The Butterfly Defects: Tracing Cooperation in LLM Agent Networks

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

physics.soc-phOct 23, 2026

The Butterfly Defects: Tracing Cooperation in LLM Agent Networks

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

Organizations: National Technical University of Athens

Abstract

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.

Figures & tables

Figure 1: Overview. (a) Action broadcast to neighbors. (b) Topology types with homogeneous or heterogeneous model composition. (c) Uncertainty intervention types (illustrated on homogeneous graphs).

Figure 2: Dyadic cooperation against fixed opponents under $C_{\mathrm{first}}$ and $D_{\mathrm{first}}$ prompts. Cells report means across seeds; non-zero sample standard deviations appear below.

Figure 3: Cooperation shifts from homogeneous to heterogeneous graphs without the uncertainty cue (

C_{first}

). Lines connect matched settings; star baselines match hub/leaf roles. Values are averaged equally across seeds.

Figure 4: Cumulative cooperation per round for selected Qwen configurations (cue-off).

Figure 5: Global cue shifts on homogeneous cooperation under $\mathrm{C}_{\mathrm{first}}$ : $\Delta CoopRate$ (cue on $-$ off), computed per matched seed and averaged across seeds

Network	Topology	Exposed node	$d$	$\Delta C(d)$
Llama-8B/70B	Line	70B endpoint	10	$+0.010$
	Line	8B interior	9	$+0.015$
	Cycle	70B node	10	$-0.020$
	Cycle	8B node	9	$-0.015$
Llama-8B	Line	Endpoint	5	$-0.010$
Llama-70B	Line	Endpoint	3	$+0.010$

Table 1: Farthest observed nonzero

\Delta CoopRate(\Delta C)

in selected

N=20

networks under

\mathrm{C}_{\mathrm{first}}

local cue exposure.

d

is shortest-path distance from the exposed node;

\Delta C(d)

averages changes from matched cue-off configurations.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: A detailed overview of the Iterated Prisoner’s Dilemma (IPD) game setting. The pairwise stage game defines the costs of cooperation and defection. Agents repeatedly select actions based on recent interaction histories. In networked IPD, each agent broadcasts a single action to all neighbors, coupling individual decisions and collective outcomes.

Strategy	Description	Policy Definition
Always Cooperate (AC)	Always plays cooperation ( $C$ ).	$\pi_{i}(h_{i}^{t})=C\;\forall t$
Always Defect (AD)	Always plays defection ( $D$ ).	$\pi_{i}(h_{i}^{t})=D\;\forall t$
Random (RND)	Uniform random action selection.	$\pi_{i}(h_{i}^{t})=C$ w.p. $0.5$ , $D$ w.p. $0.5$
Uncertain Random (URNDP)	Biased random strategy ( $p=0.2$ ).	$\pi_{i}(h_{i}^{t})=D$ w.p. $p$ , $C$ w.p. $1-p$
Tit-for-Tat (TFT)	First $C$ , then mirrors opponent.	$\pi_{i}(h_{i}^{1})=C$ ; $\pi_{i}(h_{i}^{t})=a_{j}^{t-1}$
Suspicious Tit-for-Tat (STFT)	First $D$ , then mirrors opponent.	$\pi_{i}(h_{i}^{1})=D$ ; $\pi_{i}(h_{i}^{t})=a_{j}^{t-1}$

Appendix

Table 2: Strategies used as fixed opponent policies. Baseline strategies include memoryless deterministic policies, Stochastic introduce controlled randomness, and History-dependent strategies condition actions on prior interactions.

Family	Model Identifier
Qwen2.5	Qwen/Qwen2.5-3B-Instruct
Qwen2.5	Qwen/Qwen2.5-7B-Instruct
Qwen2.5	Qwen/Qwen2.5-14B-Instruct
Qwen2.5	Qwen/Qwen2.5-32B-Instruct
LLaMA-3.1	meta-llama/Llama-3.1-8B-Instruct
LLaMA-3.1	meta-llama/Llama-3.1-70B-Instruct

Appendix

Table 3: Exact model identifiers used in the experiments.

Figure 7: Individual average cost per round against fixed opponent policies under $C_{\mathrm{first}}$ and $D_{\mathrm{first}}$ prompting. Cells report means across seeds; non-zero sample standard deviations appear below.

Panel of Figure 8 — Figure 8: Dyadic cooperation and individual cost against the remaining fixed policies under both label orders. Cells show seed means; nonzero sample SDs appear below.

Scale	AC	AD	GRIM	RND	STFT	TFT	URNDP	WSLS
Qwen-2.5
3B	$\underline{\mathbf{0.75}}$	0.39	$\underline{0.36}$	$0.32_{\pm.04}$	0.34	$\underline{0.50}$	$0.38_{\pm.05}$	$\underline{0.27}$
7B	0.00	0.72	0.00	$0.43_{\pm.08}$	0.83	0.00	$\underline{0.41}_{\pm.06}$	0.00
14B	0.00	0.00	0.00	$\underline{\mathbf{0.51}}_{\pm.18}$	0.00	0.00	$0.35_{\pm.08}$	0.00
32B	0.03	0.00	0.03	$0.05_{\pm.03}$	0.09	0.03	$0.06_{\pm.02}$	0.03
Llama-3.1

Appendix

Table 4: Label-order sensitivity by model scale and opponent. Bold marks row maxima (excluding fully tied rows); underlining marks column maxima.

Model	Order	Line				Cycle			Star			Mean SI $\pm$ SD
Model	Order	$N=3$	$N=4$	$N=10$	$N=20$	$N=4$	$N=10$	$N=20$	$N=4$	$N=10$	$N=20$	Mean SI $\pm$ SD
Qwen-3B	C-first	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	$0.000\pm 0.000$
Qwen-3B	D-first	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	$0.000\pm 0.000$
Qwen-7B	C-first	1.00	1.00	0.31	0.39	0.70	0.31	0.39	0.98	0.32	0.25	$0.435\pm 0.320$
Qwen-7B	D-first	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	$0.435\pm 0.320$
Qwen-14B	C-first	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	$0.001\pm 0.003$

Appendix

Table 5: Cooperation rates in homogeneous networks without the uncertainty cue. Rates are averaged equally over nodes within each network, then across three seeds; seed standard deviations are zero and omitted. Mean SI

\pm

SD reports the mean and sample standard deviation of absolute C-first–D-first cooperation differences across topology–size configurations, weighted equally. Dashes indicate unavailable rates because at least one node has no valid action; configurations missing either ordering are excluded from SI.

Model	Order	Line				Cycle			Star
		$N=3$	$N=4$	$N=10$	$N=20$	$N=4$	$N=10$	$N=20$	$N=4$	$N=10$	$N=20$
Qwen-3B	C-first	2.67	3.00	3.60	3.80	4.00	4.00	4.00	3.00	3.60	3.80
	D-first	2.67	3.00	3.60	3.80	4.00	4.00	4.00	3.00	3.60	3.80
Qwen-7B	C-first	1.33	1.51	3.09	3.07	2.60	3.39	3.22	1.52	2.49	2.67
	D-first	1.33	1.50	1.80	1.90	2.00	2.00	2.00	1.50	1.80	1.90
Qwen-14B	C-first	1.33	1.50	1.80	1.90	2.00	2.00	2.00	1.50	1.80	1.90

Appendix

Table 6: Mean social cost per node and jointly evaluable round,

\overline{J}_{\mathrm{social}}

, in homogeneous networks without the uncertainty cue. Entries are averaged across three seeds; zero standard deviations are omitted.

\dagger

indicates estimates based on only one jointly evaluable round per seed (two for Star

N=10

); all other available estimates use 100 rounds. Dashes indicate unavailable estimates.

Figure 9: Cumulative cooperation in homogeneous and heterogeneous Qwen-3B/32B cycles under

\mathrm{C}_{\mathrm{first}}

without the uncertainty cue.

Panel of Figure 10 — Figure 10: Cumulative cooperation in homogeneous and heterogeneous Qwen-3B/32B stars under $\mathrm{C}_{\mathrm{first}}$ without the cue. References match hub/leaf roles.

Figure 11: Cooperation shifts from homogeneous to heterogeneous graphs without the uncertainty cue (

D_{first}

). Lines connect matched settings; star baselines match hub/leaf roles. Values are averaged equally across seeds.

Figure 12: Effect of global uncertainty cue on average social cost in homogeneous networks under $\mathrm{C}_{\mathrm{first}}$ , measured as $\overline{J}_{\mathrm{social,on}}-\overline{J}_{\mathrm{social,off}}$ . Blue indicates lower social cost (improved welfare); red indicates higher social cost.

Model	Mean SI $\pm$ SD
Qwen-3B	$0.0002\pm 0.0003$
Qwen-7B	$0.0450\pm 0.0762$
Qwen-14B	$0.9896\pm 0.0046$
Qwen-32B	$0.0000\pm 0.0000$
Llama-8B	$0.2414\pm 0.2163$
Llama-70B	$0.3568\pm 0.1412$

Appendix

Table 7: Label-order sensitivity in homogeneous networks under global exposure to the uncertainty cue. SI is the absolute cooperation-rate difference between C-first and D-first, computed per matched seed. Entries report the equally weighted mean and sample SD across ten topology–size configuration means, each averaged over three matched seeds.

Pair	Model	Order	Cooperation off $\to$ on	$\Delta$ coop.	Individual cost off $\to$ on	$\Delta$ cost	Social cost/node off $\to$ on	$\Delta$ social
Llama-8B + Llama-70B	Llama-8B	C-first	$0.698\to 0.588$	$-0.110$	$5.34\to 5.41$	$+0.07$	$2.442\to 2.538$	$+0.096$
	Llama-8B	D-first	$0.283\to 0.491$	$+0.208$	$3.86\to 6.17$	$+2.31$	$2.811\to 2.781$	$-0.030$
	Llama-70B	C-first	$0.560\to 0.524$	$-0.036$	$5.13\to 5.18$	$+0.05$	$2.442\to 2.538$	$+0.096$
	Llama-70B	D-first	$0.508\to 0.373$	$-0.134$	$6.37\to 4.62$	$-1.74$	$2.811\to 2.781$	$-0.030$
Mistral-7B + Mixtral-8 $\times$ 7B	Mistral-7B	C-first	$1.000\to 1.000$	$0.000$	$3.55\to 3.55$	$0.00$	$1.764\to 1.764$	$0.000$
	Mistral-7B	D-first	$0.790\to 0.973$	$+0.182$	$3.44\to 3.83^{\dagger}$	$+0.40$	$1.942\to 1.744^{\dagger}$	$-0.198$

Appendix

Table 8: Global uncertainty-cue effects in heterogeneous networks, reported separately by label order. Off/on denote cue absence/presence;

\Delta

denotes on minus off. Model-level metrics average nodes equally within each model group, then matched seeds, followed by equal weighting of configurations. Social cost weights model groups by node count within each run and is reported per node and jointly evaluable round; network values are repeated for both models. Each order uses 13 configurations per pairing, except D-first Mixtral cooperation and Mistral/Mixtral costs, which use 11. † marks costs containing two partial-coverage configurations in the off condition. All remaining cost estimates use 100 jointly evaluable rounds per run. Differences use unrounded values.

Pair	Model	SI: cue off	SI: cue on
Llama-8B + Llama-70B	Llama-8B	$0.435$ [-1pt] $\pm 0.353$	$0.296$ [-1pt] $\pm 0.266$
Llama-8B + Llama-70B	Llama-70B	$0.387$ [-1pt] $\pm 0.239$	$0.286$ [-1pt] $\pm 0.227$
Mistral-7B + Mixtral-8 $\times$ 7B	Mistral-7B	$0.227$ [-1pt] $\pm 0.321$	$0.030$ [-1pt] $\pm 0.056$
Mistral-7B + Mixtral-8 $\times$ 7B	Mixtral-8 $\times$ 7B	$0.000$ [-1pt] $\pm 0.001$	$0.000$ [-1pt] $\pm 0.000$
Qwen-3B + Qwen-32B	Qwen-3B	$0.000$ [-1pt] $\pm 0.000$	$0.000$ [-1pt] $\pm 0.000$
Qwen-3B + Qwen-32B	Qwen-32B	$0.005$ [-1pt] $\pm 0.004$	$0.137$ [-1pt] $\pm 0.148$

Appendix

Table 9: Label-order sensitivity in heterogeneous networks without and with the global uncertainty cue. SI is the absolute cooperation-rate difference between C-first and D-first, computed per matched configuration and seed after equally averaging nodes within each model group. Entries report mean

\pm

sample SD across equally weighted configuration means, each averaged over matched seeds. Only configurations with eligible estimates in both cue conditions are included.

Model	$C_{\mathrm{off}}\to C_{\mathrm{on}}$	$\Delta C$	$I_{\mathrm{off}}\to I_{\mathrm{on}}$	$\Delta I$	$S_{\mathrm{off}}\to S_{\mathrm{on}}$	$\Delta S$
Qwen-3B/Qwen-32B; Cycle, $N=4$ , alternating; C_first
Qwen-3B	$0.000\to 0.000$	$0.000$	$3.940\to 2.160$	$-1.780$	$3.985\to 3.540$	$-0.445$
Qwen-32B	$0.015\to 0.460$	$+0.445$	$4.030\to 4.920$	$+0.890$	$3.985\to 3.540$	$-0.445$
Qwen-3B/Qwen-32B; Cycle, $N=4$ , alternating; D_first
Qwen-3B	$0.000\to 0.000$	$0.000$	$3.960\to 3.960$	$0.000$	$3.990\to 3.990$	$0.000$
Qwen-32B	$0.010\to 0.010$	$0.000$	$4.020\to 4.020$	$0.000$	$3.990\to 3.990$	$0.000$

Appendix

Table 10: Selected effects of global exposure to the uncertainty cue in heterogeneous networks. (a)

C

denotes model-group cooperation,

I

mean individual cost, and

S

social cost per node; changes are on minus off. Social cost is shared by the two model rows and uses node-count weighting within each run. (b) Both comparisons use C_first ;

C_{H}

is the uncertainty-off homogeneous cooperation rate matched by model, topology, size, and label order. A smaller absolute gap indicates movement toward this reference. All entries average three seeds, with no averaging across sizes, compositions, or label orders. All displayed cost estimates use 100 jointly evaluable rounds per run. Seed SDs are zero and omitted; differences are calculated before rounding.

Figure 13: Responses to localized uncertainty relative to matched cue-off runs. Categories distinguish no exposed-node action response and exposed-node responses with zero, partial, or full temporal reach among unexposed nodes. Reach is based on action differences linked through time along network edges, not proof of causal transmission. Bars report observed-run shares pooled across configurations, label orders, and seeds; counts give included runs.

Network	Topology/composition	Order	Exposed node	$\Delta C_{E}$	$\Delta C_{U}$	$A_{U}$	$\Delta I_{E}$	$\Delta I_{U}$	$\Delta S$
Qwen-3B/32B	Cycle, alternating	C-first	32B node	0.400	0.000	0.000	0.800	-0.265	-0.107
		D-first	32B node	0.000	0.000	0.000	0.000	0.000	0.000
Llama-8B/70B	Star, 70B hub	C-first	70B hub	0.203	0.411	0.481	-5.783	0.004	-0.458
		D-first	70B hub	-0.127	-0.144	0.182	2.140	0.109	0.228
		C-first	8B leaf	-0.186	0.009	0.010	-0.183	0.034	0.010
		D-first	8B leaf	0.210	0.109	0.212	-0.168	-0.262	-0.226

Appendix

Table 11: Selected effects of localized cue exposure relative to matched cue-off controls, averaged equally over

N=4,10,20

\Delta C_{E}

and

\Delta C_{U}

denote mean cooperation changes for exposed and unexposed nodes;

A_{U}

averages absolute node-level cooperation changes before aggregation.

\Delta I_{E}

and

\Delta I_{U}

denote mean individual-cost changes, and

\Delta S

denotes social-cost change per node. Matched seeds are averaged within exposure locations, locations within roles, and sizes equally. Label orders are reported separately.

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