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

TL;DR

This paper introduces a formal framework for coalition formation among LLM agents, providing stability guarantees and empirical validation across multiple models and protocols.

Contribution

It is the first to ground LLM coalition formation in hedonic game theory with formal stability analysis and empirical validation.

Findings

LLM coalitions achieve Nash stability in 73.2% of cases with the CoalT protocol

The framework establishes sufficient conditions for stable partitions

Empirical results validate the theoretical stability predictions

Abstract

Large Language Model (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 $\epsilon$-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.