LLM Coalition Formation Game (LCFG)

• LCFG is a formal model that partitions LLM agents into exclusive coalitions using hedonic game theory and per-capita value functions.
• It introduces a Coalition-of-Thought prompting protocol that empirically improves Nash stability and welfare compared to standard methods.
• The framework addresses practical constraints like API budgets and compute limits while leveraging bounded rationality and polynomial verification under specific conditions.

The LLM Coalition Formation Game (LCFG) is a formal model of coalition formation in multi-agent large-language-model systems in which LLM agents are treated as strategic players that partition themselves into cooperative groups for a task episode. In its explicit formulation, LCFG is a hedonic game $G=(N,v,\{\succsim_i\}_{i\in N})$ with exclusive partitions, per-capita coalition preferences, and formal stability notions such as Nash stability, individual stability, and core stability; it is paired with a prompting protocol, Coalition-of-Thought (CoalT), for coalition reasoning by LLMs (Guo et al., 15 Apr 2026).

1. Conceptual scope and problem setting

LCFG addresses a problem that differs from standard two-player or action-selection analyses. The central object is not a bilateral strategy profile, but a partition of a finite set of LLM agents into coalitions, where each agent belongs to exactly one coalition in a task episode. The explicit motivation for exclusivity is practical constraints such as API budgets, compute limits, and context-window limits (Guo et al., 15 Apr 2026).

In the formal agent model, each agent is written as

$a_i=(m_i,\theta_i,\mathbf{c}_i),$

where $m_i\in\mathcal{M}$ is the model architecture, $\theta_i\in\Theta=[0,2]\times\Sigma^*$ is the configuration consisting of temperature $\tau\in[0,2]$ and system prompt $s\in\Sigma^*$, and $\mathbf{c}_i\in[0,1]^d$ is a capability profile over $d$ skill dimensions (Guo et al., 15 Apr 2026). This makes coalition formation explicitly capability-sensitive rather than purely structural.

The framework is designed for settings in which utilities depend on coalition membership and group composition, and where the outcome is a partition of the agent set. This places LCFG within the broader coalition-formation literature, but with a specific emphasis on prompt-sensitive, boundedly rational LLM agents rather than fully rational abstract players (Guo et al., 15 Apr 2026).

2. Formal game model

An LCFG instance is defined as

$G=(N,v,\{\succsim_i\}_{i\in N}),$

where $N=\{a_1,\ldots,a_n\}$ is the set of LLM agents, $a_i=(m_i,\theta_i,\mathbf{c}_i),$0 is a coalition value function, and $a_i=(m_i,\theta_i,\mathbf{c}_i),$1 is agent $a_i=(m_i,\theta_i,\mathbf{c}_i),$2's preference relation over coalitions containing $a_i=(m_i,\theta_i,\mathbf{c}_i),$3, informed by per-capita value $a_i=(m_i,\theta_i,\mathbf{c}_i),$4 (Guo et al., 15 Apr 2026).

The coalition value function is

$a_i=(m_i,\theta_i,\mathbf{c}_i),$5

with $a_i=(m_i,\theta_i,\mathbf{c}_i),$6 as capability aggregation, $a_i=(m_i,\theta_i,\mathbf{c}_i),$7 as componentwise maximum, and $a_i=(m_i,\theta_i,\mathbf{c}_i),$8 as coordination cost. The paper instantiates

$a_i=(m_i,\theta_i,\mathbf{c}_i),$9

with default empirical calibration

$m_i\in\mathcal{M}$0

Capability aggregation by componentwise maximum means that a coalition’s effective skill vector is determined by its best member on each capability dimension, while $m_i\in\mathcal{M}$1 imposes superlinear coordination overhead (Guo et al., 15 Apr 2026).

A coalition structure is a partition

$m_i\in\mathcal{M}$2

such that the $m_i\in\mathcal{M}$3 are disjoint and $m_i\in\mathcal{M}$4. Agent preferences are hedonic because each agent’s ranking depends only on the coalition it belongs to, not on the organization of outsiders. The paper explicitly states that LCFG is a hedonic game, but does not explicitly identify it as additively separable or fractional; it is a value-based hedonic game with equal per-capita sharing (Guo et al., 15 Apr 2026).

The primary stability notion is Nash stability: $m_i\in\mathcal{M}$5 This means no agent would rather leave its current coalition to join another coalition or become a singleton. The paper also discusses individual stability, where the receiving coalition must not be harmed, and core stability, where no group of agents can jointly deviate so that all members strictly benefit (Guo et al., 15 Apr 2026).

3. Bounded rationality, consistency, and convergence

A defining feature of LCFG is that LLM preferences are not treated as perfectly rational oracle outputs. Instead, agents are modeled as $m_i\in\mathcal{M}$6-rational: $m_i\in\mathcal{M}$7 where $m_i\in\mathcal{M}$8. This means that if two coalitions differ by more than $m_i\in\mathcal{M}$9, the higher-valued coalition must be preferred; when the value gap is smaller, inconsistent or near-random behavior is permitted (Guo et al., 15 Apr 2026).

The paper reports empirical estimates

$\theta_i\in\Theta=[0,2]\times\Sigma^*$0

with the corresponding 95% confidence intervals $\theta_i\in\Theta=[0,2]\times\Sigma^*$1, $\theta_i\in\Theta=[0,2]\times\Sigma^*$2, and $\theta_i\in\Theta=[0,2]\times\Sigma^*$3 (Guo et al., 15 Apr 2026). Preference consistency is treated separately: an agent has consistency $\theta_i\in\Theta=[0,2]\times\Sigma^*$4 if repeated queries for the same coalition comparison return the same answer with probability $\theta_i\in\Theta=[0,2]\times\Sigma^*$5.

The deterministic theory uses three structural assumptions: the $\theta_i\in\Theta=[0,2]\times\Sigma^*$6-value gap condition, potential alignment, and capability monotonicity. The value-gap condition requires that for all distinct coalitions $\theta_i\in\Theta=[0,2]\times\Sigma^*$7, either $\theta_i\in\Theta=[0,2]\times\Sigma^*$8 or $\theta_i\in\Theta=[0,2]\times\Sigma^*$9. Potential alignment is defined by

$\tau\in[0,2]$0

with every improving unilateral deviation increasing $\tau\in[0,2]$1. Capability monotonicity requires that if $\tau\in[0,2]$2 componentwise, then $\tau\in[0,2]$3 (Guo et al., 15 Apr 2026).

Under these assumptions, Theorem 1 states that if $\tau\in[0,2]$4, a Nash-stable partition exists and can be found in polynomial time. Theorem 3 gives a convergence bound under asynchronous unilateral deviations: $\tau\in[0,2]$5 Theorem 4 separates verification and computation: verifying Nash stability of a given partition is in $\tau\in[0,2]$6, with $\tau\in[0,2]$7 preference queries under explicit value computation, while computing a Nash-stable partition is NP-hard in general hedonic games and remains so for LCFG when the capability dimension $\tau\in[0,2]$8 is unbounded; under capability monotonicity and potential alignment, both verification and computation are polynomial (Guo et al., 15 Apr 2026).

Because the empirical $\tau\in[0,2]$9 values exceed $s\in\Sigma^*$0, the practically relevant regime is probabilistic rather than deterministic. The paper gives a consistency-driven lower bound

$s\in\Sigma^*$1

with $s\in\Sigma^*$2 the consistency on critical decisions, $s\in\Sigma^*$3 on easy decisions, $s\in\Sigma^*$4 the number of effectively critical decisions, and $s\in\Sigma^*$5 the probability that consistent dynamics reach a Nash-stable partition (Guo et al., 15 Apr 2026). This shifts the emphasis from perfect rationality to repeatable preference elicitation.

4. Coalition-of-Thought and empirical evidence

Preferences in LCFG are elicited through prompting. The paper studies Standard prompting, Vanilla chain-of-thought, Self-Consistency, and Coalition-of-Thought (CoalT). CoalT structures each coalition comparison into five steps: Capability Analysis, Complementarity Assessment, Value Estimation, Coordination Cost Analysis, and Preference Declaration (Guo et al., 15 Apr 2026).

The empirical study uses GPT-4, Claude-3, and Llama-3 agents across 2,400 episodes. The task setting is collaborative question answering with three capability dimensions: Mathematical reasoning, Factual knowledge, and Logical analysis. The paper reports four research questions: whether LLM agents converge to stable coalition structures, whether CoalT outperforms baselines, whether heterogeneous coalitions outperform homogeneous ones, and how CoalT components contribute to performance (Guo et al., 15 Apr 2026).

Condition	Nash stability	Consistency
Standard	41.8%	0.64
Vanilla CoT	58.4%	0.74
CoalT	73.2%	0.86

CoalT also improves convergence and welfare: the reported convergence metric is $s\in\Sigma^*$6, compared with $s\in\Sigma^*$7 for Standard and $s\in\Sigma^*$8 for Vanilla CoT, while welfare rises to $s\in\Sigma^*$9 under CoalT (Guo et al., 15 Apr 2026). The paper reports $\mathbf{c}_i\in[0,1]^d$0 for the key improvements in Nash stability.

The architecture-level results introduce a stability–welfare contrast. Under CoalT, GPT-4 only coalitions achieve 78.5% stability, Claude only coalitions 81.2%, Llama only coalitions 62.3%, while mixed coalitions achieve 73.2% stability but the highest reported welfare, 0.81 (Guo et al., 15 Apr 2026). This suggests that heterogeneous coalitions can be welfare-superior even when homogeneous stronger-model groups are more stable.

The ablation results show that the largest individual contributor is Complementarity assessment: full CoalT yields 73.2%, whereas removing Complementarity drops performance to 65.4%, and removing all CoalT steps yields the Vanilla CoT result of 58.4% (Guo et al., 15 Apr 2026).

5. Relation to coalition-formation theory

LCFG draws on several strands of coalition-formation research. The generic merge-and-split framework of Apt and Witzel formalizes coalition formation through local merge and split rules over partitions, and proves that if a $\mathbf{c}_i\in[0,1]^d$1-stable partition exists, every iteration of merge and split yields the same partition; this gives a general convergence template for coalition restructuring (0709.0435). Levando’s non-cooperative formulation instead embeds coalition structure formation directly into a family of nested simultaneous games $\mathbf{c}_i\in[0,1]^d$2, with partitions described by Young diagrams, strategy sets indexed by candidate coalition structures, and a coalition structure formation mechanism $\mathbf{c}_i\in[0,1]^d$3; mixed equilibrium exists for every $\mathbf{c}_i\in[0,1]^d$4 (Levando, 2021).

Other neighboring models broaden the design space beyond LCFG’s exclusive hedonic partitions. Overlapping coalition formation allows a player to participate in multiple cooperative groups simultaneously through a resource contribution vector $\mathbf{c}_i\in[0,1]^d$5, and the resulting overlapping coalitional structure can be stabilized by local resource-unit deviations (Zhang et al., 2014). Layered coalitional games add hierarchy: an upper-layer overlapping coalition formation game in partition form can be coupled to a lower-layer coalitional graphical game, so that upper-layer coalition value is generated endogenously by lower-layer interaction (Lu et al., 2015). In partition-form games with deviation overhead, the weak and strong $\mathbf{c}_i\in[0,1]^d$6-core characterize grand-coalition stability, while merge-based distributed algorithms can yield coalition structures in a coalition structure stable set (Mochaourab et al., 2014). In congestion-style partition-form games, coalition worth $\mathbf{c}_i\in[0,1]^d$7 depends on the partition and on the Nash equilibria of the induced game among coalitions, and stability can be analyzed through robust blocking under pessimistic anticipation (Sultana et al., 2024).

Taken together, these results indicate that the LCFG of (Guo et al., 15 Apr 2026) is a specific point in a larger design space: it adopts exclusive partitions, hedonic preferences, and per-capita coalition value, whereas the broader literature also supports overlapping membership, partition-form externalities, layered architectures, and merge-and-split stability.

6. Language, dynamic coalitions, and open problems

A common simplification is to equate coalition structure with a static partition. Natural-language-based multi-agent interaction suggests a different representation. In Diplomacy, coalition structure can be modeled as a weighted undirected multigraph

$\mathbf{c}_i\in[0,1]^d$8

where bilateral agreements are edges and multiple simultaneous agreements induce a multilateral network rather than a partition (Kulkarni et al., 22 Feb 2025). The paper’s agreement weighting rule,

$\mathbf{c}_i\in[0,1]^d$9

combines strategic value with the subjective belief that the other player will honor the agreement (Kulkarni et al., 22 Feb 2025). This suggests that language-mediated LCFGs may require commitment graphs, honor probabilities, and subjective belief models rather than only static coalition partitions.

Several open issues follow directly. First, the LCFG of (Guo et al., 15 Apr 2026) assumes exclusive partitions, while overlapping and layered models show that coalition value can depend on simultaneous multi-membership and cross-layer interaction (Zhang et al., 2014). Second, exact deterministic stability is restricted to the regime $d$0, but empirical LLMs operate outside that regime, so the practically relevant notion is consistency-driven probabilistic stability rather than exact existence (Guo et al., 15 Apr 2026). Third, dynamic and language-based coalition formation raises the issue of whether coalition structure should be represented as a partition, a multigraph of agreements, or a hybrid object (Kulkarni et al., 22 Feb 2025). Fourth, the broader non-cooperative coalition-structure literature shows that efficient outcomes do not uniquely identify coalition membership; the same payoff profile can arise under different realized partitions, so answer quality alone does not identify whether LLM agents genuinely formed a coalition (Levando, 2021).

The current LCFG formulation therefore supplies formal foundations for stable coalition formation in LLM agent networks, but it does not exhaust the topic. It defines one tractable and empirically validated coalition model—hedonic, partition-based, capability-driven, and consistency-sensitive—while the surrounding literature points toward extensions with overlapping coalitions, partition-form externalities, merge-and-split dynamics, layered structures, and language-mediated commitment networks (Guo et al., 15 Apr 2026).