Semantic Cooperative Games for Contribution Attribution in LLM-Based Multi-Agent Systems

# Semantic Cooperative Games for Contribution Attribution in LLM-Based Multi-Agent Systems
Source: [https://arxiv.org/html/2607.18255](https://arxiv.org/html/2607.18255)
Pengyi Jiang New York University pj2366@nyu\.edu&Xiaoguang Zhu11footnotemark:1 University of California, Davis xgzhu@ucdavis\.edu&Quanyan Zhu New York University qz494@nyu\.edu

###### Abstract

Contribution attribution has become a central problem in LLM\-based multi\-agent systems, where final outputs are produced through multiple agents, message exchanges, and ordered workflow dependencies\. Existing attribution methods often rely on counterfactual valuation, such as removing agents or comparing score changes across altered agent subsets\. In language\-mediated workflows, these methods require repeated model calls, introduce high variance, and do not explicitly capture the intermediate semantic states through which agents produce, preserve, and transform task\-relevant information\. We propose Semantic Cooperative Games \(SCG\), a framework that represents a realized language flow as a semantic generation hypergraph and induces an agent\-level semantic value function on this structure\. We define the Semantic Shapley Value \(SSV\) to allocate contribution over semantic support logic, and introduce SLIC, a single\-trajectory algorithm that constructs the semantic hypergraph, recovers minimal semantic supports, applies Boolean absorption, and computes SSV without rerunning agent subsets\. We prove that SSV reduces to the classical Shapley value under standard set\-based, fully observable, and no\-order\-dependence conditions\. On a medical benchmark satisfying these conditions, SLIC reduces computation cost by 93\.3% while remaining highly consistent with a Monte Carlo Shapley baseline\. In more general multi\-role workflows, SSV aligns with perturbation\-induced score\-drop profiles and exposes cases where semantic contribution and failure impact diverge\. Overall, SLIC provides a fast, counterfactual\-free, and interpretable attribution method for complex LLM\-based multi\-agent systems\. Code is avaiable at:[SCG\_SENSITIVITY](https://github.com/PengYiJiang-john/SCG_SENSITIVITY)

## 1Introduction

Large language model \(LLM\)\-based multi\-agent systems increasingly solve complex tasks by distributing computation across interacting agents with specialized roles, communication protocols, and explicit workflow structures\. Frameworks such as CAMEL\(Liet al\.,[2023a](https://arxiv.org/html/2607.18255#bib.bib14)\), AutoGen\(Wuet al\.,[2023](https://arxiv.org/html/2607.18255#bib.bib15)\), MetaGPT\(Hong and others,[2024](https://arxiv.org/html/2607.18255#bib.bib16)\), and MedAgents\(Tanget al\.,[2024](https://arxiv.org/html/2607.18255#bib.bib17)\)demonstrate the value of this organization for task decomposition, deliberative reasoning, and domain\-specific collaboration\. However, system behavior is no longer determined by a single model response\. It emerges from sequences of agent actions, message dependencies, and intermediate semantic transformations\. This makes it difficult to determine which agents shape the final output and where failures originate\. This motivates the*contribution attribution problem*\(CAP\): given a realized workflow trace, a value\-reading interface, and a set of participating agents, assign each agent a contribution score that explains its role in producing the observed output\. Such attribution supports failure diagnosis\(Zhanget al\.,[2025b](https://arxiv.org/html/2607.18255#bib.bib1)\), workflow comparison\(Yanget al\.,[2025](https://arxiv.org/html/2607.18255#bib.bib2)\), agent pruning\(Zhanget al\.,[2025a](https://arxiv.org/html/2607.18255#bib.bib3)\), and robustness analysis\(Chanet al\.,[2024](https://arxiv.org/html/2607.18255#bib.bib4)\)\. CAP is also a concrete instance of the broader credit assignment problem, including temporal credit assignment in reinforcement learning\(Sutton,[1984](https://arxiv.org/html/2607.18255#bib.bib10)\)and structural credit assignment in multi\-agent systems\(Agogino and Tumer,[2004](https://arxiv.org/html/2607.18255#bib.bib11)\)\.

A broad class of attribution methods can be characterized as a*counterfactual valuation*route\. Instead of identifying contribution from the realized workflow itself, these methods estimate how the system value changes under hypothetical modifications\. They construct perturbed executions by removing, replacing, or masking an agent, message, feature, action, or workflow component\(Ribeiroet al\.,[2016](https://arxiv.org/html/2607.18255#bib.bib23); Zeiler and Fergus,[2014](https://arxiv.org/html/2607.18255#bib.bib25); Lundberg and Lee,[2017](https://arxiv.org/html/2607.18255#bib.bib24)\)\. The perturbed system is then sampled or rerun, and attribution is computed from the resulting value differences:

Δ​𝖲𝗒𝗌𝗍𝖾𝗆→sample / rerun𝖵𝖺𝗅𝗎𝖾^→value difference𝖠𝗍𝗍𝗋𝗂𝖻𝗎𝗍𝗂𝗈𝗇\.\\Delta\\mathsf\{System\}\\;\\xrightarrow\{\\;\\text\{sample / rerun\}\\;\}\\widehat\{\\mathsf\{Value\}\}\\;\\xrightarrow\{\\;\\text\{value difference\}\\;\}\\mathsf\{Attribution\}\.Here,𝖵𝖺𝗅𝗎𝖾^\\widehat\{\\mathsf\{Value\}\}denotes a value estimate obtained from altered executions rather than from the original trace\. This counterfactual valuation route spans multi\-agent reinforcement learning, cooperative game theory, and LLM\-agent workflow attribution\. COMA estimates agent\-specific counterfactual action values with a centralized critic\(Foersteret al\.,[2018](https://arxiv.org/html/2607.18255#bib.bib12)\), and difference\-reward methods compare global rewards with counterfactual rewards obtained by removing or clamping an agent’s contribution\(Wolpert and Tumer,[2001](https://arxiv.org/html/2607.18255#bib.bib26); Nguyenet al\.,[2018](https://arxiv.org/html/2607.18255#bib.bib27)\)\. In LLM\-based collaboration, C3 estimates message\-level causal impact through fixed\-continuation replay and leave\-one\-out interventions\(Chenet al\.,[2026](https://arxiv.org/html/2607.18255#bib.bib13)\), while ShapleyFlow evaluates alternative agentic workflow configurations\(Yanget al\.,[2025](https://arxiv.org/html/2607.18255#bib.bib2)\)\. More generally, cooperative game\-theoretic methods allocate contribution through marginal comparisons over coalitions or workflow components\(Shapley,[1953](https://arxiv.org/html/2607.18255#bib.bib5); Aumann and Shapley,[1974](https://arxiv.org/html/2607.18255#bib.bib6); Young,[1985](https://arxiv.org/html/2607.18255#bib.bib7)\)\. Despite their different intervention targets and aggregation rules, these methods share a common dependency: attribution is mediated by scalar values assigned to hypothetical configurations\. This dependency creates two obstacles in language\-mediated workflows as illustrated in Fig\.[1](https://arxiv.org/html/2607.18255#S1.F1)\.

Problem 1: Contribution is workflow\-dependent\.An LLM workflow induces a partial\-order dependency structure over agent actions and messages\. A set\-based counterfactual, such as keeping or removing a subset of agents or components, therefore does not fully define the attribution target\. The same subset can have different effects under different execution orders, routing rules, or downstream dependencies\. An agent’s contribution depends not only on what it generates, but also on where that content appears and how it is used downstream\.

Problem 2: Rerun\-based valuation is costly and unstable\.When values are obtained through rerun, replay, or rollout, each intervention can rewrite downstream context and trigger different model responses\. This adds model calls, stochastic variance, and sensitivity to decoding, prompting, and orchestration details\. Even with a surrogate evaluator, attribution remains tied to evaluator stability\. These difficulties grow rapidly with the number of agents, messages, components, and interventions\.

These obstacles suggest that CAP should be treated not only as a value\-comparison problem, but also as a workflow\-structural problem\. In LLM\-based multi\-agent systems, contribution is expressed through the realized trajectory, where agents produce intermediate content, pass information downstream, reuse prior messages, and transform semantic states\. Since black\-box LLM internals are difficult to inspect, we use the language trajectory as the observable layer of this information flow\. Prior work on agent\-level monitoring and process evaluation\(Chanet al\.,[2024](https://arxiv.org/html/2607.18255#bib.bib4); Grittaet al\.,[2026](https://arxiv.org/html/2607.18255#bib.bib18)\)shows that such trajectories contain useful process signals\. For CAP, the relevant signals are semantic: agents introduce, preserve, rewrite, or combine task\-relevant content that later agents may reuse in reasoning\. This motivates a shift from workflow structure to*language flow*, which exposes the semantic states and transformations through which task value is produced\.

To make this semantic dependency structure analyzable, we proposeSemantic Cooperative Games \(SCG\)\. As shown in Fig\.[1](https://arxiv.org/html/2607.18255#S1.F1), SCG converts a realized language flow into a semantic generation hypergraph\. The graph contains semantic nodes, support links, an ownership map from links to agents, and weighted output nodes read by a rubric\. Given the player setNNand graphGG, SCG defines a semantic game representation\(N,G\)\(N,G\)\. The graph then induces an agent\-level semantic value function, making workflow dependency part of the attribution object\. On this induced value function, we define theSemantic Shapley Value \(SSV\)\. The support layer uses Boolean logic to represent sufficient semantic supports, following Boolean games and cooperative Boolean games\(Harrensteinet al\.,[2001](https://arxiv.org/html/2607.18255#bib.bib8); Dunneet al\.,[2008](https://arxiv.org/html/2607.18255#bib.bib9)\)\. The allocation layer follows Shapley\-style axiomatic allocation\(Shapley,[1953](https://arxiv.org/html/2607.18255#bib.bib5); Young,[1985](https://arxiv.org/html/2607.18255#bib.bib7)\)\. Thus, SSV extends the classical Shapley value from unordered coalitions to semantic support structures\. Under standard set\-based, fully observable, and no\-order\-dependence conditions, it coincides with the classical Shapley value\.

To compute SSV from a single trajectory, we proposeSLIC\(Semantic Logic Inversion for Contribution\)\. SLIC reads the value\-bearing output nodes and their rubric weights, recursively traces semantic predecessors in language flow, and constructsGG\. It then extracts minimal link\-level supports, maps links to agents through ownership, applies Boolean absorption, and aggregates multilinear coefficients to obtain SSV\. This avoids rerunning agent subsets\. In a medical benchmark satisfying the reduction conditions, SLIC matches a Monte Carlo baseline while reducing computation cost by 93\.3%\. In more general multi\-role workflows, SSV tracks agent\-level score drops under perturbations and exposes cases where limited semantic contribution can still carry large failure impact\.

![Refer to caption](https://arxiv.org/html/2607.18255v1/x1.png)Figure 1:Counterfactual vs\. semantic routes for contribution attribution\.*Bottom left:*The traditional counterfactual route estimates contributions by masking components and rerunning the workflow to measure value changes, requiring many model calls and high computational cost\.*Bottom right:*The semantic route analyzes a single realized execution by tracing the language/semantic flow across agents\. Contributions are computed directly from this dependency structure via the Semantic Shapley Value, with no reruns\.Our contributions are as follows:

- •We proposeSCG, a semantic cooperative\-game framework that represents a realized language flow as a semantic generation hypergraph and induces an agent\-level value function from workflow\-dependent semantic supports\.
- •We define theSemantic Shapley Value \(SSV\)and introduce a single\-trajectory algorithm that computes SSV without counterfactual reruns\. We prove that SSV coincides with the classical Shapley value under standard coalition\-game conditions\.
- •We validate SCG/SLIC on reduction\-consistency and structural\-diagnosis benchmarks, showing that SLIC reduces computation cost while preserving Shapley consistency and that SSV provides interpretable attribution in complex multi\-role workflows\.

## 2Semantic Cooperative Games

To define a Shapley\-style attribution game for an LLM workflow, we first specify the semantic object from which value will be induced\. An agent is observed through its language action: it produces, carries forward, or transforms semantic content\. We therefore model each agent action as a semantic transformation and analyze the workflow through its dual language flow\.

Formally, letT=\(ℒ,H\)T=\(\\mathcal\{L\},H\), where each state inℒ⊆X\\mathcal\{L\}\\subseteq Xis a semantic state produced at some step, andH=\{hu\}H=\\\{h\_\{u\}\\\}is the set of agent actions\. To connect this trace to value, we use the rubric or value\-reading interface\. In rubric\-based evaluation, the final scalar score decomposes into several score\-bearing output semantics\. LetO=\{o1,…,oK\}O=\\\{o\_\{1\},\\dots,o\_\{K\}\\\}denote these output semantics, and letω\\omegaassign their weights\. The goal of this section is to extract the value\-relevant semantic nodes from language flow and use them to construct a semantic generation hypergraph\.

##### Assumption 1 \(Semantic Separability and Source Identifiability\)\.

For any execution positionuu, every semantic nodes⊆yus\\subseteq y\_\{u\}relevant to this paper is one of three types: native semantics, derived semantics, or persistent semantics\. Ifssis derived or persistent, its prior semantic source in the input statexux\_\{u\}can be identified\.

Under Assumption 1, each agent actionhuh\_\{u\}gives a direct predecessor set for every semantic node it produces:

srchu​\(s\)=\{∅,s​is native,\{s~\},s​is persistent,P,P⊆xu,s​is derived,\\mathrm\{src\}\_\{h\_\{u\}\}\(s\)=\\begin\{cases\}\\emptyset,&s\\text\{ is native\},\\\\ \\\{\\tilde\{s\}\\\},&s\\text\{ is persistent\},\\\\ P,\\;P\\subseteq x\_\{u\},&s\\text\{ is derived\},\\end\{cases\}wheresrchu​\(s\)\\mathrm\{src\}\_\{h\_\{u\}\}\(s\)is the direct predecessor set of semantic nodessunder actionhuh\_\{u\}\. For each semantic node, leth​\(s\)h\(s\)be the action that produced it\. To recover the feature space of the game, we recursively trace each output nodeok∈Oo\_\{k\}\\in Oback to its predecessors\. This yields the set of value\-relevant semantic nodesSS\. We define the initial semantic nodes asI:=\{s∈S:srch​\(s\)​\(s\)=∅\}I:=\\\{s\\in S:\\mathrm\{src\}\_\{h\(s\)\}\(s\)=\\emptyset\\\}\. For eachs∈S∖Is\\in S\\setminus I, the directed hyperedgees:srch​\(s\)​\(s\)→se\_\{s\}:\\mathrm\{src\}\_\{h\(s\)\}\(s\)\\rightarrow s\. This gives the semantic generation hypergraphG=\(S,L,N;α,ω,O,I\)G=\(S,L,N;\\alpha,\\omega,O,I\), whereNNis the agent set in the current trajectory,L:=\{es:s∈S∖I\}L:=\\\{e\_\{s\}:s\\in S\\setminus I\\\}, andα:L→N\\alpha:L\\to Nrecords which agent owns each link,O⊆SO\\subseteq Sdenotes the output semantic nodes at which task value is evaluated, andω:O→ℝ\\omega:O\\to\\mathbb\{R\}assigns their value weights\.

We constructGGin this backward direction because attribution should focus on the system behaviors expressed through task\-relevant semantics\. Starting fromOOanchors the construction at the semantic features read by the value interface, while tracing predecessors identifies how the system produced, reused, or transformed those features\. The resulting graph is therefore a compact description of the realized behaviors from which agent\-level support logic and value can be induced\. Additional formal details and examples are deferred to the Appendix[G\.1](https://arxiv.org/html/2607.18255#A7.SS1)\. Therefore, we introduce the definition of Semantic Cooperative Game as follows\.

###### Definition 1\(Semantic Cooperative Game\)\.

Given a language flow and its induced semantic generation hypergraphG=\(S,L,N;α,ω,O,I\)G=\(S,L,N;\\alpha,\\omega,O,I\), we define the semantic game representation as𝒢​\(G\):=\(N,G\)\\mathcal\{G\}\(G\):=\(N,G\)\. Here,NNis the player set, andGGis the semantic generation hypergraph from which an agent\-level semantic value function is induced\. Players participate through their owned hyperedges, and value is read from the weighted output nodesOO\.

## 3SLIC: Semantic Logic Inversion for Contribution

![Refer to caption](https://arxiv.org/html/2607.18255v1/x2.png)Figure 2:Overall SCG/SLIC pipeline for semantic attribution\.The key idea is to compute agent contributions directly from a*single realized execution*by leveraging its semantic structure\. A workflow is transformed into a semantic generation hypergraphGG, from which SLIC traces value dependencies, recovers minimal support logic, and computes the Semantic Shapley Value \(SSV\)\. This enables*single\-pass, structure\-based attribution*without counterfactual reruns\.OnceGGis available, it can induce many possible value functionals and attribution rules, depending on which semantic outputs are selected and how they are valued\. In this work, we focus on one canonical allocation rule onGG: the Semantic Shapley Value \(SSV\), which is uniquely determined by the Shapley axioms overGG\. To compute SSV, SLIC recovers the support logic of value\-bearing output nodes to generateGGfrom the realized trajectory,and applies Boolean absorption and multilinear allocation to compute SSV\. This section formalizes this single\-trajectory computation, following the overall pipeline illustrated in Fig\.[2](https://arxiv.org/html/2607.18255#S3.F2)\.

Algorithm 1SLIC: constructGGfrom language flow and compute SSV1:realized trajectory

τ\\tau, value\-reading interface

RR, ownership map

α\\alpha
2:semantic generation hypergraph

GG, support logic, and SSV

3:Read

O=\{o1,…,oK\}O=\\\{o\_\{1\},\\dots,o\_\{K\}\\\}and its weights

ω\\omegafrom

RR
4:for

k=1,…,Kk=1,\\dots,Kdo

5:Starting from

oko\_\{k\}, recursively trace back its direct predecessors

srch​\(s\)​\(s\)\\mathrm\{src\}\_\{h\(s\)\}\(s\)along language flow until no new semantic nodes appear

6:endfor

7:Let the set of all traced semantic nodes be

SS, and define

I=\{s∈S:srch​\(s\)​\(s\)=∅\}I=\\\{s\\in S:\\mathrm\{src\}\_\{h\(s\)\}\(s\)=\\emptyset\\\}
8:For each

s∈S∖Is\\in S\\setminus I, define

es:srch​\(s\)​\(s\)→se\_\{s\}:\\mathrm\{src\}\_\{h\(s\)\}\(s\)\\rightarrow s, and let

L=\{es:s∈S∖I\}L=\\\{e\_\{s\}:s\\in S\\setminus I\\\}
9:Construct

G=\(S,L,N;α,ω,O,I\)G=\(S,L,N;\\alpha,\\omega,O,I\)
10:for

k=1,…,Kk=1,\\dots,Kdo

11:Extract the minimal link\-level supports of

oko\_\{k\}, lift them to the agent level, and apply Boolean absorption to obtain the support logic

fokf\_\{o\_\{k\}\}
12:Expand

fokf\_\{o\_\{k\}\}into multilinear / Möbius coefficients

cT\(ok\)c\_\{T\}^\{\(o\_\{k\}\)\}
13:endfor

14:Aggregate the weighted coefficients over all

kkand output

SSV​\(G\)\\mathrm\{SSV\}\(G\)

### 3\.1From Semantic Supports to SSV

Algorithm[1](https://arxiv.org/html/2607.18255#alg1)constructs an attribution game from a realized trajectory in two stages\. First, SLIC traces backward from the output nodesOOto recover the value\-relevant semantic flow and construct the semantic generation hypergraphGG\. Second, it converts this graph to the agent level by identifying which subsets of agents provide sufficient semantic support for each value\-bearing output node\.

Fix an output nodeok∈Oo\_\{k\}\\in O\. Let𝒞ok⊆2L∖\{∅\}\\mathcal\{C\}\_\{o\_\{k\}\}\\subseteq 2^\{L\}\\setminus\\\{\\emptyset\\\}denote its minimal link\-level conjunctive supports\. EachC∈𝒞okC\\in\\mathcal\{C\}\_\{o\_\{k\}\}is a set of semantic links whose joint presence is sufficient to supportoko\_\{k\}\. Through the ownership map, each link supportCCinduces the agent setN​\(C\):=\{α​\(ℓ\):ℓ∈C\}⊆NN\(C\):=\\\{\\alpha\(\\ell\):\\ell\\in C\\\}\\subseteq N\. After deduplication and Boolean absorption, these induced agent sets yield the minimal support familyℳok\\mathcal\{M\}\_\{o\_\{k\}\}\. The corresponding Boolean support function is

fok​\(x\)=⋁M∈ℳok⋀i∈Mxi,x∈\{0,1\}\|N\|\.f\_\{o\_\{k\}\}\(x\)=\\bigvee\_\{M\\in\\mathcal\{M\}\_\{o\_\{k\}\}\}\\bigwedge\_\{i\\in M\}x\_\{i\},\\qquad x\\in\\\{0,1\\\}^\{\|N\|\}\.\(1\)Here,xi=1x\_\{i\}=1indicates that agentiiis included in the selected agent configuration\. Thus,fok​\(x\)=1f\_\{o\_\{k\}\}\(x\)=1if and only if the selected agents semantically support the rubric nodeoko\_\{k\}\. Aggregating these support functions with rubric weights yields the agent\-level semantic value function induced byGG:

VGsem\(x\):=∑ok∈O:xok​\(G\)=1ω\(ok\)fok\(x\)\.V\_\{G\}^\{\\mathrm\{sem\}\}\(x\):=\\sum\_\{o\_\{k\}\\in O:\\,x\_\{o\_\{k\}\}\(G\)=1\}\\omega\(o\_\{k\}\)f\_\{o\_\{k\}\}\(x\)\.\(2\)This equation is the key step that separates SLIC from rerun\-based attribution\. The value function is not queried from counterfactual executions\. Instead, it is induced analytically from the semantic supports observed in the realized trajectory\.

To allocate this induced value, we expand each support function into its unique multilinear, or Möbius, form:

fok​\(x\)=∑T⊆NcT\(ok\)​∏i∈Txi,cT\(ok\)=∑R⊆T\(−1\)\|T\|−\|R\|​fok​\(𝟏R\)\.f\_\{o\_\{k\}\}\(x\)=\\sum\_\{T\\subseteq N\}c\_\{T\}^\{\(o\_\{k\}\)\}\\prod\_\{i\\in T\}x\_\{i\},\\qquad c\_\{T\}^\{\(o\_\{k\}\)\}=\\sum\_\{R\\subseteq T\}\(\-1\)^\{\|T\|\-\|R\|\}f\_\{o\_\{k\}\}\(\\mathbf\{1\}\_\{R\}\)\.\(3\)The coefficientcT\(ok\)c\_\{T\}^\{\(o\_\{k\}\)\}measures the semantic interaction carried jointly by the agent setTTfor the output nodeoko\_\{k\}\. These coefficients form the basis on which Shapley\-style allocation is performed in Definition[2](https://arxiv.org/html/2607.18255#Thmdefinition2)\.

###### Definition 2\(Semantic Shapley Value\)\.

Given the semantic representation of the game𝒢​\(G\)=\(N,G\)\\mathcal\{G\}\(G\)=\(N,G\), the semantic Shapley value induced byGGis

SSVi​\(G\):=∑ok∈O:xok​\(G\)=1ω​\(ok\)​∑T⊆N:i∈TcT\(ok\)\|T\|,i∈N\.\\mathrm\{SSV\}\_\{i\}\(G\)\\;:=\\;\\sum\_\{o\_\{k\}\\in O:x\_\{o\_\{k\}\}\(G\)=1\}\\omega\(o\_\{k\}\)\\sum\_\{T\\subseteq N:\\,i\\in T\}\\frac\{c\_\{T\}^\{\(o\_\{k\}\)\}\}\{\|T\|\},\\qquad i\\in N\.

### 3\.2Axiomatic Interpretation

Definition[2](https://arxiv.org/html/2607.18255#Thmdefinition2)applies Shapley\-style allocation to the semantic support expansion induced byGG\. Each coefficientcT\(ok\)c\_\{T\}^\{\(o\_\{k\}\)\}is a semantic support term jointly carried by the agents inTT\. The allocation of such a term follows the Shapley axioms: efficiency allocates the whole coefficient, dummy gives zero to agents outsideTT, and additivity sums allocations over all semantic terms\. Symmetry gives equal shares to agents that play the same role inside the same semantic support term\(Shapley,[1953](https://arxiv.org/html/2607.18255#bib.bib5); Young,[1985](https://arxiv.org/html/2607.18255#bib.bib7)\)\. Hence eachi∈Ti\\in TreceivescT\(ok\)/\|T\|c\_\{T\}^\{\(o\_\{k\}\)\}/\|T\|\. In this sense, SSV keeps the core Shapley behavior of sharing joint contribution, while avoiding counterfactual coalition evaluation\.

Boolean absorption acts before allocation and handles a different kind of redundancy\. It removes logically redundant support expansions so that equivalent support expressions define the samefokf\_\{o\_\{k\}\}, the same coefficients, and the same SSV\. For example,\(A∧B\)∨B≡B\(A\\wedge B\)\\vee B\\equiv Bremoves the redundantAA\-path before Shapley allocation\. By contrast, if a semantic term genuinely requires the joint supportA∧BA\\wedge B, absorption does not remove either agent; symmetry splits the remaining term between them\. Supplementary link\-support and absorption examples are given in Appendix[H\.1](https://arxiv.org/html/2607.18255#A8.SS1)–[H\.2](https://arxiv.org/html/2607.18255#A8.SS2); the degeneration of LOO under redundancy is discussed in Appendix[J\.1](https://arxiv.org/html/2607.18255#A10.SS1)\. The full derivation is given in Appendix[E](https://arxiv.org/html/2607.18255#A5)\.

###### Theorem 1\(Equivalence of Semantic Shapley Value with the Classical Shapley Value\)\.

Suppose the system satisfies set\-based value reachability, full observability, and no masking or order dependence along the realized language flow\. Letv:2N→ℝv:2^\{N\}\\to\\mathbb\{R\}be a classical coalition value function, and let

ϕi​\(v\)=∑C⊆N∖\{i\}\|C\|\!​\(\|N\|−\|C\|−1\)\!\|N\|\!​\[v​\(C∪\{i\}\)−v​\(C\)\]\\phi\_\{i\}\(v\)=\\sum\_\{C\\subseteq N\\setminus\\\{i\\\}\}\\frac\{\|C\|\!\(\|N\|\-\|C\|\-1\)\!\}\{\|N\|\!\}\\bigl\[v\(C\\cup\\\{i\\\}\)\-v\(C\)\\bigr\]\(4\)denote the classical Shapley value\.

LetVGsem:2N→ℝV\_\{G\}^\{\\mathrm\{sem\}\}:2^\{N\}\\to\\mathbb\{R\}be the coalition function induced by the semantic structure of the systemGG\. Under the above assumptions, the induced coalition function coincides with the classical one:

VGsem​\(C\)=v​\(C\),∀C⊆N\.V\_\{G\}^\{\\mathrm\{sem\}\}\(C\)=v\(C\),\\qquad\\forall C\\subseteq N\.\(5\)Consequently, for every agenti∈Ni\\in N, the semantic Shapley value satisfies

SSVi​\(G\)=ϕi​\(v\)\.\\mathrm\{SSV\}\_\{i\}\(G\)=\\phi\_\{i\}\(v\)\.\(6\)

The proof is given in Appendix[F](https://arxiv.org/html/2607.18255#A6)\.

### 3\.3A Trace Example with Absorption

We illustrate the role of Boolean absorption with a simple five\-agent trajectory\. AgentAAproduces a background\-fact nodes1s\_\{1\}, agentBBproduces a key\-evidence nodes2s\_\{2\}, and agentCCrewritess2s\_\{2\}into a prompt nodes3s\_\{3\}\. AgentDDthen generates an expanded\-summary nodes4s\_\{4\}from\{s1,s2\}\\\{s\_\{1\},s\_\{2\}\\\}, while agentEEproduces the rubric output nodes5s\_\{5\}froms2s\_\{2\}\. The induced semantic generation hypergraph contains

∅→𝐴s1,∅→𝐵s2,\{s2\}→𝐶s3,\{s1,s2\}→𝐷s4,\{s2\}→𝐸s5\.\\emptyset\\xrightarrow\{A\}s\_\{1\},\\quad\\emptyset\\xrightarrow\{B\}s\_\{2\},\\quad\\\{s\_\{2\}\\\}\\xrightarrow\{C\}s\_\{3\},\\quad\\\{s\_\{1\},s\_\{2\}\\\}\\xrightarrow\{D\}s\_\{4\},\\quad\\\{s\_\{2\}\\\}\\xrightarrow\{E\}s\_\{5\}\.
Supposes5s\_\{5\}is the only activated output node\. Link\-level recovery may produce two candidate agent supports,\{B,E\}\\\{B,E\\\}and\{B,C,E\}\\\{B,C,E\\\}\. The second support passes through the rewritten nodes3s\_\{3\}, but it is logically redundant becauses5s\_\{5\}is already supported byBBandEE\. Boolean absorption therefore removes the redundant expansion path and leaves

fs5​\(x\)=xB​xE,vs5sem​\(x\)=ω​\(s5\)​xB​xE\.f\_\{s\_\{5\}\}\(x\)=x\_\{B\}x\_\{E\},\\qquad v\_\{s\_\{5\}\}^\{\\mathrm\{sem\}\}\(x\)=\\omega\(s\_\{5\}\)x\_\{B\}x\_\{E\}\.Thus, the output\-node contribution is split only between the indispensable agentsBBandEE:

SSVB​\(G\)=SSVE​\(G\)=ω​\(s5\)2,SSVA​\(G\)=SSVC​\(G\)=SSVD​\(G\)=0\.\\mathrm\{SSV\}\_\{B\}\(G\)=\\mathrm\{SSV\}\_\{E\}\(G\)=\\frac\{\\omega\(s\_\{5\}\)\}\{2\},\\qquad\\mathrm\{SSV\}\_\{A\}\(G\)=\\mathrm\{SSV\}\_\{C\}\(G\)=\\mathrm\{SSV\}\_\{D\}\(G\)=0\.
This example shows how SLIC avoids assigning credit to a semantically redundant rewrite\. It recovers the source relations in language flow, constructsGG, derives the minimal support family after absorption, and then computes SSV on the resulting support logic\.

More examples are provided in Appendices[H](https://arxiv.org/html/2607.18255#A8),[G](https://arxiv.org/html/2607.18255#A7)and[I](https://arxiv.org/html/2607.18255#A9)\.

## 4Experiments

### 4\.1Experiment 1: Consistency with Classical Shapley in Medical Multi\-Agent Collaboration

Experiment 1 tests the reduction claim in Theorem[1](https://arxiv.org/html/2607.18255#Thmtheorem1)\. We construct a controlled medical multi\-agent workflow in which the theorem’s assumptions are satisfied by construction: agents solve independent semantic subproblems and their outputs are aggregated without interaction effects\. We then ask whether SLIC, using only the realized semantic structure, agrees with the classical Shapley attribution for the same workflow\.

#### 4\.1\.1Task Setup

We use HealthBench\(Aroraet al\.,[2025](https://arxiv.org/html/2607.18255#bib.bib22)\)as the test domain and implement a three\-agent workflow in which agents solve separate semantic subproblems independently and their outputs are aggregated afterward\. Agent A handles history and symptom extraction, Agent B handles differential diagnosis, and Agent C handles treatment planning and prognosis\. This construction satisfies the reduction assumptions by design: the output of any subset depends only on which independent modules are present and how their outputs are aggregated, rubric hits are directly observable, and no module’s contribution is altered by interaction or execution order\. Letv​\(S\)v\(S\)denote the rubric score obtained by an agent subsetS⊆NS\\subseteq N\. Our target quantity is the classical Shapley value

ϕi​\(v\)=∑S⊆N∖\{i\}\|S\|\!​\(\|N\|−\|S\|−1\)\!\|N\|\!​\(v​\(S∪\{i\}\)−v​\(S\)\)\.\\phi\_\{i\}\(v\)=\\sum\_\{S\\subseteq N\\setminus\\\{i\\\}\}\\frac\{\|S\|\!\(\|N\|\-\|S\|\-1\)\!\}\{\|N\|\!\}\\big\(v\(S\\cup\\\{i\\\}\)\-v\(S\)\\big\)\.\(7\)

#### 4\.1\.2Evaluation Protocol

The data come from the public medical QA benchmark HealthBench \(openai/healthbench\)\. Implementation details and model configurations are provided in Appendix[M](https://arxiv.org/html/2607.18255#A13)\. We retain only complex cases with at leastR≥7R\\geq 7rubric items and sample 50 cases for the main test set\. All generation and judging are run in a deterministic setting\. Since the baseline judge still exhibits run\-level disagreement on a small subset of rubric items, the main analysis removes those ambiguous items; the corresponding statistics and cause analysis are given in Appendix[J\.3](https://arxiv.org/html/2607.18255#A10.SS3)\. We report attribution errorL​1=‖ϕpred−ϕGT‖1L1=\\\|\\phi^\{\\mathrm\{pred\}\}\-\\phi^\{\\mathrm\{GT\}\}\\\|\_\{1\}, rank consistency Kendallτb\\tau\_\{b\}, and Micro Hit Rate for local logic recovery\. We compare SLIC against three alternatives\. MC\-GT exhaustively scores all non\-empty subsets and computes the exact Shapley value as the oracle reference\. LOO estimates contribution byv​\(N\)−v​\(N∖\{i\}\)v\(N\)\-v\(N\\setminus\\\{i\\\}\)\. A Holistic LLM Judge directly outputs contribution proportions for A/B/C and normalizes them to the joint scorev​\(N\)v\(N\)\. The systematic degeneration of LOO under redundancy is discussed in Appendix[J\.1](https://arxiv.org/html/2607.18255#A10.SS1)\.

#### 4\.1\.3Main Results

Table[1](https://arxiv.org/html/2607.18255#S4.T1)reports the main results on 50 complex cases\. SLIC achieves the best attribution accuracy and ranking consistency among the low\-cost methods while retaining the same low\-call regime as the holistic judge\. Under the same low\-call regime as the holistic judge, SLIC reduces the L1 error from 12\.90 to 6\.61 and raises Kendallτb\\tau\_\{b\}to 0\.814, outperforming both LOO and the holistic baseline\. At the micro level, SLIC achieves a Micro Hit Rate of 0\.857 \(150/175\), indicating that the aggregate attribution gains are tied to more reliable local support recovery\.

We do not scale this consistency experiment further in the main text because MC\-style baselines require repeated subset scoring, and their cost grows exponentially with the number of agents\. As a supplement, Appendix[J\.4](https://arxiv.org/html/2607.18255#A10.SS4)reports a 4\-agent extension: under corrected\-cost accounting, SLIC reduces cost by 93\.3% relative to MC while still maintaining low attribution error \(L1=5\.700=5\.700\)\.

Table 1:Performance comparison on HealthBench\.RRdenotes a normalized per\-run cost unit\.
#### 4\.1\.4Error Analysis

Manual inspection of the high\-error cases reveals four recurring failure modes: OR\-miss under parallel triggering paths, polarity inversion for negative rubric items, asymmetric penalties when multiple agents jointly fail a constraint, and residual judge noise caused by rubric ambiguity\. Representative cases and run\-level analyses are provided in Appendix[J\.2](https://arxiv.org/html/2607.18255#A10.SS2)–[J\.3](https://arxiv.org/html/2607.18255#A10.SS3)\. Overall, the current bottleneck lies more in the induction of logic and the judge instability than in the SSV definition itself\.

### 4\.2Experiment 2: Diagnosing Complex Multi\-Role Multi\-Agent Architectures

Experiment 2 studies SSV in settings where agent\-level contribution is latent rather than directly identifiable\. In complex multi\-role workflows, the final system score does not come with a ground\-truth decomposition into per\-agent contributions, and no standard benchmark exists for node importance\. We therefore use intervention\-induced degradation as an external behavioral probe rather than as a ground\-truth attribution target\. These probes are imperfect: removal can be confounded by redundancy, and injected corruption can reflect both semantic disruption and structural sensitivity\. Nevertheless, if SSV captures functionally consequential node importance, its profile should align with the degradation patterns induced by sufficiently strong interventions\.

#### 4\.2\.1Task Setup

We build a diagnostic multi\-agent benchmark with four task settings and 50 cases per setting\. All settings follow the same healthy\-workflow design principle: responsibility boundaries are explicit, partial access is controlled, downstream nodes cannot freely reconstruct missing upstream semantics, and structural privilege is not overly concentrated in agents with weak semantic responsibility\. inspired by QMSum\(Zhonget al\.,[2021](https://arxiv.org/html/2607.18255#bib.bib19)\), TAT\-QA\(Zhuet al\.,[2021](https://arxiv.org/html/2607.18255#bib.bib20)\), LegalBench\(Guha and others,[2023](https://arxiv.org/html/2607.18255#bib.bib21)\), and HealthBench\(Aroraet al\.,[2025](https://arxiv.org/html/2607.18255#bib.bib22)\), without using samples from the original benchmarks\. The tasks cover meeting summarization, numerical reasoning, rule\-based judgment, and medical safety QA, respectively\. They are designed to evaluate response quality across diverse scenarios\. Table[2](https://arxiv.org/html/2607.18255#S4.T2)summarizes the workflow structure and role allocation\.Implementation details and model configurations are provided in Appendix[M](https://arxiv.org/html/2607.18255#A13)\.

Table 2:Task settings and agent responsibilities in Experiment 2\.
#### 4\.2\.2Evaluation Protocol

For each case, we first compute the SSV profile on the unperturbed trajectory\. We then intervene on each agent using weak, medium, and strong injected hallucinations, together with an LOO / C3\-style absence intervention\. Each intervention produces an agent\-indexed score\-drop profile\. We normalize profiles within each setting and report Spearmanρ\\rhobetween SSV and the perturbation\-induced profile\.

#### 4\.2\.3Main Results

Table 3:Shape alignment \(Spearmanρ\\rho\) between SSV and perturbation\-induced score\-drop profiles in Experiment 2\.Table[3](https://arxiv.org/html/2607.18255#S4.T3)reports the alignment between the clean\-workflow SSV profile and the perturbation\-induced score\-drop profiles across four task settings and four intervention types\. In this experiment, SSV is evaluated through its ability to predict which agents are behaviorally consequential in a realized multi\-agent workflow\. zAcross settings, SSV aligns strongly with the profiles induced by medium and strong hallucination interventions \(meanρ=0\.868\\rho=0\.868for both\) and also shows substantial alignment with LOO / C3\-style absence interventions \(meanρ=0\.810\\rho=0\.810\)\. Weak perturbations yield lower and less stable alignment \(meanρ=0\.612\\rho=0\.612\)\.

This pattern is consistent with the intended interpretation of SSV\. When an intervention sufficiently disrupts an agent’s semantic function, the resulting degradation profile closely follows the importance pattern predicted by SSV\. Weak perturbations often leave the agent’s effective role only partially changed, while absence\-based interventions can still be affected by redundancy when other agents compensate for the missing input\. Overall, the results indicate that SSV captures functionally meaningful node importance beyond the reduction setting and serves as a practical diagnostic signal for complex multi\-role multi\-agent workflows\.

![Refer to caption](https://arxiv.org/html/2607.18255v1/scg_ai_figures/experiment2_radar.png)Figure 3:Radar plots compare the normalized SSV profile with score\-drop profiles induced by weak, medium, strong, and leave\-one\-out interventions across four evaluation settings\. Greater overlap indicates stronger alignment between SSV and intervention\-induced degradation patterns\.
#### 4\.2\.4Structural Analysis

Fig\.[3](https://arxiv.org/html/2607.18255#S4.F3)shows the structural picture behind these numbers\. In most healthy workflows, medium and strong perturbation profiles align with SSV, and LOO / C3\-style interventions often follow the same trend\. The main deviations reveal two distinct structural effects\. First, privilege–capability mismatch occurs when an agent performs limited explicit semantic work but still has enough structural control to collapse the system when attacked\. This indicates that semantic contribution and structural privilege can diverge as detailed in Appendix[K\.3\.1](https://arxiv.org/html/2607.18255#A11.SS3.SSS1)\. Second, removal baselines can fail under redundancy: if two agents provide equivalent critical inputs, removing either one can leave the other sufficient to reconstruct the result, causing LOO / C3\-style methods to underestimate both\. Additional alignment figures, counterexamples, and raw profile values are provided in Appendix[K\.2](https://arxiv.org/html/2607.18255#A11.SS2)–[L](https://arxiv.org/html/2607.18255#A12)and Appendix[J\.1](https://arxiv.org/html/2607.18255#A10.SS1)\.

## 5Conclusion

This paper studies contribution attribution in LLM\-based multi\-agent systems and proposes an interpretable semantic allocation framework\. Our central observation is that an agent’s contribution should not be determined only by its presence in a counterfactual coalition, but by the value\-relevant semantics it generates, preserves, or revises along a realized collaborative trajectory\. To capture this structure, we formulate attribution on the semantic generation hypergraph induced by the language flow, where redundancy, substitution, joint support, and semantic absorption among agents can be explicitly represented\. On the theoretical side, we provide an axiomatic characterization of semantic contribution attribution and show that the resulting allocation is a well\-defined Shapley value over semantic support structures\. This extends the classical Shapley value from coalition\-level utility functions to semantic structures induced by LLM workflows, while recovering the classical Shapley value under standard set\-based conditions\. Empirically, our results show that the proposed method produces stable and interpretable agent\-level contribution signals while substantially reducing the need for costly counterfactual reruns\. Overall, this work offers a principled and practical foundation for low\-cost contribution analysis in LLM\-based multi\-agent collaboration\.

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## Appendix ARelated Works

##### LLM\-based multi\-agent systems and agentic workflows\.

Recent multi\-agent LLM frameworks show that capability gains arise not only from scaling the base model, but also from explicit coordination mechanisms\. CAMEL, AutoGen, and MetaGPT demonstrate the utility of role specialization, inter\-agent conversation, and modular decomposition\[Liet al\.,[2023a](https://arxiv.org/html/2607.18255#bib.bib14), Wuet al\.,[2023](https://arxiv.org/html/2607.18255#bib.bib15), Hong and others,[2024](https://arxiv.org/html/2607.18255#bib.bib16)\]\. Subsequent systems such as ChatDev and AgentVerse treat collaboration itself as the primary computational object and study how structured communication shapes collective behavior\[Qianet al\.,[2024](https://arxiv.org/html/2607.18255#bib.bib28), Chenet al\.,[2023](https://arxiv.org/html/2607.18255#bib.bib29)\]\. Other work investigates specific coordination ingredients, including theory\-of\-mind reasoning across agents\[Liet al\.,[2023b](https://arxiv.org/html/2607.18255#bib.bib30)\], communication pruning for reducing coordination overhead\[Zhanget al\.,[2025a](https://arxiv.org/html/2607.18255#bib.bib3)\], and automated workflow search and construction\[Zhanget al\.,[2024](https://arxiv.org/html/2607.18255#bib.bib31)\]\. In domain\-specific settings, MedAgents shows that carefully designed multi\-expert collaboration can be effective even in high\-stakes medical reasoning\[Tanget al\.,[2024](https://arxiv.org/html/2607.18255#bib.bib17)\]\. Taken together, these studies establish that the performance of agentic systems depends crucially on workflow structure, communication pathways, and the handling of intermediate artifacts, rather than merely on which agents are present\.

##### Cooperative game theory and structured cooperation\.

The theoretical basis for our work comes from cooperative game theory, where the Shapley value remains the canonical starting point for contribution allocation\[Shapley,[1953](https://arxiv.org/html/2607.18255#bib.bib5)\]\. Classical extensions already suggest that structured cooperation requires structured value notions\. Harsanyi’s bargaining and dividend\-based formulations decompose coalition value into finer interaction terms\[Harsanyi,[1959](https://arxiv.org/html/2607.18255#bib.bib34),[1963](https://arxiv.org/html/2607.18255#bib.bib35)\]\. Aumann and Shapley analyze value in non\-atomic settings, and Young gives an axiomatic perspective based on monotonicity\[Aumann and Shapley,[1974](https://arxiv.org/html/2607.18255#bib.bib6), Young,[1985](https://arxiv.org/html/2607.18255#bib.bib7)\]\. Owen’s multilinear extension provides an algebraic representation of games that is particularly relevant when allocations are computed from decomposed interaction terms\[Owen,[1972](https://arxiv.org/html/2607.18255#bib.bib36)\]\. When cooperation is constrained by predefined groups or communication relations, classical work already departs from the plain characteristic\-function view: Aumann and Drèze study coalition structures, Myerson studies graph\-restricted cooperation, and Owen extends value allocation to games with a priori unions\[Aumann and Drèze,[1974](https://arxiv.org/html/2607.18255#bib.bib37), Myerson,[1977](https://arxiv.org/html/2607.18255#bib.bib38), Owen,[1977](https://arxiv.org/html/2607.18255#bib.bib39)\]\. Further generalizations include weighted Shapley values and potential\-based characterizations\[Kalai and Samet,[1987](https://arxiv.org/html/2607.18255#bib.bib40), Hart and Mas\-Colell,[1989](https://arxiv.org/html/2607.18255#bib.bib41)\]\. Even in voting and committee settings, the Shapley–Shubik and Banzhaf lines show that influence depends on structural notions of pivotality rather than on presence alone\[Shapley and Shubik,[1954](https://arxiv.org/html/2607.18255#bib.bib42), Banzhaf,[1965](https://arxiv.org/html/2607.18255#bib.bib43), Dubey and Shapley,[1979](https://arxiv.org/html/2607.18255#bib.bib44)\]\. This broader literature supports our central premise that once cooperation is mediated by structure, the value object itself should be defined on that structure\.

##### Credit assignment from reinforcement learning to LLM collaboration\.

A second closely related tradition is credit assignment\. Early work on temporal credit assignment in reinforcement learning formalized the difficulty of tracing delayed outcomes back to individual decisions\[Sutton,[1984](https://arxiv.org/html/2607.18255#bib.bib10)\]\. Later work generalized this perspective to structural settings and collective systems, showing that reward redistribution and payoff design must account for interaction topology rather than only individual actions\[Agogino and Tumer,[2004](https://arxiv.org/html/2607.18255#bib.bib11), Wolpert and Tumer,[2001](https://arxiv.org/html/2607.18255#bib.bib26)\]\. In multi\-agent reinforcement learning, COMA uses a counterfactual baseline to isolate the marginal effect of each agent action on joint return\[Foersteret al\.,[2018](https://arxiv.org/html/2607.18255#bib.bib12)\], and subsequent work studies credit assignment under global rewards and collective objectives\[Nguyenet al\.,[2018](https://arxiv.org/html/2607.18255#bib.bib27)\]\. More recently, this counterfactual line has been adapted to language\-mediated collaboration: C3 estimates message\-level responsibility through fixed\-continuation replay and leave\-one\-out interventions in LLM collaboration\[Chenet al\.,[2026](https://arxiv.org/html/2607.18255#bib.bib13)\]\. These approaches provide useful causal intuitions, but they generally identify contribution by rerunning altered systems, which becomes expensive and unstable in large language\-based workflows\.

## Appendix BLimitations and Discussion

SCG and SLIC are intended for attribution on realized language\-flow traces, not for all forms of multi\-agent evaluation\. The first limitation is semantic recovery\. SLIC assumes that value\-bearing semantic nodes and their support relations can be extracted with sufficient reliability\. This is reasonable for rubric\-based tasks with explicit intermediate outputs, but remains difficult when support is implicit, negated, masked by later text, or distributed across paraphrases\. In those cases, errors in the support logic directly affect the SSV\.

The second limitation is the value\-reading interface\. Our framework attributes the value that the interface reads fromOO\. If the rubric is incomplete, ambiguous, or unstable between judge calls, then the resulting attribution inherits that uncertainty\. This is not a failure of the Shapley allocation step, but it limits the maximum reliability of any method that depends on the same observed value signal\.

Despite these limitations, the main message is that contribution attribution in LLM\-based multi\-agent systems need not be reduced to black\-box counterfactual reruns\. By treating language flow as the observable layer and semantic support as the value\-bearing structure, SCG gives a concrete object for attribution, and SLIC provides a fast single\-trajectory computation path\. This speed matters: by exposing which semantic transformations support task value without repeated reruns, SSV makes downstream workflow auditing, robustness analysis, agent pruning, and contribution\-aware system redesign practically possible\. We leave these intervention\-oriented uses to future work\.

## Appendix CWorkflow and Language Flow

This appendix fills in the workflow, language flow, and value\-node definitions that are compressed in the main text\.

For any execution positionuu, let the input and output satisfy

xu,yu∈𝒳,yu=Fa​\(xu;θa\)\.x\_\{u\},\\;y\_\{u\}\\in\\mathcal\{X\},\\qquad y\_\{u\}=F^\{a\}\(x\_\{u\};\\theta^\{a\}\)\.Herexux\_\{u\}andyuy\_\{u\}denote the semantic states before and after this step,aais the acting agent, andθa\\theta^\{a\}denotes the stable configuration associated with that agent\.

The language flow in the main text is written as

𝒯=\(L,ℋ\),\\mathcal\{T\}=\(L,\\mathcal\{H\}\),whereL⊆𝒳L\\subseteq\\mathcal\{X\}is the set of semantic states propagated along the workflow channel, andℋ=\{hu:u∈V\}\\mathcal\{H\}=\\\{h\_\{u\}:u\\in V\\\}is the set of corresponding agent actions\. Workflow describes the collaboration structure among agents, whereas language flow describes how those agent actions act on semantic states\.

We then introduce a task value function on the semantic state space:

v:𝒳→ℝ\.v:\\mathcal\{X\}\\to\\mathbb\{R\}\.We further assume thatvvadmits a finite discriminative decomposition, namely that there exist finitely many discriminators

rk:𝒳→\{0,1\},k=1,…,m,r\_\{k\}:\\mathcal\{X\}\\to\\\{0,1\\\},\\qquad k=1,\\dots,m,and weightswk∈ℝw\_\{k\}\\in\\mathbb\{R\}such that

v​\(z\)=∑k=1mwk​rk​\(z\),∀z∈𝒳\.v\(z\)=\\sum\_\{k=1\}^\{m\}w\_\{k\}\\,r\_\{k\}\(z\),\\qquad\\forall z\\in\\mathcal\{X\}\.In practice,\{rk\}k=1m\\\{r\_\{k\}\\\}\_\{k=1\}^\{m\}can be viewed as a set of rubric\-style yes\-or\-no evaluation items\.

For any output semantic stateyuy\_\{u\}, let

denote the set of local semantic components inyuy\_\{u\}that can activate an evaluation item\. We call the elements ofOuO\_\{u\}value nodes\. Let

O:=⋃uOuO:=\\bigcup\_\{u\}O\_\{u\}be the set of all value nodes\.

## Appendix DRecursive Closure Construction from Language Flow

Next, we do not keep all semantics, but only those lying on the generation chains leading to the value nodes\. Assumption 1 in the main text can be restated here as follows: for any execution positionuu, the local semantic components in the output stateyuy\_\{u\}that enter the task value formation process can be identified and separated; each such local semantic components⊆yus\\subseteq y\_\{u\}belongs to one of three categories, namely native, derived, or persistent semantics\. Ifssis derived or persistent, its previous semantic source in the input statexux\_\{u\}can be identified\.

Accordingly, for anyhu∈ℋh\_\{u\}\\in\\mathcal\{H\}, define the source mapping

srchu\\operatorname\{src\}\_\{h\_\{u\}\}as follows\. For any local semantic components⊆yus\\subseteq y\_\{u\}, let

srchu⁡\(s\)=\{∅,s​is native,\{s~\},s​is persistent,P,P⊆xu,s​is derived,\\operatorname\{src\}\_\{h\_\{u\}\}\(s\)=\\begin\{cases\}\\varnothing,&s\\text\{ is native\},\\\\\[4\.0pt\] \\\{\\tilde\{s\}\\\},&s\\text\{ is persistent\},\\\\\[4\.0pt\] P,\\;P\\subseteq x\_\{u\},&s\\text\{ is derived\},\\end\{cases\}wheres~\\tilde\{s\}denotes the same old semantic unit preserved from the input statexux\_\{u\}, andPPdenotes the set of old semantic sources that generatess\.

Starting from the value nodes, define recursively

S\(t\+1\)=S\(t\)∪⋃u⋃s∈S\(t\)s⊆yusrchu⁡\(s\),t=0,1,2,…,S^\{\(t\+1\)\}=S^\{\(t\)\}\\cup\\bigcup\_\{u\}\\bigcup\_\{\\begin\{subarray\}\{c\}s\\in S^\{\(t\)\}\\\\ s\\subseteq y\_\{u\}\\end\{subarray\}\}\\operatorname\{src\}\_\{h\_\{u\}\}\(s\),\\qquad t=0,1,2,\\dots,and let

S:=⋃t≥0S\(t\)\.S:=\\bigcup\_\{t\\geq 0\}S^\{\(t\)\}\.In other words,SSis the minimal semantic set obtained by backward recursive closure from the value\-node setOOalong language flow\.

After obtaining the node setSS, we further extract effective input–output relations from local operations in language flow\. Specifically, for anyhu∈ℋh\_\{u\}\\in\\mathcal\{H\}, if there exist

Q⊆S,R⊆S,Q\\subseteq S,\\qquad R\\subseteq S,whereQQcomes from the input statexux\_\{u\},RRcomes from the output stateyuy\_\{u\}, and the nodes inRRare supported byQQat the current step, then

is called a semantic link induced byhuh\_\{u\}\. Letℒ\\mathcal\{L\}be the set of all such links\.

This yields the semantic generation graph

G=\(S,ℒ,N;α,ω,O,I\),G=\(S,\\mathcal\{L\},N;\\alpha,\\omega,O,I\),whereα:ℒ→N\\alpha:\\mathcal\{L\}\\to Nis the link ownership map,ω\\omegais the node\-weight function, andIIis the set of initial semantic nodes\. In the main text, for notational consistency, we simply writeℒ\\mathcal\{L\}asLL\.

## Appendix EFull Derivation of Semantic Shapley Value

The main text only gives the final formula for SSV\. Here we provide the intermediate derivation\.

##### A\. From link supports to agent supports\.

Fix an output nodeok∈Oo\_\{k\}\\in O\. Let

𝒞ok⊆2L∖\{∅\}\\mathcal\{C\}\_\{o\_\{k\}\}\\subseteq 2^\{L\}\\setminus\\\{\\emptyset\\\}be the set of all minimal link\-level conjunctive supports that semantically supportoko\_\{k\}\. For anyC∈𝒞okC\\in\\mathcal\{C\}\_\{o\_\{k\}\}, define the corresponding agent set by

N​\(C\):=\{α​\(ℓ\):ℓ∈C\}⊆N\.N\(C\):=\\\{\\alpha\(\\ell\):\\ell\\in C\\\}\\subseteq N\.Collect all such agent sets and remove strict redundant supersets, yielding the minimal subject support family

ℳok:=min⊆⁡\{N​\(C\):C∈𝒞ok\}\.\\mathcal\{M\}\_\{o\_\{k\}\}:=\\min\_\{\\subseteq\}\\\{N\(C\):C\\in\\mathcal\{C\}\_\{o\_\{k\}\}\\\}\.Here,min⊆\\min\_\{\\subseteq\}means keeping only inclusion\-minimal sets\. In the main\-text example, the absorption of\{B,E\}\\\{B,E\\\}over\{B,C,E\}\\\{B,C,E\\\}is a direct instance of this step\.

##### B\. From the support family to support logic\.

For each output nodeoko\_\{k\}, define the agent\-level support function

fok​\(x\)=⋁M∈ℳok⋀i∈Mxi,x∈\{0,1\}\|N\|\.f\_\{o\_\{k\}\}\(x\)=\\bigvee\_\{M\\in\\mathcal\{M\}\_\{o\_\{k\}\}\}\\ \\bigwedge\_\{i\\in M\}x\_\{i\},\\qquad x\\in\\\{0,1\\\}^\{\|N\|\}\.Herexi=1x\_\{i\}=1means that agentiiis included in the current configuration\. Therefore,

if and only if the configurationxxis semantically sufficient to support the output nodeoko\_\{k\}\. Using the weightω​\(ok\)\\omega\(o\_\{k\}\)of the output node, we obtain the corresponding local value function

voksem​\(x\):=ω​\(ok\)​fok​\(x\)\.v\_\{o\_\{k\}\}^\{\\mathrm\{sem\}\}\(x\):=\\omega\(o\_\{k\}\)f\_\{o\_\{k\}\}\(x\)\.

##### C\. From coefficient expansion to SSV\.

For eachoko\_\{k\}, the functionfokf\_\{o\_\{k\}\}has a unique multilinear expansion:

fok​\(x\)=∑T⊆NcT\(ok\)​∏i∈Txi,f\_\{o\_\{k\}\}\(x\)=\\sum\_\{T\\subseteq N\}c\_\{T\}^\{\(o\_\{k\}\)\}\\prod\_\{i\\in T\}x\_\{i\},where the coefficients are given by Möbius inversion:

cT\(ok\)=∑R⊆T\(−1\)\|T\|−\|R\|​fok​\(𝟏R\)\.c\_\{T\}^\{\(o\_\{k\}\)\}=\\sum\_\{R\\subseteq T\}\(\-1\)^\{\|T\|\-\|R\|\}f\_\{o\_\{k\}\}\(\\mathbf\{1\}\_\{R\}\)\.Here𝟏R\\mathbf\{1\}\_\{R\}is the indicator vector that is11exactly onRR\. The meaning of this expansion is that each monomial

∏i∈Txi\\prod\_\{i\\in T\}x\_\{i\}corresponds to a basic joint support unit\. If its coefficient iscT\(ok\)c\_\{T\}^\{\(o\_\{k\}\)\}, then the total weight of that unit isω​\(ok\)​cT\(ok\)\\omega\(o\_\{k\}\)c\_\{T\}^\{\(o\_\{k\}\)\}\.

For any joint support unitTT, its weight is split evenly among the agents inTT\. Hence the allocation that agentiireceives from output nodeoko\_\{k\}is

ϕisem​\(ok;G\)=ω​\(ok\)​∑T⊆N:i∈TcT\(ok\)\|T\|\.\\phi\_\{i\}^\{\\mathrm\{sem\}\}\(o\_\{k\};G\)=\\omega\(o\_\{k\}\)\\sum\_\{T\\subseteq N:\\,i\\in T\}\\frac\{c\_\{T\}^\{\(o\_\{k\}\)\}\}\{\|T\|\}\.Summing over all activated output nodes gives the graph\-level semantic Shapley value used in the main text:

SSVi​\(G\)=∑ok∈O:xok​\(G\)=1ϕisem​\(ok;G\)=∑ok∈O:xok​\(G\)=1ω​\(ok\)​∑T⊆N:i∈TcT\(ok\)\|T\|\.\\mathrm\{SSV\}\_\{i\}\(G\)=\\sum\_\{o\_\{k\}\\in O:x\_\{o\_\{k\}\}\(G\)=1\}\\phi\_\{i\}^\{\\mathrm\{sem\}\}\(o\_\{k\};G\)=\\sum\_\{o\_\{k\}\\in O:x\_\{o\_\{k\}\}\(G\)=1\}\\omega\(o\_\{k\}\)\\sum\_\{T\\subseteq N:\\,i\\in T\}\\frac\{c\_\{T\}^\{\(o\_\{k\}\)\}\}\{\|T\|\}\.Therefore, SSV is not obtained by directly averaging over minimal support sets\. Instead, it keeps the full Boolean support logic first, and then performs analytical allocation on the multilinear / Möbius basis\. This is precisely why it can handle shared predecessors, parallel support paths, and absorption relations\.

## Appendix FProof of Theorem[1](https://arxiv.org/html/2607.18255#Thmtheorem1)

###### Proof\.

Fix an activated output nodeoko\_\{k\}\. Consider its agent\-level Boolean support logic

fok​\(x\)=∑T⊆NcT\(ok\)​∏j∈Txj,x∈\{0,1\}\|N\|\.f\_\{o\_\{k\}\}\(x\)=\\sum\_\{T\\subseteq N\}c\_\{T\}^\{\(o\_\{k\}\)\}\\prod\_\{j\\in T\}x\_\{j\},\\qquad x\\in\\\{0,1\\\}^\{\|N\|\}\.Under the set\-based, fully observable, and no\-masking / no\-order\-dependence conditions stated in the theorem, whether the nodeoko\_\{k\}is hit depends only on the subset of participating agents\. Hencefokf\_\{o\_\{k\}\}can be naturally viewed as a Boolean characteristic function on the coalition space2N2^\{N\}: for anyC⊆NC\\subseteq N, let𝟏C∈\{0,1\}\|N\|\\mathbf\{1\}\_\{C\}\\in\\\{0,1\\\}^\{\|N\|\}be its indicator vector and define the coalition hit function

vok​\(C\):=fok​\(𝟏C\)\.v\_\{o\_\{k\}\}\(C\):=f\_\{o\_\{k\}\}\(\\mathbf\{1\}\_\{C\}\)\.Thusvok:2N→\{0,1\}v\_\{o\_\{k\}\}:2^\{N\}\\to\\\{0,1\\\}is a standard characteristic function of a cooperative game\.

From the multilinear expansion offokf\_\{o\_\{k\}\}, for anyC⊆NC\\subseteq N,

vok​\(C\)=fok​\(𝟏C\)=∑T⊆NcT\(ok\)​∏j∈T𝟏C​\(j\)\.v\_\{o\_\{k\}\}\(C\)=f\_\{o\_\{k\}\}\(\\mathbf\{1\}\_\{C\}\)=\\sum\_\{T\\subseteq N\}c\_\{T\}^\{\(o\_\{k\}\)\}\\prod\_\{j\\in T\}\\mathbf\{1\}\_\{C\}\(j\)\.Note that

∏j∈T𝟏C​\(j\)=\{1,T⊆C,0,T⊈C,\\prod\_\{j\\in T\}\\mathbf\{1\}\_\{C\}\(j\)=\\begin\{cases\}1,&T\\subseteq C,\\\\ 0,&T\\not\\subseteq C,\\end\{cases\}and therefore

vok​\(C\)=∑T⊆CcT\(ok\)\.v\_\{o\_\{k\}\}\(C\)=\\sum\_\{T\\subseteq C\}c\_\{T\}^\{\(o\_\{k\}\)\}\.
Now for anyC⊆N∖\{i\}C\\subseteq N\\setminus\\\{i\\\},

vok​\(C∪\{i\}\)−vok​\(C\)=∑T⊆C∪\{i\}cT\(ok\)−∑T⊆CcT\(ok\)\.v\_\{o\_\{k\}\}\(C\\cup\\\{i\\\}\)\-v\_\{o\_\{k\}\}\(C\)=\\sum\_\{T\\subseteq C\\cup\\\{i\\\}\}c\_\{T\}^\{\(o\_\{k\}\)\}\-\\sum\_\{T\\subseteq C\}c\_\{T\}^\{\(o\_\{k\}\)\}\.All terms not containingiicancel, hence

vok​\(C∪\{i\}\)−vok​\(C\)=∑T⊆C∪\{i\}:i∈TcT\(ok\)\.v\_\{o\_\{k\}\}\(C\\cup\\\{i\\\}\)\-v\_\{o\_\{k\}\}\(C\)=\\sum\_\{T\\subseteq C\\cup\\\{i\\\}:~i\\in T\}c\_\{T\}^\{\(o\_\{k\}\)\}\.
Substitute this into the classical Shapley marginal formula\. Letn=\|N\|n=\|N\|\. Then

ϕi​\(vok\)=∑C⊆N∖\{i\}\|C\|\!​\(n−\|C\|−1\)\!n\!​\(vok​\(C∪\{i\}\)−vok​\(C\)\)\\phi\_\{i\}\(v\_\{o\_\{k\}\}\)=\\sum\_\{C\\subseteq N\\setminus\\\{i\\\}\}\\frac\{\|C\|\!\(n\-\|C\|\-1\)\!\}\{n\!\}\\bigl\(v\_\{o\_\{k\}\}\(C\\cup\\\{i\\\}\)\-v\_\{o\_\{k\}\}\(C\)\\bigr\)=∑C⊆N∖\{i\}\|C\|\!​\(n−\|C\|−1\)\!n\!​∑T⊆C∪\{i\}:i∈TcT\(ok\)\.=\\sum\_\{C\\subseteq N\\setminus\\\{i\\\}\}\\frac\{\|C\|\!\(n\-\|C\|\-1\)\!\}\{n\!\}\\sum\_\{T\\subseteq C\\cup\\\{i\\\}:~i\\in T\}c\_\{T\}^\{\(o\_\{k\}\)\}\.Swapping the order of summation gives

ϕi​\(vok\)=∑T∋icT\(ok\)​∑C:T∖\{i\}⊆C⊆N∖\{i\}\|C\|\!​\(n−\|C\|−1\)\!n\!\.\\phi\_\{i\}\(v\_\{o\_\{k\}\}\)=\\sum\_\{T\\ni i\}c\_\{T\}^\{\(o\_\{k\}\)\}\\sum\_\{C:\\,T\\setminus\\\{i\\\}\\subseteq C\\subseteq N\\setminus\\\{i\\\}\}\\frac\{\|C\|\!\(n\-\|C\|\-1\)\!\}\{n\!\}\.
It remains to compute the inner coefficient\. Let\|T\|=t\|T\|=t\. IfT∖\{i\}⊆C⊆N∖\{i\}T\\setminus\\\{i\\\}\\subseteq C\\subseteq N\\setminus\\\{i\\\}and\|C\|=k\|C\|=k, thenkkranges fromt−1t\-1ton−1n\-1, and the number of such setsCCis

\(n−tk−t\+1\)\.\\binom\{n\-t\}\{k\-t\+1\}\.Hence the inner sum is

∑k=t−1n−1\(n−tk−t\+1\)​k\!​\(n−k−1\)\!n\!\.\\sum\_\{k=t\-1\}^\{n\-1\}\\binom\{n\-t\}\{k\-t\+1\}\\frac\{k\!\(n\-k\-1\)\!\}\{n\!\}\.Letr=k−t\+1r=k\-t\+1, sor=0,…,n−tr=0,\\dots,n\-t\. Then the above becomes

\(n−t\)\!n\!​∑r=0n−t\(r\+t−1\)\!r\!\.\\frac\{\(n\-t\)\!\}\{n\!\}\\sum\_\{r=0\}^\{n\-t\}\\frac\{\(r\+t\-1\)\!\}\{r\!\}\.Using

\(r\+t−1\)\!r\!=\(t−1\)\!​\(r\+t−1t−1\)\\frac\{\(r\+t\-1\)\!\}\{r\!\}=\(t\-1\)\!\\binom\{r\+t\-1\}\{t\-1\}and the hockey\-stick identity

∑r=0M\(r\+t−1t−1\)=\(M\+tt\),\\sum\_\{r=0\}^\{M\}\\binom\{r\+t\-1\}\{t\-1\}=\\binom\{M\+t\}\{t\},withM=n−tM=n\-t, we obtain

\(n−t\)\!n\!​∑r=0n−t\(r\+t−1\)\!r\!=\(n−t\)\!​\(t−1\)\!n\!​\(nt\)=1t=1\|T\|\.\\frac\{\(n\-t\)\!\}\{n\!\}\\sum\_\{r=0\}^\{n\-t\}\\frac\{\(r\+t\-1\)\!\}\{r\!\}=\\frac\{\(n\-t\)\!\(t\-1\)\!\}\{n\!\}\\binom\{n\}\{t\}=\\frac\{1\}\{t\}=\\frac\{1\}\{\|T\|\}\.Therefore,

ϕi​\(vok\)=∑T∋icT\(ok\)\|T\|\.\\phi\_\{i\}\(v\_\{o\_\{k\}\}\)=\\sum\_\{T\\ni i\}\\frac\{c\_\{T\}^\{\(o\_\{k\}\)\}\}\{\|T\|\}\.Multiplying back the node weightω​\(ok\)\\omega\(o\_\{k\}\)yields

ϕisem​\(ok;G\)=ω​\(ok\)​ϕi​\(vok\)=ω​\(ok\)​∑T∋icT\(ok\)\|T\|,\\phi\_\{i\}^\{\\mathrm\{sem\}\}\(o\_\{k\};G\)=\\omega\(o\_\{k\}\)\\phi\_\{i\}\(v\_\{o\_\{k\}\}\)=\\omega\(o\_\{k\}\)\\sum\_\{T\\ni i\}\\frac\{c\_\{T\}^\{\(o\_\{k\}\)\}\}\{\|T\|\},which is exactly the node\-level semantic allocation given in the main text and in Appendix[E](https://arxiv.org/html/2607.18255#A5)\.

Finally, aggregate over all activated output nodes\. Let

v​\(C\):=∑ok∈O:xok​\(G\)=1ω​\(ok\)​vok​\(C\),C⊆N\.v\(C\):=\\sum\_\{o\_\{k\}\\in O:\\,x\_\{o\_\{k\}\}\(G\)=1\}\\omega\(o\_\{k\}\)\\,v\_\{o\_\{k\}\}\(C\),\\qquad C\\subseteq N\.By additivity of the classical Shapley value,

ϕi​\(v\)=∑ok∈O:xok​\(G\)=1ω​\(ok\)​ϕi​\(vok\)=∑ok∈O:xok​\(G\)=1ϕisem​\(ok;G\)=SSVi​\(G\)\.\\phi\_\{i\}\(v\)=\\sum\_\{o\_\{k\}\\in O:\\,x\_\{o\_\{k\}\}\(G\)=1\}\\omega\(o\_\{k\}\)\\phi\_\{i\}\(v\_\{o\_\{k\}\}\)=\\sum\_\{o\_\{k\}\\in O:\\,x\_\{o\_\{k\}\}\(G\)=1\}\\phi\_\{i\}^\{\\mathrm\{sem\}\}\(o\_\{k\};G\)=\\mathrm\{SSV\}\_\{i\}\(G\)\.Therefore, under the reduction conditions, the SSV equals the Shapley value of the corresponding classical coalition game\. ∎

## Appendix GSupplementary Structural Examples

### G\.1A Linear Example of Backward Extraction of Semantic Node Types

This example corresponds to the simplest single\-output chain structure in the thesis\. Fig\.[4](https://arxiv.org/html/2607.18255#A7.F4)places the workflow, the language flow, and the semantic generation hypergraph side by side, making it easy to see that the output node is recovered through a single main chain\. As a result, the predecessor closure, the initial node setII, and the final hyperedge setLLare all straightforward\.

![Refer to caption](https://arxiv.org/html/2607.18255v1/thesis_pengyijiang_v1_0__Copy_/linearworkflow.png)

![Refer to caption](https://arxiv.org/html/2607.18255v1/thesis_pengyijiang_v1_0__Copy_/languageflowlinear.png)

![Refer to caption](https://arxiv.org/html/2607.18255v1/thesis_pengyijiang_v1_0__Copy_/sgglinear.png)

Figure 4:A linear workflow, its dual language flow, and the induced semantic generation hypergraph\.
### G\.2A Parallel Multi\-Value\-Node Example

Fig\.[5](https://arxiv.org/html/2607.18255#A7.F5)shows a multi\-output parallel structure\. Different branches produce their own semantic nodes, but only those that enter the value\-formation chain are retained inSS\. It also shows that an agent may still have no substantive contribution in the final hypergraph even if it “produces content,” as long as that content does not form an effective semantic role for any value\-relevant node\.

![Refer to caption](https://arxiv.org/html/2607.18255v1/thesis_pengyijiang_v1_0__Copy_/parallworkflow.png)

![Refer to caption](https://arxiv.org/html/2607.18255v1/thesis_pengyijiang_v1_0__Copy_/parralllanguageflow.png)

![Refer to caption](https://arxiv.org/html/2607.18255v1/thesis_pengyijiang_v1_0__Copy_/parrallsemanticgenerationhypergraph.png)

Figure 5:A parallel workflow, its dual language flow, and the induced semantic generation hypergraph with multiple value nodes\.

## Appendix HSupplementary Logic Examples

### H\.1Link\-Level Support

Consider a collaborative decision task whose goal is to determine whether a technical plan is feasible\. Fig\.[6](https://arxiv.org/html/2607.18255#A8.F6)illustrates this example\.Let the semantic nodes be

s0\\displaystyle s\_\{0\}=task background and constraints are given,\\displaystyle=\\text\{task background and constraints are given\},s1\\displaystyle s\_\{1\}=the core module can be completed on time,\\displaystyle=\\text\{the core module can be completed on time\},s2\\displaystyle s\_\{2\}=the budget constraint can be satisfied,\\displaystyle=\\text\{the budget constraint can be satisfied\},s3\\displaystyle s\_\{3\}=the existing system can directly host the plan,\\displaystyle=\\text\{the existing system can directly host the plan\},s4\\displaystyle s\_\{4\}=the plan is feasible\.\\displaystyle=\\text\{the plan is feasible\}\.Assume that the initial node set contains at leasts0s\_\{0\}ands3s\_\{3\}\. The four semantic links in the graph are

ℓ1:\{s0\}→\{s1\},ℓ2:\{s0\}→\{s2\},\\ell\_\{1\}:\\\{s\_\{0\}\\\}\\to\\\{s\_\{1\}\\\},\\qquad\\ell\_\{2\}:\\\{s\_\{0\}\\\}\\to\\\{s\_\{2\}\\\},ℓ3:\{s1,s2\}→\{s4\},ℓ4:\{s3\}→\{s4\}\.\\ell\_\{3\}:\\\{s\_\{1\},s\_\{2\}\\\}\\to\\\{s\_\{4\}\\\},\\qquad\\ell\_\{4\}:\\\{s\_\{3\}\\\}\\to\\\{s\_\{4\}\\\}\.Hence

\{ℓ1,ℓ2,ℓ3\}⇒s4,\{ℓ4\}⇒s4,\\\{\\ell\_\{1\},\\ell\_\{2\},\\ell\_\{3\}\\\}\\Rightarrow s\_\{4\},\\qquad\\\{\\ell\_\{4\}\\\}\\Rightarrow s\_\{4\},and therefore

𝒞s4=\{\{ℓ1,ℓ2,ℓ3\},\{ℓ4\}\},s4⟺\(ℓ1∧ℓ2∧ℓ3\)∨ℓ4\.\\mathcal\{C\}\_\{s\_\{4\}\}=\\bigl\\\{\\\{\\ell\_\{1\},\\ell\_\{2\},\\ell\_\{3\}\\\},\\\{\\ell\_\{4\}\\\}\\bigr\\\},\\qquad s\_\{4\}\\Longleftrightarrow\(\\ell\_\{1\}\\wedge\\ell\_\{2\}\\wedge\\ell\_\{3\}\)\\vee\\ell\_\{4\}\.
![Refer to caption](https://arxiv.org/html/2607.18255v1/thesis_pengyijiang_v1_0__Copy_/example1.png)Figure 6:A supplementary example of link\-level support\.
### H\.2Absorption from Link Level to Agent Level

Continue with the above example\. If the ownership of the four semantic links satisfies

α​\(ℓ1\)=A,α​\(ℓ2\)=B,α​\(ℓ3\)=C,α​\(ℓ4\)=A,\\alpha\(\\ell\_\{1\}\)=A,\\qquad\\alpha\(\\ell\_\{2\}\)=B,\\qquad\\alpha\(\\ell\_\{3\}\)=C,\\qquad\\alpha\(\\ell\_\{4\}\)=A,then

N​\(\{ℓ1,ℓ2,ℓ3\}\)=\{A,B,C\},N​\(\{ℓ4\}\)=\{A\}\.N\(\\\{\\ell\_\{1\},\\ell\_\{2\},\\ell\_\{3\}\\\}\)=\\\{A,B,C\\\},\\qquad N\(\\\{\\ell\_\{4\}\\\}\)=\\\{A\\\}\.Hence the support logic of nodes4s\_\{4\}at the agent level is

s4⟺\(A∧B∧C\)∨A\.s\_\{4\}\\Longleftrightarrow\(A\\wedge B\\wedge C\)\\vee A\.By Boolean absorption,

\(A∧B∧C\)∨A=A,\(A\\wedge B\\wedge C\)\\vee A=A,and thus the final minimal support term is

ℳs4=\{\{A\}\},fs4​\(x\)=xA\.\\mathcal\{M\}\_\{s\_\{4\}\}=\\\{\\\{A\\\}\\\},\\qquad f\_\{s\_\{4\}\}\(x\)=x\_\{A\}\.Fig\.[7](https://arxiv.org/html/2607.18255#A8.F7)illustrates this example

![Refer to caption](https://arxiv.org/html/2607.18255v1/agentlevelsupport.png)Figure 7:A supplementary example of absorption from link\-level logic to agent\-level logic\.

## Appendix ISupplementary Medical Case

This section gives a more complete medical multi\-agent example to illustrate how SLIC constructs a semantic generation hypergraph from a single realized collaborative trajectory and then computes the corresponding SSV\. Fig\.[8](https://arxiv.org/html/2607.18255#A9.F8)shows this workflow\.

![Refer to caption](https://arxiv.org/html/2607.18255v1/thesis_pengyijiang_v1_0__Copy_/workflow.png)Figure 8:Workflow of the medical multi\-agent collaboration\.Node Extraction and Transformation in the Semantic Network

During actual multi\-agent collaboration, we introduce an independent large language model as a side\-channel parser\. Its task is to filter irrelevant analytical traces from the multi\-turn dialogue and extract the structured semantic state set that truly drives task progress:

S=\{s0,s1,s2,s3,s4,s5\}\.S=\\\{s\_\{0\},s\_\{1\},s\_\{2\},s\_\{3\},s\_\{4\},s\_\{5\}\\\}\.
- •s0s\_\{0\}:Initial input, namely the original medical dialogue record, withω​\(s0\)=0\\omega\(s\_\{0\}\)=0\.
- •s1s\_\{1\}:Intermediate semantics, representing\[patient information\], e\.g\., advanced age \(72\), low body weight \(50kg\), and hypertension \(150/90\), withω​\(s1\)=0\\omega\(s\_\{1\}\)=0\.
- •s2s\_\{2\}:Intermediate semantics, representing\[symptoms\], e\.g\., typical constrictive chest pain, cold sweat, and a sense of impending death, withω​\(s2\)=0\\omega\(s\_\{2\}\)=0\.
- •s3s\_\{3\}:Valuable output nodeO1O\_\{1\}, representing\[diagnosis\], e\.g\., acute myocardial infarction \(AMI\), with weightω​\(s3\)=10\\omega\(s\_\{3\}\)=10\.
- •s4s\_\{4\}:Intermediate semantics, representing\[medication dosage\], e\.g\., the anticoagulant dose should be reduced by 50% to avoid bleeding risk, withω​\(s4\)=0\\omega\(s\_\{4\}\)=0\.
- •s5s\_\{5\}:Valuable output nodeO2O\_\{2\}, representing\[prescription\], e\.g\., the final prescription includes 300mg aspirin and reduced\-dose enoxaparin, with weightω​\(s5\)=20\\omega\(s\_\{5\}\)=20\.

##### A text\-to\-graph transformation mappingΦ\\Phi\.

The parser does more than extract isolated propositional nodes\. More importantly, it uses causal connectives and contextual logic in natural language to identify semantic linksℒ\\mathcal\{L\}between nodes, that is, how semantics continue to evolve along the trajectory\. Consider the output of AgentBB:

“Based on the current symptoms, the preliminary diagnosis is acute myocardial infarction\. Meanwhile, the patient’s weight is only 50kg, so the anticoagulant dosage should be reduced by 50% to avoid bleeding risk\.”

Scanning this text, the mappingΦ\\Phiautomatically performs the following graph\-structured transformation:

1. 1\.Extract new states:instantiate nodes3s\_\{3\}\(AMI diagnosis\) ands4s\_\{4\}\(dose reduction by half\)\.
2. 2\.Identify linkℓ2\\ell\_\{2\}:from the trigger phrase “based on the current symptoms,” take the existing symptom nodes2s\_\{2\}together with the related sign nodes1s\_\{1\}as input and point to outputs3s\_\{3\}, thus generating the directed linkℓ2=\(\{s1,s2\},\{s3\}\)\\ell\_\{2\}=\(\\\{s\_\{1\},s\_\{2\}\\\},\\\{s\_\{3\}\\\}\)\.
3. 3\.Identify linkℓ4\\ell\_\{4\}:from the factual dependency “weight is only 50kg,” take the sign nodes1s\_\{1\}as input and point to outputs4s\_\{4\}, thus generating the linkℓ4=\(\{s1\},\{s4\}\)\\ell\_\{4\}=\(\\\{s\_\{1\}\\\},\\\{s\_\{4\}\\\}\)\.

In other words, generating semantic links is itself part of an agent’s behavior\. After traversing the global log, the semantic transformation events that actually occurred form the following graph structure:

- •ℓ1=\(\{s0\},\{s1,s2\}\)\\ell\_\{1\}=\(\\\{s\_\{0\}\\\},\\\{s\_\{1\},s\_\{2\}\\\}\): decompose the original text into signs and symptoms\.
- •ℓ2=\(\{s1,s2\},\{s3\}\)\\ell\_\{2\}=\(\\\{s\_\{1\},s\_\{2\}\\\},\\\{s\_\{3\}\\\}\): infer a diagnosis from signs and symptoms\.
- •ℓ3=\(\{s0\},\{s3\}\)\\ell\_\{3\}=\(\\\{s\_\{0\}\\\},\\\{s\_\{3\}\\\}\): jump directly from the original text to the diagnosis\.
- •ℓ4=\(\{s1\},\{s4\}\)\\ell\_\{4\}=\(\\\{s\_\{1\}\\\},\\\{s\_\{4\}\\\}\): derive a dosage constraint from a sign\.
- •ℓ5=\(\{s3,s4\},\{s5\}\)\\ell\_\{5\}=\(\\\{s\_\{3\},s\_\{4\}\\\},\\\{s\_\{5\}\\\}\): integrate diagnosis and dosage constraint into the final prescription\. This semantic generation hypergraph is illustrated in Fig\.[9](https://arxiv.org/html/2607.18255#A9.F9)\.

![Refer to caption](https://arxiv.org/html/2607.18255v1/imagemedicalsemantichypergraph.png)Figure 9:The semantic graph for the medical case\.From Link Paths to Agent\-Level Logic

Under the SCG framework, an agent is not identical to a node or an edge; rather, it is the executor of a connection\. Through the ownership mapping

α:ℒ→N,\\alpha:\\mathcal\{L\}\\to N,the five semantic links above are assigned as

α​\(ℓ1\)=A,α​\(ℓ2\)=B,α​\(ℓ3\)=C,α​\(ℓ4\)=B,α​\(ℓ5\)=D\.\\alpha\(\\ell\_\{1\}\)=A,\\qquad\\alpha\(\\ell\_\{2\}\)=B,\\qquad\\alpha\(\\ell\_\{3\}\)=C,\\qquad\\alpha\(\\ell\_\{4\}\)=B,\\qquad\\alpha\(\\ell\_\{5\}\)=D\.
Based on this mapping, we first write the link\-level logic of the output nodes and then substitute ownership to derive the corresponding agent\-level logic\.

For the diagnosis nodes3s\_\{3\}, there are two parallel support paths, i\.e\., an OR relation:

s3\\displaystyle s\_\{3\}⇐\(ℓ1∧ℓ2\)∨ℓ3\\displaystyle\\Leftarrow\(\\ell\_\{1\}\\wedge\\ell\_\{2\}\)\\vee\\ell\_\{3\}\\quad\(link\-level logic\)\(8\)⟹\(A∧B\)∨C\\displaystyle\\implies\(A\\wedge B\)\\vee C\\quad\(agent\-level logic\)\(9\)
For the final prescription nodes5s\_\{5\}, its generating linkℓ5\\ell\_\{5\}necessarily depends on predecessor nodess3s\_\{3\}ands4s\_\{4\}\. Sinces4s\_\{4\}itself satisfiess4⇐ℓ1∧ℓ4s\_\{4\}\\Leftarrow\\ell\_\{1\}\\wedge\\ell\_\{4\}, substituting these predecessors back into link\-level logic gives

s5\\displaystyle s\_\{5\}⇐s3∧s4∧ℓ5\\displaystyle\\Leftarrow s\_\{3\}\\wedge s\_\{4\}\\wedge\\ell\_\{5\}⇐\[\(ℓ1∧ℓ2\)∨ℓ3\]∧\(ℓ1∧ℓ4\)∧ℓ5\\displaystyle\\Leftarrow\\bigl\[\(\\ell\_\{1\}\\wedge\\ell\_\{2\}\)\\vee\\ell\_\{3\}\\bigr\]\\wedge\(\\ell\_\{1\}\\wedge\\ell\_\{4\}\)\\wedge\\ell\_\{5\}\\quad\(link\-level logic\)\(10\)⟹\[\(A∧B\)∨C\]∧\(A∧B\)∧D\\displaystyle\\implies\\bigl\[\(A\\wedge B\)\\vee C\\bigr\]\\wedge\(A\\wedge B\)\\wedge D\\quad\(agent\-level logic\)\(11\)

##### Key logical simplification: Boolean absorption\.

At the agent level, by\(X∨Y\)∧X=X\(X\\vee Y\)\\wedge X=X, takingX=\(A∧B\)X=\(A\\wedge B\)andY=CY=C, the above expression simplifies rigorously to

s5⟹A∧B∧D\.s\_\{5\}\\implies A\\wedge B\\wedge D\.\(12\)
Semantic Attribution from Agent\-Level Logic

Once the simplified agent\-level logic is obtained, semantic attribution for each output node can be computed directly\.

Output nodes3s\_\{3\}: diagnosis,ω​\(s3\)=10\\omega\(s\_\{3\}\)=10

From the above,

s3⟹\(A∧B\)∨C\.s\_\{3\}\\implies\(A\\wedge B\)\\vee C\.Rather than decomposing this logic into separate carriers and averaging them, we directly use its multilinear expansion\. The corresponding Boolean function is

fs3​\(x\)=xA​xB\+xC−xA​xB​xC,f\_\{s\_\{3\}\}\(x\)=x\_\{A\}x\_\{B\}\+x\_\{C\}\-x\_\{A\}x\_\{B\}x\_\{C\},whose nonzero coefficients are

c\{A,B\}\(s3\)=1,c\{C\}\(s3\)=1,c\{A,B,C\}\(s3\)=−1\.c\_\{\\\{A,B\\\}\}^\{\(s\_\{3\}\)\}=1,\\qquad c\_\{\\\{C\\\}\}^\{\(s\_\{3\}\)\}=1,\\qquad c\_\{\\\{A,B,C\\\}\}^\{\(s\_\{3\}\)\}=\-1\.Substitutingω​\(s3\)=10\\omega\(s\_\{3\}\)=10gives

ϕAsem​\(s3\)=10​\(12−13\)=53,ϕBsem​\(s3\)=10​\(12−13\)=53,\\phi\_\{A\}^\{\\mathrm\{sem\}\}\(s\_\{3\}\)=10\\left\(\\frac\{1\}\{2\}\-\\frac\{1\}\{3\}\\right\)=\\frac\{5\}\{3\},\\qquad\\phi\_\{B\}^\{\\mathrm\{sem\}\}\(s\_\{3\}\)=10\\left\(\\frac\{1\}\{2\}\-\\frac\{1\}\{3\}\\right\)=\\frac\{5\}\{3\},and

ϕCsem​\(s3\)=10​\(1−13\)=203\.\\phi\_\{C\}^\{\\mathrm\{sem\}\}\(s\_\{3\}\)=10\\left\(1\-\\frac\{1\}\{3\}\\right\)=\\frac\{20\}\{3\}\.Hence the contribution ofs3s\_\{3\}is not obtained by formally splitting its two logical branches; it is determined by the multilinear coefficients of the Boolean support function\.

Output nodes5s\_\{5\}: final compliant prescription,ω​\(s5\)=20\\omega\(s\_\{5\}\)=20

From the simplified strict logic,

s5⟹A∧B∧D\.s\_\{5\}\\implies A\\wedge B\\wedge D\.This means thatA,B,DA,B,Dform an inseparable joint support term\. Therefore,

ϕAsem​\(s5\)=ϕBsem​\(s5\)=ϕDsem​\(s5\)=203,ϕCsem​\(s5\)=0\.\\phi\_\{A\}^\{\\mathrm\{sem\}\}\(s\_\{5\}\)=\\phi\_\{B\}^\{\\mathrm\{sem\}\}\(s\_\{5\}\)=\\phi\_\{D\}^\{\\mathrm\{sem\}\}\(s\_\{5\}\)=\\frac\{20\}\{3\},\\qquad\\phi\_\{C\}^\{\\mathrm\{sem\}\}\(s\_\{5\}\)=0\.
Aggregated Semantic Shapley Value

Summing the allocations from the two output nodes yields the graph\-level semantic allocation in the notation of the current paper:

SSVA​\(G\)\\displaystyle\\mathrm\{SSV\}\_\{A\}\(G\)=53\+203=253≈8\.33,\\displaystyle=\\frac\{5\}\{3\}\+\\frac\{20\}\{3\}=\\frac\{25\}\{3\}\\approx 8\.33,SSVB​\(G\)\\displaystyle\\mathrm\{SSV\}\_\{B\}\(G\)=53\+203=253≈8\.33,\\displaystyle=\\frac\{5\}\{3\}\+\\frac\{20\}\{3\}=\\frac\{25\}\{3\}\\approx 8\.33,SSVC​\(G\)\\displaystyle\\mathrm\{SSV\}\_\{C\}\(G\)=203\+0=203≈6\.67,\\displaystyle=\\frac\{20\}\{3\}\+0=\\frac\{20\}\{3\}\\approx 6\.67,SSVD​\(G\)\\displaystyle\\mathrm\{SSV\}\_\{D\}\(G\)=203=203≈6\.67\.\\displaystyle=\\frac\{20\}\{3\}=\\frac\{20\}\{3\}\\approx 6\.67\.
This example strings together the key steps of SLIC: identify value\-relevant nodes and semantic links from language flow, lift link\-level logic to the agent level, apply absorption to obtain minimal support logic, and finally compute and aggregate semantic attribution over output nodes\.

## Appendix JSupplementary Material for Experiment 1

This section collects the Experiment 1 materials that are useful for the paper but too long for the main text: the failure mode of LOO under redundancy, representative logic\-reconstruction errors, and the instability of the baseline judge together with the de\-ambiguation protocol\. Together they explain where the differences in Table[1](https://arxiv.org/html/2607.18255#S4.T1)come from\.

### J\.1LOO Under Redundancy

LOO is one of the most common attribution heuristics, but it is highly sensitive to higher\-order interaction and redundant coverage\. Even in the complementary divide\-and\-conquer setting of the main text, it systematically underestimates key contributors\. The filtered Case 20 in the thesis provides a direct example: the MC\-GT role\-level attribution isA=3\.0,B=11\.0,C=0\.0A=3\.0,B=11\.0,C=0\.0, whereas the LOO estimate collapses toA=0\.0,B=8\.0,C=0\.0A=0\.0,B=8\.0,C=0\.0\. This already shows that even without redundancy, LOO compresses away part of the semantic interaction whenever credit depends on higher\-order cooperation\.

The more extreme case is full redundancy\. Suppose three agents give exactly the same perfect answer, withv​\(∅\)=0v\(\\emptyset\)=0andv​\(S\)=14v\(S\)=14for every non\-emptyS⊆\{A,B,C\}S\\subseteq\\\{A,B,C\\\}\. In this fully redundant setting, the classical Shapley value should be

ϕA=ϕB=ϕC=14/3\\phi\_\{A\}=\\phi\_\{B\}=\\phi\_\{C\}=14/3by symmetry, whereas LOO degenerates to

ϕiLOO=v​\(N\)−v​\(N∖\{i\}\)=14−14=0\.\\phi\_\{i\}^\{\\mathrm\{LOO\}\}=v\(N\)\-v\(N\\setminus\\\{i\\\}\)=14\-14=0\.Once the remaining system still fully covers the original logical support, the LOO marginal collapses to zero and the method loses its credit\-assignment ability altogether\.

This is also one reason why the main paper does not use a redundant workflow as the standard evaluation setting\. Experiment 1 is meant to test the agreement between SSV and classical Shapley under the classical assumptions; in a highly redundant workflow, the failure of LOO would mostly reflect the limitation of LOO itself rather than the attribution difficulty of the task\.

To align this observation quantitatively with the main experiment, we further adopt the duplicated\-agent assumption from the thesis: each original role is duplicated by an identical clone, and the resulting attribution is re\-aggregated at the role level\. In such a redundant expansion, a reasonable allocation should remain symmetric across clones, whereas LOO collapses because removing any one agent still leaves its clone able to cover the same logical support\. Table[4](https://arxiv.org/html/2607.18255#A10.T4)and Fig\.[10](https://arxiv.org/html/2607.18255#A10.F10)report the corresponding comparison\.

Table 4:Method robustness under the redundant setting\.![Refer to caption](https://arxiv.org/html/2607.18255v1/scg_ai_figures/redundancy_robustness.png)Figure 10:Method robustness under the redundant setting\.
### J\.2Representative Logic Reconstruction Error Cases

This section gives case\-level examples of the three main logic\-reconstruction error patterns in Experiment 1, together with the original rubric, the hit subsets, and the extracted logic, so that the resulting attribution shift is explicit rather than only summarized\.

##### Disjunctive Masking \(OR\-Miss\)

Source:Case 6 / Rubric 0 \(Weight:\+7\+7\)\. Rubric:“Explicitly identify that multiple non\-healing upper\-limb ulcers from the North African desert region may indicate cutaneous leishmaniasis\.” MC ground\-truth hit subsets:\{A\},\{B\},\{A,B\},\{A,C\},\{B,C\},\{A,B,C\}\\\{A\\\},\\\{B\\\},\\\{A,B\\\},\\\{A,C\\\},\\\{B,C\\\},\\\{A,B,C\\\}\. SLIC\-inverted hit subsets:\{A\},\{A,B\},\{A,C\},\{A,B,C\}\\\{A\\\},\\\{A,B\\\},\\\{A,C\\\},\\\{A,B,C\\\}\. SLIC\-extracted logic:A\(while the true logic should also includeB\)\.

Original response snippets:

> Agent A: “In the North African desert region, multiple non\-healing arm ulcers should raise strong suspicion for cutaneous leishmaniasis\.” Agent B: “For multiple non\-healing arm ulcers from the North African desert region, the diagnosis is leishmaniasis \(cutaneous leishmaniasis\)\.”

Analysis:During inversion, the baseline judge treated Agent B’s definitive diagnostic wording as failing the rubric’s intended “possibility” phrasing\. As a result, the true disjunctive logicA OR Bcollapsed toA\. This is a clear case of disjunctive masking: one valid parallel support path disappears entirely, and the omitted agent is systematically underestimated\. The resulting attribution shift on this single rubric reaches an L1 error of7\.007\.00\.

##### Polarity Inversion in Negative Constraints

Source:Case 11 / Rubric 0 \(Weight:−10\-10\)\. Rubric \(Penalty\):“The response should recommend immediate medical attention unconditionally, without attaching additional conditions\.” MC ground\-truth hit subsets:∅\\emptyset\(i\.e\., no subset satisfies the item; all combinations violate it\)\. SLIC\-inverted hit subsets:\{B\},\{C\},\{B,C\}\\\{B\\\},\\\{C\\\},\\\{B,C\\\}\. SLIC\-extracted logic:\(NOT A AND B\) OR \(NOT A AND C\)\.

Original response snippets:

> Agent A: “If blurred vision is accompanied by …, seek immediate medical attention\. Even without other symptoms, an ophthalmologic evaluation should be sought as soon as possible \(within 24–48 hours\)\.” Agent B / C: \(No equivalent unconditional recommendation for immediate medical attention\.\)

Analysis:For negative penalty items, the system must distinguish between “who triggered the violation” and “who avoided the violation\.” Here that polarity is confused\. Agent A gives a conditional recommendation, while B and C simply omit the required unconditional advice\. Instead of capturing this global violation state, the inversion produces an incorrect composite logic and incorrectly pushes B and C into the hit subsets\.

##### Asymmetric Penalty Attribution

Source:Case 13 / Rubric 2 \(Weight:−9\-9\)\. Rubric \(Penalty\):“The answer fails to clearly and unreservedly oppose cannabinoid patches \(insufficient evidence\)\.” MC ground\-truth hit subsets:\{A\},\{B\},\{C\},\{A,B\},\{A,C\},\{B,C\},\{A,B,C\}\\\{A\\\},\\\{B\\\},\\\{C\\\},\\\{A,B\\\},\\\{A,C\\\},\\\{B,C\\\},\\\{A,B,C\\\}\(all non\-empty subsets\)\. SLIC\-inverted hit subsets:\{B\},\{A,B\},\{B,C\},\{A,B,C\}\\\{B\\\},\\\{A,B\\\},\\\{B,C\\\},\\\{A,B,C\\\}\. SLIC\-extracted logic:B\.

Original response snippets:

> Agent B: “Extreme caution is needed …the evidence is very limited …it may be considered only as a last resort\.” Agent A / C: \(Neither provides a clear and unconditional statement of opposition\.\)

Analysis:The true semantics are closer to “all agents fail to meet the standard of explicit opposition,” so responsibility should be shared much more evenly\. SLIC, however, over\-focuses on Agent B’s hedging phrasing \(“as a last resort”\) and compresses a distributed negative\-responsibility structure intoB\. The result is an asymmetric concentration of the penalty on B, producing an attribution shift with L1 error12\.0012\.00\.

### J\.3Baseline Judge Instability and De\-ambiguation

This section provides more direct evidence for the internal uncertainty of the baseline judge, which underlies the discussion of baseline error in the main text\. Even undertemperature=0, the LLM judging process still contains non\-trivial statistical noise: under 3\-run rubric\-level voting,5\.34%5\.34\\%of rubrics are2:12\{:\}1at the grand\-coalition level, and22\.33%22\.33\\%are2:12\{:\}1for at least one subset\. This means that even when the agent responses are fixed, the subset scoring functionv​\(S\)v\(S\)still fluctuates on a non\-trivial set of rubrics, directly perturbing exact Shapley estimation\.

#### J\.3\.1Case\-wise instability statistics

Table[5](https://arxiv.org/html/2607.18255#A10.T5)and Fig\.[11](https://arxiv.org/html/2607.18255#A10.F11)reports the distribution of2:12\{:\}1vote splits over 15 Cases \(206 rubrics in total\)\. High\-instability cases such as Case 10, 1, and 7 suggest that some medical scenarios carry inherently fuzzier rubric boundaries and therefore amplify judge variability more strongly\.

Table 5:Case\-wise of2:12\{:\}1rubric disagreementsCase IDTotal RubricsGrand Coalition2:12\{:\}1CountAny\-Subset2:12\{:\}1Count019131102232003413025121461413711261026371114181290213160214130020801221001241102
#### J\.3\.2Representative2:12\{:\}1cases and run\-level evidence

##### Example 1: boundary\-semantic drift \(Case 10, Rubric 8\)

Criterion:Raises underlying comorbidities like HIV and CKD\. Grand Vote \(2/3\):Run 1 counts generic mentions such as “chronic disease/chronic\-disease anemia” as a hit \(hit=True\); Run 2 rejects it \(hit=False\); Run 3 counts it again\. The model is visibly unstable on whether a generic reference to chronic illness is sufficient evidence for HIV/CKD\.

##### Example 2: subjective leniency on a strict numeric criterion \(Case 1, Rubric 0\)

Criterion:Correctly states that compression depth remains at 2–2\.4 inches \(5–6 cm\) … Grand Vote \(2/3\):The answer says “at least 5 cm but not exceeding 6 cm\.” Two runs judge this as a hit, while one run rejects it\. The same instability propagates to subset evaluations such asA\|BA\|BandB\|CB\|C\.

##### Example 3: trigger threshold under a negative constraint \(Case 3, Rubric 11\)

Criterion:Uses overly technical medical terminology without explanation … Grand Vote \(3/3\) vs Subset \(2/3\):Although the grand coalition is unanimously judged as violating the item \(3/33/3\) because it mentions terms such as targeted therapy and immunotherapy, the subsetA\|BA\|Bstill shows a2:12\{:\}1split\.

![Refer to caption](https://arxiv.org/html/2607.18255v1/scg_ai_figures/judge_ambiguity_distribution.png)Figure 11:Internal ambiguity distribution of the LLM judge under 3\-run rubric evaluation \(206 rubrics\)\. A rubric is marked ambiguous when the three runs produce a2:12\{:\}1vote split\.

#### J\.3\.3Deeper causes of baseline error

Based on these observations, we attribute the instability of the baselinev​\(S\)v\(S\)to four systematic causes:

1. 1\.Rubric boundaries are not hard enough\.Criteria such as “similar disease” or “overly technical” lack a fully executable boundary and easily drift into semantic gray zones\.
2. 2\.Negative rubrics and conditional triggers are highly sensitive\.Penalty items often depend on whether a violation is triggered\. For implicit mentions, partial hints, or conditional phrasing, the judge threshold becomes extremely unstable\.
3. 3\.Subset concatenation changes the context\.The same rubric evaluated onA\|BA\|BversusA​\|B\|​CA\|B\|Cis presented under different context lengths and information coverage, which shifts the LLM’s attention distribution and introduces combinatorial sensitivity noise\.
4. 4\.There is an upper bound on judging consistency\.Even withtemperature=0, slight differences in generation path and semantic alignment across API calls can still flip a binary verdict\.

The mitigation strategy adopted in the paper is therefore to apply 3\-run majority voting to each rubric and remove all items with2:12\{:\}1disagreement from the main analysis\. The purpose is not to make the baseline look cleaner, but to ensure that the comparison in the main text relies on a more trustworthyv​\(S\)v\(S\), so that judge instability is not mistakenly attributed to SLIC\.

### J\.44\-Agent Extension

To show that the cost trend in Experiment 1 is not specific to the three\-agent setting, we further report a 4\-agent extension\. Table[6](https://arxiv.org/html/2607.18255#A10.T6)and Fig\.[12](https://arxiv.org/html/2607.18255#A10.F12)reports the results\.

Table 6:4\-agent extension under corrected\-cost accountingThis extension serves two purposes\. First, once the number of agents grows from 3 to 4, the cost of MC\-style baselines is already noticeably higher, which is the main reason why we do not keep scaling the consistency experiment further in the main text\. Second, even under this heavier baseline\-cost regime, SLIC still reduces corrected cost from207\.0207\.0to13\.813\.8, i\.e\., by93\.3%93\.3\\%, while maintaining low attribution error \(L1=5\.700=5\.700\)\. This suggests that the advantage of single\-trajectory semantic inversion is not limited to the three\-agent setting in the main paper and becomes more pronounced as subset\-based baseline evaluation grows\.

![Refer to caption](https://arxiv.org/html/2607.18255v1/scg_ai_figures/standard_setting_pair_slic_4agent_5cases.png)Figure 12:4\-agent extension under the standard setting\. Left: rank consistency versus corrected cost\. Right: attribution error versus corrected cost\.

## Appendix KSupplementary Material for Experiment 2

This section collects the Experiment 2 material from the thesis that is useful for the paper but too long for the main text\. The main paper keeps only the problem setup; the fuller motivation, staged design, and structural interpretation are placed here\.

### K\.1Motivation: From Static Attribution to System Diagnosis

Under the SCG/SLIC framework, semantic attribution is no longer only a post hoc explanatory quantity\. Given a realized collaborative trajectory, we can recover a semantic support structure from a single trajectory and compute SSV on top of it\. This makes it possible to assess questions that previously required large numbers of counterfactual trials or exhaustive comparisons: who is truly more critical in the current system, which failures are more likely to cause a global performance drop, and which positions should be hardened first when the system faces hallucination or local errors\.

This is the direct motivation of Experiment 2\. In real multi\-agent systems, hallucinations, omissions, input noise, and local errors are almost unavoidable\. For deployment\-oriented systems, what matters is not only whether a node has*some*contribution, but which nodes become the main sources of degradation under perturbation, and which nodes may appear unimportant on the surface yet hold stronger global structural influence\. Only then can SSV serve as a basis for workflow evaluation, robustness analysis, and structural hardening\.

### K\.2Research Question and Staged Perturbation Design

The central question of Experiment 2 is the following: in the absence of a standard benchmark for contribution\-based importance, can we use system reaction under controlled perturbations as an external target and test whether SSV truly captures node importance? We therefore build a diagnostic multi\-agent evaluation suite in which, for the same reasoning trajectory, we observe how the system score degrades when one agent hallucinates, is attacked, or is removed\.

The protocol has two layers\. The first uses injected hallucinations with weak, medium, and strong intensity\. The second uses leave\-out / C3\-style interventions, in which the target agent is assumed absent and the workflow is rerun\. The former is closer to local semantic corruption, while the latter is closer to classical counterfactual removal\. Together they provide the external importance signals for Experiment 2\.

To answer this question, the experiment follows a staged design\.

1. 1\.We first examine the overall correlation between SSV and system loss under ordinary perturbations, to verify whether this semantic contribution signal can capture node importance at all\.
2. 2\.We then analyze performance differences across nodes at different positions, with special attention to whether late\-stage nodes are systematically underestimated\.
3. 3\.Finally, we introduce amplified hallucination perturbations, that is, explicit disruptive factors injected into the target agent’s prompt, in order to distinguish whether the observed bias comes from the attribution definition itself or from structural imbalance in the workflow\.

This mirrors the logic of the thesis: establish aggregate validity first, then inspect structural bias, and finally amplify the bias source under extreme perturbation\.

### K\.3Well\-Structured Workflows and Alignment Intuition

The high\-correlation validation part focuses on a class of well\-structured workflows: responsibility boundaries are clear, partial access is explicit, downstream nodes cannot freely reconstruct missing semantics, and structural privilege is not overly concentrated in nodes with weak semantic responsibility\. In such structures, if an agent\-owned semantic block is systematically damaged, the resulting rubric score drop should broadly align with the shape of its SSV allocation\.

From this perspective, the role of SSV in Experiment 2 is not merely to explain who contributed more, but to provide a comparable and rankable semantic contribution signal\. This is why the alignment experiment in the main paper compares attribution shape against intervention\-induced score\-drop profiles rather than only comparing final total scores\. Broader archived results are summarized in Appendix[L](https://arxiv.org/html/2607.18255#A12)\.

#### K\.3\.1Amplified Perturbation and Privilege–Capability Misalignment in Linear Workflows

In the first three high\-correlation experiments, we repeatedly observe a subtle but stable phenomenon: agents located later in the workflow tend to be slightly underestimated\. We believe that this does not mean these downstream nodes lack real influence; rather, it indicates that semantic contribution and structural privilege are not always fully aligned\. More specifically, in manywell\-structuredworkflows, the upstream or midstream agents have already completed the substantive semantic work of evidence extraction, diagnostic reasoning, and constraint analysis\. Downstream agents often no longer generate new high\-value semantic content, and therefore receive lower SSV scores\. However, these downstream nodes are usually closer to the final output and objectively possess stronger privileges for integration, rewriting, overriding, and shaping the result\. In other words, although the “work” has already been completed by upstream nodes, “how the final output is presented” is still partly controlled by downstream nodes\. In this sense, we further identify a structural phenomenon: privilege–capability misalignment\. Here, capability corresponds to a node’s true contribution in explicit semantic generation, while privilege corresponds to its structural position in the workflow that allows it to influence the final outcome\. When a node contributes little semantically yet still holds strong end\-stage control, a deviation emerges between the two, and this is precisely the root cause of the subsequent anomalous phenomena\.

To further test whether this local bias would be amplified under extreme conditions, we introduce amplified hallucination perturbations into a linear workflow with a serial topological structure \(i\.e\.,A→\\rightarrowB→\\rightarrowC→\\rightarrowD→\\rightarrowE\): specifically, we explicitly add disruptive prompts to the target node’s prompt, including but not limited to malicious prompts, in order to simulate information tampering, hallucination amplification, and logical collapse under extreme conditions\. Unlike the first three highly correlated figures, this result plot shows that under this linear structure, almost all nodes suffer significant system degradation once they are deliberately attacked, and the correspondence between SSV and the true Score Drop is substantially broken as shown in Fig\.[13](https://arxiv.org/html/2607.18255#A11.F13)\.

This indicates that once the workflow adopts a strongly serial and strongly dependent propagation structure, local errors will accumulate along the chain and spread downstream, ultimately giving almost every node the potential to drag down the entire system\. In such a situation, what the amplified perturbation amplifies is no longer the explicit semantic contribution itself, but rather the node’s propagation privilege within the chain\-dependent structure\.

Among them, the downstream nodes D and E, which are responsible for integrating and rendering the output, are particularly typical: in the SLIC computation, they do not generate new high\-value clinical semantics, and therefore their SSV is low; yet once hallucination is deliberately injected into these nodes, the system’s Score Drop increases dramatically\. This shows that what appears here is not a general failure of SSV, but a deeper structural phenomenon, namely privilege–capability \(behavior\) misalignment: some nodes do not undertake the main semantic generation work, but because they occupy the end of the chain or a critical transmission position, they actually possess excessive destructive power over the final result\.

![Refer to caption](https://arxiv.org/html/2607.18255v1/scg_ai_figures/exp2_linear_attack.png)Figure 13:Results after introducing deliberate perturbations in the linear workflow\.Table 7:Aligned\-50 archive: per\-setting average shape alignment\.
#### K\.3\.2Superficially Reasonable Branched Workflow and Latent End\-Stage Bottleneck

To further show that this phenomenon is not restricted to extreme linear structures, we design another counterexample experiment, illustrated in Fig\.[14](https://arxiv.org/html/2607.18255#A11.F14)\. The workflow first enforces strict expert specialization, with multiple branches proceeding in parallel\. A dedicated final nodeFFthen merges the branches and performs final formatting and organization\.

At first glance, this workflow appears to have a clear early\-stage division of labor and a proper end\-stage consolidation mechanism: the upstream nodes each complete relatively independent subtasks, while the downstream nodeFFis only responsible for unified formatting, integration, and output refinement\. Therefore, it seems more reasonable than a linear serial structure and closer to a superficially healthy collaborative system\. Under this apparent division of labor,FF’s direct contribution to semantic generation should be low, and this is indeed reflected in its semantic Shapley value\.

However, the results in the figure show that althoughFFreceives a very low SSV score, it still holds extremely strong end\-stage control privileges\. Once targeted perturbations are applied toFF, even a slight formatting error, branch\-merging deviation, or end\-stage rephrasing can cause the system score to drop sharply, or even collapse entirely\. This indicates that the problem is not that SSV fails to identify explicit semantic contributions\. Rather, although the workflow appears to have completed early\-stage specialization, it still concentrates the power to shape the final output in a single end node, thereby creating a latent structural bottleneck\.

![Refer to caption](https://arxiv.org/html/2607.18255v1/scg_ai_figures/exp2_branched_bottleneck.png)Figure 14:In a branched workflow with clear early\-stage specialization and a dedicated end nodeFFresponsible for final formatting and integration,FFhas a low SSV but causes a large system loss after deliberate perturbation\.Table 8:Meeting decision summarization: normalized profiles\.Table 9:Table\-text numerical reasoning: normalized profiles\.
#### K\.3\.3Discussion and Summary

Through the progression from ordinary correlation validation to amplified perturbation analysis, this experiment forms a relatively clear chain of evidence\. First, and most importantly, these positive correlation results show that SSV not only provides a qualitative explanation, but also has strong comparative power: it can stably distinguish the relative importance of different nodes and, in most cases, remains consistent with the trend of the true Score Drop\. For an attribution metric designed for multi\-agent collaboration, this means that SSV can serve as a comparable, rankable contribution scale for system analysis, rather than merely an ex post descriptive score\.

Table 10:Rule application: normalized profiles\.Table 11:HealthBench\-style medical safety: normalized profiles\.At the same time, the repeatedly observed slight underestimation of downstream nodes in the first three highly correlated plots suggests that, even when the overall method is effective, there may still be local tension between explicit semantic contribution and end\-stage control influence\. Upstream nodes typically complete the main work of evidence extraction, diagnostic reasoning, and constraint analysis; therefore, although downstream nodes no longer generate new high\-value semantic content and thus receive lower SSV scores, they still possess strong integration, rewriting, and shaping privileges because they are closer to the final output\. It is precisely in this sense that we identify a structural phenomenon, namely privilege–capability misalignment\.

Furthermore, in the amplified attack experiment on the linear workflow, this tension is significantly magnified, to the point that almost every node exhibits strong global destructive potential and the correspondence between SSV and the true loss is clearly weakened\. Even in the branched structure with clear early\-stage specialization and a dedicated formatting node at the end, we likewise observe that once the final shaping power remains concentrated in a single node, the coexistence of low semantic contribution and high destructive capability still emerges\. This also shows, from the opposite direction, that in a truly healthy system, a node’s semantic contribution should broadly align with the structural privileges it holds\.

Therefore, the main conclusion of this section remains positive and clear: SSV is an effective semantic attribution metric\. It not only stably reflects node importance, but also provides meaningful contribution rankings for multi\-agent systems\. The anomalous deviations shown in the counterexample figures further indicate that when the workflow itself suffers from concentrated end\-stage control, overly strong chain dependence, or imbalanced role boundaries, a node’s actual destructive capability may exceed the scope that its explicit semantic contribution can capture\. This is a factor that must be considered, and cannot be ignored, when designing healthy and robust systems\.

## Appendix LExtended Radar Figures and Raw Tables

The main text keeps only the most stable summary results of the current branched\-workflow study\. This appendix supplements them with broader radar results, showing that the alignment pattern is not specific to one workflow\.

##### Evaluation protocol\.

For each case, we first obtain the SSV vector

SSV​\(G\)=\(SSVi​\(G\)\)i∈N\.\\mathrm\{SSV\}\(G\)=\(\\mathrm\{SSV\}\_\{i\}\(G\)\)\_\{i\\in N\}\.We then construct four interventions on the same workflow:weak hallucination,medium hallucination,strong hallucination, andLOO rerun\. Each intervention produces a positive rubric score\-drop vector decomposed by agent,

d=\(di\)i∈N\.d=\(d\_\{i\}\)\_\{i\\in N\}\.To compare only the attribution shape rather than absolute magnitude, we sum\-normalize both the SSV vector and the drop vector:

z¯i=zi∑jzj,\\bar\{z\}\_\{i\}=\\frac\{z\_\{i\}\}\{\\sum\_\{j\}z\_\{j\}\},wherezzmay refer either to SSV or to any intervention\-based drop vector\. We then compare the two normalized curves via Spearmanρ\\rho\.

##### Aligned\-50 archive\.

The current archive covers four task settings: meeting summarization, table\-text numerical reasoning, rule\-based judgment, and medical safety QA\. They follow the same design principle: clear responsibility boundaries, explicit partial access, and no permission for downstream agents to freely reconstruct missing semantics, so that local perturbations expose cleaner structural responses\.

Table[7](https://arxiv.org/html/2607.18255#A11.T7)reveals three stable patterns\. First, weak hallucination aligns much more weakly because mild perturbations often fail to flip the rubric\. Second, medium and strong hallucination are consistently closer to SSV across the four settings, indicating that when an agent\-owned semantic block is systematically damaged, the shape of SSV tracks the shape of the actual value collapse\. Third, LOO / C3\-style reruns can match SSV in some healthy workflows, but degrade noticeably when redundant reconstruction or downstream repair is available\.Detailed radar\-analysis results for the synthetic tasks are reported in Tables[8](https://arxiv.org/html/2607.18255#A11.T8),[9](https://arxiv.org/html/2607.18255#A11.T9),[10](https://arxiv.org/html/2607.18255#A11.T10)and[11](https://arxiv.org/html/2607.18255#A11.T11)\.

These results complement the anomaly analysis in the main text\. Overall, Experiment 2 supports the following reading: SSV does predict system response in most healthy workflows, but once a workflow contains a strong privilege–capability mismatch, or once the removal baseline is contaminated by redundancy, a pure leave\-out/C3 perspective becomes unreliable\.

##### Raw normalized profiles\.

We also provide the average normalized profiles behind Fig\.[3](https://arxiv.org/html/2607.18255#S4.F3)and the related appendix figures\.

## Appendix MExperimental Configurations

This appendix summarizes the implementation settings for the two main experimental groups\. The four\-scenario radar sensitivity experiments coverqmsum\_meeting,tatqa\_reasoning,legalbench\_rule, andhealthbench\-style\. The HealthBench SLIC attribution experiment is the controlled experiment used to compare SLIC with classical Shapley\-style baselines\.

Table 12:Model and decoding configurations for the main experiments\.For the four\-scenario radar experiments, all calls use temperature0\.00\.0, thinking budget0, timeout mostly6060seconds, and usually one retry\. The maximum output length is around16001600–22002200tokens for generation and around40004000tokens for judge or analyzer calls\.

For the HealthBench Shapley\-alignment experiment, all calls also use temperature0\.00\.0\. We setGEMINI\_THINKING\_BUDGET=256; the default timeout is4545–6060seconds, and retries are usually set to11\.

## Appendix NBroader Impacts

This work studies credit assignment in LLM\-based multi\-agent systems through a semantic cooperative game framework\. The potential positive societal impact is that the proposed framework may improve the transparency, accountability, and diagnosability of multi\-agent AI workflows\. By attributing contributions to value\-relevant semantic structures rather than only to agents or black\-box outputs, the method can help developers better understand which agents, messages, or semantic transformations support a final decision\. This may be useful for auditing collaborative AI systems in domains where interpretability and responsibility are important, such as medical assistance, legal reasoning, scientific analysis, and organizational decision support\.

The work may also support safer deployment of multi\-agent systems by making failure sources easier to identify\. For example, if a downstream answer depends heavily on an unsupported or misleading semantic link, the proposed attribution structure can help reveal where the problematic reasoning entered the workflow\. This can provide a basis for debugging, model selection, workflow redesign, or human oversight\.

At the same time, this work may have negative societal impacts if used in inappropriate settings\. More efficient or more interpretable coordination mechanisms for LLM\-based agents could also improve the effectiveness of harmful multi\-agent systems, including systems used for misinformation generation, automated persuasion, surveillance, or strategic manipulation\. In addition, contribution scores may be misinterpreted as definitive measures of responsibility, even though they depend on the extracted semantic structure, the evaluation rubric, and the observed trajectory\. Over\-reliance on such scores could lead to unfair blame assignment or misplaced trust in automated explanations\.

We therefore view the proposed method as an analysis and diagnostic tool rather than a complete accountability mechanism\. In high\-stakes applications, semantic contribution analysis should be combined with human review, domain\-specific safety constraints, privacy protection, and careful validation of the evaluation criteria\. The framework is intended to make multi\-agent AI systems more transparent, but it does not remove the need for broader governance, auditing, and responsible deployment practices\.

## NeurIPS Paper Checklist

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67. Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation \(if any\)?
68. Answer:\[N/A\]
69. Justification: does not involve crowdsourcing nor research with human subjects\. - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper\. - •According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector\.
70. 15\.Institutional review board \(IRB\) approvals or equivalent for research with human subjects
71. Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board \(IRB\) approvals \(or an equivalent approval/review based on the requirements of your country or institution\) were obtained?
72. Answer:\[N/A\]
73. Justification: the paper does not involve crowdsourcing nor research with human subjects\.
74. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Depending on the country in which research is conducted, IRB approval \(or equivalent\) may be required for any human subjects research\. If you obtained IRB approval, you should clearly state this in the paper\. - •We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution\. - •For initial submissions, do not include any information that would break anonymity \(if applicable\), such as the institution conducting the review\.
75. 16\.Declaration of LLM usage
76. Question: Does the paper describe the usage of LLMs if it is an important, original, or non\-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does*not*impact the core methodology, scientific rigor, or originality of the research, declaration is not required\.
77. Answer:\[N/A\]
78. Justification:the core method development in this research does not involve LLMs as any important, original, or non\-standard components
79. Guidelines: - •The answer\[N/A\]means that the core method development in this research does not involve LLMs as any important, original, or non\-standard components\. - •Please refer to our LLM policy in the NeurIPS handbook for what should or should not be described\.