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The Butterfly Defects: Tracing Cooperation in LLM Agent Networks

Maria Sampani, Maria Lymperaiou, Angeliki Dimitriou, Giorgos Filandrianos, Chrysoula Zerva, Giorgos Stamou

arXiv:2610.32726v1physics.soc-phcs.CL

TL;DR

The paper asks whether cooperation in individual LLM interactions carries over to networks, where agents’ behavior may depend on partners, structural roles, and language. It studies this question in dyadic and networked IPD, finding that cooperation shifts with model composition, uncertainty cues, and action-label order.

  • Problem

    How cooperation emerges and persists in LLM networks remains unclear when interaction partners, structural roles, and prompt language vary.

    Source: §1 · pp. 1–2

  • Method

    The paper uses a controlled networked IPD framework, varying topology, network size, model composition, uncertainty-cue exposure, and action-label ordering.

    Source: §1 · p. 2

  • Results

    Cooperation shifts with model composition and network position; a single uncertainty cue can affect distant agents or redistribute costs, and reversing action-label order can change outcomes.

    Source: §1 · p. 2, §5.3 · p. 6, §6 · p. 8

  • Takeaways & Limitations

    Cooperative behavior in the evaluated settings is not a stable model-level property, so multi-agent LLM systems should be evaluated across relevant populations, structures, and prompt formulations.

    Source: §6 · p. 8

    The study uses selected within-family pairings and networks of up to 20 agents under a controlled protocol, limiting generalization to larger, more diverse, or differently structured environments.

    Source: Limitations · p. 9

Abstract

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As large language models (LLMs) increasingly operate as interconnected agents, a fundamental question emerges: does individual cooperation translate into collective cooperation? We investigate this question through the Iterated Prisoner's Dilemma (IPD), moving from dyadic interactions to homogeneous and heterogeneous networks of open-weight LLMs. Our findings reveal that cooperation is highly sensitive to model composition and seemingly minor linguistic changes. Models that cooperate with identical peers can shift toward defection in mixed populations. A simple uncertainty cue can increase or decrease cooperation, redistribute costs, and affect unexposed agents several hops away. Even reversing action-label order can dramatically alter behavior, with uncertainty framing sometimes amplifying this sensitivity. Together, these findings challenge the notion of cooperation as a stable model-level property, revealing how strategic behavior depends on interaction partners, network positions, and the language used to frame decisions.

1 Introduction

The paper asks how cooperation emerges and changes as LLM agents move from dyadic play into networks, where model composition, structural roles, and language may shape collective outcomes. It introduces a controlled networked IPD framework to study these factors and reports that cooperation can shift with population composition, uncertainty cues, and action-label order.

1 Introduction

  • Networked cooperation matters because decentralized local decisions become interdependent and can shape collective outcomes beyond individual capabilities.The IPD provides a controlled setting for studying cooperation and strategic adaptation under repeated conflicting incentives.Source: §1 · p. 1
  • Network topology and structural roles interact with model composition because each agent broadcasts one action to all neighbors, coupling simultaneous interactions.An agent may behave differently in a heterogeneous network than against fixed opponents or identical peers.Source: §1 · p. 1
  • A linguistic cue may reshape cooperation across a network because an exposed agent’s actions enter neighbors’ histories and can influence their later decisions.The paper asks how far effects can reach when only one agent receives a cue.Source: §1 · p. 2
  • The framework varies topology, network size, model composition, global or local uncertainty-cue exposure, and semantically equivalent action-label ordering in networked IPD.The study also evaluates cooperation rates and resulting individual and social costs.Source: §1 · p. 2
  • The study targets gaps in understanding how localized cues interact with model composition, structural roles, and action-label order, including effects on unexposed agents and collective welfare.Its questions include shifts from homogeneous baselines, cue effects under global and local exposure, and changes from swapping C and D label order.Source: §1 · p. 2
  • Cooperation among identical agents can deteriorate in heterogeneous networks, while a single uncertainty cue can affect distant agents or redistribute costs, and reversing action-label order can change outcomes.These findings indicate that cooperation depends on interaction partners and how the game is presented.Source: §1 · p. 2

2 Related work

Prior work studies LLM strategic behavior in repeated dilemmas, networked cooperation, and linguistic sensitivity. This paper connects these strands by testing how model composition and localized linguistic interventions interact in networked IPD.

LLMs as Strategic Agents

  • Prior IPD studies examine cooperation, retaliation, and forgiveness against artificial and human opponents, as well as LLM-designed strategies and behavior under varying incentives, horizons, and framing.These studies report discrepancies between game-theoretic incentives and LLM behavior.Source: §2 · p. 2

Networked Cooperation among LLM Agents

Linguistic Sensitivity and Behavioral Propagation

  • Prior work links strategic decisions to contextual framing and option ordering, and finds that minor simulation-design changes can alter cooperation (Zheng et al., 2024; Wei et al., 2024; Ye et al., 2026).The paper examines whether uncertainty-cue effects extend beyond exposed agents, affect costs, and depend on action-label ordering.Source: §2 · pp. 2–3

3 Method

The method models repeated Prisoner’s Dilemma interactions on networks, where each LLM broadcasts one action to all its neighbors. It varies agents, network structure, and prompt conditions, then evaluates cooperation and costs.

3.1 Game-Theoretic Environment

  • Each agent broadcasts one action to all neighbors, coupling the pairwise Prisoner’s Dilemma interactions across the network.The one-shot game favors defection, while mutual cooperation minimizes joint cost, creating the tension the experiments examine.Source: §3.1 · p. 3

3.2 Agent Design

  • LLM agents receive game rules, costs, and local histories, then choose an action without instructions to cooperate, defect, or optimize a specified cost.Fixed-policy opponents provide controlled references in dyadic experiments.Source: §3.2 · p. 3

3.3 Network Topologies

  • Line, cycle, and star graphs provide contrasting boundary, symmetric, and hub–leaf structural roles for comparing network behavior.The evaluated topologies support comparisons across network size and model assignments.Source: §3.3 · p. 4

3.4 Experimental Conditions

  • Experiments vary model composition and exposure to a neighbor-predictability cue while holding game payoffs, rules, and generation settings fixed.The cue is textual; it does not measure subjective uncertainty or add environmental randomness, and local exposure informs only one designated agent.Source: §3.4 · p. 4

3.5 Evaluation

  • Cooperation rates measure valid actions, while individual and social costs are computed over jointly evaluable rounds.For heterogeneous networks, cooperation is reported by model group and, when relevant, by star-graph role; fewer-than-full-round cost estimates have reduced coverage.Source: §3.5 · pp. 4–5
  • The Sensitivity Index measures the absolute cooperation-rate difference between matched C-first and D-first prompts, averaged over seeds.In networks it uses averaged node-group cooperation, so opposing individual responses may cancel and identical rates need not imply identical action trajectories.Source: §3.5 · p. 5

4 Experimental Setup

The experiments use open-weight instruction-tuned models from three families in dyadic and network settings. They vary network size, composition, prompt ordering, and uncertainty exposure, with repeated runs across a 100-round horizon.

4 Experimental Setup

  • Eight open-weight instruction-tuned models come from the Qwen2.5, LLaMA-3.1, and Mistral families.The heterogeneous pairings include small and large models from the same family.Source: §4 · p. 5
  • Line networks use 3, 4, 10, or 20 nodes, while cycle and star networks use 4, 10, or 20 nodes.Homogeneous networks are built for each model; heterogeneous configurations use four within-family small–large pairs.Source: §4 · p. 5
  • Each network configuration combines both action-label orders with baseline, global-cue, and local-cue conditions.Every configuration is repeated with three seeds; dyadic runs against stochastic opponents use ten seeds.Source: §4 · p. 5
  • Interactions last 100 rounds, and agents receive at most the ten most recent rounds in their prompts.The game horizon is not disclosed, avoiding endgame effects from a known final round.Source: §4 · p. 5

5 Results and Analysis

Cooperation shifts across dyadic and networked settings, depending on opponent, model composition, topology, and prompt wording. Uncertainty cues and label order can also change cooperation and redistribute costs, including among agents that did not receive the cue.

5.1 Dyadic behavior

  • Dyadic cooperation varies by model and opponent, with no monotonic relationship between model scale and cooperation.Qwen-14B cooperates at 1.00 against AC and TFT but 0.01 against AD; Llama-70B predominantly defects against all four tested opponents.Source: §5.1 · p. 5, Figure 2 · p. 6

5.2 Homogeneous networks.

  • Homogeneous-network cooperation varies with topology and size: Llama-70B shifts from 0.96 to 0.05 in cycles but from 0.51 to 0.91 in stars as N rises from 4 to 20.Qwen-3B consistently defects, while Qwen-14B and Mixtral remain nearly fully cooperative in configurations with available estimates.Source: §5.2 · pp. 5–6

5.3 Heterogeneous networks.

  • Heterogeneous interaction can shift both models from their homogeneous baselines, with effects depending on pairing and network configuration.In the Qwen-3B/32B pairing, Qwen-32B loses much of its cooperation while Qwen-3B remains predominantly defecting; in stars, hub assignment also matters.Source: §5.3 · p. 6

5.4 Uncertainty cues.

  • Global uncertainty cues can leave cooperation unchanged, reinforce it, or weaken it, while redistributing costs across agents.Responses vary by model, size, and topology; for example, Llama-70B’s cooperation response reverses between larger lines or cycles and stars.Source: §5.4 · p. 6
  • A cue given to one agent can affect unexposed peers several hops away, but cooperation changes and cost spillovers need not coincide.At N = 20, effects reach ten hops in mixed Llama-8B/70B networks; reach and direction depend on model composition, topology, and exposure location.Source: §5.4 · pp. 7–8

5.5 Label order

  • Reversing action-label order can dramatically change cooperation, qualify network effects, and alter uncertainty-cue outcomes.The direction and magnitude vary by model and setting; some cue effects on cooperation and social cost reverse across label orders.Source: §5.5 · p. 8

6 Conclusion

The study finds that cooperation among LLM agents is fragile across population composition, network position, uncertainty cues, and action-label order. Cooperative individuals therefore do not guarantee cooperative networks.

6 Conclusion

  • Cooperation among identical agents can deteriorate in heterogeneous networks, with behavior shifting according to population composition and network position.Source: §6 · p. 8
  • A single uncertainty cue can shift behavior in distant unexposed agents or redistribute costs without changing their cooperation.Source: §6 · p. 8
  • Reversing action-label order changes outcomes, and uncertainty framing can sometimes amplify that sensitivity.Source: §6 · p. 8

Limitations

The findings are bounded by the evaluated models and a controlled networked IPD protocol. The study also uses a narrow uncertainty intervention and evaluates behavior under specific decoding and validity conditions.

Limitations

  • The evaluation covers three canonical topologies, networks of up to 20 agents, and selected within-family model pairings.Agents broadcast one action to all neighbors, without communication or neighbor-specific decisions; interactions last 100 rounds with a ten-round history window.Source: Limitations · p. 9
  • The findings may not generalize to larger, more diverse, or differently structured multi-agent environments.Source: Limitations · p. 9
  • Uncertainty framing uses one neighbor-predictability cue, while label reversal tests only one form of prompt sensitivity; greedy decoding and repeated seeds primarily assess deterministic reproducibility.Metrics exclude invalid outputs, and cost estimates based on few jointly valid rounds provide limited evidence about full-interaction outcomes.Source: Limitations · p. 9

Ethical Considerations

The appendix describes the IPD cost framework, fixed-policy opponents, model families, and prompt variants used in the experiments. It also notes ethical considerations and limits on interpreting or generalizing the results.

Ethical Considerations

  • The study has no human participants or real-world decision-making, but its findings could inform attempts to manipulate deployed LLM agent networks; safeguards and independent robustness evaluation are warranted.The computational and environmental costs of the experiments were not quantified.Source: Ethical Considerations · p. 9

A Background in Prisoners’ Dilemma

  • The background frames the IPD as a cost-based social dilemma in which defection can benefit an agent immediately while imposing costs on neighbors and collective welfare.In repeated play, actions enter neighbors’ histories and may prompt later strategic responses.Source: A Background in Prisoners’ Dilemma · p. 10

B Fixed Strategies

  • Fixed-policy opponents cover unconditional cooperation or defection, stochastic play, and history-dependent adaptation, providing controlled reference behaviors before LLM-only networks.Source: B Fixed Strategies · p. 10

C Model Identifiers

D Prompts

  • Network prompts specify one action applied simultaneously to all neighbors, with individual cost summed across those interactions.Dyadic and network experiments use different opponent types and prompts, so their differences cannot be attributed exclusively to network structure.Source: D Prompts · pp. 11–12

D.1 Prompt for dyadic fixed-policy opponents

  • Cfirst and Dfirst reverse the presentation order of cooperation and defection while preserving action meanings and game incentives.Source: §D.1 · p. 12

D.2 Prompt for network topologies

  • The network prompt provides node and neighborhood information, recent self-actions, neighboring action histories, and an optional uncertainty cue.Invalid responses are retained in interaction histories; cooperation rates exclude them, while cost estimates use only rounds with valid actions from all agents.Source: §D.2 · p. 12

E Dyadic Results

Dyadic results distinguish cooperation against a particular opponent from the resulting individual cost, which also depends on that opponent’s responses. The figures report seed means and show that nonzero seed variation occurs only against stochastic opponents.

E Dyadic Results

  • Against GRIM, Mixtral’s cooperation rate under C_first is 0.86 while its average cost is 2.84, showing that higher cooperation need not mean lower cost because outcomes also depend on opponent responses.Source: E Dyadic Results · p. 12
  • Cooperation rates and individual costs capture distinct outcomes, since higher cooperation need not imply lower cost when opponent responses differ.Source: E Dyadic Results · pp. 12–13
  • Nonzero standard deviations across seeds occur only against stochastic opponents, consistent with variation in their sampled action sequences.Source: E Dyadic Results · p. 13, Figure 7 · p. 13, Figure 8 · p. 13

F Homogeneous Results

Homogeneous baselines vary by model, topology, network size, and action-label order; averaging across configurations can conceal these differences. Social-cost comparisons also require care because costs are not normalized by edge count.

F Homogeneous Results

  • Qwen-32B stays nearly fully cooperative in lines and cycles under both label orders, but falls to 0.02 in larger stars under D_first.The homogeneous baselines are configuration-specific; Llama-70B also shows different order effects across topologies.Source: F Homogeneous Results · p. 13
  • Order effects differ across topologies, with larger lines and cycles showing different patterns from stars.Keeping configurations separate can reveal effects hidden by averages across network structures.Source: F Homogeneous Results · p. 13
  • Social-cost differences across topologies reflect both strategic behavior and connectivity because costs are not normalized by edge count.Matched topology–size comparisons better isolate behavioral effects; partial-round estimates describe only their evaluable rounds.Source: F Homogeneous Results · p. 13

G Heterogeneous Results

Heterogeneous-network comparisons show that cooperation can depart from matched homogeneous behavior, with patterns considered under both action-label orders. The section also describes how global uncertainty-cue comparisons are made in homogeneous networks.

G Heterogeneous Results

  • Qwen-3B remains predominantly defecting, while Qwen-32B’s cooperation departs from its homogeneous reference in heterogeneous interaction.The figures compare cumulative cooperation over the first 20 of 100 rounds; those trajectories alone do not establish whether differences persist throughout all rounds.Source: G Heterogeneous Results · p. 13, Figure 9 · p. 14
  • Under D_first without the uncertainty cue, matched homogeneous-to-heterogeneous comparisons show which cooperation departures persist or change after label-order reversal.References are matched by model, topology, network size, seed, and hub/leaf role in stars.Source: G Heterogeneous Results · pp. 13–14

H.1 Homogeneous Networks

  • For homogeneous networks, the global-cue analysis compares social-cost changes against matched cue-off configurations and reports label-order sensitivity in Table 7.Figure 12 covers Cfirst social-cost changes; Table 7 reports the sensitivity measure under global cue exposure.Source: §H.1 · p. 14

H.2 Heterogeneous Networks

In heterogeneous networks, uncertainty-cue effects on cooperation and costs depend on label order and can be uneven across model groups and configurations. Localized cues also accompany responses beyond the exposed node, with outcomes varying by model and node position.

H.2 Heterogeneous Networks

  • Under Cfirst, Qwen-7B/32B cooperation increases in both model groups as social cost falls by 0.268, but individual costs move in opposite directions.Qwen-7B individual cost rises by 0.12 while Qwen-32B’s falls by 2.75; Llama-8B/70B cooperation declines and social cost rises by 0.096 under Cfirst, while under Dfirst their cooperation shifts have opposite signs and social cost falls by 0.030.Source: §H.2 · p. 14, Table 8 · p. 17
  • Collective improvements can coincide with unequal individual consequences, and label order can change the direction of cue effects.Configuration-specific comparisons also show reversals across sizes and effects that disappear under a different label order or hub assignment.Source: §H.2 · p. 14, Table 10 · p. 18

I Local Cue Exposure

  • In localized-cue runs, heterogeneous Llama-8B/70B networks show partial temporal reach in about 40% and full reach in another 41% of observed runs.These pooled proportions are descriptive; response categories do not directly establish overall spillover prevalence.Source: I Local Cue Exposure · p. 16, Figure 13 · p. 21
  • Localized cue effects vary with model and exposed-node position, and unexposed agents can have lower costs without changing cooperation.A small mean response at an exposed Qwen-7B hub can accompany substantial changes elsewhere; collective gains can coexist with higher individual costs for some agents.Source: I Local Cue Exposure · p. 16, Table 11 · p. 19
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