Coalition-of-Thought (CoalT)

• Coalition-of-Thought (CoalT) is a five-step prompting protocol designed for decentralized coalition formation, where agents analyze capabilities, complementarity, task value, and coordination cost.
• It utilizes a formal hedonic game theory framework to evaluate Nash stability and per-capita payoff, yielding higher consistency and faster convergence compared to standard and chain-of-thought methods.
• Empirical results with GPT-4, Claude-3, and Llama-3 demonstrate CoalT's effectiveness, achieving 73.2% Nash stability and improved social welfare over 2,400 experimental episodes.

Searching arXiv for the cited papers to ground the article in current literature. Coalition-of-Thought (CoalT) is a five-step prompting protocol and a concrete multi-agent interaction pattern for coalition formation decisions in LLM agent networks, introduced in "Coalition Formation in LLM Agent Networks: Stability Analysis and Convergence Guarantees" (Guo et al., 15 Apr 2026). It is defined for settings in which $n$ LLM agents repeatedly decide whether to remain in a current coalition or move to another, and it is grounded in a formal hedonic game model, the LLM Coalition Formation Game (LCFG). CoalT differs from generic chain-of-thought prompting by making coalition membership itself the object of structured reasoning: agents explicitly analyze capability composition, complementarity, expected coalition value, and coordination cost before declaring a preference. In experiments with GPT-4, Claude-3, and Llama-3 across 2,400 episodes, CoalT achieved Nash stability in 73.2% of cases, compared with 58.4% under chain-of-thought and 41.8% under standard prompting (Guo et al., 15 Apr 2026).

1. Definition and protocol structure

In its primary formulation, CoalT is a five-step prompting protocol used whenever an agent compares its current coalition $C$ with a candidate coalition $C'$. The input is an agent $a_i$, its current coalition $C$, and a candidate coalition $C'$; the output is a preference $C \succ C'$, $C' \succ C$, or $C \sim C'$ (Guo et al., 15 Apr 2026).

The five steps are:

1. Capability Analysis: “List capabilities of members in $C$ and $C$0.”
2. Complementarity Assessment: “Identify capability gaps and overlaps.”
3. Value Estimation: “Estimate task performance for each coalition.”
4. Coordination Cost Analysis: “Assess communication/coordination overhead.”
5. Preference Declaration: “Based on analysis, declare preference.”

The protocol is local and individual rather than centralized. Each agent separately runs CoalT to decide whether to deviate; there is no central controller. CoalT is also explicitly game-aware. Its reasoning targets capability coverage, complementarity, per-capita payoff, and coordination cost, which are the same primitives used in the formal game model. The prompt template described in the appendix of the introducing work makes this alignment explicit by referencing the agent’s own capability profile, coalition capability coverage, and coordination cost (Guo et al., 15 Apr 2026).

CoalT is not identical to generic chain-of-thought. Standard prompting asks an agent to choose directly between coalitions. Vanilla chain-of-thought asks the model to “think step by step” but does not constrain the content of reasoning toward coalition-game concepts. CoalT instead specifies what the model should reason about. The reported comparison therefore isolates structured coalition reasoning from mere verbosity or longer reasoning traces. The protocol also uses one structured prompt per decision, whereas Self-Consistency CoT improves reasoning quality through multiple CoT samples with majority voting (Guo et al., 15 Apr 2026).

2. Formalization in the LLM Coalition Formation Game

CoalT is embedded in the LLM Coalition Formation Game, an $C$1-agent hedonic game in which each agent is represented as

$C$2

where $C$3 is the model architecture, $C$4 is the configuration, and $C$5 is a capability profile over $C$6 skill dimensions. In the experiments, the architectures are GPT-4, Claude-3, and Llama-3; the skill dimensions are Math, Facts, and Logic (Guo et al., 15 Apr 2026).

An LCFG is defined as

$C$7

where $C$8, $C$9 is a coalition value function, and $C'$0 is agent $C'$1’s preference relation over coalitions containing $C'$2. The value function is

$C'$3

with $C'$4 the componentwise maximum of capability vectors. The aggregation function is

$C'$5

and the coordination cost is modeled as

$C'$6

Each agent’s coalition utility is the per-capita value

$C'$7

This formulation makes coalition formation a problem of balancing skill coverage against superlinear coordination cost. Because $C'$8 takes the componentwise maximum, coalition value depends on coverage rather than additive duplication; complementarity matters more than simple redundancy. This is why CoalT’s explicit complementarity step is formally matched to the value model rather than added as an informal heuristic (Guo et al., 15 Apr 2026).

Coalitions are nonempty subsets $C'$9, and a partition $a_i$0 covers all agents with disjoint coalitions. The hedonic assumption is central: each agent evaluates only the membership of its own coalition and not the grouping of agents elsewhere. CoalT therefore operates in a unilateral-deviation regime: agents compare their current coalition with a prospective destination coalition and express a local preference, after which the environment updates the global partition (Guo et al., 15 Apr 2026).

3. Stability, bounded rationality, and convergence

The principal stability notion is Nash stability. A partition $a_i$1 is Nash-stable if no agent prefers to move unilaterally to another coalition or to become a singleton: $a_i$2 The same framework also defines individual stability and core stability, but the analysis concentrates on Nash stability because the dynamics allow at most one deviating agent per round and because Nash stability is polynomial-time checkable under the structural assumptions adopted in the model (Guo et al., 15 Apr 2026).

The theory does not assume perfect rationality. Instead, preferences are modeled as $a_i$3-rational: $a_i$4 If the per-capita value difference is larger than $a_i$5, the agent must prefer the better coalition; if the difference falls within $a_i$6, inconsistent behavior is permitted. This provides a bounded-rationality model for LLM coalition choice. Empirically, the estimated thresholds are $a_i$7 for GPT-4, $a_i$8 for Claude-3, and $a_i$9 for Llama-3-70B, whereas the empirically observed value gap in the 6-agent setting is $C$0, so $C$1. The realistic regime is therefore one in which $C$2 for all models (Guo et al., 15 Apr 2026).

Under ideal conditions, the introducing work proves a deterministic existence theorem. If the game satisfies a $C$3-value gap condition, potential alignment, and capability monotonicity, and if all agents have $C$4-rational preferences with

$C$5

then a Nash-stable partition exists and can be found in polynomial time. The associated potential function is

$C$6

Potential alignment requires that any improving deviation in per-capita value also increases $C$7. Under these assumptions, improving dynamics terminate at a Nash-stable partition (Guo et al., 15 Apr 2026).

Because the empirical $C$8 regime violates the deterministic condition, the main realistic guarantee is probabilistic. The theory introduces preference consistency $C$9, defined as the probability that an agent gives the same preference if asked again about the same coalition comparison. The probability of reaching a Nash-stable partition is lower bounded by a product of consistency terms over critical and easy decisions, multiplied by $C'$0, the probability that consistent preferences lead the dynamics to a Nash-stable partition. Under logit dynamics with precision $C'$1,

$C'$2

The central implication is that stability depends less on perfect payoff optimization than on repeated preference consistency. CoalT is designed specifically to increase that consistency (Guo et al., 15 Apr 2026).

A further convergence theorem states that, under the ideal conditions and temporal consistency of preferences, improving dynamics with at most one deviating agent per round converge in at most

$C'$3

rounds, where $C'$4. This is a worst-case bound; the empirical dynamics are substantially faster (Guo et al., 15 Apr 2026).

4. Dynamics, verification, and empirical results

The empirical setting uses cooperative question answering tasks requiring mathematical reasoning, factual knowledge, and logical analysis. There are $C'$5 agents: 2 GPT-4, 2 Claude-3-Opus, and 2 Llama-3-70B-Instruct instances. Each agent has a three-dimensional capability profile estimated from MATH, an MMLU knowledge subset, and LogiQA, with temperature $C'$6 and three runs per model. The experiments use 200 questions stratified by difficulty and capability mix, with 400 episodes per condition and 6 conditions for a total of 2,400 episodes (Guo et al., 15 Apr 2026).

Each episode begins from an initial partition $C'$7. In each round, agents observe the question, the current partition, and candidate coalition summaries. Under a specified prompting protocol, they decide whether to deviate. The environment implements at most one improving deviation per round. The process stops when no agent wants to deviate, yielding a candidate Nash-stable partition, or after 30 rounds, in which case the episode is classified as unstable. Stability verification is then performed by querying each agent three times for each relevant coalition comparison and taking a majority vote (Guo et al., 15 Apr 2026).

The main results are summarized below.

Protocol	Nash stability	Preference consistency
Standard prompting	41.8%	0.64
Vanilla CoT	58.4%	0.74
Self-Consistency CoT	62.7%	0.79
CoalT	73.2%	0.86

CoalT also outperforms Greedy, which achieves 52.1% Nash stability and 0.71 consistency. Relative to vanilla CoT, CoalT improves Nash stability by $C'$8 percentage points, with $C'$9. The paper attributes this to the structure of reasoning rather than to the mere presence of multi-step reasoning (Guo et al., 15 Apr 2026).

The convergence and welfare results follow the same pattern. Mean convergence rounds are $C \succ C'$0 for CoalT, $C \succ C'$1 for CoT, and $C \succ C'$2 for standard prompting. Social welfare, measured as average task accuracy, is $C \succ C'$3 for CoalT, compared with $C \succ C'$4 for Self-Consistency CoT, $C \succ C'$5 for CoT, and $C \succ C'$6 for standard prompting. The improvements are statistically significant with $C \succ C'$7, and the reported effect size versus standard prompting is Cohen’s $C \succ C'$8 (Guo et al., 15 Apr 2026).

Ablations further identify the role of individual CoalT steps. Removing capability analysis reduces Nash stability from 73.2% to 68.9%; removing complementarity assessment reduces it to 65.4%; removing value estimation to 67.1%; removing coordination cost to 70.8%; and removing all specialized steps, thereby reducing the protocol to vanilla CoT, yields 58.4%. The largest single drop is observed when complementarity assessment is removed (Guo et al., 15 Apr 2026).

5. Position within coalition-formation research

CoalT belongs to a broader research line on coalition formation and coalition inference in language-mediated multi-agent systems, but its contribution is specific. The introducing work states that recent research had analyzed LLM behavior in two-player games, while coalition formation in $C \succ C'$9-agent systems remained theoretically uncharacterized; CoalT therefore appears as part of the first framework grounding coalition formation in LLM agent networks in hedonic game theory with formal stability guarantees (Guo et al., 15 Apr 2026).

A useful comparison arises with "Dynamic Coalition Structure Detection in Natural Language-based Interactions" (Kulkarni et al., 22 Feb 2025). That work studies coalition structure prediction in Diplomacy rather than coalition decision-making in LLM agent networks. Its method is explicitly two-stage: agreement extraction from dialogue, followed by strategic evaluation using subjective rationalizability and hypergame theory. CoalT differs in both ontology and mechanism. In the Diplomacy setting, a coalition structure is a weighted multigraph of bilateral agreements, and private natural-language negotiation is itself the object of inference. In CoalT, by contrast, interaction is implicit: agents do not directly negotiate, there is no debate-style protocol with explicit argument exchange, and the environment updates the partition according to local coalition preferences (Kulkarni et al., 22 Feb 2025).

A second comparison is with "United for Change: Deliberative Coalition Formation to Change the Status Quo" (Elkind et al., 2020). That work models myopic, consensus-seeking coalition formation around proposals in metric spaces and studies transitions such as follow, merge, compromise, and $C' \succ C$0-compromise. Its guarantees depend strongly on the geometry of the proposal space: in Euclidean spaces, pairwise compromise transitions suffice for success, while in hypercubes higher-arity compromise may be necessary. CoalT does not adopt this proposal-centric metric-space model. Instead, it uses a hedonic coalition-value formulation in which coverage and coordination cost determine per-capita utility. A plausible implication is that CoalT occupies a different point in the coalition-formation design space: it is agent-centric and payoff-based rather than proposal-centric and status-quo-relative (Elkind et al., 2020).

A third adjacent line is "Cognitive Insights and Stable Coalition Matching for Fostering Multi-Agent Cooperation" (Shao et al., 2024). That work forms coalitions by stable many-to-many matching using belief-action alignment and specialized abilities, motivated by Theory of Mind. CoalT shares the concern with stable cooperation and heterogeneous agents, but it does not derive preferences from ToM-based alignment. Its preference model is instead grounded in coalition value $C' \succ C$1, per-capita payoff $C' \succ C$2, and $C' \succ C$3-rational deviations. This suggests two distinct but compatible perspectives on multi-agent coordination: one built around alignment-sensitive matching, the other around hedonic coalition values and unilateral deviation dynamics (Shao et al., 2024).

6. Scope, limitations, and interpretation

CoalT is best understood as a coalition-aware reasoning protocol for cooperative multi-agent LLM systems. It is designed for settings with explicit or implicit team formation, access to rough capability profiles, and a need to avoid thrashing caused by repeated partner switching. The most direct use cases named in the source are research assistants, negotiation teams, and planning agents, especially when collaboration patterns must remain stable over the course of a task episode (Guo et al., 15 Apr 2026).

Several misconceptions are explicitly ruled out by its formulation. CoalT is not generic chain-of-thought with more steps; the protocol is specialized to coalition decisions. It is not a debate framework, since the reported experiments do not include explicit argument exchange among agents. It is not a centralized coalition planner, because each agent runs the protocol independently and coalition changes occur through iterative local deviations. It is also not a mechanism-design solution to strategic manipulation: the analysis assumes truthful preference reporting, and the authors state that incentive-compatible mechanisms are not designed in the framework (Guo et al., 15 Apr 2026).

The limitations are correspondingly precise. The hedonic assumption means that an agent cares only about its own coalition and not about the structure of coalitions elsewhere. The coordination cost function

$C' \succ C$4

is chosen and calibrated rather than derived from first principles. The $C' \succ C$5-rational and logit models are stylized approximations of LLM behavior. Experimentally, the study uses only 6 agents, 3 model architectures, and 3 capability dimensions, all within cooperative question answering. Stability verification depends on repeated querying of the same models under no drift or retraining. The authors also note that stability decreases as the number of agents grows, roughly like $C' \succ C$6, which suggests that large agent societies may require hierarchical or clustered coalition formation procedures (Guo et al., 15 Apr 2026).

Within those boundaries, CoalT’s main significance lies in the link it establishes between structured prompting and formal coalition dynamics. The theoretical claim is not that LLMs are perfectly rational coalition players, but that stability can still be analyzed when their bounded rationality is modeled explicitly and when prompting raises preference consistency. The empirical result that CoalT raises consistency from 0.64 under standard prompting and 0.74 under vanilla CoT to 0.86, alongside the increase in Nash stability from 41.8% and 58.4% to 73.2%, is the central evidence for that interpretation (Guo et al., 15 Apr 2026).