Coalition Formation in LLM Agent Networks: Stability Analysis and Convergence Guarantees

Abstract: LLM agents are increasingly deployed in multi-agent systems requiring strategic coordination. While recent work has analyzed LLM behavior in two-player games, coalition formation, where $n$ agents dynamically form cooperative groups, remains theoretically uncharacterized. We present the first framework grounding coalition formation in LLM agent networks in hedonic game theory with formal stability guarantees. We introduce the LLM Coalition Formation Game (LCFG), establish sufficient conditions for Nash-stable partitions, and prove complexity results. Our analysis reveals that LLM agents exhibit bounded rationality characterized by $ε$-rational preferences; we provide both deterministic existence guarantees and consistency-driven stability bounds whose predictions are consistent with empirical outcomes. Experiments with GPT-4, Claude-3, and Llama-3 across 2,400 episodes validate our framework: LLM coalitions achieve Nash stability in 73.2% of cases under our Coalition-of-Thought (CoalT) protocol, compared to 58.4% under chain-of-thought and 41.8% under standard prompting ($p < 0.001$). Our framework provides theoretical foundations for designing stable multi-agent LLM systems.

• The paper introduces LLM agent coalition formation anchored in hedonic game theory, establishing stability and convergence guarantees.
• It employs bounded rationality and potential dynamics to prove Nash-stable partitions and quantify preference consistency across multiple LLM architectures.
• Empirical evaluations show that the Coalition-of-Thought protocol significantly improves coalition stability over standard prompting.

Stability and Convergence in LLM Agent Coalition Formation

Introduction

The paper "Coalition Formation in LLM Agent Networks: Stability Analysis and Convergence Guarantees" (2604.14386) delivers a rigorous theoretical and empirical treatment of coalition formation among LLM agents in multi-agent systems. By situating LLM agents within the framework of hedonic cooperative game theory, the authors establish the first formal and experimentally validated foundation for analyzing coalition stability, rationality bounds, and computational complexity within networks of LLM agents. Their analysis spans theoretical models, algorithmic developments, and quantitative evaluations across diverse LLM architectures and coalition tasks.

Theoretical Framework: Hedonic Games for LLM Agents

The paper introduces the LLM Coalition Formation Game (LCFG), extending classical hedonic game theory to encapsulate distinctive properties of LLM agents:

• Agent Model: Each agent is a tuple $(m_i, \theta_i, \mathbf{c}_i)$, representing model architecture, configuration, and a task-specific capability profile. Capability profiles empirically capture agent strengths across skills such as mathematical reasoning, factual recall, and logical analysis.
• Coalition Value Function: The collective value of any agent subset $S$ is modeled as $$v(S) = \phi(\bigoplus_{a_i \in S} \mathbf{c}_i) - \psi(|S|)$$, where $\phi$ is an aggregation operator (componentwise-max), and $\psi$ is a superlinear coordination cost. This formulation naturally accounts for both capability complementarity and increasing coordination friction with coalition size.
• Bounded Rationality: LLM agents are formalized as possessing $\epsilon$-rational preferences—agents reliably choose higher-value coalitions only if payoff gaps exceed a threshold $\epsilon$. This introduces quantifiable and empirically measurable deviation from perfect rationality, crucial for LLM agents where stochasticity and context dependency are prevalent.
• Stability Concepts: Nash stability is prioritized as the principal solution concept, providing robustness against unilateral deviations and computational tractability under the paper's structural assumptions.

Existence, Stability, and Convergence Guarantees

The authors derive both deterministic and probabilistic results for coalition stability:

• Deterministic Existence: Under sufficiently large value gaps ($\delta$-value gap) and potential alignment (deviations increase global potential), and with agent error less than half the gap ($\epsilon < \delta/2$), Nash-stable partitions provably exist and are reachable by potential-improving dynamics. This extends the classic result to bounded-rational LLM agents and aligns coalition stability with structural properties of the value function and agent heterogeneity.
• Consistency-Driven Stability: For practical settings where $\epsilon > \delta/2$, which is typical for contemporary LLMs, perfect rationality cannot be assumed. The stability rate then becomes a function of preference consistency $S$0 across $S$1 critical coalition decisions. The explicit bound $S$2 predicts that even when LLM preferences are inconsistent under small payoff gaps, overall coalition stability can remain high if consistency on critical decisions is sufficiently robust.
• Convergence Analysis: Under the above assumptions, improving-dynamics converge to Nash-stable partitions in a polynomial number of steps, and both verification and computation of stable partitions are polynomial-time solvable under capability monotonicity. In general, without this structure, the computation of stable partitions in hedonic games is NP-hard.

Coalition-of-Thought Prompting Protocol

Recognizing the essential role of protocol design in mitigating agent inconsistency, the authors propose the Coalition-of-Thought (CoalT) prompting strategy. CoalT sequences LLM agent reasoning through explicit game-theoretic steps—capability enumeration, complementarity analysis, value estimation, coordination cost assessment, and structured preference declaration. This protocol demonstrably increases both the quality and consistency of coalition choices compared to standard and chain-of-thought baselines.

Empirical Evaluation and Numerical Results

The framework is empirically validated through 2,400 coalition episodes across GPT-4, Claude-3, and Llama-3 architectures. Key findings include:

• Coalition Stability: CoalT achieves Nash-stable partitions in 73.2% of episodes, significantly outperforming chain-of-thought (58.4%) and standard prompting (41.8%) ($S$3). The increase of 14.8 percentage points over vanilla CoT underscores the impact of explicit coalition reasoning.
• Preference Consistency: CoalT improves preference consistency from 0.64 (standard) to 0.86; stability closely tracks this metric, quantitatively confirming the theoretical $S$4 scaling law.
• Mixed vs. Homogeneous Teams: Heterogeneous coalitions composed of different LLM architectures yield higher welfare (0.81) than homogeneous groups, despite mildly lower stability, validating the value of capability complementarity.
• Component Ablations: Removing complementarity or value estimation reasoning from CoalT causes the largest decrements in stability, establishing the centrality of explicit game-theoretic reasoning.

Notably, estimated $S$5-rationality parameters for GPT-4 ($S$6), Claude-3 ($S$7), and Llama-3 ($S$8) exceed observed value gaps ($S$9), making deterministic guarantees inapplicable but validating the necessity of the consistency-driven framework.

Computational Complexity and Scalability

The general case of coalition partition generation remains NP-hard in hedonic games, and LCFGs preserve this hardness at unbounded capability dimension. However, when capability monotonicity holds (empirically validated in practical deployments), polynomial-time coalition verification and construction are feasible via potential-based improving dynamics.

Scalability experiments confirm that as the number of agents grows, the Nash stability rate degrades as roughly $$v(S) = \phi(\bigoplus_{a_i \in S} \mathbf{c}_i) - \psi(|S|)$$0. This observation indicates that large-scale multi-agent LLM deployments may require hierarchical coalition organization to maintain desirable stability.

Implications and Future Directions

From a practical perspective, the results provide actionable guidance for multi-agent LLM system design:

• Stable, high-welfare coalitions prefer mixed-architecture teams using prompting strategies that target preference consistency and explicit reasoning about complementarities and tradeoffs.
• Monitoring consistency metrics serves as a predictive signal of system-wide stability, enabling early intervention.
• Theoretical connections to Quantal Response Equilibria underpin future extensions into more granular and expressive behavioral game-theoretic models of LLM agent populations.

Theoretically, this work situates the design and control of LLM agent coalitions on a rigorous game-theoretic foundation, opening avenues for incentive-compatible protocol design, dynamic coalition restructuring under evolving capabilities, and empirical extension to next-generation LLMs. The finding that stability is fundamentally bounded by agent consistency, rather than theoretical utility optimality, invites further study of model training and alignment methods to directly optimize for consistent cooperative behavior.

Conclusion

This paper provides a comprehensive theoretical and empirical foundation for coalition formation among LLM agents via hedonic cooperative game theory, with new contributions in model formalization, stability guarantees, convergence analysis, complex protocol engineering, and extensive validation across multiple LLM architectures (2604.14386). The work demonstrates that in practical settings, preference consistency—rather than perfect rationality—is the critical determinant of stable and high-performing coalition outcomes in multi-agent LLM systems. The proposed Coalition-of-Thought protocol exemplifies how explicit, game-theoretic reasoning can be systematically elicited from LLM agents to achieve superior stability, and the theoretical analysis provides a blueprint for the scalable, robust deployment of LLM agent societies. Future research directions include incentivizing truthful preference reporting, handling dynamic agent capability evolution, and extending scalability to even larger agent populations through hierarchical coalition formation.