Shapley-Coop: Credit Assignment for Emergent Cooperation in Self-Interested LLM Agents

We sincerely thank the reviewer for their thoughtful and constructive feedback. Your expertise in cooperative game theory and MARL has provided invaluable insights that will significantly strengthen our paper.

The central point of confusion appears to be whether our methodology merely evaluates existing trajectories or actively influences the decision-making process. We would like to clarify this upfront: Shapley-Coop is an online framework designed to actively shape agent decisions in real-time. It achieves this by integrating forward-looking reasoning to guide immediate actions with retrospective analysis for fair reward distribution.

We address your specific questions below:

1. How Shapley-Coop Influences Decision-Making (vs. Only Evaluating Trajectories)

You are right to question this, as it is the core of our contribution. The methodology operates in two interconnected phases, influencing behavior both during and after a task:

• Short-Term, Forward-Looking Influence (Online Decision-Making): This component directly impacts an agent's immediate actions. At each decision point, agents use a qualitative form of Shapley reasoning (our "Shapley-CoT") to prospectively estimate the marginal value of their potential contributions to a coalition. This reasoning process directly informs how Eq. (7) is realized: agents use these estimations to negotiate and decide whether to cooperate on the next action and on what terms. This is not a retrospective evaluation but a real-time mechanism for forming agreements.

• Long-Term, Retrospective Influence (Post-Task Credit Assignment): This component evaluates completed action trajectories (from the current task or historical data) to guide post-task reward redistribution. Its purpose is to create a robust, long-term incentive structure. By ensuring fair credit is assigned after the fact, it encourages agents to engage honestly in the short-term, forward-looking negotiations in future interactions.

We acknowledge this workflow could be explained more clearly. We will revise the methodology section to provide a much clearer high-level overview of this dual process, explicitly separating the online decision-influencing part from the post-task reward mechanism.

2. Relationship with MARL

We appreciate this valuable suggestion to better connect our work with the MARL literature. You are correct that this context is crucial, and our initial description was imprecise. We will revise the paper to draw a clearer distinction based on the fundamental differences in agent types and learning paradigms.

A fundamental distinction lies in the learning paradigm itself:

• MARL Credit Assignment (e.g., Shapley Q-value, ShaQ): These methods are designed for agents that learn policies from scratch through extensive trial-and-error interaction with an environment. The core challenge is to solve the credit assignment problem during this training-intensive process to shape cooperative behavior.

• Shapley-Coop: Our framework is designed for pre-trained LLM agents that already possess vast world knowledge and reasoning capabilities. Therefore, our goal is not to train a policy, but to guide the existing reasoning process of these agents towards cooperation in zero-shot or few-shot scenarios. It is a training-free, guidance-based process.

This leads to a different mechanism for achieving cooperation:

• MARL methods typically learn implicit cooperation through mechanisms like value decomposition or opponent modeling integrated into the learning algorithm.
• Shapley-Coop facilitates explicit cooperation through structured, language-based negotiation, leveraging the agents' innate ability to communicate and reason about intent and fairness.

The practical implication of this paradigm shift is a dramatic difference in efficiency and adaptability:

• For instance, in the Escape Room task, traditional MARL approaches might require tens of thousands of training episodes to converge on a cooperative policy. In contrast, LLM agents using Shapley-Coop can reason through the social dilemma and successfully cooperate from the very first attempt. This demonstrates a significant advantage in novel, cognitively complex environments where extensive training is impractical.

We will incorporate this detailed comparison into the revised manuscript to better situate our contribution.

3. Clarification of the "LLM-Only" Baseline

The "LLM-only" baseline was designed as a diagnostic tool. In this setup, agents operate without any Shapley-Coop framework; they rely solely on the base LLM's inherent, unguided capabilities to reason about cooperation and credit. This baseline helps measure how much of the observed cooperation is attributable to our structured method versus the raw intelligence of the LLM.

4. Typo in Eq. (9)

You are correct. The symbol in Eq. (9) is indeed a typo.

5. Acknowledging Limitations (Approximation & Overhead)

We agree completely. A principled discussion of limitations is essential. We will expand the limitations section to explicitly address:

• The trade-off between scalability and theoretical purity, acknowledging that our approximation may not satisfy the efficiency property, which we view as a necessary compromise for practical application.
• The communication overhead (i.e., token cost) introduced by the negotiation process, framing it as a key challenge for future optimization research.

Thank you once again for your valuable feedback. It has provided us with a clear roadmap for improving the paper's clarity, rigor, and contribution.

We sincerely thank you for your insightful and constructive feedback. Your comments accurately capture the core of our work and provide invaluable guidance for our future research. We are delighted that you recognized the novelty and originality of our framework. We address your questions below:

---

1. Scalability of the Shapley-Coop Framework

Shapley-Coop scales effectively to larger agent populations by avoiding exact Shapley value computation, which indeed has factorial complexity. Instead, we employ:

• Context-based approximations (via Chain-of-Thought reasoning)
• A structured negotiation protocol

Shapley values serve as qualitative guides rather than precise metrics, making real-time estimation efficient. The negotiation process scales linearly with the number of agents, as it relies on pairwise or group dialogues.

To demonstrate scalability, we conducted an additional experiment with 10 agents in the Raid Battle environment (setting the boss HP to 4000 and 8000), and measured running times for varying agent population sizes. You can find the results in the rebuttal to reviewers 8Y2r and 4hz6.

While scalability is not the primary focus of this paper, we will provide these results in the supplementary material for further validation. The following table contains a summary of the results.

---

2. Theoretical Guarantees and Convergence Behavior

Current LLM-related work faces inherent challenges in providing formal theoretical guarantees due to the black-box nature of large language models. We acknowledge that our work shares this limitation, which is difficult to fully resolve.

However, Shapley-Coop addresses convergence through its negotiation process by establishing appropriate stopping conditions—specifically, when agents reach mutually acceptable consensus while being constrained by prompts to exhibit rational behavior. This design provides reasonable convergence guarantees.

As demonstrated in Figure 3(a), agents using the complete Shapley-Coop framework consistently converge to theoretically optimal outcomes.

---

3. Handling Imperfect or Noisy Contribution Estimation

We acknowledge this as an important consideration. However, our current work focuses on developing a workflow that enables emergent collaboration among self-interested LLM agents in open-ended environments characterized by social dilemmas—where individual interests may conflict, yet collective success requires cooperation.

While we have not extensively explored this issue, we observed that agents may exhibit bias in self-contribution evaluation, including assessment errors. Nevertheless, the iterative negotiation process and LLM contextual corrections help mitigate these issues to some extent.

Due to rebuttal character limitations, we cannot provide complete prompt examples here, but we will include relevant content in the appendix.

---

4. Comparison with Alternative Credit Assignment Mechanisms

Thank you for this valuable suggestion. Contextualizing our work against established MARL methods helps highlight its unique contributions.

• Different Target Agents:
  A fundamental difference lies in the type of agents each approach is designed for:

  • MARL credit assignment mechanisms (e.g., LIO, LOPT) are built for RL agents trained from scratch, typically in simulated environments using trial-and-error.
  • Shapley-Coop, in contrast, targets pretrained LLM agents that already possess rich world knowledge and reasoning abilities. This enables us to operate in zero-shot or few-shot settings, focusing on coordination and negotiation rather than learning from raw interaction.

• A Paradigm Comparison, Not a Direct Benchmark:
  Our work represents a distinct technical paradigm compared to traditional MARL methods.

• MARL Methods:
  These approaches learn implicit cooperation policies via extensive training (often tens of thousands of steps). While powerful in controlled environments, they tend to be less generalizable and require significant engineering effort. It is a training-intensive process.

• Shapley-Coop:
  Our framework utilizes the reasoning, language understanding, and commonsense knowledge of pretrained LLMs. It focuses on enabling structured interaction among agents through prompts and negotiation strategies. It is a guidance-based process that requires no additional training on the target task.

• Efficiency and Adaptability Advantage:
  For example, in the Escape Room task, MARL methods like LIO or LOPT require ~10⁴ training steps to learn cooperative behavior. In stark contrast, LLM agents using Shapley-Coop can reason about the social dilemma and achieve cooperation on the first attempt, demonstrating strong efficiency and adaptability—especially in novel, cognitively demanding environments.

---

Clarifications on Assumptions and Limitations

We acknowledge that transferable utility is a simplifying assumption. We view it as a reasonable abstraction of real-world scenarios like project-based bonus distributions or freelance contracts. We will discuss this in the limitations section and believe that adapting Shapley-Coop to scenarios with constrained reward transfer is a promising direction for future work.

---

We believe these responses address the key concerns and further reinforce the significance and originality of our contributions.
Thank you for your time and consideration.

We sincerely appreciate your thoughtful and constructive feedback. We are especially encouraged that you recognized the strong quality of our paper, the significance of the under-explored research problem, and the diversity of our empirical evaluations. Your insightful comments helped us realize the need to better clarify the core novelty of our work, which indeed goes well beyond a simple application of Shapley values.

---

Clarification of Motivation and Core Novelty

Our key motivation is to develop a scalable workflow that enables emergent cooperation among self-interested LLM agents in open-ended environments characterized by social dilemmas—settings where individual incentives may conflict with group success, yet collaboration is critical. This is essential for general-purpose human-AI interaction in domains like multi-agent software development or real-time crisis response.

While Shapley values offer a fair and principled method for credit assignment, exact computation is exponential in the number of agents, and impractical in dynamic, open-ended settings where agent roles are fluid and contributions cannot be cleanly quantified.

Shapley-Coop addresses this challenge by introducing a LLM-specific approximation framework that integrates Chain-of-Thought (CoT) reasoning with structured, language-based negotiation. Unlike prior work that assumes goal alignment or imposes fixed roles, our method enables dynamic cooperation through two key components:

• Short-term Shapley-CoT (Eq. 4): Provides qualitative, CoT-based guidance during task execution by prompting agents to reflect on their marginal contributions in real-time.
• Long-term Shapley-CoT (Eq. 6): Enables post-task reward redistribution via multi-turn negotiation protocols informed by contribution approximations.

This approach resembles a "TextGrad"-style approximation—leveraging LLMs' contextual reasoning—combined with principles from cooperative game theory. Importantly, precise Shapley values are not computed; instead, they serve as an interpretable, theory-grounded reference to guide agent discussions toward fair outcomes.

Our contribution lies in designing a full, practical workflow—not just a set of prompts—that integrates LLM-driven reasoning, game-theoretic bargaining, and emergent credit assignment in socially complex, unstructured environments.

---

Responses to Specific Reviewer Questions

1. Does the method only work in environments with dense reward functions?

No. Shapley-Coop is designed to function effectively in sparse or even reward-free settings. Instead of relying on numerical reward signals, it uses LLM-based contextual reasoning to approximate agent contributions through CoT reflection and negotiation. For example, in our Escape Room and Raid Battle experiments, rewards are sparse and delayed, yet agents can still collaboratively estimate contributions and reach equitable agreements.

---

2. How does Shapley-Coop perform with a larger number of agents?

We conducted additional experiments with 10 agents in the Raid Battle environment to evaluate scalability. Since our method avoids exact Shapley computation, which is exponential, and instead uses CoT-based estimation and negotiation, the framework scales linearly in practice, as negotiations occur via pairwise or group dialogues.

Below are results from two large-scale experiments:

Example: 4000 HP Case (10 Agents)

8000 HP Case: Similar results with slightly longer convergence times. Full data provided above.

These results demonstrate that final negotiated allocations closely track actual contributions, even with 10 agents, and the framework remains tractable in terms of runtime.

We also report detailed runtime breakdowns in 4-agent settings with increasing task complexity (boss HP = 2000–3000), as summarized below:

(Full breakdowns of negotiation/action times per round will be added to the supplementary material because of the limited space.)

---

3. How does the framework handle real-world fairness perceptions that differ from Shapley-optimality?

Shapley-Coop is designed to be modular and theory-informed, not rigid. It uses Shapley values as a guiding norm, but the actual credit assignment is determined through agent negotiation—allowing for subjective and context-sensitive adjustments.

In real-world scenarios where Shapley may not align with fairness perceptions (e.g., adversarial, political, or asymmetric-power settings), alternative principles like Banzhaf values could easily be substituted into the same CoT + negotiation workflow. This generality supports our broader motivation: enabling flexible, self-interested cooperation in non-aligned, social dilemma environments.

---

4. Figure 2 shows a relatively simple negotiation. How does the method handle tasks that are not easily decomposable?

Figure 2 illustrates a clear-cut case for exposition (Escape Room). However, our method is not limited to decomposable tasks. In more entangled, non-modular environments like Raid Battle (interdependent combat sequences) and ChatDEV (open-ended software collaboration), agents still use contextual CoT reflection to reason about their impact, without explicit decomposition.

Shapley-Coop does not require manual task structuring—instead, it allows LLM agents to infer contributions from contextual cues during both execution and negotiation. We provide full prompts in Appendix C, showing how this scales to complex, creative, and cooperative tasks.

---

Addressing Other Reviewer Concerns

Evaluation Breadth: We agree that evaluating in adversarial or deceptive contexts would be a valuable future direction. Our current focus is on social dilemma settings, where agents are self-interested but not explicitly malicious, and cooperation must emerge through interaction and negotiation. We now explicitly note the absence of adversarial agents as a limitation in our revised conclusion.

To highlight the diversity already present in our current evaluation, we emphasize that our three main tasks are carefully selected to represent a spectrum of cooperative challenges:

• Escape Room – a single-step coordination task with sparse rewards, testing agents' ability to align actions in a one-shot setting.
• Raid Battle – a multi-step, temporally extended task requiring real-time cooperation under delayed rewards and interdependent actions.
• ChatDEV – a complex, open-ended creative collaboration where agents must coordinate iteratively and adaptively in software design and implementation.

This progression reflects increasing task complexity and cooperation depth, and shows that our framework generalizes across diverse cooperative scenarios. While adversarial dynamics remain outside this paper’s scope, the current tasks already cover a broad range of non-trivial cooperative structures.

• Limitations Section: We appreciate your feedback here and will expand our discussion beyond the single limitation previously noted (inability to adjust prices dynamically). We’ll add discussion of adversarial robustness, negotiation deadlocks, and domain generality in future iterations.

---

We hope this clarification reinforces the originality, scalability, and practical impact of our contribution.

Thank you once again for your thoughtful and constructive review.

We would like to take this opportunity to unequivocally clarify our core contributions, research scope, and the supplementary experiments we have conducted. We hope these clarifications will directly address and resolve any misunderstandings you may have regarding our work.

Core Contribution: An LLM-Native Cooperation Workflow for Open-Ended Environments

We acknowledge that cooperative game theory and LLM agent negotiation, in isolation, are not entirely novel concepts. However, our core technical contribution lies in designing and implementing the first complete, scalable, and LLM-native workflow (Shapley-Coop) that translates a principle from cooperative game theory (the Shapley value) into an operational framework.
This is not mere prompt engineering; it is a deeply integrated framework that combines real-time marginal contribution reasoning (Short-term Shapley-CoT) with post-hoc negotiated allocation (Long-term Shapley-CoT). Its novelty lies in:

• Bridging Theory and Practice: Exact Shapley value computation is infeasible in dynamic, open-ended settings. Our work demonstrates how to leverage LLMs’ contextual reasoning capabilities to approximate this fairness principle, making it practical for real-world, unstructured tasks.
• Generality and Flexibility: Unlike SOTA methods that depend heavily on domain-specific training data or rigid protocols, our framework is zero-shot and enables agents to dynamically and autonomously construct cooperation protocols.

Defining Our Scope: A Deliberate Focus on Cooperative Social Dilemmas, Not Adversarial Interaction

We fully agree that testing agents in strategic or adversarial contexts is an important direction for multi-agent research. However, the scope of our work is intentionally focused on cooperative social dilemmas. In this class of problems, the central challenge is not defeating an opponent but overcoming the conflict between individual self-interest and collective benefit to achieve cooperation. We address two key challenges: how to enable LLM agents to exhibit emergent cooperation in such scenarios, and how to fairly allocate rewards based on performance after task completion.
This focus is deliberate because:

• It is a critical domain: In real-world human-AI collaborations, such as software development or crisis response, cooperative dilemmas are far more common than purely zero-sum adversarial encounters. (In particular, the issue of contribution pricing for LLMs is, to the best of our knowledge, a novel and highly significant research problem).
• Methodological Alignment: The Shapley value is a theoretical tool specifically designed for cooperative games. Forcing its application into adversarial settings would misalign with the core theoretical foundation of our work.

Comparison with State-of-the-Art (SOTA) Cooperation Methods

The most critical distinction lies in the target agents. Current SOTA methods, such as those in MARL, focus on RL agents requiring extensive training, whereas we target LLM agents. While Shapley value-based allocation is a classic and effective approach, its computation is inherently challenging. Our primary contribution is in establishing the workflow, not merely introducing Shapley values.
Our framework is uniquely positioned relative to other approaches:

• vs. Multi-Agent Reinforcement Learning (MARL): Many SOTA MARL methods excel in specific games but typically require well-defined state-action spaces and extensive simulation training. Our method focuses on open-ended, unstructured text-based worlds, where defining and training MARL policies is highly impractical.
• vs. Fixed Negotiation Protocols (e.g., Contract-Net): While efficient, such protocols are rigid and must be custom-designed for each domain. Shapley-Coop leverages LLMs’ general capabilities to enable agents to negotiate dynamically without pre-defined rules, adapting to a much broader and more dynamic range of tasks.

Additional Experiment

Regarding the scope, our work is clearly focused on cooperative social dilemmas—specifically, the emergence of cooperation among self-interested LLM agents and the fair allocation of payoffs after task completion. Theoretically, we have established that the Shapley value is the most appropriate metric for guiding payoff allocation in these settings, a conclusion further supported by our additional experiments comparing it with the Banzhaf value. For the detailed experimental results, please refer specifically to our latest response to Reviewer 4hz6 We would like to emphasize that these supplementary experiments are provided not as an admission of limitation, but to further clarify and justify our methodological choices, in line with both established theory and empirical evidence.

We respectfully note that the reviewer’s comment focuses on Escape Room and Raid Battle, but our evaluation is not limited to these two domains. As described in Lines 298–320 of the main paper and in Appendix B.2, we also conducted experiments on the ChatDEV task, which models collaborative software development. In this setting, we compared our approach against widely recognized value allocation standards in the software industry. We believe ChatDEV represents a more complex, open-ended, and realistic cooperation scenario—closer to the real-world applications described in our motivation—than the aforementioned games, and it aligns well with our goal of studying the emergence of cooperation among self-interested agents.

We would also like to clarify that the Melting Pot task mentioned by Reviewer 4hz6 is designed primarily to study credit assignment in a fully cooperative setting, rather than to explore the emergence of cooperation in social dilemmas involving self-interested agents, which is the focus of our work.

We acknowledge the importance of comprehensive evaluation, and our current set of experiments already covers multiple domains and includes analyses using both Shapley and Banzhaf values. Given the limited time and resources during the review stage, it is not feasible to indefinitely expand the experimental scope, but we will consider exploring additional domains and baselines in future work.

Thank you for your thoughtful and detailed review. Your feedback has been invaluable in helping us sharpen the core message of our paper. Below, we address your concerns point by point. We clarify the novelty of our framework, provide new empirical results on scalability, and outline the specific, actionable revisions we will make to the manuscript.

---

Clarification of Motivation and Core Novelty

Our key motivation is to develop a scalable workflow that enables emergent cooperation among self-interested LLM agents in open-ended environments characterized by social dilemmas—settings where individual incentives may conflict with group success, yet collaboration is critical. This is essential for general-purpose human-AI interaction in domains like multi-agent software development or real-time crisis response.

While Shapley values offer a fair and principled method for credit assignment, exact computation is exponential in the number of agents, and impractical in dynamic, open-ended settings where agent roles are fluid and contributions cannot be cleanly quantified.

Shapley-Coop addresses this challenge by introducing a LLM-specific approximation framework that integrates Chain-of-Thought (CoT) reasoning with structured, language-based negotiation. Unlike prior work that assumes goal alignment or imposes fixed roles, our method enables dynamic cooperation through two key components:

• Short-term Shapley-CoT (Eq. 4): Provides qualitative, CoT-based guidance during task execution by prompting agents to reflect on their marginal contributions in real-time.
• Long-term Shapley-CoT (Eq. 6): Enables post-task reward redistribution via multi-turn negotiation protocols informed by contribution approximations.

This approach resembles a "TextGrad"-style approximation—leveraging LLMs' contextual reasoning—combined with principles from cooperative game theory. Importantly, precise Shapley values are not computed; instead, they serve as an interpretable, theory-grounded reference to guide agent discussions toward fair outcomes.

Our contribution lies in designing a full, practical workflow—not just a set of prompts—that integrates LLM-driven reasoning, game-theoretic bargaining, and emergent credit assignment in socially complex, unstructured environments.

---

Response to Specific Questions and Suggestions

1. Motivation for Using LLMs Over Bespoke Protocols (e.g., Contract-Net)

This is a crucial point. Our goal is to design a cooperation framework that is native to the capabilities of LLM agents.

• Flexibility and Generality: Bespoke protocols like Contract-Net are powerful but rigid and domain-specific. Our approach leverages LLMs’ natural language understanding and reasoning capabilities to allow agents to dynamically construct fair and effective protocols on the fly. This flexibility is essential for the open-ended, unstructured environments we target.

• A General Workflow: The Shapley value is a foundational principle often used in the theoretical design of bespoke protocols. In this light, Shapley-Coop is not merely an alternative to a fixed protocol—it is a general workflow that guides LLMs to instantiate fair protocols through reasoned negotiation. We will explicitly incorporate this discussion into the paper to better frame our contribution.

---

2. Including More Strategic Interaction Examples

We appreciate the suggestion to evaluate our work in more diverse environments. Our three selected experimental scenarios were chosen to provide meaningful coverage across a spectrum of cooperation tasks (simple, complex, and creative), with ChatDEV demonstrating the practical need for our method in a real-world production pipeline.

Due to the time constraints of the rebuttal period, integrating a completely new and complex environment was not feasible.

However, we took your feedback on strengthening our empirical evaluation seriously. To address concerns about robustness, we performed a new scalability experiment on our most complex task, demonstrating performance with 10 agents in the Raid Battle scenario (Boss HP: 4000 and 8000). Below are results from two large-scale experiments:

Example: 4000 HP Case (10 Agents)

8000 HP Case: Similar results with slightly longer convergence times. Full data provided above.

---

3. Justification for Shapley Value and Ablation with Alternatives (e.g., Banzhaf, Deegan-Packel)

Our work focuses on enabling emergent collaboration in social dilemmas. Alternatives like the Banzhaf value measure influence in coalition-adversarial scenarios (e.g., voting), which is not aligned with our cooperative focus.

• The Shapley value is uniquely suited due to its efficiency axiom, which ensures the entire value is distributed across the coalition.
• While Banzhaf and others are useful in adversarial or voting contexts, they lack this property, making them conceptually misaligned with our framework.

That said, Shapley-Coop is generalizable—it could be adapted to incorporate other value concepts like Banzhaf to address different domains. We will mention this extensibility more explicitly in the revised discussion section.

---

Additional Questions (Runtime, LLM Queries, Shapley Computation)

Scalability: Shapley-Coop scales efficiently to larger agent populations by avoiding exact Shapley computation (which is exponentially complex). Instead, it uses context-based approximations via CoT and negotiation to estimate contributions.

Negotiation is performed per-agent in real-time and scales linearly through pairwise or group dialogues.

Runtime in Raid Battle Task (The task in the main paper with 4 heroes)

• Negotiation time increases with agent count and complexity but remains far below the cost of exact Shapley computation.
• Shapley-Coop demonstrates tractable scaling via decentralized, parallelizable reasoning and structured dialogue.

We acknowledge that negotiation costs grow with the number of agents. However, the approach remains practical and outperforms exact Shapley computation methods in scalability.

We hope these clarifications have addressed the reviewer's questions. Thank you once again for your valuable and constructive feedback.

Thank you for your response. We must reiterate that our work addresses the cooperation problem in open environments, where self-interested agents—such as LLM-based agents—form temporary coalitions to jointly complete tasks and share rewards. Each agent has its own independent motivation, and cooperation is often established through explicit negotiation or contract-like mechanisms, resembling settings common in social dilemma games.

Given this setup, the Shapley value is a natural and principled solution, as it is specifically designed to handle cooperative payoff allocation under arbitrary coalition structures and individual incentives [1]. Its axiomatic foundation enables fair and consistent marginal contribution attribution, making it particularly well-suited for agents coordinating to complete open-ended tasks.

By contrast, indices such as Banzhaf or Deegan-Packel were originally developed for analyzing voting power in binary decision games [2][3]. These indices lack some of the essential properties required for fair utility sharing in diverse, open-ended teams, such as full reward allocation and responsiveness to marginal contributions [4]. Importantly, adopting such indices would fundamentally deviate from our motivation, which is to ensure fair and principled attribution of value among self-interested agents in open cooperative environments. Using these indices in our context risks misrepresenting each agent’s contribution and compromising the core fairness objective that our work is grounded upon.

Supporting Experimental Results

To illustrate this point concretely, we followed your suggestion and integrated the Banzhaf value into our framework for testing. The results align perfectly with theoretical expectations:

Performance in Simple Scenarios is As Expected

In the "Escape Room" task, where agent roles are simple and mutually critical, the Banzhaf value performs adequately, yielding results similar to the Shapley value. This is consistent with its theoretical nature of measuring "criticality".

Its Unsuitability is Exposed in Our Core Scenarios

However, in the "Raid Battle" and "ChatDEV" scenarios—which are more complex and central to our research—contributions are continuous and quantifiable. As predicted by theory, the Banzhaf value, by ignoring the dynamics of marginal contributions, leads to demonstrably unfair allocations.

• Raid Battle (Level 2): The Banzhaf value assigns excessive credit to low-contributing agents while severely penalizing another, creating a large gap with actual contributions. This proves its inadequacy for tasks requiring precise contribution accounting.

• ChatDEV (ArtCanvas): The Banzhaf value erroneously allocates a large share of rewards to the "CEO" due to its managerial position, while significantly undervaluing the "Programmer" who produced the most quantifiable work. This violates the fundamental principle of fair allocation.

In summary, the Shapley value is not simply a convention, but a well-founded and empirically supported solution for our cooperative payoff allocation setting. We will add this comparative analysis to the paper—not as an admission of limitation, but to further clarify and justify our methodological choices in line with both established theory and experimental evidence.

We hope that the additional experiments and explanations help provide a clearer understanding and support a more informed evaluation of our contributions.

References

[1] Chalkiadakis, G., et al. (2011). Computational Aspects of Cooperative Game Theory.
[2] Laruelle, A., & Valenciano, F. (1999). On the choice of a power index.
[3] Deegan, J., & Packel, E. W. (1978). A new index of power.
[4] Elkind, E., et al. (2009). Computational Aspects of Cooperative Game Theory.