^\{\\mathchar 28950\\relax\}\_\{\\mathchar 28953\\relax\_\{\\mathchar 28950\\relax\}\\delimiter 67273472\\mathchar 29034\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\delimiter 84054785\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785for every1≤j<n\\mathchar 28721\\relax\\mathchar 12820\\relax\\mathchar 29034\\relax\\mathchar 12604\\relax\\mathchar 29038\\relax\. Then,π¯\\mathchar 28953\\relax\_\{\\mathchar 28950\\relax\}is an optimal solution to

maximizeß∈ΠnAUC​\[ß;An​\(⋅;𝐱¯∗​\(Un\),b¯∗​\(Un\)\)\]\.\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29025\\relax\\mathchar 29048\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathchar 28953\\relax\\mathchar 12850\\relax\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\}\}\\ \\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}\\delimiter 67482370\\mathchar 28953\\relax\\mathchar 24635\\relax\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 8705\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28950\\relax\}^\{\\mathchar 8707\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\mathchar 29026\\relax\_\{\\mathchar 28950\\relax\}^\{\\mathchar 8707\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\delimiter 84267779\\mathchar 314\\relax\\end\{gathered\}\(15\)

This corollary is immediate by observing that the induced player rankings ofϕ¯​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 28950\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785and𝐱¯∗\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28950\\relax\}^\{\\mathchar 8707\\relax\}are the same\.

In summary, many popular axiomatic attribution methods for ranking players \(e\.g\., Beta Shapley values, weighted Banzhaf values, and the egalitarian least core\) can be viewed as implicitly learning an additive approximation to the true utility functionUn\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}, which is then used to derive an approximate optimal ranking\. This motivates us to next explore nonlinear, non\-efficient attribution methods through the same lens of approximation\.

## 4Main Results

### 4\.1How Unreliable Is the Shapley Value

Before proceeding, we demonstrate that the Shapley value may fail to distinguish between utility functions in terms of approximately maximizing the inclusion AUC\.

###### Theorem 4\.1\.

LetΠn\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\}be the set of all permutations of\[n\]\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779withn≥3\\mathchar 29038\\relax\\mathchar 12821\\relax\\mathchar 28723\\relax, and letϕShap\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 29011\\relax\\mathchar 29032\\relax\\mathchar 29025\\relax\\mathchar 29040\\relax\}\}denote the Shapley value\. Then,Πn\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\}can be partitioned into\{Πn,j\}j∈𝒥\\\{\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\}\\\}\_\{\\mathchar 29034\\relax\\mathchar 12850\\relax\\mathcal\{\\mathchar 29002\\relax\}\}with\|𝒥\|≥2\\delimiter 69640972\\mathcal\{\\mathchar 29002\\relax\}\\delimiter 69640972\\mathchar 12821\\relax\\mathchar 28722\\relaxsuch that, for everyΠn,j\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\}and everyUn∈𝒢n\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 29038\\relax\}, there exists a perturbationUn′∈𝒢n\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}^\{\\mathchar 560\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 29038\\relax\}satisfying

ŒShap​\(Un\)=ŒShap​\(Un\+Un′\),yet​arg​maxß∈Πn⁡AUC​\(ß;Un\+Un′\)⊆Πn,j\.\\begin\{gathered\}\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 29011\\relax\\mathchar 29032\\relax\\mathchar 29025\\relax\\mathchar 29040\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 29011\\relax\\mathchar 29032\\relax\\mathchar 29025\\relax\\mathchar 29040\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 8235\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}^\{\\mathchar 560\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\\\ \\text\{yet \}\\ \\operatorname\\mathchar 8707\\relax\{\\mathchar 29025\\relax\\mathchar 29042\\relax\\mathchar 29031\\relax\\,\\mathchar 29037\\relax\\mathchar 29025\\relax\\mathchar 29048\\relax\}\_\{\\mathchar 28953\\relax\\mathchar 12850\\relax\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\}\}\\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 28953\\relax\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 8235\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}^\{\\mathchar 560\\relax\}\\delimiter 84054785\\mathchar 12818\\relax\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(16\)

Theorem[4\.1](https://arxiv.org/html/2607.09869#S4.Thmtheorem1)states that, preserving the same Shapley value, the AUC\-optimal permutation can be manipulated to lie in anyΠn,j\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\}\. Specifically, the partition in Theorem[4\.1](https://arxiv.org/html/2607.09869#S4.Thmtheorem1)is constructed by exploiting the null space of the Shapley value\. In other words, the linearity axiom induces an excessively large null space that can be detrimental\. We note that this is not the only negative perspective established so far\. For example,wang2024rethinkingshowed that the Shapley value is not reliable for maximizing the utilityU​\(S\)\\mathchar 29013\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\. On the other hand,bilodeau2024impossibilitydemonstrated that the Shapley value fails to distinguish local behaviors of different functions\. In particular, all these negative results rely on exploiting the null space induced by the linearity axiom\.

### 4\.2A Pathological Example Induced by Linearity

The inarguable merit of linearity is that it divides the calculation ofϕ​\(U\+U′\)\\mathchar 28958\\relax\\delimiter 67273472\\mathchar 29013\\relax\\mathchar 8235\\relax\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\\delimiter 84054785intoϕ​\(U\)\+ϕ​\(U′\)\\mathchar 28958\\relax\\delimiter 67273472\\mathchar 29013\\relax\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28958\\relax\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\\delimiter 84054785\. However, we demonstrate that such a convenience comes with pathological examples in maximizing the problem \([10](https://arxiv.org/html/2607.09869#S3.E10)\)\. Let2\[4\]\\mathchar 28722\\relax^\{\\delimiter 67482370\\mathchar 28724\\relax\\delimiter 84267779\}be ordered as

∅,\{1\},\{2\},\{1,2\},\{3\},\{1,3\},\{2,3\},\{1,2,3\},\{4\},\{1,4\},\{2,4\},\{1,2,4\},\{3,4\},\{1,3,4\},\{2,3,4\},\{1,2,3,4\},\\begin\{gathered\}\\mathchar 571\\relax\\mathchar 24891\\relax\\\{\\mathchar 28721\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28722\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28724\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28724\\relax\\\}\\mathchar 24891\\relax\\\\ \\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28724\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28724\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 28724\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 28724\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 28724\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 28724\\relax\\\}\\mathchar 24891\\relax\\end\{gathered\}\(17\)which is the binary ordering used ingrabisch2000equivalent\. Then, every utility function in𝒢4\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 28724\\relax\}can be expressed as a vector\. Let

U4=\(\\displaystyle\\mathchar 29013\\relax\_\{\\mathchar 28724\\relax\}\\mathchar 12349\\relax\\delimiter 672734724,−3,−4,−8,2,6,3,−8,\\displaystyle\\mathchar 28724\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28724\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28728\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28726\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28728\\relax\\mathchar 24891\\relax\(18\)4,−1Γ,−3,8,5,1,4,−3\),\\displaystyle\\mathchar 28724\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax 0\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 28728\\relax\\mathchar 24891\\relax\\mathchar 28725\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28724\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28723\\relax\\delimiter 84054785\\mathchar 24891\\relaxU4′=\(\\displaystyle\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28724\\relax\}\\mathchar 12349\\relax\\delimiter 67273472−5,−2,−7,7,−1​Γ,4,−4,9,\\displaystyle\\mathchar 8704\\relax\\mathchar 28725\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28727\\relax\\mathchar 24891\\relax\\mathchar 28727\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax 0\\mathchar 24891\\relax\\mathchar 28724\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28724\\relax\\mathchar 24891\\relax\\mathchar 28729\\relax\\mathchar 24891\\relax−1,−6,−1,1,−3,−2,−1Γ,−5\)\.\\displaystyle\\mathchar 8704\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28726\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax 0\\mathchar 24891\\relax\\mathchar 8704\\relax\\mathchar 28725\\relax\\delimiter 84054785\\mathchar 314\\relaxParticularly, they are constructed such thatπShap​\(U4\)\\mathchar 28953\\relax^\{\\mathrm\{\\mathchar 29011\\relax\\mathchar 29032\\relax\\mathchar 29025\\relax\\mathchar 29040\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28724\\relax\}\\delimiter 84054785andπShap​\(U4′\)\\mathchar 28953\\relax^\{\\mathrm\{\\mathchar 29011\\relax\\mathchar 29032\\relax\\mathchar 29025\\relax\\mathchar 29040\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28724\\relax\}\\delimiter 84054785each exactly maximize the problem \([10](https://arxiv.org/html/2607.09869#S3.E10)\), whereπShap\\mathchar 28953\\relax^\{\\mathrm\{\\mathchar 29011\\relax\\mathchar 29032\\relax\\mathchar 29025\\relax\\mathchar 29040\\relax\}\}denotes the player ranking induced by the Shapley value\. However, one can verify thatπShap​\(U4\+U4′\)\\mathchar 28953\\relax^\{\\mathrm\{\\mathchar 29011\\relax\\mathchar 29032\\relax\\mathchar 29025\\relax\\mathchar 29040\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28724\\relax\}\\mathchar 8235\\relax\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28724\\relax\}\\delimiter 84054785instead*minimizes*the inclusion AUC\. This pathological example clearly demonstrates the unexpected behavior that arises from simply adding the Shapley values ofU4\\mathchar 29013\\relax\_\{\\mathchar 28724\\relax\}andU4′\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28724\\relax\}when maximizing the inclusion AUC\. Moreover, there exist many such examples, as the one presented here was instantly found via random search\.

The above example raises a concern about the common practice of using the sum of contribution vectors of different utility functions as the contribution vector of their sum\. This phenomenon also serves as our motivation to explore what nonlinear axiomatic attribution methods would look like\.

### 4\.3Nonlinear Attribution Methods

Looking at the problem \([6](https://arxiv.org/html/2607.09869#S3.E6)\) of the least core, the termB\\mathchar 28994\\relaxappears to hinder faithful approximation ofUn\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\. Recall that this term arises from enforcing the efficiency axiom\. To relax this axiom, it is natural to allow two\-sided approximation errors for each subsetS\\mathchar 29011\\relax, i\.e\., boundingAn​\(S;𝐱,b\)−Un​\(S\)\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785from both sides\. Moreover, to better fitUn\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}, we treat the bias termb\\mathchar 29026\\relaxas a variable to be optimized, rather than fixing it straightforwardly toUn​\(∅\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\. All in all, we consider the minimization problem below that attempts to faithfully approximate the utility function using an additive one:

minimize𝐱∈ℝn,b∈ℝ‖𝐀𝐱\+b⋅𝟏2n−𝐕​\(Un\)‖p\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\ \\delimiter 69645069\\mathbf\{\\mathchar 28993\\relax\}\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 8235\\relax\\mathchar 29026\\relax\\mathchar 8705\\relax\\mathbf\{\\mathchar 28721\\relax\}\_\{\\mathchar 28722\\relax^\{\\mathchar 29038\\relax\}\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 29040\\relax\}\\end\{gathered\}\(19\)where𝐀∈ℝ2n×n\\mathbf\{\\mathchar 28993\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 28722\\relax^\{\\mathchar 29038\\relax\}\\mathchar 8706\\relax\\mathchar 29038\\relax\}, and𝐕​\(Un\)∈ℝ2n\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 28722\\relax^\{\\mathchar 29038\\relax\}\}containsUn​\(S\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785for all subsetsS⊆\[n\]\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\. Each rowAS\\mathchar 28993\\relax\_\{\\mathchar 29011\\relax\}of𝐀\\mathbf\{\\mathchar 28993\\relax\}is the indicator vector of the subsetS\\mathchar 29011\\relax, i\.e\.,AS,i=1\\mathchar 28993\\relax\_\{\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 29033\\relax\}\\mathchar 12349\\relax\\mathchar 28721\\relaxifi∈S\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax, andΓotherwise\. Note that whenp=∞\\mathchar 29040\\relax\\mathchar 12349\\relax\\mathchar 561\\relax, the problem \([19](https://arxiv.org/html/2607.09869#S4.E19)\) coincides with the problem \([6](https://arxiv.org/html/2607.09869#S3.E6)\) with the efficiency axiom removed andB​\(S\)≡Γ\\mathchar 28994\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 12817\\relax 0\. We note thatCharnesGKR88considered a similar approximation withp∈\{1,2,∞\}\\mathchar 29040\\relax\\mathchar 12850\\relax\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\\}\.

LetOSp​\(Un\)\\mathrm\{\\mathchar 29007\\relax\\mathchar 29011\\relax\}\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785denote the set of all optimal solutions of𝐱\\mathbf\{\\mathchar 29048\\relax\}to the minimization problem \([19](https://arxiv.org/html/2607.09869#S4.E19)\)\. Then, We define thep\\mathchar 29040\\relax\-norm attribution methodϕp​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785by selecting the minimumℓ2\\mathchar 352\\relax\_\{\\mathchar 28722\\relax\}\-norm solution amongOSp​\(Un\)\\mathrm\{\\mathchar 29007\\relax\\mathchar 29011\\relax\}\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785:

Œp​\(Un\)≔arg​min𝐱∈OSp​\(Un\)⁡‖𝐱‖2\.\\begin\{gathered\}\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\coloneq\\operatorname\\mathchar 8707\\relax\{\\mathchar 29025\\relax\\mathchar 29042\\relax\\mathchar 29031\\relax\\,\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathrm\{\\mathchar 29007\\relax\\mathchar 29011\\relax\}\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(20\)
Forp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785, the problem admits a unique solution since the objective is strictly convex\. However, forp∈\{1,∞\}\\mathchar 29040\\relax\\mathchar 12850\\relax\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\\}, multiple optimal solutions may exist\. As an example forp=∞\\mathchar 29040\\relax\\mathchar 12349\\relax\\mathchar 561\\relax, considern=3\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28723\\relaxand defineU3\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}such that

U​\(S\)≔\{Γ,S=∅,Γ​\.5,S∈\{\{2\},\{3\}\},1,otherwise\.\\begin\{gathered\}\\mathchar 29013\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\coloneq\\begin\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}0\\mathchar 24891\\relax&\\mathchar 29011\\relax\\mathchar 12349\\relax\\mathchar 571\\relax\\mathchar 24891\\relax\\\\ 0\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 24891\\relax&\\mathchar 29011\\relax\\mathchar 12850\\relax\\\{\\\{\\mathchar 28722\\relax\\\}\\mathchar 24891\\relax\\\{\\mathchar 28723\\relax\\\}\\\}\\mathchar 24891\\relax\\\\ \\mathchar 28721\\relax\\mathchar 24891\\relax&\\text\{otherwise\.\}\\end\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\end\{gathered\}\(21\)Then, one can verify that the solution ofb=Γ​\.25\\mathchar 29026\\relax\\mathchar 12349\\relax 0\\mathchar 314\\relax\\mathchar 28722\\relax\\mathchar 28725\\relaxand𝐱=\(Γ​\.5,Γ​\.25\+t,Γ​\.25−t\)\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12349\\relax\\delimiter 672734720\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 24891\\relax 0\\mathchar 314\\relax\\mathchar 28722\\relax\\mathchar 28725\\relax\\mathchar 8235\\relax\\mathchar 29044\\relax\\mathchar 24891\\relax 0\\mathchar 314\\relax\\mathchar 28722\\relax\\mathchar 28725\\relax\\mathchar 8704\\relax\\mathchar 29044\\relax\\delimiter 84054785is optimal for everyt∈\[−Γ​\.25,Γ​\.25\]\\mathchar 29044\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 8704\\relax 0\\mathchar 314\\relax\\mathchar 28722\\relax\\mathchar 28725\\relax\\mathchar 24891\\relax 0\\mathchar 314\\relax\\mathchar 28722\\relax\\mathchar 28725\\relax\\delimiter 84267779\.

Input:Utility function

Un\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}, regularization parameter

η=1​Γ−12\\mathchar 28945\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax 0^\{\\mathchar 8704\\relax\\mathchar 28721\\relax\\mathchar 28722\\relax\},

p∈\(1,∞\)∪\{∞,ELS\}\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785\\mathchar 8795\\relax\\\{\\mathchar 561\\relax\\mathchar 24891\\relax\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 29011\\relax\}\\\}, and sample budget

B\\mathchar 28994\\relax
Output:Approximation

𝐱^∗∈ℝn\\hat\{\\mathbf\{\\mathchar 29048\\relax\}\}^\{\\mathchar 8707\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}of

ϕp​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785
1

2

𝒮←\[\]\\mathcal\{\\mathchar 29011\\relax\}\\mathchar 12832\\relax\\delimiter 67482370\\ \\delimiter 84267779
3for*b=1,2,…,B\\mathchar 29026\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\dots\\mathchar 24891\\relax\\mathchar 28994\\relax*do

4Uniformly sample a subset

Sb⊆\[n\]\\mathchar 29011\\relax\_\{\\mathchar 29026\\relax\}\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779not already in

𝒮\\mathcal\{\\mathchar 29011\\relax\};

5if*p=ELS\\mathchar 29040\\relax\\mathchar 12349\\relax\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 29011\\relax\}*then

6Resample

Sb\\mathchar 29011\\relax\_\{\\mathchar 29026\\relax\}if

Sb∈\{∅,\[n\]\}\\mathchar 29011\\relax\_\{\\mathchar 29026\\relax\}\\mathchar 12850\\relax\\\{\\mathchar 571\\relax\\mathchar 24891\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\\}
7

8

𝒮\.append​\(Sb\)\\mathcal\{\\mathchar 29011\\relax\}\\mathchar 314\\relax\\mathrm\{\\mathchar 29025\\relax\\mathchar 29040\\relax\\mathchar 29040\\relax\\mathchar 29029\\relax\\mathchar 29038\\relax\\mathchar 29028\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\_\{\\mathchar 29026\\relax\}\\delimiter 84054785
9if*p∈\(1,∞\)∪\{∞\}\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785\\mathchar 8795\\relax\\\{\\mathchar 561\\relax\\\}*then

10

11Construct

𝐀∈ℝB×n\\mathbf\{\\mathchar 28993\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 28994\\relax\\mathchar 8706\\relax\\mathchar 29038\\relax\}and

𝐕∈ℝB\\mathbf\{\\mathchar 29014\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 28994\\relax\}using

𝒮\\mathcal\{\\mathchar 29011\\relax\}and

Un\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}
12

η←Γ\\mathchar 28945\\relax\\mathchar 12832\\relax 0if

p∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785
13

𝐱^∗←minimize𝐱∈ℝn,b∈ℝ‖𝐀𝐱\+b⋅𝟏B−𝐕‖p\+η⋅‖𝐱‖22\\hat\{\\mathbf\{\\mathchar 29048\\relax\}\}^\{\\mathchar 8707\\relax\}\\mathchar 12832\\relax\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\ \\delimiter 69645069\\mathbf\{\\mathchar 28993\\relax\\mathchar 29048\\relax\}\\mathchar 8235\\relax\\mathchar 29026\\relax\\mathchar 8705\\relax\\mathbf\{\\mathchar 28721\\relax\}\_\{\\mathchar 28994\\relax\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 69645069\_\{\\mathchar 29040\\relax\}\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}
14else if*p=ELC\\mathchar 29040\\relax\\mathchar 12349\\relax\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}*then

15

𝐱^∗←minimize𝐱∈ℝn,e∈ℝe\+η⋅‖𝐱‖22\\hat\{\\mathbf\{\\mathchar 29048\\relax\}\}^\{\\mathchar 8707\\relax\}\\mathchar 12832\\relax\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29029\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\ \\mathchar 29029\\relax\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\,subject to

Un​\(S\)−Un​\(∅\)−∑i∈Sxi≤e\\,\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 29048\\relax\_\{\\mathchar 29033\\relax\}\\mathchar 12820\\relax\\mathchar 29029\\relaxfor every

S∈𝒮\\mathchar 29011\\relax\\mathchar 12850\\relax\\mathcal\{\\mathchar 29011\\relax\},

Γ≤e0\\mathchar 12820\\relax\\mathchar 29029\\relax, and

∑i∈\[n\]xi=Un​\(\[n\]\)−Un​\(∅\)\\,\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 29048\\relax\_\{\\mathchar 29033\\relax\}\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785
16

return*𝐱^∗\\hat\{\\mathbf\{\\mathchar 29048\\relax\}\}^\{\\mathchar 8707\\relax\}*

Algorithm 1Approximation ofϕp​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785![Refer to caption](https://arxiv.org/html/2607.09869v1/x1.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x2.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x3.png)GPSP \(n=32\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28723\\relax\\mathchar 28722\\relax\)FOTP \(n=51\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28725\\relax\\mathchar 28721\\relax\)wave\_energy \(n=48\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28724\\relax\\mathchar 28728\\relax\)![Refer to caption](https://arxiv.org/html/2607.09869v1/x4.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x5.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x6.png)jannis \(n=54\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28725\\relax\\mathchar 28724\\relax\)spambase \(n=57\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28725\\relax\\mathchar 28727\\relax\)superconduct \(n=81\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28728\\relax\\mathchar 28721\\relax\)Figure 1:Comparison of attribution methods on six datasets for player ranking\. A larger area under the curve indicates better performance\. Since all utility functions contain more than3​Γ\\mathchar 28723\\relax 0players, nonlinear attribution methods are approximated by sampling\#​players×1,Γ​Γ​Γ\\\#\\mathrm\{\\mathchar 29040\\relax\\mathchar 29036\\relax\\mathchar 29025\\relax\\mathchar 29049\\relax\\mathchar 29029\\relax\\mathchar 29042\\relax\\mathchar 29043\\relax\}\\mathchar 8706\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax 000subsets, with mean and standard deviation reported over five random seeds\. Linear methods are computed exactly\. The linear and nonlinear curves shown are the best among11\\mathchar 28721\\relax\\mathchar 28721\\relaxand6\\mathchar 28726\\relaxcandidates respectively under the inclusion AUC metric\.###### Proposition 4\.2\.

The1\\mathchar 28721\\relax\-norm attribution methodϕ1\\mathchar 28958\\relax^\{\\mathchar 28721\\relax\}does not satisfy the consistency axiom\.

Such a failure of the1\\mathchar 28721\\relax\-norm attribution methodϕ1\\mathchar 28958\\relax^\{\\mathchar 28721\\relax\}arises becauseminimizeb∈ℝ​∑j=1m\|b−yj\|\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathchar 29026\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29034\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 29037\\relax\}\\delimiter 69640972\\mathchar 29026\\relax\\mathchar 8704\\relax\\mathchar 29049\\relax\_\{\\mathchar 29034\\relax\}\\delimiter 69640972does not necessarily yield a unique solution\. We provide a concrete counterexample to prove Proposition[4\.2](https://arxiv.org/html/2607.09869#S4.Thmtheorem2)\.

###### Proof\.

DefineU2∈𝒢2\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 28722\\relax\}by letting

U2​\(∅\)=Γ,U2​\(\{1\}\)=3,U2​\(\{2\}\)=2,U2​\(\{1,2\}\)=1\.\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 24891\\relax\\ \\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28723\\relax\\mathchar 24891\\relax\\ \\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\ \\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 314\\relax\\end\{gathered\}\(22\)Then,ϕ1​\(U2\)=\(Γ,Γ\)\\mathchar 28958\\relax^\{\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\delimiter 672734720\\mathchar 24891\\relax 0\\delimiter 84054785as the solution ofb=1\.5\\mathchar 29026\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 314\\relax\\mathchar 28725\\relaxand𝐱=\(Γ,Γ\)\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12349\\relax\\delimiter 672734720\\mathchar 24891\\relax 0\\delimiter 84054785is optimal to the problem

minimize𝐱∈ℝ2,b∈ℝ​∑S⊆\[2\]\|U2​\(S\)−b−∑i∈Sxi\|\.\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\\delimiter 69640972\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29026\\relax\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 29048\\relax\_\{\\mathchar 29033\\relax\}\\delimiter 69640972\\mathchar 314\\relax\\end\{gathered\}\(23\)The optimality is due to that the problem is convex andΓ\\mathbf\{0\}is a subgradient at that point\. In particular,\(b,Γ,Γ\)\\delimiter 67273472\\mathchar 29026\\relax\\mathchar 24891\\relax 0\\mathchar 24891\\relax 0\\delimiter 84054785withb∈\[1,2\]\\mathchar 29026\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\delimiter 84267779is optimal to the problem\. Then, we construct a utility functionU3∈𝒢3\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 28723\\relax\}by letting

U3​\(S\)≔\{U2​\(S\),3/⁣∈S,U2​\(S\)\+1,otherwise\.\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\coloneq\\begin\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 24891\\relax&\\mathchar 28723\\relax\\mathrel\{\{\\mathchar 566\\relax\\mathchar 562\\relax\}\}\\mathchar 29011\\relax\\mathchar 24891\\relax\\\\ \\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax&\\text\{otherwise\.\}\\end\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\end\{gathered\}\(24\)Note that

min𝐱∈ℝ3,b∈ℝ​∑S⊆\[3\]\|U3​\(S\)−b−∑i∈Sxi\|=2​e\.\\begin\{gathered\}\\min\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 28723\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\delimiter 69640972\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29026\\relax\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 29048\\relax\_\{\\mathchar 29033\\relax\}\\delimiter 69640972\\ \\mathchar 12349\\relax\\mathchar 28722\\relax\\mathchar 29029\\relax\\mathchar 314\\relax\\end\{gathered\}\(25\)Then, one can verify that the solution ofb=2\\mathchar 29026\\relax\\mathchar 12349\\relax\\mathchar 28722\\relaxand𝐱=\(Γ,Γ,Γ\)\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12349\\relax\\delimiter 672734720\\mathchar 24891\\relax 0\\mathchar 24891\\relax 0\\delimiter 84054785is optimal, which suggests thatϕ1​\(U3\)=\(Γ,Γ,Γ\)\\mathchar 28958\\relax^\{\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\delimiter 672734720\\mathchar 24891\\relax 0\\mathchar 24891\\relax 0\\delimiter 84054785\. With this example, it is clear thatϕ1\\mathchar 28958\\relax^\{\\mathchar 28721\\relax\}violates the axiom of consistency\. ∎

Nevertheless, for the remaining values ofp\\mathchar 29040\\relax, the resulting attribution methods are axiomatically sound\.

###### Theorem 4\.3\.

Forp∈\(1,∞\)∪\{∞\}\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785\\mathchar 8795\\relax\\\{\\mathchar 561\\relax\\\}, thep\\mathchar 29040\\relax\-norm attribution methodϕp\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}satisfies the axioms of consistency, equal treatment, monotonicity and translation invariance\. Whenn\>2\\mathchar 29038\\relax\\mathchar 12606\\relax\\mathchar 28722\\relax,ϕp\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}is linear if and only ifp=2\\mathchar 29040\\relax\\mathchar 12349\\relax\\mathchar 28722\\relax\.

#### Possible Weighted Extensions\.

Forp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785, we can define weighted versions of thep\\mathchar 29040\\relax\-norm objective while retaining the axiomatic guarantees of Theorem[4\.3](https://arxiv.org/html/2607.09869#S4.Thmtheorem3)\. Givenω∈\(Γ,1\)\\mathchar 28961\\relax\\mathchar 12850\\relax\\delimiter 672734720\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84054785andp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785, defineϕp,\!​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\\mathchar 24891\\relax\\mathchar 28961\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785as the unique optimal solution of𝐱\\mathbf\{\\mathchar 29048\\relax\}to the problem

minimize𝐱∈ℝn,b∈ℝ∑S⊆\[n\]\!\|S\|\(1−\!\)n−\|S\|\|An\(S;𝐱,b\)−Un\(S\)\|p\.\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 28961\\relax^\{\\delimiter 69640972\\mathchar 29011\\relax\\delimiter 69640972\}\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 28961\\relax\\delimiter 84054785^\{\\mathchar 29038\\relax\\mathchar 8704\\relax\\delimiter 69640972\\mathchar 29011\\relax\\delimiter 69640972\}\\left\\delimiter 69640972\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\right\\delimiter 69640972^\{\\mathchar 29040\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(26\)Then,ϕ2,\!\\mathchar 28958\\relax^\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28961\\relax\}is the weighted Banzhaf value parameterized byω\\mathchar 28961\\relax\(marichal2011weighted;li2023robust\)\. Though we did not find a rigorous way to prove thatϕp,\!\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\\mathchar 24891\\relax\\mathchar 28961\\relax\}is linear if and only ifp=2\\mathchar 29040\\relax\\mathchar 12349\\relax\\mathchar 28722\\relax, we conjecture that this claim holds true\.

![Refer to caption](https://arxiv.org/html/2607.09869v1/x7.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x8.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x9.png)letter \(n=16\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 28726\\relax\)pendigits\(n=16\)\\delimiter 67273472\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 28726\\relax\\delimiter 84054785EES\(n=14\)\\delimiter 67273472\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 28724\\relax\\delimiter 84054785![Refer to caption](https://arxiv.org/html/2607.09869v1/x10.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x11.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x12.png)WQW \(n=11\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 28721\\relax\)elevators \(n=18\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 28728\\relax\)credit \(n=1​Γ\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax 0\)Figure 2:Comparison of attribution methods on six datasets for feature ranking\. A larger area under the curve indicates better performance\. Since all utility functions contain fewer than2​Γ\\mathchar 28722\\relax 0players, all subsets are used for nonlinear methods and exact computations are performed for linear methods\. The linear and nonlinear curves shown are the best among11\\mathchar 28721\\relax\\mathchar 28721\\relaxand6\\mathchar 28726\\relaxcandidates respectively under the inclusion AUC metric\.###### Corollary 4\.4\.

For everyω∈\(Γ,1\)\\mathchar 28961\\relax\\mathchar 12850\\relax\\delimiter 672734720\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84054785and everyp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785, the weightedp\\mathchar 29040\\relax\-norm attribution methodϕp,\!\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\\mathchar 24891\\relax\\mathchar 28961\\relax\}satisfies the axioms of consistency, equal treatment, monotonicity and translation invariance\.

### 4\.4Optimization

Note that whenn\\mathchar 29038\\relaxis sufficiently large, all of the mentioned attribution methods can only be approximated, unless the utility functionUn\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}exhibits certain favorable properties\(e\.g\.,jia2019efficient;lundberg2020local\)\.

Forp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785,ϕp​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785can be effectively approximated by solving the minimization problem \([19](https://arxiv.org/html/2607.09869#S4.E19)\), since its solution is unique\. However, this is not the case forϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}, as it is defined as the optimal solution with the smallestℓ2\\mathchar 352\\relax\_\{\\mathchar 28722\\relax\}\-norm among potentially many minimizers\. To facilitate the approximation ofϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}, we establish the following theoretical result\.

###### Theorem 4\.5\.

For everyη\>Γ\\mathchar 28945\\relax\\mathchar 12606\\relax 0, letϕ∞​\(Un;η\)\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24635\\relax\\mathchar 28945\\relax\\delimiter 84054785denote an optimal solution of𝐱\\mathbf\{\\mathchar 29048\\relax\}to

minimize𝐱∈ℝn,b∈ℝ‖𝐀𝐱\+b⋅𝟏2n−𝐕​\(Un\)‖∞\+ȷ⋅‖𝐱‖22\.\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\ \\delimiter 69645069\\mathbf\{\\mathchar 28993\\relax\\mathchar 29048\\relax\}\\mathchar 8235\\relax\\mathchar 29026\\relax\\mathchar 8705\\relax\\mathbf\{\\mathchar 28721\\relax\}\_\{\\mathchar 28722\\relax^\{\\mathchar 29038\\relax\}\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(27\)Then, we haveϕ∞​\(Un;η\)→ϕ∞​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24635\\relax\\mathchar 28945\\relax\\delimiter 84054785\\mathchar 12833\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785asη→Γ\\mathchar 28945\\relax\\mathchar 12833\\relax 0\.

The key to proving this result is that the constraint‖𝐀𝐱−b⋅𝟏n−𝐕​\(Un\)‖∞≤e\\delimiter 69645069\\mathbf\{\\mathchar 28993\\relax\}\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 8704\\relax\\mathchar 29026\\relax\\mathchar 8705\\relax\\mathbf\{\\mathchar 28721\\relax\}\_\{\\mathchar 29038\\relax\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 12820\\relax\\mathchar 29029\\relaximplies boundedness of\(𝐱,b\)\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\delimiter 84054785\. While previous studies have focused on approximating the least core, it remains unclear how to approximate the egalitarian least coreϕELC\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\. Nevertheless, the regularization technique used for approximatingϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}can be extended toϕELC\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}as well\.

###### Theorem 4\.6\.

For everyη\>Γ\\mathchar 28945\\relax\\mathchar 12606\\relax 0, letϕELC​\(Un;η\)\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24635\\relax\\mathchar 28945\\relax\\delimiter 84054785denote the unique optimal solution of𝐱\\mathbf\{\\mathchar 29048\\relax\}to

minimize𝐱∈ℝn,e∈ℝe\+ȷ⋅‖𝐱‖22subject to​Un​\(S\)−An​\(S;𝐱,Un​\(∅\)\)≤e​for every​S⊊\[n\]and​An​\(\[n\];𝐱,Un​\(∅\)\)=Un​\(\[n\]\)\.\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29029\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\ \\mathchar 29029\\relax\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\\\ \\text\{subject to \}\\ \\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\mathchar 12820\\relax\\mathchar 29029\\relax\\ \\text\{ for every \}\\mathchar 29011\\relax\\subsetneq\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\\\ \\text\{and \}\\ \\mathchar 28993\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 24635\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24891\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(28\)Then, we haveϕELC​\(Un;η\)→ϕELC​\(Un\)\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24635\\relax\\mathchar 28945\\relax\\delimiter 84054785\\mathchar 12833\\relax\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785asη→Γ\\mathchar 28945\\relax\\mathchar 12833\\relax 0\.

Accordingly, we propose Algorithm[1](https://arxiv.org/html/2607.09869#algorithm1)to approximate the attribution methods of interest when exact computation is infeasible\. Specifically, all the minimization problems we solve are convex, and thus off\-the\-shelf convex solvers can be directly applied\.

Table 1:Average inclusion AUC scores of various attribution methods\. The best results are shown in bold, and the second\-best results are underlined\. Note that Beta Shapley reports the best one over1​Γ\\mathchar 28721\\relax 0candidates, as described in \([29](https://arxiv.org/html/2607.09869#S5.E29)\)\.

## 5Experiments

The goal of our experiments is to evaluate whether the nonlinear attribution methods introduced in this work outperform semi\-values in ranking players\. According to Theorem[4\.3](https://arxiv.org/html/2607.09869#S4.Thmtheorem3), the considered nonlinear attribution methods includeϕp\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}withp∈\{1\.5,2\.5,5,1​Γ,∞\}\\mathchar 29040\\relax\\mathchar 12850\\relax\\\{\\mathchar 28721\\relax\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 24891\\relax\\mathchar 28725\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax 0\\mathchar 24891\\relax\\mathchar 561\\relax\\\}as well asϕELC\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\. We use the CVXPY library with the CLARABEL solver to solve the minimization problems in Algorithm[1](https://arxiv.org/html/2607.09869#algorithm1)\(diamond2016cvxpy;agrawal2018rewriting\)\.

#### Utility functions\.

The utility function we employ isUf𝐱​\(S\)≔𝔼𝐗\[n\]\\S​\[f​\(𝐱S,𝐗\[n\]\\S\)\]\\mathchar 29013\\relax\_\{\\mathchar 29030\\relax\}^\{\\mathbf\{\\mathchar 29048\\relax\}\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\coloneq\\mathbb\{\\mathchar 28997\\relax\}\_\{\\mathbf\{\\mathchar 29016\\relax\}\_\{\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\mathchar 29011\\relax\}\}\\delimiter 67482370\\mathchar 29030\\relax\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 29011\\relax\}\\mathchar 24891\\relax\\mathbf\{\\mathchar 29016\\relax\}\_\{\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\mathchar 29011\\relax\}\\delimiter 84054785\\delimiter 84267779, wheref:ℝn→ℝ\\mathchar 29030\\relax\\mathchar 12346\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 12833\\relax\\mathbb\{\\mathchar 29010\\relax\}is a trained gradient boosted tree \(GBT\) with five estimators,𝐱∈ℝn\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}is an input instance, and𝐱S\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 29011\\relax\}is the restriction of𝐱\\mathbf\{\\mathchar 29048\\relax\}onS\\mathchar 29011\\relax\. Then, the number of players corresponds to the number of features\. We follow the path\-dependent definition proposed bylundberg2020local\. For this type of utility functions, the corresponding semi\-values can be computed in polynomial time\(karczmarz2022improved;Yu2022linear;muschalik2024beyond\), and we use the numerically stable version developed byli2026treegrad\. In general,ϕ¯​\(f𝐱\)\\mathchar 28958\\relax^\{\\mathchar 28950\\relax\}\\delimiter 67273472\\mathchar 29030\\relax\_\{\\mathbf\{\\mathchar 29048\\relax\}\}\\delimiter 84054785can only be approximated, and our rationale is to avoid such approximation error\. All GBTs are trained using the scikit\-learn library\(pedregosa2011scikit\)\. We train one GBT for each dataset\. To account for the variability ofUf𝐱\\mathchar 29013\\relax\_\{\\mathchar 29030\\relax\}^\{\\mathbf\{\\mathchar 29048\\relax\}\}, for each trained GBTf\\mathchar 29030\\relax, the reported curve is averaged over1​Γ​Γ\\mathchar 28721\\relax 00different instances of𝐱\\mathbf\{\\mathchar 29048\\relax\}\. For classification datasets,f\\mathchar 29030\\relaxoutputs the logit of a randomly selected class\.

#### Task\.

The task is to rank players, where performance is evaluated using the inclusion AUC metric defined in problem \([10](https://arxiv.org/html/2607.09869#S3.E10)\)\. A higher value indicates better performance\. According to Lemma[3\.3](https://arxiv.org/html/2607.09869#S3.Thmtheorem3), the player rankingπ\\mathchar 28953\\relaxinduced by a specific contribution vectorŒ\\bm\{\\mathchar 28958\\relax\}satisfies thatϕß​\(j\)≥ϕß​\(j\+1\)\\mathchar 28958\\relax\_\{\\mathchar 28953\\relax\\delimiter 67273472\\mathchar 29034\\relax\\delimiter 84054785\}\\mathchar 12821\\relax\\mathchar 28958\\relax\_\{\\mathchar 28953\\relax\\delimiter 67273472\\mathchar 29034\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\delimiter 84054785\}for every1≤j<n\\mathchar 28721\\relax\\mathchar 12820\\relax\\mathchar 29034\\relax\\mathchar 12604\\relax\\mathchar 29038\\relax\. Intuitively,Uf𝐱\\mathchar 29013\\relax\_\{\\mathchar 29030\\relax\}^\{\\mathbf\{\\mathchar 29048\\relax\}\}is supposed to increase sharply after including the players that contribute the most\.

#### Datasets\.

We employ12\\mathchar 28721\\relax\\mathchar 28722\\relaxdatasets in total, which are summarized in Table[2](https://arxiv.org/html/2607.09869#A2.T2)\. In particular, the datasets we selected are equally divided into two group\. Each dataset in the first group contains fewer than 20 features, whereas those in the second group contain more than 30 features\. For decision trees trained using the first group of datasets, we solve the exact minimization problem for each considered nonlinear attribution method\. For the other group of datasets, the budgetB\\mathchar 28994\\relaxin Algorithm[1](https://arxiv.org/html/2607.09869#algorithm1)is set to\#​players×1,Γ​Γ​Γ\\\#\\mathrm\{\\mathchar 29040\\relax\\mathchar 29036\\relax\\mathchar 29025\\relax\\mathchar 29049\\relax\\mathchar 29029\\relax\\mathchar 29042\\relax\\mathchar 29043\\relax\}\\mathchar 8706\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax 000\. For the two regression datasets, we scale the predicted values for better presentation\.

#### Baselines\.

We compare the selected nonlinear attribution methods to Beta Shapley values\(kwon2022beta\)\. The considered candidates include Beta\(α,β\)\\delimiter 67273472\\mathchar 28939\\relax\\mathchar 24891\\relax\\mathchar 28940\\relax\\delimiter 84054785with

\(ff,fi\)∈\{\\displaystyle\\delimiter 67273472\\mathchar 28939\\relax\\mathchar 24891\\relax\\mathchar 28940\\relax\\delimiter 84054785\\mathchar 12850\\relax\\\{\(16,1\),\(8,1\),\(4,1\),\(2,1\),\(1,1\),\\displaystyle\\delimiter 67273472\\mathchar 28726\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84054785\\mathchar 24891\\relax\\delimiter 67273472\\mathchar 28728\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84054785\\mathchar 24891\\relax\\delimiter 67273472\\mathchar 28724\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84054785\\mathchar 24891\\relax\\delimiter 67273472\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84054785\\mathchar 24891\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84054785\\mathchar 24891\\relax\(29\)\(1,2\),\(1,4\),\(1,8\),\(1,16\),\(1,32\)\},\\displaystyle\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\delimiter 84054785\\mathchar 24891\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28724\\relax\\delimiter 84054785\\mathchar 24891\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28728\\relax\\delimiter 84054785\\mathchar 24891\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28726\\relax\\delimiter 84054785\\mathchar 24891\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\delimiter 84054785\\\}\\mathchar 24891\\relaxwhich were used in\(kwon2022weightedshap\)\. Meanwhile, we also compare with another well\-known semi\-value, the Banzhaf value\(banzhaf1965weighted\)\. Therefore, a total of11\\mathchar 28721\\relax\\mathchar 28721\\relaxlinear attribution methods are used\.

For reproducibility, we fix the random seed for training GBTs and selecting logits\. When performing approximation, we average the results over five different random seeds and report the mean along with the corresponding standard deviation\. More experimental details and statistical results can be found in Appendix[B](https://arxiv.org/html/2607.09869#A2)\.

### 5\.1Empirical Observations

Results for utility functions containing fewer than2​Γ\\mathchar 28722\\relax 0players are shown in Figure[2](https://arxiv.org/html/2607.09869#S4.F2), while those for datasets with more than3​Γ\\mathchar 28723\\relax 0players are presented in Figure[1](https://arxiv.org/html/2607.09869#S4.F1)\. For clarity, we plot only the inclusion curves of the best linear and nonlinear candidates that maximize the inclusion AUC\. In Appendix[B](https://arxiv.org/html/2607.09869#A2), we also plot the inclusion curves for all employed nonlinear attribution methods\. The average inclusion AUC scores are reported in Table[1](https://arxiv.org/html/2607.09869#S4.T1)\. We make the following observations: \(i\) nonlinear attribution methods tend to outperform linear attribution methods under the inclusion AUC metric; and \(ii\) no single attribution method consistently outperforms others across all datasets\.

## 6Conclusion

While the Shapley value is widely adopted for its axiomatic properties, we demonstrate that its linearity in utility functions can lead to unreliable/undesirable player ranking under the inclusion AUC metric\. Moreover, the efficiency axiom has also been empirically proved unfavorable\. Accordingly, we introduce a class of attribution methods by relaxing the linearity and efficiency axioms while preserving other desirable axioms such as consistency, equal treatment, monotonicity, and translation invariance\. In particular, these introduced nonlinear attribution methods are based on faithful additive approximation of utility functions\. We view our work as the first step to establish nonlinear axiomatic attribution methods through the lens of additive approximation\.

###### Acknowledgements\.

This research is supported by the National Research Foundation Singapore and the Singapore Ministry of Digital Development and Innovation, National AI Group under the AI Visiting Professorship Programme \(award number AIVP\-

2​Γ​24\\mathchar 28722\\relax 0\\mathchar 28722\\relax\\mathchar 28724\\relax\-

Γ​Γ​100\\mathchar 28721\\relax\)\. YY gratefully acknowledges NSERC and CIFAR for funding support\.

## References

Nonlinear Axiomatic Attribution for Cooperative Games \(Supplementary Material\)

## Appendix AMissing Proofs

See[3\.1](https://arxiv.org/html/2607.09869#S3.Thmtheorem1)

###### Proof\.

Efficiency\. This is imposed by the constraint∑i∈\[n\]xi=Un​\(\[n\]\)−Un​\(∅\)\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 29048\\relax\_\{\\mathchar 29033\\relax\}\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785that appears in the problem \([4](https://arxiv.org/html/2607.09869#S3.E4)\)\.

Translation Invariance\. This is clear as it is\{Un​\(S1\)−Un​\(S2\)\}\\\{\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\_\{\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\\}that matters in the problem \([4](https://arxiv.org/html/2607.09869#S3.E4)\)\.

Equal Treatment\. Suppose there existi,j∈\[n\]\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779such thatUn​\(S∪\{i\}\)=Un​\(S∪\{j\}\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785for everyS⊆\[n\]\\\{i,j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\\}\. For the sake of contradiction, assumeϕiELC​\(Un\)\>ϕjELC​\(Un\)\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12606\\relax\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\_\{\\mathchar 29034\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\. Define a contribution vectorŒ∈ℝn\\bm\{\\mathchar 28958\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}by letting

Œk≔\{ŒkELC​\(Un\),k∈\[n\]\\\{i,j\}ŒiELC​\(Un\)\+ŒjELC​\(Un\)2,otherwise\.\\begin\{gathered\}\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}\\coloneq\\begin\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\_\{\\mathchar 29035\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax&\\mathchar 29035\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\\}\\\\ \{\{\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\over\\mathchar 28722\\relax\}\}\\mathchar 24891\\relax&\\text\{otherwise\.\}\\end\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\end\{gathered\}\(30\)Let

F​\(S\)≔Un​\(S∪\{i\}\)−Un​\(∅\)−∑k∈SŒkELC​\(Un\)\.\\begin\{gathered\}\\mathchar 28998\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\coloneq\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29035\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(31\)Since

F​\(S\)−Œi≤max⁡\(F​\(S\)−ŒiELC​\(Un\),F​\(S\)−ŒjELC​\(Un\),Γ\)\\begin\{gathered\}\\mathchar 28998\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\mathchar 12820\\relax\\max\\delimiter 67273472\\mathchar 28998\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\ \\mathchar 28998\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\ 0\\delimiter 84054785\\end\{gathered\}\(32\)for everyS⊆\[n\]\\\{i,j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\\},Œ\\bm\{\\mathchar 28958\\relax\}withe=maxS⊆\[n\]⁡Un​\(S\)−∑i∈SϕiELC​\(Un\)\\mathchar 29029\\relax\\mathchar 12349\\relax\\max\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785is another optimal solution to the problem \([4](https://arxiv.org/html/2607.09869#S3.E4)\) \. However,

2×\(ŒiELC​\(Un\)\+ŒjELC​\(Un\)2\)2<ŒiELC​\(Un\)2\+ŒjELC​\(Un\)2,\\begin\{gathered\}\\mathchar 28722\\relax\\mathchar 8706\\relax\\left\\delimiter 67273472\{\{\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\over\\mathchar 28722\\relax\}\}\\right\\delimiter 84054785^\{\\mathchar 28722\\relax\}\\mathchar 12604\\relax\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785^\{\\mathchar 28722\\relax\}\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785^\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\end\{gathered\}\(33\)a contradiction to thatϕELC​\(Un\)\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785is the least\-norm optimal solution\.

Consistency\. Suppose there existsC∈ℝ\\mathchar 28995\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}such thatUn\+1​\(S\)=Un​\(S∩\[n\]\)\+C⋅𝟙n\+1∈S\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8796\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28995\\relax\\mathchar 8705\\relax\\mathds\{\\mathchar 28721\\relax\}\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}for everyS⊆\[n\+1\]\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\delimiter 84267779\. First, we prove thatϕn\+1ELC​\(Un\+1\)≥C\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 12821\\relax\\mathchar 28995\\relax\. For convenience, we write

d​\(S\)≔Un\+1​\(S\)−Un\+1​\(∅\)−∑i∈SŒiELC​\(Un\+1\)​for every​S⊆\[n\+1\]and​e=maxS⊆\[n\]⁡d​\(S\)≥C−Un\+1ELC​\(Un\+1\)\>Γ\.\\begin\{gathered\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\coloneqq\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\ \\text\{ for every \}\\ \\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\delimiter 84267779\\\\ \\text\{and \}\\ \\mathchar 29029\\relax\\mathchar 12349\\relax\\max\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 12821\\relax\\mathchar 28995\\relax\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 12606\\relax 0\\mathchar 314\\relax\\end\{gathered\}\(34\)For the sake of contradiction, assumeϕn\+1ELC​\(Un\+1\)<C\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 12604\\relax\\mathchar 28995\\relax\. Then, we have

d​\(S\)<d​\(S\)\+\(C−Un\+1ELC​\(Un\+1\)\)=d​\(S∪\{n\+1\}\)​for every​S⊆\[n\]\.\\begin\{gathered\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 12604\\relax\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8235\\relax\\delimiter 67273472\\mathchar 28995\\relax\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\\}\\delimiter 84054785\\ \\text\{ for every \}\\ \\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 314\\relax\\end\{gathered\}\(35\)It indicates that

d​\(S\)=e⟹n\+1∈S\.\\begin\{gathered\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29029\\relax\\implies\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\\mathchar 314\\relax\\end\{gathered\}\(36\)Define a contributionŒ∈ℝn\+1\\bm\{\\mathchar 28958\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}by letting

Œk≔\{ŒkELC​\(Un\+1\)\+C−Œn\+1ELC​\(Un\+1\)3,k=n\+1,ŒkELC​\(Un\+1\)−C−Œn\+1ELC​\(Un\+1\)3​n,otherwise\.\\begin\{gathered\}\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}\\coloneq\\begin\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\{\{\\mathchar 28995\\relax\\mathchar 8704\\relax\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\over\\mathchar 28723\\relax\}\}\\mathchar 24891\\relax&\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\\\\[3\.0pt\] \\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\{\{\\mathchar 28995\\relax\\mathchar 8704\\relax\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\over\\mathchar 28723\\relax\\mathchar 29038\\relax\}\}\\mathchar 24891\\relax&\\text\{otherwise\.\}\\end\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\end\{gathered\}\(37\)Then,Œ\\bm\{\\mathchar 28958\\relax\}satisfies the efficiency constraint, and for everyS⊆\[n\]\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779

Un\+1​\(S\)−Un\+1​\(∅\)−∑i∈SŒi\\displaystyle\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}≤dn\+1​\(S\)\+C−Œn\+1ELC​\(Un\+1\)3\\displaystyle\\mathchar 12820\\relax\\mathchar 29028\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8235\\relax\{\{\\mathchar 28995\\relax\\mathchar 8704\\relax\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\over\\mathchar 28723\\relax\}\}\(38\)<dn\+1​\(S∪\{n\+1\}\)−C−Œn\+1ELC​\(Un\+1\)3\\displaystyle\\mathchar 12604\\relax\\mathchar 29028\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\{\{\\mathchar 28995\\relax\\mathchar 8704\\relax\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\over\\mathchar 28723\\relax\}\}≤Un\+1​\(S∪\{n\+1\}\)−Un\+1​\(∅\)−∑i∈S∪\{n\+1\}Œi\.\\displaystyle\\mathchar 12820\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\\}\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\mathchar 314\\relaxIn particular,

Un\+1​\(S∪\{n\+1\}\)−Un\+1​\(∅\)−∑i∈S∪\{n\+1\}Œi<d​\(S∪\{n\+1\}\)\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\\}\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\mathchar 12604\\relax\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\\}\\delimiter 84054785\\end\{gathered\}\(39\)for everyS⊊\[n\]\\mathchar 29011\\relax\\subsetneq\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779, which contradicts the optimality of\(ϕELC​\(Un\+1\),e\)\\delimiter 67273472\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\,\\mathchar 29029\\relax\\delimiter 84054785\. Therefore, we haveϕn\+1ELC​\(Un\+1\)≥C\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 12821\\relax\\mathchar 28995\\relax\.

To haveϕn\+1ELC​\(Un\+1\)≤C\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 12820\\relax\\mathchar 28995\\relax, the same argument applies by noticing that

e≥d​\(\[n\]\)\>d​\(\[n\+1\]\)=Γ\\begin\{gathered\}\\mathchar 29029\\relax\\mathchar 12821\\relax\\mathchar 29028\\relax\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 12606\\relax\\mathchar 29028\\relax\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 12349\\relax 0\\end\{gathered\}\(40\)if we assumeϕn\+1ELC​\(Un\+1\)\>C\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 12606\\relax\\mathchar 28995\\relax\. As a result, we haveϕn\+1ELC​\(Un\+1\)=C\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28995\\relax\. With a moment’s thought,ϕn\+1ELC​\(Un\+1\)=C\\mathchar 28958\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28995\\relaximplies

ŒkELC​\(Un\+1\)=ŒkELC​\(Un\)​for every​k∈\[n\]\.\\begin\{gathered\}\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\ \\text\{ for every \}\\ \\mathchar 29035\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 314\\relax\\end\{gathered\}\(41\)
Monotonicity\. Suppose there existsj∈\[n\]\\mathchar 29034\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779such thatUn​\(S∪\{j\}\)≥Un​\(S\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 12821\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785for everyS⊆\[n\]\\\{j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\. Assume, for the sake of contradiction, thatϕjELC<Γ\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\mathchar 12604\\relax 0\. Then,

r≔maxS⊆\[n\]⁡Un​\(S\)−Un​\(∅\)−∑i∈SŒiELC​\(Un\)≥Un​\(\{j\}\)−Un​\(∅\)−ŒjELC​\(Un\)\>Γ,\\begin\{gathered\}\\mathchar 29042\\relax\\coloneq\\max\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12821\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12606\\relax 0\\mathchar 24891\\relax\\end\{gathered\}\(42\)and, for everyS⊆\[n\]\\\{j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\},

Un​\(S\)−Un​\(∅\)−∑i∈SŒiELC​\(Un\)<Un​\(S∪\{j\}\)−Un​\(∅\)−∑i∈S∪\{j\}ŒiELC​\(Un\),\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12604\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\end\{gathered\}\(43\)which indicates

Un​\(S\)−Un​\(∅\)−∑i∈SŒiELC​\(Un\)=r⟹j∈S\.\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29042\\relax\\implies\\mathchar 29034\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\\mathchar 314\\relax\\end\{gathered\}\(44\)Selectδ\>Γ\\mathchar 28942\\relax\\mathchar 12606\\relax 0such that

ffi<minS⊆\[n\]\\\{j\}⁡Un​\(S∪\{j\}\)−Un​\(S\)−ŒjELC​\(Un\)\.\\begin\{gathered\}\\mathchar 28942\\relax\\mathchar 12604\\relax\\min\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(45\)DefineŒ∈ℝn\\bm\{\\mathchar 28958\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}by letting

Œk≔\{ŒkELC​\(Un\)\+ffi3,k=jŒkELC​\(Un\)−ffi3​\(n−1\),otherwise\.\\begin\{gathered\}\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}\\coloneq\\begin\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\{\{\\mathchar 28942\\relax\\over\\mathchar 28723\\relax\}\}\\mathchar 24891\\relax&\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 29034\\relax\\\\\[3\.0pt\] \\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\{\{\\mathchar 28942\\relax\\over\\mathchar 28723\\relax\\delimiter 67273472\\mathchar 29038\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\\delimiter 84054785\}\}\\mathchar 24891\\relax&\\text\{otherwise\.\}\\end\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\end\{gathered\}\(46\)Observe thatŒ\\bm\{\\mathchar 28958\\relax\}satisfies the efficiency constraint, and for everyS⊆\[n\]\\\{j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}

Un​\(S\)−Un​\(∅\)−∑i∈SŒi\\displaystyle\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}≤Un​\(S\)−Un​\(∅\)−∑i∈SŒiELC​\(Un\)\+ffi3\\displaystyle\\mathchar 12820\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\{\{\\mathchar 28942\\relax\\over\\mathchar 28723\\relax\}\}\(47\)<Un​\(S∪\{j\}\)−Un​\(∅\)−∑i∈S∪\{j\}ŒiELC​\(Un\)−ffi3\\displaystyle\\mathchar 12604\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\{\{\\mathchar 28942\\relax\\over\\mathchar 28723\\relax\}\}≤Un​\(S∪\{j\}\)−Un​\(∅\)−∑i∈S∪\{j\}Œi\.\\displaystyle\\mathchar 12820\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\mathchar 314\\relaxSpecifically,

Un​\(S∪\{j\}\)−Un​\(∅\)−∑S∪\{j\}Œi<Un​\(S∪\{j\}\)−Un​\(∅\)−∑S∪\{j\}ŒiELC​\(Un\)\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}\\mathchar 12604\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\end\{gathered\}\(48\)for everyS⊊\[n\]\\\{j\}\\mathchar 29011\\relax\\subsetneq\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}, contradicting the optimality of\(ϕELC​\(Un\),r\)\\delimiter 67273472\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\,\\mathchar 29042\\relax\\delimiter 84054785\. Consequently, we haveϕjELC​\(Un\)≥Γ\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12821\\relax 0\.

On the other hand, suppose there existsj∈\[n\]\\mathchar 29034\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779such thatUn​\(S∪\{j\}\)≤Un​\(S\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 12820\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785for everyS⊆\[n\]\\\{j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\. Then,ϕjELC​\(Un\)≤Γ\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12820\\relax 0can be proved similarly by noting that

r≥Un​\(\[n\]\\\{j\}\)−Un​\(∅\)−∑i∈\[n\]\\\{j\}ŒiELC​\(Un\)\>Un​\(\[n\]\)−Un​\(∅\)−∑i∈\[n\]ŒiELC​\(Un\)=Γ\\begin\{gathered\}\\mathchar 29042\\relax\\mathchar 12821\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12606\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax 0\\end\{gathered\}\(49\)if we assumeϕjELC​\(Un\)\>Γ\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12606\\relax 0\.

Non\-Linearity\. DefineU2∈𝒢2\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 28722\\relax\}by letting

U2​\(∅\)=U2​\(\{1,2\}\)=Γ,U2​\(\{1\}\)=−1,and​U2​\(\{2\}\)=−2\.\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 24891\\relax\\ \\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\,\\text\{ and \}\\,\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 8704\\relax\\mathchar 28722\\relax\\mathchar 314\\relax\\end\{gathered\}\(50\)Then, it is clear thatϕELC​\(U2\)=\(Γ,Γ\)⊤\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\delimiter 672734720\\mathchar 24891\\relax 0\\delimiter 84054785^\{\\mathchar 574\\relax\}\. IfϕELC\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}is linear, it would reduce to the Shapley value\[shapley1953value\], as we have proved thatϕELC\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}satisfies the axioms of equal treatment, consistency, and efficiency\. However, the Shapley value ofU2\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}is\(12,−12\)⊤\\delimiter 67273472\{\{\\mathchar 28721\\relax\\over\\mathchar 28722\\relax\}\}\\mathchar 24891\\relax\\mathchar 8704\\relax\{\{\\mathchar 28721\\relax\\over\\mathchar 28722\\relax\}\}\\delimiter 84054785^\{\\mathchar 574\\relax\}, which indicates thatϕELC\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}must be non\-linear\. ∎

See[4\.1](https://arxiv.org/html/2607.09869#S4.Thmtheorem1)

###### Proof\.

According toyokote2016new, a basis of the null space of the Shapley operator is\{UT\}T∈𝒩\\\{\\mathchar 29013\\relax\_\{\\mathchar 29012\\relax\}\\\}\_\{\\mathchar 29012\\relax\\mathchar 12850\\relax\\mathcal\{\\mathchar 29006\\relax\}\}where𝒩≔\{S⊆\[n\]\|\|S\|/⁣=1\}\\mathcal\{\\mathchar 29006\\relax\}\\coloneq\\\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 12906\\relax\\delimiter 69640972\\mathchar 29011\\relax\\delimiter 69640972\\mathrel\{\{\\mathchar 566\\relax\\mathchar 61\\relax\}\}\\mathchar 28721\\relax\\\}\. Specifically,

U∅​\(S\)=1​for every​S⊆\[n\],and​UT​\(S\)=\{1,\|T∩S\|=1,Γ,otherwise\.\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 571\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28721\\relax\\ \\text\{ for every \}\\ \\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 24891\\relax\\\\ \\text\{and \}\\ \\mathchar 29013\\relax\_\{\\mathchar 29012\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 12349\\relax\\begin\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\mathchar 28721\\relax\\mathchar 24891\\relax&\\delimiter 69640972\\mathchar 29012\\relax\\mathchar 8796\\relax\\mathchar 29011\\relax\\delimiter 69640972\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\\\ 0\\mathchar 24891\\relax&\\text\{otherwise\.\}\\end\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\end\{gathered\}\(51\)Let2\[n\]\\mathchar 28722\\relax^\{\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}and𝒩\\mathcal\{\\mathchar 29006\\relax\}be ordered so that everyUT\\mathchar 29013\\relax\_\{\\mathchar 29012\\relax\}can be written as a vector𝐚T∈\{Γ,1\}2n\\mathbf\{\\mathchar 29025\\relax\}\_\{\\mathchar 29012\\relax\}\\mathchar 12850\\relax\\\{0\\mathchar 24891\\relax\\mathchar 28721\\relax\\\}^\{\\mathchar 28722\\relax^\{\\mathchar 29038\\relax\}\}, all of which constitute a matrix𝐀∈\{Γ,1\}2n×\|𝒩\|\\mathbf\{\\mathchar 28993\\relax\}\\mathchar 12850\\relax\\\{0\\mathchar 24891\\relax\\mathchar 28721\\relax\\\}^\{\\mathchar 28722\\relax^\{\\mathchar 29038\\relax\}\\mathchar 8706\\relax\\delimiter 69640972\\mathcal\{\\mathchar 29006\\relax\}\\delimiter 69640972\}\. Then, every utility functionUn′\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 29038\\relax\}in this null space can be expressed asUn′=𝐀​ff\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 29038\\relax\}\\mathchar 12349\\relax\\mathbf\{\\mathchar 28993\\relax\}\\bm\{\\mathchar 28939\\relax\}whereff∈ℝ\|𝒩\|\\bm\{\\mathchar 28939\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\delimiter 69640972\\mathcal\{\\mathchar 29006\\relax\}\\delimiter 69640972\}\.

Given a permutationπ∈Πn\\mathchar 28953\\relax\\mathchar 12850\\relax\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\}, we write

Sk​\(ß\)=\{ß1,ß2,…,ßk\}​for every​Γ≤k≤n\.\\begin\{gathered\}\\mathchar 29011\\relax\_\{\\mathchar 29035\\relax\}\\delimiter 67273472\\mathchar 28953\\relax\\delimiter 84054785\\mathchar 12349\\relax\\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\dots\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 29035\\relax\}\\\}\\ \\text\{ for every \}0\\mathchar 12820\\relax\\mathchar 29035\\relax\\mathchar 12820\\relax\\mathchar 29038\\relax\\mathchar 314\\relax\\end\{gathered\}\(52\)For each permutationπ∈Πn\\mathchar 28953\\relax\\mathchar 12850\\relax\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\}, it can be represented by a vector𝐩ß∈\{Γ,1\}2n\\mathbf\{\\mathchar 29040\\relax\}\_\{\\mathchar 28953\\relax\}\\mathchar 12850\\relax\\\{0\\mathchar 24891\\relax\\mathchar 28721\\relax\\\}^\{\\mathchar 28722\\relax^\{\\mathchar 29038\\relax\}\}whoseS\\mathchar 29011\\relax\-th entry is1\\mathchar 28721\\relaxifS∈\{Sk​\(π\):Γ≤k≤n\}\\mathchar 29011\\relax\\mathchar 12850\\relax\\\{\\mathchar 29011\\relax\_\{\\mathchar 29035\\relax\}\\delimiter 67273472\\mathchar 28953\\relax\\delimiter 84054785\\mathchar 24634\\relax 0\\mathchar 12820\\relax\\mathchar 29035\\relax\\mathchar 12820\\relax\\mathchar 29038\\relax\\\}andΓotherwise\. Then, we have

AUC​\(ß;Un′\)=𝐩ß⊤​𝐀​ff−Un′​\(∅\)\.\\begin\{gathered\}\\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 28953\\relax\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}^\{\\mathchar 560\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathbf\{\\mathchar 29040\\relax\}\_\{\\mathchar 28953\\relax\}^\{\\mathchar 574\\relax\}\\mathbf\{\\mathchar 28993\\relax\}\\bm\{\\mathchar 28939\\relax\}\\mathchar 8704\\relax\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(53\)For convenience, we write𝐝ß⊤=𝐩ß⊤​𝐀\\mathbf\{\\mathchar 29028\\relax\}\_\{\\mathchar 28953\\relax\}^\{\\mathchar 574\\relax\}\\mathchar 12349\\relax\\mathbf\{\\mathchar 29040\\relax\}\_\{\\mathchar 28953\\relax\}^\{\\mathchar 574\\relax\}\\mathbf\{\\mathchar 28993\\relax\}for everyπ∈Πn\\mathchar 28953\\relax\\mathchar 12850\\relax\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\}\. Next, we prove that every𝐝ß⊤\\mathbf\{\\mathchar 29028\\relax\}\_\{\\mathchar 28953\\relax\}^\{\\mathchar 574\\relax\}can be re\-ordered to the same vector\. Note that this is equivalent to ordering𝒩\\mathcal\{\\mathchar 29006\\relax\}according to the givenπ\\mathchar 28953\\relax\. Without loss of generality, we demonstrate this ordering process by usingn=4\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28724\\relaxas an example\.

For subsets that includeπ1\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}, their order is generated by

\{ß1\}\\displaystyle\\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\\}\(54\)\{ß1\},\{ß1,ß4\},\\displaystyle\\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\\}\\mathchar 24891\\relax\{ß1\},\{ß1,ß4\},\{ß1,ß3\},\{ß1,ß4,ß3\}\\displaystyle\\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\{ß1\},\{ß1,ß4\},\{ß1,ß3\},\{ß1,ß4,ß3\},\{ß1,ß2\},\{ß1,ß4,ß2\},\{ß1,ß3,ß2\},\{ß1,ß4,ß3,ß2\}\.\\displaystyle\\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\\}\\mathchar 314\\relaxFollowing these ordered subsets, the subsets that includeπ2\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}but notπ1\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}are ordered as

\{ß2\}\\displaystyle\\\{\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\\}\(55\)\{ß2\},\{ß2,ß4\}\\displaystyle\\\{\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\\}\{ß2\},\{ß2,ß4\},\{ß2,ß3\},\{ß2,ß4,ß3\}\.\\displaystyle\\\{\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\\mathchar 314\\relaxFinally, it is

\{ß3\}\\displaystyle\\\{\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\(56\)\{ß3\},\{ß3,ß4\}\.\\displaystyle\\\{\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\\}\\mathchar 314\\relaxThe empty set is appended to these ordered non\-empty subsets\. The order is finalized by removing all the singleton subsets\. Precisely, theπ\\mathchar 28953\\relax\-specific order for𝒩\\mathcal\{\\mathchar 29006\\relax\}is

\{ß1,ß4\},\{ß1,ß3\},\{ß1,ß4,ß3\},\{ß1,ß2\},\{ß1,ß4,ß2\},\{ß1,ß3,ß2\},\{ß1,ß4,ß3,ß2\},\{ß2,ß4\},\{ß2,ß3\},\{ß2,ß4,ß3\},\{ß3,ß4\},∅\.\\begin\{gathered\}\\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\\}\\mathchar 24891\\relax\\\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\\}\\mathchar 24891\\relax\\ \\\{\\mathchar 28953\\relax\_\{\\mathchar 28723\\relax\}\\mathchar 24891\\relax\\mathchar 28953\\relax\_\{\\mathchar 28724\\relax\}\\\}\\mathchar 24891\\relax\\ \\mathchar 571\\relax\\mathchar 314\\relax\\end\{gathered\}\(57\)One can verify that, using this order for𝒩\\mathcal\{\\mathchar 29006\\relax\},𝐝ß⊤=\(3,2,2,1,1,1,1,2,1,1,1,5\)\\mathbf\{\\mathchar 29028\\relax\}\_\{\\mathchar 28953\\relax\}^\{\\mathchar 574\\relax\}\\mathchar 12349\\relax\\delimiter 67273472\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28725\\relax\\delimiter 84054785\. In other words, for1≤i≤2n−1\\mathchar 28721\\relax\\mathchar 12820\\relax\\mathchar 29033\\relax\\mathchar 12820\\relax\\mathchar 28722\\relax^\{\\mathchar 29038\\relax\}\\mathchar 8704\\relax\\mathchar 28721\\relax, thei\\mathchar 29033\\relax\-th entry of𝐝ß\\mathbf\{\\mathchar 29028\\relax\}\_\{\\mathchar 28953\\relax\}is equal to the number of the occurrences of the corresponding subset in the ordering process, whereas the last entry is equal ton\+1\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\.

From now on, assume the order of𝒩\\mathcal\{\\mathchar 29006\\relax\}is fixed\. Consequently, we can partitionΠn\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\}into\{Πn,j\}j\\\{\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\}\\\}\_\{\\mathchar 29034\\relax\}such that, for everyπ,π′∈Πn,j\\mathchar 28953\\relax\\mathchar 24891\\relax\\mathchar 28953\\relax^\{\\mathchar 560\\relax\}\\mathchar 12850\\relax\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\},𝐝ß=𝐝ß′\\mathbf\{\\mathchar 29028\\relax\}\_\{\\mathchar 28953\\relax\}\\mathchar 12349\\relax\\mathbf\{\\mathchar 29028\\relax\}\_\{\\mathchar 28953\\relax^\{\\mathchar 560\\relax\}\}\. From the ordering process, it is clear that\|\{Πn,j\}j\|≥2\\delimiter 69640972\\\{\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\}\\\}\_\{\\mathchar 29034\\relax\}\\delimiter 69640972\\mathchar 12821\\relax\\mathchar 28722\\relaxifn≥3\\mathchar 29038\\relax\\mathchar 12821\\relax\\mathchar 28723\\relax\.

For eachΠn,j\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\}, letUn′=𝐀𝐝ß\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 29038\\relax\}\\mathchar 12349\\relax\\mathbf\{\\mathchar 28993\\relax\}\\mathbf\{\\mathchar 29028\\relax\}\_\{\\mathchar 28953\\relax\}whereπ∈Πn,j\\mathchar 28953\\relax\\mathchar 12850\\relax\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\}\. It is clear thatAUC​\(π;Un′\)\>AUC​\(π′;Un′\)\\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 28953\\relax\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}^\{\\mathchar 560\\relax\}\\delimiter 84054785\\mathchar 12606\\relax\\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}\\delimiter 67273472\\mathchar 28953\\relax^\{\\mathchar 560\\relax\}\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}^\{\\mathchar 560\\relax\}\\delimiter 84054785for everyπ′/⁣∈Πn,j\\mathchar 28953\\relax^\{\\mathchar 560\\relax\}\\mathrel\{\{\\mathchar 566\\relax\\mathchar 562\\relax\}\}\\mathchar 28677\\relax\_\{\\mathchar 29038\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\}\. By choosing an appropriate scalarc\>Γ\\mathchar 29027\\relax\\mathchar 12606\\relax 0, we have the result by settingUn′=c⋅𝐀𝐝ß\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 29038\\relax\}\\mathchar 12349\\relax\\mathchar 29027\\relax\\mathchar 8705\\relax\\mathbf\{\\mathchar 28993\\relax\}\\mathbf\{\\mathchar 29028\\relax\}\_\{\\mathchar 28953\\relax\}\.

∎

See[4\.3](https://arxiv.org/html/2607.09869#S4.Thmtheorem3)

###### Proof\.

We divide our proof into two parts\. The first part is to demonstrate the properties ofϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}, whereas the second part goes forp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785\. Obviously,ϕp\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}withp∈\(Γ,∞\)∪\{∞\}\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 672734720\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785\\mathchar 8795\\relax\\\{\\mathchar 561\\relax\\\}satisfies the axiom of translation invariance due to the bias variableb\\mathchar 29026\\relax\.

For convenience, we write

d​\(S,𝐱;Un\)≔Un​\(S\)−bp​\(Un\)−∑i∈Sxi,and​ep​\(Un\)≔min𝐱∈ℝn,b∈ℝ⁡‖𝐀𝐱\+b⋅𝟏2n−𝐕​\(Un\)‖p\.\\begin\{gathered\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\coloneq\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 29048\\relax\_\{\\mathchar 29033\\relax\}\\mathchar 24891\\relax\\\\ \\text\{and \}\\ \\mathchar 29029\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\coloneq\\min\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\delimiter 69645069\\mathbf\{\\mathchar 28993\\relax\\mathchar 29048\\relax\}\\mathchar 8235\\relax\\mathchar 29026\\relax\\mathchar 8705\\relax\\mathbf\{\\mathchar 28721\\relax\}\_\{\\mathchar 28722\\relax^\{\\mathchar 29038\\relax\}\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 29040\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(58\)wherebp​\(Un\)\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785is the optimal value ofb\\mathchar 29026\\relaxin then minimization problem with𝐱=ϕ∞​\(Un\)\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12349\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\.

Note that

e∞​\(Un\)=maxS⊆\[n\]⁡d​\(S,Œ∞​\(Un\);Un\)−minS⊆\[n\]⁡d​\(S,Œ∞​\(Un\);Un\)2\.\\begin\{gathered\}\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\{\{\\max\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\\min\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\over\\mathchar 28722\\relax\}\}\\mathchar 314\\relax\\end\{gathered\}\(59\)
Besides, forp∈\(Γ,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 672734720\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785, defineψp:ℝ→ℝ\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\mathchar 12346\\relax\\mathbb\{\\mathchar 29010\\relax\}\\mathchar 12833\\relax\\mathbb\{\\mathchar 29010\\relax\}by letting

̵p​\(t\)=sign​\(t\)⋅\|t\|p−1\.\\begin\{gathered\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29044\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathrm\{\\mathchar 29043\\relax\\mathchar 29033\\relax\\mathchar 29031\\relax\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29044\\relax\\delimiter 84054785\\mathchar 8705\\relax\\delimiter 69640972\\mathchar 29044\\relax\\delimiter 69640972^\{\\mathchar 29040\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(60\)
ϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}satisfies the axiom of consistency\.

This is straightforward by observing that the bias variableb\\mathchar 29026\\relaxensures that there exist two subsetsS,T⊆\[n\]\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 29012\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779such that

d​\(S,Œ∞​\(Un\);Un\)=e∞​\(Un\)​and​d​\(T,Œ∞​\(Un\);Un\)=−e∞​\(Un\)\.\\begin\{gathered\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\ \\text\{ and \}\\ \\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29012\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 8704\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(61\)
ϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}satisfies the axiom of monotonicity\. Suppose there existsj∈\[n\]\\mathchar 29034\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779such thatUn​\(S∪\{j\}\)≥Un​\(S\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 12821\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785for everyS⊆\[n\]\\\{j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\. Assume, for the sake of contradiction, thatϕj∞​\(Un\)<Γ\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\_\{\\mathchar 29034\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12604\\relax 0\. Then, for every subset𝐒⊆\[n\]\\\{j\}\\mathbf\{\\mathchar 29011\\relax\}\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}

d​\(S,Œ∞​\(Un\);Un\)<d​\(S∪\{j\},Œ∞​\(Un\);Un\)\.\\begin\{gathered\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12604\\relax\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(62\)It indicates that

d​\(S,Œ∞​\(Un\);Un\)=e∞​\(Un\)⟹j∈S​and​d​\(S,Œ∞​\(Un\);Un\)=−e∞​\(Un\)⟹j/⁣∈S\.\\begin\{gathered\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\implies\\mathchar 29034\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\\ \\text\{ and \}\\ \\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 8704\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\implies\\mathchar 29034\\relax\\mathrel\{\{\\mathchar 566\\relax\\mathchar 562\\relax\}\}\\mathchar 29011\\relax\\mathchar 314\\relax\\end\{gathered\}\(63\)Select

Γ<ffi<minS⊆\[n\]\\\{j\}⁡d​\(S∪\{j\},Œ∞​\(Un\);Un\)−minS⊆\[n\]\\\{j\}⁡d​\(S,Œ∞​\(Un\);Un\)\.\\begin\{gathered\}0\\mathchar 12604\\relax\\mathchar 28942\\relax\\mathchar 12604\\relax\\min\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\\min\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(64\)DefineŒ∈ℝn\\bm\{\\mathchar 28958\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}by letting

Œk≔\{Œk∞​\(Un\),k/⁣=j,Œk∞​\(Un\)\+ffi,otherwise\.\\begin\{gathered\}\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}\\coloneq\\begin\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax&\\mathchar 29035\\relax\\mathrel\{\{\\mathchar 566\\relax\\mathchar 61\\relax\}\}\\mathchar 29034\\relax\\mathchar 24891\\relax\\\\ \\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28942\\relax\\mathchar 24891\\relax&\\text\{otherwise\.\}\\end\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\end\{gathered\}\(65\)Clearly,

minS⊆\[n\]\\\{j\}⁡d​\(S∪\{j\},Œ;Un\)<minS⊆\[n\]\\\{j\}⁡d​\(S∪\{j\},Œ∞​\(Un\);Un\),\\begin\{gathered\}\\min\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\mathchar 24891\\relax\\bm\{\\mathchar 28958\\relax\}\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12604\\relax\\min\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\end\{gathered\}\(66\)which, by Eq\. \([59](https://arxiv.org/html/2607.09869#A1.E59)\), contradicts the optimality of\(ϕ∞​\(Un\),b∞​\(Un\)\)\\delimiter 67273472\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\,\\mathchar 29026\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\. Therefore, we must haveϕj∞​\(Un\)≥Γ\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12821\\relax 0\. The other case can be proved similarly\.

ϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}satisfies the axiom of equal treatment\. Suppose there existi,j∈\[n\]\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779such thatUn​\(S∪\{i\}\)=Un​\(S∪\{j\}\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785for everyS⊆\[n\]\\\{i,j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\\}\. For the sake of contradiction, assume thatϕi∞​\(Un\)\>ϕ∞​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12606\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\. DefineŒ∈ℝn\\bm\{\\mathchar 28958\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}by letting

Œk≔\{Œi∞​\(Un\)\+Œj∞​\(Un\)2,k∈\{i,j\},Œk∞​\(Un\),otherwise\.\\begin\{gathered\}\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}\\coloneq\\begin\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\{\{\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\over\\mathchar 28722\\relax\}\}\\mathchar 24891\\relax&\\mathchar 29035\\relax\\mathchar 12850\\relax\\\{\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\\}\\mathchar 24891\\relax\\\\ \\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax&\\text\{otherwise\.\}\\end\{\\mathchar 29027\\relax\\mathchar 29025\\relax\\mathchar 29043\\relax\\mathchar 29029\\relax\\mathchar 29043\\relax\}\\end\{gathered\}\(67\)Observe that for everyS⊆\[n\]\\\{i,j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\\}

\|d​\(S∪\{i\},Œ;Un\)\|<max⁡\(d​\(S∪\{i\},Œ∞​\(Un\);Un\),d​\(S∪\{j\},Œ∞​\(Un\);Un\)\)\.\\begin\{gathered\}\\delimiter 69640972\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\mathchar 24891\\relax\\bm\{\\mathchar 28958\\relax\}\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69640972\\mathchar 12604\\relax\\max\\left\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\,\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\right\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(68\)It means that the solution of\(Œ,b∗\)\\delimiter 67273472\\bm\{\\mathchar 28958\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax^\{\\mathchar 8707\\relax\}\\delimiter 84054785is also

2×\(Œi∞​\(Un\)\+Œj∞​\(Un\)2\)2<Œi∞​\(Un\)2\+Œj∞​\(Un\)2,\\begin\{gathered\}\\mathchar 28722\\relax\\mathchar 8706\\relax\\left\\delimiter 67273472\{\{\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\over\\mathchar 28722\\relax\}\}\\right\\delimiter 84054785^\{\\mathchar 28722\\relax\}\\mathchar 12604\\relax\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785^\{\\mathchar 28722\\relax\}\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785^\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\end\{gathered\}\(69\)a contradiction to thatϕ∞​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785is the least\-norm optimal solution\.

ϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}is not linear\. Assume, for the sake of contradiction, thatϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}is linear\. Sinceϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}satisfies the axioms of linearity, consistency, equal treatment, and monotonicity, according toweber1988probabilistic,ϕ∞​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785can be expressed as

Œi∞​\(Un\)=∑S⊆\[n\]\\\{i\}q\|S\|\+1​\[Un​\(S∪\{i\}\)−Un​\(S\)\]\\begin\{gathered\}\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\\}\}\\mathchar 29041\\relax\_\{\\delimiter 69640972\\mathchar 29011\\relax\\delimiter 69640972\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67482370\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\delimiter 84267779\\end\{gathered\}\(70\)where𝐪∈ℝn\\mathbf\{\\mathchar 29041\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}is non\-negative and satisfies that∑k=1n\(n−1\)Γptk−1​qk=1\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 29038\\relax\}\{\{\\mathchar 29038\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\\abovewithdelims\( 0\.0pt\\delimiter 840547850\\mathchar 29040\\relax\\mathchar 29044\\relax\\mathchar 29035\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\}\}\\mathchar 29041\\relax\_\{\\mathchar 29035\\relax\}\\mathchar 12349\\relax\\mathchar 28721\\relax\. Fort∈\[3,5\]\\mathchar 29044\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 28725\\relax\\delimiter 84267779, defineU3t∈𝒢3\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 28723\\relax\}by letting

U3t​\(∅\)=Γ,U3t​\(\{1\}\)=U3t​\(\{2,3\}\)=U3t​\(\{1,2,3\}\)=4,U3t​\(\{2\}\)=U3t​\(\{3\}\)=3,and​U3t​\(\{1,2\}\)=U3t​\(\{1,3\}\)=t\.\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 24891\\relax\\ \\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 67273472\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28724\\relax\\mathchar 24891\\relax\\ \\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 67273472\\\{\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 67273472\\\{\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28723\\relax\\mathchar 24891\\relax\\\\ \\text\{and \}\\ \\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29044\\relax\\mathchar 314\\relax\\end\{gathered\}\(71\)Using the constraints for∅\\mathchar 571\\relaxand\{1\}\\\{\\mathchar 28721\\relax\\\}, we have

b∞​\(U3t\)∈\[−e∞​\(U3t\),e∞​\(U3t\)\]​and​b\+Œ1∞​\(U3t\)∈\[4−e∞​\(U3t\),4\+e∞​\(U3t\)\],\\begin\{gathered\}\\mathchar 29026\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 8704\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\,\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\delimiter 84267779\\ \\text\{ and \}\\ \\mathchar 29026\\relax\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 28724\\relax\\mathchar 8704\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\,\\mathchar 28724\\relax\\mathchar 8235\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\delimiter 84267779\\mathchar 24891\\relax\\end\{gathered\}\(72\)which leads to

Œ1∞​\(U3t\)∈\[4−2​e∞​\(U3t\),4\+2​e∞​\(U3t\)\]\.\\begin\{gathered\}\\mathchar 28958\\relax\_\{\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 28724\\relax\\mathchar 8704\\relax\\mathchar 28722\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\,\\mathchar 28724\\relax\\mathchar 8235\\relax\\mathchar 28722\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\delimiter 84267779\\mathchar 314\\relax\\end\{gathered\}\(73\)Using the constraint for\{2,3\}\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}, there is

b∗\+Œ2∞​\(U3t\)\+Œ3∞​\(U3t\)∈\[4−e∞​\(U3t\),4\+e∞​\(U3t\)\]\\begin\{gathered\}\\mathchar 29026\\relax^\{\\mathchar 8707\\relax\}\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 28722\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 28724\\relax\\mathchar 8704\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\,\\mathchar 28724\\relax\\mathchar 8235\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\delimiter 84267779\\end\{gathered\}\(74\)Combining Eqs\. \([73](https://arxiv.org/html/2607.09869#A1.E73)\) and \([74](https://arxiv.org/html/2607.09869#A1.E74)\) yields

b∗\+Œ1∞​\(U3t\)\+Œ2∞​\(U3t\)\+Œ3∞​\(U3t\)∈\[8−3​e∞​\(U3t\),8\+3​e∞​\(U3t\)\]\.\\begin\{gathered\}\\mathchar 29026\\relax^\{\\mathchar 8707\\relax\}\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 28722\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 28728\\relax\\mathchar 8704\\relax\\mathchar 28723\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\,\\mathchar 28728\\relax\\mathchar 8235\\relax\\mathchar 28723\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\delimiter 84267779\\mathchar 314\\relax\\end\{gathered\}\(75\)On the other hand, from the constraint for\{1,2,3\}\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}, there is

b∗\+Œ1∞​\(U3t\)\+Œ2∞​\(U3t\)\+Œ3∞​\(U3t\)∈\[4−e∞​\(U3t\),4\+e∞​\(U3t\)\]\.\\begin\{gathered\}\\mathchar 29026\\relax^\{\\mathchar 8707\\relax\}\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 28722\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28958\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 28724\\relax\\mathchar 8704\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 24891\\relax\\,\\mathchar 28724\\relax\\mathchar 8235\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\delimiter 84267779\\mathchar 314\\relax\\end\{gathered\}\(76\)Solving4\+e∞​\(U3t\)=8−3​e∞​\(U3t\)\\mathchar 28724\\relax\\mathchar 8235\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28728\\relax\\mathchar 8704\\relax\\mathchar 28723\\relax\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785yields an lower bound fore∞​\(U3t\)\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785, i\.e\.,e∞​\(U3t\)≥1\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 12821\\relax\\mathchar 28721\\relax\. Ife∗=1\\mathchar 29029\\relax^\{\\mathchar 8707\\relax\}\\mathchar 12349\\relax\\mathchar 28721\\relax, following the derivation of Eq\. \([73](https://arxiv.org/html/2607.09869#A1.E73)\), the lower bounds forϕ1∞​\(U3t\)\\mathchar 28958\\relax\_\{\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785,ϕ2∞​\(U3t\)\\mathchar 28958\\relax\_\{\\mathchar 28722\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785, andϕ3∞​\(U3t\)\\mathchar 28958\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785are2\\mathchar 28722\\relax,1\\mathchar 28721\\relax, and1\\mathchar 28721\\relax, respectively\. Then, one can verify that the solution ofb=1\\mathchar 29026\\relax\\mathchar 12349\\relax\\mathchar 28721\\relaxand𝐱=\(2,1,1\)\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12349\\relax\\delimiter 67273472\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84054785is indeed optimal, leading toe∞​\(U3t\)=1\\mathchar 29029\\relax\_\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28721\\relax\. In other words,ϕ∞​\(U3t\)=\(2,1,1\)\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\delimiter 67273472\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28721\\relax\\delimiter 84054785for everyt∈\[3,5\]\\mathchar 29044\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 28725\\relax\\delimiter 84267779\.

Then, it indicates that

q1=1,q2=q3=Γ\.\\begin\{gathered\}\\mathchar 29041\\relax\_\{\\mathchar 28721\\relax\}\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\ \\mathchar 29041\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 12349\\relax\\mathchar 29041\\relax\_\{\\mathchar 28723\\relax\}\\mathchar 12349\\relax 0\\mathchar 314\\relax\\end\{gathered\}\(77\)However, Eq\. \([70](https://arxiv.org/html/2607.09869#A1.E70)\) producesϕ∞​\(U3t\)=\(4,3,3\)\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\delimiter 67273472\\mathchar 28724\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\delimiter 84054785, contradicting to the uniqueness ofϕ∞​\(U3t\)\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}^\{\\mathchar 29044\\relax\}\\delimiter 84054785\. Therefore,ϕ∞\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}must be nonlinear\.

ϕp\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}withp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785satisfies the axiom of consistency\. Suppose there existsC∈ℝ\\mathchar 28995\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}such thatUn\+1​\(S\)=Un​\(S∩\[n\]\)\+C⋅𝟙n\+1∈S\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8796\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28995\\relax\\mathchar 8705\\relax\\mathds\{\\mathchar 28721\\relax\}\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}for everyS⊆\[n\+1\]\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\delimiter 84267779\. Since the corresponding minimization problem is strictly convex, using the first order condition for optimality, we have

∑S⊆\[n\]̵p​\(d​\(S,Œp​\(Un\);Un\)\)=Γand​∑S⊆\[n\]\\\{i\}̵p​\(d​\(S∪\{i\},Œp​\(Un\);Un\)\)=Γ​for every​i∈\[n\]\.\\begin\{gathered\}\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax 0\\\\ \\text\{and \}\\ \\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\\}\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax 0\\ \\text\{ for every \}\\ \\mathchar 29033\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 314\\relax\\end\{gathered\}\(78\)DefineŒ∈ℝn\+1\\bm\{\\mathchar 28958\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}by lettingϕk=ϕkp​\(Un\)\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}\\mathchar 12349\\relax\\mathchar 28958\\relax\_\{\\mathchar 29035\\relax\}^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785ifk∈\[n\]\\mathchar 29035\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779andC\\mathchar 28995\\relaxotherwise\. Define

d′​\(S,𝐱\)≔Un\+1​\(S\)−bp​\(Un\)−∑i∈Sxi\.\\begin\{gathered\}\\mathchar 29028\\relax^\{\\mathchar 560\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathbf\{\\mathchar 29048\\relax\}\\delimiter 84054785\\coloneq\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 29048\\relax\_\{\\mathchar 29033\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(79\)Verifying the first order condition forUn\+1\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\},

∑S⊆\[n\+1\]̵p​\(d′​\(S,Œ\)\)=2​∑S⊆\[n\]̵p​\(d​\(S,Œp​\(Un\);Un\)\)=Γ,∑S⊆\[n\+1\]\\\{i\}̵d​\(d′​\(S∪\{i\},Œ\)\)=2​∑S⊆\[n\]\\\{i\}̵p​\(d​\(S∪\{i\},Œp​\(Un\);Un\)\)=Γ​for every​i∈\[n\],∑S⊆\[n\]̵d​\(d′​\(S∪\{n\+1\},Œ\)\)=∑S⊆\[n\]̵d​\(d​\(S,Œp​\(Un\);Un\)\)=Γ\.\\begin\{gathered\}\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\delimiter 84267779\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax^\{\\mathchar 560\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\bm\{\\mathchar 28958\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28722\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 24891\\relax\\\\ \\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\\}\}\\mathchar 28960\\relax\_\{\\mathchar 29028\\relax\}\\delimiter 67273472\\mathchar 29028\\relax^\{\\mathchar 560\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\mathchar 24891\\relax\\bm\{\\mathchar 28958\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28722\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\\}\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax 0\\ \\text\{ for every \}\\mathchar 29033\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 24891\\relax\\\\ \\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 28960\\relax\_\{\\mathchar 29028\\relax\}\\delimiter 67273472\\mathchar 29028\\relax^\{\\mathchar 560\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\\}\\mathchar 24891\\relax\\bm\{\\mathchar 28958\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 28960\\relax\_\{\\mathchar 29028\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 314\\relax\\end\{gathered\}\(80\)
ϕp\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}withp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785satisfies the axiom of monotonicity\. Suppose there existsj∈\[n\]\\mathchar 29034\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779such thatUn​\(S∪\{j\}\)≥Un​\(S\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785\\mathchar 12821\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785for everyS⊆\[n\]\\\{j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\. Using Eq\. \([78](https://arxiv.org/html/2607.09869#A1.E78)\), we have

∑S⊆\[n\]\\\{j\}̵d​\(d​\(S,Œp​\(Un\);Un\)\)=∑S⊆\[n\]\\\{j\}̵d​\(d​\(S∪\{j\},Œp​\(Un\);Un\)\)=Γ\\begin\{gathered\}\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 28960\\relax\_\{\\mathchar 29028\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 28960\\relax\_\{\\mathchar 29028\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax 0\\end\{gathered\}\(81\)Notice that the functionψd\\mathchar 28960\\relax\_\{\\mathchar 29028\\relax\}is strictly increasing\. Ifϕjp​\(Un\)<Γ\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12604\\relax 0, for everyS⊆\[n\]\\\{j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}, we have

d​\(S,Œp​\(Un\);Un\)<d​\(S∪\{j\},Œp​\(Un\);Un\)\.\\begin\{gathered\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12604\\relax\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(82\)Then,

∑S⊆\[n\]\\\{j\}̵d​\(d​\(S,Œp​\(Un\);Un\)\)<∑S⊆\[n\]\\\{j\}̵d​\(d​\(S∪\{j\},Œp​\(Un\);Un\)\),\\begin\{gathered\}\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 28960\\relax\_\{\\mathchar 29028\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12604\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29034\\relax\\\}\}\\mathchar 28960\\relax\_\{\\mathchar 29028\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 24891\\relax\\end\{gathered\}\(83\)which makes Eq\. \([81](https://arxiv.org/html/2607.09869#A1.E81)\) impossible\. Therefore,ϕjp​\(Un\)≥Γ\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12821\\relax 0\. The other case can be proved similarly\.

ϕp\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}withp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785satisfies the axiom of equal treatment\. Suppose there existi,j∈\[n\]\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779such thatUn​\(S∪\{i\}\)=Un​\(S∪\{j\}\)\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\delimiter 84054785for everyS⊆\[n\]\\\{i,j\}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\\}\. Using Eq\. \([78](https://arxiv.org/html/2607.09869#A1.E78)\), we have

∑S⊆\[n\]\\\{i,j\}Œp​\(d​\(S∪\{i\},Œp​\(Un\);Un\)\)=∑S⊆\[n\]\\\{i,j\}Œp​\(d​\(S∪\{j\},Œp​\(Un\);Un\)\)\.\\begin\{gathered\}\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\\}\}\\mathchar 28958\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 8814\\relax\\\{\\mathchar 29033\\relax\\mathchar 24891\\relax\\mathchar 29034\\relax\\\}\}\\mathchar 28958\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29034\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(84\)Since the functionϕp\\mathchar 28958\\relax\_\{\\mathchar 29040\\relax\}is strictly increasing, this equality implies thatϕip​\(Un\)=ϕjp​\(Un\)\\mathchar 28958\\relax\_\{\\mathchar 29033\\relax\}^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28958\\relax\_\{\\mathchar 29034\\relax\}^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\.

ϕp\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}withp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785is linear if and only ifp=2\\mathchar 29040\\relax\\mathchar 12349\\relax\\mathchar 28722\\relax\. It is obvious thatϕ2\\mathchar 28958\\relax^\{\\mathchar 28722\\relax\}is linear; therefore, we focus on proving the converse\. Supposeϕp\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}withp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785is linear\. Sinceϕp\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}satisfies the axioms of linearity, consistency, equal treatment, and monotonicity, according toweber1988probabilistic, we have

Œip​\(Un\)=∑S⊆\[n\]⊆\{i\}fl\|S\|\+1\[n\]​\[Un​\(S∪\{i\}\)−Un​\(S\)\]​for every​i∈\[n\]\\begin\{gathered\}\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\_\{\\mathchar 29033\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 12818\\relax\\\{\\mathchar 29033\\relax\\\}\}\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\_\{\\delimiter 69640972\\mathchar 29011\\relax\\delimiter 69640972\\mathchar 8235\\relax\\mathchar 28721\\relax\}\\delimiter 67482370\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\mathchar 8795\\relax\\\{\\mathchar 29033\\relax\\\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\delimiter 84267779\\ \\text\{ for every \}\\mathchar 29033\\relax\\mathchar 12850\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\end\{gathered\}\(85\)wherefl\[n\]∈ℝn\\bm\{\\mathchar 28941\\relax\}^\{\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}is non\-negative and satisfies that∑k=1n\(n−1\)Γptk−1​γk\[n\]=1\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 29038\\relax\}\{\{\\mathchar 29038\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\\abovewithdelims\( 0\.0pt\\delimiter 840547850\\mathchar 29040\\relax\\mathchar 29044\\relax\\mathchar 29035\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\}\}\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\}\_\{\\mathchar 29035\\relax\}\\mathchar 12349\\relax\\mathchar 28721\\relax\. DefineU2∈𝒢2\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 28722\\relax\}by letting

U2​\(∅\)=Γ​and​U2​\(\{1\}\)=U2​\(\{2\}\)=U2​\(\{1,2\}\)=1\.\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 12349\\relax 0\\ \\text\{ and \}\\ \\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 314\\relax\\end\{gathered\}\(86\)Then,

Œ1p​\(U2\)=Œ2p​\(U2\)=fl1\[2\]\.\\begin\{gathered\}\\mathchar 28958\\relax\_\{\\mathchar 28721\\relax\}^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28958\\relax\_\{\\mathchar 28722\\relax\}^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\_\{\\mathchar 28721\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(87\)Using the first order condition, as described in Eq\. \([78](https://arxiv.org/html/2607.09869#A1.E78)\), there is

̵p​\(−bp​\(U2\)\)=̵p​\(fl1\[2\]\+bp​\(U2\)−1\)​and​̵p​\(1−fl1\[2\]−bp​\(U2\)\)=̵p​\(2​fl1\[2\]\+bp​\(U2\)−1\)\.\\begin\{gathered\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 8704\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\_\{\\mathchar 28721\\relax\}\\mathchar 8235\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28721\\relax\\delimiter 84054785\\ \\text\{ and \}\\ \\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\_\{\\mathchar 28721\\relax\}\\mathchar 8704\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 28722\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\_\{\\mathchar 28721\\relax\}\\mathchar 8235\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28721\\relax\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(88\)Sinceψp\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}is strictly increasing, it leads to

−bp​\(U2\)=fl1\[2\]\+bp​\(U2\)−1​and​1−fl1\[2\]−bp​\(U2\)=2​fl1\[2\]\+bp​\(U2\)−1,\\begin\{gathered\}\\mathchar 8704\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\_\{\\mathchar 28721\\relax\}\\mathchar 8235\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28721\\relax\\ \\text\{ and \}\\ \\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\_\{\\mathchar 28721\\relax\}\\mathchar 8704\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28722\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\_\{\\mathchar 28721\\relax\}\\mathchar 8235\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\end\{gathered\}\(89\)which, combined withγ1\[2\]\+γ2\[2\]=1\\mathchar 28941\\relax\_\{\\mathchar 28721\\relax\}^\{\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\\mathchar 8235\\relax\\mathchar 28941\\relax\_\{\\mathchar 28722\\relax\}^\{\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\\mathchar 12349\\relax\\mathchar 28721\\relax, yields

fl1\[2\]=fl2\[2\]=Γ​\.5\.\\begin\{gathered\}\\mathchar 28941\\relax\_\{\\mathchar 28721\\relax\}^\{\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\\mathchar 12349\\relax\\mathchar 28941\\relax\_\{\\mathchar 28722\\relax\}^\{\\delimiter 67482370\\mathchar 28722\\relax\\delimiter 84267779\}\\mathchar 12349\\relax 0\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 314\\relax\\end\{gathered\}\(90\)Then, employing the axiom of consistency, we obtain

fl1\[3\]\+fl2\[3\]=fl2\[3\]\+fl3\[3\]=Γ​\.5\.\\begin\{gathered\}\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\_\{\\mathchar 28721\\relax\}\\mathchar 8235\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\_\{\\mathchar 28722\\relax\}\\mathchar 12349\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\_\{\\mathchar 28722\\relax\}\\mathchar 8235\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\_\{\\mathchar 28723\\relax\}\\mathchar 12349\\relax 0\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 314\\relax\\end\{gathered\}\(91\)It is worth pointing out thatϕp\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}is linear for everyp∈\(1,∞\)\\mathchar 29040\\relax\\mathchar 12850\\relax\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 561\\relax\\delimiter 84054785ifn=2\\mathchar 29038\\relax\\mathchar 12349\\relax\\mathchar 28722\\relax, and thus we requiren≥3\\mathchar 29038\\relax\\mathchar 12821\\relax\\mathchar 28723\\relax\. DefineU3∈𝒢3\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 28723\\relax\}by letting

U3​\(∅\)=U3​\(\{2\}\)=U2​\(\{1,2\}\)=U2​\(\{1,3\}\)=Γ,U3​\(\{1\}\)=U3​\(\{1,2,3\}\)=1,U2​\(\{3\}\)=−2​fl1\[3\],and​U2​\(\{2,3\}\)=1−2​fl1\[3\]\.\\begin\{gathered\}\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\\{\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 24891\\relax\\ \\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax\\\\ \\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 8704\\relax\\mathchar 28722\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\_\{\\mathchar 28721\\relax\}\\mathchar 24891\\relax\\,\\text\{ and \}\\,\\mathchar 29013\\relax\_\{\\mathchar 28722\\relax\}\\delimiter 67273472\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 28722\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\_\{\\mathchar 28721\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(92\)Using the first order condition, as described in Eq\. \([78](https://arxiv.org/html/2607.09869#A1.E78)\), there is

̵p​\(d​\(\{3\},Œp​\(U3\);U3\)\)\+̵p​\(d​\(\{1,3\},Œp​\(U3\);U3\)\)\+̵p​\(d​\(\{2,3\},Œp​\(U3\);U3\)\)\+̵p​\(d​\(\{1,2,3\},Œp​\(U3\);U3\)\)=Γ\.\\begin\{gathered\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\\\ \\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 314\\relax\\end\{gathered\}\(93\)Using Eqs\. \([85](https://arxiv.org/html/2607.09869#A1.E85)\) and \([91](https://arxiv.org/html/2607.09869#A1.E91)\), one can verify that

d​\(\{3\},Œp​\(U3\);U3\)=d​\(\{1,3\},Œp​\(U3\);U3\)=−bp​\(U3\)−fl1\[3\]−fl3\[3\],and​d​\(\{2,3\},Œp​\(U3\);U3\)=d​\(\{1,2,3\},Œp​\(U3\);U3\)=1−3​fl3\[3\]−bp​\(U3\)\.\\begin\{gathered\}\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 8704\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\_\{\\mathchar 28721\\relax\}\\mathchar 8704\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\_\{\\mathchar 28723\\relax\}\\mathchar 24891\\relax\\\\ \\text\{and \}\\ \\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 28723\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\_\{\\mathchar 28723\\relax\}\\mathchar 8704\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(94\)Thus, Eq\. \([93](https://arxiv.org/html/2607.09869#A1.E93)\) can be simplified as

̵p​\(1−3​fl3\[3\]−bp​\(U3\)\)=̵p​\(bp​\(U3\)\+fl1\[3\]\+fl3\[3\]\)\.\\begin\{gathered\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 28723\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\_\{\\mathchar 28723\\relax\}\\mathchar 8704\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28941\\relax\_\{\\mathchar 28721\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 8235\\relax\\mathchar 28941\\relax\_\{\\mathchar 28723\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(95\)Sinceψp\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}is strictly increasing, it leads to

1−3​fl3\[3\]−bp​\(U3\)=bp​\(U3\)\+fl1\[3\]\+fl3\[3\],\\begin\{gathered\}\\mathchar 28721\\relax\\mathchar 8704\\relax\\mathchar 28723\\relax\\mathchar 28941\\relax^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\_\{\\mathchar 28723\\relax\}\\mathchar 8704\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28941\\relax\_\{\\mathchar 28721\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 8235\\relax\\mathchar 28941\\relax\_\{\\mathchar 28723\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 24891\\relax\\end\{gathered\}\(96\)which yields

bp​\(U3\)=Γ​\.5−2​fl3\[3\]−Γ​\.5​fl1\[3\]\.\\begin\{gathered\}\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 8704\\relax\\mathchar 28722\\relax\\mathchar 28941\\relax\_\{\\mathchar 28723\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 8704\\relax 0\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 28941\\relax\_\{\\mathchar 28721\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 314\\relax\\end\{gathered\}\(97\)Using the first order condition again,

̵p​\(d​\(∅,Œp​\(U3\);U3\)\)\+̵p​\(d​\(\{2\},Œp​\(U3\);U3\)\)\+̵p​\(d​\(\{3\},Œp​\(U3\);U3\)\)\+̵p​\(d​\(\{2,3\},Œp​\(U3\);U3\)\)=Γ\.\\begin\{gathered\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 571\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28722\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\\\ \\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 314\\relax\\end\{gathered\}\(98\)Using Eqs \([85](https://arxiv.org/html/2607.09869#A1.E85)\), \([91](https://arxiv.org/html/2607.09869#A1.E91)\) and \([97](https://arxiv.org/html/2607.09869#A1.E97)\),

d​\(∅,Œp​\(U3\);U3\)=Γ​\.75−2\.5​fl2\[3\],d​\(\{2\},Œp​\(U3\);U3\)=Γ​\.25−1\.5​fl2\[3\],d​\(\{3\},Œp​\(U3\);U3\)=−Γ​\.5​fl2\[3\]−Γ​\.25,and​d​\(\{2,3\},Œp​\(U3\);U3\)=Γ​\.5​fl2\[3\]\+Γ​\.25\.\\begin\{gathered\}\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 571\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 314\\relax\\mathchar 28727\\relax\\mathchar 28725\\relax\\mathchar 8704\\relax\\mathchar 28722\\relax\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 28941\\relax\_\{\\mathchar 28722\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 24891\\relax\\\\ \\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28722\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 314\\relax\\mathchar 28722\\relax\\mathchar 28725\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 28941\\relax\_\{\\mathchar 28722\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 24891\\relax\\\\ \\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 8704\\relax 0\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 28941\\relax\_\{\\mathchar 28722\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 8704\\relax 0\\mathchar 314\\relax\\mathchar 28722\\relax\\mathchar 28725\\relax\\mathchar 24891\\relax\\\\ \\text\{and \}\\ \\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 28941\\relax\_\{\\mathchar 28722\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 8235\\relax 0\\mathchar 314\\relax\\mathchar 28722\\relax\\mathchar 28725\\relax\\mathchar 314\\relax\\end\{gathered\}\(99\)Since the functionψp\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}is odd and strictly increasing, we have

Γ​\.75−2\.5​fl2\[3\]=−Γ​\.25\+1\.5​fl2\[3\],\\begin\{gathered\}0\\mathchar 314\\relax\\mathchar 28727\\relax\\mathchar 28725\\relax\\mathchar 8704\\relax\\mathchar 28722\\relax\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 28941\\relax\_\{\\mathchar 28722\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 12349\\relax\\mathchar 8704\\relax 0\\mathchar 314\\relax\\mathchar 28722\\relax\\mathchar 28725\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\\mathchar 314\\relax\\mathchar 28725\\relax\\mathchar 28941\\relax\_\{\\mathchar 28722\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 24891\\relax\\end\{gathered\}\(100\)which, combined with Eq\. \([91](https://arxiv.org/html/2607.09869#A1.E91)\), yields

fl1\[3\]=fl2\[3\]=fl3\[3\]=Γ​\.25\.\\begin\{gathered\}\\mathchar 28941\\relax\_\{\\mathchar 28721\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 12349\\relax\\mathchar 28941\\relax\_\{\\mathchar 28722\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 12349\\relax\\mathchar 28941\\relax\_\{\\mathchar 28723\\relax\}^\{\\delimiter 67482370\\mathchar 28723\\relax\\delimiter 84267779\}\\mathchar 12349\\relax 0\\mathchar 314\\relax\\mathchar 28722\\relax\\mathchar 28725\\relax\\mathchar 314\\relax\\end\{gathered\}\(101\)DefineU3′∈𝒢3\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\mathchar 12850\\relax\\mathcal\{\\mathchar 28999\\relax\}\_\{\\mathchar 28723\\relax\}by letting

U3′​\(\{1\}\)=U3′​\(\{1,3\}\)=U3′​\(\{2,3\}\)=U3′​\(\{1,2,3\}\)=U3′​\(\{2\}\)=Γ,U3′​\(\{3\}\)=U3′​\(∅\)=−4,and​U3′​\(\{1,2\}\)=8\.\\begin\{gathered\}\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\\{\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 24891\\relax\\\\ \\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\\{\\mathchar 28723\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\mathchar 571\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 8704\\relax\\mathchar 28724\\relax\\mathchar 24891\\relax\\,\\text\{ and \}\\,\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28728\\relax\\mathchar 314\\relax\\end\{gathered\}\(102\)Using the first order condition,

̵p​\(d​\(∅,Œp​\(U3′\);U3\)\)\+̵p​\(d​\(\{2\},Œp​\(U3′\);U3\)\)\+̵p​\(d​\(\{3\},Œp​\(U3′\);U3\)\)\+̵p​\(d​\(\{2,3\},Œp​\(U3′\);U3\)\)=Γ,\\begin\{gathered\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\mathchar 571\\relax\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28722\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\\\ \\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 24891\\relax\\end\{gathered\}\(103\)which, combined with Eqs\. \([85](https://arxiv.org/html/2607.09869#A1.E85)\) and \([101](https://arxiv.org/html/2607.09869#A1.E101)\), is equal to

̵p​\(−bp​\(U3′\)−4\)=̵p​\(bp​\(U3′\)\+2\)\.\\begin\{gathered\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 8704\\relax\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 28724\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28722\\relax\\delimiter 84054785\\mathchar 314\\relax\\end\{gathered\}\(104\)Sinceψp\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}is strictly increasing,

bp​\(U3′\)=−3\.\\begin\{gathered\}\\mathchar 29026\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 8704\\relax\\mathchar 28723\\relax\\mathchar 314\\relax\\end\{gathered\}\(105\)Using the first order condition again,

̵p​\(d​\(\{3\},Œp​\(U3′\);U3\)\)\+̵p​\(d​\(\{1,3\},Œp​\(U3′\);U3\)\)\+̵p​\(d​\(\{2,3\},Œp​\(U3′\);U3\)\)\+̵p​\(d​\(\{1,2,3\},Œp​\(U3′\);U3\)\)=Γ,\\begin\{gathered\}\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\\\ \\mathchar 8235\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29028\\relax\\delimiter 67273472\\\{\\mathchar 28721\\relax\\mathchar 24891\\relax\\mathchar 28722\\relax\\mathchar 24891\\relax\\mathchar 28723\\relax\\\}\\mathchar 24891\\relax\\mathchar 28958\\relax^\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 29013\\relax^\{\\mathchar 560\\relax\}\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\mathchar 24635\\relax\\mathchar 29013\\relax\_\{\\mathchar 28723\\relax\}\\delimiter 84054785\\delimiter 84054785\\mathchar 12349\\relax 0\\mathchar 24891\\relax\\end\{gathered\}\(106\)which, combined with Eqs\. \([85](https://arxiv.org/html/2607.09869#A1.E85)\), \([101](https://arxiv.org/html/2607.09869#A1.E101)\) and \([105](https://arxiv.org/html/2607.09869#A1.E105)\), is equal to

3⋅1p−1=3​̵p​\(1\)=̵p​\(3\)=3p−1\.\\begin\{gathered\}\\mathchar 28723\\relax\\mathchar 8705\\relax\\mathchar 28721\\relax^\{\\mathchar 29040\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\}\\mathchar 12349\\relax\\mathchar 28723\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 28721\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28960\\relax\_\{\\mathchar 29040\\relax\}\\delimiter 67273472\\mathchar 28723\\relax\\delimiter 84054785\\mathchar 12349\\relax\\mathchar 28723\\relax^\{\\mathchar 29040\\relax\\mathchar 8704\\relax\\mathchar 28721\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(107\)Solving3p−2=1\\mathchar 28723\\relax^\{\\mathchar 29040\\relax\\mathchar 8704\\relax\\mathchar 28722\\relax\}\\mathchar 12349\\relax\\mathchar 28721\\relaxyieldsp=2\\mathchar 29040\\relax\\mathchar 12349\\relax\\mathchar 28722\\relax\. ∎

See[4\.5](https://arxiv.org/html/2607.09869#S4.Thmtheorem5)

###### Proof\.

The minimization problem can be equivalently rewritten as

minimize𝐱∈ℝn,b,e∈ℝe\+ȷ⋅‖𝐱‖22subject to​\|Un​\(S\)−b−∑i∈Sxi\|≤e​for every​S⊆\[n\]\.\\begin\{gathered\}\\operatorname\\mathchar 8707\\relax\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\\mathchar 29033\\relax\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29050\\relax\\mathchar 29029\\relax\}\_\{\\mathbf\{\\mathchar 29048\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\\mathchar 24891\\relax\\mathchar 29029\\relax\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}\}\\ \\mathchar 29029\\relax\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\\\ \\text\{subject to \}\\ \\delimiter 69640972\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 67273472\\mathchar 29011\\relax\\delimiter 84054785\\mathchar 8704\\relax\\mathchar 29026\\relax\\mathchar 8704\\relax\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12850\\relax\\mathchar 29011\\relax\}\\mathchar 29048\\relax\_\{\\mathchar 29033\\relax\}\\delimiter 69640972\\mathchar 12820\\relax\\mathchar 29029\\relax\\ \\text\{ for every \}\\mathchar 29011\\relax\\mathchar 12818\\relax\\delimiter 67482370\\mathchar 29038\\relax\\delimiter 84267779\\mathchar 314\\relax\\end\{gathered\}\(108\)Then, according toblum1972direct,ϕ∞​\(Un;η\)\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24635\\relax\\mathchar 28945\\relax\\delimiter 84054785exists\. To proveϕ∞​\(Un;η\)→ϕ∞​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24635\\relax\\mathchar 28945\\relax\\delimiter 84054785\\mathchar 12833\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785asη→Γ\\mathchar 28945\\relax\\mathchar 12833\\relax 0, it is sufficient to prove that for every sequence\(ϕ∞​\(Un;ηk\)\)k=1∞\\delimiter 67273472\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24635\\relax\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\\delimiter 84054785\\delimiter 84054785\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}such thatηk→Γ\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\\mathchar 12833\\relax 0ask→∞\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax, there isϕ∞​\(Un;ηk\)→ϕ∞​\(Un\)\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24635\\relax\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\\delimiter 84054785\\mathchar 12833\\relax\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785ask→∞\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\.

For convenience, we write

𝐱ȷ≔Œ∞​\(Un;ȷ\),\\begin\{gathered\}\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\coloneq\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24635\\relax\\mathchar 28945\\relax\\delimiter 84054785\\mathchar 24891\\relax\\end\{gathered\}\(109\)and letbȷ\\mathchar 29026\\relax\_\{\\mathchar 28945\\relax\}be the optimal solution ofb\\mathchar 29026\\relaxassociated with𝐱ȷ\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\. Besides, write𝐱Γ≔ϕ∞​\(Un\)\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\coloneq\\mathchar 28958\\relax^\{\\mathchar 561\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785, and definebΓ\\mathchar 29026\\relax\_\{0\}similarly\. Additionally, we append𝟏2n\\mathbf\{\\mathchar 28721\\relax\}\_\{\\mathchar 28722\\relax^\{\\mathchar 29038\\relax\}\}as the last column to𝐀\\mathbf\{\\mathchar 28993\\relax\}, the result of which is denoted by𝐁\\mathbf\{\\mathchar 28994\\relax\}, and write𝐲ȷ≔\(𝐱ȷ,bȷ\)∈ℝn\+1\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\}\\coloneq\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\mathchar 24891\\relax\\mathchar 29026\\relax\_\{\\mathchar 28945\\relax\}\\delimiter 84054785\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}\.

Observe that

‖𝐀𝐱ȷ\+bȷ⋅𝟏2n−𝐕​\(Un\)‖∞\\displaystyle\\delimiter 69645069\\mathbf\{\\mathchar 28993\\relax\}\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\mathchar 8235\\relax\\mathchar 29026\\relax\_\{\\mathchar 28945\\relax\}\\mathchar 8705\\relax\\mathbf\{\\mathchar 28721\\relax\}\_\{\\mathchar 28722\\relax^\{\\mathchar 29038\\relax\}\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}=‖𝐁𝐲ȷ−𝐕​\(Un\)‖∞\\displaystyle\\mathchar 12349\\relax\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\(110\)≤‖𝐁𝐲ȷ−𝐕​\(Un\)‖∞\+ȷ⋅‖𝐱ȷ‖22\\displaystyle\\mathchar 12820\\relax\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\,\\mathchar 8235\\relax\\,\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}≤‖𝐁𝐲Γ−𝐕​\(Un\)‖∞\+ȷ⋅‖𝐱Γ‖22\.\\displaystyle\\mathchar 12820\\relax\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{0\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\,\\mathchar 8235\\relax\\,\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 314\\relaxIt indicates that\(𝐲ȷk\)k=1∞\\delimiter 67273472\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 84054785\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}is eventually bounded\. Precisely,

‖𝐲ȷ‖2\\displaystyle\\delimiter 69645069\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}≤1œmin​\(𝐁\)​‖𝐁𝐲ȷ‖2\\displaystyle\\mathchar 12820\\relax\{\{\\mathchar 28721\\relax\\over\\mathchar 28955\\relax\_\{\\mathrm\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\}\}\\delimiter 67273472\\mathbf\{\\mathchar 28994\\relax\}\\delimiter 84054785\}\}\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}\(111\)≤C∞œmin​\(𝐁\)​‖𝐁𝐲ȷ‖∞\\displaystyle\\mathchar 12820\\relax\{\{\\mathchar 28995\\relax\_\{\\mathchar 561\\relax\}\\over\\mathchar 28955\\relax\_\{\\mathrm\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\}\}\\delimiter 67273472\\mathbf\{\\mathchar 28994\\relax\}\\delimiter 84054785\}\}\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\}\\delimiter 69645069\_\{\\mathchar 561\\relax\}≤C∞œmin​\(𝐁\)​\(‖𝐁𝐲ȷ−𝐕​\(Un\)‖∞\+‖𝐕​\(Un\)‖∞\)\\displaystyle\\mathchar 12820\\relax\{\{\\mathchar 28995\\relax\_\{\\mathchar 561\\relax\}\\over\\mathchar 28955\\relax\_\{\\mathrm\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\}\}\\delimiter 67273472\\mathbf\{\\mathchar 28994\\relax\}\\delimiter 84054785\}\}\\left\\delimiter 67273472\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\,\\mathchar 8235\\relax\\,\\delimiter 69645069\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\right\\delimiter 84054785≤C∞œmin​\(𝐁\)​\(‖𝐁𝐲Γ−𝐕​\(Un\)‖∞\+ȷ⋅‖𝐱Γ‖22\+‖𝐕​\(Un\)‖∞\)\.\\displaystyle\\mathchar 12820\\relax\{\{\\mathchar 28995\\relax\_\{\\mathchar 561\\relax\}\\over\\mathchar 28955\\relax\_\{\\mathrm\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\}\}\\delimiter 67273472\\mathbf\{\\mathchar 28994\\relax\}\\delimiter 84054785\}\}\\left\\delimiter 67273472\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{0\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\,\\mathchar 8235\\relax\\,\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\,\\mathchar 8235\\relax\\,\\delimiter 69645069\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\right\\delimiter 84054785\\mathchar 314\\relaxFor the first inequality,σmin​\(𝐁\)\\mathchar 28955\\relax\_\{\\mathrm\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\}\}\\delimiter 67273472\\mathbf\{\\mathchar 28994\\relax\}\\delimiter 84054785denotes the smallest singular value of𝐁\\mathbf\{\\mathchar 28994\\relax\};σmin​\(𝐁\)\>Γ\\mathchar 28955\\relax\_\{\\mathrm\{\\mathchar 29037\\relax\\mathchar 29033\\relax\\mathchar 29038\\relax\}\}\\delimiter 67273472\\mathbf\{\\mathchar 28994\\relax\}\\delimiter 84054785\\mathchar 12606\\relax 0as𝐁\\mathbf\{\\mathchar 28994\\relax\}has full column rank\.444We have proved that𝐁⊤​𝐁\\mathbf\{\\mathchar 28994\\relax\}^\{\\mathchar 574\\relax\}\\mathbf\{\\mathchar 28994\\relax\}is invertible in\[li2024one, Theorem 2\]The second inequality follows from the fact that all norms onℝn\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}are equivalent\.

Since the sequence\(𝐲ȷk\)k=1∞\\delimiter 67273472\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 84054785\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}is eventually bounded, it is valid to select some convergent subsequence\. Therefore, suffice it to prove that every convergent subsequence of\(𝐱ȷk\)k=1∞\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 84054785\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}converges to the same point𝐱Γ\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\.555This is equivalent to proving thatlim infk→∞ak=lim supk→∞ak\\liminf\_\{\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\}\\mathchar 29025\\relax\_\{\\mathchar 29035\\relax\}\\mathchar 12349\\relax\\limsup\_\{\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\}\\mathchar 29025\\relax\_\{\\mathchar 29035\\relax\}for a sequence\(ak\)k=1∞\\delimiter 67273472\\mathchar 29025\\relax\_\{\\mathchar 29035\\relax\}\\delimiter 84054785\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}\.

For simplicity, we abuse the notation\(𝐲ȷk\)k=1∞\\delimiter 67273472\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 84054785\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}to refer to a convergent subsequence of\(𝐲ȷk\)k=1∞\\delimiter 67273472\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 84054785\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}\.666Given a subsequence\(𝐱ȷkj\)j=1∞\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\_\{\\mathchar 29034\\relax\}\}\}\\delimiter 84054785\_\{\\mathchar 29034\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}that converges to𝐱′\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 560\\relax\}, it may happen thatbȷkj\\mathchar 29026\\relax\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\_\{\\mathchar 29034\\relax\}\}\}does not converge\. Nevertheless, we can further select a subsequence of it such that the corresponding bias term converges, and𝐱′\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 560\\relax\}is still the limit\.Thus, we have𝐲ȷk→𝐲′\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\mathchar 12833\\relax\\mathbf\{\\mathchar 29049\\relax\}^\{\\mathchar 560\\relax\}ask→∞\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relaxfor some point𝐲′∈ℝn\+1\\mathbf\{\\mathchar 29049\\relax\}^\{\\mathchar 560\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\\mathchar 8235\\relax\\mathchar 28721\\relax\}, the firstn\\mathchar 29038\\relaxentries of which is denoted by𝐱′\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 560\\relax\}\. Since

‖𝐁𝐲Γ−𝐕​\(Un\)‖∞≤‖𝐁𝐲ȷ−𝐕​\(Un\)‖∞,\\begin\{gathered\}\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{0\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 12820\\relax\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 24891\\relax\\end\{gathered\}\(112\)combining this with Eq\. \([110](https://arxiv.org/html/2607.09869#A1.E110)\), we have

Γ≤‖𝐁𝐲ȷ−𝐕​\(Un\)‖∞−‖𝐁𝐲Γ−𝐕​\(Un\)‖∞≤ȷ⋅‖𝐱Γ‖22\.\\begin\{gathered\}0\\mathchar 12820\\relax\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 8704\\relax\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{0\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 12820\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(113\)This suggests that‖𝐁𝐲ȷk−𝐕​\(Un\)‖∞→‖𝐁𝐲Γ−𝐕​\(Un\)‖∞\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 12833\\relax\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{0\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}ask→∞\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\. Since the function𝐲↦→‖𝐁𝐲−𝐕​\(Un\)‖∞\\mathbf\{\\mathchar 29049\\relax\}\\mathrel\{\\mathchar 567\\relax\\mathchar 545\\relax\}\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\\mathchar 29049\\relax\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}is continuous, we have

‖𝐁𝐲′−𝐕​\(Un\)‖∞=‖𝐁𝐲Γ−𝐕​\(Un\)‖∞\.\\begin\{gathered\}\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}^\{\\mathchar 560\\relax\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 12349\\relax\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{0\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(114\)
On the other hand, since𝐱↦→‖𝐱‖22\\mathbf\{\\mathchar 29048\\relax\}\\mathrel\{\\mathchar 567\\relax\\mathchar 545\\relax\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}is continuous, there is

limk→∞‖𝐱ȷk‖22→‖𝐱′‖22\.\\begin\{gathered\}\\lim\_\{\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12833\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 560\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(115\)Besides, note that

‖𝐁𝐲Γ−𝐕​\(Un\)‖∞\+ȷ⋅‖𝐱ȷ‖22\\displaystyle\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{0\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}≤‖𝐁𝐲ȷ−𝐕​\(Un\)‖∞\+ȷ⋅‖𝐱ȷ‖22\\displaystyle\\mathchar 12820\\relax\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{\\mathchar 28945\\relax\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\(116\)≤‖𝐁𝐲Γ−𝐕​\(Un\)‖∞\+ȷ⋅‖𝐱Γ‖22,\\displaystyle\\mathchar 12820\\relax\\delimiter 69645069\\mathbf\{\\mathchar 28994\\relax\}\\mathbf\{\\mathchar 29049\\relax\}\_\{0\}\\mathchar 8704\\relax\\mathbf\{\\mathchar 29014\\relax\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\\delimiter 69645069\_\{\\mathchar 561\\relax\}\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 24891\\relaxwhich implies

‖𝐱ȷ‖22≤‖𝐱Γ‖22\.\\begin\{gathered\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12820\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(117\)Then,

‖𝐱′‖22≤‖𝐱Γ‖22\.\\begin\{gathered\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 560\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12820\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(118\)Using Eqs\. \([114](https://arxiv.org/html/2607.09869#A1.E114)\) and \([118](https://arxiv.org/html/2607.09869#A1.E118)\), and the uniqueness of𝐱Γ\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}, we obtain

𝐱′=𝐱Γ\.\\begin\{gathered\}\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 560\\relax\}\\mathchar 12349\\relax\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\mathchar 314\\relax\\end\{gathered\}\(119\)∎

See[4\.6](https://arxiv.org/html/2607.09869#S4.Thmtheorem6)

###### Proof\.

According toblum1972direct,ϕELC​\(Un;η\)\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24635\\relax\\mathchar 28945\\relax\\delimiter 84054785exists\. For everyη\>Γ\\mathchar 28945\\relax\\mathchar 12606\\relax 0, let the produced optimal solution be denoted by\(𝐱ȷ,eȷ\)\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\mathchar 24891\\relax\\mathchar 29029\\relax\_\{\\mathchar 28945\\relax\}\\delimiter 84054785, and similarly we define\(𝐱Γ,eΓ\)\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\mathchar 24891\\relax\\mathchar 29029\\relax\_\{0\}\\delimiter 84054785as the optimal solution corresponding to the egalitarian least core\. In other words,𝐱ȷ=ϕELC​\(Un;η\)\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\mathchar 12349\\relax\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\mathchar 24635\\relax\\mathchar 28945\\relax\\delimiter 84054785and𝐱Γ=ϕELC​\(Un\)\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\mathchar 12349\\relax\\mathchar 28958\\relax^\{\\mathrm\{\\mathchar 28997\\relax\\mathchar 29004\\relax\\mathchar 28995\\relax\}\}\\delimiter 67273472\\mathchar 29013\\relax\_\{\\mathchar 29038\\relax\}\\delimiter 84054785\. Then, suffice it to prove that for every sequence\(𝐱ȷk\)k=1n\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 84054785\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 29038\\relax\}such thatηk→Γ\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\\mathchar 12833\\relax 0ask→∞\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax, there is𝐱ȷk→𝐱Γ\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\mathchar 12833\\relax\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}ask→∞\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\.

Notice that

eΓ≤eȷ≤eȷ\+ȷ⋅‖𝐱ȷ‖22≤eΓ\+ȷ⋅‖𝐱Γ‖22,\\begin\{gathered\}\\mathchar 29029\\relax\_\{0\}\\mathchar 12820\\relax\\mathchar 29029\\relax\_\{\\mathchar 28945\\relax\}\\mathchar 12820\\relax\\mathchar 29029\\relax\_\{\\mathchar 28945\\relax\}\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\,\\mathchar 12820\\relax\\mathchar 29029\\relax\_\{0\}\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\end\{gathered\}\(120\)which yields

Γ≤eȷ−eΓ≤ȷ⋅‖𝐱Γ‖22\.\\begin\{gathered\}0\\mathchar 12820\\relax\\mathchar 29029\\relax\_\{\\mathchar 28945\\relax\}\\mathchar 8704\\relax\\mathchar 29029\\relax\_\{0\}\\mathchar 12820\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(121\)Therefore,eȷk→eΓ\\mathchar 29029\\relax\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\mathchar 12833\\relax\\mathchar 29029\\relax\_\{0\}ask→∞\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\. On the other hand,

eΓ\+ȷ⋅‖𝐱ȷ‖22≤eȷ\+ȷ⋅‖𝐱ȷ‖22≤eΓ\+ȷ⋅‖𝐱Γ‖22,\\begin\{gathered\}\\mathchar 29029\\relax\_\{0\}\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12820\\relax\\mathchar 29029\\relax\_\{\\mathchar 28945\\relax\}\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12820\\relax\\mathchar 29029\\relax\_\{0\}\\mathchar 8235\\relax\\mathchar 28945\\relax\\mathchar 8705\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\end\{gathered\}\(122\)which leads to

‖𝐱ȷ‖22≤‖𝐱Γ‖22\.\\begin\{gathered\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12820\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(123\)Therefore, the sequence\(𝐱ȷk\)k=1∞\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 84054785\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}is eventually bounded, and thus we can select a convergent subsequence\. In particular, there must be

lim infk→∞‖𝐱ȷk‖22=lim supk→∞‖𝐱ȷk‖22=‖𝐱Γ‖22\.\\begin\{gathered\}\\liminf\_\{\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12349\\relax\\limsup\_\{\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12349\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(124\)Otherwise, there would exist a subsequence of\(𝐱ȷk\)k=1∞\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 84054785\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}converging to some point𝐱′∈ℝn\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 560\\relax\}\\mathchar 12850\\relax\\mathbb\{\\mathchar 29010\\relax\}^\{\\mathchar 29038\\relax\}such that

‖𝐱′‖22=lim infk→∞‖𝐱ȷk‖22<‖𝐱2‖22,\\begin\{gathered\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 560\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12349\\relax\\liminf\_\{\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12604\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28722\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 24891\\relax\\end\{gathered\}\(125\)which, together with Eq\. \([121](https://arxiv.org/html/2607.09869#A1.E121)\), contradicts the uniqueness of𝐱Γ\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\. Note that\(𝐱′,eΓ\)\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}^\{\\mathchar 560\\relax\}\\mathchar 24891\\relax\\mathchar 29029\\relax\_\{0\}\\delimiter 84054785is a feasible solution, since the domain is closed\. Consequently,

limk→∞‖𝐱ȷk‖22=‖𝐱Γ‖22\.\\begin\{gathered\}\\lim\_\{\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\}\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12349\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 314\\relax\\end\{gathered\}\(126\)
Then, every convergent subsequence of\(𝐱ȷk\)k=1∞\\delimiter 67273472\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\delimiter 84054785\_\{\\mathchar 29035\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 561\\relax\}must converge to the same point𝐱Γ\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}, since𝐱Γ\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}is the unique vector such that \(i\)\(eΓ,𝐱Γ\)\\delimiter 67273472\\mathchar 29029\\relax\_\{0\}\\mathchar 24891\\relax\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 84054785is feasible to the problem of least core, and \(ii\) for any feasible\(eΓ,𝐱\)\\delimiter 67273472\\mathchar 29029\\relax\_\{0\}\\mathchar 24891\\relax\\mathbf\{\\mathchar 29048\\relax\}\\delimiter 84054785, it holds that‖𝐱Γ‖22≤‖𝐱‖22\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\\mathchar 12820\\relax\\delimiter 69645069\\mathbf\{\\mathchar 29048\\relax\}\\delimiter 69645069\_\{\\mathchar 28722\\relax\}^\{\\mathchar 28722\\relax\}\. As a result, we have𝐱ȷk→𝐱Γ\\mathbf\{\\mathchar 29048\\relax\}\_\{\\mathchar 28945\\relax\_\{\\mathchar 29035\\relax\}\}\\mathchar 12833\\relax\\mathbf\{\\mathchar 29048\\relax\}\_\{0\}ask→∞\\mathchar 29035\\relax\\mathchar 12833\\relax\\mathchar 561\\relax\. ∎

## Appendix BExperiments

The datasets used are summarized in Table[2](https://arxiv.org/html/2607.09869#A2.T2)\. More experimental results on the inclusion curves are presented in Figures[3](https://arxiv.org/html/2607.09869#A2.F3)and[4](https://arxiv.org/html/2607.09869#A2.F4)\.

#### Statistical comparison\.

As shown in Figures[5](https://arxiv.org/html/2607.09869#A2.F5)and[6](https://arxiv.org/html/2607.09869#A2.F6), we also compare these methods statistically\. For the value associated with methodsA\\mathchar 28993\\relax\(row\) andB\\mathchar 28994\\relax\(column\), it is computed as

∑i=1m\[𝟙​\{AUCiA\>AUCiB\}−𝟙​\{AUCiA<AUCiB\}\]m\\begin\{gathered\}\{\{\\mathchar 4944\\relax\\displaylimits\_\{\\mathchar 29033\\relax\\mathchar 12349\\relax\\mathchar 28721\\relax\}^\{\\mathchar 29037\\relax\}\\delimiter 67482370\\mathds\{\\mathchar 28721\\relax\}\\\{\\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}^\{\\mathchar 28993\\relax\}\_\{\\mathchar 29033\\relax\}\\mathchar 12606\\relax\\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}^\{\\mathchar 28994\\relax\}\_\{\\mathchar 29033\\relax\}\\\}\\mathchar 8704\\relax\\mathds\{\\mathchar 28721\\relax\}\\\{\\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}^\{\\mathchar 28993\\relax\}\_\{\\mathchar 29033\\relax\}\\mathchar 12604\\relax\\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}^\{\\mathchar 28994\\relax\}\_\{\\mathchar 29033\\relax\}\\\}\\delimiter 84267779\\over\\mathchar 29037\\relax\}\}\\end\{gathered\}\(127\)wherem\\mathchar 29037\\relaxis the total number of instances used andAUCiA\\mathrm\{\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relax\}\_\{\\mathchar 29033\\relax\}^\{\\mathchar 28993\\relax\}is the AUC achieved by methodA\\mathchar 28993\\relaxon the utility function created using thei\\mathchar 29033\\relax\-th instance\. Therefore, the larger it is, the better methodA\\mathchar 28993\\relaxis\.

Table 2:Summary of the used datasets\.![Refer to caption](https://arxiv.org/html/2607.09869v1/x13.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x14.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x15.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x16.png)GPSPFOTPwave\_energy![Refer to caption](https://arxiv.org/html/2607.09869v1/x17.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x18.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x19.png)jannisspambasesuperconductFigure 3:Comparison of attribution methods on six datasets for player ranking\. A larger area under the curve indicates better performance\. Since all utility functions contain more than3​Γ\\mathchar 28723\\relax 0players, nonlinear methods are approximated by sampling\#​players×1,Γ​Γ​Γ\\\#\\mathrm\{\\mathchar 29040\\relax\\mathchar 29036\\relax\\mathchar 29025\\relax\\mathchar 29049\\relax\\mathchar 29029\\relax\\mathchar 29042\\relax\\mathchar 29043\\relax\}\\mathchar 8706\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax 000subsets, with mean and standard deviation reported over five random seeds\. Linear methods are computed exactly\. Beta Shapley achieves the best inclusion AUC among the1​Γ\\mathchar 28721\\relax 0candidates\.![Refer to caption](https://arxiv.org/html/2607.09869v1/x20.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x21.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x22.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x23.png)letterpendigitsEES![Refer to caption](https://arxiv.org/html/2607.09869v1/x24.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x25.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x26.png)WQWelevatorscreditFigure 4:Comparison of attribution methods on six datasets for player ranking\. A larger area under the curve indicates better performance\. Since all utility functions have fewer than2​Γ\\mathchar 28722\\relax 0players, all subsets are enumerated exactly for all methods\. Beta Shapley achieves the best inclusion AUC among the1​Γ\\mathchar 28721\\relax 0candidates\.![Refer to caption](https://arxiv.org/html/2607.09869v1/x27.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x28.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x29.png)GPSPFOTPwave\_energy![Refer to caption](https://arxiv.org/html/2607.09869v1/x30.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x31.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x32.png)jannisspambasesuperconductFigure 5:Statistical comparison of attribution methods on six datasets for player ranking\. Since all utility functions contain more than3​Γ\\mathchar 28723\\relax 0players, nonlinear methods are approximated by sampling\#​players×1,Γ​Γ​Γ\\\#\\mathrm\{\\mathchar 29040\\relax\\mathchar 29036\\relax\\mathchar 29025\\relax\\mathchar 29049\\relax\\mathchar 29029\\relax\\mathchar 29042\\relax\\mathchar 29043\\relax\}\\mathchar 8706\\relax\\mathchar 28721\\relax\\mathchar 24891\\relax 000subsets, withA​U​C\\mathchar 28993\\relax\\mathchar 29013\\relax\\mathchar 28995\\relaxaveraged over five random seeds\. Linear methods are computed exactly\. The blue lines separate the linear and nonlinear methods\.![Refer to caption](https://arxiv.org/html/2607.09869v1/x33.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x34.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x35.png)letterpendigitsEES![Refer to caption](https://arxiv.org/html/2607.09869v1/x36.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x37.png)![Refer to caption](https://arxiv.org/html/2607.09869v1/x38.png)WQWelevatorscreditFigure 6:Statistical comparison of attribution methods on six datasets for player ranking\. Since all utility functions have fewer than2​Γ\\mathchar 28722\\relax 0players, all subsets are enumerated exactly for all methods\. The blue lines separate the linear and nonlinear methods\.