Internal Pluralism and the Limits of Pairwise Comparisons

# Internal Pluralism and the Limits of Pairwise Comparisons
Source: [https://arxiv.org/html/2607.02672](https://arxiv.org/html/2607.02672)
###### Abstract\.

Local pairwise comparisons are a standard tool for learning how people want decision rules to work, e\.g\., in participatory design or alignment\. However, their use builds in two strong assumptions: that local comparisons are sufficient evidence about how a person wants an automated decision rule to behave, and that people can always answer those comparisons decisively\. We investigate how these assumptions may be compromised underinternal pluralism: the idea that an individual evaluates decision rules according to multiple authoritative priorities about how the rule should behave\. We provide a formal model of such pluralistic preferences over decision rules, which then lets us identify two distinct failures of forced local pairwise comparison data\. First, priorities such as proportionality, egalitarianism, and equal treatment are inherentlyglobal: what they imply in one case can depend on what happens elsewhere, so local comparisons may fail to capture them\. Second, even when prioritiesarerepresentable locally, tension between strongly\-held priorities can generate internal conflict, producing potentially costly behavioral distortions when comparisons are forced\. We then use our model to investigate the alternative — allowing people to report indecision — and our findings suggest that doing so can considerably reduce the number of queries needed to learn preferences accurately\. We conclude by describing how our model points toward preference\-learning methods that elicit these priorities directly, yielding more faithful and interpretable accounts of what people value\.

††conference:; ;###### Contents

1. [1Introduction](https://arxiv.org/html/2607.02672#S1)
2. [2Model](https://arxiv.org/html/2607.02672#S2)
3. [3Generalization \#1: Inseparability](https://arxiv.org/html/2607.02672#S3)
4. [4Generalization \#2: Latent Indecision](https://arxiv.org/html/2607.02672#S4)
5. [5Discussion](https://arxiv.org/html/2607.02672#S5)
6. [References](https://arxiv.org/html/2607.02672#bib)
7. [ASupplemental Materials fromSection2](https://arxiv.org/html/2607.02672#A1)
8. [BSupplemental Materials fromSection3](https://arxiv.org/html/2607.02672#A2)
9. [CSupplemental Materials forSection4](https://arxiv.org/html/2607.02672#A3)

Acknowledgements\.We thank Serena Wang, Kate Donahue, and Chara Podimata for their detailed feedback on the paper\. We thank Finale Doshi\-Velez, Charis Pipis, Daniel Lee, Jakob de Raaij, Marina Mancoridis, Sarah Bentley, Colin Camerer, and Maxon Rubin\-Toles, for their feedback on presentations related to the paper\. We thank Bernardo Zacka, Carmel Baharav, Elias Bareinboim, and Andre Ye for helpful conversations, and we also thank Vijay Keswani and Min Kyung Lee for providing access to their data\.

## 1\.Introduction

Across areas like participatory design and AI alignment, there is increasing interest in trying to embed human values in automated systems for making societal decisions\. Existing work has considered how to design algorithms emulating human judgments in, e\.g\., the trolley problem, motivated by automated vehicles\(Awadet al\.,[2018](https://arxiv.org/html/2607.02672#bib.bib122); Noothigattuet al\.,[2018](https://arxiv.org/html/2607.02672#bib.bib121)\); how to prioritize who should receive kidneys\(Freedmanet al\.,[2020](https://arxiv.org/html/2607.02672#bib.bib127); Keswaniet al\.,[2024](https://arxiv.org/html/2607.02672#bib.bib220),[2025a](https://arxiv.org/html/2607.02672#bib.bib212); Boerstleret al\.,[2024](https://arxiv.org/html/2607.02672#bib.bib169); Cousinset al\.,[2025](https://arxiv.org/html/2607.02672#bib.bib192)\); how to allocate food to food banks\(Leeet al\.,[2019](https://arxiv.org/html/2607.02672#bib.bib244),[2017](https://arxiv.org/html/2607.02672#bib.bib298)\); how LLMs should respond to prompts\(Geet al\.,[2024a](https://arxiv.org/html/2607.02672#bib.bib99); Huanget al\.,[2024](https://arxiv.org/html/2607.02672#bib.bib168)\); how to hypothetically allocate life jackets\(Mohsinet al\.,[2022](https://arxiv.org/html/2607.02672#bib.bib207)\); and how to allocate resources to medical patients\(Johnstonet al\.,[2023](https://arxiv.org/html/2607.02672#bib.bib213); Dieterenet al\.,[2022](https://arxiv.org/html/2607.02672#bib.bib296); Gattoet al\.,[2026](https://arxiv.org/html/2607.02672#bib.bib1)\)\.

We will refer to a generic such setting as adecision task, consisting of a fixed set of decision inputs𝒳\\mathcal\{X\}and decision outputs𝒴\\mathcal\{Y\}, with the valid decision outputs atx∈𝒳x\\in\\mathcal\{X\}being𝒴​\(x\)⊆𝒴\\mathcal\{Y\}\(x\)\\subseteq\\mathcal\{Y\}\. Adecision ruleF:𝒳→𝒴F:\\mathcal\{X\}\\to\\mathcal\{Y\}is then any automated rule for selecting an output given any input\. For example, in a simple version of the trolley problem \(our running example\), an inputx∈𝒳x\\in\\mathcal\{X\}consists of a trolley scenario and two groups of peopleN1,N2N\_\{1\},N\_\{2\}; the decision rule must choose a group in𝒴​\(x\)=\{N1,N2\}\\mathcal\{Y\}\(x\)=\\\{N\_\{1\},N\_\{2\}\\\}to be spared\. Let the set of decision rulesℱ\\mathcal\{F\}be the set of all mappings from𝒳→𝒴\\mathcal\{X\}\\to\\mathcal\{Y\}\. The goal is to learn a decision ruleF∈ℱF\\in\\mathcal\{F\}that is most “aligned” with either an individual or a group; for the purposes of this paper, we focus on aligning a decision rule to asingle individual\.

Across the applications above, a common way to make such alignment problems empirically tractable is to elicitlocal pairwise comparisons, which ask: given an inputx∈𝒳x\\in\\mathcal\{X\}, would you prefer the rule outputyyory′y^\{\\prime\}\(wherey,y′∈𝒴​\(x\)y,y^\{\\prime\}\\in\\mathcal\{Y\}\(x\)\)?111Pairwise comparisons are a dominant modality in preference learning for alignment of language models and simpler decision rules \(e\.g\.,\(Jianget al\.,[2024](https://arxiv.org/html/2607.02672#bib.bib167)\)\)\. Local pairwise comparisons are also part of a broader tradition of discrete\-choice elicitation methods in economics, political science, healthcare, marketing, and transportation\(Ben\-Akiva and Lerman,[1985](https://arxiv.org/html/2607.02672#bib.bib34); Green and Srinivasan,[1978](https://arxiv.org/html/2607.02672#bib.bib92); Hainmuelleret al\.,[2014](https://arxiv.org/html/2607.02672#bib.bib110); Lancsar and Louviere,[2008](https://arxiv.org/html/2607.02672#bib.bib91)\)\.These comparisons are often forced, i\.e\., the person must pick one option or the other\. After many such comparisons are collected, the learner fits a choice model—often based on Bradley\-Terry\(Bradley and Terry,[1952](https://arxiv.org/html/2607.02672#bib.bib368)\)or related random\-utility models—whose induced choices reproduce the person’s responses as faithfully as possible\.

This approach reflects two important and potentially non\-neutral assumptions\. First, it treats local pairwise comparisons as sufficient evidence for learning preferences over the actual decision space,ℱ\\mathcal\{F\}\. However, it could easily be that people’s beliefs about how the rule should work sometimes operate on theentire rule, in a way that cannot be decomposed over individual inputs: for example, someone might want the rule to avoid imposing harms disproportionately on protected groups \(we call thisproportionality\)\. According to proportionality, who should be spared atxxdepends crucially on who is spared at otherx′∈𝒳∖xx^\{\\prime\}\\in\\mathcal\{X\}\\setminus\{x\}— a dependence that must be somehow eliminated when the individual responds to a local pairwise comparison\. We refer to such beliefs asinseparable, as the response they dictate at an individualxxcannot be separated from what is done at other inputs\.

The second key assumption is that people can always give decisive answers to local pairwise comparison queries\. We question this assumption on the grounds ofinternal pluralism: the widely\-discussed idea that an individual’s judgments may derive from multiple core values or objectives, which can be brought into irreducible conflict by hard trade\-offs\. For instance, if asked how the decision rule should work in the trolley case, the individual may articulate several priorities: in addition to proportionality, they may want the rule to save as many people as possible \(call this prioritysize\) and to save members of their immediate family \(call this priorityfamily\)\. These priorities can easily be brought into conflict: for example, in the classic dilemma whereN1N\_\{1\}contains 1000 people butN2N\_\{2\}contains the individual’s mother, the size and family priorities strongly advocate opposite responses, and the individual may not be able to produce a morally authoritative decision\. In such cases, forcing a decisive response erases this latent internal conflict at best, and at worst may prompt the individual to respond arbitrarily, leading to response inconsistency and incorrect conclusions by the learner\.

Using a formal model that captures such pluralistic priorities, we formalize these two concerns: we examine how inseparable beliefs can be erased by local pairwise comparisons, and how even absent this problem, internal conflict between these priorities can lead to misspecification under forced comparisons\. Our aim is not to reject pairwise comparisons as an elicitation tool, but to understand when they can and cannot faithfully reflect how people think the rule should work\. These three contributions — plus how our model can be applied as a learning tool — are described below in I\-IV\.

Contribution I: A Pluralistic Model of Priorities \([Section2](https://arxiv.org/html/2607.02672#S2)\)\.Our first contribution is to introduce this formal model, shown in[Figure1](https://arxiv.org/html/2607.02672#S1.F1)\. This model’s core feature is that it representsinternal pluralism:the individual’s beliefs about how the rule should work are not represented by a single preference ordering over rules, but rather by acollectionof individually authoritative preference orderings over rules\. We call these orderingspriorities\. Each priority represents one coherent way the individual evaluates rules, such as saving as many people as possible, protecting family members, or avoiding disproportionate harm to protected groups\.

Formally, let there bem∈ℕm\\in\\mathbb\{N\}priorities, each represented by a complete, transitive, and weak preference relation overℱ\\mathcal\{F\}\. For a given priorityj∈\[m\]j\\in\[m\], we model this relation with a utility functionuj:ℱ→ℝu\_\{j\}:\\mathcal\{F\}\\to\\mathbb\{R\}\. Note that this captures the single\-objective case when there is one priority\.

![Refer to caption](https://arxiv.org/html/2607.02672v1/x1.png)Figure 1\.Depiction of priority model over decision rules\. Each priority is represented by a weak, complete, transitive preference relation over the rule space\. In the diagram, circles represent subsets ofℱ\\mathcal\{F\}over which the individual is indifferent; because the rankings are complete, each row of circles represents a partition overℱ\\mathcal\{F\}, e\.g\.,ℱ1∪ℱ2∪ℱ3=ℱ\\mathcal\{F\}\_\{1\}\\cup\\mathcal\{F\}\_\{2\}\\cup\\mathcal\{F\}\_\{3\}=\\mathcal\{F\}\. In thesizepriority applied to the trolley example,ℱ1\\mathcal\{F\}\_\{1\}could, e\.g\., represent the set of all rules that choose to spare the maximum number of people whenever the two sets of people are of different size\.With this pluralistic priority model in hand, we then formally model how such rule\-level priorities are translated into responses to local pairwise comparison queries\. This is where the two concerns above formally arise\.

First, an individual priority may beinseparable: what it says about outputyyversusy′y^\{\\prime\}at inputxxmay depend on what the rule does at other inputs\. Then, to determine what local query response dictates, the individual must somehow collapse competing evidence depending on what the rule could do at other inputs\. Second, even if each priority can individually evaluate the local query, the resulting evidenceacross prioritiesmay still fail to support a decisive answer\. Some priorities may provide evidence for choosingyyatxx, while others provide evidence for choosingy′y^\{\\prime\}, producingconflict\(e\.g\., a choice between the trolley sparing your mother and 1000 people\)\. Alternatively, no priority may provide meaningful evidence for eitheryyory′y^\{\\prime\}, producingindifference\(e\.g\., a choice between the trolley sparing an old sandwich or a plastic bag\)\. We refer to these states of conflict and indifference aslatent states, allowing how they manifest behaviorally to vary\.

In its full generality, our pluralistic priority model thus captures both inseparable priorities and indecisive latent states — the two components we worried may not be captured by standard pairwise learning pipelines\. We use our model to formalize this intuition, too: we show that standardscore\-based random utility models\(S\-RUMs\), including Bradley\-Terry, correspond exactly to the special case of our framework in which both complications disappear\. In this special case, priorities are perfectlyseparable, so local evidence atxxis independent of the rule’s behavior elsewhere; moreover, all evidence is reduced to a single score difference, and any “indecision” collapses to noise between decisive responses\. Formally, we show that these restrictions of our model render latent indifference and conflict behaviorally irrelevant, and forced comparisons are consequently justified mechanically\.

For Contributions 2 and 3, we start from this special case of our model that recovers S\-RUMs, and then we study the effect of relaxing each restriction separately: first we allow priorities to be inseparable, and then we allow local queries to generate latent indifference or conflict\.

Contribution II \([Section3](https://arxiv.org/html/2607.02672#S3)\)\.We first isolate the consequences of allowing priorities to be inseparable\. We show that natural classes of priorities, including proportionality, egalitarianism, and equal treatment, satisfy strong notions of inseparability\. We then show that in standard pipelines, these inseparable priorities can be erased or misinterpreted, leading the learning system to infer highly suboptimal rules\. More specifically, we first demonstrate that the most strongly inseparable priorities are erased without a trace when perfect separability is assumed, and moreover, their existence is not identifiable by local pairwise comparisons at all, even if the possibility of their existence is known ahead of time\. For inseparable priorities that can in principle be identified, we show that inferences by learners assuming separability can be systematically distorted, due to the misinterpretation of this inseparable evidence\. In this way, inseparability offers one formal explanation for apparent inconsistency in pairwise\-comparison data: the inconsistency may reflect a mismatch between inseparable rule\-level priorities and local elicitation\.

Contribution III \([Section4](https://arxiv.org/html/2607.02672#S4)\)\.We then isolate the consequences of allowing local queries to generate latent indifference and conflict\. We do this in a linear special case of our model, where priorities exist in a known feature space, and each priority advocates the importance of a single feature\. This separable restriction of our model captures linear social choice\(Geet al\.,[2024a](https://arxiv.org/html/2607.02672#bib.bib99),[2026](https://arxiv.org/html/2607.02672#bib.bib22)\)and related feature\-based preference\-learning models\(Leeet al\.,[2019](https://arxiv.org/html/2607.02672#bib.bib244); Freedmanet al\.,[2020](https://arxiv.org/html/2607.02672#bib.bib127); Geet al\.,[2024b](https://arxiv.org/html/2607.02672#bib.bib188); Noothigattuet al\.,[2018](https://arxiv.org/html/2607.02672#bib.bib121); Boerstleret al\.,[2024](https://arxiv.org/html/2607.02672#bib.bib169)\), but strictly generalizes them by keeping the evidence from each feature separate: some priorities may decisively supportyyand othersy′y^\{\\prime\}, and these trade\-offs need not be resolvable\. In simulated settings, we learn our model via Bayesian active learning under various conditions\. We first analyze losses in learning accuracy when individuals resolve indecision by deviating from the learner’s assumed response model, e\.g\., in lexicographic, biased, or random ways\. We find large learning losses on certain metrics and modest losses on others\. One potential way to avoid such losses would be to allow individuals to directly report their indecision\. After providing evidence that such richer responses are plausible to elicit, we generalize our learner to use indecision reports and demonstrate that allowing such responses has an additional advantage: when individuals can report conflict and indifference — or even just generic indecision conflating these two states — such reports can substantially accelerate learning\.

Contribution IV \([Section5](https://arxiv.org/html/2607.02672#S5)\)\.Although we apply our pluralistic priority model to formalize intuitions about the limitations of local pairwise comparisons, this model opens up a much broader possibility: rather than inferring an entire preference structure from local comparisons alone, one can learn our model to capture richer elements of preferences about how decision rules should work\. In[Section5](https://arxiv.org/html/2607.02672#S5)we build on our results to propose an approach to doing so:priority\-aware learning, which elicits priorities directly—through text and other structured queries—and then interprets all comparison responses through them\. We discuss how such priority\-based preference learning methods — applied either within individuals or across them — could help make preference learning more faithful, more efficient in complex decision spaces, and more interpretable, as the resulting model is composed of priorities the individual\(s\) themselves specified\.

### 1\.1\.Related Work

Model of Pluralistic Priorities \(Contribution I\)\.At a conceptual level, our model reflects many behavioral\-science accounts of value pluralism, moral conflict, and ambivalence, which we discuss in[Section2\.2\.3](https://arxiv.org/html/2607.02672#S2.SS2.SSS3)and Appendix[A\.1](https://arxiv.org/html/2607.02672#A1.SS1)\. From the elicitation literature, our model is conceptually closest to recent work on value inference\(Siebertet al\.,[2022](https://arxiv.org/html/2607.02672#bib.bib203); Liscioet al\.,[2025](https://arxiv.org/html/2607.02672#bib.bib202)\), whose model similarly supposes that individuals make choices by aggregating over multiple values \(priorities\), and that the goal of learning should be to recover values’ relative importance\. However, their model is formally quite different: in their work, a ‘value” is a textual tag \(e\.g\., “Cost\-Effectiveness”\) rather than a preference model over the decision space; these values are of ranked importance, and they are applied linearly to imply a choice over a small set of policy options\. This setup eliminates the two worries our model aims to capture\. First, because participants can express preferences directly over the decision space \(rather than answering local questions about global decision rules, as in our case\), the possibility of inseparability is removed\. Second, linear aggregation over values automatically resolves trade\-offs, thereby smoothing over the conflict between values we explicitly model\. In the same vein, our pluralistic priority model is similar in spirit to multiattribute value and utility theory, which represents preferences as trade\-offs among multiple objectives\(Keeney and Raiffa,[1976](https://arxiv.org/html/2607.02672#bib.bib74)\)\. However, whereas this tradition ultimately models an overall preference or utility representation, our model keeps the underlying priorities separate and allows them to conflict\.

More broadly, our model is one of many recent attempts to learn more cognitively faithful models of moral preferences infeature\-based decision settings, where individuals must trade off morally salient features\(Kimet al\.,[2018](https://arxiv.org/html/2607.02672#bib.bib318); Mohsinet al\.,[2022](https://arxiv.org/html/2607.02672#bib.bib207); Cousinset al\.,[2025](https://arxiv.org/html/2607.02672#bib.bib192)\)\. This work is complementary to ours, presenting decision models in the restricted setting where priorities exist over predefined features \(a setting our model formally captures\), and exploring different methods of resolving trade\-offs between features, including linear utility models, heuristic or lexicographic rules, and feature\-processing rules followed by fixed comparison rules\. These different ways of aggregating over features represent different instantiations of ourpriority aggregators, as defined in[Section2](https://arxiv.org/html/2607.02672#S2)\.

Separability and Inseparability \(Contribution II\)\.To our knowledge, there has not been a paper formally describing the problem of inseparability in the domain of decision rule design\.222Another recent alignment paper may initially appear related — it shows that sparse pairwise comparisons cannot identify higher\-order population information needed for inequality\-aware objectives\(Geet al\.,[2026](https://arxiv.org/html/2607.02672#bib.bib22)\)\. However, theirs is a distinct concern from ours: their inequality\-aware objective is not inseparable in our sense, since it is evaluated on a single candidate/input\.The closest is a paper in participatory design of fair division outcomes\(Shiraliet al\.,[2024](https://arxiv.org/html/2607.02672#bib.bib126)\), which is motivated by a conceptually similar concern: that social preferences like inequality aversion cannot be “expressed” by standard cardinal utilities\. They state this observation in passing, while our work formalizes the analogous concern as an inseparability problem\. Their model is also conceptually related to ours: they propose to let people want to optimize a linear combination of multiple objectives, where these objectives correspond in spirit to our priorities\. They apply their model toward a different goal;giveneach agent’s desired weighting over objectives, they analyze the welfare of the allocation produced by optimizing an aggregate objective\.

Outside of alignment and participatory design, the importance of separability assumptions — and their potential lack of realism, is well\-recognized in the distinct setting of committee selection / multi\-winner voting\. In this domain, separability is a standard assumption, in that case meaning that a voter’s preference over one alternative does not depend on the rest of the set of winners\(Lang and Xia,[2016](https://arxiv.org/html/2607.02672#bib.bib16)\)\. This literature also points out that when preferences are inseparable, local voting becomes misspecified\(Lang and Xia,[2016](https://arxiv.org/html/2607.02672#bib.bib16); Barrot and Lang,[2016](https://arxiv.org/html/2607.02672#bib.bib18); Ratliff,[2006](https://arxiv.org/html/2607.02672#bib.bib19); Ratliff and Saari,[2014](https://arxiv.org/html/2607.02672#bib.bib20); Uckelman,[2010](https://arxiv.org/html/2607.02672#bib.bib21)\); our model captures this same concern when the decision space consists of decision rules instead of committees\. Our work also differs in its goal: while these papers aim to document the problem with examples and/or develop richer ballots and aggregation methods, we formalize what is lost downstream when inseparability forces individuals to compress their inseparable priorities in response to local ballots\.

Conflict and Indifference \(Contribution III\)\.The importance of indecision has been broadly highlighted by empirical work showing that pairwise comparisons can be difficult, unstable, or low\-confidence, and suggesting that forced binary responses may obscure meaningful non\-decisive states\(Boerstleret al\.,[2024](https://arxiv.org/html/2607.02672#bib.bib169); Keswaniet al\.,[2025a](https://arxiv.org/html/2607.02672#bib.bib212),[b](https://arxiv.org/html/2607.02672#bib.bib190)\)\. The paper that comes the closest to our model of indecision isMcElfreshet al\.\([2021](https://arxiv.org/html/2607.02672#bib.bib302)\), which presents several formal models of indecision in discrete choices\. Our indifference notion is close to their desirability\-based modelMin\-UU, and our notion of conflict is a hybrid of their desirability\-based and difference\-based modelsMax\-UUandMin\-δ\\delta\. The more substantial difference is that their indecision state is decided by comparing exogenous utilities over each local option, while indecision in our model is microfounded as resulting from multiple competing priorities evidencing different choices\. Our work has a similar relationship to the broader set of work extending classical pairwise comparison models like Bradley\-Terry to permit ties\(Rao and Kupper,[1967](https://arxiv.org/html/2607.02672#bib.bib113); Davidson,[1970](https://arxiv.org/html/2607.02672#bib.bib114)\)\.

These tie\-permitting extensions of Bradley\-Terry have been applied in LLM alignment byLiuet al\.\([2024](https://arxiv.org/html/2607.02672#bib.bib278)\); as we do, they find that ignoring ties comes with learning costs\. The intuition for why utilizing indecision helps in our setting is closely related to active\-learning results showing that information about closeness to the decision boundary can improve learning\(Kaneet al\.,[2017](https://arxiv.org/html/2607.02672#bib.bib189)\)\. In learning our model, we apply standard tools from Bayesian Active Learning by Disagreement\(Houlsbyet al\.,[2011](https://arxiv.org/html/2607.02672#bib.bib292)\)\.

Relationship to LLM alignment\.Our model can in principle represent decision rules as simple as linear models and as complicated as large language models, where𝒳\\mathcal\{X\}is the space of prompts and𝒴\\mathcal\{Y\}is the space of responses\. Our approach, examples, and intuitions are built primarily around simpler high\-stakes decision tasks because these settings make the rule\-level object more concrete: one can describe the inputs, outputs, and counterfactual rule behavior, abstracting away from the separate challenge of controlling a large generative model\. However, our results concern the information content of local pairwise comparisons, not the tractability of optimizing overℱ\\mathcal\{F\}, so they are relevant whenever pairwise comparisons are used as evidence about human values, including in LLM alignment\.

Relationship to Pluralistic Alignment\.Our work is also related to the growing literature on pluralistic alignment, which argues that aligned AI systems should account for the diversity of values, preferences, and perspectives across people and groups\(Conitzer and others,[2024](https://arxiv.org/html/2607.02672#bib.bib287); Sorensenet al\.,[2024](https://arxiv.org/html/2607.02672#bib.bib171)\)\. This literature is similar in spirit to ours in rejecting the idea that alignment can be reduced to a single objective, but our work differs in that we study the consequences of pluralitywithinrather thanacrossindividuals\. This distinction matters because reconciling plurality within versus across individuals pose different challenges: across individuals, the central challenge is defining an external rule for fairly making trade\-offs; within an individual, the challenge is learning howthey want toreconcile trade\-offs\. Our results complement the pluralistic alignment literature, showing that failures due to value pluralism can arise even before the social\-choice problem of aggregating across people\. We discuss this further in[Section5](https://arxiv.org/html/2607.02672#S5)\.

## 2\.Model

Let adecision taskbe defined by an input space𝒳\\mathcal\{X\}and output space𝒴\\mathcal\{Y\}\. We denote a single input and output byx∈Xx\\in Xandy∈Yy\\in Y, respectively\. Not all outputs may be valid for a given inputxx, so we let𝒴​\(x\)⊆𝒴\\mathcal\{Y\}\(x\)\\subseteq\\mathcal\{Y\}denote the set of permissible outputs on inputxx\. While𝒳,𝒴\\mathcal\{X\},\\mathcal\{Y\}can be discrete or continuous, for technical convenience we take𝒳\\mathcal\{X\}and𝒴\\mathcal\{Y\}to be discrete and finite unless otherwise stated\. Adecision ruleis a functionF:𝒳→𝒴F:\\mathcal\{X\}\\to\\mathcal\{Y\}such thatF​\(x\)∈𝒴​\(x\)F\(x\)\\in\\mathcal\{Y\}\(x\)for everyx∈𝒳x\\in\\mathcal\{X\}\. We focus on deterministic rules, but randomized rules could also be of interest\. Letℱ\\mathcal\{F\}denote the set of all decision rules for a given𝒳,𝒴\\mathcal\{X\},\\mathcal\{Y\}\(where the input/output spaces are left implicit, as they will be clear from context\)\.

For concreteness, we will often use the running example ofallocation decision tasks\. In an allocation task, each decision allocates a set of goods or bads to a set of possible recipients\. Formally, there is a universe of recipientsNNand a universe of goods or badsKK\. Each inputx∈𝒳x\\in\\mathcal\{X\}consists of a set of goods or badsK​\(x\)⊆KK\(x\)\\subseteq Kand a set of recipientsN​\(x\)⊆NN\(x\)\\subseteq N, sox=\(K​\(x\),N​\(x\)\)x=\(K\(x\),N\(x\)\)\. Each outputy∈𝒴​\(x\)y\\in\\mathcal\{Y\}\(x\)is an assignment involving the elements ofK​\(x\)K\(x\)andN​\(x\)N\(x\)\.

###### Example 0 \(Trolley problem\)\.

In a simple version of the trolley problem, an inputxxconsists of two groups of individualsN1​\(x\),N2​\(x\)N\_\{1\}\(x\),N\_\{2\}\(x\)which partitionN​\(x\)N\(x\)\. The decision rule must choose which group is spared\. Thus𝒴​\(x\)=\{1,2\}\\mathcal\{Y\}\(x\)=\\\{1,2\\\}, where outputℓ∈\{1,2\}\\ell\\in\\\{1,2\\\}means that groupNℓ​\(x\)N\_\{\\ell\}\(x\)is spared\.

Finally, we define a local pairwise comparison query in the context of decision task𝒳,𝒴\\mathcal\{X\},\\mathcal\{Y\}\. Aquery formatis itself a decision task, with query space𝒬\\mathcal\{Q\}and response space𝒲\\mathcal\{W\}\. A query is denotedq∈𝒬q\\in\\mathcal\{Q\}, and a response is denotedw∈𝒲w\\in\\mathcal\{W\}\. The set of valid responses to queryqqis the response alphabet𝒲​\(q\)⊆𝒲\\mathcal\{W\}\(q\)\\subseteq\\mathcal\{W\}\.

###### Definition 2\.0 \(Local Pairwise Comparison Query Format\)\.

A local pairwise comparison query consists of an inputx∈𝒳x\\in\\mathcal\{X\}and two feasible outputsy,y′∈𝒴​\(x\)y,y^\{\\prime\}\\in\\mathcal\{Y\}\(x\):

𝒬pc=\{\(y,y′;x\):x∈𝒳,y,y′∈𝒴​\(x\)\}\.\\mathcal\{Q\}^\{\\mathrm\{pc\}\}=\\\{\(y,y^\{\\prime\};x\):x\\in\\mathcal\{X\},\\ y,y^\{\\prime\}\\in\\mathcal\{Y\}\(x\)\\\}\.We write a generic query asq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)\. We allow there to be four possible responses to a local pairwise comparison query:

𝒲pc​\(q\)=\{≻,≺,∼,⋈\}∀q∈𝒬pc\.\\mathcal\{W\}^\{\\mathrm\{pc\}\}\(q\)=\\\{\\succ,\\prec,\\sim,\\bowtie\\\}\\qquad\\forall q\\in\\mathcal\{Q\}^\{\\mathrm\{pc\}\}\.As we will formalize later, responses≻\\succand≺\\precindicate thatyyis preferred toy′y^\{\\prime\}andy′y^\{\\prime\}is preferred toyy, respectively,∼\\simindicates indifference betweenyyandy′y^\{\\prime\}, and⋈\\bowtieindicates conflict or incomparability, which we will describe later\.

### 2\.1\.Priority Model

The priority model is composed ofmmpriorities and their relative weights\. Formally, apriorityj∈\[m\]j\\in\[m\]is a complete and transitive preference relation≿j\\succsim\_\{j\}overℱ\\mathcal\{F\}\. As usual,F≿jF′F\\succsim\_\{j\}F^\{\\prime\}means that ruleFFis weakly preferred to ruleF′F^\{\\prime\}according to priorityjj\. Because each≿j\\succsim\_\{j\}is complete and transitive, it admits a utility representationuj:ℱ→ℝu\_\{j\}:\\mathcal\{F\}\\to\\mathbb\{R\}, so that

F≿jF′⇔uj​\(F\)≥uj​\(F′\)∀F,F′∈ℱ\.F\\succsim\_\{j\}F^\{\\prime\}\\iff u\_\{j\}\(F\)\\geq u\_\{j\}\(F^\{\\prime\}\)\\qquad\\forall F,F^\{\\prime\}\\in\\mathcal\{F\}\.We assume utilities are normalized so that\|uj​\(F\)−uj​\(F′\)\|≤1\|u\_\{j\}\(F\)\-u\_\{j\}\(F^\{\\prime\}\)\|\\leq 1for allj∈\[m\],F,F′∈ℱ\.j\\in\[m\],\\ F,F^\{\\prime\}\\in\\mathcal\{F\}\.Letu=\(uj\)j∈\[m\]u=\(u\_\{j\}\)\_\{j\\in\[m\]\}\. When we write a natural utility function whose range is not explicitly bounded by one, we implicitly take its positive affine normalization to satisfy this convention; we suppress this normalization in the notation whenever it plays no substantive role\.

We additionally let each priorityjjhave aweightωj∈\[0,1\]\\omega\_\{j\}\\in\[0,1\], interpreted as its relative importance to the individual\. We assume these weights are normalized to 1; that is, lettingΔm−1\\Delta^\{m\-1\}denote the standardm−1m\-1simplex,ω=\(ωj\)j∈\[m\]∈Δm−1\.\\omega=\(\\omega\_\{j\}\)\_\{j\\in\[m\]\}\\in\\Delta^\{m\-1\}\.

Then, apriority modelis a tupleM=\(u,ω\),M=\\big\(u,\\omega\\big\),consisting of themmpriorities’ utility representations and relative weights\. Whenu,ωu,\\omegaare clear we will simply writeMM; when not, we will writeM=\(u,ω\)M=\(u,\\omega\)\. For a fixedℱ\\mathcal\{F\}, letℳ\\mathcal\{M\}be the class of all such priority models\. Before proceeding, we give intuition via an example illustrating how natural priorities can be formalized into utility functions\.

###### Example 0 \(Formalization of the priorities in[Figure1](https://arxiv.org/html/2607.02672#S1.F1)\)\.

Here we formalize the three priorities from[Figure1](https://arxiv.org/html/2607.02672#S1.F1), as applied to the trolley problem example \([Example2\.1](https://arxiv.org/html/2607.02672#S2.Thmtheorem1)\), where atxxthe decision rule must choose between sparing groupN1​\(x\)N\_\{1\}\(x\)orN2​\(x\)N\_\{2\}\(x\)\. Here,𝒟∈Δ​\(𝒳\)\\mathcal\{D\}\\in\\Delta\(\\mathcal\{X\}\)is a reference distribution over inputs, and in the Proportionality priority,G1,…,GgG\_\{1\},\\dots,G\_\{g\}are protected groups of potential recipients andαℓ\\alpha\_\{\\ell\}is the ideal fraction of harm borne by groupℓ∈\[g\]\\ell\\in\[g\]:

usize​\(F\)\\displaystyle u\_\{\\mathrm\{size\}\}\(F\)=Prx∼𝒟⁡\[F​\(x\)∈arg⁡maxℓ∈\{1,2\}⁡\|Nℓ​\(x\)\|\],\\displaystyle=\\Pr\_\{x\\sim\\mathcal\{D\}\}\\left\[F\(x\)\\in\\arg\\max\_\{\\ell\\in\\\{1,2\\\}\}\|N\_\{\\ell\}\(x\)\|\\right\],ufamily​\(F\)\\displaystyle u\_\{\\mathrm\{family\}\}\(F\)=Prx∼𝒟⁡\[F​\(x\)∈arg⁡maxℓ∈\{1,2\}⁡𝟏​\{Nℓ​\(x\)​contains a family member\}\],\\displaystyle=\\Pr\_\{x\\sim\\mathcal\{D\}\}\\left\[F\(x\)\\in\\arg\\max\_\{\\ell\\in\\\{1,2\\\}\}\\mathbf\{1\}\\\{N\_\{\\ell\}\(x\)\\text\{ contains a family member\}\\\}\\right\],uprop​\(F\)\\displaystyle u\_\{\\mathrm\{prop\}\}\(F\)=−∑ℓ∈\[g\]\(𝔼x∼𝒟​\[𝟏​\{F​\(x\)​fails to spare a member of group​Gℓ\}\]−αℓ\)2\.\\displaystyle=\-\\sum\_\{\\ell\\in\[g\]\}\\left\(\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\}\[\\mathbf\{1\}\\\{F\(x\)\\text\{ fails to spare a member of group \}G\_\{\\ell\}\\\}\]\-\\alpha\_\{\\ell\}\\right\)^\{2\}\.

Of course, a textual priority can often be formalized in more than one reasonable way\.333For instance, one could instead formulate these priorities as binary constraints, sorting rules by whether or not they satisfy the priority perfectly\. For example, thesizepriority could alternatively be formulated asusize​\(F\)=𝟏​\{F​\(x\)∈arg​maxℓ∈\{1,2\}⁡\|Nℓ​\(x\)\|​∀x∈𝒳\}u\_\{\\mathrm\{size\}\}\(F\)=\\mathbf\{1\}\\left\\\{F\(x\)\\in\\operatorname\*\{arg\\,max\}\_\{\\ell\\in\\\{1,2\\\}\}\|N\_\{\\ell\}\(x\)\|\\,\\forall x\\in\\mathcal\{X\}\\right\\\}\.Which formalization is “true” may be ambiguous not only to the learner, but to the individual themselves\. In this paper, we do not study the problem of translating textually\-articulated priorities into formal priorities, but we do discuss the possibility of doing so in[Section5](https://arxiv.org/html/2607.02672#S5)\.

Aggregate Utility and Regret\.Given a priority modelM=\(u,ω\)M=\(u,\\omega\), we define a rule’saggregate utilityUM:ℱ→ℝU\_\{M\}:\\mathcal\{F\}\\to\\mathbb\{R\}as the weighted sum of its utilities over the priorities:

UM​\(F\):=∑j∈\[m\]ωj​uj​\(F\)\.U\_\{M\}\(F\):=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}u\_\{j\}\(F\)\.We will refer to the set of rules that maximize the aggregate utilityarg⁡maxF∈ℱ′⁡UM​\(F\)\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}U\_\{M\}\(F\)asaggregate\-optimal rules\.

Importantly, we do not claim that aggregate\-optimal rules are most preferred by the individual — in fact, we explicitly abstain from defining global preferences over rules \(though we discuss a natural way of doing so in[Section4](https://arxiv.org/html/2607.02672#S4)\)\. We define the aggregate utility because we find that it corresponds to notions used in the existing models we capture\. We will use it as a convenient way to measure learning losses in the form of theregret:

###### Definition 2\.0 \(Regret\)\.

LettingF∗F^\{\*\}be any aggregate\-optimal rule inℱ\\mathcal\{F\}, the regret of ruleF^\\widehat\{F\}is

RegretM​\(F^\):=UM​\(F∗\)−UM​\(F^\)\.\\mathrm\{Regret\}\_\{M\}\(\\widehat\{F\}\):=U\_\{M\}\(F^\{\*\}\)\-U\_\{M\}\(\\widehat\{F\}\)\.

### 2\.2\.Query\-Level Preferences: theLatent State

Here, we model how rule\-level priorities are used to construct beliefs about local pairwise comparison queries\. The key primitive in defining this translation is a projection from rule space onto the query\(y,y′;x\)\(y,y^\{\\prime\};x\): For any ruleF∈ℱF\\in\\mathcal\{F\}, inputx∈𝒳x\\in\\mathcal\{X\}, and outputy∈𝒴​\(x\)y\\in\\mathcal\{Y\}\(x\), define thelocal projectionofFFontox,yx,y, calledFx→yF\_\{x\\to y\}, as the rule obtained by surgically changingFF’s output atxxtoyy, while leaving its behavior elsewhere fixed\.

\(1\)Fx→y​\(x′\)=\{yif​x′=x,F​\(x′\)if​x′≠x\.∀x′∈𝒳\.F\_\{x\\to y\}\(x^\{\\prime\}\)=\\begin\{cases\}y&\\text\{if \}x^\{\\prime\}=x,\\\\ F\(x^\{\\prime\}\)&\\text\{if \}x^\{\\prime\}\\neq x\.\\end\{cases\}\\quad\\forall x^\{\\prime\}\\in\\mathcal\{X\}\.When it is used in a projection,FFis known as a “background” rule, because it is defining what is being done in the “background” ofxx, i\.e\., at all other inputs\. Now, for any local pairwise comparison queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)and priorityjj, we apply the projection to define the marginal gain of choosingyyovery′y^\{\\prime\}, with respect to background ruleFF:

\(2\)ΔjF​\(q\)=uj​\(Fx→y\)−uj​\(Fx→y′\)∀F∈ℱ\.\\Delta\_\{j\}^\{F\}\(q\)=u\_\{j\}\(F\_\{x\\to y\}\)\-u\_\{j\}\(F\_\{x\\to y^\{\\prime\}\}\)\\qquad\\forall F\\in\\mathcal\{F\}\.We call this theprojection gapfor queryqqwith respect toFF\. IfΔjF​\(q\)\>0\\Delta\_\{j\}^\{F\}\(q\)\>0, then, given the background ruleFF, priorityjjfavors outputyyovery′y^\{\\prime\}atxx\. IfΔjF​\(q\)<0\\Delta\_\{j\}^\{F\}\(q\)<0, it favorsy′y^\{\\prime\}overyy\. IfΔjF​\(q\)=0\\Delta\_\{j\}^\{F\}\(q\)=0, it is indifferent between the two projected rules\. Conceptually, this is saying: if at allx′x^\{\\prime\}outside ofxx, the rule behaves according toFF, would priorityjjprefer the rule outputy′y^\{\\prime\}oryy?

Here arises the first main complication: in answering a queryqq, the individual must give aglobal judgmentabout whether to chooseyyory′y^\{\\prime\}\. In contrast, the above quantity depends on the background ruleFFand the priorityjj, meaning we need to aggregate over both background rulesF∈ℱF\\in\\mathcal\{F\}and prioritiesj∈\[m\]j\\in\[m\]to produce a global judgment\.

We will assume aggregation happens over rules first, and then over priorities\. This is natural because if priority aggregation instead came first, the individual would be implicitly tracking a quantity for every background rule inℱ\\mathcal\{F\}\(enormous\) rather than one for every priority\[m\]\[m\]\(small\)\. The core intuition is captured fully under this assumption; we leave the extensions of our results to other aggregation orders to future work\.

Over the next subsections, we define rule aggregation \([Section2\.2\.1](https://arxiv.org/html/2607.02672#S2.SS2.SSS1)\), priority aggregation \([Section2\.2\.2](https://arxiv.org/html/2607.02672#S2.SS2.SSS2)\), and the individual’s resulting global query preference, known as theirlatent state\([Section2\.2\.3](https://arxiv.org/html/2607.02672#S2.SS2.SSS3)\)\. We depict these three steps from left to right in[Figure2](https://arxiv.org/html/2607.02672#S2.F2)\.

![Refer to caption](https://arxiv.org/html/2607.02672v1/x2.png)Figure 2\.Example of how we go from the priority modelMM, to the directional evidence scoressM\+​\(q\),sM−​\(q\)s^\{\+\}\_\{M\}\(q\),s^\{\-\}\_\{M\}\(q\)by aggregating over rules and then priorities, to how these scores translate to latent states\.#### 2\.2\.1\.Rule Aggregation

As shown in[Figure2](https://arxiv.org/html/2607.02672#S2.F2), rule aggregation happens within each priorityjj\. It is done by arule aggregator, which aggregates the projection gaps across rules\{ΔjF​\(y,y′;x\)\|F∈ℱ\}\\\{\\Delta\_\{j\}^\{F\}\(y,y^\{\\prime\};x\)\|F\\in\\mathcal\{F\}\\\}into a single number describing how strongly priorityjjprefersyyovery′y^\{\\prime\}\. In the definition below, this corresponds to pluggingΔjF\\Delta\_\{j\}^\{F\}in forzFz^\{F\}\.

###### Definition 2\.0 \(Rule Aggregator\)\.

A rule aggregator is a functionϕrules:ℝ\|ℱ\|→ℝ\\phi^\{\\text\{rules\}\}:\\mathbb\{R\}^\{\|\\mathcal\{F\}\|\}\\to\\mathbb\{R\}\.

We assume two regularity conditions on rule aggregators:

1. \(1\)ϕrules\\phi^\{\\text\{rules\}\}is permutation\-invariant iff the aggregator depends only on the multiset of evidence values, but not their order; that is, for every vectorz∈ℝ\|ℱ\|z\\in\\mathbb\{R\}^\{\|\\mathcal\{F\}\|\}and every bijectionσ:ℱ→ℱ\\sigma:\\mathcal\{F\}\\to\\mathcal\{F\}, ϕrules​\(\(zF\)F∈ℱ\)=ϕrules​\(\(zσ​\(F\)\)F∈ℱ\)\.\\phi^\{\\text\{rules\}\}\\big\(\(z^\{F\}\)\_\{F\\in\\mathcal\{F\}\}\\big\)=\\phi^\{\\text\{rules\}\}\\big\(\(z^\{\\sigma\(F\)\}\)\_\{F\\in\\mathcal\{F\}\}\\big\)\.
2. \(2\)ϕrules\\phi^\{\\text\{rules\}\}is unanimous iff, for any constantz∈ℝz\\in\\mathbb\{R\}, ϕrules​\(\(z\)F∈ℱ\)=z\.\\phi^\{\\text\{rules\}\}\\big\(\(z\)\_\{F\\in\\mathcal\{F\}\}\\big\)=z\.

Natural examples of rule aggregators satisfying these regularity conditions include the maximum and average:

ϕavgrules​\(\(zF\)F∈ℱ\)=1\|ℱ\|​∑F∈ℱzF,ϕmaxrules​\(\(zF\)F∈ℱ\)=maxF∈ℱ⁡zF,ϕp​\-Pctrules​\(\{zF\}F∈ℱ\)=Pctp​\(\{zF\}F∈ℱ\)\.\\phi^\{\\text\{rules\}\}\_\{\\mathrm\{avg\}\}\\big\(\(z^\{F\}\)\_\{F\\in\\mathcal\{F\}\}\\big\)=\\frac\{1\}\{\|\\mathcal\{F\}\|\}\\sum\_\{F\\in\\mathcal\{F\}\}z\_\{F\},\\qquad\\phi^\{\\text\{rules\}\}\_\{\\max\}\\big\(\(z\_\{F\}\)\_\{F\\in\\mathcal\{F\}\}\\big\)=\\max\_\{F\\in\\mathcal\{F\}\}z\_\{F\},\\quad\\phi^\{\\text\{rules\}\}\_\{p\\text\{\-Pct\}\}\\big\(\\\{z\_\{F\}\\\}\_\{F\\in\\mathcal\{F\}\}\\big\)=\\mathrm\{Pct\}\_\{p\}\\big\(\\\{z\_\{F\}\\\}\_\{F\\in\\mathcal\{F\}\}\\big\)\.Except in simulations and some examples where instantiation is necessary or useful for clarity, we will reason about genericϕrules\\phi^\{\\text\{rules\}\}\.

TheSeparableSpecial Case\.As we will see, this rule aggregation step can erase competing directional evidence across background rules\. We now define a special kind of priority — aperfectly separablepriority — within which this loss doesnotoccur, so long asϕrules\\phi^\{\\text\{rules\}\}is unanimous\.

###### Definition 2\.0 \(Perfect Separability\)\.

A priorityjjis perfectly separable at queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)iff there exists some constantδj​\(q\)∈ℝ\\delta\_\{j\}\(q\)\\in\\mathbb\{R\}such that

ΔjF​\(y,y′;x\)=δj​\(q\)for all​F∈ℱ\.\\Delta\_\{j\}^\{F\}\(y,y^\{\\prime\};x\)=\\delta\_\{j\}\(q\)\\qquad\\text\{for all \}F\\in\\mathcal\{F\}\.

Note that whenjjis perfectly separable, for anyqqand any unanimousϕrules\\phi^\{\\text\{rules\}\}, it holds that

ϕrules​\(\{ΔjF​\(q\)\}F∈ℱ\)=δj​\(q\)\.\\phi^\{\\text\{rules\}\}\(\\\{\\Delta\_\{j\}^\{F\}\(q\)\\\}\_\{F\\in\\mathcal\{F\}\}\)=\\delta\_\{j\}\(q\)\.Conceptually, what this is saying is that according to priorityjj, what should be donexxiscompletely independentof what decisions are made at other inputs\.444For intuition, the natural weakening of this notion, which we will not formally use, is its directional \(i\.e\., non\-cardinal\) analog,separability\. A priorityjjis separable at queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)iffSign​\(ΔjF​\(y,y′;x\)\)\\mathrm\{Sign\}\(\\Delta^\{F\}\_\{j\}\(y,y^\{\\prime\};x\)\)is constant over allF∈ℱF\\in\\mathcal\{F\}, i\.e\., the preference betweenFx→yF\_\{x\\to y\}andFx→y′F\_\{x\\to y^\{\\prime\}\}dictated byuju\_\{j\}must go in the same direction for all background rulesFF\.If a priority is perfectly separable at every queryq∈𝒬pcq\\in\\mathcal\{Q\}^\{\\mathrm\{pc\}\}, we say that priority is perfectly separable\. If every priorityj∈\[m\]j\\in\[m\]in a modelMMis perfectly separable, we say that model is perfectly separable\. We letℳsep⊆ℳ\\mathcal\{M\}\_\{\\mathrm\{sep\}\}\\subseteq\\mathcal\{M\}be the class of perfectly separable pluralistic models\.

#### 2\.2\.2\.Priority Aggregation

Once evidence is tallied across ruleswithineach priority, we must then aggregate evidenceacrosspriorities\. The key intuition we want to capture in this aggregation is that the individual may not be able to resolve evidence across priorities that conflicts, i\.e\., cases where certain priorities advocate foryyand others fory′y^\{\\prime\}\. We thus define our priority aggregation process to produce two directional evidence scoressM\+​\(q\)s^\{\+\}\_\{M\}\(q\)andsM−​\(q\)s^\{\-\}\_\{M\}\(q\), which respectively tally evidence in favor ofyyovery′y^\{\\prime\}andy′y^\{\\prime\}overyyseparately,rather than collapsing cross\-priority evidence into a single score\.

For underlying priority modelMM, rule aggregatorϕrules\\phi^\{\\text\{rules\}\}, queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)and notation\[a\]\+=max⁡\{a,0\}\[a\]\_\{\+\}=\\max\\\{a,0\\\}, let thedirectional evidence scoresbe

\(3\)sM\+​\(q\)=∑j=1mωj​\[ϕrules​\(\(ΔjF​\(y,y′;x\)\)F∈ℱ\)\]\+,sM−​\(q\)=∑j=1mωj​\[ϕrules​\(\(ΔjF​\(y′,y;x\)\)F∈ℱ\)\]\+\.s^\{\+\}\_\{M\}\(q\)=\\sum\_\{j=1\}^\{m\}\\omega\_\{j\}\[\\phi^\{\\text\{rules\}\}\\big\(\(\\Delta\_\{j\}^\{F\}\(y,y^\{\\prime\};x\)\)\_\{F\\in\\mathcal\{F\}\}\\big\)\]\_\{\+\},\\quad s^\{\-\}\_\{M\}\(q\)=\\sum\_\{j=1\}^\{m\}\\omega\_\{j\}\[\\phi^\{\\text\{rules\}\}\\big\(\(\\Delta\_\{j\}^\{F\}\(y^\{\\prime\},y;x\)\)\_\{F\\in\\mathcal\{F\}\}\\big\)\]\_\{\+\}\.Here, we are assuming that evidence is tallied across priorities linearly inω\\omega; this will be sufficient to capture classical models in the literature, and corresponds conveniently with our notions of aggregate utility and regret\. Some results will hold for more general priority aggregation methods, which we discuss where relevant\.

#### 2\.2\.3\.Latent State

The directional evidence scoressM\+​\(q\)s^\{\+\}\_\{M\}\(q\)andsM−​\(q\)s^\{\-\}\_\{M\}\(q\)represent global evidence, i\.e\., evidence aggregated over all elements of the modelMM\. We summarize the relative values of these scores into two key intermediates: thevalencerM​\(q\)r\_\{M\}\(q\)describes the strength of the total evidence generated across priorities, and thedecisivenessκM​\(q\)\\kappa\_\{M\}\(q\)describes the extent to which there is a clear choice:

\(4\)rM​\(q\):=sM\+​\(q\)\+sM−​\(q\),andκM​\(q\)=sM\+​\(q\)−sM−​\(q\)\.r\_\{M\}\(q\):=s\_\{M\}^\{\+\}\(q\)\+s\_\{M\}^\{\-\}\(q\),\\qquad\\text\{and\}\\qquad\\kappa\_\{M\}\(q\)=s\_\{M\}^\{\+\}\(q\)\-s\_\{M\}^\{\-\}\(q\)\.WhenMMis clear from context, we will drop it from the notation\. The latent states are derived from these quantities via thresholdsτr∈\[0,2\]\\tau\_\{r\}\\in\[0,2\]andτκ∈\[0,1\]\\tau\_\{\\kappa\}\\in\[0,1\], which respectively dictate how much valence is required to avoid latent indifference, and how much decisiveness is required to avoid latent conflict:555Note thatsM\+,sM−∈\[0,1\]s^\{\+\}\_\{M\},s^\{\-\}\_\{M\}\\in\[0,1\]because for alljj,\|uj​\(F\)−uj​\(F′\)\|≤1\|u\_\{j\}\(F\)\-u\_\{j\}\(F^\{\\prime\}\)\|\\leq 1\. Thus, these thresholds operate on the correct scale\.

###### Definition 2\.0 \(Latent State\)\.

Fix a priority modelMM, queryq∈𝒬pcq\\in\\mathcal\{Q\}^\{\\mathrm\{pc\}\}, valence and decisivenessrM​\(q\)r\_\{M\}\(q\)andκM​\(q\)\\kappa\_\{M\}\(q\)\(droppingMMsubscripts onrMr\_\{M\}andκM\\kappa\_\{M\}for readability\), and thresholdsτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}\. The individual’s latent stateLM​\(q\)L\_\{M\}\(q\)is

LM​\(q\)=\{y≻∗y′if​r​\(q\)−τr≥0andκ​\(q\)−τκ​r​\(q\)≥0,y≺∗y′if​r​\(q\)−τr≥0and−κ​\(q\)−τκ​r​\(q\)≥0,y⋈∗y′if​r​\(q\)−τr≥0and\|κ​\(q\)\|−τκ​r​\(q\)<0y∼∗y′if​r​\(q\)−τr<0\.L\_\{M\}\(q\)=\\begin\{cases\}y\\succ^\{\*\}y^\{\\prime\}&\\text\{if \}\\ r\(q\)\-\\tau\_\{r\}\\geq 0\\ \\ \\text\{ and \}\\ \\ \\ \\ \\ \\kappa\(q\)\-\\tau\_\{\\kappa\}r\(q\)\\geq 0,\\\\\[3\.99994pt\] y\\prec^\{\*\}y^\{\\prime\}&\\text\{if \}\\ r\(q\)\-\\tau\_\{r\}\\geq 0\\ \\ \\text\{ and \}\\ \-\\kappa\(q\)\-\\tau\_\{\\kappa\}r\(q\)\\geq 0,\\\\\[3\.99994pt\] y\\bowtie^\{\*\}y^\{\\prime\}&\\text\{if \}\\ r\(q\)\-\\tau\_\{r\}\\geq 0\\ \\ \\text\{ and \}\\ \\ \\ \|\\kappa\(q\)\|\-\\tau\_\{\\kappa\}r\(q\)<0\\\\\[3\.99994pt\] y\\sim^\{\*\}y^\{\\prime\}&\\text\{if \}\\ r\(q\)\-\\tau\_\{r\}<0\.\\end\{cases\}Note that in the special case whereτκ=κ​\(q\)=0\\tau\_\{\\kappa\}=\\kappa\(q\)=0, this definition is technically improper as it assigns two states:y≻∗y′y\\succ^\{\*\}y^\{\\prime\}andy≺∗y′y\\prec^\{\*\}y^\{\\prime\}\. In this case, arbitrarily assign the state to either≻∗\\succ^\{\*\}or≺∗\\prec^\{\*\}\.

We diagram these states in[Figure2](https://arxiv.org/html/2607.02672#S2.F2), which are most easily understood by considering the relative values ofsM\+,sM−s\_\{M\}^\{\+\},s\_\{M\}^\{\-\}they reflect\. Decisive states≻∗\\succ^\{\*\}and≺∗\\prec^\{\*\}occur when one score is high and the other is low, i\.e\., overwhelming evidence in favor of one outcome and not the other\. The other two states,∼∗\\sim^\{\*\}and⋈∗\\bowtie^\{\*\}, correspond to distinct indecisive states:indifference∼∗\\sim^\{\*\}arises when both scores are too low, i\.e\., no priority offers strong evidence in favor of either response over the other\.Conflict⋈∗\\bowtie^\{\*\}, in contrast, arises when both scores are high and are similar, i\.e\., there is strong evidence forbothoutcomes\. To illustrate this difference in the trolley example, indifference might occur when the individual must decide whether the trolley should hit a plastic bag or an old sandwich; conflict might occur when they must decide between their mother and 1000 strangers\.

### 2\.3\.Query Response Model

In contrast to the latent stateLM​\(q\)L\_\{M\}\(q\), which is unobserved, aquery response modeldescribes the query response that is actually observed\. These two elements may come apart, e\.g\., due to noise and/or distortions due to restrictions in the permissible responses\. Formally, a query response model is a mapping

R:ℳ→\(𝒬p​c→Δ​\(𝒲p​c\)\)\.R:\\mathcal\{M\}\\to\\left\(\\mathcal\{Q\}^\{pc\}\\to\\Delta\(\\mathcal\{W\}^\{pc\}\)\\right\)\.Then, given a queryqqand a modelMM, the resulting objectR​\(q;M\)R\(q;M\)is a distribution over𝒲p​c\\mathcal\{W\}^\{pc\}\. To model randomness in responses, we consider query response models that are parameterized by alink functionhh, which describes the functional form of the noise:

###### Definition 2\.0 \(Link Function\)\.

A link functionh:ℝ→\(0,1\)h:\\mathbb\{R\}\\to\(0,1\)is a continuous, strictly increasing function that satisfies

limt→−∞h​\(t\)=0,limt→\+∞h​\(t\)=1,h​\(−t\)=1−h​\(t\)∀t∈ℝ\.\\lim\_\{t\\to\-\\infty\}h\(t\)=0,\\qquad\\lim\_\{t\\to\+\\infty\}h\(t\)=1,\\qquad h\(\-t\)=1\-h\(t\)\\quad\\forall t\\in\\mathbb\{R\}\.Lethβ​\(t\):=h​\(β​t\)h\_\{\\beta\}\(t\):=h\(\\beta t\)be the link function with scalar inverse temperature parameterβ\>0\\beta\>0, which makes the responses less noisy asβ\\betagets larger\.

Now, we will define the class of local pairwise comparison query response models we will study, calledbaselinequery response models and denoted asRhβ;τr,τκ∘R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\. The∘\\circdenotes the baseline designation; the response models in this class vary over the choice of link functionhhandβ\\beta, as well as the thresholdsτr,τκ\.\\tau\_\{r\},\\tau\_\{\\kappa\}\.This response model corresponds to the individual reporting a noisy version of their latent state\.666Here, we only apply noise at the boundaries between≻∗,≺∗,\\succ^\{\*\},\\prec^\{\*\},and⋈∗\\bowtie^\{\*\}because this is sufficient for our investigation, but one could noise the latent state in many ways\. The only property of this noising required for our results is that whenτr=τκ=0\\tau\_\{r\}=\\tau\_\{\\kappa\}=0, the model collapses to the zero\-threshold case described in[Lemma2\.11](https://arxiv.org/html/2607.02672#S2.Thmtheorem11)\.

###### Definition 2\.0 \(Baseline Query Response Model\)\.

Fix thresholdsτr∈\[0,2\]\\tau\_\{r\}\\in\[0,2\],τκ∈\[0,1\]\\tau\_\{\\kappa\}\\in\[0,1\], a link functionhh, andβ\>0\\beta\>0\. Given a pluralistic modelM∈ℳM\\in\\mathcal\{M\}and queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\), the baseline response modelRhβ;τr,τκ∘R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}gives the response distribution \(dropping theMMsubscripts for readability\):

Rhβ;τr,τκ∘​\(q;M\)​\(y⊳y′\)=\{𝟏​\{r​\(q\)−τr≥0\}⋅hβ​\(κ​\(q\)−τκ​r​\(q\)\)⊳⁣=⁣≻,𝟏​\{r​\(q\)−τr≥0\}⋅hβ​\(−κ​\(q\)−τκ​r​\(q\)\)⊳⁣=⁣≺,𝟏​\{r​\(q\)−τr≥0\}⋅\(1−hβ​\(κ​\(q\)−τκ​r​\(q\)\)−hβ​\(−κ​\(q\)−τκ​r​\(q\)\)\)⊳⁣=⁣⋈,𝟏​\{r​\(q\)−τr<0\}⊳⁣=⁣∼\.R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\)\(y\\triangleright y^\{\\prime\}\)=\\begin\{cases\}\\mathbf\{1\}\\\{r\(q\)\-\\tau\_\{r\}\\geq 0\\\}\\cdot h\_\{\\beta\}\(\\kappa\(q\)\-\\tau\_\{\\kappa\}r\(q\)\)&\\triangleright=\\,\\succ,\\\\\[6\.0pt\] \\mathbf\{1\}\\\{r\(q\)\-\\tau\_\{r\}\\geq 0\\\}\\cdot h\_\{\\beta\}\(\-\\kappa\(q\)\-\\tau\_\{\\kappa\}r\(q\)\)&\\triangleright=\\,\\prec,\\\\\[6\.0pt\] \\mathbf\{1\}\\\{r\(q\)\-\\tau\_\{r\}\\geq 0\\\}\\cdot\\left\(1\-h\_\{\\beta\}\(\\kappa\(q\)\-\\tau\_\{\\kappa\}r\(q\)\)\-h\_\{\\beta\}\(\-\\kappa\(q\)\-\\tau\_\{\\kappa\}r\(q\)\)\\right\)&\\triangleright=\\,\\bowtie,\\\\\[6\.0pt\] \\mathbf\{1\}\\\{r\(q\)\-\\tau\_\{r\}<0\\\}&\\triangleright=\\,\\sim\.\\end\{cases\}

Note that for any link functionhh, asβ→∞\\beta\\to\\inftythis response model corresponds to the individual deterministically reporting their latent state\.777This is true except at points exactly on the response boundary whereκ​\(q\)=τκ​r​\(q\)\\kappa\(q\)=\\tau\_\{\\kappa\}r\(q\); then, byh​\(t\)=1−h​\(−t\)h\(t\)=1\-h\(\-t\), the individual must randomize 50/50 between the relevant decisive response \(≻\\succor≺\\prec\) and conflict \(⋈\\bowtie\)\.Note also that we distinguish the latent state from the query response with an∗, where the∗designates the latent state and its absence designates the reported relation\.

TheZero\-ThresholdSpecial Case\.One important special case occurs whenτr=τκ=0\\tau\_\{r\}=\\tau\_\{\\kappa\}=0\. The key observation is that in this case, indecisive latent states and query responses become impossible:

###### Lemma 2\.11 \(Zero\-threshold case\)\.

Whenτr=τκ=0\\tau\_\{r\}=\\tau\_\{\\kappa\}=0, the latent states reduce to

LM​\(q\)=\{y≻∗y′if​κ​\(q\)≥0,y≺∗y′if−κ​\(q\)≥0L\_\{M\}\(q\)=\\begin\{cases\}y\\succ^\{\*\}y^\{\\prime\}&\\text\{if \}\\kappa\(q\)\\geq 0,\\\\\[3\.99994pt\] y\\prec^\{\*\}y^\{\\prime\}&\\text\{if \}\\ \-\\kappa\(q\)\\geq 0\\end\{cases\}and for any link functionhhandβ\>0\\beta\>0, the baseline query model reduces to

Rhβ;0,0∘​\(q;M\)​\(y⊳y′\)=\{hβ​\(κ​\(q\)\)if⊳=≻hβ​\(−κ​\(q\)\)if⊳=≺0if⊳∈\{∼,⋈\}\.R\_\{h\_\{\\beta\};0,0\}^\{\\circ\}\(q;M\)\(y\\triangleright y^\{\\prime\}\)=\\begin\{cases\}h\_\{\\beta\}\(\\kappa\(q\)\)&\\text\{if \}\\ \\triangleright=\\,\\succ\\\\ h\_\{\\beta\}\(\-\\kappa\(q\)\)&\\text\{if \}\\ \\triangleright=\\,\\prec\\\\ 0&\\text\{if \}\\ \\triangleright\\in\\\{\\sim,\\bowtie\\\}\.\\end\{cases\}

###### Proof\.

The latent states follow by definition\. In the response model the first two probabilities are by definition; then by the fact thath​\(−t\)=1−h​\(t\)h\(\-t\)=1\-h\(t\), it follows that the probability of≻\\succand≺\\precresponses must add to 1, and thus the remaining possible responses occur with 0 probability\. ∎

### 2\.4\.Key Preliminary: S\-RUMs as thePerfectly Separable, Zero\-ThresholdSpecial Case

We now relate our model toscore\-based random utility models\(S\-RUMs\), a popular class of choice models that assume that every feasible local output can be assigned a single latent score, and that pairwise comparisons are generated by noisily comparing these scores:

###### Definition 2\.0\.

A score\-based random utility model \(S\-RUM\) is defined by a local score function

V:X×Y→ℝ≥0\.V:X\\times Y\\to\\mathbb\{R\}\_\{\\geq 0\}\.Let𝒱\\mathcal\{V\}be the class of feasible local score functions\. For any𝒱\\mathcal\{V\}, a link functionhh, andβ\>0\\beta\>0, the S\-RUM is a query response modelShβ:𝒱→\(𝒬p​c→Δ​\(𝒲p​c\)\)S\_\{h\_\{\\beta\}\}:\\mathcal\{V\}\\to\\left\(\\mathcal\{Q\}^\{pc\}\\to\\Delta\(\\mathcal\{W\}^\{pc\}\)\\right\)such that, for any queryqqand anyV∈𝒱V\\in\\mathcal\{V\},Shβ​\(q;V\)S\_\{h\_\{\\beta\}\}\(q;V\)is the following distribution over possible responses:

Shβ​\(q;V\)​\(y≻y′\)\\displaystyle S\_\{h\_\{\\beta\}\}\(q;V\)\(y\\succ y^\{\\prime\}\)=hβ​\(V​\(x,y\)−V​\(x,y′\)\)andShβ​\(q;V\)​\(y′≻y\)=1−Shβ​\(q;V\)​\(y≻y′\)\.\\displaystyle=h\_\{\\beta\}\\bigl\(V\(x,y\)\-V\(x,y^\{\\prime\}\)\\bigr\)\\quad\\text\{and\}\\quad S\_\{h\_\{\\beta\}\}\(q;V\)\(y^\{\\prime\}\\succ y\)=1\-S\_\{h\_\{\\beta\}\}\(q;V\)\(y\\succ y^\{\\prime\}\)\.

Note that S\-RUMs capture several choice models popular in alignment, most notably Bradley\-Terry with the logit link functionhβlogit​\(t\)=\(1\+exp⁡\(−β​t\)\)−1h\_\{\\beta\}^\{\\mathrm\{logit\}\}\(t\)=\(1\+\\exp\(\-\\beta t\)\)^\{\-1\}, and Thurstone\-Mosteller\(Thurstone,[1927](https://arxiv.org/html/2607.02672#bib.bib369); Mosteller,[1951](https://arxiv.org/html/2607.02672#bib.bib48)\)with the probit link functionhβprobit​\(t\)=Φ​\(β​t\)h\_\{\\beta\}^\{\\mathrm\{probit\}\}\(t\)=\\Phi\(\\beta t\)\. We now show that S\-RUMs correspond exactly to a version of our model that is heavily restricted on two key dimensions: that all priorities areperfectly separable, and that the thresholdsτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}are zero, i\.e\., indifference and conflict are impossible\. In this reduction,VVis the analog ofMMand the response modelShβS\_\{h\_\{\\beta\}\}is the analog ofRhβ;0,0∘R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\. We show this correspondence in both the response behaviorShβS\_\{h\_\{\\beta\}\}andRhβ;0,0∘R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}, and the natural choice of optimal decision rule underVVandMM\.888Technically, the utilities defined in this construction may violate the normalization convention that\|uj​\(F\)−uj​\(F′\)\|≤1​∀F,F′∈ℱ\|u\_\{j\}\(F\)\-u\_\{j\}\(F^\{\\prime\}\)\|\\leq 1\\ \\forall F,F^\{\\prime\}\\in\\mathcal\{F\}\. This is not a substantive issue here because that normalization is just to ensure the thresholdsτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}are appropriately scaled, and here they are zero\. If desired, one can instead normalize these utilities and absorb the resulting rescaling into the inverse\-temperature parameterβ\\beta, yielding the scale\-free version of the correspondence formalized later in[Definition3\.7](https://arxiv.org/html/2607.02672#S3.Thmtheorem7)\.

###### Theorem 2\.13 \(S\-RUMs as the Separable, Zero\-Threshold Case\)\.

Fix any link functionhhandβ\>0\\beta\>0\. For every local score functionV∈𝒱V\\in\\mathcal\{V\}, there exists a perfectly separable priority modelMV∈ℳsepM\_\{V\}\\in\\mathcal\{M\}\_\{\\mathrm\{sep\}\}such that

Rhβ;0,0∘​\(q;MV\)=Shβ​\(q;V\)∀q∈𝒬p​c,R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\(q;M\_\{V\}\)=S\_\{h\_\{\\beta\}\}\(q;V\)\\qquad\\forall q\\in\\mathcal\{Q\}^\{pc\},and for everyM∈ℳsepM\\in\\mathcal\{M\}\_\{\\mathrm\{sep\}\}, there exists a local score functionVM∈𝒱V\_\{M\}\\in\\mathcal\{V\}such that

Rhβ;0,0∘​\(q;M\)=Shβ​\(q;VM\)∀q∈𝒬p​c\.R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\(q;M\)=S\_\{h\_\{\\beta\}\}\(q;V\_\{M\}\)\\qquad\\forall q\\in\\mathcal\{Q\}^\{pc\}\.
Moreover, for any corresponding pair\(V′,M′\)∈\{\(V,MV\),\(VM,M\)\}\(V^\{\\prime\},M^\{\\prime\}\)\\in\\\{\(V,M\_\{V\}\),\(V\_\{M\},M\)\\\}, for any subset of rulesℱ′⊆ℱ\\mathcal\{F\}^\{\\prime\}\\subseteq\\mathcal\{F\}the learner might consider, the aggregate\-optimal rule is the same:

arg⁡maxF∈ℱ′​∑x∈𝒳V′​\(x,F​\(x\)\)=arg⁡maxF∈ℱ′⁡UM′​\(F\)\.\\arg\\max\_\{F\\in\\mathcal\{\\mathcal\{F\}\}^\{\\prime\}\}\\sum\_\{x\\in\\mathcal\{X\}\}V^\{\\prime\}\(x,F\(x\)\)=\\arg\\max\_\{F\\in\\mathcal\{\\mathcal\{F\}\}^\{\\prime\}\}U\_\{M^\{\\prime\}\}\(F\)\.

###### Proof Sketch\.

The proof is deferred to[SectionA\.2](https://arxiv.org/html/2607.02672#A1.SS2)\. The construction in both directions is simple: GivenVV,MVM\_\{V\}is constructed with a single priority with utility functionuV​\(F\):=∑x∈𝒳V​\(x,F​\(x\)\)u\_\{V\}\(F\):=\\sum\_\{x\\in\\mathcal\{X\}\}V\(x,F\(x\)\)\. Given perfectly separableMM,VMV\_\{M\}is constructed such thatV​\(x,y\)=∑j∈\[m\]ωj​Vj​\(x,y\)V\(x,y\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}V\_\{j\}\(x,y\), whereVj​\(x,y\)=uj​\(Fx→y\)V\_\{j\}\(x,y\)=u\_\{j\}\(F\_\{x\\to y\}\)for an arbitrary background ruleFF\. The key in both cases is showing that for all queriesq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\),V​\(x,y\)−V​\(x,y′\)=s\+​\(q\)−s−​\(q\)V\(x,y\)\-V\(x,y^\{\\prime\}\)=s^\{\+\}\(q\)\-s^\{\-\}\(q\), as these are the differences on which the probability link functions depend in the respective response models\. The optimal rule correspondence holds by perfect separability of the priorities inM′M^\{\\prime\}, which allows the utility impact of the rule’s behavior to decompose across allxx, making the two sums equivalent for everyFF\(up to a shift, which is irrelevant to the optimization\)\. ∎

## 3\.Generalization \#1: Inseparability

Now, we ask: what happens when the standard assumption of perfect separability no longer holds — i\.e\., when the individual’s priority model can be inℳ∖ℳsep\\mathcal\{M\}\\setminus\\mathcal\{M\}^\{\\mathrm\{sep\}\}? To isolate this generalization all else held equal, we keep the standard restriction thatτr,τκ=0\\tau\_\{r\},\\tau\_\{\\kappa\}=0throughout this section\.

We begin by illustrating that this generalization fromℳsep\\mathcal\{M\}^\{\\mathrm\{sep\}\}toℳ\\mathcal\{M\}is practically relevant — that realistic priorities violate separability, and can do so to the maximum possible degree\. We formalize such violations asinseparabilityandperfect inseparability, defined below\.

###### Definition 3\.0 \(\(Perfect\) Inseparability\)\.

A priorityjjis inseparable at queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)iff there existsF,F′∈ℱF,F^\{\\prime\}\\in\\mathcal\{F\}such that

uj​\(Fx→y\)\>uj​\(Fx→y′\)anduj​\(Fx→y′\)<uj​\(Fx→y′′\)\.u\_\{j\}\(F\_\{x\\to y\}\)\>u\_\{j\}\(F\_\{x\\to y^\{\\prime\}\}\)\\qquad\\text\{and\}\\qquad u\_\{j\}\(F^\{\\prime\}\_\{x\\to y\}\)<u\_\{j\}\(F^\{\\prime\}\_\{x\\to y^\{\\prime\}\}\)\.A priorityjjis perfectly inseparable at queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)iff there existsδ\>0\\delta\>0and a partitionℱy,ℱy′\\mathcal\{F\}\_\{y\},\\mathcal\{F\}\_\{y^\{\\prime\}\}ofℱ\\mathcal\{F\}such that\|ℱy\|=\|ℱy′\|\|\\mathcal\{F\}\_\{y\}\|=\|\\mathcal\{F\}\_\{y^\{\\prime\}\}\|and

ΔjF​\(y,y′;x\)=δ∀F∈ℱy,ΔjF​\(y,y′;x\)=−δ∀F∈ℱy′\.\\Delta^\{F\}\_\{j\}\(y,y^\{\\prime\};x\)=\\delta\\quad\\forall F\\in\\mathcal\{F\}\_\{y\},\\qquad\\Delta^\{F\}\_\{j\}\(y,y^\{\\prime\};x\)=\-\\delta\\quad\\forall F\\in\\mathcal\{F\}\_\{y^\{\\prime\}\}\.A priorityjjis \(perfectly\) inseparable when it is \(perfectly\) inseparable at every queryq∈𝒬p​cq\\in\\mathcal\{Q\}^\{pc\}\.

[Figure3](https://arxiv.org/html/2607.02672#S3.F3)illustrates the four separability\-related notions we define, from most to least separable\. As shown, perfect inseparability represents the strongest kind of inseparability, where the priority yieldsexactlyequal and opposing evidence for each responseyyandy′y^\{\\prime\}\.

![Refer to caption](https://arxiv.org/html/2607.02672v1/x3.png)

Figure 3\.Illustrations of Separability/Inseparability notions\. Each dot’s horizontal position represents the value ofΔjF​\(q\)\\Delta\_\{j\}^\{F\}\(q\)for anF∈ℱF\\in\\mathcal\{F\}\. The defining characteristic of separability is that all dots appear on the same side of 0 \(and for perfect separability, in exactly the same horizontal position\)\. Oppositely, inseparability occurs when dots appear on opposite sides of 0, and perfect inseparability reflects the case where they appear in symmetric clusters equidistant from 0\.\.

We now show that in fact, many natural priorities are inseparable, because inseparability simply requires the priority to evaluate a ruleacross inputs\. We point out two types of inseparable priorities recurring in the literature, though there are almost certainly others\. First,distributional prioritiesevaluate the allocation of benefits or harms across individuals or groups\. This class includes canonical priorities likeegalitarianismandproportionality, which we formalize below\. Second,axiomatic prioritiesrequire the rule to satisfy consistency conditions across related inputs, of whichequal treatmentis our canonical example\. The salience of these issues to people is supported by substantial research on distributive fairness and procedural justice \(e\.g\.,\(Cappelenet al\.,[2007](https://arxiv.org/html/2607.02672#bib.bib29); Fismanet al\.,[2007](https://arxiv.org/html/2607.02672#bib.bib240); Colquitt,[2001](https://arxiv.org/html/2607.02672#bib.bib239); Leventhal,[1980](https://arxiv.org/html/2607.02672#bib.bib31)\)\)\.

We will use egalitarianism as the main illustrative example\. To formalize “benefit” and “harm”, letvi:𝒳×𝒴→ℝv\_\{i\}:\\mathcal\{X\}\\times\\mathcal\{Y\}\\to\\mathbb\{R\}describeii’s benefit associated with the outcome, sovi​\(x,y\)v\_\{i\}\(x,y\)is their benefit \(or if negative, harm\) from the rule selectingyyatxx\. Because distributional priorities measure how benefits are distributedover inputs, we let𝒟∈Δ​\(𝒳\)\\mathcal\{D\}\\in\\Delta\(\\mathcal\{X\}\)be a generic distribution over inputs\.

###### Definition 3\.0 \(Egalitarianism\)\.

Egalitarianismreflects the priority that “the rule should make sure no one receives too little benefit in the long run\.” For a generic input distribution𝒟∈Δ​\(𝒳\)\\mathcal\{D\}\\in\\Delta\(\\mathcal\{X\}\),999The 2\-scaling is just to make the numbers more convenient\. This is just one possible formulation of this priority, but conceptually, egalitarian’s structural inseparability is not due to the specific formulation, but because it considers the distribution of benefits over inputs\.

uegal​\(F\)=2⋅mini∈N⁡𝔼x∼𝒟​\[vi​\(x,F​\(x\)\)\]∀F∈ℱ\.u\_\{\\mathrm\{egal\}\}\(F\)=2\\cdot\\min\_\{i\\in N\}\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\}\\\!\\left\[v\_\{i\}\\bigl\(x,F\(x\)\\bigr\)\\right\]\\qquad\\forall\\,F\\in\\mathcal\{F\}\.

To see why egalitarianism is structurally inseparable, consider an allocation problem in which each input asks whether a good should be allocated to recipientaaor recipientbb\. The egalitarian value of this local choice depends on who is worst off, which depends centrally on the rule’s behavior elsewhere\. Concretely, for a query comparingaaandbb, there are background rules under whichaahas been maximally shortchanged at all other inputs, in which case egalitarianism favors allocating the good toaa\. There are also background rules under whichbbhas been maximally shortchanged, in which case egalitarianism favors allocating the good tobb\. We formalize this intuition in the proposition below; the proof is in[SectionB\.1](https://arxiv.org/html/2607.02672#A2.SS1)\.

###### Proposition 3\.3 \(Egalitarianism is inseparable \(in non\-degenerate cases\)\)\.

Define an allocation task with recipientsNNand itemsKK\. Let\|N\|≥3\|N\|\\geq 3and fix input space𝒳=\{\(a,b;k\):a,b∈N,a≠b,k∈K\}\\mathcal\{X\}=\\\{\(a,b;k\):a,b\\in N,\\ a\\neq b,\\ k\\in K\\\}and output space𝒴​\(\(a,b;k\)\)=\{a,b\}\.\\mathcal\{Y\}\(\(a,b;k\)\)=\\\{a,b\\\}\.Let𝒟\\mathcal\{D\}have full support, i\.e\.,𝒟​\(x\)\>0\\mathcal\{D\}\(x\)\>0for allx∈𝒳x\\in\\mathcal\{X\}\. Define the benefits such that they are nonnegative, and each recipient benefits only when they receive a good:

vj​\(\(a,b;k\),y\)\>0⇔y=jfor all​j∈N,a,b∈N,a≠b,k∈K,y∈\{a,b\}\.v\_\{j\}\(\(a,b;k\),y\)\>0\\iff y=j\\qquad\\text\{for all \}j\\in N,\\ a,b\\in N,\\ a\\neq b,\\ k\\in K,\\ y\\in\\\{a,b\\\}\.Then, the egalitarian priority is inseparable at every local pairwise queryqq\.

This result is proven for an allocation task over individual recipients, but similar propositions can be proven for other types of allocation tasks; in each setup, one simply needs to avoid degeneracy in𝒟\\mathcal\{D\}\. For example, suppose we are allocating aid oversetsof recipients, and let𝒟\\mathcal\{D\}be restricted such that recipientiiappears rarely under𝒟\\mathcal\{D\}\(soiican be worst off\), but wheneveriidoes appear, they appear in both possible sets of recipients\. Then, on such queries, the egalitarian priority cannot generate evidence in favor of one side onii’s behalf, and its inseparability does not bind\.

Now, we show something stronger: that in some instances, egalitarianism can beperfectlyinseparable at every query\. We use an extremely simple example for illustrative purposes\. We will re\-use this example in later results\.

###### Proposition 3\.4 \(Egalitarianism can be perfectly inseparable\)\.

For certain𝒳,𝒴\\mathcal\{X\},\\mathcal\{Y\},Egalitarianismcan be perfectly inseparable on all local pairwise comparison queriesq∈𝒬pcq\\in\\mathcal\{Q\}^\{\\text\{pc\}\}\.

###### Proof\.

Define an allocation task with two recipients,N=\{a,b\}N=\\\{a,b\\\}and two goodsK=\{k1,k2\}K=\\\{k\_\{1\},k\_\{2\}\\\}\. Suppose the task is to allocate one good to one recipient; thus there are two possible inputs,x1=\(a,b;k1\)x\_\{1\}=\(a,b;k\_\{1\}\)andx2=\(a,b;k2\)x\_\{2\}=\(a,b;k\_\{2\}\)and𝒴​\(x\)=\{a,b\}\\mathcal\{Y\}\(x\)=\\\{a,b\\\}forx∈\{x1,x2\}x\\in\\\{x\_\{1\},x\_\{2\}\\\}\. Suppose𝒟\\mathcal\{D\}is uniform, soPrx∼𝒟⁡\[x=x1\]=Prx∼𝒟⁡\[x=x2\]=1/2\\Pr\_\{x\\sim\\mathcal\{D\}\}\[x=x\_\{1\}\]=\\Pr\_\{x\\sim\\mathcal\{D\}\}\[x=x\_\{2\}\]=1/2\. Suppose both recipients have the same benefit for either good, and no benefit if they aren’t given a good:

vi​\(x,j\)=𝟏​\{j=i\}for all​i,j∈\{a,b\}\.v\_\{i\}\(x,j\)=\\mathbf\{1\}\\\{j=i\\\}\\qquad\\text\{for all \}\\ i,j\\in\\\{a,b\\\}\.In this instance, there are four possible deterministic rules:ℱ=\{Fa​a,Fa​b,Fb​a,Fb​b\}\\mathcal\{F\}=\\\{F^\{aa\},F^\{ab\},F^\{ba\},F^\{bb\}\\\}, whereFi​jF^\{ij\}is the rule that assignsx1→ix\_\{1\}\\to iandx2→jx\_\{2\}\\to jfor alli,j∈\{a,b\}i,j\\in\\\{a,b\\\}\. As expected, the egalitarian priority prefers rules that spread out the goods over recipients:

uegal​\(Fa​b\)=uegal​\(Fb​a\)=1;uegal​\(Fa​a\)=uegal​\(Fb​b\)=0\.u\_\{\\text\{egal\}\}\(F^\{ab\}\)=u\_\{\\text\{egal\}\}\(F^\{ba\}\)=1;\\quad u\_\{\\text\{egal\}\}\(F^\{aa\}\)=u\_\{\\text\{egal\}\}\(F^\{bb\}\)=0\.We immediately see that this priority is perfectly inseparable at both possible queries: applying[Definition3\.1](https://arxiv.org/html/2607.02672#S3.Thmtheorem1), at each query we have thatδ=1\\delta=1, and the following equal\-size partitions of rules:

q1=\(a,b;x1\):\\displaystyle q\_\{1\}=\(a,b;x\_\{1\}\):ℱa=\{Fb​b,Fa​b\},ℱb=\{Fa​a,Fb​a\},\\displaystyle\\quad\\mathcal\{F\}\_\{a\}=\\\{F^\{bb\},F^\{ab\}\\\},\\ \\ \\mathcal\{F\}\_\{b\}=\\\{F^\{aa\},F^\{ba\}\\\},q2=\(a,b;x2\):\\displaystyle q\_\{2\}=\(a,b;x\_\{2\}\):ℱa=\{Fb​b,Fb​a\},ℱb=\{Fa​a,Fa​b\}\.\\displaystyle\\quad\\mathcal\{F\}\_\{a\}=\\\{F^\{bb\},F^\{ba\}\\\},\\ \\ \\mathcal\{F\}\_\{b\}=\\\{F^\{aa\},F^\{ab\}\\\}\.In words, at each query, the setℱj\\mathcal\{F\}\_\{j\}consists of the rules for whichjjis the more advantageous projection at that query’s input, always with a margin of 1\. ∎

To further illustrate the space of inseparable priorities, we now give our two additional examples:proportionality, another distributional priority, andequal treatment, an axiomatic priority\. We show in Appendix[B\.2](https://arxiv.org/html/2607.02672#A2.SS2)that, like Egalitarianism, both of these priorities can be perfectly inseparable\.

###### Definition 3\.0 \(Proportionality\)\.

Proportionality describes the intuition that “groupGℓG\_\{\\ell\}should receive anαℓ\\alpha\_\{\\ell\}\-share of the overall benefit\.” We can formalize this as follows, for a generic distribution of inputs𝒟\\mathcal\{D\}over𝒳\\mathcal\{X\}\. LetGℓ⊆NG\_\{\\ell\}\\subseteq Nfor allℓ∈\[g\]\\ell\\in\[g\]be the set of protected groups, with ideal fractionαℓ∈\[0,1\]\\alpha\_\{\\ell\}\\in\[0,1\]\. Then,

uprop​\(F\)=−∑ℓ∈\[g\]\(𝔼x∼𝒟​\[𝟏​\{F​\(x\)​allocates to a member of group​Gℓ\}\]−αℓ\)2\.u\_\{\\mathrm\{prop\}\}\(F\)=\-\\sum\_\{\\ell\\in\[g\]\}\\left\(\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\}\[\\mathbf\{1\}\\\{F\(x\)\\text\{ allocates to a member of group \}G\_\{\\ell\}\\\}\]\-\\alpha\_\{\\ell\}\\right\)^\{2\}\.

###### Definition 3\.0 \(Equal Treatment\)\.

The equal treatment priority describes the intuition that counterparts should be treated equally\. Formally, let𝒞\\mathcal\{C\}be a collection of “counterpart constraints” of the form\(x,x′,η\)\(x,x^\{\\prime\},\\eta\), where each\(x,x′,η\)∈𝒞\(x,x^\{\\prime\},\\eta\)\\in\\mathcal\{C\}consists of two inputsx,x′∈𝒳x,x^\{\\prime\}\\in\\mathcal\{X\}that differ only in some protected\-group identity of otherwise comparable recipients, together with a bijectionη:𝒴​\(x\)→𝒴​\(x′\)\\eta:\\mathcal\{Y\}\(x\)\\to\\mathcal\{Y\}\(x^\{\\prime\}\)that describes what it means for the outputs atxxandx′x^\{\\prime\}to correspond appropriately\. Then,

ueq​\(F\)=1\|𝒞\|​∑\(x,x′,η\)∈𝒞𝟏​\{F​\(x′\)=η​\(F​\(x\)\)\}\.u\_\{\\mathrm\{eq\}\}\(F\)=\\frac\{1\}\{\|\\mathcal\{C\}\|\}\\sum\_\{\(x,x^\{\\prime\},\\eta\)\\in\\mathcal\{C\}\}\\mathbf\{1\}\\\{F\(x^\{\\prime\}\)=\\eta\(F\(x\)\)\\\}\.

### 3\.1\.Technical Preliminaries

In the next two subsections, we consider the consequences of assuming separability when it does not hold\. In order to formalize the consequences of this misspecification, we define some key preliminaries\. First, let

ℛh;0,0∘=\{Rhβ;0,0∘​\(⋅;M\):β\>0\}\\mathcal\{R\}^\{\\circ\}\_\{h;0,0\}=\\\{R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\(\\cdot;M\):\\beta\>0\\\}be the class of baseline response models with link functionhh, varied over all possible values ofβ\\beta\. The core definition we will use isscale\-free indistinguishabilityof two underlying priority models, which means that these two models’ induced baseline response models are behaviorally indistinguishable up to the inverse temperature parameterβ\>0\\beta\>0\.

###### Definition 3\.0 \(Scale\-Free Indistinguishability\)\.

Two priority modelsM,M′∈ℳM,M^\{\\prime\}\\in\\mathcal\{M\}are scale\-free indistinguishable with respect to \(w\.r\.t\.\) link functionhhiff

ℛh;0,0∘​\(q;M\)=ℛh;0,0∘​\(q;M′\)∀q∈𝒬p​c\.\\mathcal\{R\}^\{\\circ\}\_\{h;0,0\}\(q;M\)=\\mathcal\{R\}^\{\\circ\}\_\{h;0,0\}\(q;M^\{\\prime\}\)\\quad\\forall q\\in\\mathcal\{Q\}^\{pc\}\.In other words, inhβh\_\{\\beta\},β\\betais a nuisance parameter: for everyβ\>0\\beta\>0, there is aβ′\>0\\beta^\{\\prime\}\>0such thatRhβ;0,0∘​\(q;M\)=Rhβ′;0,0∘​\(q,M′\)R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\(q;M\)=R^\{\\circ\}\_\{h\_\{\\beta^\{\\prime\}\};0,0\}\(q,M^\{\\prime\}\), and likewise for allβ′\>0\\beta^\{\\prime\}\>0there exists some suchβ\\beta\.

Scale\-free indistinguishability is closely related to identifiability of the priority weights\.

###### Definition 3\.0 \(Scale\-Free Identifiability\)\.

FixingM=\(u,ω\)M=\(u,\\omega\), weightωj\\omega\_\{j\}is scale\-free identifiable from local pairwise comparison queries w\.r\.t\.hhif there do not exist two weight vectorsω,ω′∈Δm−1\\omega,\\omega^\{\\prime\}\\in\\Delta^\{m\-1\}such thatωj≠ωj′\\omega\_\{j\}\\neq\\omega^\{\\prime\}\_\{j\}and the modelsM=\(u,ω\)M=\(u,\\omega\)andM′=\(u,ω′\)M^\{\\prime\}=\(u,\\omega^\{\\prime\}\)are scale\-free indistinguishable w\.r\.t\.hh\.

We say thatωj\\omega\_\{j\}is scale\-free non\-identifiable overS⊆\[0,1\]S\\subseteq\[0,1\]if, for everyγ∈S\\gamma\\in S, there exists a scale\-free indistinguishable modelM′=\(u,ω′\)M^\{\\prime\}=\(u,\\omega^\{\\prime\}\)withωj′=γ\\omega^\{\\prime\}\_\{j\}=\\gamma\.

To consider what might be learned under the assumption of perfect separability, we define aperfect separable rationalizationof a generic modelM∈ℳM\\in\\mathcal\{M\}, which is a perfectly separable modelM~∈ℳsep\\widetilde\{M\}\\in\\mathcal\{M\}^\{\\mathrm\{sep\}\}whose resulting response behavior is indistinguishable from that produced byMM:

###### Definition 3\.0 \(Perfect Separable Rationalization\)\.

M~∈ℳsep\\widetilde\{M\}\\in\\mathcal\{M\}^\{\\mathrm\{sep\}\}is a perfect separable rationalization ofM∈ℳM\\in\\mathcal\{M\}w\.r\.t\.hhiffMMandM~\\widetilde\{M\}are scale\-free indistinguishable w\.r\.t\.hh\.

Let the set of all perfectly separable rationalizations ofMMw\.r\.t\.hhbe defined as follows\. Note that this set could be empty, i\.e\., an inseparable model may not have a perfectly separable rationalization\.

PSRh​\(M\)=\{M~∈ℳsep:M​and​M~​are scale\-free indistinguishable w\.r\.t\.​h\}\.\\mathrm\{PSR\}\_\{h\}\(M\)=\\left\\\{\\widetilde\{M\}\\in\\mathcal\{M\}^\{\\mathrm\{sep\}\}:M\\text\{ and \}\\widetilde\{M\}\\text\{ are scale\-free indistinguishable w\.r\.t\.~\}h\\right\\\}\.
Finally, we formalize a class of decoders that are standard when behavior is assumed to arise from an S\-RUM\. Atranscriptis a sequence𝒯=\(qt,wt\)t≥1\\mathcal\{T\}=\(q\_\{t\},w\_\{t\}\)\_\{t\\geq 1\}of query, response pairs, and adecoderDh,ℳ′D\_\{h,\\mathcal\{M\}^\{\\prime\}\}maps a transcript to a modelM^∈ℳ′\\widehat\{M\}\\in\\mathcal\{M\}^\{\\prime\}\. Here, the subscripts respectively reflect the decoder’s assumptions that the baseline response model has link functionhhand the true model lies inℳ′⊆ℳ\\mathcal\{M\}^\{\\prime\}\\subseteq\\mathcal\{M\}\. Then,Dh,ℳ′​\(𝒯\)D\_\{h,\\mathcal\{M\}^\{\\prime\}\}\(\\mathcal\{T\}\)is the model inℳ′\\mathcal\{M\}^\{\\prime\}the decoder returns on𝒯\\mathcal\{T\}\. We study decoders in the infinite\-data limit, so that identifiability rather than sampling noise is the binding constraint: call a transcriptexhaustiveif every queryq∈𝒬pcq\\in\\mathcal\{Q\}^\{\\mathrm\{pc\}\}recurs infinitely often\. Then, along an exhaustive transcript generated by true response modelR∗​\(⋅,M∗\)R^\{\*\}\(\\cdot,M^\{\*\}\), the empirical response frequencies converge toR∗​\(⋅;M∗\)R^\{\*\}\(\\cdot\\,;M^\{\*\}\)at every query\.

###### Definition 3\.0 \(Separable\-consistent decoder\)\.

A decoderDh,ℳsepD\_\{h,\\mathcal\{M\}^\{\\mathrm\{sep\}\}\}is separable\-consistent iff for everyM∗∈ℳM^\{\*\}\\in\\mathcal\{M\}withPSRh​\(M∗\)≠∅\\mathrm\{PSR\}\_\{h\}\(M^\{\*\}\)\\neq\\emptysetand every exhaustive transcript𝒯\\mathcal\{T\}generated underRhβ;0,0∘​\(⋅;M∗\)R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\(\\cdot;M^\{\*\}\)for anyβ\>0\\beta\>0,

Dh,ℳsep​\(𝒯\)∈PSRh​\(M∗\)\.D\_\{h,\\mathcal\{M\}^\{\\mathrm\{sep\}\}\}\(\\mathcal\{T\}\)\\in\\mathrm\{PSR\}\_\{h\}\(M^\{\*\}\)\.

Intuitively, a separable\-consistent decoder “assumes perfect separability” in the sense that, when the responsescould havecome from a perfectly separable model, it commits to such an explanation\. This is an extremely weak requirement, and it is automatically satisfied by standard preference\-learning estimators \(e\.g\., maximum\-likelihood estimation\) that assume S\-RUMs: any decoder that fits an S\-RUM reports a fitted score functionV^\\widehat\{V\}, which is equivalent to a perfectly separable modelMV^M\_\{\\widehat\{V\}\}by Theorem[2\.13](https://arxiv.org/html/2607.02672#S2.Thmtheorem13)\. WhenPSR​\(M∗\)≠∅\\mathrm\{PSR\}\(M^\{\*\}\)\\neq\\emptyset, the fit is exact on exhaustive data, soMV^∈PSR​\(M∗\)M\_\{\\widehat\{V\}\}\\in\\mathrm\{PSR\}\(M^\{\*\}\)\.

In the next two subsections, we illustrate the risks of using separable\-consistent decoders\. In particular, we will show that both perfectly inseparable \([Section3\.2](https://arxiv.org/html/2607.02672#S3.SS2)\) and general inseparable \([Section3\.3](https://arxiv.org/html/2607.02672#S3.SS3)\) priority modelsM∗M^\{\*\}can admit perfectly separable rationalizations whose aggregate\-optimal rules are highly suboptimal in the true modelM∗M^\{\*\}\. Our results will also clarify when there is hope for other kinds of decoders to avoid this issue\.

### 3\.2\.Perfectly Inseparable Priorities

Now, we first show that when perfect separability is assumed erroneously, perfectly separable priorities are erased with no trace \([Corollary3\.12](https://arxiv.org/html/2607.02672#S3.Thmtheorem12)\)\. Then, we show that perfectly inseparable priorities cannot be identified by local pairwise comparisons at all, meaning that no learner restricted to such queries can rectify this issue \([Corollary3\.14](https://arxiv.org/html/2607.02672#S3.Thmtheorem14)\)\. Both of these conclusions follow from the following key theorem below\. This result and its subsequent corollaries are not actually dependent on the use of the linear priority aggregator: they hold for a more general class of priority aggregators that arescale\-preserving\([DefinitionB\.4](https://arxiv.org/html/2607.02672#A2.Thmtheorem4)\), which just requires that adding priorityj∗j^\{\*\}does not change how the remaining priorities trade off against each other\. We give the full proof in[SectionB\.3](https://arxiv.org/html/2607.02672#A2.SS3)\.

Here, we letM−J′M^\{\-J^\{\\prime\}\}denote the modelM∈ℳM\\in\\mathcal\{M\}with priorities inJ′⊂JJ^\{\\prime\}\\subset Jremoved, and with the priority weights rescaled so thatω−J′=\(ωj\)j∈\[m\]∖\{J′\}‖\(ωj\)j∈\[m\]∖\{J′\}‖1\.\\omega^\{\-J^\{\\prime\}\}=\\frac\{\(\\omega\_\{j\}\)\_\{j\\in\[m\]\\setminus\\\{J^\{\\prime\}\\\}\}\}\{\\\|\(\\omega\_\{j\}\)\_\{j\\in\[m\]\\setminus\\\{J^\{\\prime\}\\\}\}\\\|\_\{1\}\}\.

###### Theorem 3\.11\.

LetMinsep∈ℳM\_\{\\mathrm\{insep\}\}\\in\\mathcal\{M\}be a priority model containing a nonzero number of perfectly inseparable prioritiesJinsep⊂\[m\]J\_\{\\mathrm\{insep\}\}\\subset\[m\]\.101010One can also allowJinsep⊆\[m\]J\_\{\\mathrm\{insep\}\}\\subseteq\[m\], and handle the case ofJinsep=\[m\]J\_\{\\mathrm\{insep\}\}=\[m\]by defining the null model consisting of one priority in whichuj​\(F\)=0​∀F∈ℱu\_\{j\}\(F\)=0\\ \\forall F\\in\\mathcal\{F\}\.LetM=\(Minsep\)−JinsepM=\(M\_\{\\mathrm\{insep\}\}\)^\{\-J\_\{\\mathrm\{insep\}\}\}\. Then,MinsepM\_\{\\mathrm\{insep\}\}andMMare scale\-free indistinguishable w\.r\.t\. any link functionhh\.

###### Proof sketch\.

The key to the proof is that no reasonable rule aggregator can extract directional evidence from a perfectly inseparable priority\. Fix a perfectly inseparable priorityj∈Jinsepj\\in J\_\{\\mathrm\{insep\}\}and a queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)\. By perfect inseparability, there is someδ\>0\\delta\>0and an equal\-size partitionℱy,ℱy′\\mathcal\{F\}\_\{y\},\\mathcal\{F\}\_\{y^\{\\prime\}\}ofℱ\\mathcal\{F\}such that

\{ΔjF​\(y,y′;x\)\}F∈ℱ\\displaystyle\\left\\\{\\Delta^\{F\}\_\{j\}\(y,y^\{\\prime\};\\,x\)\\right\\\}\_\{F\\in\\mathcal\{F\}\}=\{δ,…,δ⏟\|ℱy\|,−δ,…,−δ⏟\|ℱy′\|\},\{ΔjF​\(y′,y;x\)\}F∈ℱ=\{−δ,…,−δ⏟\|ℱy\|,δ,…,δ⏟\|ℱy′\|\}\.\\displaystyle\\;=\\;\\\{\\underbrace\{\\delta,\\ldots,\\delta\}\_\{\|\\mathcal\{F\}\_\{y\}\|\},\\;\\underbrace\{\-\\delta,\\ldots,\-\\delta\}\_\{\|\\mathcal\{F\}\_\{y^\{\\prime\}\}\|\}\\\},\\qquad\\left\\\{\\Delta^\{F\}\_\{j\}\(y^\{\\prime\},y;\\,x\)\\right\\\}\_\{F\\in\\mathcal\{F\}\}\\;=\\;\\\{\\underbrace\{\-\\delta,\\ldots,\-\\delta\}\_\{\|\\mathcal\{F\}\_\{y\}\|\},\\;\\underbrace\{\\delta,\\ldots,\\delta\}\_\{\|\\mathcal\{F\}\_\{y^\{\\prime\}\}\|\}\\\}\.Since\|ℱy\|=\|ℱy′\|\|\\mathcal\{F\}\_\{y\}\|=\|\\mathcal\{F\}\_\{y^\{\\prime\}\}\|, these two multisets are identical up to permutation\. Therefore, by the permutation invariance ofϕrules\\phi^\{\\text\{rules\}\},

ϕrules​\(\{ΔjF​\(y,y′;x\)\}F∈ℱ\)=ϕrules​\(\{ΔjF​\(y′,y;x\)\}F∈ℱ\)\.\\phi^\{\\mathrm\{rules\}\}\\left\(\\left\\\{\\Delta^\{F\}\_\{j\}\(y,y^\{\\prime\};x\)\\right\\\}\_\{F\\in\\mathcal\{F\}\}\\right\)=\\phi^\{\\mathrm\{rules\}\}\\left\(\\left\\\{\\Delta^\{F\}\_\{j\}\(y^\{\\prime\},y;x\)\\right\\\}\_\{F\\in\\mathcal\{F\}\}\\right\)\.Thus, the output ofϕrules\\phi^\{\\text\{rules\}\}affects both directional evidence scoressM\+s^\{\+\}\_\{M\}andsM−s^\{\-\}\_\{M\}symmetrically, so after applying a scale\-preserving priority aggregator, its contribution cancels fromκ​\(q\)\\kappa\(q\), up to a constant \(over queries\) rescaling of the remaining priorities\. It follows that removing such priorities can only rescale the total contribution of the remaining priorities but does not affect their relative importance, and thus can be absorbed by the inverse temperature parameterβ\\beta\. Hence,MinsepM\_\{\\mathrm\{insep\}\}andMMare scale\-free indistinguishable\. ∎

We now consider what a separable\-consistent decoder will do in the presence of perfectly inseparable priorities\. We assume here what is arguably the best\-case scenario: that all priorities that are not perfectly inseparable are perfectly separable\. We show that such decoders erase the perfectly inseparable priorities without a trace \([Corollary3\.12](https://arxiv.org/html/2607.02672#S3.Thmtheorem12)\)\.

###### Corollary 3\.12 \(Erasure by Separable\-Consistent Decoders\)\.

Fix any link functionhh\. LetMinsepM\_\{\\mathrm\{insep\}\}be as in[Theorem3\.11](https://arxiv.org/html/2607.02672#S3.Thmtheorem11), and suppose that every priority in\[m\]∖Jinsep\[m\]\\setminus J\_\{\\mathrm\{insep\}\}is perfectly separable\. Again, letM=\(Minsep\)−JinsepM=\(M\_\{\\mathrm\{insep\}\}\)^\{\-J\_\{\\mathrm\{insep\}\}\}\. ThenM∈PSRh​\(Minsep\)\.M\\in\\mathrm\{PSR\}\_\{h\}\(M\_\{\\mathrm\{insep\}\}\)\.Consequently, given an exhaustive transcript generated underMinsepM\_\{\\mathrm\{insep\}\}, any separable\-consistent decoder returns a perfectly separable model that exactly rationalizes the observed response behavior\.

As shown by the following example, it is not hard to construct cases where the erasure of perfectly inseparable priorities leads to highly suboptimal inferred rules \(formal details in[SectionB\.4](https://arxiv.org/html/2607.02672#A2.SS4)\)\.

###### Example 0 \(Erasure can lead to high\-regret rules\)\.

Assume we are in the allocation task from[Proposition3\.4](https://arxiv.org/html/2607.02672#S3.Thmtheorem4), withN=\{a,b\}N=\\\{a,b\\\}, two inputsx1,x2x\_\{1\},x\_\{2\}that occur with equal probability, and any rule must assign each input to eitheraaorbb\. Suppose the true priority modelMinsepM\_\{\\mathrm\{insep\}\}contains two priorities,Egalitarianism\([Definition3\.2](https://arxiv.org/html/2607.02672#S3.Thmtheorem2)\) andFamily\([Example2\.3](https://arxiv.org/html/2607.02672#S2.Thmtheorem3)\), where the individual is primarily egalitarian: forϵ∈\(0,1/2\)\\epsilon\\in\(0,1/2\), let these priorities have respective weightsωegal=1−ϵ\\omega\_\{\\mathrm\{egal\}\}=1\-\\epsilonandωfamily=ϵ\\omega\_\{\\mathrm\{family\}\}=\\epsilon\. In this example, we know egalitarianism is perfectly inseparable; on the other hand, it is not hard to show that the family priority is perfectly separable\.

Then, by[Corollary3\.12](https://arxiv.org/html/2607.02672#S3.Thmtheorem12), any separable\-consistent decoder will effectively111111Technically, this requires one extra lemma to reason about the aggregate\-optimal rule when there are multiple perfect separable rationalizations\. See[SectionB\.4](https://arxiv.org/html/2607.02672#A2.SS4)\.return the priority model containing only the family priority, call itMfamilyM\_\{\\mathrm\{family\}\}\. The aggregate\-optimal rule according toMfamilyM\_\{\\mathrm\{family\}\}is far from egalitarian, always prioritizing the individual’s family members and giving zero benefit to anyone else\. Asϵ→0\\epsilon\\to 0and the individual truly becomes more egalitarian, the aggregate utility of the resulting rule approaches the worst possible rule according to the true modelMinsepM\_\{\\mathrm\{insep\}\}: the regret of the chosen rule is1−3​ϵ/21\-\\nicefrac\{\{3\\epsilon\}\}\{\{2\}\}, where the regret of the worst possible rule is1−ϵ1\-\\epsilon\.

One may wonder whether a more sophisticated decoder might be able to avoid the problem illustrated above\. Unfortunately, the answer isno:[Theorem3\.11](https://arxiv.org/html/2607.02672#S3.Thmtheorem11)implies that the weights on perfectly separable priorities are not at all identifiable when the learner only has access to local pairwise comparison queries\.

###### Corollary 3\.14 \(Non\-identification\)\.

Letjjbe any perfectly inseparable priority in any priority modelMM\. Then,ωj\\omega\_\{j\}is scale\-free non\-identifiable w\.r\.t\. any link functionhhover\[0,1\)\[0,1\)from local pairwise comparison queries\.

### 3\.3\.General Inseparable Priorities

Now we consider the more general class of inseparable priorities which, unlike perfectly inseparable priorities, can generate some directional evidence in response to queries\. Unfortunately, because these priorities still fall outside of any model inℳsep\\mathcal\{M\}^\{\\mathrm\{sep\}\}, we now illustrate how this evidence is misinterpreted by separable\-consistent decoders — again, sometimes without a trace, and with major consequences for the resulting rule quality\. We fully formalize this argument in[SectionB\.6](https://arxiv.org/html/2607.02672#A2.SS6)\.

###### Example 0 \(Misinterpretation by Separable\-Consistent Decoders\)\.

Fix any link functionhh\. We again assume we are in the allocation task from[Proposition3\.4](https://arxiv.org/html/2607.02672#S3.Thmtheorem4), withN=\{a,b\}N=\\\{a,b\\\}, two inputsx1,x2x\_\{1\},x\_\{2\}that occur with equal probability, and any rule assigning each input to eitheraaorbb\. We impose an additional, very mild restriction onϕrules\\phi^\{\\text\{rules\}\}just for this example \(see[SectionB\.6](https://arxiv.org/html/2607.02672#A2.SS6)\)\.

LetMMcontain a single priority,Proportionality, with target sharesαa=1/2\+ϵ\\alpha\_\{a\}=1/2\+\\epsilonandαb=1/2−ϵ\\alpha\_\{b\}=1/2\-\\epsilon, where0<ϵ<1/40<\\epsilon<1/4\. That is, this priority wants to avoid shortchanging bothaaandbb, but with a bias towardaa\.

Now, examine the evidence produced across background rules on the queryq1=\(a,b;x1\)q\_\{1\}=\(a,b;x\_\{1\}\)\. If the other inputx2x\_\{2\}is assigned tobb, then assigningx1x\_\{1\}toaamoves the rule fromFb​bF^\{bb\}toFa​bF^\{ab\}, which improves proportionality\. If the other inputx2x\_\{2\}is assigned toaa, then assigningx1x\_\{1\}tobbmoves the rule fromFa​aF^\{aa\}toFb​aF^\{ba\}, which also improves proportionality\. However, because the target share foraais slightly above one half,Fb​bF^\{bb\}is worse thanFa​aF^\{aa\}\.

Thus the improvement fromFb​bF^\{bb\}toFa​bF^\{ab\}is larger than the improvement fromFa​aF^\{aa\}toFb​aF^\{ba\}, and the benefit of choosingaaatx1x\_\{1\}outweighs that of choosingbb, after aggregating over these background rules\. Thus, the response model favorsaaon the queryq1q\_\{1\}\. The same argument applies to the queryq2=\(a,b;x2\)q\_\{2\}=\(a,b;x\_\{2\}\)\. Consequently, the response model favors assigning each input toaa, even though the proportionality priority itself is maximized by the balanced rulesFa​bF^\{ab\}andFb​aF^\{ba\}\.

ThusMMadmits a perfect separable rationalizationM~∈PSRh​\(M\)\\widetilde\{M\}\\in\\mathrm\{PSR\}\_\{h\}\(M\)consisting of the single priority

ua​\(F\)=𝔼X∼𝒟​\[𝟏​\{F​\(X\)=a\}\]\.u\_\{a\}\(F\)=\\mathbb\{E\}\_\{X\\sim\\mathcal\{D\}\}\\left\[\\mathbf\{1\}\\\{F\(X\)=a\\\}\\right\]\.A separable\-consistent decoder can therefore fit the exhaustive transcript exactly withM~\\widetilde\{M\}\. According toM~\\widetilde\{M\}, the resulting aggregate\-optimal rule isFa​aF^\{aa\}\. However, this rule is highly disproportional, and accordingly, asϵ→0\\epsilon\\to 0, its aggregate utility approaches that of the worst possible rule under the true modelMM, i\.e\.,

UM​\(Fa​b\)=UM​\(Fb​a\)→0,UM​\(Fa​a\)→−1/2,UM​\(Fb​b\)→−1/2\.U\_\{M\}\(F^\{ab\}\)=U\_\{M\}\(F^\{ba\}\)\\to 0,\\qquad U\_\{M\}\(F^\{aa\}\)\\to\-\\nicefrac\{\{1\}\}\{\{2\}\},\\qquad U\_\{M\}\(F^\{bb\}\)\\to\-\\nicefrac\{\{1\}\}\{\{2\}\}\.

The same phenomenon can persist when the true model contains additional priorities\. An inseparable priority can mask, distort, or even reverse the local signal generated by perfectly separable priorities\. In some cases, this means that no perfect separable rationalization exists at all\.121212For example, inseparable priorities can generate score differences whose magnitudes are inconsistent with any separable model, or can even induce cyclic pairwise comparison patterns\.Then the assumption of perfect separability leads to irreducible population misfit: even with unlimited data from every local query, the best separable model cannot perfectly explain the response distribution\. This offers one possible source of observed inconsistency, i\.e\., the failure of any preference model within the assumed class to fit people’s responses\.

Finally, we return to whether a different decoder could avoid this problem\. Unlike the perfect\-inseparability case, where the obstacle was non\-identifiability, the obstacle here ismisinterpretation: general inseparable priorities may be identifiable from local pairwise comparisons, but a separable\-consistent decoder interprets their signal through the wrong structural lens\. This points to a possible alternative:priority\-aware learning, where the decoder is given the individual’s priority utility functions up front and interprets query responses while accounting for both separable and inseparable priority structures\. We expand on this proposal in[Section5](https://arxiv.org/html/2607.02672#S5), but formalizing this idea constitutes rich future work\.

## 4\.Generalization \#2: Latent Indecision

In the previous section, we considered the generalization fromℳsep\\mathcal\{M\}^\{\\mathrm\{sep\}\}toℳ\\mathcal\{M\}, holdingτr,τκ=0\\tau\_\{r\},\\tau\_\{\\kappa\}=0\. Now, we consider the opposite generalization: restricting toℳsep\\mathcal\{M\}^\{\\mathrm\{sep\}\}but allowingτr,τκ\>0\\tau\_\{r\},\\tau\_\{\\kappa\}\>0and thus permitting latent indecision\. This generalization allows us to go beyond the mechanical justification of forced comparisons under zero\-thresholds, where only decisive latent states≻∗\\succ^\{\*\}and≺∗\\prec^\{\*\}can arise\.

The practical relevance of this generalization is demonstrated by mounting evidence that individuals struggle to answer forced comparisons, acting inconsistently, hesitating to choose, expressing difficulty or anguish, or expressing low confidence in their answer\.

For example, in interviews conducted byKeswaniet al\.\([2025a](https://arxiv.org/html/2607.02672#bib.bib212)\), a participant was asked to decide whether to give a kidney to a recipient who would live 10 more years with one dependent, versus a recipient who would live 20 more years with no dependents\. In their response, they said:

> “…If this person has a child or something, 10 years is going to be the difference between leaving a child and leaving a young adult \[…\] I wouldn’t sleep well after making this decision\.”

In the language of our model, this quote seems to express that the decision is of high moral valence with no clear decision, in line with our notion of conflict\. This is not an isolated incident in this dataset: we hand\- and LLM\-coded the 20 interview transcripts underlying the study, each of which contains three narrated pairwise comparisons\. We coded for language markers of indecision, including back\-and\-forth reasoning, explicit statements of difficulty, hedging, self\-correction, and discomfort with choosing, details are in Appendix[C\.1](https://arxiv.org/html/2607.02672#A3.SS1)\. We find these indecision markers to be common: 18 of 20 participants displayed at least one such marker, and 8 of 20 displayed them on at least two of the three queries\.

This interview data — along with other studies documenting participants voluntarily reporting indecision and conflict\(McElfreshet al\.,[2021](https://arxiv.org/html/2607.02672#bib.bib302); Rosaset al\.,[2019](https://arxiv.org/html/2607.02672#bib.bib178)\)131313In pairwise kidney\-allocation experiments, participants frequently used an explicit indecision option when it was available\(McElfreshet al\.,[2021](https://arxiv.org/html/2607.02672#bib.bib302)\), and in sacrificial moral\-dilemma experiments, many participants directly reported conflict while making each judgment\(Rosaset al\.,[2019](https://arxiv.org/html/2607.02672#bib.bib178)\)\.— supports the possibility that people \(a\) experience indecision when answering pairwise comparisons, and \(b\) can report it\. Our model allows us to investigate the technical consequences of these possibilities by giving a formal notion of “true” indecision: the latent states∼∗\\sim^\{\*\}and⋈∗\\bowtie^\{\*\}, which arise exactly in the generalization of S\-RUMs whereτr,τκ\>0\\tau\_\{r\},\\tau\_\{\\kappa\}\>0\. We use this generalization to ask two questions\. First, in[Section4\.2](https://arxiv.org/html/2607.02672#S4.SS2), we ask: To what extent can forced decisive responses compromise learning accuracy when individualsdistort their behavioron queries where they are latently indecisive? Second, in[Section4\.3](https://arxiv.org/html/2607.02672#S4.SS3), we ask: Can allowing individuals to report conflict and indifference, or even just general indecision, not only avoid this distortion but improve learning speed by providing additional information about their underlying beliefs?

Both of these questions examine how indecision can potentially compromise or aid the technical task of learningω∗\\omega^\{\*\}\. Before investigating these reasons for considering indecision, we formalize another intuition about why indecision at the query level is useful: that indecision at the level of queries can tell us something about indecision at the level ofrules\.

Now, we turn our attention to the two questions above\. In our simulations, we will specifically analyzelinearpriority models, in which each priority advocates for the importance of a single feature in a latent feature space; we will show these are perfectly separable, as needed\. While this is a substantial restriction of our model, it is of independent interest: it captures the popular approach of assuming that S\-RUM scores have linear structure over a fixed feature space, as in the emerging paradigm of linear social choice\(Geet al\.,[2024a](https://arxiv.org/html/2607.02672#bib.bib99),[2026](https://arxiv.org/html/2607.02672#bib.bib22)\)and many other papers in preference learning\(Leeet al\.,[2019](https://arxiv.org/html/2607.02672#bib.bib244); Freedmanet al\.,[2020](https://arxiv.org/html/2607.02672#bib.bib127); Geet al\.,[2024b](https://arxiv.org/html/2607.02672#bib.bib188); Noothigattuet al\.,[2018](https://arxiv.org/html/2607.02672#bib.bib121); Boerstleret al\.,[2024](https://arxiv.org/html/2607.02672#bib.bib169)\)\. By allowingτr,τκ\>0\\tau\_\{r\},\\tau\_\{\\kappa\}\>0, we obtain a strict generalization of these linear settings in which the individual can experience conflict when salient features give evidence for different responses, or indifference when no salient feature is relevant to a given comparison\. This setting also gives a simple first test case for learning within our model\. We introduce this model, along with additional technical preliminaries, in[Section4\.1](https://arxiv.org/html/2607.02672#S4.SS1)\.

### 4\.1\.Technical Setup

#### 4\.1\.1\.Linear Priority Models

We restrict toMMwhose priorities are linear in a known feature space, defined by feature map

ψ:\{\(x,y\):x∈𝒳,y∈𝒴​\(x\)\}→\[0,1\]d\.\\psi:\\\{\(x,y\):x\\in\\mathcal\{X\},\\ y\\in\\mathcal\{Y\}\(x\)\\\}\\to\[0,1\]^\{d\}\.For example, in the kidney allocation task,ψ​\(x,y\)\\psi\(x,y\)could describe the features of the chosen recipientyy, e\.g\., their age, gender, and whether they have dependents\. It could also encompass features that spanxxandyy, such as a feature capturing the match quality of kidneyxxwith recipientyy\.

Accordingly, the priority modelMMconsists ofddpriorities, where each priorityj∈\[d\]j\\in\[d\]advocates for the importance of a single feature\. This produces utility functions as follows, whereψj​\(x,y\)\\psi\_\{j\}\(x,y\)is thejj\-th feature in the feature vector:

###### Definition 4\.0 \(Linear Priority Model\)\.

Fix a feature mapψ\\psiwith dimensiondd\. A linear priority modelMω=\(u,ω\)M\_\{\\omega\}=\(u,\\omega\)is any model withddpriorities and utilities

uj​\(F\):=1\|𝒳\|​∑x∈𝒳ψj​\(x,F​\(x\)\)∀j∈\[d\]\.u\_\{j\}\(F\):=\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\sum\_\{x\\in\\mathcal\{X\}\}\\psi\_\{j\}\(x,F\(x\)\)\\qquad\\forall j\\in\[d\]\.

Because the utility functions are fixed byψ\\psi, given aψ\\psithe only varying element of a priority model isω\\omega; accordingly, letMωM\_\{\\omega\}be the priority model with vectorω\\omegaunder \(implicit\) feature mapψ\\psi\. Letℳψ:=\{Mω:ω∈Ω\}\\mathcal\{M\}\_\{\\psi\}:=\\\{M\_\{\\omega\}:\\omega\\in\\Omega\\\}be the class of all linear priority models over feature mapψ\\psi\. EveryMω∈ℳψM\_\{\\omega\}\\in\\mathcal\{M\}\_\{\\psi\}is perfectly separable: for any queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)and priorityjj, the projected rulesFx→yF\_\{x\\to y\}andFx→y′F\_\{x\\to y^\{\\prime\}\}differ only atxx, so all other feature terms cancel andΔjF​\(q\)\\Delta^\{F\}\_\{j\}\(q\)is independent ofFF:

ΔjF​\(q\)=uj​\(Fx→y\)−uj​\(Fx→y′\)=1\|𝒳\|​\(ψj​\(x,y\)−ψj​\(x,y′\)\)∀F∈ℱ\.\\Delta^\{F\}\_\{j\}\(q\)=u\_\{j\}\(F\_\{x\\to y\}\)\-u\_\{j\}\(F\_\{x\\to y^\{\\prime\}\}\)=\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\big\(\\psi\_\{j\}\(x,y\)\-\\psi\_\{j\}\(x,y^\{\\prime\}\)\\big\)\\qquad\\forall F\\in\\mathcal\{F\}\.
Now, let thelinear ruleFω∈ℱF\_\{\\omega\}\\in\\mathcal\{F\}be the rule that at any givenxxoutputs theyywith the highest linear\-weighted score:

Fω​\(x\)=arg⁡maxy∈𝒴​\(x\)⁡⟨ω,ψ​\(x,y\)⟩\.F\_\{\\omega\}\(x\)=\\arg\\max\_\{y\\in\\mathcal\{Y\}\(x\)\}\\langle\\omega,\\psi\(x,y\)\\rangle\.We now show that, given any linear priority modelMω∗M\_\{\\omega^\{\*\}\}, its aggregate\-optimal rule is the linear ruleFω∗F\_\{\\omega^\{\*\}\}, i\.e\., the linear rule defined byω∗\\omega^\{\*\}:

###### Proposition 4\.3\.

For every feature mapψ\\psiand every linear priority modelMω∗∈ℳψM\_\{\\omega^\{\*\}\}\\in\\mathcal\{M\}\_\{\\psi\},

Fω∗∈arg⁡maxF∈ℱ⁡UMω∗​\(F\)\.F\_\{\\omega^\{\\ast\}\}\\in\\arg\\max\_\{F\\in\\mathcal\{F\}\}U\_\{M\_\{\\omega^\{\\ast\}\}\}\(F\)\.

We prove this in[SectionC\.2](https://arxiv.org/html/2607.02672#A3.SS2); the intuition is that the linear rule maximizes the weighted score pointwise at eachxx, and because aggregate utility decomposes additively over inputs under perfect separability, a pointwise\-optimal rule is globally optimal\.

Now, we formalize the claim that in the special case whereτr=τκ=0\\tau\_\{r\}=\\tau\_\{\\kappa\}=0, linear priority models exactly capture the linear models used in the literature, as cited above\. We call these modelsLinear S\-RUMs, because they are S\-RUMs whose local score function is structured according to a linear weight vectorθ\\theta\. We defer the proof to[SectionC\.3](https://arxiv.org/html/2607.02672#A3.SS3)\.

###### Theorem 4\.4 \(Capture of Linear S\-RUMs\)\.

Fix any link functionhh\. Letθ∈ℝ≥0d\\theta\\in\\mathbb\{R\}^\{d\}\_\{\\geq 0\}satisfy‖θ‖1\>0\\\|\\theta\\\|\_\{1\}\>0, and define the linear S\-RUM score function

Vθ​\(x,y\)=⟨θ,ψ​\(x,y\)⟩\.V\_\{\\theta\}\(x,y\)=\\langle\\theta,\\psi\(x,y\)\\rangle\.Letω=θ/‖θ‖1\\omega=\\theta/\\\|\\theta\\\|\_\{1\}, and letMωM\_\{\\omega\}be the linear priority model from[Definition4\.2](https://arxiv.org/html/2607.02672#S4.Thmtheorem2)\. ThenVθV\_\{\\theta\}under the S\-RUM response model is scale\-free indistinguishable fromMωM\_\{\\omega\}under the zero threshold response model: that is, for everyβ\>0\\beta\>0, there existsβ′=β​\|𝒳\|​‖θ‖1\\beta^\{\\prime\}=\\beta\\,\|\\mathcal\{X\}\|\\,\\\|\\theta\\\|\_\{1\}such that

Shβ​\(⋅;Vθ\)≡Rhβ′;0,0∘​\(⋅;Mω\)\.S\_\{h\_\{\\beta\}\}\(\\cdot;V\_\{\\theta\}\)\\equiv R^\{\\circ\}\_\{h\_\{\\beta^\{\\prime\}\};0,0\}\(\\cdot;M\_\{\\omega\}\)\.Moreover, they induce the same learning target: for everyℱ′⊆ℱ\\mathcal\{F\}^\{\\prime\}\\subseteq\\mathcal\{F\},

arg⁡maxF∈ℱ′⁡UMω​\(F\)=arg⁡maxF∈ℱ′​∑x∈𝒳Vθ​\(x,F​\(x\)\)\.\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}U\_\{M\_\{\\omega\}\}\(F\)=\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}\\sum\_\{x\\in\\mathcal\{X\}\}V\_\{\\theta\}\(x,F\(x\)\)\.

#### 4\.1\.2\.Simulation and Learning Setup

Decision Task\.In our simulation experiments, each decision instance consists of five candidate recipients\. Each candidate is represented byd=5d=5normalized features: we write an input asx=\(z\(1\),…,z\(5\)\)∈\(\[0,1\]d\)5x=\(z^\{\(1\)\},\\ldots,z^\{\(5\)\}\)\\in\(\[0,1\]^\{d\}\)^\{5\}, wherez\(ℓ\)∈\[0,1\]dz^\{\(\\ell\)\}\\in\[0,1\]^\{d\}is the feature vector of candidateℓ\\ell\.141414Technically, inputs are being drawn from a continuous region\[0,1\]d\[0,1\]^\{d\}, while the sums above treat the input space as discrete\. Because each simulation uses finite samples, we are effectively in a finite and discrete𝒳\\mathcal\{X\}regime; one could simply discretize the sampling space to an arbitrarily fine degree to make𝒳\\mathcal\{X\}formally discrete, if desired\.On inputxx, the rule must select one of the five candidates, so the feasible output set is𝒴​\(x\)=\{1,…,5\}\\mathcal\{Y\}\(x\)=\\\{1,\\ldots,5\\\}\. The feature map returns the features of the selected candidate, soψ​\(x,ℓ\)=z\(ℓ\)\\psi\(x,\\ell\)=z^\{\(\\ell\)\}\. Thus, a local pairwise queryq=\(ℓ,ℓ′;x\)q=\(\\ell,\\ell^\{\\prime\};x\)asks whether, in decision instancexx, the rule should select candidateℓ\\ellor candidateℓ′\\ell^\{\\prime\}\. We assume that inputs are drawn uniformly at random; this is a natural and non\-degenerate choice, giving an unbiased sample of the array of trade\-offs that can occur\.

True priority weights\.The underlying linear priority model must then havem=5m=5priorities, one per feature, and the priority utilitiesuj∗u\_\{j\}^\{\*\}are thus fixed\. We repeat all tests for 40 ground\-truth choices ofω∗\\omega^\{\*\}, where the true weights are drawn randomly asω∗∼Dirichlet​\(0\.2\)\\omega^\{\*\}\\sim\\text\{Dirichlet\}\(0\.2\)\. We report means and standard errors over random choices ofω∗\.\\omega^\{\*\}\.As is standard in linear S\-RUMs, we assume the feature map, and hence the priority utilities, are known to the learner, so the learner’s task is then to recover these priorities’ true importanceω∗\.\\omega^\{\*\}\.

##### Bayesian active learning\.

For each ground\-truth weight vectorω∗\\omega^\{\*\}, we learnω∗\\omega^\{\*\}by Bayesian active learning\. We use active learning to be fair to all response conditions; if we in contrast fixed a query sequence across all algorithms, it could happen to ask queries that are far more useful for one response condition over another, confounding our ability to compare the actual information value of difference response conditions\. With active learning, the query algorithm explicitly seeks the most useful queries given the assumed response condition\. At a high level, our active learning methods work as follows\.

At roundtt, the learner maintains a posteriorπt\\pi\_\{t\}over plausible weightsω\\omega, selects a queryqtq\_\{t\}, observes a response drawn from the true response modelR∗R^\{\*\}, and updates its posterior using its assumed response modelRR\. Upon reaching a stopping condition,151515In[Section4\.3](https://arxiv.org/html/2607.02672#S4.SS3), we run all algorithms for a fixedT=100T=100rounds to compare learning speed\. In[Section4\.2](https://arxiv.org/html/2607.02672#S4.SS2), we run to convergence: lettingω^t\\widehat\{\\omega\}\_\{t\}be the posterior mean attt, the convergence condition is that\|ω^t\+1−ω^t\|<0\.01\|\\widehat\{\\omega\}\_\{t\+1\}\-\\widehat\{\\omega\}\_\{t\}\|<0\.01for 5 consecutive iterations\.the algorithm outputs the posterior meanω^T=𝔼ω∼πT​\[ω\]\\widehat\{\\omega\}\_\{T\}=\\mathbb\{E\}\_\{\\omega\\sim\\pi\_\{T\}\}\[\\omega\]\. This general algorithmic template is formally specified in Algorithm[1](https://arxiv.org/html/2607.02672#alg1), with its variants formally specified in the appendix \(Appendix[C\.6](https://arxiv.org/html/2607.02672#A3.SS6)\)\. Queries are chosen using Bayesian Active Learning by Disagreement \(BALD\), which scores a query by how much its possible answers would reduce uncertainty about the true value ofω\\omega\.

The ideal implementation of this algorithmic method would calculate all objects exactly\. However, as is standard, for tractability we use sampling to approximate the objects above\. First, we would ideally compute the BALD score for every possible pairwise query in𝒬p​c\\mathcal\{Q\}^\{pc\}and select the query with the largest score\. We approximate this by computing the BALD score for each of \(C=50\) randomly\-sampled queries from𝒬p​c\\mathcal\{Q\}^\{pc\}, and then asking the highest\-scoring query in this sample\. Each score estimate is computed using an estimate of the posterior, based onNBALD=50N\_\{\\mathrm\{BALD\}\}=50samples ofω\\omegadrawn from the learner’s maintained posterior sample set of sizeNpost=200N\_\{\\mathrm\{post\}\}=200\.[SectionC\.6](https://arxiv.org/html/2607.02672#A3.SS6.SSS0.Px5)gives the standard consistency statement: as the candidate\-pool size and number of posterior samples grow, this approximation returns a near\-optimal BALD query with high probability\.

Response models\.Across the simulations, we distinguish between the true response modelR∗R^\{\*\}, which generates the individual’s observed answers, and the learner’s assumed response modelRR, which is used for posterior updates\. We vary these models across experiments starting from the baseline familyRhβ;τr,τκ∘R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}; we specify the details in the respective sections\. Throughout we use the logistic linkhβ​\(t\)=\(1\+exp⁡\(−β​t\)\)−1h\_\{\\beta\}\(t\)=\(1\+\\exp\(\-\\beta t\)\)^\{\-1\}withβ=10\\beta=10\. Thus, whenτr=τκ=0\\tau\_\{r\}=\\tau\_\{\\kappa\}=0, the baseline model reduces to the standard Bradley\-Terry model\. Unless otherwise stated, the thresholdsτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}are known to the learner; in one variant, we instead learnτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}jointly withω∗\\omega^\{\*\}, using an extension of our active learning procedure described in[SectionC\.7](https://arxiv.org/html/2607.02672#A3.SS7)\. Because the priority models in this section are perfectly separable, the choice of rule aggregatorϕrules\\phi^\{\\text\{rules\}\}does not affect the directional evidence scores, providedϕrules\\phi^\{\\text\{rules\}\}is unanimous\. To avoid arbitrary scaling due to the discretization of𝒳\\mathcal\{X\}, in a minor departure from our model we drop the1/\|𝒳\|1/\|\\mathcal\{X\}\|from the utility gap, so forq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\), we letϕrules​\(\(ΔjF​\(q\)\)F∈ℱ\)=ψj​\(x,y\)−ψj​\(x,y′\)\\phi^\{\\text\{rules\}\}\(\(\\Delta^\{F\}\_\{j\}\(q\)\)\_\{F\\in\\mathcal\{F\}\}\)=\\psi\_\{j\}\(x,y\)\-\\psi\_\{j\}\(x,y^\{\\prime\}\)\.

#### 4\.1\.3\.Evaluation Metrics

We evaluate learning at two levels: whether the learner recovers the individual’s priority weights, and whether the learned weights induce low\-regret decisions\. For weight recovery, we report theℓ1\\ell\_\{1\}loss‖ω^−ω∗‖1\.\\\|\\widehat\{\\omega\}\-\\omega^\{\*\}\\\|\_\{1\}\.Such errors are an issue if we are interpretingω∗\\omega^\{\*\}as substantively meaningful importance weights on features, or if we are measuring the welfare of the resulting weights according to their cardinal distance from the true weights\. This distance also has significance for the regret: by[LemmaC\.1](https://arxiv.org/html/2607.02672#A3.Thmtheorem1), if‖ω^−ω∗‖1≤ϵ\\\|\\widehat\{\\omega\}\-\\omega^\{\*\}\\\|\_\{1\}\\leq\\epsilon, then the rule induced byω^\\widehat\{\\omega\}has aggregate\-utility regret at most2​ϵ2\\epsilonunder the true model\.

To evaluate the quality of the actual choices made by the learned ruleFω^F\_\{\\widehat\{\\omega\}\}, we use the regret \([Definition2\.4](https://arxiv.org/html/2607.02672#S2.Thmtheorem4)\)\. For a learned vectorω^\\widehat\{\\omega\}, define the regret at inputxxby

Regretx⁡\(ω^;ω∗\):=⟨ω∗,ψ​\(x,Fω∗​\(x\)\)⟩−⟨ω∗,ψ​\(x,Fω^​\(x\)\)⟩,\\operatorname\{Regret\}\_\{x\}\(\\widehat\{\\omega\};\\omega^\{\*\}\):=\\langle\\omega^\{\*\},\\psi\(x,F\_\{\\omega^\{\*\}\}\(x\)\)\\rangle\-\\langle\\omega^\{\*\},\\psi\(x,F\_\{\\widehat\{\\omega\}\}\(x\)\)\\rangle,and define the utility range atxxas the utility gap between the best and worst choice atxx:

Rangex⁡\(ω∗\):=⟨ω∗,ψ​\(x,Fω∗​\(x\)\)⟩−miny∈𝒴​\(x\)⁡⟨ω∗,ψ​\(x,y\)⟩\.\\operatorname\{Range\}\_\{x\}\(\\omega^\{\*\}\):=\\langle\\omega^\{\*\},\\psi\(x,F\_\{\\omega^\{\*\}\}\(x\)\)\\rangle\-\\min\_\{y\\in\\mathcal\{Y\}\(x\)\}\\langle\\omega^\{\*\},\\psi\(x,y\)\\rangle\.We then define the normalizedaverage regretas the expected regret normalized by the range, where this normalization accounts for the fact that certain settings will produce lower\-stakes choices\. The average regret is then interpretable as the fraction of the typical available utility range lost by the learned rule:161616Note that the numerator ofAvg−Regret\\operatorname\{Avg\-Regret\}is equivalent to the regret in[Definition2\.4](https://arxiv.org/html/2607.02672#S2.Thmtheorem4), reduced to the linear case, where the aggregate\-optimal rule under modelMω∗M\_\{\\omega^\{\*\}\}is the linear ruleFω∗F\_\{\\omega^\{\*\}\}\([Proposition4\.3](https://arxiv.org/html/2607.02672#S4.Thmtheorem3)\)\.

Avg−Regret⁡\(ω^;ω∗\):=1/\|𝒳\|​∑x∈𝒳\[Regretx⁡\(ω^;ω∗\)\]1/\|𝒳\|​∑x∈𝒳\[Rangex⁡\(ω∗\)\]\.\\operatorname\{Avg\-Regret\}\(\\widehat\{\\omega\};\\omega^\{\*\}\):=\\frac\{\\nicefrac\{\{1\}\}\{\{\|\\mathcal\{X\}\|\}\}\\sum\_\{x\\in\\mathcal\{X\}\}\\left\[\\operatorname\{Regret\}\_\{x\}\(\\widehat\{\\omega\};\\omega^\{\*\}\)\\right\]\}\{\\nicefrac\{\{1\}\}\{\{\|\\mathcal\{X\}\|\}\}\\sum\_\{x\\in\\mathcal\{X\}\}\\left\[\\operatorname\{Range\}\_\{x\}\(\\omega^\{\*\}\)\\right\]\}\.We define theworst\-caseregret analogously, the fraction of the maximum utility range lost on the input where the learned rule makes its lossiest mistake:

WC−Regret⁡\(ω^;ω∗\):=supx∈𝒳Regretx⁡\(ω^;ω∗\)supx∈𝒳Rangex⁡\(ω∗\)\.\\operatorname\{WC\-Regret\}\(\\widehat\{\\omega\};\\omega^\{\*\}\):=\\frac\{\\sup\_\{x\\in\\mathcal\{X\}\}\\operatorname\{Regret\}\_\{x\}\(\\widehat\{\\omega\};\\omega^\{\*\}\)\}\{\\sup\_\{x\\in\\mathcal\{X\}\}\\operatorname\{Range\}\_\{x\}\(\\omega^\{\*\}\)\}\.For any fixedω∗\\omega^\{\*\}and learnedω^\\widehat\{\\omega\}, we estimate the average regret using a fixed set of300300i\.i\.d\. uniformly sampled inputs, shared across all response conditions and random choices ofω∗\\omega^\{\*\}\. We compute the worst\-case regret exactly via linear programming, as described in[SectionC\.5](https://arxiv.org/html/2607.02672#A3.SS5)\.

### 4\.2\.What if the individual’s response behavior changes when they are indecisive, but forced to decide?

To formally test this question, we let the learner assume the standard response modelRhβ;0,0∘R\_\{h\_\{\\beta\};0,0\}^\{\\circ\}, as this is the assumption under which forced comparisons are justified\. In contrast, we allow the true response modelR∗R^\{\*\}to vary such that when the individual is truly conflicted or indifferent \(i\.e\., whenLMω∗​\(q\)∈\{∼∗,⋈∗\}L\_\{M\_\{\\omega^\{\*\}\}\(q\)\}\\in\\\{\\sim^\{\*\},\\bowtie^\{\*\}\\\}\), they may deviate from the assumed behavior in order to coerce their response into the permitted response alphabet\{≻,≺\}\\\{\\succ,\\prec\\\}\. To be maximally friendly to the learner, we assume that outside indecisive latent states \(i\.e\., whenLMω∗​\(q\)∈\{≻∗,≺∗\}L\_\{M\_\{\\omega^\{\*\}\}\(q\)\}\\in\\\{\\succ^\{\*\},\\prec^\{\*\}\\\}\), the individual responds according toRhβ;0,0∘R\_\{h\_\{\\beta\};0,0\}^\{\\circ\}\. We describe the four behavioral conditions we test here, and formally define them in Appendix[C\.8](https://arxiv.org/html/2607.02672#A3.SS8)\.

1. \(1\)Correct: The individual resolves latent indecision according to the baseline model — i\.e\., regardless ofLMω∗​\(q\)L\_\{M\_\{\\omega^\{\*\}\}\}\(q\), the individual responds according toRhβ;0,0∘R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\. This is the idealized benchmark\.
2. \(2\)50/50: WhenLMω∗​\(q\)∈\{∼∗,⋈∗\}L\_\{M\_\{\\omega^\{\*\}\}\}\(q\)\\in\\\{\\sim^\{\*\},\\bowtie^\{\*\}\\\}, the individual decides betweenyyandy′y^\{\\prime\}by flipping an unbiased coin\.
3. \(3\)Lexicographic: WhenLMω∗​\(q\)∈\{∼∗,⋈∗\}L\_\{M\_\{\\omega^\{\*\}\}\}\(q\)\\in\\\{\\sim^\{\*\},\\bowtie^\{\*\}\\\}, the individual decides betweenyyandy′y^\{\\prime\}lexicographically: they choose deterministically based on the highest\-weight priority \(perω∗\\omega^\{\*\}\) that, in isolation, produces a decisive response between the two options\.
4. \(4\)Self\-similarity: Here, we suppose the individual themselves has features that belong inψ\\psi\(this would make sense, e\.g\., in any allocation problem where goods/bads are being allocated to people\)\. In each run, the individual’s feature vectorvvis drawn uniformly randomly from\[0,1\]5\[0,1\]^\{5\}\. WhenLMω∗​\(q\)∈\{∼∗,⋈∗\}L\_\{M\_\{\\omega^\{\*\}\}\}\(q\)\\in\\\{\\sim^\{\*\},\\bowtie^\{\*\}\\\}, the individual deterministically chooses the alternative that is most similar to them in feature spaceψ\\psi, reflecting self\-similarity bias\.

In the following results, we setτr=τκ=0\.25\\tau\_\{r\}=\\tau\_\{\\kappa\}=0\.25so that latent indecision occurs at nontrivial frequency but does not overwhelm directional signal\.

![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/fig_threepanel_envnorm.png)Figure 4\.Learner performance under the four models of resolving indecision\. Error bars are±1\\pm 1standard errors, reflecting randomness over 40 random choices ofω∗\\omega^\{\*\}\.\(a\)ℓ1\\ell\_\{1\}error ofω^\\widehat\{\\omega\}\.\(b\)Average regret, estimated via 300 sampledx∈𝒳x\\in\\mathcal\{X\}\.\(c\)Worst\-case regret\.ℓ1\\ell\_\{1\}errors\.In[Figure4](https://arxiv.org/html/2607.02672#S4.F4)\(a\), we examine theℓ1\\ell\_\{1\}error in the estimatedω^\\widehat\{\\omega\}relative toω∗\\omega^\{\*\}\. Noting that the maximum possible value of thisℓ1\\ell\_\{1\}error is 2 \(by the triangle inequality\), theℓ1\\ell\_\{1\}error across the three deviating response models is nontrivial, reaching between 14%\-24% of the worst case\. As expected, the idealized baselineCorrecthas negligible error\.

Under each behavioral deviation, the driver of theℓ1\\ell\_\{1\}error in the learnedω^\\widehat\{\\omega\}reflects the corresponding structure of the forced\-response rule\. We illustrate these findings in Figure[8](https://arxiv.org/html/2607.02672#A3.F8), Appendix[C\.9](https://arxiv.org/html/2607.02672#A3.SS9)\. UnderLexicographic\([Figure8](https://arxiv.org/html/2607.02672#A3.F8)\(a\)\),ω^\\widehat\{\\omega\}is overly concentrated on the highest\-weight priority: across runs it assigns an extra0\.13±0\.010\.13\\pm 0\.01mass to the true top feature relative toω∗\\omega^\{\*\}, exceeding the true top\-feature mass in 38 of 40 runs\. UnderSelf\-Similarity\([Figure8](https://arxiv.org/html/2607.02672#A3.F8)\(b\)\), the distortion is instead directed toward the individual’s own feature vectorvv: the projection⟨ω^−ω∗,v⟩\\langle\\widehat\{\\omega\}\-\\omega^\{\*\},v\\rangleis positive in 35 of 40 runs, with a cross\-run mean of0\.10±0\.010\.10\\pm 0\.01\. In contrast, these sub\-figures together show that50/50distortsω^\\widehat\{\\omega\}in neither direction, because its forced labels are an unbiased coin flip\.

Average regret\.If one simply cares about making the optimal choice over𝒴​\(x\)\\mathcal\{Y\}\(x\)at eachxx, the story is more positive, at least in the average case\.[Figure4](https://arxiv.org/html/2607.02672#S4.F4)\(b\), displayingAvg−Regret⁡\(ω^,ω∗\)\\operatorname\{Avg\-Regret\}\(\\widehat\{\\omega\},\\omega^\{\*\}\), shows that over randomxx, the expected choice loss from forced\-response behavior is small across deviations\. Normalized average regret is highest forSelf\-Similarity, which loses2\.6%2\.6\\%of the typical utility range available across decision instances;50/50andLexicographiclose0\.5%0\.5\\%and1\.1%1\.1\\%, respectively\. Thus, on average, the choices induced by the learned weights nearly optimally recover utility, even when the weights themselves are substantially misestimated\.

These regret levels are largely explained by whether the learned vectorω^\\widehat\{\\omega\}points in the right direction\.Self\-Similaritysystematically biasesω^\\widehat\{\\omega\}toward the individual’s own feature vectorvv, which can point in a substantially different direction fromω∗\\omega^\{\*\}, producing the largest average regret\. By contrast, underLexicographic, behavior remains anchored to the direction ofω∗\\omega^\{\*\}, even though the largest priorities are overweighted; under50/50, deviations aroundω∗\\omega^\{\*\}arise from noise rather than from a systematic directional bias\. We illustrate this by showing the cosine similaritycos⁡\(ω^,ω∗\)\\cos\(\\widehat\{\\omega\},\\omega^\{\*\}\)in Figure[9](https://arxiv.org/html/2607.02672#A3.F9)\(Appendix[C\.9](https://arxiv.org/html/2607.02672#A3.SS9)\)\. The fact that cosine similarity closely tracks average regret is consistent with prior work on linear decision rules showing that angular alignment is theoretically related to good downstream choice behavior\(Fefferet al\.,[2023](https://arxiv.org/html/2607.02672#bib.bib5); Baharavet al\.,[2026](https://arxiv.org/html/2607.02672#bib.bib6)\)\.

Worst\-case regret\.The picture is less reassuring in the worst case\.[Figure4](https://arxiv.org/html/2607.02672#S4.F4)\(c\) shows worst\-case regretWC−Regret⁡\(ω^,ω∗\)\\operatorname\{WC\-Regret\}\(\\widehat\{\\omega\},\\omega^\{\*\}\), i\.e\., largest choice loss the learned rule can incur on any input\. On this metric,50/50,Lexicographic, andSelf\-Similarityrespectively reach21%21\\%,17%17\\%, and39%39\\%of the maximum utility range available over choices, on average across runs\.

The changed ordering in relative performance ofLexicographicand50/50from average to worst\-case regret is not spurious — it occurs becauseLexicographichas less directional distortion, overweighting priorities that are already highly weighted underω∗\\omega^\{\*\}\. As a result, when it makes mistakes, it often does so by leaning too heavily on features thatω∗\\omega^\{\*\}also regards as important\. By contrast, the noisy behavior produced under50/50produces aω^\\widehat\{\\omega\}that is less directionally tied toω∗\\omega^\{\*\}, and this can be exploited by a worst\-case adversary\.

### 4\.3\.If the learner allows andutilizesself\-reported indecision, can they learn faster?

One natural solution to the problem described above is to allow the individual to report their indecision\. Using a learner that canutilizethis indecision, we now investigate the extent to which this also allows faster learning\. To test this question, we compare how quickly four different learning approaches converge to the trueω∗\\omega^\{\*\}\. In all cases, the learner assumes the correct underlying response model; what varies is the extent to which they receive or can utilize indecision information\.

The first two response conditions below represent forced comparison benchmarks\. The latter two represent cases where the learner actively solicits reports of conflict and indecision \(Utilize\-4\) or generic indecision \(Utilize\-3\)\. Here, the individual can report a richer alphabet of responses\.

1. \(1\)Correct: The true response model isRhβ;0,0∘R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}, so the individual only reports responses in\{≻,≺\}\\\{\\succ,\\prec\\\}\. The learner assumes the same response model\. This is standard Bradley\-Terry, and serves as the baseline testing how fast we can learn from forced comparisons under perfect conditions\.
2. \(2\)Ignore: The true response model isRhβ;τr,τκ∘R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}, the learner assumesRhβ;0,0∘R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}, and any query response in\{∼,⋈\}\\\{\\sim,\\bowtie\\\}is dropped from the transcript\. This reflects a learner who forces decisive comparisons by dropping queries in which the individual was unable to respond due to indecision\.
3. \(3\)Utilize\-3: Here, the true response model is a version ofRhβ;τr,τκ∘R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}where indecisive responses\{∼,⋈\}\\\{\\sim,\\bowtie\\\}are both reported as generic indecision⊘\\oslash\(see[C\.8](https://arxiv.org/html/2607.02672#A3.Thmtheorem8)for the formal specification\)\. The learner assumes the same\. This case lets us examine the relative benefit of distinguishing∼\\simand⋈\\bowtieversus letting the individual report general indecision\.
4. \(4\)Utilize\-4: Here, the true response model isRhβ;τr,τκ∘R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}and the learner assumes the same\. This represents the case where the individual responds in a way that reflects \(a potentially noisy version of\) their latent state, and the learner can utilize all possible responses\.

We also include a version of \(4\) where we learnτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}alongsideω\\omega, to demonstrate the possibility of doing this efficiently; we denote this variant with a⋄\\diamondand detail the methodology in Algorithm[2](https://arxiv.org/html/2607.02672#alg2), Appendix[C\.7](https://arxiv.org/html/2607.02672#A3.SS7)\.

We now compare the convergence rates of all five learning setups \([Figure5](https://arxiv.org/html/2607.02672#S4.F5)\)\. To give underlying intuition about what types of queries are useful, we additionally show which query types were sought by the active learning algorithm in each setting \(Figure[6](https://arxiv.org/html/2607.02672#S4.F6)\)\. We discuss our key findings below\.

![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/bald_diag_l1_avgreg_wcr_bars_5methods_warm.png)Figure 5\.Performance of five methods versus query number across the diagonal threshold regimes\.Top:ℓ1\\ell\_\{1\}error‖ω^−ω∗‖1\\\|\\widehat\{\\omega\}\-\\omega^\{\\ast\}\\\|\_\{1\}\.Middle:Average regret on the uniform\-\[0,1\]5\[0,1\]^\{5\}distribution, all features independent\.Bottom:Worst\-case single\-decision regret\. See performance for moreτ\\tauregimes and extensions in Appendix[C\.10](https://arxiv.org/html/2607.02672#A3.SS10)\.![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/response_distribution_main_warm.png)Figure 6\.Fraction of each response type \(Left≻\\succ, Right≺\\prec, Indifferent∼\\sim, Conflict⋈\\bowtie\) for three representative methods —Utilize\-4,Ignore, andCorrect— across the full5×55\\times 5threshold grid\.Utilizing indecision allows faster learning\.First, comparingCorrectandIgnoreto theUtilizeconditions in[Figure5](https://arxiv.org/html/2607.02672#S4.F5), we see that all three indecision\-aware learners converge much faster to lowℓ1\\ell\_\{1\}error and worst\-case regret\. As the indecision region grows, the gap also widens in the average regret\. Only in the most severe indecision regime doesCorrectoutperformUtilizemethods, where it is simply benefiting from its independence of the latent state in a regime where the latent states are saturated with indecision\. However, this is perhaps the least behaviorally plausible regime for the forced benchmark: it assumes that at maximal true indecision, individuals can still resolve all indecision via unbiased, well\-structured noise\. Making finer comparisons, we see that asking people to distinguish between indifference and conflict makes almost no difference, withUtilize\-3andUtilize\-4performing almost identically relative to the forced\-comparison gap\.

Utilizemethods outperformingIgnoreis partly mechanical:Ignoreis dropping data, and it is doing so in a non\-random way, meaning it may never converge to the correct weight vector\.171717Whenτr\>0\\tau\_\{r\}\>0, dropping∼\\simresponses removes low\-valence queries and can eliminate identifying variation\. Whenτκ\>0\\tau\_\{\\kappa\}\>0, dropping⋈\\bowtieresponses conditions on being outside the conflict band, so the remaining≻,≺\\succ,\\precresponses follow a conditional response distribution rather than the original link model\.This comparison is practically interesting, because dropping queries when the individual cannot answer them isa prioria natural solution to indecision\. It is much more striking that theUtilizeimproves so substantially overCorrect, which is the response condition under which decisive responses are behaviorally justified\.

This strongly supports the idea that there is distinct and useful information content in indecision\. This claim is bolstered by balance of the queries actively sought by theUtilizemethods: as shown in[Figure6](https://arxiv.org/html/2607.02672#S4.F6)\(and Figure[15](https://arxiv.org/html/2607.02672#A3.F15)in Appendix[C\.10](https://arxiv.org/html/2607.02672#A3.SS10)forUtilize\-3, Utilize\-4⋄\),Utilizelearners do not simply avoid indecisive regions, nor do they simply inherit the latent state distribution\. Compared withIgnore, whose queried responses become quickly dominated by∼\\simasτr\\tau\_\{r\}grows and⋈\\bowtieasτκ\\tau\_\{\\kappa\}grows,Utilizeseems to “fight against” the latent state distribution and seeks a mix of decisive and indecisive queries through most of the threshold grid\. This is consistent with the mechanism suggested by the model: whenτr\\tau\_\{r\}andτκ\\tau\_\{\\kappa\}are small or intermediate,∼\\simand⋈\\bowtieresponses are highly informative because they impose tight constraints onω\\omega\. As the thresholds get higher, these queries may become less informative but more unavoidable\.

Extensions\.An important possibility we need to rule out is thatUtilize\-4’s advantage over forced comparison settings is coming from our assumption that it knowsτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}up front\. ComparingUtilize\-4andUtilize\-4⋄, we see that this concern is unwarranted —Utilize\-4⋄learns almost exactly as fast asUtilize\-4despite having to learn the thresholds alongsideω^\\widehat\{\\omega\}\. We note that whileUtilize\-4⋄should not generally outperformUtilize\-4\(as it is solving a harder problem\), our results atτr=0\.8\\tau\_\{r\}=0\.8appear to contradict this at lowtt\. This early edge likely comes from the fact that by virtue of not knowing the thresholds,Utilize\-4⋄does not update too strongly based on responses that contradict the latent state due to noise, because early on, such responses can be partly attributed to uncertainty inτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}\.

Additionally, in[Figure10](https://arxiv.org/html/2607.02672#A3.F10)\([SectionC\.10](https://arxiv.org/html/2607.02672#A3.SS10)\), we investigate whether the richer response alphabet is powerful enough to retain gains in learning speed under weaker assumptions about noise model\. To test this, we compareUtilize\-4,Ignore, andCorrectlearners that know the logistic link function to otherwise identical learners that instead fit a flexible mixture\-of\-Gaussians noise model, denoted by†\\dagger\. We find that generalizing the assumed noise model has essentially no effect on the learning speed underUtilize\-4, while further compromising learning rates forIgnoreandCorrect\. This suggests that permitting indecisive responses can reduce risk of model misspecification in an additional way, allowing weaker assumptions about the functional form of noise\.

Magnitude of gains are practically significant\.The magnitude of the gains described above matter because the relevant query budgets are realistic: a few tens of comparisons per individual are plausible in elicitation studies, while many tens or hundreds quickly become burdensome\. In the intermediate threshold regimes,Utilizemoves accurate learning into this plausible range\. For example, at\(τr,τκ\)=\(0\.4,0\.4\)\(\\tau\_\{r\},\\tau\_\{\\kappa\}\)=\(0\.4,0\.4\),Utilize\-4drops belowℓ1=0\.05\\ell\_\{1\}=0\.05after roughly2525queries and reachesℓ1=0\.014\\ell\_\{1\}=0\.014byT=100T=100, compared with0\.160\.16forCorrectand0\.340\.34forIgnore\. At\(0\.6,0\.6\)\(0\.6,0\.6\), the contrast withIgnoreis even larger:Utilize\-4reachesℓ1=0\.012\\ell\_\{1\}=0\.012, whileIgnoreremains at0\.620\.62\. As shown in[Figure5](https://arxiv.org/html/2607.02672#S4.F5), the regret advantage ofUtilizemethods is also decisive within a realistic budget: byT=25T=25queries,Utilize\-4’s average regret is0\.03%0\.03\\%of the per\-decision utility range at thresholds\(0\.4,0\.4\)\(0\.4,0\.4\)and\(0\.6,0\.6\)\(0\.6,0\.6\), versus1\.2%1\.2\\%/1\.8%1\.8\\%\(Correct/Ignore\) at\(0\.4,0\.4\)\(0\.4,0\.4\)and0\.8%0\.8\\%/5\.4%5\.4\\%at\(0\.6,0\.6\)\(0\.6,0\.6\)— a multiplicative gap of3030–180180\. Gaps are even more pronounced for worst\-case regret\. By way of explanation, we again demonstrate that these regret gains correspond to more quickly learning the correct direction ofω∗\\omega^\{\*\}, as quantified by the cosine similarity \(Figure[12](https://arxiv.org/html/2607.02672#A3.F12)\)\.

## 5\.Discussion

Taken together, our results in[Section4](https://arxiv.org/html/2607.02672#S4)— despite their limitations as stylized simulations181818First, we assume that the individual reports indecision accurately with respect to their latent states, up to some noise\. This assumption could be unrealistically favorable to richer response alphabets, e\.g\., under the realistic possibility that people will instead overuse the indecision response to avoid exerting effort to decide\. Second, the benefits of richer responses may be much greater when queries are chosen via active learning, given our results showing that under richer response alphabets, the learner seems to be seeking a certain ideal balance of query types\. Under passive or randomly sampled queries, indecision reports may arise less often, or in less informative regions of the query space\.— suggest that permitting individuals to report indecision in local pairwise comparisons \(even without distinguishing between indifference and conflict\) can sidestep issues with deviating response behavior due to forcing comparisons, and dramatically improve learning speed, with gains occurring exactly within the practical query complexity regime\. These technical reasons for eliciting indecision add to the reasons articulated in[Section4](https://arxiv.org/html/2607.02672#S4): learning indecision over queries can help us learn about the individual’s indecision overrules, and this information has both normative value for understanding the moral authority of the individual’s judgments within the decision space, and instrumental value for finding consensus\.

While our learning methods from[Section4](https://arxiv.org/html/2607.02672#S4)illustrate how a learner can utilize indecision and learnτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}inlinearpriority models,[Section3](https://arxiv.org/html/2607.02672#S3)demonstrates that this is actually a highly restricted case relative to the universe of priorities the individual might care about\. A complete solution is out of scope, because[Section3\.2](https://arxiv.org/html/2607.02672#S3.SS2)shows that perfectly inseparable priorities are not identifiable from local pairwise comparison queries alone, so richer query design is required\. However, our results do suggest a learning approach under the restriction thatMM’s weights that are identifiable\. We call itpriority\-aware learning: if the learner isaware of the underlying priority structure, they can interpret responses according to this information and consequently learn the weights\. We discuss this approach below\.

### 5\.1\.Toward learning generalMM:Priority\-Aware Learning

Fix a ground\-truth modelM=\(u∗,ω∗\)M=\(u^\{\*\},\\omega^\{\*\}\)in which everyωj∗\\omega^\{\*\}\_\{j\}is scale\-free identifiable with respect tohh\. For now, let’s assume the learner knowsu∗u^\{\*\}in advance, so the goal is to learn isω∗\\omega^\{\*\}— we will address how to elicitu∗u^\{\*\}below\. We also assume, as usual, that the learner knows the response modelRhβ;τr,τκ∘R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}, for genericτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}\. For intuition, we describe the decoder in the deterministic limitβ→∞\\beta\\to\\infty, where an individual acting according toRhβ;τr,τκ∘R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}reports their latent state exactly; the noisy case is the Bayesian analog\.

Given a queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\), each possible response imposes a constraint onω\\omega:

w=≻\\displaystyle w=\{\\succ\}⟹rM∗​\(q\)≥τr​and​κM∗​\(q\)≥τκ​rM∗​\(q\),\\displaystyle\\implies r\_\{M^\{\*\}\}\(q\)\\geq\\tau\_\{r\}\\ \\text\{ and \}\\ \\kappa\_\{M^\{\*\}\}\(q\)\\geq\\tau\_\{\\kappa\}\\,r\_\{M^\{\*\}\}\(q\),w=≺\\displaystyle w=\{\\prec\}⟹rM∗​\(q\)≥τr​and−κM∗​\(q\)≥τκ​rM∗​\(q\),\\displaystyle\\implies r\_\{M^\{\*\}\}\(q\)\\geq\\tau\_\{r\}\\ \\text\{ and \}\\ \-\\kappa\_\{M^\{\*\}\}\(q\)\\geq\\tau\_\{\\kappa\}\\,r\_\{M^\{\*\}\}\(q\),w=⋈\\displaystyle w=\{\\bowtie\}⟹rM∗​\(q\)≥τr​and​\|κM∗​\(q\)\|<τκ​rM∗​\(q\),\\displaystyle\\implies r\_\{M^\{\*\}\}\(q\)\\geq\\tau\_\{r\}\\ \\text\{ and \}\\ \|\\kappa\_\{M^\{\*\}\}\(q\)\|<\\tau\_\{\\kappa\}\\,r\_\{M^\{\*\}\}\(q\),w=∼\\displaystyle w=\{\\sim\}⟹rM∗​\(q\)<τr,\\displaystyle\\implies r\_\{M^\{\*\}\}\(q\)<\\tau\_\{r\},where bothrM∗​\(q\)r\_\{M^\{\*\}\}\(q\)andκM∗​\(q\)\\kappa\_\{M^\{\*\}\}\(q\)are linear inω\\omega:

rM∗​\(q\):=∑j∈\[m\]ωj​\|ϕrules​\(\(ΔjF​\(q\)\)F∈ℱ\)\|,κM∗​\(q\):=∑j∈\[m\]ωj​ϕrules​\(\(ΔjF​\(q\)\)F∈ℱ\)\.r\_\{M^\{\*\}\}\(q\):=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\,\\bigl\|\\phi^\{\\mathrm\{rules\}\}\\bigl\(\(\\Delta^\{F\}\_\{j\}\(q\)\)\_\{F\\in\\mathcal\{F\}\}\\bigr\)\\bigr\|,\\qquad\\kappa\_\{M^\{\*\}\}\(q\):=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\,\\phi^\{\\mathrm\{rules\}\}\\bigl\(\(\\Delta^\{F\}\_\{j\}\(q\)\)\_\{F\\in\\mathcal\{F\}\}\\bigr\)\.Given the priority utilitiesuj∗u\_\{j\}^\{\*\}, the learner can compute the coefficientsϕrules​\(\(ΔjF​\(q\)\)F∈ℱ\)\\phi^\{\\text\{rules\}\}\\big\(\(\\Delta\_\{j\}^\{F\}\(q\)\)\_\{F\\in\\mathcal\{F\}\}\\big\)above,191919One may wonder whether this is computationally feasible whenℱ\\mathcal\{F\}is large\. This depends on the priority structure: ifℱ\\mathcal\{F\}is parametrized and the utility gapsF↦uj​\(Fx→y\)−uj​\(Fx→y′\)F\\mapsto u\_\{j\}\(F\_\{x\\to y\}\)\-u\_\{j\}\(F\_\{x\\to y^\{\\prime\}\}\)are tractable, computing the scores becomes an optimization problem over that parameter space\. If exact optimization is difficult, many rule aggregators can be approximated by sampling over background rules\.making these linear constraints on the unknownω\\omega\.202020If the thresholdsτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}are also unknown, learning them can be folded into the same procedure, just as in Section[4\.3](https://arxiv.org/html/2607.02672#S4.SS3)\. SincerM∗​\(q\)r\_\{M^\{\*\}\}\(q\)andκM∗​\(q\)\\kappa\_\{M^\{\*\}\}\(q\)are affine inω\\omega, the thresholdτr\\tau\_\{r\}enters the constraints linearly and can be learned jointly withω\\omega; the thresholdτκ\\tau\_\{\\kappa\}enters bilinearly, through the productτκ​rM∗​\(q\)\\tau\_\{\\kappa\}\\,r\_\{M^\{\*\}\}\(q\), so it can be handled by grid search\.Given that theω∗\\omega^\{\*\}are scale\-free identifiable, the constraints onω\\omegaaccumulated along an exhaustive query sequence eventually pin the feasible set down toω∗\\omega^\{\*\}\. The query sequence can also be actively derived: due to the linearity of the constraints, active query choice reduces to the standard problem of choosing halfspace constraints that maximally shrink the remaining feasible set ofω\\omega, and by Lemma C\.1, approximatingω∗\\omega^\{\*\}ensures that we closely approximate the aggregate\-optimal rule\.212121One may instead want to directly shrink uncertainty over the rule space, possibly aiming for a rule other than the aggregate\-optimal rule; this is more complex, requiring a tractable connection between the remaining space ofω\\omegaand the remaining rule space\.

Eliciting priorities\.Above we assumed the prioritiesu∗u^\{\*\}are known before learning their weights\. How precisely to elicit these priorities requires empirical study, but one possible procedure could be interactive and text\-based: First, the learner could ask the individual to describe textually how they think the rule should and should not behave, to get an initial set of priorities\. Then, as the individual is asked comparison queries, the learner could askwhythey answered as they did, using these explanations to surface additional priorities\. Each time, the prior query responses could be reinterpreted over the expanded priority list\. While this approach may not recover all priorities, it is reasonable to assume that the most important priorities surface more readily, in which case the recovered priorities will approximately recover the total weight inω∗\\omega^\{\*\}\.

Of course, a key point of slippage in this approach is the process of translating textually\-articulated priorities into utility functions\. However, we argue that knowing the exact cardinal utilities attached to each priority is not the point — and in fact, such precise values over such a large decision space probably don’t even exist within the individual\. Rather, we suggest that even an approximate priority basis may be more faithful than other approaches to making learning tractable—e\.g\., forcing preferences into a highly restricted separable score model, or reducing to highly restricted rule classes that structurally cannot serve certain priorities well\. The risk of mistranslating priorities could also be managed by the learner maintaining uncertainty over these representations, and seeking further information when that uncertainty is decision\-relevant\.

At a higher level, tractable learning over a massive rule spaceℱ′⊆ℱ\\mathcal\{F\}^\{\\prime\}\\subseteq\\mathcal\{F\}generally requires some kind of simplification or down\-projection of preferences into a lower\-dimensional representation\. The approach proposed here — of eliciting priorities textually then using their known structure to learn the weights — can be seen as a version of learning a lower\-dimensional representation in whichthe dimensions are chosen by the individual themself\. This gets the learning\-speed benefits of a lower dimensional representation while using text for what it is especially good at: eliciting the qualitative, non\-numeric structure of a person’s judgments\. As an added bonus, the resulting representation is also interpretable to the individual, since its dimensions correspond to considerations they can recognize and revise\.

### 5\.2\.Applications of the priority model

In this paper, we used our model as a theoretical tool to formally investigate intuitions about why forced, local pairwise comparisons — despite being behaviorally justified by standard models \(S\-RUMs\) — feel as if they might be missing important elements of our values\. However, as alluded to by[Section5\.1](https://arxiv.org/html/2607.02672#S5.SS1), the fact that our model is potentially tractable to learn means that it can also be useful as a preference\-learning tool\. In thinking about its applications, it is important to note that while this model is formalized for the decision space of decision rules, its fundamental elements can be applied to many decision tasks, and its key insights apply especially when the decision space is large, e\.g\., combinatorial \(as in committee selection or fair division\) or continuous \(as with decision rules, prediction algorithms, or policies\)\.

One immediate application is deliberation\. In AI\-mediated deliberative processes, it is tempting to let an LLM implicitly infer what participants value and use that latent representation to summarize disagreement or guide the conversation through complex policy spaces\(Fishet al\.,[2026](https://arxiv.org/html/2607.02672#bib.bib177); Tessleret al\.,[2024](https://arxiv.org/html/2607.02672#bib.bib176)\)\. Our model suggests a different architecture: learn an explicit modelMiM\_\{i\}for each participantii, and use the LLM only as an interface to help elicit the priorities within the model\. Instead of using the LLM as a black\-box interpreter of opinion, this keeps the preference representation outside the LLM as an interpretable object, allowing the individual to inspect whether their priorities, importance weights, and regions of conflict and indifference have been learned correctly\.

Such individual models could also serve as richer inputs in decentralized public input processes, e\.g\., in maxipublics: instead of simply submitting a local vote, people could submit their priorities — possibly along with a few additional questions to get at relative importance — leading to much richer information about what trade\-offs they would prefer than is captured by a single vote\. This suggests a different way to think about platforms such as Polis\(Smallet al\.,[2021](https://arxiv.org/html/2607.02672#bib.bib175)\)or Remesh\(Konyaet al\.,[2023](https://arxiv.org/html/2607.02672#bib.bib174)\), which accept short, free\-form comments and then cluster them\. Using similar text snippets, one could instead elicit priorities, which could be used as richer inputs to downstream deliberative processes, particularly those designed as described in the paragraph above\.

Although we study pluralitywithina single individual, our priority model can also be applied to capture pluralityacrossindividuals\. This could be done, e\.g\., by learning a single shared model across individuals in a deliberative context, or by learning a model for each individual and then aggregating them\. Doing this would make clear which priorities are fundamentally at odds across people, allowing people to more directly consent to compromises\. In the likely case where people’s preferences contain large equivalence classes, this approach could also help identify where people are broadly indifferent, potentially leading to greater opportunities for consensus\.

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## Appendix ASupplemental Materials from[Section2](https://arxiv.org/html/2607.02672#S2)

### A\.1\.Foundations in Behavioral and Decision Sciences

#### A\.1\.1\.Preferences over rules

One key difference between our model and S\-RUMs is that we define preferences at the level ofrules, from which preferences over outcomes at individual inputs are constructed\. This approach lets the model encompass the possibility that the individual evaluates not only realized outcomes but also the rules and procedures that generate them\. Research on procedural justice emphasizes that individuals care about the fairness of processes, and that process judgments shape attitudes such as trust and willingness to comply\(Lind and Tyler,[1988](https://arxiv.org/html/2607.02672#bib.bib366); Lind,[2001](https://arxiv.org/html/2607.02672#bib.bib149); Leviet al\.,[2009](https://arxiv.org/html/2607.02672#bib.bib367)\)\. Distributive justice work likewise highlights that judgments about what is fair invoke multiple competing values \(e\.g\., equity, equality, need\) that reflect how a rule behavesacross outcomes, making rules the natural locus for studying value conflict\(Deutsch,[1975](https://arxiv.org/html/2607.02672#bib.bib146)\)\.

#### A\.1\.2\.Internal pluralism

The second core feature of our model is internal pluralism, as captured in the multiple priorities definingMM\. At a high level, our priority model is closest in spirit to Tetlock’s influentialvalue pluralism model of ideological reasoning, which was originally proposed to explain individual differences in political reasoning\(Tetlock,[1986](https://arxiv.org/html/2607.02672#bib.bib135)\)\. Tetlock’s proposal was articulated textually and operationalized as a measurement tool, rather than specified as a formal mathematical model of judgment, as we do\. As described in[Section2\.2\.3](https://arxiv.org/html/2607.02672#S2.SS2.SSS3), our priority model captures many of the intuitions described by Tetlock, including decisions selectively activating certain values and/or bringing values into conflict\. Tetlock also alludes to the possibility that some values are more important to an individual than others, a concept modeled by ourω\\omegavector\.

Given that our model so closely imitates Tetlock’s setup, it is natural that our model must also engage closely with Tetlock’s two core elements of moral reasoning:differentiationandintegration\.Differentiationis the extent to which a person recognizes multiple relevant considerations in a decision, captured by the multiplicity of priorities inMM\.Integrationrefers to how people actually adjudicate trade\-offsbetweenvalues when faced with dilemmas\. Our response model assumes people can linearly make trade\-offs between priorities when the trade\-offs are sufficiently weak \(i\.e\., when the decisiveness is high\), but also allows certain trade\-offs to be irreducible, when both sides are strongly\-held or there is no evidence in either direction\.

There is considerable support in the literature for these modeling decisions, as well as robust challenges\. We discuss this competing evidence in the next two subsections, and where relevant we discuss how countervailing theories can be captured via alternative instantiations of our model\.

#### A\.1\.3\.Differentiation

Another grounding example of value differentiation is in Moral Foundations Theory, which conjectures that people are influenced by multiple moral “foundations” \(roughly, concerns – like harm, fairness, loyalty, respect, etc\.\)\. There is empirical evidence applying this theory, finding that moral judgments cluster into multiple dissociable dimensions rather than collapsing to a single factor\(Grahamet al\.,[2011](https://arxiv.org/html/2607.02672#bib.bib46)\)\. Likewise, in the literature exploring how people respond to moral dilemmas \(e\.g\., the trolley problem\), experiments show that variation in responses is explained by participants drawing upon multiple priorities, including sensitivity to both consequences and moral norms\(Gawronskiet al\.,[2017](https://arxiv.org/html/2607.02672#bib.bib53)\)\.

On the other hand, even if many values can matter to an individual in principle, the individual may not be able toretrieveall of them when making any given judgment, especially in settings where there is low accountability or low deliberative demand\. Tetlock \(building on prior work\) explicitly frames people as “cognitive misers,” becoming more complex only when the decision context induces deeper tradeoff reasoning, e\.g\., due to accountability\(Tetlock,[1986](https://arxiv.org/html/2607.02672#bib.bib135); Fiske and Taylor,[1984](https://arxiv.org/html/2607.02672#bib.bib313); Tetlock and Kim,[1987](https://arxiv.org/html/2607.02672#bib.bib50)\)\. Complementarily, intuitionist and dual\-process traditions emphasize that judgments are frequently driven by quick intuitions, with reasons supplied post hoc; this predicts limited explicit differentiation unless prompted or socially required\(Haidt,[2001](https://arxiv.org/html/2607.02672#bib.bib51); Haidtet al\.,[2000](https://arxiv.org/html/2607.02672#bib.bib52); Kahneman,[2011](https://arxiv.org/html/2607.02672#bib.bib345)\)\.

Our modelMMalready captures selective value retrieval to some degree: if a priority is not active on a given query, it does not contribute to the comparison scoress\+​\(q\)s^\{\+\}\(q\)ands−​\(q\)s^\{\-\}\(q\), and is effectively ignored in the latent stateLLand response modelRR\. Our model can be extended naturally to capture richer limitations on value retrieval, too\. For example, to capture the recognition of only the frames most important to a decision, one could slightly modify our instantiation ofMMandLLand defines\+​\(q\)s^\{\+\}\(q\)ands−​\(q\)s^\{\-\}\(q\)to account only for active prioritiesjjwithωj\\omega\_\{j\}above a certain threshold, or thekkpriorities with the largest values ofωj\\omega\_\{j\}\.

#### A\.1\.4\.Integration

Our model of integration appears primarily in two places, our latent states, and our query response model\. Secondarily, our notion of regret — formulated primarily for convenience of evaluating rules rather than as a behavioral assumption — does also encode a form of integration, taking the “best rule” to be the rule with the highest value of theω\\omega\-weighted sum of priority utilities\.

Conflict and indifference when facing dilemmas\.Our model’s permission of⋈\\bowtie\(conflict\) versus∼\\sim\(indifference\) in both latent states and responses is closely aligned with the work ofCacioppo and Berntson \([1994](https://arxiv.org/html/2607.02672#bib.bib254)\)onevaluative space, as we discuss in[Section2\.2\.3](https://arxiv.org/html/2607.02672#S2.SS2.SSS3)\. In short, they distinguish conflict and indifference in a conceptually similar way — the distinction resulting from decoupling positive and negative evidence\.

Our explicit modeling of conflict — rather than just indifference, which would indicate no strong reaction — is also consistent with other research showing that conflict is often experienced as ambivalence and yields discomfort, deferral, and decision avoidance\(van Harreveldet al\.,[2009](https://arxiv.org/html/2607.02672#bib.bib256); Schneider and Schwarz,[2017](https://arxiv.org/html/2607.02672#bib.bib257); Anderson,[2003](https://arxiv.org/html/2607.02672#bib.bib261)\)\. Classic experiments show that conflict is behaviorally relevant, increasing the tendency to defer choice, to choose a default/no\-choice option, or to search for additional alternatives or information \(i\.e\., to expand the option set rather than force a premature commitment\)\(Tversky and Shafir,[1992](https://arxiv.org/html/2607.02672#bib.bib264); Dhar,[1997](https://arxiv.org/html/2607.02672#bib.bib262)\)\.

Compensatory value integration\.Although our model permits indecision whens\+​\(q\)s^\{\+\}\(q\)ands−​\(q\)s^\{\-\}\(q\)are both very high or both very low, over the remaining regimes of these scores, trade\-offscanbe adjudicated by compensatory aggregation: in such cases, the winning alternative is simply the one with the higher directional evidence score\. This captures the idea that individuals canto some degreeintegrate considerations in a compensatory way\. In the special case of zero thresholds and perfect separability, this reduces exactly to compensatory aggregation — simply deciding based on the weighted utility\-gap comparison\.

The assumption that people can compensatorily aggregate is standard in normative and empirical decision theory: multiattribute utility theory explicitly represents preferences over alternatives with multiple conflicting objectives via additive \(or additively separable\) value functions with attribute weights\(Keeney and Raiffa,[1976](https://arxiv.org/html/2607.02672#bib.bib74)\)\. In moral psychology, Guzmán et al\.’s “moral trade\-off system” similarly models judgments as selecting the most “right” option among feasible alternatives, where rightness is determined by a weighted function over moral values; they argue this framework captures coherent compromise in dilemma\-like settings\(Guzmánet al\.,[2022](https://arxiv.org/html/2607.02672#bib.bib73)\)\.

Non\-compensatory value integration\.At the same time, influential alternative theories emphasize that moral integration is oftennon\-compensatory: rather than continuously trading off all considerations via weights, agents may treat some constraints as deontological prohibitions, “protected values,” or hard side\-constraints that behave like vetoes or lexicographic priorities\(Baron and Spranca,[1997](https://arxiv.org/html/2607.02672#bib.bib78); Greene,[2008](https://arxiv.org/html/2607.02672#bib.bib314)\)\. Tetlock’s work onsacred valuesandtaboo trade\-offsalso presents a version of this idea by arguing that, in some domains, even reasoning about certain trade\-offs is socially and psychologically prohibited, because certain values are inviolable\(Tetlock,[2003](https://arxiv.org/html/2607.02672#bib.bib56)\)\. Additionally,Cushman and Greene \([2012](https://arxiv.org/html/2607.02672#bib.bib315)\)make the case that the moral domain is fundamentally different than other motivational domains due to the non\-negotiability over certain values, like not harming others – leading to compromise intractability\.

While we allow non\-compensatory value integration through conflict, our model does not represent taboo/non\-negotiable regions as categorically different objects, instead treating all priorities as commensurable objects of identical structure\. While high weights are insufficient to guarantee constraint behavior, one can approximate lexicographic reasoning by setting the priority weights to beω=\(1−∑j∈\[m−1\]ϵj,ϵ,ϵ2,…,ϵm−1\)\\omega=\(1\-\\sum\_\{j\\in\[m\-1\]\}\\epsilon^\{j\},\\epsilon,\\epsilon^\{2\},\.\.\.,\\epsilon^\{m\-1\}\)\(when the relevant nonzero utility gaps between rules are bounded away from zero, and withϵ\\epsilonchosen sufficiently small relative to that gap\), decreasing in the lexicographic order one wants to achieve\. Of course, one can always instead capture lexicographic reasoning or taboo trade\-offs in the response modelRR, the former of which we do in[DefinitionC\.6](https://arxiv.org/html/2607.02672#A3.Thmtheorem6); however, this is a non\-neutral choice, encoding the assumption that this non\-negotiability is just behavioral, and not part of one’s fundamental commitments as modeled byMM\.

#### A\.1\.5\.Separating underlying priorities and contextual response behavior

The preceding discussion examines how agents construct beliefs from priorities\. An additional challenge, documented across survey methodology and behavioral decision research, is that the judgments we elicit are often the product of interactions between such internal reasoningand the elicitation environment\. By definingMMandRRseparately, our model explicitly decouples these two elements, making clear the distinction between the individual’s underlying commitments, as represented byMM, and the context\- or procedure\-dependent process through which those commitments are expressed in observed responses, as represented byRR\. This also decouples the goals of learning a behavioral emulator versus learning the individual’s underlying beliefs,despitebehavioral distortions\.

There are some documented sources of contextual variation that could be particularly interesting to capture in extensions of our model using the distinction betweenMMandRR\. First, how agents integrate values can vary across elicitation settings: work on constructed preferences suggests that effective trade\-off weights may not be invariant across elicitation formats\(Slovic,[1995](https://arxiv.org/html/2607.02672#bib.bib141); Tverskyet al\.,[1990](https://arxiv.org/html/2607.02672#bib.bib140)\)\. In our terms, such variation could reflect changes in the commitments represented byMM, changes in the response processRR, or both\. Second, differentiation — i\.e\.,whichconsiderations enter into a judgment — can also be context\-sensitive: framing and task demands can shift which attributes are attended to or treated as relevant\(Tversky and Kahneman,[1981](https://arxiv.org/html/2607.02672#bib.bib246); Payneet al\.,[1993](https://arxiv.org/html/2607.02672#bib.bib138); Tversky,[1972](https://arxiv.org/html/2607.02672#bib.bib128); Tversky and Kahneman,[1974](https://arxiv.org/html/2607.02672#bib.bib342)\)\. Finally, underlying values and priorities can evolve through learning, persuasion, or deliberation\(Petty and Cacioppo,[1986](https://arxiv.org/html/2607.02672#bib.bib82); Fishkin and Luskin,[2005](https://arxiv.org/html/2607.02672#bib.bib83)\)\.

### A\.2\.Proof of[Theorem2\.13](https://arxiv.org/html/2607.02672#S2.Thmtheorem13)

We first derive a simplification that will be used in both directions\.

###### Lemma A\.1\.

Fix a perfectly separable pluralistic modelM∈ℳsepM\\in\\mathcal\{M\}^\{\\mathrm\{sep\}\}with constant utility gapΔjF​\(q\)=δj​\(q\)​∀F∈ℱ\\Delta^\{F\}\_\{j\}\(q\)=\\delta\_\{j\}\(q\)\\ \\forall F\\in\\mathcal\{F\}for any priorityjjand queryqq\. Then \(droppingMMfrom the notation for clarity\),

s\+​\(q\)−s−​\(q\)=∑j∈\[m\]ωj​δj​\(q\)\.s^\{\+\}\(q\)\-s^\{\-\}\(q\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\delta\_\{j\}\(q\)\.

###### Proof\.

By perfect separability, it holds that

ΔjF​\(y,y′;x\)=uj​\(Fx→y\)−uj​\(Fx→y′\)=δj​\(q\)∀F∈ℱ\.\\Delta\_\{j\}^\{F\}\(y,y^\{\\prime\};x\)=u\_\{j\}\(F\_\{x\\to y\}\)\-u\_\{j\}\(F\_\{x\\to y^\{\\prime\}\}\)=\\delta\_\{j\}\(q\)\\qquad\\forall F\\in\\mathcal\{F\}\.Likewise,

ΔjF​\(y′,y;x\)=uj​\(Fx→y′\)−uj​\(Fx→y\)=−δj​\(q\)∀F∈ℱ\.\\Delta\_\{j\}^\{F\}\(y^\{\\prime\},y;x\)=u\_\{j\}\(F\_\{x\\to y^\{\\prime\}\}\)\-u\_\{j\}\(F\_\{x\\to y\}\)=\-\\delta\_\{j\}\(q\)\\qquad\\forall F\\in\\mathcal\{F\}\.Therefore, by unanimity ofϕrules\\phi^\{\\mathrm\{rules\}\}, defining shorthandsj\+,sj−s\_\{j\}^\{\+\},s\_\{j\}^\{\-\},

sj\+​\(q\)=ϕrules​\(\{ΔjF​\(y,y′;x\)\}F∈ℱ\)=δj​\(q\),s\_\{j\}^\{\+\}\(q\)=\\phi^\{\\mathrm\{rules\}\}\\left\(\\\{\\Delta\_\{j\}^\{F\}\(y,y^\{\\prime\};x\)\\\}\_\{F\\in\\mathcal\{F\}\}\\right\)=\\delta\_\{j\}\(q\),and similarly,

sj−​\(q\)=ϕrules​\(\{ΔjF​\(y′,y;x\)\}F∈ℱ\)=−δj​\(q\)\.s\_\{j\}^\{\-\}\(q\)=\\phi^\{\\mathrm\{rules\}\}\\left\(\\\{\\Delta\_\{j\}^\{F\}\(y^\{\\prime\},y;x\)\\\}\_\{F\\in\\mathcal\{F\}\}\\right\)=\-\\delta\_\{j\}\(q\)\.Then, aggregating over priorities,

s\+​\(q\)=∑j∈\[m\]ωj​\(sj\+​\(q\)\)\+=∑j∈\[m\]ωj​\(δj​\(q\)\)\+,s^\{\+\}\(q\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\bigl\(s\_\{j\}^\{\+\}\(q\)\\bigr\)\_\{\+\}=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\bigl\(\\delta\_\{j\}\(q\)\\bigr\)\_\{\+\},while

s−​\(q\)=∑j∈\[m\]ωj​\(sj−​\(q\)\)\+=∑j∈\[m\]ωj​\(−δj​\(q\)\)\+\.s^\{\-\}\(q\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\bigl\(s\_\{j\}^\{\-\}\(q\)\\bigr\)\_\{\+\}=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\bigl\(\-\\delta\_\{j\}\(q\)\\bigr\)\_\{\+\}\.Hence,

s\+​\(q\)−s−​\(q\)=∑j∈\[m\]ωj​\[\(δj​\(q\)\)\+−\(−δj​\(q\)\)\+\]=∑j∈\[m\]ωj​δj​\(q\),\\displaystyle s^\{\+\}\(q\)\-s^\{\-\}\(q\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\left\[\\bigl\(\\delta\_\{j\}\(q\)\\bigr\)\_\{\+\}\-\\bigl\(\-\\delta\_\{j\}\(q\)\\bigr\)\_\{\+\}\\right\]=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\delta\_\{j\}\(q\),where the final equality uses\(a\)\+−\(−a\)\+=a\(a\)\_\{\+\}\-\(\-a\)\_\{\+\}=a\. ∎

###### Proof of[Theorem2\.13](https://arxiv.org/html/2607.02672#S2.Thmtheorem13)\.

Forward direction:Fix any local score functionV:X×Y→ℝV:X\\times Y\\to\\mathbb\{R\}\. Construct a priority modelℳV\\mathcal\{M\}\_\{V\}with a single priority, weight11, and utility function

uV​\(F\):=∑x∈XV​\(x,F​\(x\)\)\.u\_\{V\}\(F\):=\\sum\_\{x\\in X\}V\(x,F\(x\)\)\.For any queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)and any background ruleF∈ℱF\\in\\mathcal\{F\}, the projected rulesFx→yF\_\{x\\to y\}andFx→y′F\_\{x\\to y^\{\\prime\}\}agree at every inputx′≠xx^\{\\prime\}\\neq x\. Therefore,

ΔVF​\(y,y′;x\)=uV​\(Fx→y\)−uV​\(Fx→y′\)=V​\(x,y\)−V​\(x,y′\)\.\\displaystyle\\Delta\_\{V\}^\{F\}\(y,y^\{\\prime\};x\)=u\_\{V\}\(F\_\{x\\to y\}\)\-u\_\{V\}\(F\_\{x\\to y^\{\\prime\}\}\)=V\(x,y\)\-V\(x,y^\{\\prime\}\)\.This expression is independent ofFF, soℳV\\mathcal\{M\}\_\{V\}is perfectly separable with constant gap

δV​\(q\)=V​\(x,y\)−V​\(x,y′\)\.\\delta\_\{V\}\(q\)=V\(x,y\)\-V\(x,y^\{\\prime\}\)\.SinceℳV\\mathcal\{M\}\_\{V\}has one priority with weight11,[LemmaA\.1](https://arxiv.org/html/2607.02672#A1.Thmtheorem1)implies

s\+​\(q\)−s−​\(q\)=δV​\(q\)=V​\(x,y\)−V​\(x,y′\)\.s^\{\+\}\(q\)\-s^\{\-\}\(q\)=\\delta\_\{V\}\(q\)=V\(x,y\)\-V\(x,y^\{\\prime\}\)\.Note that whenτr=τκ=0\\tau\_\{r\}=\\tau\_\{\\kappa\}=0, it is guaranteed thatr​\(q\)≥τrr\(q\)\\geq\\tau\_\{r\}\. By the above,κ​\(q\)=s\+​\(q\)−s−​\(q\)=V​\(x,y\)−V​\(x,y′\)\\kappa\(q\)=s^\{\+\}\(q\)\-s^\{\-\}\(q\)=V\(x,y\)\-V\(x,y^\{\\prime\}\)\. Then, by[Lemma2\.11](https://arxiv.org/html/2607.02672#S2.Thmtheorem11), it holds that

Pr⁡\[Rhβ;0,0​\(q;MV\)=y≻y′\]\\displaystyle\\Pr\[R\_\{h\_\{\\beta\};\\,0,0\}\(q;M\_\{V\}\)=y\\succ y^\{\\prime\}\]=hβ​\(s\+​\(q\)−s−​\(q\)\)\\displaystyle=h\_\{\\beta\}\\\!\\left\(s^\{\+\}\(q\)\-s^\{\-\}\(q\)\\right\)=hβ​\(V​\(x,y\)−V​\(x,y′\)\)\\displaystyle=h\_\{\\beta\}\\\!\\left\(V\(x,y\)\-V\(x,y^\{\\prime\}\)\\right\)=Pr⁡\[Shβ​\(q;V\)=y≻y′\]\.\\displaystyle=\\Pr\[S\_\{h\_\{\\beta\}\}\(q;V\)=y\\succ y^\{\\prime\}\]\.The complementary response probability is therefore also identical\. The learning target correspondence holds by the fact that the following \(lettingℱ′⊆ℱ\\mathcal\{F\}^\{\\prime\}\\subseteq\\mathcal\{F\}be any subset of rules\):

∑x∈𝒳V​\(x,F​\(x\)\)=uV​\(F\)=∑j∈\[m\]ωj​uj∀F∈ℱ⟹arg⁡maxF∈ℱ′​∑x∈𝒳V​\(x,F​\(x\)\)=arg⁡maxF∈ℱ′⁡UM​\(F\)\.\\displaystyle\\sum\_\{x\\in\\mathcal\{X\}\}V\(x,F\(x\)\)=u\_\{V\}\(F\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}u\_\{j\}\\quad\\forall F\\in\\mathcal\{F\}\\implies\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}\\sum\_\{x\\in\\mathcal\{X\}\}V\(x,F\(x\)\)=\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}U\_\{M\}\(F\)\.
Reverse direction:A perfectly separable priority modelMM, and an arbitrary background ruleF0∈ℱF^\{0\}\\in\\mathcal\{F\}\. For each priorityjjand each inputxx, define

Vj​\(x,y\):=uj​\(Fx→y0\)\.V\_\{j\}\(x,y\):=u\_\{j\}\(F^\{0\}\_\{x\\to y\}\)\.Then, define the local score function as

VM​\(x,y\):=∑j∈\[m\]ωj​Vj​\(x,y\)\.V\_\{M\}\(x,y\):=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}V\_\{j\}\(x,y\)\.We will show that this Score\-RUM induces the same score gap asMMon every query\. Fix any queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)\. Since priorityjjis perfectly separable atqq, there exists a constantδj​\(q\)\\delta\_\{j\}\(q\)such that

uj​\(Fx→y\)−uj​\(Fx→y′\)=δj​\(q\)∀F∈ℱ\.u\_\{j\}\(F\_\{x\\to y\}\)\-u\_\{j\}\(F\_\{x\\to y^\{\\prime\}\}\)=\\delta\_\{j\}\(q\)\\qquad\\forall F\\in\\mathcal\{F\}\.Applying this equality to the particular background ruleF0F^\{0\}gives

Vj​\(x,y\)−Vj​\(x,y′\)\\displaystyle V\_\{j\}\(x,y\)\-V\_\{j\}\(x,y^\{\\prime\}\)=uj​\(Fx→y0\)−uj​\(Fx→y′0\)=δj​\(q\)\.\\displaystyle=u\_\{j\}\(F^\{0\}\_\{x\\to y\}\)\-u\_\{j\}\(F^\{0\}\_\{x\\to y^\{\\prime\}\}\)=\\delta\_\{j\}\(q\)\.Therefore,

VM​\(x,y\)−VM​\(x,y′\)=∑j∈\[m\]ωj​\(Vj​\(x,y\)−Vj​\(x,y′\)\)=∑j∈\[m\]ωj​δj​\(q\)=s\+​\(q\)−s−​\(q\)\.\\displaystyle V\_\{M\}\(x,y\)\-V\_\{M\}\(x,y^\{\\prime\}\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\left\(V\_\{j\}\(x,y\)\-V\_\{j\}\(x,y^\{\\prime\}\)\\right\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\delta\_\{j\}\(q\)=s^\{\+\}\(q\)\-s^\{\-\}\(q\)\.where the last step is by[LemmaA\.1](https://arxiv.org/html/2607.02672#A1.Thmtheorem1)\. Hence,

VM​\(x,y\)−VM​\(x,y′\)=s\+​\(q\)−s−​\(q\)\.V\_\{M\}\(x,y\)\-V\_\{M\}\(x,y^\{\\prime\}\)=s^\{\+\}\(q\)\-s^\{\-\}\(q\)\.This equality implies behavioral equivalence exactly as it did in the forward direction\.

Learning target correspondence\.We will show the claim that for everyjj, there exists somecj∈ℝc\_\{j\}\\in\\mathbb\{R\}such that the following is true\.

\(5\)∑x∈𝒳VM​\(x,F​\(x\)\)=∑j∈\[m\]∑x∈𝒳ωj​uj​\(Fx→F​\(x\)0\)=∑j∈\[m\]cj\+ωj​uj​\(F\)∀F∈ℱ\.\\displaystyle\\sum\_\{x\\in\\mathcal\{X\}\}V\_\{M\}\(x,F\(x\)\)=\\sum\_\{j\\in\[m\]\}\\sum\_\{x\\in\\mathcal\{X\}\}\\omega\_\{j\}u\_\{j\}\(F^\{0\}\_\{x\\to F\(x\)\}\)=\\sum\_\{j\\in\[m\]\}c\_\{j\}\+\\omega\_\{j\}u\_\{j\}\(F\)\\qquad\\forall F\\in\\mathcal\{F\}\.Which will implied the desired claim because the constant shift of∑jcj\\sum\_\{j\}c\_\{j\}doesn’t affect the optimization\.

We will construct thecjc\_\{j\}so that for alljjand for anyF∈ℱF\\in\\mathcal\{F\},

∑x∈𝒳uj​\(Fx→F​\(x\)0\)=uj​\(F\)\+cj/ωj\.\\sum\_\{x\\in\\mathcal\{X\}\}u\_\{j\}\(F^\{0\}\_\{x\\to F\(x\)\}\)=u\_\{j\}\(F\)\+c\_\{j\}/\\omega\_\{j\}\.In words, we are saying that the utility ofFFcan be constructed by starting from ruleF0F^\{0\}, and then adding up all the utility gaps for each individual output replacement at thexx’s whereFFandF0F^\{0\}differ, plus some constant shift\. Naturally, this will be a telescoping sum argument\.

Fix anyF∈ℱF\\in\\mathcal\{F\}, and let\{x1,…,xℓ\}⊆𝒳\\\{x\_\{1\},\\dots,x\_\{\\ell\}\\\}\\subseteq\\mathcal\{X\}be the subset of inputs on whichF0F^\{0\}andFFdiffer, enumerated arbitrarily \(if this set is empty, we are done\)\. Construct a sequence of rulesF0,F1,…,FℓF^\{0\},F^\{1\},\\dots,F^\{\\ell\}, whereFℓ=FF^\{\\ell\}=F, andFtF^\{t\}agrees withFFon\{x1,…,xt\}\\\{x\_\{1\},\\ldots,x\_\{t\}\\\}andF0F^\{0\}on\{xt\+1,…,xℓ\}\\\{x\_\{t\+1\},\\dots,x\_\{\\ell\}\\\}\. ThusFtF^\{t\}is obtained fromFt−1F^\{t\-1\}by changing only the value at inputxtx\_\{t\}fromF0​\(xt\)F^\{0\}\(x\_\{t\}\)toF​\(xt\)F\(x\_\{t\}\)\.

By telescoping,

uj​\(F\)−uj​\(F0\)=∑t=1ℓ\(uj​\(Ft\)−uj​\(Ft−1\)\)\.u\_\{j\}\(F\)\-u\_\{j\}\(F^\{0\}\)=\\sum\_\{t=1\}^\{\\ell\}\\left\(u\_\{j\}\(F^\{t\}\)\-u\_\{j\}\(F^\{t\-1\}\)\\right\)\.For eachtt, perfect separability implies that the effect of changing the value atxtx\_\{t\}fromF0​\(xt\)F^\{0\}\(x\_\{t\}\)toF​\(xt\)F\(x\_\{t\}\)is independent of the background rule\. Hence, we can equivalently write this utility gap as the one obtained by modifyingF0F^\{0\}at that individual input:

uj​\(Ft\)−uj​\(Ft−1\)=uj​\(Fxt→F​\(xt\)0\)−uj​\(F0\)\.u\_\{j\}\(F^\{t\}\)\-u\_\{j\}\(F^\{t\-1\}\)=u\_\{j\}\(F^\{0\}\_\{x\_\{t\}\\to F\(x\_\{t\}\)\}\)\-u\_\{j\}\(F^\{0\}\)\.Therefore,

uj​\(F\)−uj​\(F0\)=∑t=1ℓ\(uj​\(Fxt→F​\(xt\)0\)−uj​\(F0\)\)=∑x∈𝒳uj​\(Fx→F​\(x\)0\)−\|𝒳\|​uj​\(F0\)\.\\displaystyle u\_\{j\}\(F\)\-u\_\{j\}\(F^\{0\}\)=\\sum\_\{t=1\}^\{\\ell\}\\left\(u\_\{j\}\(F^\{0\}\_\{x\_\{t\}\\to F\(x\_\{t\}\)\}\)\-u\_\{j\}\(F^\{0\}\)\\right\)=\\sum\_\{x\\in\\mathcal\{X\}\}u\_\{j\}\(F^\{0\}\_\{x\\to F\(x\)\}\)\-\|\\mathcal\{X\}\|u\_\{j\}\(F^\{0\}\)\.Rearranging gives

∑x∈𝒳uj​\(Fx→F​\(x\)0\)=uj​\(F\)\+\(\|𝒳\|−1\)​uj​\(F0\)\.\\sum\_\{x\\in\\mathcal\{X\}\}u\_\{j\}\(F^\{0\}\_\{x\\to F\(x\)\}\)=u\_\{j\}\(F\)\+\(\|\\mathcal\{X\}\|\-1\)u\_\{j\}\(F^\{0\}\)\.Thus, we setcj=\(\|𝒳\|−1\)​uj​\(F0\)​ωjc\_\{j\}=\(\|\\mathcal\{X\}\|\-1\)u\_\{j\}\(F^\{0\}\)\\omega\_\{j\}, which is a constant inFFas desired, and we conclude the desired equality\.

∑x∈𝒳Vj​\(x,F​\(x\)\)=uj​\(F\)\+cj/ωj\.\\sum\_\{x\\in\\mathcal\{X\}\}V\_\{j\}\(x,F\(x\)\)=u\_\{j\}\(F\)\+c\_\{j\}/\\omega\_\{j\}\.Putting it all together, we get that

∑x∈𝒳VM​\(x,F​\(x\)\)=∑j∈\[m\]ωj​uj​\(F\)\+∑j∈\[m\]ωj​cj,\\displaystyle\\sum\_\{x\\in\\mathcal\{X\}\}V\_\{M\}\(x,F\(x\)\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}u\_\{j\}\(F\)\+\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}c\_\{j\},where the final term is constant inFF, and thus does not affect the optimization\. We conclude that for anyℱ′⊆ℱ\\mathcal\{F\}^\{\\prime\}\\subseteq\\mathcal\{F\},

arg⁡maxF∈ℱ′​∑x∈𝒳VM​\(x,F​\(x\)\)=arg⁡maxF∈ℱ′​∑j∈\[m\]ωj​uj​\(F\)\.∎\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}\\sum\_\{x\\in\\mathcal\{X\}\}V\_\{M\}\(x,F\(x\)\)=\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}u\_\{j\}\(F\)\.\\qed

## Appendix BSupplemental Materials from[Section3](https://arxiv.org/html/2607.02672#S3)

### B\.1\.Proof of[Proposition3\.3](https://arxiv.org/html/2607.02672#S3.Thmtheorem3)

###### Proof\.

Fix any twoa,b∈Na,b\\in Nwherea≠ba\\neq b, and fix an arbitrary queryq=\(a,b;\(a,b;k\)\)q=\(a,b;\(a,b;k\)\), wherex∗=\(a,b;k\)x^\{\*\}=\(a,b;k\)\. To prove inseparability, we must simply construct two background rulesFaF^\{a\},FbF^\{b\}such that

Fx∗→aa≻egalFx∗→baandFx∗→bb≻egalFx∗→ab,F^\{a\}\_\{x^\{\*\}\\to a\}\\succ\_\{\\mathrm\{egal\}\}F^\{a\}\_\{x^\{\*\}\\to b\}\\qquad\\text\{and\}\\qquad F^\{b\}\_\{x^\{\*\}\\to b\}\\succ\_\{\\mathrm\{egal\}\}F^\{b\}\_\{x^\{\*\}\\to a\},i\.e\., who we allocate to atx∗x^\{\*\}according to the egalitarian priority depends on the background rule\. We will prove that we can construct such anFaF^\{a\}; by symmetry, the same proof demonstrates that we can construct such anFbF^\{b\}\.

LetFaF^\{a\}be any rule that never allocates toaaat any otherx∈𝒳∖\{x∗\}x\\in\\mathcal\{X\}\\setminus\\\{x^\{\*\}\\\}, but does allocate to allc∈N∖\{a\}c\\in N\\setminus\\\{a\\\}at least once\. This is possible because\|N\|≥3\|N\|\\geq 3: in all other queries, eitheraadoes not appear, oraadoes appear butFaF^\{a\}can allocate to another recipient\.

By the assumption that𝒟\\mathcal\{D\}has full support, i\.e\., everyxxoccurs with positive probability, it must be that under either projectionFx∗→aaF^\{a\}\_\{x^\{\*\}\\to a\}orFx∗→baF^\{a\}\_\{x^\{\*\}\\to b\}, all other recipients receive nonzero benefit:

𝔼x∼𝒟​\[vc​\(x,Fx∗→ja​\(x\)\)\]\>0for all​c∈N∖\{a\},j∈\{a,b\}\.\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\}\\left\[v\_\{c\}\\left\(x,F^\{a\}\_\{x^\{\*\}\\to j\}\(x\)\\right\)\\right\]\>0\\qquad\\text\{for all \}c\\in N\\setminus\\\{a\\\},\\ j\\in\\\{a,b\\\}\.However, becauseaanever receives a good at anyx≠x∗x\\neq x^\{\*\}, whetheraa’s benefit is0is entirely dictated byFaF^\{a\}’s behavior atx∗x^\{\*\}:

𝔼x∼𝒟​\[va​\(x,Fx∗→ba​\(x\)\)\]=0and𝔼x∼𝒟​\[va​\(x,Fx∗→aa​\(x\)\)\]\>0\.\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\}\\left\[v\_\{a\}\\left\(x,F^\{a\}\_\{x^\{\*\}\\to b\}\(x\)\\right\)\\right\]=0\\qquad\\text\{and\}\\qquad\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\}\\left\[v\_\{a\}\\left\(x,F^\{a\}\_\{x^\{\*\}\\to a\}\(x\)\\right\)\\right\]\>0\.This means thatFaF^\{a\}makesaathe most shortchanged recipient at the query, and thus whether they receive the good atx∗x^\{\*\}is binding for the egalitarian priority:

uegal​\(Fx∗→ba\)=0anduegal​\(Fx∗→aa\)\>0\.u\_\{\\mathrm\{egal\}\}\\left\(F^\{a\}\_\{x^\{\*\}\\to b\}\\right\)=0\\qquad\\text\{and\}\\qquad u\_\{\\mathrm\{egal\}\}\\left\(F^\{a\}\_\{x^\{\*\}\\to a\}\\right\)\>0\.This implies the following, as needed:

Fx∗→aa≻egalFx∗→ba\.∎F^\{a\}\_\{x^\{\*\}\\to a\}\\succ\_\{\\mathrm\{egal\}\}F^\{a\}\_\{x^\{\*\}\\to b\}\.\\qed

### B\.2\.Perfect Inseparability of Proportionality and Equal Treatment

###### Proposition B\.1\.

For certain𝒳,𝒴\\mathcal\{X\},\\mathcal\{Y\},Proportionalitycan be perfectly inseparable on all local pairwise comparison queriesq∈𝒬pcq\\in\\mathcal\{Q\}^\{\\text\{pc\}\}\.

###### Proof\.

Consider the example from the proof of[Proposition3\.4](https://arxiv.org/html/2607.02672#S3.Thmtheorem4), where there are two groups: group 1 is\{a\}\\\{a\\\}, group 2 is\{b\}\\\{b\\\}andα1=α2=1/2\\alpha\_\{1\}=\\alpha\_\{2\}=1/2\. Then,

uprop​\(Fa​b\)=uprop​\(Fb​a\)=0,uprop​\(Fa​a\)=uprop​\(Fb​b\)=−1/2\.u\_\{\\text\{prop\}\}\(F^\{ab\}\)=u\_\{\\text\{prop\}\}\(F^\{ba\}\)=0,\\quad u\_\{\\text\{prop\}\}\(F^\{aa\}\)=u\_\{\\text\{prop\}\}\(F^\{bb\}\)=\-1/2\.These utilities have exactly the same structure as the utilities in[Proposition3\.4](https://arxiv.org/html/2607.02672#S3.Thmtheorem4), creating exactly the same implementation of perfect inseparability \([Definition3\.1](https://arxiv.org/html/2607.02672#S3.Thmtheorem1)\) withδ=1/2\\delta=1/2\. ∎

###### Proposition B\.2\.

For certain𝒳,𝒴\\mathcal\{X\},\\mathcal\{Y\},Equal Treatmentcan be perfectly inseparable on all local pairwise comparison queriesq∈𝒬pcq\\in\\mathcal\{Q\}^\{\\text\{pc\}\}\.

###### Proof\.

Fix three recipientsN=\{a,a′,b\}N=\\\{a,a^\{\\prime\},b\\\}, wherea∈G1a\\in G\_\{1\},a′∈G2a^\{\\prime\}\\in G\_\{2\}, anda,a′a,a^\{\\prime\}are otherwise identical\. Recipientbbis not the protected counterpart of eitheraaora′a^\{\\prime\}\. Consider an allocation task with two inputs,

𝒳=\{xa​b,xa′​b\},\\mathcal\{X\}=\\\{x^\{ab\},x^\{a^\{\\prime\}b\}\\\},where

𝒴​\(xa​b\)=\{a,b\}and𝒴​\(xa′​b\)=\{a′,b\}\.\\mathcal\{Y\}\(x^\{ab\}\)=\\\{a,b\\\}\\qquad\\text\{and\}\\qquad\\mathcal\{Y\}\(x^\{a^\{\\prime\}b\}\)=\\\{a^\{\\prime\},b\\\}\.The inputsxa​bx^\{ab\}andxa′​bx^\{a^\{\\prime\}b\}are counterparts: the former contains recipientaa, while the latter replacesaaby its counterparta′a^\{\\prime\}, leavingbbunchanged\.

GivenG1,G2G\_\{1\},G\_\{2\}, the equal\-treatment priority is then defined by

ueq​\(F\)=𝟏​\{𝟏​\{F​\(xa​b\)=b\}=𝟏​\{F​\(xa′​b\)=b\}\}\.u\_\{\\mathrm\{eq\}\}\(F\)=\\mathbf\{1\}\\left\\\{\\mathbf\{1\}\\\{F\(x^\{ab\}\)=b\\\}=\\mathbf\{1\}\\\{F\(x^\{a^\{\\prime\}b\}\)=b\\\}\\right\\\}\.Thus the priority is satisfied exactly when the rule either choosesaaatxa​bx^\{ab\}anda′a^\{\\prime\}atxa′​bx^\{a^\{\\prime\}b\}, or choosesbbat both inputs\.

There are four deterministic rules, where the superscript records the outputs atxa​bx^\{ab\}andxa′​bx^\{a^\{\\prime\}b\}, respectively:

ℱ=\{Fa​a′,Fa​b,Fb​a′,Fb​b\},\\mathcal\{F\}=\\\{F^\{aa^\{\\prime\}\},F^\{ab\},F^\{ba^\{\\prime\}\},F^\{bb\}\\\},Hence,

ueq​\(Fa​a′\)=ueq​\(Fb​b\)=1,ueq​\(Fa​b\)=ueq​\(Fb​a′\)=0\.u\_\{\\mathrm\{eq\}\}\(F^\{aa^\{\\prime\}\}\)=u\_\{\\mathrm\{eq\}\}\(F^\{bb\}\)=1,\\qquad u\_\{\\mathrm\{eq\}\}\(F^\{ab\}\)=u\_\{\\mathrm\{eq\}\}\(F^\{ba^\{\\prime\}\}\)=0\.
We first consider the query

q1=\(a,b;xa​b\)\.q\_\{1\}=\(a,b;x^\{ab\}\)\.For any background ruleFF, the value of the projection atxa​bx^\{ab\}depends only on the rule’s output at the counterpart inputxa′​bx^\{a^\{\\prime\}b\}\. IfF​\(xa′​b\)=a′F\(x^\{a^\{\\prime\}b\}\)=a^\{\\prime\}, then choosingaaatxa​bx^\{ab\}satisfies equal treatment, while choosingbbviolates it\. Thus

ueq​\(Fxa​b→a\)−ueq​\(Fxa​b→b\)=1\.u\_\{\\mathrm\{eq\}\}\(F\_\{x^\{ab\}\\to a\}\)\-u\_\{\\mathrm\{eq\}\}\(F\_\{x^\{ab\}\\to b\}\)=1\.IfF​\(xa′​b\)=bF\(x^\{a^\{\\prime\}b\}\)=b, then choosingbbatxa​bx^\{ab\}satisfies equal treatment, while choosingaaviolates it\. Thus

ueq​\(Fxa​b→a\)−ueq​\(Fxa​b→b\)=−1\.u\_\{\\mathrm\{eq\}\}\(F\_\{x^\{ab\}\\to a\}\)\-u\_\{\\mathrm\{eq\}\}\(F\_\{x^\{ab\}\\to b\}\)=\-1\.Therefore, applying[Definition3\.1](https://arxiv.org/html/2607.02672#S3.Thmtheorem1), this priority is perfectly inseparable atq1q\_\{1\}withδ=1\\delta=1and the equal\-size partition

ℱa=\{Fa​a′,Fb​a′\},ℱb=\{Fa​b,Fb​b\}\.\\mathcal\{F\}\_\{a\}=\\\{F^\{aa^\{\\prime\}\},F^\{ba^\{\\prime\}\}\\\},\\qquad\\mathcal\{F\}\_\{b\}=\\\{F^\{ab\},F^\{bb\}\\\}\.
The argument for the query

q2=\(a′,b;xa′​b\)q\_\{2\}=\(a^\{\\prime\},b;x^\{a^\{\\prime\}b\}\)is symmetric\. IfF​\(xa​b\)=aF\(x^\{ab\}\)=a, then choosinga′a^\{\\prime\}atxa′​bx^\{a^\{\\prime\}b\}satisfies equal treatment, while choosingbbviolates it\. IfF​\(xa​b\)=bF\(x^\{ab\}\)=b, then choosingbbsatisfies equal treatment, while choosinga′a^\{\\prime\}violates it\. Henceq2q\_\{2\}hasδ=1\\delta=1and the equal\-size partition

ℱa′=\{Fa​a′,Fa​b\},ℱb=\{Fb​a′,Fb​b\}\.\\mathcal\{F\}\_\{a^\{\\prime\}\}=\\\{F^\{aa^\{\\prime\}\},F^\{ab\}\\\},\\qquad\\mathcal\{F\}\_\{b\}=\\\{F^\{ba^\{\\prime\}\},F^\{bb\}\\\}\.Thus equal treatment is perfectly inseparable at every local pairwise comparison query in this instance\. ∎

### B\.3\.Proof of[Theorem3\.11](https://arxiv.org/html/2607.02672#S3.Thmtheorem11)

We begin by stating a useful lemma:

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

Suppose priorityjjis perfectly inseparable atq=\(y,y′;x\)q=\(y,y^\{\\prime\};\\,x\)\. Letϕrules\\phi^\{\\text\{rules\}\}be permutation\-invariant; thenjjcontributes equal evidence in both directions:

ϕrules​\(\{ΔjF​\(y,y′;x\)\}F∈ℱ\)=ϕrules​\(\{ΔjF​\(y′,y;x\)\}F∈ℱ\)\.\\phi^\{\\text\{rules\}\}\(\\\{\\Delta\_\{j\}^\{F\}\(y,y^\{\\prime\};x\)\\\}\_\{F\\in\\mathcal\{F\}\}\)=\\phi^\{\\text\{rules\}\}\(\\\{\\Delta\_\{j\}^\{F\}\(y^\{\\prime\},y;x\)\\\}\_\{F\\in\\mathcal\{F\}\}\)\.

###### Proof\.

By perfect inseparability, the evidence multisets in the two directions are

\{ΔjF​\(y,y′;x\)\}F∈ℱ\\displaystyle\\left\\\{\\Delta^\{F\}\_\{j\}\(y,y^\{\\prime\};\\,x\)\\right\\\}\_\{F\\in\\mathcal\{F\}\}=\{δ,…,δ⏟\|ℱy\|,−δ,…,−δ⏟\|ℱy′\|\},\{ΔjF​\(y′,y;x\)\}F∈ℱ=\{−δ,…,−δ⏟\|ℱy\|,δ,…,δ⏟\|ℱy′\|\}\.\\displaystyle\\;=\\;\\\{\\underbrace\{\\delta,\\ldots,\\delta\}\_\{\|\\mathcal\{F\}\_\{y\}\|\},\\;\\underbrace\{\-\\delta,\\ldots,\-\\delta\}\_\{\|\\mathcal\{F\}\_\{y^\{\\prime\}\}\|\}\\\},\\qquad\\left\\\{\\Delta^\{F\}\_\{j\}\(y^\{\\prime\},y;\\,x\)\\right\\\}\_\{F\\in\\mathcal\{F\}\}\\;=\\;\\\{\\underbrace\{\-\\delta,\\ldots,\-\\delta\}\_\{\|\\mathcal\{F\}\_\{y\}\|\},\\;\\underbrace\{\\delta,\\ldots,\\delta\}\_\{\|\\mathcal\{F\}\_\{y^\{\\prime\}\}\|\}\\\}\.Since\|ℱy\|=\|ℱy′\|\|\\mathcal\{F\}\_\{y\}\|=\|\\mathcal\{F\}\_\{y^\{\\prime\}\}\|, these two multisets are identical, so the permutation\-invariance ofϕrules\\phi^\{\\text\{rules\}\}implies the claim\. ∎

General priority aggregators\.We will prove the claim for more generic priority aggregators here, so now we define them formally\. Apriority aggregatoris a map

ϕpriorities:ℝm×Δm−1⟶ℝ,\\phi^\{\\text\{priorities\}\}:\\ \\mathbb\{R\}^\{m\}\\times\\Delta^\{m\-1\}\\ \\longrightarrow\\ \\mathbb\{R\},which takes in a vector ofmmper\-priority evidence values \(the output ofϕrules\\phi^\{\\text\{rules\}\}on every priority\) and a weight vector and outputs a single nonnegative directional score\. Formally, given a rule aggregatorϕrules\\phi^\{\\mathrm\{rules\}\}, a modelM=\(u,ω\)M=\(u,\\omega\), and a queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\), define the per\-priority rule\-aggregated evidence in each direction as

sj\+​\(q\):=ϕrules​\(\(ΔjF​\(y,y′;x\)\)F∈ℱ\),sj−​\(q\):=ϕrules​\(\(ΔjF​\(y′,y;x\)\)F∈ℱ\),s^\{\+\}\_\{j\}\(q\)\\ :=\\ \\phi^\{\\mathrm\{rules\}\}\\\!\\big\(\(\\Delta^\{F\}\_\{j\}\(y,y^\{\\prime\};x\)\)\_\{F\\in\\mathcal\{F\}\}\\big\),\\qquad s^\{\-\}\_\{j\}\(q\)\\ :=\\ \\phi^\{\\mathrm\{rules\}\}\\\!\\big\(\(\\Delta^\{F\}\_\{j\}\(y^\{\\prime\},y;x\)\)\_\{F\\in\\mathcal\{F\}\}\\big\),and collect𝐬±​\(q\):=\(sj±​\(q\)\)j∈\[m\]\\mathbf\{s\}^\{\\pm\}\(q\):=\(s^\{\\pm\}\_\{j\}\(q\)\)\_\{j\\in\[m\]\}\. The two directional evidence scores are obtained by applyingϕpriorities\\phi^\{\\text\{priorities\}\}in each direction:

sM\+​\(q\):=ϕpriorities​\(𝐬\+​\(q\),ω\),sM−​\(q\):=ϕpriorities​\(𝐬−​\(q\),ω\)\.s^\{\+\}\_\{M\}\(q\)\\ :=\\ \\phi^\{\\text\{priorities\}\}\\\!\\big\(\\mathbf\{s\}^\{\+\}\(q\),\\,\\omega\\big\),\\qquad s^\{\-\}\_\{M\}\(q\)\\ :=\\ \\phi^\{\\text\{priorities\}\}\\\!\\big\(\\mathbf\{s\}^\{\-\}\(q\),\\,\\omega\\big\)\.Note that this construction is a generalization of the linear aggregator, in whichϕpriorities\\phi^\{\\text\{priorities\}\}is linear inω\\omega\.

Now we define the more general class of priority aggregators for which the theorem will hold:

###### Definition B\.0 \(Scale\-preservation\)\.

A priority aggregatorϕpriorities\\phi^\{\\text\{priorities\}\}satisfies scale\-preservation iff, for every priorityj∗j^\{\*\}and every weight vectorω∈Δm−1\\omega\\in\\Delta^\{m\-1\}, there exist a constantc​\(ω,j∗\)\>0c\(\\omega,j^\{\*\}\)\>0and a functionAω,j∗:ℝ→ℝA\_\{\\omega,j^\{\*\}\}:\\mathbb\{R\}\\to\\mathbb\{R\}such that, for every evidence vector𝐳∈ℝm\\mathbf\{z\}\\in\\mathbb\{R\}^\{m\},

ϕpriorities​\(z,ω\)=c​\(ω,j∗\)​ϕpriorities​\(𝐳−j∗,ω−j∗\)\+Aω,j∗​\(zj∗\)\.\\phi^\{\\text\{priorities\}\}\(z,\\omega\)=c\(\\omega,j^\{\*\}\)\\,\\phi^\{\\text\{priorities\}\}\(\\mathbf\{z\}\_\{\-j^\{\*\}\},\\omega\_\{\-j^\{\*\}\}\)\+A\_\{\\omega,j^\{\*\}\}\(z\_\{j^\{\*\}\}\)\.In words, the effect of a priority can be separated from the aggregate contribution of the remaining priorities: adding priorityj∗j^\{\*\}may add its own evidence term and may even rescale the contribution of all other priorities, but it cannot change how the remaining priorities trade off against one another\. Note that by linearity, the linear priority aggregator satisfies this criterion\.

Now, we prove the theorem for scale\-preserving priority aggregators\.

###### Proof of[Theorem3\.11](https://arxiv.org/html/2607.02672#S3.Thmtheorem11)\.

We first prove the claim for the removal of a single perfectly inseparable priorityj∗∈Jinsepj^\{\*\}\\in J\_\{\\mathrm\{insep\}\}\. The result for the full setJinsepJ\_\{\\mathrm\{insep\}\}follows by iterating the same argument\.

Fix any local pairwise queryq=\(y,y′;x\)∈𝒬p​cq=\(y,y^\{\\prime\};x\)\\in\\mathcal\{Q\}^\{pc\}\. For each priorityjj, define shorthand for the output ofϕrules\\phi^\{\\text\{rules\}\}:

sj\+​\(q\):=ϕrules​\(\{ΔjF​\(y,y′;x\)\}F∈ℱ\),sj−​\(q\):=ϕrules​\(\{ΔjF​\(y′,y;x\)\}F∈ℱ\)\.s^\{\+\}\_\{j\}\(q\):=\\phi^\{\\text\{rules\}\}\\\!\\left\(\\\{\\Delta^\{F\}\_\{j\}\(y,y^\{\\prime\};x\)\\\}\_\{F\\in\\mathcal\{F\}\}\\right\),\\qquad s^\{\-\}\_\{j\}\(q\):=\\phi^\{\\text\{rules\}\}\\\!\\left\(\\\{\\Delta^\{F\}\_\{j\}\(y^\{\\prime\},y;x\)\\\}\_\{F\\in\\mathcal\{F\}\}\\right\)\.Summarize these values in

𝐬\+​\(q\):=\(sj\+​\(q\)\)j∈J,𝐬−​\(q\):=\(sj−​\(q\)\)j∈J\.\\mathbf\{s\}^\{\+\}\(q\):=\(s^\{\+\}\_\{j\}\(q\)\)\_\{j\\in J\},\\qquad\\mathbf\{s\}^\{\-\}\(q\):=\(s^\{\-\}\_\{j\}\(q\)\)\_\{j\\in J\}\.Then the two directional evidence scores induced byMinsepM\_\{\\mathrm\{insep\}\}are

sMinsep​\(y,y′;x\)=ϕpriorities​\(𝐬\+​\(q\),ω\),sMinsep​\(y′,y;x\)=ϕpriorities​\(𝐬−​\(q\),ω\)\.s\_\{M\_\{\\mathrm\{insep\}\}\}\(y,y^\{\\prime\};x\)=\\phi^\{\\text\{priorities\}\}\(\\mathbf\{s\}^\{\+\}\(q\),\\omega\),\\qquad s\_\{M\_\{\\mathrm\{insep\}\}\}\(y^\{\\prime\},y;x\)=\\phi^\{\\text\{priorities\}\}\(\\mathbf\{s\}^\{\-\}\(q\),\\omega\)\.
By scale preservation ofϕpriorities\\phi^\{\\text\{priorities\}\}, there exist a constantc​\(ω,j∗\)\>0c\(\\omega,j^\{\*\}\)\>0and a functionAω,j∗:ℝ→ℝA\_\{\\omega,j^\{\*\}\}:\\mathbb\{R\}\\to\\mathbb\{R\}such that, writingc=c​\(ω,j∗\)c=c\(\\omega,j^\{\*\}\)andA=Aω,j∗A=A\_\{\\omega,j^\{\*\}\},

ϕpriorities​\(𝐬\+​\(q\),ω\)=c​ϕpriorities​\(𝐬−j∗\+​\(q\),ω−j∗\)\+A​\(sj∗\+​\(q\)\),\\phi^\{\\text\{priorities\}\}\(\\mathbf\{s\}^\{\+\}\(q\),\\omega\)=c\\,\\phi^\{\\text\{priorities\}\}\(\\mathbf\{s\}^\{\+\}\_\{\-j^\{\*\}\}\(q\),\\omega\_\{\-j^\{\*\}\}\)\+A\(s^\{\+\}\_\{j^\{\*\}\}\(q\)\),and

ϕpriorities​\(𝐬−​\(q\),ω\)=c​ϕpriorities​\(𝐬−j∗−​\(q\),ω−j∗\)\+A​\(sj∗−​\(q\)\)\.\\phi^\{\\text\{priorities\}\}\(\\mathbf\{s\}^\{\-\}\(q\),\\omega\)=c\\,\\phi^\{\\text\{priorities\}\}\(\\mathbf\{s\}^\{\-\}\_\{\-j^\{\*\}\}\(q\),\\omega\_\{\-j^\{\*\}\}\)\+A\(s^\{\-\}\_\{j^\{\*\}\}\(q\)\)\.By[LemmaB\.3](https://arxiv.org/html/2607.02672#A2.Thmtheorem3), sincej∗j^\{\*\}is perfectly inseparable atqq,

sj∗\+​\(q\)=sj∗−​\(q\)\.s^\{\+\}\_\{j^\{\*\}\}\(q\)=s^\{\-\}\_\{j^\{\*\}\}\(q\)\.Thus the twoA​\(⋅\)A\(\\cdot\)terms cancel from the score gap, giving

sMinsep​\(y,y′;x\)−sMinsep​\(y′,y;x\)=c​\(ϕpriorities​\(𝐬−j∗\+​\(q\),ω−j∗\)−ϕpriorities​\(𝐬−j∗−​\(q\),ω−j∗\)\)\.\\displaystyle s\_\{M\_\{\\mathrm\{insep\}\}\}\(y,y^\{\\prime\};x\)\-s\_\{M\_\{\\mathrm\{insep\}\}\}\(y^\{\\prime\},y;x\)=c\\left\(\\phi^\{\\text\{priorities\}\}\(\\mathbf\{s\}^\{\+\}\_\{\-j^\{\*\}\}\(q\),\\omega\_\{\-j^\{\*\}\}\)\-\\phi^\{\\text\{priorities\}\}\(\\mathbf\{s\}^\{\-\}\_\{\-j^\{\*\}\}\(q\),\\omega\_\{\-j^\{\*\}\}\)\\right\)\.The expression in parentheses is exactly the score gap induced by the modelM−j∗M^\{\-j^\{\*\}\}\. Hence, for every local pairwise queryqq,

sMinsep​\(y,y′;x\)−sMinsep​\(y′,y;x\)=c​\(sM−j∗​\(y,y′;x\)−sM−j∗​\(y′,y;x\)\)\.s\_\{M\_\{\\mathrm\{insep\}\}\}\(y,y^\{\\prime\};x\)\-s\_\{M\_\{\\mathrm\{insep\}\}\}\(y^\{\\prime\},y;x\)=c\\left\(s\_\{M^\{\-j^\{\*\}\}\}\(y,y^\{\\prime\};x\)\-s\_\{M^\{\-j^\{\*\}\}\}\(y^\{\\prime\},y;x\)\\right\)\.
Now fix any inverse\-temperature parameterβ\>0\\beta\>0forMinsepM\_\{\\mathrm\{insep\}\}, and define

β′:=β​c\.\\beta^\{\\prime\}:=\\beta\\,c\.Sincec\>0c\>0, we haveβ′\>0\\beta^\{\\prime\}\>0\. Therefore, for anyhhand every local pairwise queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\),

Rhβ;0,0∘​\(q;Minsep\)​\(y≻y′\)\\displaystyle R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\(q;M\_\{\\mathrm\{insep\}\}\)\(y\\succ y^\{\\prime\}\)=hβ​\(sMinsep​\(y,y′;x\)−sMinsep​\(y′,y;x\)\)\\displaystyle=h\_\{\\beta\}\\\!\\left\(s\_\{M\_\{\\mathrm\{insep\}\}\}\(y,y^\{\\prime\};x\)\-s\_\{M\_\{\\mathrm\{insep\}\}\}\(y^\{\\prime\},y;x\)\\right\)=h​\(β​c​\[sM−j∗​\(y,y′;x\)−sM−j∗​\(y′,y;x\)\]\)\\displaystyle=h\\\!\\left\(\\beta c\\left\[s\_\{M^\{\-j^\{\*\}\}\}\(y,y^\{\\prime\};x\)\-s\_\{M^\{\-j^\{\*\}\}\}\(y^\{\\prime\},y;x\)\\right\]\\right\)=hβ′​\(sM−j∗​\(y,y′;x\)−sM−j∗​\(y′,y;x\)\)\\displaystyle=h\_\{\\beta^\{\\prime\}\}\\\!\\left\(s\_\{M^\{\-j^\{\*\}\}\}\(y,y^\{\\prime\};x\)\-s\_\{M^\{\-j^\{\*\}\}\}\(y^\{\\prime\},y;x\)\\right\)=Rhβ′;0,0∘​\(q;M−j∗\)​\(y≻y′\)\.\\displaystyle=R^\{\\circ\}\_\{h\_\{\\beta^\{\\prime\}\};0,0\}\(q;M^\{\-j^\{\*\}\}\)\(y\\succ y^\{\\prime\}\)\.Under the zero\-threshold response model, the remaining probability mass is assigned to the complementary responsey′≻yy^\{\\prime\}\\succ y, so the full response distributions are identical at every local pairwise query\. HenceMinsepM\_\{\\mathrm\{insep\}\}andM−j∗M^\{\-j^\{\*\}\}are scale\-free indistinguishable\.

Finally, repeat the argument for each priority inJinsepJ\_\{\\mathrm\{insep\}\}\. This yields a positive constantC\>0C\>0such that, for every local pairwise queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\),

sMinsep​\(y,y′;x\)−sMinsep​\(y′,y;x\)=C​\(sM​\(y,y′;x\)−sM​\(y′,y;x\)\),s\_\{M\_\{\\mathrm\{insep\}\}\}\(y,y^\{\\prime\};x\)\-s\_\{M\_\{\\mathrm\{insep\}\}\}\(y^\{\\prime\},y;x\)=C\\left\(s\_\{M\}\(y,y^\{\\prime\};x\)\-s\_\{M\}\(y^\{\\prime\},y;x\)\\right\),whereM=M−JinsepM=M^\{\-J\_\{\\mathrm\{insep\}\}\}\. Takingβ′′=β​C\\beta^\{\\prime\\prime\}=\\beta C, we obtain

Rhβ;0,0∘​\(q;Minsep\)≡Rhβ′′;0,0∘​\(q;M\)∀q∈𝒬p​c\.R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\(q;M\_\{\\mathrm\{insep\}\}\)\\equiv R^\{\\circ\}\_\{h\_\{\\beta^\{\\prime\\prime\}\};0,0\}\(q;M\)\\qquad\\forall q\\in\\mathcal\{Q\}^\{pc\}\.ThusMinsepM\_\{\\mathrm\{insep\}\}andMMare scale\-free indistinguishable\. ∎

### B\.4\.Formalization of[Example3\.13](https://arxiv.org/html/2607.02672#S3.Thmtheorem13)

Consider again the two\-input allocation task from[Proposition3\.4](https://arxiv.org/html/2607.02672#S3.Thmtheorem4): there are two recipientsN=\{a,b\}N=\\\{a,b\\\}, two inputsx1=\(a,b;k1\)x\_\{1\}=\(a,b;k\_\{1\}\)andx2=\(a,b;k2\)x\_\{2\}=\(a,b;k\_\{2\}\), and𝒴​\(x1\)=𝒴​\(x2\)=\{a,b\}\\mathcal\{Y\}\(x\_\{1\}\)=\\mathcal\{Y\}\(x\_\{2\}\)=\\\{a,b\\\}\.𝒟\\mathcal\{D\}is such that both inputs occur with equal probability, and suppose both recipients have the same benefit for either good, and no benefit if they aren’t given a good:

vi​\(x,j\)=𝟏​\{j=i\}for all​i,j∈\{a,b\}\.v\_\{i\}\(x,j\)=\\mathbf\{1\}\\\{j=i\\\}\\qquad\\text\{for all \}\\ i,j\\in\\\{a,b\\\}\.As usual, writeℱ=\{Fa​a,Fa​b,Fb​a,Fb​b\},\\mathcal\{F\}=\\\{F^\{aa\},F^\{ab\},F^\{ba\},F^\{bb\}\\\},whereFi​j​\(x1\)=iF^\{ij\}\(x\_\{1\}\)=iandFi​j​\(x2\)=jF^\{ij\}\(x\_\{2\}\)=j\.

Now, we letMinsepM\_\{\\mathrm\{insep\}\}have two priorities\. The first isEgalitarianism\([Definition3\.2](https://arxiv.org/html/2607.02672#S3.Thmtheorem2)\), which we know is perfectly inseparable in this example\. The second is theFamilypriority \([Example2\.3](https://arxiv.org/html/2607.02672#S2.Thmtheorem3)\), which rewards giving the good to family members\. Lettingaabe a family member andbbbe a non\-family member, the utility function for the family priority becomes

ufamily​\(F\)=𝔼x∼𝒟​\[𝟏​\{F​\(x\)=a\}\]\.u\_\{\\mathrm\{family\}\}\(F\)=\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\}\\left\[\\mathbf\{1\}\\\{F\(x\)=a\\\}\\right\]\.Note that the family priority is perfectly separable\.222222For any queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\),ΔfamilyF​\(q\)=𝒟​\(x\)​\(𝟏​\{y=a\}−𝟏​\{y′=a\}\)\\Delta^\{F\}\_\{\\mathrm\{family\}\}\(q\)=\\mathcal\{D\}\(x\)\\left\(\\mathbf\{1\}\\\{y=a\\\}\-\\mathbf\{1\}\\\{y^\{\\prime\}=a\\\}\\right\)is independent of the background ruleFF\.Then, by[Corollary3\.12](https://arxiv.org/html/2607.02672#S3.Thmtheorem12), the reduced modelMfamily=\(Minsep\)−egalM\_\{\\mathrm\{family\}\}=\(M\_\{\\mathrm\{insep\}\}\)^\{\-\\mathrm\{egal\}\}, which consists only of the family priority with weight11, is a perfect separable rationalization ofMinsepM\_\{\\mathrm\{insep\}\}\. Thus, the set of perfectly separable rationalizations is non\-empty, and on an exhaustive transcript generated byRhβ;0,0∘​\(⋅;Minsep\)R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\(\\cdot;M\_\{\\mathrm\{insep\}\}\), any separable\-consistent decoder returns some modelM~∈PSR​\(Minsep\)\.\\widetilde\{M\}\\in\\mathrm\{PSR\}\(M\_\{\\mathrm\{insep\}\}\)\.

Finally, suppose that the system outputs an aggregate\-optimal rule according toM~\\widetilde\{M\}, which must coincide with the aggregate\-optimal rules forMfamilyM\_\{\\mathrm\{family\}\}\([LemmaB\.5](https://arxiv.org/html/2607.02672#A2.Thmtheorem5)\)\.232323By[LemmaB\.5](https://arxiv.org/html/2607.02672#A2.Thmtheorem5)\(below\), every perfect separable rationalization ofMinsepM\_\{\\mathrm\{insep\}\}induces the same aggregate\-optimal rules\.SinceMfamilyM\_\{\\mathrm\{family\}\}contains only the family priority,

ufamily​\(Fa​a\)=1,ufamily​\(Fa​b\)=ufamily​\(Fb​a\)=12,ufamily​\(Fb​b\)=0\.u\_\{\\mathrm\{family\}\}\(F^\{aa\}\)=1,\\qquad u\_\{\\mathrm\{family\}\}\(F^\{ab\}\)=u\_\{\\mathrm\{family\}\}\(F^\{ba\}\)=\\frac\{1\}\{2\},\\qquad u\_\{\\mathrm\{family\}\}\(F^\{bb\}\)=0\.Therefore the system uniquely selectsFa​aF^\{aa\}, the rule that always allocates to the family memberaa\.

Whenωegal\\omega\_\{\\mathrm\{egal\}\}is dominant this rule is suboptimal, and it approaches the worst rule asωegal\\omega\_\{\\mathrm\{egal\}\}becomes more dominant\. Formally, letϵ∈\(0,1/2\)\\epsilon\\in\(0,1/2\)and letωegal=1−ϵ\\omega\_\{\\mathrm\{egal\}\}=1\-\\epsilonandωfamily=ϵ\\omega\_\{\\mathrm\{family\}\}=\\epsilon; then,

UMinsep​\(Fa​b\)=UMinsep​\(Fb​a\)=\(1−ϵ\)\+ϵ/2=1−ϵ/2,UMinsep​\(Fa​a\)=ϵ,UMinsep​\(Fb​b\)=0\.\\displaystyle U\_\{M\_\{\\mathrm\{insep\}\}\}\(F^\{ab\}\)=U\_\{M\_\{\\mathrm\{insep\}\}\}\(F^\{ba\}\)=\(1\-\\epsilon\)\+\\epsilon/2=1\-\\epsilon/2,\\ \\ \\ \\ \\ \\ U\_\{M\_\{\\mathrm\{insep\}\}\}\(F^\{aa\}\)=\\epsilon,\\ \\ \\ \\ \\ \\ U\_\{M\_\{\\mathrm\{insep\}\}\}\(F^\{bb\}\)=0\.Then, the regret is

UMinsep​\(Fa​b\)−UMinsep​\(Fa​a\)=1−ϵ/2−ϵ=1−3​ϵ/2\.U\_\{M\_\{\\mathrm\{insep\}\}\}\(F^\{ab\}\)\-U\_\{M\_\{\\mathrm\{insep\}\}\}\(F^\{aa\}\)=1\-\\epsilon/2\-\\epsilon=1\-3\\epsilon/2\.Asϵ→0\\epsilon\\to 0, this gets arbitrarily close to the regret of the worst rule, which is

UMinsep​\(Fa​b\)−UMinsep​\(Fb​b\)=1−ϵ/2\.U\_\{M\_\{\\mathrm\{insep\}\}\}\(F^\{ab\}\)\-U\_\{M\_\{\\mathrm\{insep\}\}\}\(F^\{bb\}\)=1\-\\epsilon/2\.

### B\.5\.[LemmaB\.5](https://arxiv.org/html/2607.02672#A2.Thmtheorem5)

###### Lemma B\.5\.

Fix a link functionhhand letM,M′∈ℳsepM,M^\{\\prime\}\\in\\mathcal\{M\}\_\{\\mathrm\{sep\}\}be scale\-free indistinguishable w\.r\.t\.hh\. Then, for any subset of rulesℱ′⊆ℱ\\mathcal\{F\}^\{\\prime\}\\subseteq\\mathcal\{F\},ℱ⊆ℱ\\mathcal\{\\mathcal\{F\}\}\\subseteq\\mathcal\{F\},

arg⁡maxF∈ℱ′⁡UM​\(F\)=arg⁡maxF∈ℱ′⁡UM′​\(F\)\.\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}U\_\{M\}\(F\)=\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}U\_\{M^\{\\prime\}\}\(F\)\.

###### Proof\.

BecauseMMandM′M^\{\\prime\}are perfectly separable,[Theorem2\.13](https://arxiv.org/html/2607.02672#S2.Thmtheorem13)gives local score functionsVMV\_\{M\}andVM′V\_\{M^\{\\prime\}\}such that, for every queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\),

κM​\(q\)=VM​\(x,y\)−VM​\(x,y′\)\\kappa\_\{M\}\(q\)=V\_\{M\}\(x,y\)\-V\_\{M\}\(x,y^\{\\prime\}\)and

κM′​\(q\)=VM′​\(x,y\)−VM′​\(x,y′\)\.\\kappa\_\{M^\{\\prime\}\}\(q\)=V\_\{M^\{\\prime\}\}\(x,y\)\-V\_\{M^\{\\prime\}\}\(x,y^\{\\prime\}\)\.Moreover,[Theorem2\.13](https://arxiv.org/html/2607.02672#S2.Thmtheorem13)shows that maximizingUMU\_\{M\}over any subset of rules is equivalent to maximizing

∑x∈XVM​\(x,F​\(x\)\),\\sum\_\{x\\in X\}V\_\{M\}\(x,F\(x\)\),and analogously forM′M^\{\\prime\}\.

SinceMMandM′M^\{\\prime\}are scale\-free indistinguishable, there existβ,β′\>0\\beta,\\beta^\{\\prime\}\>0such that, for every queryqq,

h​\(β​κM​\(q\)\)=h​\(β′​κM′​\(q\)\)\.h\(\\beta\\kappa\_\{M\}\(q\)\)=h\(\\beta^\{\\prime\}\\kappa\_\{M^\{\\prime\}\}\(q\)\)\.Becausehhis strictly increasing, it is injective\. Therefore,

β​κM​\(q\)=β′​κM′​\(q\)∀q∈Qpc\.\\beta\\kappa\_\{M\}\(q\)=\\beta^\{\\prime\}\\kappa\_\{M^\{\\prime\}\}\(q\)\\qquad\\forall q\\in Q^\{\\mathrm\{pc\}\}\.Letc:=β/β′\>0c:=\\beta/\\beta^\{\\prime\}\>0\. Then, for everyx∈Xx\\in Xand everyy,y′∈𝒴​\(x\)y,y^\{\\prime\}\\in\\mathcal\{Y\}\(x\),

VM′​\(x,y\)−VM′​\(x,y′\)=c​\(VM​\(x,y\)−VM​\(x,y′\)\)\.V\_\{M^\{\\prime\}\}\(x,y\)\-V\_\{M^\{\\prime\}\}\(x,y^\{\\prime\}\)=c\\bigl\(V\_\{M\}\(x,y\)\-V\_\{M\}\(x,y^\{\\prime\}\)\\bigr\)\.Fixx∈Xx\\in X, and choose an arbitrary reference outputy0∈𝒴​\(x\)y\_\{0\}\\in\\mathcal\{Y\}\(x\)\. The above equality implies that, for everyy∈𝒴​\(x\)y\\in\\mathcal\{Y\}\(x\),

VM′​\(x,y\)−VM′​\(x,y0\)=c​\(VM​\(x,y\)−VM​\(x,y0\)\)\.V\_\{M^\{\\prime\}\}\(x,y\)\-V\_\{M^\{\\prime\}\}\(x,y\_\{0\}\)=c\\bigl\(V\_\{M\}\(x,y\)\-V\_\{M\}\(x,y\_\{0\}\)\\bigr\)\.Equivalently,

VM′​\(x,y\)−c​VM​\(x,y\)=VM′​\(x,y0\)−c​VM​\(x,y0\)\.V\_\{M^\{\\prime\}\}\(x,y\)\-cV\_\{M\}\(x,y\)=V\_\{M^\{\\prime\}\}\(x,y\_\{0\}\)\-cV\_\{M\}\(x,y\_\{0\}\)\.Thus there exists a constantax=VM′​\(x,y0\)−c​VM​\(x,y0\)a\_\{x\}=V\_\{M^\{\\prime\}\}\(x,y\_\{0\}\)\-cV\_\{M\}\(x,y\_\{0\}\), depending onxxbut not onyy, such that

VM′​\(x,y\)=c​VM​\(x,y\)\+ax∀y∈𝒴​\(x\)\.V\_\{M^\{\\prime\}\}\(x,y\)=cV\_\{M\}\(x,y\)\+a\_\{x\}\\qquad\\forall y\\in\\mathcal\{Y\}\(x\)\.Therefore, for every ruleF∈ℱF\\in\\mathcal\{F\},

∑x∈XVM′​\(x,F​\(x\)\)=c​∑x∈XVM​\(x,F​\(x\)\)\+∑x∈Xax\.\\sum\_\{x\\in X\}V\_\{M^\{\\prime\}\}\(x,F\(x\)\)=c\\sum\_\{x\\in X\}V\_\{M\}\(x,F\(x\)\)\+\\sum\_\{x\\in X\}a\_\{x\}\.The additive term∑x∈Xax\\sum\_\{x\\in X\}a\_\{x\}is independent ofFF, andc\>0c\>0\. Hence the two local\-score objectives have the same maximizers over everyℱ′⊆ℱ\\mathcal\{F\}^\{\\prime\}\\subseteq\\mathcal\{F\}\. Translating back through the perfect\-separability correspondence gives

arg⁡maxF∈ℱ′⁡UM​\(F\)=arg⁡maxF∈ℱ′⁡UM′​\(F\)\.∎\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}U\_\{M\}\(F\)=\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}U\_\{M^\{\\prime\}\}\(F\)\.\\qed

### B\.6\.Formalization of[Example3\.15](https://arxiv.org/html/2607.02672#S3.Thmtheorem15)

Setup\.Consider again the two\-input allocation task from[Proposition3\.4](https://arxiv.org/html/2607.02672#S3.Thmtheorem4): there are two recipientsN=\{a,b\}N=\\\{a,b\\\}, two inputsx1=\(a,b;k1\)x\_\{1\}=\(a,b;k\_\{1\}\)andx2=\(a,b;k2\)x\_\{2\}=\(a,b;k\_\{2\}\), and𝒴​\(x1\)=𝒴​\(x2\)=\{a,b\}\\mathcal\{Y\}\(x\_\{1\}\)=\\mathcal\{Y\}\(x\_\{2\}\)=\\\{a,b\\\}\.𝒟\\mathcal\{D\}is such that both inputs occur with equal probability, and suppose both recipients have the same benefit for either good, and no benefit if they aren’t given a good:

vi​\(x,j\)=𝟏​\{j=i\}for all​i,j∈\{a,b\}\.v\_\{i\}\(x,j\)=\\mathbf\{1\}\\\{j=i\\\}\\qquad\\text\{for all \}\\ i,j\\in\\\{a,b\\\}\.As usual, writeℱ=\{Fa​a,Fa​b,Fb​a,Fb​b\},\\mathcal\{F\}=\\\{F^\{aa\},F^\{ab\},F^\{ba\},F^\{bb\}\\\},whereFi​j​\(x1\)=iF^\{ij\}\(x\_\{1\}\)=iandFi​j​\(x2\)=jF^\{ij\}\(x\_\{2\}\)=j\.

Additional \(mild\) restriction onϕrules\\phi^\{\\text\{rules\}\}\.Assume thatϕrules\\phi^\{\\text\{rules\}\}isbalanced sign\-responsive: that is, for every evidence profilezzthat is exactly balanced between the two responses,

\{zF:F∈ℱ\}=\{−zF:F∈ℱ\},\\\{z\_\{F\}:F\\in\\mathcal\{F\}\\\}=\\\{\-z\_\{F\}:F\\in\\mathcal\{F\}\\\},any positive uniform shift at every index ofzzmakes the aggregate evidence positive: for everyc\>0c\>0,

ϕrules​\(z\+c​𝟏\)\>0\.\\phi^\{\\text\{rules\}\}\(z\+c\\mathbf\{1\}\)\>0\.This condition is satisfied by the average and maximum aggregators, but not by lower\-percentile aggregators such as the minimum\.

Model\.LetMMcontain the single proportionality priority

uprop​\(F\)=−∑i∈\{a,b\}\(PrX∼𝒟⁡\[F​\(X\)=i\]−αi\)2,u\_\{\\mathrm\{prop\}\}\(F\)=\-\\sum\_\{i\\in\\\{a,b\\\}\}\\left\(\\Pr\_\{X\\sim\\mathcal\{D\}\}\[F\(X\)=i\]\-\\alpha\_\{i\}\\right\)^\{2\},with

αa=12\+ϵ,αb=12−ϵ,0<ϵ<14\.\\alpha\_\{a\}=\\frac\{1\}\{2\}\+\\epsilon,\\qquad\\alpha\_\{b\}=\\frac\{1\}\{2\}\-\\epsilon,\\qquad 0<\\epsilon<\\frac\{1\}\{4\}\.In words, this priority reflects the intuition that the individual wants the goods to be roughly evenly split, with a slight bias toward individualaa\(or, conceptually, individuals of typeaa\)\.

True rule utilities\.Define the shorthandpa​\(F\)=PrX∼𝒟⁡\[F​\(X\)=a\],p\_\{a\}\(F\)=\\Pr\_\{X\\sim\\mathcal\{D\}\}\[F\(X\)=a\],for the probability thataareceives a good underFF\. Noting thatpb​\(F\)=1−pa​\(F\)p\_\{b\}\(F\)=1\-p\_\{a\}\(F\), it follows that

uprop​\(F\)=−\(\(pa​\(F\)−\(1/2\+ϵ\)\)2\+\(1−pa​\(F\)−\(1/2−ϵ\)\)2\)=−2​\(pa​\(F\)−\(1/2\+ϵ\)\)2\.u\_\{\\mathrm\{prop\}\}\(F\)=\-\\bigg\(\\big\(p\_\{a\}\(F\)\-\(1/2\+\\epsilon\)\\big\)^\{2\}\+\\big\(1\-p\_\{a\}\(F\)\-\(1/2\-\\epsilon\)\\big\)^\{2\}\\bigg\)=\-2\\left\(p\_\{a\}\(F\)\-\\left\(1/2\+\\epsilon\\right\)\\right\)^\{2\}\.Therefore,

uprop​\(Fa​a\)=−2​\(1/2−ϵ\)2=−1/2\+2​ϵ−2​ϵ2,u\_\{\\mathrm\{prop\}\}\(F^\{aa\}\)=\-2\\left\(1/2\-\\epsilon\\right\)^\{2\}=\-1/2\+2\\epsilon\-2\\epsilon^\{2\},uprop​\(Fb​b\)=−2​\(1/2\+ϵ\)2=−1/2−2​ϵ−2​ϵ2,u\_\{\\mathrm\{prop\}\}\(F^\{bb\}\)=\-2\\left\(1/2\+\\epsilon\\right\)^\{2\}=\-1/2\-2\\epsilon\-2\\epsilon^\{2\},and

uprop​\(Fa​b\)=uprop​\(Fb​a\)=−2​ϵ2\.u\_\{\\mathrm\{prop\}\}\(F^\{ab\}\)=u\_\{\\mathrm\{prop\}\}\(F^\{ba\}\)=\-2\\epsilon^\{2\}\.
Thus the proportionality priority is maximized by the balanced rulesFa​bF^\{ab\}andFb​aF^\{ba\}\.

Observed Query Behavior\.Consider the queryq1=\(a,b;x1\)q\_\{1\}=\(a,b;x\_\{1\}\)\. For each background ruleF∈ℱF\\in\\mathcal\{F\}, define

ΔpropF​\(q1\):=uprop​\(Fx1→a\)−uprop​\(Fx1→b\)\.\\Delta^\{F\}\_\{\\mathrm\{prop\}\}\(q\_\{1\}\):=u\_\{\\mathrm\{prop\}\}\(F\_\{x\_\{1\}\\to a\}\)\-u\_\{\\mathrm\{prop\}\}\(F\_\{x\_\{1\}\\to b\}\)\.IfF​\(x2\)=bF\(x\_\{2\}\)=b, thenFx1→a=Fa​bF\_\{x\_\{1\}\\to a\}=F^\{ab\}andFx1→b=Fb​bF\_\{x\_\{1\}\\to b\}=F^\{bb\}, so

ΔpropF​\(q1\)=uprop​\(Fa​b\)−uprop​\(Fb​b\)=−2​ϵ2−\(−12−2​ϵ−2​ϵ2\)=12\+2​ϵ\.\\Delta^\{F\}\_\{\\mathrm\{prop\}\}\(q\_\{1\}\)=u\_\{\\mathrm\{prop\}\}\(F^\{ab\}\)\-u\_\{\\mathrm\{prop\}\}\(F^\{bb\}\)=\-2\\epsilon^\{2\}\-\\left\(\-\\frac\{1\}\{2\}\-2\\epsilon\-2\\epsilon^\{2\}\\right\)=\\frac\{1\}\{2\}\+2\\epsilon\.IfF​\(x2\)=aF\(x\_\{2\}\)=a, thenFx1→a=Fa​aF\_\{x\_\{1\}\\to a\}=F^\{aa\}andFx1→b=Fb​aF\_\{x\_\{1\}\\to b\}=F^\{ba\}, so

ΔpropF​\(q1\)=uprop​\(Fa​a\)−uprop​\(Fb​a\)=\(−12\+2​ϵ−2​ϵ2\)−\(−2​ϵ2\)=−12\+2​ϵ\.\\Delta^\{F\}\_\{\\mathrm\{prop\}\}\(q\_\{1\}\)=u\_\{\\mathrm\{prop\}\}\(F^\{aa\}\)\-u\_\{\\mathrm\{prop\}\}\(F^\{ba\}\)=\\left\(\-\\frac\{1\}\{2\}\+2\\epsilon\-2\\epsilon^\{2\}\\right\)\-\\left\(\-2\\epsilon^\{2\}\\right\)=\-\\frac\{1\}\{2\}\+2\\epsilon\.Thus the rule\-indexed evidence profile atq1q\_\{1\}is, up to permutation,

z\+=\(12\+2​ϵ,12\+2​ϵ,−12\+2​ϵ,−12\+2​ϵ\)=z0\+2​ϵ​𝟏,z^\{\+\}=\\left\(\\frac\{1\}\{2\}\+2\\epsilon,\\frac\{1\}\{2\}\+2\\epsilon,\-\\frac\{1\}\{2\}\+2\\epsilon,\-\\frac\{1\}\{2\}\+2\\epsilon\\right\)=z^\{0\}\+2\\epsilon\\mathbf\{1\},where

z0=\(12,12,−12,−12\)z^\{0\}=\\left\(\\frac\{1\}\{2\},\\frac\{1\}\{2\},\-\\frac\{1\}\{2\},\-\\frac\{1\}\{2\}\\right\)is balanced around zero\. Then, by balanced sign\-responsiveness ofϕrules\\phi^\{\\text\{rules\}\},ϕrules​\(z\+\)\>0\.\\phi^\{\\text\{rules\}\}\(z^\{\+\}\)\>0\.Since the model has only the proportionality priority with weight11, the linear priority aggregator gives

sM\+​\(q1\)=\[ϕrules​\(z\+\)\]\+,sM−​\(q1\)=\[−ϕrules​\(z\+\)\]\+\.s\_\{M\}^\{\+\}\(q\_\{1\}\)=\[\\phi^\{\\text\{rules\}\}\(z^\{\+\}\)\]\_\{\+\},\\qquad s\_\{M\}^\{\-\}\(q\_\{1\}\)=\[\-\\phi^\{\\text\{rules\}\}\(z^\{\+\}\)\]\_\{\+\}\.Therefore,

κM​\(q1\)=sM\+​\(q1\)−sM−​\(q1\)=ϕrules​\(z\+\)\>0\.\\kappa\_\{M\}\(q\_\{1\}\)=s\_\{M\}^\{\+\}\(q\_\{1\}\)\-s\_\{M\}^\{\-\}\(q\_\{1\}\)=\\phi^\{\\text\{rules\}\}\(z^\{\+\}\)\>0\.The same calculation applies toq2=\(a,b;x2\)q\_\{2\}=\(a,b;x\_\{2\}\)\. Hence, letting

γ:=ϕrules​\(z\+\)\>0,\\gamma:=\\phi^\{\\text\{rules\}\}\(z^\{\+\}\)\>0,we have

κM​\(a,b;xi\)=γfor​i∈\{1,2\}\.\\kappa\_\{M\}\(a,b;x\_\{i\}\)=\\gamma\\qquad\\text\{for \}i\\in\\\{1,2\\\}\.
Thus, under the zero\-threshold response model,

Rhβ;0,0∘​\(\(a,b;xi\);M\)​\(a≻b\)=h​\(β​γ\)\>h​\(−β​γ\)=Rhβ;0,0∘​\(\(a,b;xi\);M\)​\(b≻a\),R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\\bigl\(\(a,b;x\_\{i\}\);M\\bigr\)\(a\\succ b\)=h\(\\beta\\gamma\)\>h\(\-\\beta\\gamma\)=R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\\bigl\(\(a,b;x\_\{i\}\);M\\bigr\)\(b\\succ a\),where the strict inequality follows becausehhis strictly increasing\. Hence the response model induced byMMfavorsaaoverbbat both inputs\.

Construction of the perfectly separable rationalization\.Now define the perfectly separable priority

ua​\(F\)=PrX∼𝒟⁡\[F​\(X\)=a\]\.u\_\{a\}\(F\)=\\Pr\_\{X\\sim\\mathcal\{D\}\}\[F\(X\)=a\]\.For every background ruleFFand every inputxix\_\{i\},

ua​\(Fxi→a\)−ua​\(Fxi→b\)=12,u\_\{a\}\(F\_\{x\_\{i\}\\to a\}\)\-u\_\{a\}\(F\_\{x\_\{i\}\\to b\}\)=\\frac\{1\}\{2\},which is independent ofFF\. Thusuau\_\{a\}is perfectly separable\. LetM~\\widetilde\{M\}be the model consisting of this single priority\. By unanimity ofϕrules\\phi^\{\\text\{rules\}\},

κM~​\(a,b;xi\)=12andκM~​\(b,a;xi\)=−12\\kappa\_\{\\widetilde\{M\}\}\(a,b;x\_\{i\}\)=\\frac\{1\}\{2\}\\qquad\\text\{and\}\\qquad\\kappa\_\{\\widetilde\{M\}\}\(b,a;x\_\{i\}\)=\-\\frac\{1\}\{2\}for eachi∈\{1,2\}i\\in\\\{1,2\\\}\.

We now show thatMMandM~\\widetilde\{M\}are scale\-free indistinguishable\. Fix anyβ\>0\\beta\>0, and set

β~:=2​β​γ\>0\.\\widetilde\{\\beta\}:=2\\beta\\gamma\>0\.Then, for eachi∈\{1,2\}i\\in\\\{1,2\\\},

β​κM​\(a,b;xi\)=β​γ=β~⋅12=β~​κM~​\(a,b;xi\)\.\\beta\\kappa\_\{M\}\(a,b;x\_\{i\}\)=\\beta\\gamma=\\widetilde\{\\beta\}\\cdot\\frac\{1\}\{2\}=\\widetilde\{\\beta\}\\kappa\_\{\\widetilde\{M\}\}\(a,b;x\_\{i\}\)\.Thus the scaled directional gaps agree on every local pairwise query, and therefore

Rhβ;0,0∘​\(⋅;M\)=Rhβ~;0,0∘​\(⋅;M~\)\.R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\(\\cdot;M\)=R^\{\\circ\}\_\{h\_\{\\widetilde\{\\beta\}\};0,0\}\(\\cdot;\\widetilde\{M\}\)\.Hence

M~∈PSRh​\(M\)\.\\widetilde\{M\}\\in\\mathrm\{PSR\}\_\{h\}\(M\)\.
Chosen rule underM~\\widetilde\{M\}\.UnderM~\\widetilde\{M\},

ua​\(Fa​a\)=1,ua​\(Fa​b\)=ua​\(Fb​a\)=12,ua​\(Fb​b\)=0\.u\_\{a\}\(F^\{aa\}\)=1,\\qquad u\_\{a\}\(F^\{ab\}\)=u\_\{a\}\(F^\{ba\}\)=\\frac\{1\}\{2\},\\qquad u\_\{a\}\(F^\{bb\}\)=0\.Therefore,

arg⁡maxF∈ℱ⁡UM~​\(F\)=\{Fa​a\}\.\\arg\\max\_\{F\\in\\mathcal\{F\}\}U\_\{\\widetilde\{M\}\}\(F\)=\\\{F^\{aa\}\\\}\.By[LemmaB\.5](https://arxiv.org/html/2607.02672#A2.Thmtheorem5), under the linear priority aggregator all elements ofPSRh​\(M\)\\mathrm\{PSR\}\_\{h\}\(M\)induce the same aggregate\-optimal rules\. Hence every perfect separable rationalization ofMMselectsFa​aF^\{aa\}\.

Regret\.The selected ruleFa​aF^\{aa\}is highly suboptimal\. Its regret is

UM​\(Fa​b\)−UM​\(Fa​a\)=−2​ϵ2−\(−12\+2​ϵ−2​ϵ2\)=12−2​ϵ\.U\_\{M\}\(F^\{ab\}\)\-U\_\{M\}\(F^\{aa\}\)=\-2\\epsilon^\{2\}\-\\left\(\-\\frac\{1\}\{2\}\+2\\epsilon\-2\\epsilon^\{2\}\\right\)=\\frac\{1\}\{2\}\-2\\epsilon\.Asϵ→0\\epsilon\\to 0, this becomes arbitrarily close to the regret of the worst possible ruleFb​bF^\{bb\}:

UM​\(Fa​b\)−UM​\(Fb​b\)=−2​ϵ2−\(−12−2​ϵ−2​ϵ2\)=12\+2​ϵ\.U\_\{M\}\(F^\{ab\}\)\-U\_\{M\}\(F^\{bb\}\)=\-2\\epsilon^\{2\}\-\\left\(\-\\frac\{1\}\{2\}\-2\\epsilon\-2\\epsilon^\{2\}\\right\)=\\frac\{1\}\{2\}\+2\\epsilon\.

## Appendix CSupplemental Materials for[Section4](https://arxiv.org/html/2607.02672#S4)

### C\.1\.Interview Coding Methods and Results

[Figure7](https://arxiv.org/html/2607.02672#A3.F7)summarizes the frequency of various signals of indecision detected in each of2020interview transcripts fromKeswaniet al\.\([2025a](https://arxiv.org/html/2607.02672#bib.bib212)\), in which each participant made exactly33pairwise kidney\-allocation comparisons \(6060comparisons total\)\. Below, we describe our coding methods\.

![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/fig_criteria_per_transcript_all6.png)Figure 7\.Frequency of language markers of indecision across2020interview transcripts, each containing33pairwise kidney\-allocation comparisons \(60 total comparisons\)\. We count for each transcript how many of its33comparisons exhibit each of five conflict criteria, and report the mean per transcripts \(±1\\pm 1SE\)\.We separately hand and LLM242424LLM coding was done by Claude\-opus\-4\-6 using a structured\-output schema that enforces valid JSON\.coded each comparison according to whether they fit any element of a multi\-label list of the five indecision criteria described in[Figure7](https://arxiv.org/html/2607.02672#A3.F7)\. The codebook is in Table[1](https://arxiv.org/html/2607.02672#A3.T1)\.

Table 1\.Indecision codebook, as given verbatim to the coding model\.
### C\.2\.Proof of Proposition[4\.3](https://arxiv.org/html/2607.02672#S4.Thmtheorem3)

###### Proof\.

Fix anyψ\\psiand anyω∗∈Δm−1\\omega^\{\*\}\\in\\Delta^\{m\-1\}\. By definition ofFω∗F\_\{\\omega^\{\*\}\}, at everyxxit chooses the maximizer of⟨ω∗,ψ​\(x,Fω∗​\(x\)\)⟩\\langle\\omega^\{\*\},\\psi\(x,F\_\{\\omega^\{\*\}\}\(x\)\)\\rangle; thus, for everyx∈𝒳x\\in\\mathcal\{X\}and everyF∈ℱF\\in\\mathcal\{F\},

⟨ω∗,ψ​\(x,Fω∗​\(x\)\)⟩≥⟨ω∗,ψ​\(x,F​\(x\)\)⟩\.\\langle\\omega^\{\*\},\\psi\(x,F\_\{\\omega^\{\*\}\}\(x\)\)\\rangle\\geq\\langle\\omega^\{\*\},\\psi\(x,F\(x\)\)\\rangle\.Fix anyF∈ℱF\\in\\mathcal\{F\}\. Then, summing overxxand dividing by\|𝒳\|\|\\mathcal\{X\}\|gives

1\|𝒳\|​∑x∈𝒳⟨ω∗,ψ​\(x,Fω∗​\(x\)\)⟩≥1\|𝒳\|​∑x∈𝒳⟨ω∗,ψ​\(x,F​\(x\)\)⟩\.\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\sum\_\{x\\in\\mathcal\{X\}\}\\langle\\omega^\{\*\},\\psi\(x,F\_\{\\omega^\{\*\}\}\(x\)\)\\rangle\\geq\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\sum\_\{x\\in\\mathcal\{X\}\}\\langle\\omega^\{\*\},\\psi\(x,F\(x\)\)\\rangle\.By Definition 4\.1, the left\-hand side is exactlyUMω∗​\(Fω∗\)U\_\{M\_\{\\omega^\{\*\}\}\}\(F\_\{\\omega^\{\*\}\}\)and the right\-hand side isUMω∗​\(F\)U\_\{M\_\{\\omega^\{\*\}\}\}\(F\)\. SinceFFwas arbitrary,

Fω∗∈arg⁡maxF∈ℱ⁡UMω∗​\(F\)\.∎F\_\{\\omega^\{\*\}\}\\in\\arg\\max\_\{F\\in\\mathcal\{F\}\}U\_\{M\_\{\\omega^\{\*\}\}\}\(F\)\.\\qed

### C\.3\.Proof of Theorem[4\.4](https://arxiv.org/html/2607.02672#S4.Thmtheorem4)

###### Proof\.

Fixψ,h,β\>0\\psi,h,\\beta\>0, and a queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)\. Fix any linear S\-RUM defined byθ\\theta\. Then, the probability of reportingy≻y′y\\succ y^\{\\prime\}is

Shβ​\(q;Vθ\)​\(y≻y′\)=h​\(β​\(Vθ​\(x,y\)−Vθ​\(x,y′\)\)\)=h​\(β​⟨θ,ψ​\(x,y\)−ψ​\(x,y′\)⟩\)\.S\_\{h\_\{\\beta\}\}\(q;V\_\{\\theta\}\)\(y\\succ y^\{\\prime\}\)=h\\\!\\left\(\\beta\\left\(V\_\{\\theta\}\(x,y\)\-V\_\{\\theta\}\(x,y^\{\\prime\}\)\\right\)\\right\)=h\\\!\\left\(\\beta\\langle\\theta,\\psi\(x,y\)\-\\psi\(x,y^\{\\prime\}\)\\rangle\\right\)\.
Now consider the linear priority modelMωM\_\{\\omega\}withω=θ/‖θ‖1\\omega=\\theta/\\\|\\theta\\\|\_\{1\}\. By[Definition4\.2](https://arxiv.org/html/2607.02672#S4.Thmtheorem2), for each priorityjjand any background ruleFF,

ΔjF​\(q\)=uj​\(Fx→y\)−uj​\(Fx→y′\)=1\|𝒳\|​\(ψj​\(x,y\)−ψj​\(x,y′\)\)\.\\Delta^\{F\}\_\{j\}\(q\)=u\_\{j\}\(F\_\{x\\to y\}\)\-u\_\{j\}\(F\_\{x\\to y^\{\\prime\}\}\)=\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\left\(\\psi\_\{j\}\(x,y\)\-\\psi\_\{j\}\(x,y^\{\\prime\}\)\\right\)\.This quantity is independent of the background ruleFF, so any unanimous rule aggregator returns this same value\. Hence the directional score gap is

κMω​\(q\)=sMω\+​\(q\)−sMω−​\(q\)=∑j=1dωj​ΔjF​\(q\)=1\|𝒳\|​⟨ω,ψ​\(x,y\)−ψ​\(x,y′\)⟩\.\\kappa\_\{M\_\{\\omega\}\}\(q\)=s^\{\+\}\_\{M\_\{\\omega\}\}\(q\)\-s^\{\-\}\_\{M\_\{\\omega\}\}\(q\)=\\sum\_\{j=1\}^\{d\}\\omega\_\{j\}\\Delta^\{F\}\_\{j\}\(q\)=\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\left\\langle\\omega,\\psi\(x,y\)\-\\psi\(x,y^\{\\prime\}\)\\right\\rangle\.Substitutingω=θ/‖θ‖1\\omega=\\theta/\\\|\\theta\\\|\_\{1\}gives

κMω​\(q\)=1\|𝒳\|​‖θ‖1​⟨θ,ψ​\(x,y\)−ψ​\(x,y′\)⟩\.\\kappa\_\{M\_\{\\omega\}\}\(q\)=\\frac\{1\}\{\|\\mathcal\{X\}\|\\\|\\theta\\\|\_\{1\}\}\\left\\langle\\theta,\\psi\(x,y\)\-\\psi\(x,y^\{\\prime\}\)\\right\\rangle\.Therefore, ifβ′=β​\|𝒳\|​‖θ‖1\\beta^\{\\prime\}=\\beta\|\\mathcal\{X\}\|\\\|\\theta\\\|\_\{1\}, then

hβ′​\(κMω​\(q\)\)=h​\(β′​κMω​\(q\)\)=h​\(β​⟨θ,ψ​\(x,y\)−ψ​\(x,y′\)⟩\),h\_\{\\beta^\{\\prime\}\}\(\\kappa\_\{M\_\{\\omega\}\}\(q\)\)=h\\\!\\left\(\\beta^\{\\prime\}\\kappa\_\{M\_\{\\omega\}\}\(q\)\\right\)=h\\\!\\left\(\\beta\\langle\\theta,\\psi\(x,y\)\-\\psi\(x,y^\{\\prime\}\)\\rangle\\right\),which equals the S\-RUM response probability fory≻y′y\\succ y^\{\\prime\}\. The probability ofy′≻yy^\{\\prime\}\\succ yalso matches, since in the zero\-threshold case both models assign it the complementary probability\.

It remains to show that the induced aggregate\-optimal rules agree\. For any ruleFF,

UMω​\(F\)=∑j=1dωj​uj​\(F\)=1\|𝒳\|​∑x∈𝒳⟨ω,ψ​\(x,F​\(x\)\)⟩\.U\_\{M\_\{\\omega\}\}\(F\)=\\sum\_\{j=1\}^\{d\}\\omega\_\{j\}u\_\{j\}\(F\)=\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\sum\_\{x\\in\\mathcal\{X\}\}\\left\\langle\\omega,\\psi\(x,F\(x\)\)\\right\\rangle\.Usingω=θ/‖θ‖1\\omega=\\theta/\\\|\\theta\\\|\_\{1\}, this becomes

UMω​\(F\)=1\|𝒳\|​‖θ‖1​∑x∈𝒳⟨θ,ψ​\(x,F​\(x\)\)⟩=1\|𝒳\|​‖θ‖1​∑x∈𝒳Vθ​\(x,F​\(x\)\)\.U\_\{M\_\{\\omega\}\}\(F\)=\\frac\{1\}\{\|\\mathcal\{X\}\|\\\|\\theta\\\|\_\{1\}\}\\sum\_\{x\\in\\mathcal\{X\}\}\\left\\langle\\theta,\\psi\(x,F\(x\)\)\\right\\rangle=\\frac\{1\}\{\|\\mathcal\{X\}\|\\\|\\theta\\\|\_\{1\}\}\\sum\_\{x\\in\\mathcal\{X\}\}V\_\{\\theta\}\(x,F\(x\)\)\.ThusUMω​\(F\)U\_\{M\_\{\\omega\}\}\(F\)is a positive scalar multiple of the linear S\-RUM objective for everyFF\. Therefore, the two objectives have the same maximizers over anyℱ′⊆ℱ\\mathcal\{F\}^\{\\prime\}\\subseteq\\mathcal\{F\}\. ∎

### C\.4\.Proof that weight recovery implies rule recovery

###### Lemma C\.1\.

Letℱ′⊆ℱ\\mathcal\{F\}^\{\\prime\}\\subseteq\\mathcal\{F\}be any rule class\. Suppose‖ω^−ω∗‖1≤ε\\\|\\widehat\{\\omega\}\-\\omega^\{\*\}\\\|\_\{1\}\\leq\\varepsilon, and let

F^∈arg⁡maxF∈ℱ′⁡Uω^​\(F\),F∗∈arg⁡maxF∈ℱ′⁡Uω∗​\(F\)\.\\widehat\{F\}\\in\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}U\_\{\\widehat\{\\omega\}\}\(F\),\\qquad F^\{\*\}\\in\\arg\\max\_\{F\\in\\mathcal\{F\}^\{\\prime\}\}U\_\{\\omega^\{\*\}\}\(F\)\.Then

Uω∗​\(F^\)≥Uω∗​\(F∗\)−2​ε\.U\_\{\\omega^\{\*\}\}\(\\widehat\{F\}\)\\geq U\_\{\\omega^\{\*\}\}\(F^\{\*\}\)\-2\\varepsilon\.

###### Proof\.

Note that we may shift eachuju\_\{j\}by a constant without changing any utility differences or aggregate\-optimal rules\. In particular, because\|uj​\(F\)−uj​\(F′\)\|≤1\|u\_\{j\}\(F\)\-u\_\{j\}\(F^\{\\prime\}\)\|\\leq 1for allF∈ℱF\\in\\mathcal\{F\}, we can shift these utilities so that\|uj​\(F\)\|∈\[0,1\]\|u\_\{j\}\(F\)\|\\in\[0,1\]for allj,Fj,F, and hence\|uj​\(F\)\|≤1\|u\_\{j\}\(F\)\|\\leq 1\. For every ruleFF,

\|Uω^​\(F\)−Uω∗​\(F\)\|=\|∑j=1d\(ω^j−ωj∗\)​uj​\(F\)\|≤∑j=1d\|ω^j−ωj∗\|​\|uj​\(F\)\|≤‖ω^−ω∗‖1≤ε\.\|U\_\{\\widehat\{\\omega\}\}\(F\)\-U\_\{\\omega^\{\\ast\}\}\(F\)\|=\\left\|\\sum\_\{j=1\}^\{d\}\(\\widehat\{\\omega\}\_\{j\}\-\\omega^\{\\ast\}\_\{j\}\)u\_\{j\}\(F\)\\right\|\\leq\\sum\_\{j=1\}^\{d\}\|\\widehat\{\\omega\}\_\{j\}\-\\omega^\{\\ast\}\_\{j\}\|\\,\|u\_\{j\}\(F\)\|\\leq\\\|\\widehat\{\\omega\}\-\\omega^\{\\ast\}\\\|\_\{1\}\\leq\\varepsilon\.SinceF^\\widehat\{F\}maximizesUω^U\_\{\\widehat\{\\omega\}\}overℱ′\\mathcal\{F\}^\{\\prime\},

Uω^​\(F^\)≥Uω^​\(F∗\)\.U\_\{\\widehat\{\\omega\}\}\(\\widehat\{F\}\)\\geq U\_\{\\widehat\{\\omega\}\}\(F^\{\\ast\}\)\.Therefore,

Uω∗​\(F^\)≥Uω^​\(F^\)−ε≥Uω^​\(F∗\)−ε≥Uω∗​\(F∗\)−2​ε\.U\_\{\\omega^\{\\ast\}\}\(\\widehat\{F\}\)\\geq U\_\{\\widehat\{\\omega\}\}\(\\widehat\{F\}\)\-\\varepsilon\\geq U\_\{\\widehat\{\\omega\}\}\(F^\{\\ast\}\)\-\\varepsilon\\geq U\_\{\\omega^\{\\ast\}\}\(F^\{\\ast\}\)\-2\\varepsilon\.ThusF^\\widehat\{F\}is2​ε2\\varepsilon\-optimal\. ∎

### C\.5\.Regret Computation Methods

#### C\.5\.1\.Empirical estimate ofAvg−Regret\\operatorname\{Avg\-Regret\}

The input distribution is fixed once and shared across all methods and oracles, so that differences in regret reflect the methods rather than the sampled inputs\. It consists ofM=300M=300inputsxx, each with\|𝒴​\(x\)\|=5\|\\mathcal\{Y\}\(x\)\|=5candidate outputs whose feature vectorsψ​\(x,y\)∈ℝ5\\psi\(x,y\)\\in\\mathbb\{R\}^\{5\}are drawn i\.i\.d\. fromUniform​\[0,1\]\\mathrm\{Uniform\}\[0,1\]; the same sample is reused for every\(ω∗,ω^\)\(\\omega^\{\*\},\\widehat\{\\omega\}\)pair\.

For an inputxx, the learned rule selectsFω^​\(x\)=arg⁡maxy∈𝒴​\(x\)⁡⟨ω^,ψ​\(x,y\)⟩F\_\{\\widehat\{\\omega\}\}\(x\)=\\arg\\max\_\{y\\in\\mathcal\{Y\}\(x\)\}\\langle\\widehat\{\\omega\},\\psi\(x,y\)\\rangle, and we define

Regretx=maxy∈𝒴​\(x\)⁡⟨ω∗,ψ​\(x,y\)⟩−⟨ω∗,ψ​\(x,Fω^​\(x\)\)⟩,Rangex=maxy∈𝒴​\(x\)⁡⟨ω∗,ψ​\(x,y\)⟩−miny∈𝒴​\(x\)⁡⟨ω∗,ψ​\(x,y\)⟩\.\\operatorname\{Regret\}\_\{x\}=\\max\_\{y\\in\\mathcal\{Y\}\(x\)\}\\langle\\omega^\{\*\},\\psi\(x,y\)\\rangle\-\\langle\\omega^\{\*\},\\psi\\\!\\left\(x,F\_\{\\widehat\{\\omega\}\}\(x\)\\right\)\\rangle,\\qquad\\operatorname\{Range\}\_\{x\}=\\max\_\{y\\in\\mathcal\{Y\}\(x\)\}\\langle\\omega^\{\*\},\\psi\(x,y\)\\rangle\-\\min\_\{y\\in\\mathcal\{Y\}\(x\)\}\\langle\\omega^\{\*\},\\psi\(x,y\)\\rangle\.NoteRangex\\operatorname\{Range\}\_\{x\}depends onω∗\\omega^\{\*\}alone\. We estimate the average regret as the*ratio of two sample means*over the shared inputs\{x1,…,xM\}\\\{x\_\{1\},\\dots,x\_\{M\}\\\},

Avg​\-​Regret^​\(ω^;ω∗\)=1M​∑i=1MRegretxi1M​∑i=1MRangexi=∑i=1MRegretxi∑i=1MRangexi,\\widehat\{\\operatorname\{Avg\\text\{\-\}Regret\}\}\(\\widehat\{\\omega\};\\omega^\{\*\}\)=\\frac\{\\tfrac\{1\}\{M\}\\sum\_\{i=1\}^\{M\}\\operatorname\{Regret\}\_\{x\_\{i\}\}\}\{\\tfrac\{1\}\{M\}\\sum\_\{i=1\}^\{M\}\\operatorname\{Range\}\_\{x\_\{i\}\}\}=\\frac\{\\sum\_\{i=1\}^\{M\}\\operatorname\{Regret\}\_\{x\_\{i\}\}\}\{\\sum\_\{i=1\}^\{M\}\\operatorname\{Range\}\_\{x\_\{i\}\}\},i\.e\. bothRegretx\\operatorname\{Regret\}\_\{x\}andRangex\\operatorname\{Range\}\_\{x\}are evaluated on the same300300Monte\-Carlo inputs \(no closed form for the range\), and we pool numerator and denominator before dividing\. Bars in Figure[5](https://arxiv.org/html/2607.02672#S4.F5)report the mean±1\\pm 1SE of this quantity over the oracle ground truths\.

#### C\.5\.2\.Computing theWC−Regret\\operatorname\{WC\-Regret\}

We derive the closed form used to compute the worst\-case regretWC−RegretMω∗⁡\(Fω^\)\\operatorname\{WC\-Regret\}\_\{M\_\{\\omega^\{\*\}\}\}\(F\_\{\\widehat\{\\omega\}\}\)\. Recall that

WC−RegretMω∗⁡\(Fω^\)=supxRegretx⁡\(ω^;ω∗\)supxRangex⁡\(ω∗\),\\operatorname\{WC\-Regret\}\_\{M\_\{\\omega^\{\*\}\}\}\(F\_\{\\widehat\{\\omega\}\}\)=\\frac\{\\sup\_\{x\}\\operatorname\{Regret\}\_\{x\}\(\\widehat\{\\omega\};\\omega^\{\*\}\)\}\{\\sup\_\{x\}\\operatorname\{Range\}\_\{x\}\(\\omega^\{\*\}\)\},where both suprema are taken over the worst\-case input domainx∈\(\[0,1\]d\)5x\\in\(\[0,1\]^\{d\}\)^\{5\}\.

##### Computing the numerator\.

For fixedω∗\\omega^\{\*\}andω^\\widehat\{\\omega\}, the worst\-case regret can be computed by the linear program

maxδ∈\[−1,1\]d⁡\{⟨ω∗,δ⟩:⟨ω^,δ⟩≤0\}\.\\max\_\{\\delta\\in\[\-1,1\]^\{d\}\}\\left\\\{\\langle\\omega^\{\*\},\\delta\\rangle:\\langle\\widehat\{\\omega\},\\delta\\rangle\\leq 0\\right\\\}\.The vectorδ\\deltarepresents the feature differencez−z′z\-z^\{\\prime\}between the true\-optimal outputzzand the outputz′z^\{\\prime\}selected by the learned rule\. The constraint⟨ω^,δ⟩≤0\\langle\\widehat\{\\omega\},\\delta\\rangle\\leq 0says that the learned rule weakly prefersz′z^\{\\prime\}tozz, while the objective⟨ω∗,δ⟩\\langle\\omega^\{\*\},\\delta\\rangleis the regret under the true weights\.

###### Lemma C\.2 \(Worst\-case regret as a linear program\)\.

Let𝒵=\[0,1\]d\\mathcal\{Z\}=\[0,1\]^\{d\}\. Suppose each inputxxhas a feasible output set𝒴​\(x\)⊆𝒵\\mathcal\{Y\}\(x\)\\subseteq\\mathcal\{Z\}consisting ofkkdistinct feature vectors\. Forω∈Δd−1\\omega\\in\\Delta^\{d\-1\}, letFω​\(x\)∈arg⁡maxz∈𝒴​\(x\)⁡⟨ω,z⟩F\_\{\\omega\}\(x\)\\in\\arg\\max\_\{z\\in\\mathcal\{Y\}\(x\)\}\\langle\\omega,z\\rangle, with arbitrary fixed tie\-breaking\. Suppose the worst\-case domain ranges over all possible collections ofkkdistinct feature vectors in\[0,1\]d\[0,1\]^\{d\}\. Then, for any fixedω^,ω∗\\widehat\{\\omega\},\\omega^\{\*\},

supxRegretx⁡\(ω^;ω∗\)=supz,z′∈\[0,1\]d\{⟨ω∗,z−z′⟩:⟨ω^,z′⟩≥⟨ω^,z⟩\}=maxδ∈\[−1,1\]d⁡\{⟨ω∗,δ⟩:⟨ω^,δ⟩≤0\}\.\\sup\_\{x\}\\operatorname\{Regret\}\_\{x\}\(\\widehat\{\\omega\};\\omega^\{\*\}\)=\\sup\_\{z,z^\{\\prime\}\\in\[0,1\]^\{d\}\}\\left\\\{\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle:\\langle\\widehat\{\\omega\},z^\{\\prime\}\\rangle\\geq\\langle\\widehat\{\\omega\},z\\rangle\\right\\\}=\\max\_\{\\delta\\in\[\-1,1\]^\{d\}\}\\left\\\{\\langle\\omega^\{\*\},\\delta\\rangle:\\langle\\widehat\{\\omega\},\\delta\\rangle\\leq 0\\right\\\}\.

###### Proof\.

We first prove the equality between the worst\-case regret and the pairwise optimization\. Fix any inputxx, and letz=Fω∗​\(x\)z=F\_\{\\omega^\{\*\}\}\(x\)andz′=Fω^​\(x\)z^\{\\prime\}=F\_\{\\widehat\{\\omega\}\}\(x\)\. SinceFω^​\(x\)F\_\{\\widehat\{\\omega\}\}\(x\)selectsz′z^\{\\prime\}from𝒴​\(x\)\\mathcal\{Y\}\(x\), we have⟨ω^,z′⟩≥⟨ω^,z⟩\\langle\\widehat\{\\omega\},z^\{\\prime\}\\rangle\\geq\\langle\\widehat\{\\omega\},z\\rangle\. Moreover,Regretx⁡\(ω^;ω∗\)=⟨ω∗,z⟩−⟨ω∗,z′⟩=⟨ω∗,z−z′⟩\\operatorname\{Regret\}\_\{x\}\(\\widehat\{\\omega\};\\omega^\{\*\}\)=\\langle\\omega^\{\*\},z\\rangle\-\\langle\\omega^\{\*\},z^\{\\prime\}\\rangle=\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle\. Thus every regret value achieved by some inputxxis feasible for the pairwise optimization, so

supxRegretx⁡\(ω^;ω∗\)≤supz,z′∈\[0,1\]d\{⟨ω∗,z−z′⟩:⟨ω^,z′⟩≥⟨ω^,z⟩\}\.\\sup\_\{x\}\\operatorname\{Regret\}\_\{x\}\(\\widehat\{\\omega\};\\omega^\{\*\}\)\\leq\\sup\_\{z,z^\{\\prime\}\\in\[0,1\]^\{d\}\}\\left\\\{\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle:\\langle\\widehat\{\\omega\},z^\{\\prime\}\\rangle\\geq\\langle\\widehat\{\\omega\},z\\rangle\\right\\\}\.
For the reverse inequality, define

B:=supz,z′∈\[0,1\]d\{⟨ω∗,z−z′⟩:⟨ω^,z′⟩≥⟨ω^,z⟩\}\.B:=\\sup\_\{z,z^\{\\prime\}\\in\[0,1\]^\{d\}\}\\left\\\{\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle:\\langle\\widehat\{\\omega\},z^\{\\prime\}\\rangle\\geq\\langle\\widehat\{\\omega\},z\\rangle\\right\\\}\.We show thatsupxRegretx⁡\(ω^;ω∗\)≥B\\sup\_\{x\}\\operatorname\{Regret\}\_\{x\}\(\\widehat\{\\omega\};\\omega^\{\*\}\)\\geq B\. IfB=0B=0, this is immediate because regret is nonnegative\. Now supposeB\>0B\>0\. Fixε\>0\\varepsilon\>0\. By the definition of supremum, there existz,z′∈\[0,1\]dz,z^\{\\prime\}\\in\[0,1\]^\{d\}such that⟨ω^,z′⟩≥⟨ω^,z⟩\\langle\\widehat\{\\omega\},z^\{\\prime\}\\rangle\\geq\\langle\\widehat\{\\omega\},z\\rangleand⟨ω∗,z−z′⟩\>B−ε/3\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle\>B\-\\varepsilon/3\. Takingε\\varepsilonsmall enough, this implies⟨ω∗,z⟩\>⟨ω∗,z′⟩\\langle\\omega^\{\*\},z\\rangle\>\\langle\\omega^\{\*\},z^\{\\prime\}\\rangle\.

The learned rule may be indifferent betweenzzandz′z^\{\\prime\}, so we perturb the pair to make the learned rule’s preference strict\. Forρ∈\(0,1\)\\rho\\in\(0,1\), definez~=\(1−ρ\)​z\\tilde\{z\}=\(1\-\\rho\)zandz~′=\(1−ρ\)​z′\+ρ​𝟏\\tilde\{z\}^\{\\prime\}=\(1\-\\rho\)z^\{\\prime\}\+\\rho\\mathbf\{1\}\. Thenz~,z~′∈\[0,1\]d\\tilde\{z\},\\tilde\{z\}^\{\\prime\}\\in\[0,1\]^\{d\}\. Sinceω^∈Δd−1\\widehat\{\\omega\}\\in\\Delta^\{d\-1\},

⟨ω^,z~′⟩−⟨ω^,z~⟩=\(1−ρ\)​⟨ω^,z′−z⟩\+ρ\>0\.\\langle\\widehat\{\\omega\},\\tilde\{z\}^\{\\prime\}\\rangle\-\\langle\\widehat\{\\omega\},\\tilde\{z\}\\rangle=\(1\-\\rho\)\\langle\\widehat\{\\omega\},z^\{\\prime\}\-z\\rangle\+\\rho\>0\.Thus the learned rule strictly prefersz~′\\tilde\{z\}^\{\\prime\}toz~\\tilde\{z\}\. The true\-regret objective changes to⟨ω∗,z~−z~′⟩=\(1−ρ\)​⟨ω∗,z−z′⟩−ρ\\langle\\omega^\{\*\},\\tilde\{z\}\-\\tilde\{z\}^\{\\prime\}\\rangle=\(1\-\\rho\)\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle\-\\rho, which converges to⟨ω∗,z−z′⟩\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangleasρ↓0\\rho\\downarrow 0\. Hence we can chooseρ\>0\\rho\>0small enough that⟨ω∗,z~−z~′⟩\>B−2​ε/3\\langle\\omega^\{\*\},\\tilde\{z\}\-\\tilde\{z\}^\{\\prime\}\\rangle\>B\-2\\varepsilon/3\. In particular,⟨ω∗,z~⟩\>⟨ω∗,z~′⟩\\langle\\omega^\{\*\},\\tilde\{z\}\\rangle\>\\langle\\omega^\{\*\},\\tilde\{z\}^\{\\prime\}\\rangle\.

Now choosek−2k\-2additional distinct outputsr1,…,rk−2∈\[0,1\]dr\_\{1\},\\ldots,r\_\{k\-2\}\\in\[0,1\]^\{d\}, distinct fromz~\\tilde\{z\}andz~′\\tilde\{z\}^\{\\prime\}, with sufficiently small coordinates so that, for everyℓ\\ell,⟨ω∗,rℓ⟩<⟨ω∗,z~⟩\\langle\\omega^\{\*\},r\_\{\\ell\}\\rangle<\\langle\\omega^\{\*\},\\tilde\{z\}\\rangleand⟨ω^,rℓ⟩<⟨ω^,z~′⟩\\langle\\widehat\{\\omega\},r\_\{\\ell\}\\rangle<\\langle\\widehat\{\\omega\},\\tilde\{z\}^\{\\prime\}\\rangle\. Such outputs exist because\[0,1\]d\[0,1\]^\{d\}contains infinitely many distinct points arbitrarily close to the origin, while⟨ω∗,z~⟩\>0\\langle\\omega^\{\*\},\\tilde\{z\}\\rangle\>0and⟨ω^,z~′⟩\>0\\langle\\widehat\{\\omega\},\\tilde\{z\}^\{\\prime\}\\rangle\>0\.

By the worst\-case\-domain assumption, there is an inputxxwhose feasible output set is𝒴​\(x\)=\{z~,z~′,r1,…,rk−2\}\\mathcal\{Y\}\(x\)=\\\{\\tilde\{z\},\\tilde\{z\}^\{\\prime\},r\_\{1\},\\ldots,r\_\{k\-2\}\\\}\. By construction,Fω∗​\(x\)=z~F\_\{\\omega^\{\*\}\}\(x\)=\\tilde\{z\}andFω^​\(x\)=z~′F\_\{\\widehat\{\\omega\}\}\(x\)=\\tilde\{z\}^\{\\prime\}\. Therefore,Regretx⁡\(ω^;ω∗\)=⟨ω∗,z~−z~′⟩\>B−2​ε/3\\operatorname\{Regret\}\_\{x\}\(\\widehat\{\\omega\};\\omega^\{\*\}\)=\\langle\\omega^\{\*\},\\tilde\{z\}\-\\tilde\{z\}^\{\\prime\}\\rangle\>B\-2\\varepsilon/3\. Sinceε\>0\\varepsilon\>0was arbitrary, we havesupxRegretx⁡\(ω^;ω∗\)≥B\\sup\_\{x\}\\operatorname\{Regret\}\_\{x\}\(\\widehat\{\\omega\};\\omega^\{\*\}\)\\geq B\. Combining this with the forward inequality proves the first equality\.

It remains to convert the pairwise optimization to the linear program inδ\\delta\. Letδ=z−z′\\delta=z\-z^\{\\prime\}\. Thenz,z′∈\[0,1\]dz,z^\{\\prime\}\\in\[0,1\]^\{d\}impliesδ∈\[−1,1\]d\\delta\\in\[\-1,1\]^\{d\}, and the learned\-rule constraint becomes⟨ω^,z′⟩≥⟨ω^,z⟩\\langle\\widehat\{\\omega\},z^\{\\prime\}\\rangle\\geq\\langle\\widehat\{\\omega\},z\\rangle, equivalently⟨ω^,δ⟩≤0\\langle\\widehat\{\\omega\},\\delta\\rangle\\leq 0\. The objective becomes⟨ω∗,z−z′⟩=⟨ω∗,δ⟩\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle=\\langle\\omega^\{\*\},\\delta\\rangle\. Thus every feasible pair\(z,z′\)\(z,z^\{\\prime\}\)induces a feasibleδ\\deltawith the same objective value\.

Conversely, every feasibleδ∈\[−1,1\]d\\delta\\in\[\-1,1\]^\{d\}satisfying⟨ω^,δ⟩≤0\\langle\\widehat\{\\omega\},\\delta\\rangle\\leq 0can be written asδ=z−z′\\delta=z\-z^\{\\prime\}for somez,z′∈\[0,1\]dz,z^\{\\prime\}\\in\[0,1\]^\{d\}: setzj=max⁡\{δj,0\}z\_\{j\}=\\max\\\{\\delta\_\{j\},0\\\}andzj′=max⁡\{−δj,0\}z^\{\\prime\}\_\{j\}=\\max\\\{\-\\delta\_\{j\},0\\\}for each coordinatejj\. Thenz,z′∈\[0,1\]dz,z^\{\\prime\}\\in\[0,1\]^\{d\},z−z′=δz\-z^\{\\prime\}=\\delta, and⟨ω^,z′⟩≥⟨ω^,z⟩\\langle\\widehat\{\\omega\},z^\{\\prime\}\\rangle\\geq\\langle\\widehat\{\\omega\},z\\rangle\. Hence this pair is feasible for the pairwise problem and achieves the same objective value asδ\\delta\. Therefore,

supz,z′∈\[0,1\]d\{⟨ω∗,z−z′⟩:⟨ω^,z′⟩≥⟨ω^,z⟩\}=maxδ∈\[−1,1\]d⁡\{⟨ω∗,δ⟩:⟨ω^,δ⟩≤0\}\.∎\\sup\_\{z,z^\{\\prime\}\\in\[0,1\]^\{d\}\}\\left\\\{\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle:\\langle\\widehat\{\\omega\},z^\{\\prime\}\\rangle\\geq\\langle\\widehat\{\\omega\},z\\rangle\\right\\\}=\\max\_\{\\delta\\in\[\-1,1\]^\{d\}\}\\left\\\{\\langle\\omega^\{\*\},\\delta\\rangle:\\langle\\widehat\{\\omega\},\\delta\\rangle\\leq 0\\right\\\}\.\\qed

We now show that the worst\-case utility range is11\.

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

Fix anyω∗∈Δd−1\\omega^\{\*\}\\in\\Delta^\{d\-1\}\. ThensupxRangex⁡\(ω∗\)=1\\sup\_\{x\}\\operatorname\{Range\}\_\{x\}\(\\omega^\{\*\}\)=1\.

###### Proof\.

The normalizing range is the largest utility spread attainable on a single inputx∈\(\[0,1\]d\)5x\\in\(\[0,1\]^\{d\}\)^\{5\}:

supxRangex⁡\(ω∗\)=supx\(maxy∈𝒴​\(x\)⁡⟨ω∗,ψ​\(x,y\)⟩−miny∈𝒴​\(x\)⁡⟨ω∗,ψ​\(x,y\)⟩\)\.\\sup\_\{x\}\\operatorname\{Range\}\_\{x\}\(\\omega^\{\*\}\)=\\sup\_\{x\}\\left\(\\max\_\{y\\in\\mathcal\{Y\}\(x\)\}\\langle\\omega^\{\*\},\\psi\(x,y\)\\rangle\-\\min\_\{y\\in\\mathcal\{Y\}\(x\)\}\\langle\\omega^\{\*\},\\psi\(x,y\)\\rangle\\right\)\.Equivalently, this is

supxRangex⁡\(ω∗\)=supxmaxy,y′∈𝒴​\(x\)⁡⟨ω∗,ψ​\(x,y\)−ψ​\(x,y′\)⟩\.\\sup\_\{x\}\\operatorname\{Range\}\_\{x\}\(\\omega^\{\*\}\)=\\sup\_\{x\}\\max\_\{y,y^\{\\prime\}\\in\\mathcal\{Y\}\(x\)\}\\left\\langle\\omega^\{\*\},\\psi\(x,y\)\-\\psi\(x,y^\{\\prime\}\)\\right\\rangle\.
We first upper bound this quantity\. Fix any inputxxand any pairy,y′∈𝒴​\(x\)y,y^\{\\prime\}\\in\\mathcal\{Y\}\(x\)\. Sincex∈\(\[0,1\]d\)5x\\in\(\[0,1\]^\{d\}\)^\{5\}, both feature vectorsψ​\(x,y\)\\psi\(x,y\)andψ​\(x,y′\)\\psi\(x,y^\{\\prime\}\)lie in\[0,1\]d\[0,1\]^\{d\}\. Therefore

⟨ω∗,ψ​\(x,y\)−ψ​\(x,y′\)⟩≤maxz,z′∈\[0,1\]d⁡⟨ω∗,z−z′⟩\.\\left\\langle\\omega^\{\*\},\\psi\(x,y\)\-\\psi\(x,y^\{\\prime\}\)\\right\\rangle\\leq\\max\_\{z,z^\{\\prime\}\\in\[0,1\]^\{d\}\}\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle\.Since this holds for every inputxxand every pairy,y′∈𝒴​\(x\)y,y^\{\\prime\}\\in\\mathcal\{Y\}\(x\),supxRangex⁡\(ω∗\)≤maxz,z′∈\[0,1\]d⁡⟨ω∗,z−z′⟩\\sup\_\{x\}\\operatorname\{Range\}\_\{x\}\(\\omega^\{\*\}\)\\leq\\max\_\{z,z^\{\\prime\}\\in\[0,1\]^\{d\}\}\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle\.

Becauseω∗∈Δd−1\\omega^\{\*\}\\in\\Delta^\{d\-1\}, all coordinates ofω∗\\omega^\{\*\}are nonnegative\. Thus, for anyz,z′∈\[0,1\]dz,z^\{\\prime\}\\in\[0,1\]^\{d\},⟨ω∗,z−z′⟩≤⟨ω∗,𝟏−𝟎⟩=‖ω∗‖1\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle\\leq\\langle\\omega^\{\*\},\\mathbf\{1\}\-\\mathbf\{0\}\\rangle=\\\|\\omega^\{\*\}\\\|\_\{1\}\. This upper bound is attained byz=𝟏z=\\mathbf\{1\}andz′=𝟎z^\{\\prime\}=\\mathbf\{0\}, somaxz,z′∈\[0,1\]d⁡⟨ω∗,z−z′⟩=‖ω∗‖1\\max\_\{z,z^\{\\prime\}\\in\[0,1\]^\{d\}\}\\langle\\omega^\{\*\},z\-z^\{\\prime\}\\rangle=\\\|\\omega^\{\*\}\\\|\_\{1\}\.

Finally, this upper bound is attainable in the worst\-case input domain\. Since the supremum ranges over allx∈\(\[0,1\]d\)5x\\in\(\[0,1\]^\{d\}\)^\{5\}, consider an input whose feasible output set contains one candidate with feature vector𝟏\\mathbf\{1\}and another with feature vector𝟎\\mathbf\{0\}\. For this input,Rangex⁡\(ω∗\)≥⟨ω∗,𝟏−𝟎⟩=‖ω∗‖1\\operatorname\{Range\}\_\{x\}\(\\omega^\{\*\}\)\\geq\\langle\\omega^\{\*\},\\mathbf\{1\}\-\\mathbf\{0\}\\rangle=\\\|\\omega^\{\*\}\\\|\_\{1\}\. Combining the upper and lower bounds givessupxRangex⁡\(ω∗\)=‖ω∗‖1\\sup\_\{x\}\\operatorname\{Range\}\_\{x\}\(\\omega^\{\*\}\)=\\\|\\omega^\{\*\}\\\|\_\{1\}\. Sinceω∗∈Δd−1\\omega^\{\*\}\\in\\Delta^\{d\-1\},‖ω∗‖1=1\\\|\\omega^\{\*\}\\\|\_\{1\}=1\. Therefore the denominator ofWC−RegretMω∗⁡\(Fω^\)\\operatorname\{WC\-Regret\}\_\{M\_\{\\omega^\{\*\}\}\}\(F\_\{\\widehat\{\\omega\}\}\)is11\. ∎

Combining the two lemmas gives the linear program we solve:

WC−RegretMω∗⁡\(Fω^\)=maxδ∈\[−1,1\]d⁡\{⟨ω∗,δ⟩:⟨ω^,δ⟩≤0\}\.\\operatorname\{WC\-Regret\}\_\{M\_\{\\omega^\{\*\}\}\}\(F\_\{\\widehat\{\\omega\}\}\)=\\max\_\{\\delta\\in\[\-1,1\]^\{d\}\}\\left\\\{\\langle\\omega^\{\*\},\\delta\\rangle:\\langle\\widehat\{\\omega\},\\delta\\rangle\\leq 0\\right\\\}\.

### C\.6\.Active Learning Algorithm Details

This appendix gives the algorithmic details for the active learning procedures used in[Section4\.3](https://arxiv.org/html/2607.02672#S4.SS3)\. Throughout,R∗R^\{\*\}denotes the true response model that generates the individual’s observed response, andRRdenotes the learner’s assumed response model used for posterior updates\. The learner’s unknown parameter vector is denoted byϑ\\vartheta\. In the simplest methods,ϑ=ω\\vartheta=\\omega; in the variants that learn thresholds or a flexible noise model,ϑ\\varthetaalso contains those additional parameters\.

##### Queries and candidate pool\.

At each active learning round, the learner does not optimize over all possible pairwise queries\. Instead, it draws a finite*candidate pool*

𝒞t=\{qt\(1\),…,qt\(C\)\}\\mathcal\{C\}\_\{t\}=\\\{q\_\{t\}^\{\(1\)\},\\ldots,q\_\{t\}^\{\(C\)\}\\\}ofCCrandomly sampled pairwise queries, scores each query in this pool, and asks the query with the highest score\. In the simulations, a query is generated by independently sampling two candidate feature vectors from\[0,1\]d\[0,1\]^\{d\}withd=5d=5\.

Let the feature difference beδ​\(q\)=ψ​\(x,y\)−ψ​\(x,y′\)\\delta\(q\)=\\psi\(x,y\)\-\\psi\(x,y^\{\\prime\}\)\. As established for the linear perfectly separable model \(after dropping the1/\|𝒳\|1/\|\\mathcal\{X\}\|factor\), the decisiveness and total evidence are respectivelyκMω​\(q\)=⟨ω,δ​\(q\)⟩\\kappa\_\{M\_\{\\omega\}\}\(q\)=\\langle\\omega,\\delta\(q\)\\rangleandrMω​\(q\)=∑j=1dωj​\|δj​\(q\)\|r\_\{M\_\{\\omega\}\}\(q\)=\\sum\_\{j=1\}^\{d\}\\omega\_\{j\}\\,\|\\delta\_\{j\}\(q\)\|\. In the linear simulationsrM​\(q\)≤1r\_\{M\}\(q\)\\leq 1, so we restrictτr∈\[0,1\]\\tau\_\{r\}\\in\[0,1\]\.

##### BALD score\.

Queries are selected using Bayesian Active Learning by Disagreement \(BALD\), a standard acquisition rule that scores a query by the expected information its response provides about the unknown parameter\(Houlsbyet al\.,[2011](https://arxiv.org/html/2607.02672#bib.bib292)\)\. At each round, the learner maintainsNpostN\_\{\\mathrm\{post\}\}posterior samples\{ϑt−1\(i\)\}i=1Npost\\\{\\vartheta\_\{t\-1\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\_\{\\mathrm\{post\}\}\}from the current posterior after transcript𝒯t−1\\mathcal\{T\}\_\{t\-1\}\. To estimate the BALD score of a candidate query, the learner uses a subsample of sizeNBALD≤NpostN\_\{\\mathrm\{BALD\}\}\\leq N\_\{\\mathrm\{post\}\}from this posterior sample set\. For a candidate queryqq, the learner evaluates the response distributionR\(⋅∣q,ϑt−1\(i\)\)R\(\\cdot\\mid q,\\vartheta\_\{t\-1\}^\{\(i\)\}\)under each sampled posterior draw\. The Monte Carlo BALD estimate is

\(6\)BALD^t\(q\)=Ent\(1NBALD∑i=1NBALDR\(⋅∣q,ϑt−1\(i\)\)\)−1NBALD∑i=1NBALDEnt\(R\(⋅∣q,ϑt−1\(i\)\)\),\\widehat\{\\mathrm\{BALD\}\}\_\{t\}\(q\)=\\mathrm\{Ent\}\\\!\\left\(\\frac\{1\}\{N\_\{\\mathrm\{BALD\}\}\}\\sum\_\{i=1\}^\{N\_\{\\mathrm\{BALD\}\}\}R\(\\cdot\\mid q,\\vartheta\_\{t\-1\}^\{\(i\)\}\)\\right\)\-\\frac\{1\}\{N\_\{\\mathrm\{BALD\}\}\}\\sum\_\{i=1\}^\{N\_\{\\mathrm\{BALD\}\}\}\\mathrm\{Ent\}\\\!\\left\(R\(\\cdot\\mid q,\\vartheta\_\{t\-1\}^\{\(i\)\}\)\\right\),whereEnt\\mathrm\{Ent\}is Shannon entropy\. The first term is the learner’s marginal uncertainty about the response toqq, averaging over posterior uncertainty aboutϑ\\vartheta; the second term subtracts the expected response noise that would remain ifϑ\\varthetawere known\. Thus, BALD rewards queries where different values ofϑ\\varthetagive highly different,confidentanswers—these are exactly the questions that can help pin downϑ\\varthetathe quickest\. In the main experiments, we useNpost=200N\_\{\\mathrm\{post\}\}=200andNBALD=50\.N\_\{\\mathrm\{BALD\}\}=50\.In the flexible\-noise variants \(Utilize\-4†,Ignore†,Correct†\), we useNBALD=30N\_\{\\mathrm\{BALD\}\}=30for computational efficiency\.

##### Posterior updating and MCMC

After each observed response, the learner must update its posterior over the unknown parameter vectorϑ\\vartheta\. Because this posterior generally cannot be sampled from exactly, we approximate it using Markov\-chain Monte Carlo \(MCMC\): a standard family of sampling methods that constructs a random walk over parameter values whose long\-run distribution is the desired posterior\. In each round, after observing a new response, the learner needs a fresh sample from the updated posterior\. We initialize the MCMC random walk using information from the previous round’s posterior, which we call a warm start\. The firstBBsteps of the chain are discarded and called “burn\-in,” because these early steps can still depend strongly on the chain’s initialization\. We then keep the nextNpostN\_\{\\mathrm\{post\}\}states of the chain and use them as the posterior sample set for the next BALD step\.

ALGORITHM 1BALD Active Learning TemplateInput:query budgetTT, candidate\-pool sizeCC, posterior\-sample countNpostN\_\{\\mathrm\{post\}\},BALD subsample sizeNBALDN\_\{\\mathrm\{BALD\}\}, MCMC burn\-inBB, priorπ0\\pi\_\{0\},query distribution𝒟q\\mathcal\{D\}\_\{q\}, learner response modelRR, true response modelR∗R^\{\*\}\.Initialize transcript𝒯0←∅\\mathcal\{T\}\_\{0\}\\leftarrow\\emptyset\.Draw posterior samples\{ϑ0\(i\)\}i=1Npost∼π0\\\{\\vartheta\_\{0\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\_\{\\mathrm\{post\}\}\}\\sim\\pi\_\{0\}\.fort=1,…,Tt=1,\\ldots,Tdo\(1\) Draw candidate pool:𝒞t=\{qt\(1\),…,qt\(C\)\},qt\(c\)∼i\.i\.d\.𝒟q\.\\mathcal\{C\}\_\{t\}=\\\{q\_\{t\}^\{\(1\)\},\\ldots,q\_\{t\}^\{\(C\)\}\\\},\\quad q\_\{t\}^\{\(c\)\}\\stackrel\{\{\\scriptstyle\\mathrm\{i\.i\.d\.\}\}\}\{\{\\sim\}\}\\mathcal\{D\}\_\{q\}\.\(2\) EstimateBALD^t​\(qt\(c\)\)\\widehat\{\\mathrm\{BALD\}\}\_\{t\}\(q\_\{t\}^\{\(c\)\}\)for eachqt\(c\)∈𝒞tq\_\{t\}^\{\(c\)\}\\in\\mathcal\{C\}\_\{t\}using equation \([6](https://arxiv.org/html/2607.02672#A3.E6)\) withNBALDN\_\{\\mathrm\{BALD\}\}posterior samples\.\(3\) Choose the next query:qt∈arg⁡maxq∈𝒞t⁡BALD^t​\(q\)\.q\_\{t\}\\in\\arg\\max\_\{q\\in\\mathcal\{C\}\_\{t\}\}\\widehat\{\\mathrm\{BALD\}\}\_\{t\}\(q\)\.\(4\) Observe response:⊳t∼R∗\(⋅∣qt;Mω∗\)\.\\triangleright\_\{t\}\\sim R^\{\*\}\(\\cdot\\mid q\_\{t\};M\_\{\\omega^\{\*\}\}\)\.\(5\) Update transcript according to whether the method wants to skip or retain that query\.For most methods,𝒯t←𝒯t−1∪\{\(qt,⊳t\)\}\\mathcal\{T\}\_\{t\}\\leftarrow\\mathcal\{T\}\_\{t\-1\}\\cup\\\{\(q\_\{t\},\\triangleright\_\{t\}\)\\\}\.ForIgnoremethods, indecisive responses are not retained\.\(6\) Sample from the learner posteriorπt\(ϑ\)∝π0\(ϑ\)∏\(q,⊳\)∈𝒯tR\(⊳∣q,ϑ\)\\pi\_\{t\}\(\\vartheta\)\\propto\\pi\_\{0\}\(\\vartheta\)\\prod\_\{\(q,\\triangleright\)\\in\\mathcal\{T\}\_\{t\}\}R\(\\triangleright\\mid q,\\vartheta\)using MCMC withBBburn\-in iterations andNpostN\_\{\\mathrm\{post\}\}retained samples, obtaining\{ϑt\(i\)\}i=1Npost\.\\\{\\vartheta\_\{t\}^\{\(i\)\}\\\}\_\{i=1\}^\{N\_\{\\mathrm\{post\}\}\}\.Letωt\(i\):=ω​\(ϑt\(i\)\)\\omega\_\{t\}^\{\(i\)\}:=\\omega\(\\vartheta\_\{t\}^\{\(i\)\}\)denote the priority\-weight component of theii\-th posterior draw\.end forOutput:ω^T=1Npost​∑i=1NpostωT\(i\)\.\\widehat\{\\omega\}\_\{T\}=\\frac\{1\}\{N\_\{\\mathrm\{post\}\}\}\\sum\_\{i=1\}^\{N\_\{\\mathrm\{post\}\}\}\\omega\_\{T\}^\{\(i\)\}\.
##### Sampling methods used\.

All posterior updates in Line \(6\) of Algorithm[1](https://arxiv.org/html/2607.02672#alg1)are performed using MCMC\. We use three standard MCMC methods\. Metropolis–Hastings is a basic MCMC update: the sampler proposes a random change to the current parameter value and accepts or rejects that proposal based on how plausible it is under the posterior\(Metropoliset al\.,[1953](https://arxiv.org/html/2607.02672#bib.bib180); Hastings,[1970](https://arxiv.org/html/2607.02672#bib.bib181)\)\. Metropolis\-within\-Gibbs is the version used when the parameter vector has several components\(Gelfand and Smith,[1990](https://arxiv.org/html/2607.02672#bib.bib182); Tierney,[1994](https://arxiv.org/html/2607.02672#bib.bib184); Chib and Greenberg,[1995](https://arxiv.org/html/2607.02672#bib.bib185)\)\. Instead of proposing a change to the entire vector at once, the sampler updates one component at a time, such asω\\omega, thenτr\\tau\_\{r\}, thenγ\\gamma, holding the others fixed during each update\. When a component cannot be sampled exactly, that component is updated using a Metropolis–Hastings step\. Third, for parameters constrained to lie on a simplex, such as the priority weightsω\\omega, we use hit\-and\-run proposals, which are designed for random\-walk sampling in bounded convex sets and therefore avoid leaving the simplex\(Smith,[1984](https://arxiv.org/html/2607.02672#bib.bib186)\)\.

##### Method specifications\.

All methods instantiate Algorithm[1](https://arxiv.org/html/2607.02672#alg1)\. Unless otherwise stated, the query distribution𝒟q\\mathcal\{D\}\_\{q\}, candidate\-pool sizeCC, posterior sample sizeNpostN\_\{\\mathrm\{post\}\}, BALD subsample sizeNBALDN\_\{\\mathrm\{BALD\}\}, and MCMC burn\-inBBare shared across methods and given in[Table2](https://arxiv.org/html/2607.02672#A3.T2)\. The methods differ only in five parts of the algorithm: the learned parameter vector and prior, the learner response modelRR, the true response modelR∗R^\{\*\}, the transcript retention rule, and the posterior sampler:

1. \(1\)Learned parameter vector and prior\.The methods use one of three learned parameterizations\. - •Known\-noise, known\-threshold methods\(Utilize\-4,Utilize\-3,Ignore, andCorrect\)\. The learner estimates only the priority weights:ϑ=ω∈Δd−1\\vartheta=\\omega\\in\\Delta^\{d\-1\}, withω∼Dirichlet​\(𝟏\)\\omega\\sim\\mathrm\{Dirichlet\}\(\\mathbf\{1\}\)\. The thresholds\(τr,τκ\)\(\\tau\_\{r\},\\tau\_\{\\kappa\}\), the link functionhh, and the logistic inverse temperatureβ\\betaare known\. - •Unknown\-noise methods\(Utilize\-4†,Ignore†, andCorrect†\)\. The learner estimates both the priority weights and a flexible noise model: ϑ=\(ω,η\),η=\(α,μ,σ\)\.\\vartheta=\(\\omega,\\eta\),\\qquad\\eta=\(\\alpha,\\mu,\\sigma\)\.Hereα=\(α1,α2,α3\)∈Δ2\\alpha=\(\\alpha\_\{1\},\\alpha\_\{2\},\\alpha\_\{3\}\)\\in\\Delta^\{2\}are the weights of a three\-component Gaussian mixture,μ=\(μ1,μ2,μ3\)\\mu=\(\\mu\_\{1\},\\mu\_\{2\},\\mu\_\{3\}\)are the component means, andσ=\(σ1,σ2,σ3\)\\sigma=\(\\sigma\_\{1\},\\sigma\_\{2\},\\sigma\_\{3\}\)are the component standard deviations\. These parameters define the Gaussian\-mixture CDF FGMM​\(t;η\)=∑k=13αk​Φ​\(t−μkσk\),F\_\{\\mathrm\{GMM\}\}\(t;\\eta\)=\\sum\_\{k=1\}^\{3\}\\alpha\_\{k\}\\Phi\\\!\\left\(\\frac\{t\-\\mu\_\{k\}\}\{\\sigma\_\{k\}\}\\right\),whereΦ\\Phiis the standard normal CDF\. The flexible\-noise learner usesFGMMF\_\{\\mathrm\{GMM\}\}as the CDF of an additive noise term, not as a symmetric two\-sided link\. In the four\-response model below, conflict is computed as the probability that a noisy latent margin falls inside the conflict band\. This gives a CDF difference, which is nonnegative by monotonicity ofFGMMF\_\{\\mathrm\{GMM\}\}, even when the Gaussian mixture is asymmetric\. We use the same uniform Dirichlet prior onω\\omegaas above\. For the mixture parameters, we use weakly informative priors: the component weights have a uniform Dirichlet prior, the component means have centered Gaussian priors, and the log standard deviations have Gaussian priors\. Eachσk\\sigma\_\{k\}is constrained to lie in\[σmin,σmax\]\[\\sigma\_\{\\min\},\\sigma\_\{\\max\}\]to avoid numerically degenerate mixture components\. The exact hyperparameter values are listed in[Table2](https://arxiv.org/html/2607.02672#A3.T2)\. - •Threshold\-learning method\(Utilize\-4⋄\)\. This method is the version ofUtilize\-4that does not assume the response thresholds are known\. The learner estimatesϑ=\(ω,τr,γ\)\\vartheta=\(\\omega,\\tau\_\{r\},\\gamma\), whereω\\omegais the priority\-weight vector,τr\\tau\_\{r\}is the total\-evidence threshold, andγ\\gammais the learner’s estimate of the conflict thresholdτκ\\tau\_\{\\kappa\}\. We allowτr\\tau\_\{r\}to vary continuously over\[0,1\]\[0,1\]\. For the conflict threshold, we use a finite grid of possible values, Γ=\{0,γmaxKγ−1,2​γmaxKγ−1,…,γmax\}\.\\Gamma=\\left\\\{0,\\frac\{\\gamma\_\{\\max\}\}\{K\_\{\\gamma\}\-1\},\\frac\{2\\gamma\_\{\\max\}\}\{K\_\{\\gamma\}\-1\},\\ldots,\\gamma\_\{\\max\}\\right\\\}\.HereKγK\_\{\\gamma\}is the number of grid points, andγmax=0\.95\\gamma\_\{\\max\}=0\.95is the largest allowed value\. Thus, instead of learning an arbitrary real\-valued conflict threshold, the learner chooses amongKγK\_\{\\gamma\}evenly spaced candidate values between0and0\.950\.95\. The prior onω\\omegais uniform over the simplex,ω∼Dirichlet​\(𝟏\)\\omega\\sim\\mathrm\{Dirichlet\}\(\\mathbf\{1\}\), and the prior onτr\\tau\_\{r\}is uniform over\[0,1\]\[0,1\],τr∼Uniform​\[0,1\]\\tau\_\{r\}\\sim\\mathrm\{Uniform\}\[0,1\]\. Forγ\\gamma, we put a prior directly on the gridΓ\\Gamma\. This prior assigns zero mass to the two grid endpoints and positive mass to interior grid values\. Thus, it excludes the degenerate boundary casesγ=0\\gamma=0andγ=γmax\\gamma=\\gamma\_\{\\max\}while placing a weak symmetric prior over the remaining interior grid values\. See details in Appendix[C\.7](https://arxiv.org/html/2607.02672#A3.SS7)\.
2. \(2\)Learner’s assumed response modelRRused in Line \(2\) and Line \(6\)\.The learner response model is the model used both to score candidate queries by BALD and to compute the likelihood in the posterior update\. - •Four\-response logistic learner\(Utilize\-4\)\. The learner uses the four\-response model R\(⋅∣q,ω\)=Rhβlogit;τr,τκ∘\(⋅∣q;Mω\),R\(\\cdot\\mid q,\\omega\)=R^\{\\circ\}\_\{h^\{\\mathrm\{logit\}\}\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(\\cdot\\mid q;M\_\{\\omega\}\),wherehβlogit​\(t\)=\(1\+exp⁡\(−β​t\)\)−1h^\{\\mathrm\{logit\}\}\_\{\\beta\}\(t\)=\(1\+\\exp\(\-\\beta t\)\)^\{\-1\}\. - •Three\-response generic\-indecision learner\(Utilize\-3\)\. The learner uses the collapsed three\-response model R\(⋅∣q,ω\)=Rhβlogit;τr,τκ⊘\(⋅∣q;Mω\),R\(\\cdot\\mid q,\\omega\)=R^\{\\oslash\}\_\{h^\{\\mathrm\{logit\}\}\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(\\cdot\\mid q;M\_\{\\omega\}\),over alphabet\{≻,≺,⊘\}\\\{\\succ,\\prec,\\oslash\\\}, where⊘\\oslashdenotes generic indecision\. This model is formalized in[DefinitionC\.8](https://arxiv.org/html/2607.02672#A3.Thmtheorem8)\. - •Four\-response flexible\-noise learner\(Utilize\-4†\)\. The learner uses a flexible latent\-noise model for the four\-response probabilities\. For a queryqq, letδ=κMω​\(q\)\\delta=\\kappa\_\{M\_\{\\omega\}\}\(q\)be the signed aggregate evidence and letr=rMω​\(q\)r=r\_\{M\_\{\\omega\}\}\(q\)be its total\-evidence magnitude\. The learner models the response as depending on a noisy latent marginδ\+ε\\delta\+\\varepsilon, whereε\\varepsilonhas CDFFGMM​\(⋅;η\)F\_\{\\mathrm\{GMM\}\}\(\\cdot;\\eta\)\. Ifr<τrr<\\tau\_\{r\}, the query has insufficient total evidence, soR\(∼∣q,ω,η\)=1R\(\\sim\\mid q,\\omega,\\eta\)=1\. Ifr≥τrr\\geq\\tau\_\{r\}, the conflict band is\[−τκ​r,τκ​r\]\[\-\\tau\_\{\\kappa\}r,\\tau\_\{\\kappa\}r\], and the response probabilities are R\(≻∣q,ω,η\)\\displaystyle R\(\\succ\\mid q,\\omega,\\eta\)=1−FGMM​\(τκ​r−δ;η\),\\displaystyle=1\-F\_\{\\mathrm\{GMM\}\}\(\\tau\_\{\\kappa\}r\-\\delta;\\eta\),R\(≺∣q,ω,η\)\\displaystyle R\(\\prec\\mid q,\\omega,\\eta\)=FGMM​\(−τκ​r−δ;η\),\\displaystyle=F\_\{\\mathrm\{GMM\}\}\(\-\\tau\_\{\\kappa\}r\-\\delta;\\eta\),R\(⋈∣q,ω,η\)\\displaystyle R\(\\bowtie\\mid q,\\omega,\\eta\)=FGMM​\(τκ​r−δ;η\)−FGMM​\(−τκ​r−δ;η\),\\displaystyle=F\_\{\\mathrm\{GMM\}\}\(\\tau\_\{\\kappa\}r\-\\delta;\\eta\)\-F\_\{\\mathrm\{GMM\}\}\(\-\\tau\_\{\\kappa\}r\-\\delta;\\eta\),R\(∼∣q,ω,η\)\\displaystyle R\(\\sim\\mid q,\\omega,\\eta\)=0\.\\displaystyle=0\.These are exactly the probabilities that the noisy marginδ\+ε\\delta\+\\varepsilonlies above the conflict band, below the conflict band, or inside the conflict band\. This construction gives valid probabilities for any Gaussian\-mixture parametersη\\eta, including asymmetric mixtures\. Sinceτκ​r≥0\\tau\_\{\\kappa\}r\\geq 0, the upper endpointτκ​r−δ\\tau\_\{\\kappa\}r\-\\deltais at least the lower endpoint−τκ​r−δ\-\\tau\_\{\\kappa\}r\-\\delta\. BecauseFGMMF\_\{\\mathrm\{GMM\}\}is a CDF and hence monotone, the conflict probabilityFGMM​\(τκ​r−δ;η\)−FGMM​\(−τκ​r−δ;η\)F\_\{\\mathrm\{GMM\}\}\(\\tau\_\{\\kappa\}r\-\\delta;\\eta\)\-F\_\{\\mathrm\{GMM\}\}\(\-\\tau\_\{\\kappa\}r\-\\delta;\\eta\)is nonnegative\. The other two probabilities are a lower\-tail probability and an upper\-tail probability, and the three terms sum to one\. - •Four\-response threshold\-learning learner\(Utilize\-4⋄\)\. The learner uses R\(⋅∣q,ω,τr,γ\)=Rhβlogit;τr,γ∘\(⋅∣q;Mω\)\.R\(\\cdot\\mid q,\\omega,\\tau\_\{r\},\\gamma\)=R^\{\\circ\}\_\{h^\{\\mathrm\{logit\}\}\_\{\\beta\};\\tau\_\{r\},\\gamma\}\(\\cdot\\mid q;M\_\{\\omega\}\)\.Further details are given in[SectionC\.7](https://arxiv.org/html/2607.02672#A3.SS7)\. - •Binary logistic learner\(IgnoreandCorrect\)\. The learner assumes a binary Bradley–Terry response model: R\(≻∣q,ω\)=hβlogit\(κMω\(q\)\),R\(≺∣q,ω\)=hβlogit\(−κMω\(q\)\)\.R\(\\succ\\mid q,\\omega\)=h^\{\\mathrm\{logit\}\}\_\{\\beta\}\(\\kappa\_\{M\_\{\\omega\}\}\(q\)\),\\qquad R\(\\prec\\mid q,\\omega\)=h^\{\\mathrm\{logit\}\}\_\{\\beta\}\(\-\\kappa\_\{M\_\{\\omega\}\}\(q\)\)\.BALD scores are computed under this binary model, and the posterior is updated only on retained binary labels\. - •Binary flexible\-noise learner\(Ignore†andCorrect†\)\. The learner uses the corresponding binary latent\-noise likelihood\. Forδ=κMω​\(q\)\\delta=\\kappa\_\{M\_\{\\omega\}\}\(q\), the noisy latent margin is againδ\+ε\\delta\+\\varepsilon, withε\\varepsilondistributed according toFGMM​\(⋅;η\)F\_\{\\mathrm\{GMM\}\}\(\\cdot;\\eta\)\. The binary response probabilities are R\(≻∣q,ω,η\)=Pr\(δ\+ε\>0\)=1−FGMM\(−δ;η\),R\(≺∣q,ω,η\)=Pr\(δ\+ε<0\)=FGMM\(−δ;η\)\.R\(\\succ\\mid q,\\omega,\\eta\)=\\Pr\(\\delta\+\\varepsilon\>0\)=1\-F\_\{\\mathrm\{GMM\}\}\(\-\\delta;\\eta\),\\qquad R\(\\prec\\mid q,\\omega,\\eta\)=\\Pr\(\\delta\+\\varepsilon<0\)=F\_\{\\mathrm\{GMM\}\}\(\-\\delta;\\eta\)\.These probabilities are valid for any Gaussian\-mixture parameters because they are a CDF tail probability and its complement\. As in the four\-response case, the learner does not require the Gaussian mixture to be symmetric\.
3. \(3\)True response modelR∗R^\{\*\}used in Line \(4\)\.The true response model is the data\-generating model used to sample the observed response⊳t\\triangleright\_\{t\}\. - •Four\-response logistic individual\(Utilize\-4,Utilize\-4†,Utilize\-4⋄,Ignore, andIgnore†\)\. Responses are generated from the four\-response model at the true weights and true thresholds: R∗\(⋅∣q\)=Rhβlogit;τr∗,τκ∗∘\(⋅∣q;Mω∗\)\.R^\{\*\}\(\\cdot\\mid q\)=R^\{\\circ\}\_\{h^\{\\mathrm\{logit\}\}\_\{\\beta\};\\tau\_\{r\}^\{\*\},\\tau\_\{\\kappa\}^\{\*\}\}\(\\cdot\\mid q;M\_\{\\omega^\{\*\}\}\)\.For methods that treat thresholds as known, we set\(τr,τκ\)=\(τr∗,τκ∗\)\(\\tau\_\{r\},\\tau\_\{\\kappa\}\)=\(\\tau\_\{r\}^\{\*\},\\tau\_\{\\kappa\}^\{\*\}\)\. - •Collapsed three\-response individual\(Utilize\-3\)\. Responses are first generated from the four\-response logistic model at the true weights and true thresholds, and then collapsed to the alphabet\{≻,≺,⊘\}\\\{\\succ,\\prec,\\oslash\\\}by mapping both indecisive responses to generic indecision\. Equivalently, R∗\(⋅∣q\)=Rhβlogit;τr∗,τκ∗⊘\(⋅∣q;Mω∗\),R^\{\*\}\(\\cdot\\mid q\)=R^\{\\oslash\}\_\{h^\{\\mathrm\{logit\}\}\_\{\\beta\};\\tau\_\{r\}^\{\*\},\\tau\_\{\\kappa\}^\{\*\}\}\(\\cdot\\mid q;M\_\{\\omega^\{\*\}\}\),as formalized in[DefinitionC\.8](https://arxiv.org/html/2607.02672#A3.Thmtheorem8)\. - •Binary forced\-choice individual\(CorrectandCorrect†\)\. Responses are generated directly from the binary zero\-threshold logistic model at the true weights: R∗\(≻∣q\)=hβlogit\(κMω∗\(q\)\),R∗\(≺∣q\)=hβlogit\(−κMω∗\(q\)\)\.R^\{\*\}\(\\succ\\mid q\)=h^\{\\mathrm\{logit\}\}\_\{\\beta\}\(\\kappa\_\{M\_\{\\omega^\{\*\}\}\}\(q\)\),\\qquad R^\{\*\}\(\\prec\\mid q\)=h^\{\\mathrm\{logit\}\}\_\{\\beta\}\(\-\\kappa\_\{M\_\{\\omega^\{\*\}\}\}\(q\)\)\.These methods are best\-case forced\-choice benchmarks: unlikeIgnore, the data are generated by the same binary response family that the learner assumes\.
4. \(4\)Transcript retention rule in Line \(5\)\.The transcript retention rule specifies which observed responses are used in the posterior update\. - •Retain all responses\(Utilize\-4,Utilize\-3,Utilize\-4†,Utilize\-4⋄,Correct, andCorrect†\)\. Every observed response is retained:𝒯t=𝒯t−1∪\{\(qt,⊳t\)\}\\mathcal\{T\}\_\{t\}=\\mathcal\{T\}\_\{t\-1\}\\cup\\\{\(q\_\{t\},\\triangleright\_\{t\}\)\\\}\. ForUtilize\-3, this includes generic\-indecision responses⊘\\oslash\. ForCorrectandCorrect†, every response is already binary\. - •Discard indecisive responses\(IgnoreandIgnore†\)\. Decisive responses are retained:𝒯t=𝒯t−1∪\{\(qt,⊳t\)\}\\mathcal\{T\}\_\{t\}=\\mathcal\{T\}\_\{t\-1\}\\cup\\\{\(q\_\{t\},\\triangleright\_\{t\}\)\\\}if⊳t∈\{≻,≺\}\\triangleright\_\{t\}\\in\\\{\\succ,\\prec\\\}\. Indecisive responses are discarded:𝒯t=𝒯t−1\\mathcal\{T\}\_\{t\}=\\mathcal\{T\}\_\{t\-1\}if⊳t∈\{∼,⋈\}\\triangleright\_\{t\}\\in\\\{\\sim,\\bowtie\\\}\. The query still counts against the query budget even when its response is discarded\.
5. \(5\)Posterior sampler in Line \(6\)\.The sampler is chosen according to what parameters the learner must learn\. - •Simplex\-only samplers\(Utilize\-4,Utilize\-3,Ignore, andCorrect\)\. The posterior is overω∈Δd−1\\omega\\in\\Delta^\{d\-1\}\. We use Metropolis–Hastings with hit\-and\-run proposals on the simplex\. - •Flexible\-noise samplers\(Utilize\-4†,Ignore†, andCorrect†\)\. The posterior is over\(ω,η\)\(\\omega,\\eta\), whereη=\(α,μ,σ\)\\eta=\(\\alpha,\\mu,\\sigma\)parameterizes the Gaussian\-mixture noise CDFFGMM​\(⋅;η\)F\_\{\\mathrm\{GMM\}\}\(\\cdot;\\eta\)\. We update the different parts of\(ω,η\)\(\\omega,\\eta\)separately\. We use hit\-and\-run proposals forω\\omegaandα\\alpha, because both are vectors on the simplex\. For the remaining mixture parameters,μ\\muandlog⁡σ\\log\\sigma, we use random\-walk Metropolis updates: the sampler proposes a small random change to the current value and accepts the change if it is sufficiently plausible under the posterior\. - •Threshold\-learning sampler\(Utilize\-4⋄\)\. In this variant, the learner is uncertain not only about the priority weightsω\\omega, but also about the two response thresholds\. Thus, after each new response, the learner updates a posterior over\(ω,τr,γ\)\(\\omega,\\tau\_\{r\},\\gamma\), whereτr\\tau\_\{r\}is the total\-evidence threshold andγ\\gammais the learner’s discretized version of the conflict thresholdτκ\\tau\_\{\\kappa\}\. Because multiple parameters are being learned, the algorithm uses Metropolis\-within\-Gibbs\. The sampler updates these three quantities one at a time\. First, it updatesω\\omega, the priority\-weight vector\. Sinceω\\omegamust remain in the simplex, we use a hit\-and\-run proposal\. Second, it updatesτr\\tau\_\{r\}\. Sinceτr\\tau\_\{r\}is a single number in\[0,1\]\[0,1\], we propose a small random change to its current value and reflect proposals that fall outside\[0,1\]\[0,1\]back into the interval\. Third, it updatesγ\\gamma\. Sinceγ\\gammais restricted to the finite gridΓ\\Gamma, we compute the posterior probability of each grid value conditional on the currentω\\omegaandτr\\tau\_\{r\}, and then sample a new value ofγ\\gammafrom this finite distribution\.

Table 2\.Hyperparameters for the BALD experiments\.The proposal\-size hyperparameters control the size of the random moves used by the MCMC sampler\. For example, when updatingτr\\tau\_\{r\}, the sampler proposes a new value by adding a small random perturbation to the current value:τr′=reflect\[0,1\]​\(τr\+ϵ\)\\tau\_\{r\}^\{\\prime\}=\\mathrm\{reflect\}\_\{\[0,1\]\}\(\\tau\_\{r\}\+\\epsilon\), whereϵ∼𝒩​\(0,sdτr2\)\\epsilon\\sim\\mathcal\{N\}\(0,\\mathrm\{sd\}\_\{\\tau\_\{r\}\}^\{2\}\)\. Herereflect\[0,1\]\\mathrm\{reflect\}\_\{\[0,1\]\}maps a real number back into\[0,1\]\[0,1\]by bouncing it off the endpoints: for example,1\.051\.05is mapped to0\.950\.95, and−0\.05\-0\.05is mapped to0\.050\.05\. This reflection only ensures that the proposed value is valid\. The proposal is still accepted or rejected afterward using the usual Metropolis acceptance probability\. Thus, reflection handles feasibility, while Metropolis rejection handles posterior plausibility\. The valuesdτr\\mathrm\{sd\}\_\{\\tau\_\{r\}\}controls how large the proposed moves inτr\\tau\_\{r\}tend to be\. Similarly, the GMM proposal sizes control the size of the random moves proposed for the mixture meansμ\\muand log standard deviationslog⁡σ\\log\\sigma\.

All experiments in Sections[4\.2](https://arxiv.org/html/2607.02672#S4.SS2)and[4\.3](https://arxiv.org/html/2607.02672#S4.SS3), unless otherwise noted, use4040independently sampled true oraclesω∗∼Dirichlet​\(0\.2​1\)\\omega^\{\\ast\}\\sim\\mathrm\{Dirichlet\}\(0\.2\\,\\mathbf\{1\}\)ind=5d=5, logistic inverse temperatureβ=10\\beta=10, candidate\-pool sizeC=50C=50,Npost=200N\_\{\\mathrm\{post\}\}=200retained posterior samples,B=200B=200warm\-started burn\-in samples, and BALD acquisition\. In Section[4\.2](https://arxiv.org/html/2607.02672#S4.SS2), we learn theω^\\widehat\{\\omega\}’s using theCorrectversion of the BALD algorithm detailed in Appendix[C\.6](https://arxiv.org/html/2607.02672#A3.SS6)\. Because we are trying to test how bad our learning and regret are when we assume the typical Bradley\-Terry response model, the learner assumes

R\(⋅∣q,ω\)=Rhβlogit;τr,τκ∘\(⋅∣q;Mω\),R\(\\cdot\\mid q,\\omega\)=R^\{\\circ\}\_\{h^\{\\mathrm\{logit\}\}\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(\\cdot\\mid q;M\_\{\\omega\}\),The true response modelsR∗R^\{\*\}are detailed in the above section\. To ensure indecision is present but not overwhelming, we learn allω^\\widehat\{\\omega\}’s under the regimeτr=τκ=0\.25\\tau\_\{r\}=\\tau\_\{\\kappa\}=0\.25\. Since we are not testing speed of learning in this section, we run all learners to convergence:\|ω^t\+1−ω^t\|<ε=0\.01\|\\widehat\{\\omega\}\_\{t\+1\}\-\\widehat\{\\omega\}\_\{t\}\|<\\varepsilon=0\.01for 5 consecutive queries\. The same random seeds for the BALD acquisition steps are used across all four true response modelsR∗R^\{\*\}’s for everyω∗\\omega^\{\*\}in the 40 runs, so the four methods are directly comparable\.

### C\.7\.Learningτr\\tau\_\{r\}andτκ\\tau\_\{\\kappa\}

This section details the posterior update used byUtilize\-4⋄, the version ofUtilize\-4that does not assume the response thresholds are known\. Instead of learning only the priority weightsω\\omega, the learner jointly estimatesϑ=\(ω,τr,γ\)\.\\vartheta=\(\\omega,\\tau\_\{r\},\\gamma\)\.Hereτr\\tau\_\{r\}is the total\-evidence threshold, andγ\\gammais the learner’s estimate of the conflict thresholdτκ\\tau\_\{\\kappa\}\.

For a queryqq, recall that

rMω​\(q\)=∑j=1dωj​\|δj​\(q\)\|,κMω​\(q\)=∑j=1dωj​δj​\(q\)\.r\_\{M\_\{\\omega\}\}\(q\)=\\sum\_\{j=1\}^\{d\}\\omega\_\{j\}\|\\delta\_\{j\}\(q\)\|,\\qquad\\kappa\_\{M\_\{\\omega\}\}\(q\)=\\sum\_\{j=1\}^\{d\}\\omega\_\{j\}\\delta\_\{j\}\(q\)\.Both quantities are linear functions ofω\\omega\. The two thresholds enter the response model differently\. The thresholdτr\\tau\_\{r\}is a single cutoff for total evidence, so we learn it as a continuous parameter in\[0,1\]\[0,1\]\. By contrast,τκ\\tau\_\{\\kappa\}appears in comparisons of the form\|κMω​\(q\)\|<τκ​rMω​\(q\)\.\|\\kappa\_\{M\_\{\\omega\}\}\(q\)\|<\\tau\_\{\\kappa\}r\_\{M\_\{\\omega\}\}\(q\)\.Becauseτκ\\tau\_\{\\kappa\}is multiplied byrMω​\(q\)r\_\{M\_\{\\omega\}\}\(q\), learningτκ\\tau\_\{\\kappa\}continuously together withω\\omegawould introduce bilinear terms\. To keep the posterior update simple and stable, we learn the conflict threshold over a finite grid\. Specifically, the learner choosesγ∈Γ,\\gamma\\in\\Gamma,where

Γ=\{0,γmaxKγ−1,2​γmaxKγ−1,…,γmax\}\.\\Gamma=\\left\\\{0,\\frac\{\\gamma\_\{\\max\}\}\{K\_\{\\gamma\}\-1\},\\frac\{2\\gamma\_\{\\max\}\}\{K\_\{\\gamma\}\-1\},\\ldots,\\gamma\_\{\\max\}\\right\\\}\.In the experiments,γmax=0\.95\\gamma\_\{\\max\}=0\.95andKγ=10K\_\{\\gamma\}=10\. Thus,γ\\gammais chosen from ten evenly spaced candidate values between0and0\.950\.95\. Note that this grid does not actually perfectly overlap with the true values ofτκ\\tau\_\{\\kappa\}\.

At active\-learning roundtt, the learner has observed a transcript𝒯t=\{\(qs,⊳s\)\}s=1t,\\mathcal\{T\}\_\{t\}=\\\{\(q\_\{s\},\\triangleright\_\{s\}\)\\\}\_\{s=1\}^\{t\},whereqsq\_\{s\}is the query asked in roundssand⊳s\\triangleright\_\{s\}is the observed response\. For any candidate parameter value\(ω,τr,γ\)\(\\omega,\\tau\_\{r\},\\gamma\), the likelihood of this transcript is

ℒt\(ω,τr,γ\)=∏s=1tRhβ;τr,γ∘\(⊳s∣qs;Mω\)\.\\mathcal\{L\}\_\{t\}\(\\omega,\\tau\_\{r\},\\gamma\)=\\prod\_\{s=1\}^\{t\}R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\gamma\}\(\\triangleright\_\{s\}\\mid q\_\{s\};M\_\{\\omega\}\)\.The prior factorizes across the three learned quantities:π0​\(ω,τr,γ\)=π0​\(ω\)​π0​\(τr\)​π0​\(γ\)\.\\pi\_\{0\}\(\\omega,\\tau\_\{r\},\\gamma\)=\\pi\_\{0\}\(\\omega\)\\pi\_\{0\}\(\\tau\_\{r\}\)\\pi\_\{0\}\(\\gamma\)\.We use a uniform prior over the simplex forω\\omegaand a uniform prior over\[0,1\]\[0,1\]forτr\\tau\_\{r\}\. Forγ\\gamma, we put a prior directly on the gridΓ\\Gamma:

π0​\(γ\)∝\[u​\(1−u\)\]1/2,u=γ/γmax\.\\pi\_\{0\}\(\\gamma\)\\propto\[u\(1\-u\)\]^\{1/2\},\\qquad u=\\gamma/\\gamma\_\{\\max\}\.This prior assigns zero mass to the two grid endpoints and positive mass to interior grid values\. Thus, it excludes the degenerate boundary casesγ=0\\gamma=0andγ=γmax\\gamma=\\gamma\_\{\\max\}while placing a weak symmetric prior over the remaining interior grid values\.

The posterior after transcript𝒯t\\mathcal\{T\}\_\{t\}is therefore

πt​\(ω,τr,γ\)∝π0​\(ω\)​π0​\(τr\)​π0​\(γ\)​ℒt​\(ω,τr,γ\)\.\\pi\_\{t\}\(\\omega,\\tau\_\{r\},\\gamma\)\\propto\\pi\_\{0\}\(\\omega\)\\pi\_\{0\}\(\\tau\_\{r\}\)\\pi\_\{0\}\(\\gamma\)\\mathcal\{L\}\_\{t\}\(\\omega,\\tau\_\{r\},\\gamma\)\.Algorithm[2](https://arxiv.org/html/2607.02672#alg2)describes one MCMC sweep \(a single increment intt\) for sampling from this posterior\. A sweep means one pass through the three components of the parameter: first updateω\\omega, then updateτr\\tau\_\{r\}, then updateγ\\gamma\. In the BALD template, Line \(6\) runs many such sweeps: the firstBBsweeps are discarded as burn\-in, and the nextNpostN\_\{\\mathrm\{post\}\}states are retained as posterior samples\.

ALGORITHM 2Threshold\-Learning Posterior Update \(one MCMC sweep\)Input:transcript𝒯t=\{\(qs,⊳s\)\}s=1t\\mathcal\{T\}\_\{t\}=\\\{\(q\_\{s\},\\triangleright\_\{s\}\)\\\}\_\{s=1\}^\{t\}; current state\(ω,τr,γ\)t\(\\omega,\\tau\_\{r\},\\gamma\)\_\{t\}; gridΓ=\{γ1,…,γKγ\}\\Gamma=\\\{\\gamma\_\{1\},\\ldots,\\gamma\_\{K\_\{\\gamma\}\}\\\};priorπ0\\pi\_\{0\}; linkhβh\_\{\\beta\}; proposal size sdτr\{\}\_\{\\tau\_\{r\}\}\(1\) Updateω\\omega\.Propose a new priority\-weight vectorω′\\omega^\{\\prime\}using a hit\-and\-run step onΩ=\{ω∈ℝ≥0d:‖ω‖1=1\}\\Omega=\\\{\\omega\\in\\mathbb\{R\}^\{d\}\_\{\\geq 0\}:\\\|\\omega\\\|\_\{1\}=1\\\}\.Acceptω′\\omega^\{\\prime\}with probabilityaω=min⁡\{1,π0​\(ω′\)​ℒt​\(ω′,τr,γ\)π0​\(ω\)​ℒt​\(ω,τr,γ\)\}\.\\displaystyle a\_\{\\omega\}=\\min\\\!\\left\\\{1,\\ \\frac\{\\pi\_\{0\}\(\\omega^\{\\prime\}\)\\mathcal\{L\}\_\{t\}\(\\omega^\{\\prime\},\\tau\_\{r\},\\gamma\)\}\{\\pi\_\{0\}\(\\omega\)\\mathcal\{L\}\_\{t\}\(\\omega,\\tau\_\{r\},\\gamma\)\}\\right\\\}\.If the proposal is rejected, keep the current value ofω\\omega\.\(2\) Updateτr\\tau\_\{r\}\.Propose a small random change toτr\\tau\_\{r\}:τr′=reflect\[0,1\]​\(τr\+ϵ\),ϵ∼𝒩​\(0,sdτr2\)\.\\displaystyle\\tau\_\{r\}^\{\\prime\}=\\mathrm\{reflect\}\_\{\[0,1\]\}\(\\tau\_\{r\}\+\\epsilon\),\\qquad\\epsilon\\sim\\mathcal\{N\}\(0,\\mathrm\{sd\}\_\{\\tau\_\{r\}\}^\{2\}\)\.The reflection keeps the proposal inside the interval\[0,1\]\[0,1\]\.Acceptτr′\\tau\_\{r\}^\{\\prime\}with probabilityaτ=min⁡\{1,π0​\(τr′\)​ℒt​\(ω,τr′,γ\)π0​\(τr\)​ℒt​\(ω,τr,γ\)\}\.\\displaystyle a\_\{\\tau\}=\\min\\\!\\left\\\{1,\\ \\frac\{\\pi\_\{0\}\(\\tau\_\{r\}^\{\\prime\}\)\\mathcal\{L\}\_\{t\}\(\\omega,\\tau\_\{r\}^\{\\prime\},\\gamma\)\}\{\\pi\_\{0\}\(\\tau\_\{r\}\)\\mathcal\{L\}\_\{t\}\(\\omega,\\tau\_\{r\},\\gamma\)\}\\right\\\}\.If the proposal is rejected, keep the current value ofτr\\tau\_\{r\}\.\(3\) Updateγ\\gamma\.Sinceγ\\gammalies on the finite gridΓ\\Gamma, compute one posterior weight per grid point:ak=π0​\(γk\)​ℒt​\(ω,τr,γk\),k=1,…,Kγ\.\\displaystyle a\_\{k\}=\\pi\_\{0\}\(\\gamma\_\{k\}\)\\mathcal\{L\}\_\{t\}\(\\omega,\\tau\_\{r\},\\gamma\_\{k\}\),\\qquad k=1,\\ldots,K\_\{\\gamma\}\.Normalize these weights:pk=ak∑ℓ=1Kγaℓ\.\\displaystyle p\_\{k\}=\\frac\{a\_\{k\}\}\{\\sum\_\{\\ell=1\}^\{K\_\{\\gamma\}\}a\_\{\\ell\}\}\.Draw the new value ofγ\\gammafrom the grid distributionPr⁡\(γ=γk∣ω,τr,𝒯t\)=pk\.\\Pr\(\\gamma=\\gamma\_\{k\}\\mid\\omega,\\tau\_\{r\},\\mathcal\{T\}\_\{t\}\)=p\_\{k\}\.Output:updated state\(ω,τr,γ\)t\+1\(\\omega,\\tau\_\{r\},\\gamma\)\_\{t\+1\}\.The three updates have simple roles\. The update forω\\omegamoves around the simplex of valid priority weights\. The update forτr\\tau\_\{r\}makes a small continuous move in the interval\[0,1\]\[0,1\]\. The update forγ\\gammachecks all grid values and resamplesγ\\gammaaccording to their posterior probabilities\. Becauseγ\\gammais resampled inside the same Markov chain asω\\omegaandτr\\tau\_\{r\}, the posterior samples represent uncertainty about all three quantities jointly\.

In the Bayesian active learning step, the learner uses these posterior samples exactly as in Algorithm[1](https://arxiv.org/html/2607.02672#alg1), except that each posterior draw now contains\(ω,τr,γ\)\(\\omega,\\tau\_\{r\},\\gamma\)rather than onlyω\\omega\. For a candidate queryqq, define

ρω,τr,γ,q\(⊳\)=Rhβ;τr,γ∘\(⊳∣q;Mω\)\.\\rho\_\{\\omega,\\tau\_\{r\},\\gamma,q\}\(\\triangleright\)=R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\gamma\}\(\\triangleright\\mid q;M\_\{\\omega\}\)\.The BALD score is

BALDt​\(q\)=Ent​\(𝔼\(ω,τr,γ\)∼πt​\[ρω,τr,γ,q\]\)−𝔼\(ω,τr,γ\)∼πt​\[Ent​\(ρω,τr,γ,q\)\]\.\\mathrm\{BALD\}\_\{t\}\(q\)=\\mathrm\{Ent\}\\\!\\left\(\\mathbb\{E\}\_\{\(\\omega,\\tau\_\{r\},\\gamma\)\\sim\\pi\_\{t\}\}\[\\rho\_\{\\omega,\\tau\_\{r\},\\gamma,q\}\]\\right\)\-\\mathbb\{E\}\_\{\(\\omega,\\tau\_\{r\},\\gamma\)\\sim\\pi\_\{t\}\}\\left\[\\mathrm\{Ent\}\(\\rho\_\{\\omega,\\tau\_\{r\},\\gamma,q\}\)\\right\]\.In implementation, these expectations are approximated by averaging over the posterior samples retained from the MCMC chain, as described in Algorithm[1](https://arxiv.org/html/2607.02672#alg1)\.

AfterTTrounds, the learner reports the posterior means

ω^T=𝔼πT​\[ω\],τ^r,T=𝔼πT​\[τr\],τ^κ,T=𝔼πT​\[γ\]\.\\widehat\{\\omega\}\_\{T\}=\\mathbb\{E\}\_\{\\pi\_\{T\}\}\[\\omega\],\\qquad\\widehat\{\\tau\}\_\{r,T\}=\\mathbb\{E\}\_\{\\pi\_\{T\}\}\[\\tau\_\{r\}\],\\qquad\\widehat\{\\tau\}\_\{\\kappa,T\}=\\mathbb\{E\}\_\{\\pi\_\{T\}\}\[\\gamma\]\.

### C\.8\.Formal Indecision Response Models

In all these definitions, fix thresholdsτr∈\[0,1\]\\tau\_\{r\}\\in\[0,1\],τκ∈\[0,1\)\\tau\_\{\\kappa\}\\in\[0,1\), and a link functionhβh\_\{\\beta\}\.

###### Definition C\.0 \(50/50 Response Model\)\.

The 50/50 query response model

Rhβ;τr,τκ50/50:ℳ→\(𝒬p​c→Δ​\(𝒲p​c\)\)R^\{\\mathrm\{50/50\}\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}:\\mathcal\{M\}\\to\\left\(\\mathcal\{Q\}^\{pc\}\\to\\Delta\(\\mathcal\{W\}^\{pc\}\)\\right\)is defined by

Rhβ;τr,τκ50/50​\(q;M\)=\{Rhβ;τr,τκ∘​\(q;M\),if​LM​\(q\)∈\{y≻∗y′,y≺∗y′\},μ50/50​\(q\),if​LM​\(q\)∈\{y∼∗y′,y⋈∗y′\},R^\{\\mathrm\{50/50\}\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\)=\\begin\{cases\}R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\),&\\text\{if \}L\_\{M\}\(q\)\\in\\\{y\\succ^\{\\ast\}y^\{\\prime\},\\,y\\prec^\{\\ast\}y^\{\\prime\}\\\},\\\\\[4\.0pt\] \\mu^\{\\mathrm\{50/50\}\}\(q\),&\\text\{if \}L\_\{M\}\(q\)\\in\\\{y\\sim^\{\\ast\}y^\{\\prime\},\\,y\\bowtie^\{\\ast\}y^\{\\prime\}\\\},\\end\{cases\}whereμ50/50​\(q\)∈Δ​\(𝒲p​c​\(q\)\)\\mu^\{\\mathrm\{50/50\}\}\(q\)\\in\\Delta\(\\mathcal\{W\}^\{pc\}\(q\)\)is the distribution given by

μ50/50​\(q\)​\(y≻y′\)=12,μ50/50​\(q\)​\(y≺y′\)=12,μ50/50​\(q\)​\(y∼y′\)=μ50/50​\(q\)​\(y⋈y′\)=0\.\\mu^\{\\mathrm\{50/50\}\}\(q\)\(y\\succ y^\{\\prime\}\)=\\frac\{1\}\{2\},\\qquad\\mu^\{\\mathrm\{50/50\}\}\(q\)\(y\\prec y^\{\\prime\}\)=\\frac\{1\}\{2\},\\qquad\\mu^\{\\mathrm\{50/50\}\}\(q\)\(y\\sim y^\{\\prime\}\)=\\mu^\{\\mathrm\{50/50\}\}\(q\)\(y\\bowtie y^\{\\prime\}\)=0\.Thus, when the latent state is indecisive, the respondent chooses between the two decisive responses uniformly at random\.

###### Definition C\.0 \(Lexicographic Response Model\)\.

The lexicographic query response model

Rhβ;τr,τκlex:ℳ→\(𝒬p​c→Δ​\(𝒲p​c\)\)R^\{\\mathrm\{lex\}\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}:\\mathcal\{M\}\\to\\left\(\\mathcal\{Q\}^\{pc\}\\to\\Delta\(\\mathcal\{W\}^\{pc\}\)\\right\)is defined by

Rhβ;τr,τκlex​\(q;M\)=\{Rhβ;τr,τκ∘​\(q;M\),if​LM​\(q\)∈\{y≻∗y′,y≺∗y′\},μlex​\(q\),if​LM​\(q\)∈\{y∼∗y′,y⋈∗y′\}\.R^\{\\mathrm\{lex\}\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\)=\\begin\{cases\}R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\),&\\text\{if \}L\_\{M\}\(q\)\\in\\\{y\\succ^\{\\ast\}y^\{\\prime\},\\,y\\prec^\{\\ast\}y^\{\\prime\}\\\},\\\\\[4\.0pt\] \\mu^\{\\mathrm\{lex\}\}\(q\),&\\text\{if \}L\_\{M\}\(q\)\\in\\\{y\\sim^\{\\ast\}y^\{\\prime\},\\,y\\bowtie^\{\\ast\}y^\{\\prime\}\\\}\.\\end\{cases\}The lexicographic distributionμlex​\(q\)∈Δ​\(𝒲p​c​\(q\)\)\\mu^\{\\mathrm\{lex\}\}\(q\)\\in\\Delta\(\\mathcal\{W\}^\{pc\}\(q\)\)is defined as follows\.

Fix a true modelM∗=\(u,ω∗\)M^\{\*\}=\(u,\\omega^\{\*\}\)\. Letλω∗=\(λ1,…,λm\)\\lambda^\{\\omega^\{\*\}\}=\(\\lambda\_\{1\},\\ldots,\\lambda\_\{m\}\)be any ordering of priorities such that

ωλ1∗≥ωλ2∗≥⋯≥ωλm∗,\\omega^\{\*\}\_\{\\lambda\_\{1\}\}\\geq\\omega^\{\*\}\_\{\\lambda\_\{2\}\}\\geq\\cdots\\geq\\omega^\{\*\}\_\{\\lambda\_\{m\}\},with ties broken arbitrarily but fixed throughout\.

For any priorityjjand queryqq, letsj​\(q\)=ϕrules​\(\(ΔjF​\(q\)\)F∈ℱ\)s\_\{j\}\(q\)=\\phi^\{\\mathrm\{rules\}\}\\\!\\left\(\(\\Delta^\{F\}\_\{j\}\(q\)\)\_\{F\\in\\mathcal\{F\}\}\\right\)be the evidence from priorityjjonqq\.

We say that priorityjjis*decisive onqq*ifsj​\(q\)≠0s\_\{j\}\(q\)\\neq 0\. The*highest\-ranked decisive priority*onqqis the first priority in the weight\-induced orderingλω∗\\lambda^\{\\omega^\{\*\}\}that is decisive onqq\. Formally, define

ℓMlex​\(q\):=min⁡\{ℓ∈\[m\]:sλℓ​\(q\)≠0\}\.\\ell\_\{M\}^\{\\mathrm\{lex\}\}\(q\):=\\min\\left\\\{\\ell\\in\[m\]:s\_\{\\lambda\_\{\\ell\}\}\(q\)\\neq 0\\right\\\}\.WhenℓMlex​\(q\)\\ell\_\{M\}^\{\\mathrm\{lex\}\}\(q\)is defined \(i\.e\., the set is non\-empty\), letj∗​\(q\):=λℓMlex​\(q\)j^\{\*\}\(q\):=\\lambda\_\{\\ell\_\{M\}^\{\\mathrm\{lex\}\}\(q\)\}denote the highest\-ranked decisive priority\. Then, the response distribution follows the sign of the aggregate directional evidence of the highest\-ranked decisive priority:

μlex​\(q\)​\(y≻y′\)=\{1if​sj∗​\(q\)​\(q\)\>0,0if​sj∗​\(q\)​\(q\)<0,μlex​\(q\)​\(y≺y′\)=\{0if​sj∗​\(q\)​\(q\)\>0,1if​sj∗​\(q\)​\(q\)<0\.\\mu^\{\\mathrm\{lex\}\}\(q\)\(y\\succ y^\{\\prime\}\)=\\begin\{cases\}1&\\text\{if \}s\_\{j^\{\*\}\(q\)\}\(q\)\>0,\\\\ 0&\\text\{if \}s\_\{j^\{\*\}\(q\)\}\(q\)<0,\\end\{cases\}\\qquad\\mu^\{\\mathrm\{lex\}\}\(q\)\(y\\prec y^\{\\prime\}\)=\\begin\{cases\}0&\\text\{if \}s\_\{j^\{\*\}\(q\)\}\(q\)\>0,\\\\ 1&\\text\{if \}s\_\{j^\{\*\}\(q\)\}\(q\)<0\.\\end\{cases\}IfℓMlex​\(q\)\\ell\_\{M\}^\{\\mathrm\{lex\}\}\(q\)is undefined, meaning no priority is decisive onqq, then the lexicographic response distribution chooses randomly between the two decisive responses:

μlex​\(q\)​\(y≻y′\)=12,μlex​\(q\)​\(y≺y′\)=12\.\\mu^\{\\mathrm\{lex\}\}\(q\)\(y\\succ y^\{\\prime\}\)=\\frac\{1\}\{2\},\\qquad\\mu^\{\\mathrm\{lex\}\}\(q\)\(y\\prec y^\{\\prime\}\)=\\frac\{1\}\{2\}\.In all cases,

μlex​\(q\)​\(y∼y′\)=μlex​\(q\)​\(y⋈y′\)=0\.\\mu^\{\\mathrm\{lex\}\}\(q\)\(y\\sim y^\{\\prime\}\)=\\mu^\{\\mathrm\{lex\}\}\(q\)\(y\\bowtie y^\{\\prime\}\)=0\.
Thus, on latently decisive queries, the individual follows the baseline response model\. On latently indecisive queries, the individual falls back to the decisive recommendation of the highest\-weight priority that has nonzero aggregate directional evidence on the query; if no priority has nonzero evidence, the individual chooses uniformly at random betweeny≻y′y\\succ y^\{\\prime\}andy≺y′y\\prec y^\{\\prime\}\.

###### Definition C\.0 \(Self\-Similarity Response Model\)\.

Let the individual’s true feature vector bev∈\[0,1\]dv\\in\[0,1\]^\{d\}and fix a queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)\. Then, the self\-similarity query response model is

Rhβ;τr,τκ,vself:ℳ→\(𝒬p​c→Δ​\(𝒲p​c\)\)R^\{\\mathrm\{self\}\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\},v\}:\\mathcal\{M\}\\to\\left\(\\mathcal\{Q\}^\{pc\}\\to\\Delta\(\\mathcal\{W\}^\{pc\}\)\\right\)is defined by

Rhβ;τr,τκ,vself​\(q;M\)=\{Rhβ;τr,τκ∘​\(q;M\),if​LM​\(q\)∈\{y≻∗y′,y≺∗y′\},μv​\(q\),if​LM​\(q\)∈\{y∼∗y′,y⋈∗y′\}\.R^\{\\mathrm\{self\}\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\},v\}\(q;M\)=\\begin\{cases\}R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\),&\\text\{if \}L\_\{M\}\(q\)\\in\\\{y\\succ^\{\\ast\}y^\{\\prime\},\\,y\\prec^\{\\ast\}y^\{\\prime\}\\\},\\\\\[4\.0pt\] \\mu^\{v\}\(q\),&\\text\{if \}L\_\{M\}\(q\)\\in\\\{y\\sim^\{\\ast\}y^\{\\prime\},\\,y\\bowtie^\{\\ast\}y^\{\\prime\}\\\}\.\\end\{cases\}Whereμv\\mu^\{v\}is defined as follows\. Let the profile\-distance gap be

bv​\(q\):=‖ψ​\(x,y′\)−v‖2−‖ψ​\(x,y\)−v‖2\.b\_\{v\}\(q\):=\\\|\\psi\(x,y^\{\\prime\}\)\-v\\\|\_\{2\}\-\\\|\\psi\(x,y\)\-v\\\|\_\{2\}\.In words,yyis closer to the respondent’s profile thany′y^\{\\prime\}whenbv​\(q\)\>0b\_\{v\}\(q\)\>0and farther whenbv​\(q\)<0b\_\{v\}\(q\)<0\. Letμv​\(q\)∈Δ​\(𝒲p​c​\(q\)\)\\mu^\{v\}\(q\)\\in\\Delta\(\\mathcal\{W\}^\{pc\}\(q\)\)be the distribution

μv​\(q\)​\(y≻y′\)=\{1,if​bv​\(q\)\>0,12,if​bv​\(q\)=0,0,if​bv​\(q\)<0,μv​\(q\)​\(y≺y′\)=\{0,if​bv​\(q\)\>0,12,if​bv​\(q\)=0,1,if​bv​\(q\)<0,\\mu^\{v\}\(q\)\(y\\succ y^\{\\prime\}\)=\\begin\{cases\}1,&\\text\{if \}b\_\{v\}\(q\)\>0,\\\\ \\frac\{1\}\{2\},&\\text\{if \}b\_\{v\}\(q\)=0,\\\\ 0,&\\text\{if \}b\_\{v\}\(q\)<0,\\end\{cases\}\\qquad\\mu^\{v\}\(q\)\(y\\prec y^\{\\prime\}\)=\\begin\{cases\}0,&\\text\{if \}b\_\{v\}\(q\)\>0,\\\\ \\frac\{1\}\{2\},&\\text\{if \}b\_\{v\}\(q\)=0,\\\\ 1,&\\text\{if \}b\_\{v\}\(q\)<0,\\end\{cases\}and

μv​\(q\)​\(y∼y′\)=μv​\(q\)​\(y⋈y′\)=0\.\\mu^\{v\}\(q\)\(y\\sim y^\{\\prime\}\)=\\mu^\{v\}\(q\)\(y\\bowtie y^\{\\prime\}\)=0\.In words, when the latent state is indecisive, the respondent deterministically chooses the alternative whose feature representation is closest in Euclidean distance to their assigned profile, breaking exact ties uniformly at random\.

###### Definition C\.0 \(Utilize\-3 Response Model\)\.

Let the coarsened pairwise\-comparison response alphabet be

𝒲p​c,⊘​\(q\)=\{≻,≺,⊘\},\\mathcal\{W\}^\{pc,\\oslash\}\(q\)=\\\{\\succ,\\prec,\\oslash\\\},where⊘\\oslashdenotes generic indecision, without distinguishing between indifference and conflict\. TheUtilize\-3query response model

Rhβ;τr,τκ⊘:ℳ→\(𝒬p​c→Δ​\(𝒲p​c,⊘\)\)R^\{\\oslash\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}:\\mathcal\{M\}\\to\\left\(\\mathcal\{Q\}^\{pc\}\\to\\Delta\(\\mathcal\{W\}^\{pc,\\oslash\}\)\\right\)is obtained from the baseline modelRhβ;τr,τκ∘R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}by coarsening the two indecisive responses into a single generic\-indecision response\. The decisive probabilities are unchanged,

Rhβ;τr,τκ⊘​\(q;M\)​\(y≻y′\)=Rhβ;τr,τκ∘​\(q;M\)​\(y≻y′\),Rhβ;τr,τκ⊘​\(q;M\)​\(y≺y′\)=Rhβ;τr,τκ∘​\(q;M\)​\(y≺y′\),R^\{\\oslash\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\)\(y\\succ y^\{\\prime\}\)=R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\)\(y\\succ y^\{\\prime\}\),\\quad R^\{\\oslash\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\)\(y\\prec y^\{\\prime\}\)=R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\)\(y\\prec y^\{\\prime\}\),while the generic\-indecision response pools the two indecisive states,

Rhβ;τr,τκ⊘​\(q;M\)​\(⊘\)=Rhβ;τr,τκ∘​\(q;M\)​\(y∼y′\)\+Rhβ;τr,τκ∘​\(q;M\)​\(y⋈y′\)\.R^\{\\oslash\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\)\(\\oslash\)=R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\)\(y\\sim y^\{\\prime\}\)\+R^\{\\circ\}\_\{h\_\{\\beta\};\\tau\_\{r\},\\tau\_\{\\kappa\}\}\(q;M\)\(y\\bowtie y^\{\\prime\}\)\.

### C\.9\.Additional Experiments for Section[4\.2](https://arxiv.org/html/2607.02672#S4.SS2)

#### C\.9\.1\.Exploring directional biases inω^\\widehat\{\\omega\}

[Figure8](https://arxiv.org/html/2607.02672#A3.F8)shows the plots demonstrating the expected biases in learnedω^\\widehat\{\\omega\}under the three deviating response conditions from[Section4\.2](https://arxiv.org/html/2607.02672#S4.SS2)\. Each plotted quantity is first computed within a single simulation run, using that run’s true weightsω∗\\omega^\{\*\}, learned weightsω^\\widehat\{\\omega\}, and, in one case, the self vectorvv\. We then report the mean and standard error over the 40 runs\.

![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/fig_bias_decomposition.png)Figure 8\.Decomposing theℓ1\\ell\_\{1\}weight\-recovery error across response conditions\.Means and standard errors computed over 40 runs\.\(a\)Bias toward own top feature,\(b\)Bias toward own self vector\.Panel \(a\) measures bias toward the true top priority: Letj∗∈arg⁡maxj⁡ωj∗j^\{\*\}\\in\\arg\\max\_\{j\}\\omega^\{\*\}\_\{j\}\. For each run, we computeω^j∗−ωj∗∗\\widehat\{\\omega\}\_\{j^\{\*\}\}\-\\omega^\{\*\}\_\{j^\{\*\}\}; positive values mean that the learned model overweights the priority that was already most important underω∗\\omega^\{\*\}\. This is the distortion expected underLexicographicbehavior\.

Panel \(b\) measures bias toward the respondent’s self vector\. For each run, we⟨ω^−ω∗,v⟩\.\\langle\\widehat\{\\omega\}\-\\omega^\{\*\},v\\rangle\.Positive values mean that the learned model shifts weight toward features that are large in the respondent’s own self vector\. This is the distortion expected underSelf\-Similaritybehavior\.

#### C\.9\.2\.Cosine Similarity

In[Figure9](https://arxiv.org/html/2607.02672#A3.F9), we show the cosine similarity ofω^\\widehat\{\\omega\}andω∗\\omega^\{\*\}to demonstrate the directional recovery conditioned on the four types of true response models\.

![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/fig_cosine.png)Figure 9\.Cosine similarity between the learned weightsω^\\widehat\{\\omega\}and true weightsω∗\\omega^\{\*\}\.Error bars show±1\\pm 1standard error over 40 runs\.

### C\.10\.Additional Experimental Results for Section[4\.3](https://arxiv.org/html/2607.02672#S4.SS3)

We now present the fullτ\\tau\-grids behind the learning speed results of[Section4\.3](https://arxiv.org/html/2607.02672#S4.SS3)\. The eight methods areUtilize\-4,Utilize\-3,Utilize\-4†,Utilize\-4⋄,Ignore,Ignore†,Correct, andCorrect†\.

#### C\.10\.1\.Unknown Noise Variants

We include unknown\-noise variants ofUtilizeandIgnore, denoted with a†\\dagger\. In these variants the individual’s true response model still uses logistic noise, but the learner is not given the logistic link or its scale; it knows only that the noise distribution is a three\-component mixture of Gaussians,ε∼∑k=13wk​𝒩​\(μk,σk2\)\\varepsilon\\sim\\sum\_\{k=1\}^\{3\}w\_\{k\}\\,\\mathcal\{N\}\(\\mu\_\{k\},\\sigma\_\{k\}^\{2\}\)where∑k=13wk=1\.\\sum\_\{k=1\}^\{3\}w\_\{k\}=1\.This family is far more flexible than the one\-parameter logistic familyFlogistic​\(t\)=1/\(1\+exp⁡\(−β​t\)\)F\_\{\\mathrm\{logistic\}\}\(t\)=1/\(1\+\\exp\(\-\\beta t\)\)\. It contains the probit link exactly — a single Gaussian component recoversΦ​\(t/σ\)\\Phi\(t/\\sigma\)— and, because finite Gaussian mixtures approximate any continuous noise CDF arbitrarily well as the number of components grows, it approximates the logistic link\. It does not reproduce the logistic CDF exactly: a finite mixture has Gaussian tails, lighter than the logistic’s exponential tails\. The point of the†\\daggervariants is exactly this misspecification — the oracle’s noise is logistic while the learner fits a Gaussian mixture — so that the experiment tests whether the added flexibility lets the learner recoverω\\omegawithout being told the true noise family\. The unknown\-noise learner thus jointly infers the preference weights and a model of the response noise\. We test these variants to demonstrate that the learning gains achieved byUtilize\-3andUtilize\-4remaineven if we weaken the learner’s assumptions about the noise structure, which is potentially useful if people are not responding according to the specific noise assumptions assumed by Bradley\-Terry\.

![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/bald_diag_l1_avgreg_wcr_bars_8methods_warm.png)Figure 10\.Performance of all methods across the diagonal threshold regimes versus number of queries\.Top:ℓ1\\ell\_\{1\}error‖ω^−ω∗‖1\\\|\\widehat\{\\omega\}\-\\omega^\{\\ast\}\\\|\_\{1\}\.Middle:Average regret on the uniform\-\[0,1\]5\[0,1\]^\{5\}distribution with independently drawn features\.Bottom:Worst\-case single\-decision regret\.![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/bald_grid_l1_7methods_warm.png)Figure 11\.ℓ1\\ell\_\{1\}error‖ω^−ω∗‖1\\\|\\widehat\{\\omega\}\-\\omega^\{\\ast\}\\\|\_\{1\}versus number of queries, across the full5×55\\times 5grid of thresholds\.![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/bald_grid_cos_7methods_warm.png)Figure 12\.Cosine similaritycos⁡\(ω^,ω∗\)\\cos\(\\widehat\{\\omega\},\\omega^\{\\ast\}\)versus number of queries, across the full5×55\\times 5threshold grid\.![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/bald_grid_avgreg_7methods_warm.png)Figure 13\.Average regret versus query number on the uniform\-\[0,1\]5\[0,1\]^\{5\}distribution where features are all independent, across the full5×55\\times 5threshold grid\.![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/bald_grid_wcr_7methods_warm.png)Figure 14\.Worst\-case single\-decision regret versus query number, across the full5×55\\times 5threshold grid\.![Refer to caption](https://arxiv.org/html/2607.02672v1/00_arxiv-new/response_distribution_7methods_warm.png)Figure 15\.Fraction of each response type \(Left≻\\succ, Right≺\\prec, Indifferent∼\\sim, Conflict⋈\\bowtie\) elicited by the active learner, across the full5×55\\times 5threshold grid, for all eight methods\. Within each method’s block, rows index the indifference thresholdτr\\tau\_\{r\}and columns the incomparability thresholdτκ\\tau\_\{\\kappa\}\.

### C\.11\.Rule\-Level Preferences

The preceding results focus on the importance of measuring indecision in the technical task of learning the modelMM: forced comparisons can leave information on the table, or worse, distort what a learner recovers about the individual’s priorities if the individual deviates from the assumed model to resolve indecision\. But even when the individual resolves indecision via the zero\-threshold response modelRhβ;0,0∘R^\{\\circ\}\_\{h\_\{\\beta\};0,0\}\(thus avoiding learning errors\), forced comparisons still erase information that may be normatively important: truly decisive responses and forced responses from an indecisive latent state are recorded as identical, but their moral authority may not be, varying depending on whether the individual regarded the comparison as clear, morally weighty but conflicted, or too low\-stakes to warrant a meaningful distinction\.

When we learn a single rule that best imitates the individual’s forced\-choice behavior, we can also lose something else: information about where the individual is indifferent or conflicted betweenrules\. Making this claim requires us to define a global preference relation over rules that encompasses these forms of indecision, and then show that local pairwise comparison queries are informative about these rule\-level relations\. To do so, we define a global rule\-level preference relation analogously to our latent states model at the query level\.

###### Definition C\.0 \(Rule\-Level Latent Relation\)\.

Fix a priority modelM=\(u,ω\)M=\(u,\\omega\)and thresholdsτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}\. For two rulesF,F′∈ℱF,F^\{\\prime\}\\in\\mathcal\{F\}, define the rule\-level directional evidence scores

sM\+​\(F,F′\)=∑j∈\[m\]ωj​\[uj​\(F\)−uj​\(F′\)\]\+,sM−​\(F,F′\)=∑j∈\[m\]ωj​\[uj​\(F′\)−uj​\(F\)\]\+\.s^\{\+\}\_\{M\}\(F,F^\{\\prime\}\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\[u\_\{j\}\(F\)\-u\_\{j\}\(F^\{\\prime\}\)\]\_\{\+\},\\qquad s^\{\-\}\_\{M\}\(F,F^\{\\prime\}\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\[u\_\{j\}\(F^\{\\prime\}\)\-u\_\{j\}\(F\)\]\_\{\+\}\.Let

rM​\(F,F′\)=sM\+​\(F,F′\)\+sM−​\(F,F′\),κM​\(F,F′\)=sM\+​\(F,F′\)−sM−​\(F,F′\)\.r\_\{M\}\(F,F^\{\\prime\}\)=s^\{\+\}\_\{M\}\(F,F^\{\\prime\}\)\+s^\{\-\}\_\{M\}\(F,F^\{\\prime\}\),\\qquad\\kappa\_\{M\}\(F,F^\{\\prime\}\)=s^\{\+\}\_\{M\}\(F,F^\{\\prime\}\)\-s^\{\-\}\_\{M\}\(F,F^\{\\prime\}\)\.The rule\-level latent relation betweenFFandF′F^\{\\prime\}, denotedF⊳M∗F′F\\triangleright^\{\*\}\_\{M\}F^\{\\prime\}, is defined by

F⊳M∗F′=\{F≻M∗F′if​rM​\(F,F′\)−τr≥0​and​κM​\(F,F′\)−τκ​rM​\(F,F′\)≥0,F≺M∗F′if​rM​\(F,F′\)−τr≥0​and−κM​\(F,F′\)−τκ​rM​\(F,F′\)≥0,F⋈M∗F′if​rM​\(F,F′\)−τr≥0​and​\|κM​\(F,F′\)\|−τκ​rM​\(F,F′\)<0,F∼M∗F′if​rM​\(F,F′\)−τr<0\.F\\triangleright^\{\*\}\_\{M\}F^\{\\prime\}=\\begin\{cases\}F\\succ^\{\*\}\_\{M\}F^\{\\prime\}&\\text\{if \}r\_\{M\}\(F,F^\{\\prime\}\)\-\\tau\_\{r\}\\geq 0\\text\{ and \}\\kappa\_\{M\}\(F,F^\{\\prime\}\)\-\\tau\_\{\\kappa\}r\_\{M\}\(F,F^\{\\prime\}\)\\geq 0,\\\\\[3\.99994pt\] F\\prec^\{\*\}\_\{M\}F^\{\\prime\}&\\text\{if \}r\_\{M\}\(F,F^\{\\prime\}\)\-\\tau\_\{r\}\\geq 0\\text\{ and \}\-\\kappa\_\{M\}\(F,F^\{\\prime\}\)\-\\tau\_\{\\kappa\}r\_\{M\}\(F,F^\{\\prime\}\)\\geq 0,\\\\\[3\.99994pt\] F\\bowtie^\{\*\}\_\{M\}F^\{\\prime\}&\\text\{if \}r\_\{M\}\(F,F^\{\\prime\}\)\-\\tau\_\{r\}\\geq 0\\text\{ and \}\|\\kappa\_\{M\}\(F,F^\{\\prime\}\)\|\-\\tau\_\{\\kappa\}r\_\{M\}\(F,F^\{\\prime\}\)<0,\\\\\[3\.99994pt\] F\\sim^\{\*\}\_\{M\}F^\{\\prime\}&\\text\{if \}r\_\{M\}\(F,F^\{\\prime\}\)\-\\tau\_\{r\}<0\.\\end\{cases\}As in[Definition2\.7](https://arxiv.org/html/2607.02672#S2.Thmtheorem7), whenτκ=κM​\(F,F′\)=0\\tau\_\{\\kappa\}=\\kappa\_\{M\}\(F,F^\{\\prime\}\)=0, the decisive cases overlap; in that degenerate case, either decisive relation may be assigned arbitrarily\.

Now, we show that in perfectly separable models, latent states at the query level correspond exactly to pairwise preference relations between the corresponding projections\. This means that if we can learn the latent state thresholdsτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}— which are erased under forced choice — we simultaneously learn the global preference relation described above, including regions of conflict and indifference between rules\.252525This equivalence also leads to another interpretation: that latent query states arise from an underlying rule\-level preference relation, where the person picks an arbitrary background rule and compares the two projections, and then simply answers according to the relation between rules\.The proof follows directly from perfect separability\.262626WhenMMis not perfectly separable, one can still derive a relationship between rule preferences and query responses, one has to aggregate over background rules, which is exactly the role played byϕrules\\phi^\{\\mathrm\{rules\}\}\.

###### Lemma C\.10 \(Latent query states as rule\-level relations\)\.

Fix a perfectly separable modelM∈ℳsepM\\in\\mathcal\{M\}^\{\\mathrm\{sep\}\}\. For every queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\), everyF∈ℱF\\in\\mathcal\{F\}, and every latent state⊳∗∈\{≻∗,≺∗,∼∗,⋈∗\}\\triangleright^\{\*\}\\in\\\{\\succ^\{\*\},\\prec^\{\*\},\\sim^\{\*\},\\bowtie^\{\*\}\\\},

LM​\(y,y′;x\)=y⊳∗y′⟺Fx→y⊳M∗Fx→y′\.L\_\{M\}\(y,y^\{\\prime\};x\)=y\\triangleright^\{\*\}y^\{\\prime\}\\quad\\Longleftrightarrow\\quad F\_\{x\\to y\}\\triangleright^\{\*\}\_\{M\}F\_\{x\\to y^\{\\prime\}\}\.

###### Proof\.

Query\-level directional evidence scores\.Fix a queryq=\(y,y′;x\)q=\(y,y^\{\\prime\};x\)\. SinceMMis perfectly separable, for each priorityj∈\[m\]j\\in\[m\], there exists a constantδj​\(q\)\\delta\_\{j\}\(q\)such that

ΔjG​\(q\)=uj​\(Gx→y\)−uj​\(Gx→y′\)=δj​\(q\)∀G∈ℱ\.\\Delta^\{G\}\_\{j\}\(q\)=u\_\{j\}\(G\_\{x\\to y\}\)\-u\_\{j\}\(G\_\{x\\to y^\{\\prime\}\}\)=\\delta\_\{j\}\(q\)\\qquad\\forall G\\in\\mathcal\{F\}\.Becauseϕrules\\phi^\{\\text\{rules\}\}is unanimous,

ϕrules​\(\(ΔjG​\(q\)\)G∈ℱ\)=δj​\(q\)\.\\phi^\{\\text\{rules\}\}\\bigl\(\(\\Delta^\{G\}\_\{j\}\(q\)\)\_\{G\\in\\mathcal\{F\}\}\\bigr\)=\\delta\_\{j\}\(q\)\.Therefore the query\-level directional scores satisfy

sM\+​\(q\)=∑j∈\[m\]ωj​\[ϕrules​\(\(ΔjG​\(q\)\)G∈ℱ\)\]\+=∑j∈\[m\]ωj​\[δj​\(q\)\]\+,s^\{\+\}\_\{M\}\(q\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\left\[\\phi^\{\\text\{rules\}\}\\bigl\(\(\\Delta^\{G\}\_\{j\}\(q\)\)\_\{G\\in\\mathcal\{F\}\}\\bigr\)\\right\]\_\{\+\}=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\[\\delta\_\{j\}\(q\)\]\_\{\+\},and

sM−​\(q\)=∑j∈\[m\]ωj​\[−ϕrules​\(\(ΔjG​\(q\)\)G∈ℱ\)\]\+=∑j∈\[m\]ωj​\[−δj​\(q\)\]\+\.s^\{\-\}\_\{M\}\(q\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\\left\[\-\\phi^\{\\text\{rules\}\}\\bigl\(\(\\Delta^\{G\}\_\{j\}\(q\)\)\_\{G\\in\\mathcal\{F\}\}\\bigr\)\\right\]\_\{\+\}=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\[\-\\delta\_\{j\}\(q\)\]\_\{\+\}\.Rule\-Level Directional Evidence Scores\.Fix a background ruleF∈ℱF\\in\\mathcal\{F\}; by perfect separability,uj​\(Fx→y\)−uj​\(Fx→y′\)=δj​\(q\)\.u\_\{j\}\(F\_\{x\\to y\}\)\-u\_\{j\}\(F\_\{x\\to y^\{\\prime\}\}\)=\\delta\_\{j\}\(q\)\.Then, the rule\-level directional scores betweenFx→yF\_\{x\\to y\}andFx→y′F\_\{x\\to y^\{\\prime\}\}are

sM\+​\(Fx→y,Fx→y′\)=∑j∈\[m\]ωj​\[uj​\(Fx→y\)−uj​\(Fx→y′\)\]\+=∑j∈\[m\]ωj​\[δj​\(q\)\]\+,s^\{\+\}\_\{M\}\(F\_\{x\\to y\},F\_\{x\\to y^\{\\prime\}\}\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\[u\_\{j\}\(F\_\{x\\to y\}\)\-u\_\{j\}\(F\_\{x\\to y^\{\\prime\}\}\)\]\_\{\+\}=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\[\\delta\_\{j\}\(q\)\]\_\{\+\},and

sM−​\(Fx→y,Fx→y′\)=∑j∈\[m\]ωj​\[uj​\(Fx→y′\)−uj​\(Fx→y\)\]\+=∑j∈\[m\]ωj​\[−δj​\(q\)\]\+\.s^\{\-\}\_\{M\}\(F\_\{x\\to y\},F\_\{x\\to y^\{\\prime\}\}\)=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\[u\_\{j\}\(F\_\{x\\to y^\{\\prime\}\}\)\-u\_\{j\}\(F\_\{x\\to y\}\)\]\_\{\+\}=\\sum\_\{j\\in\[m\]\}\\omega\_\{j\}\[\-\\delta\_\{j\}\(q\)\]\_\{\+\}\.It follows that

sM\+​\(q\)=sM\+​\(Fx→y,Fx→y′\),sM−​\(q\)=sM−​\(Fx→y,Fx→y′\)\.s^\{\+\}\_\{M\}\(q\)=s^\{\+\}\_\{M\}\(F\_\{x\\to y\},F\_\{x\\to y^\{\\prime\}\}\),\\qquad s^\{\-\}\_\{M\}\(q\)=s^\{\-\}\_\{M\}\(F\_\{x\\to y\},F\_\{x\\to y^\{\\prime\}\}\)\.and thus the query\-level and rule\-level values ofrMr\_\{M\}andκM\\kappa\_\{M\}are identical\. Applying the same thresholdsτr,τκ\\tau\_\{r\},\\tau\_\{\\kappa\}therefore yields the same latent state\. ∎