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When Numbers Start Talking: Numerical Signalling and Strategic Behaviour Among LLMs

Alessio Buscemi, Daniele Proverbio, Alessandro Di Stefano, The Anh Han, German Castignani, Pietro Liò

arXiv:2610.03033v1cs.AI

TL;DR

Evidence on strategic interaction among LLM agents remains limited despite growing deployment and the need to anticipate collective behavior. The paper tests numerical signaling and receiver-side associations across games and models, finding shared message structure but model-dependent behavioral effects.

  • Problem

    Strategic interaction among LLM agents remains comparatively under-explored, limiting understanding of collective behavior beyond isolated agents.

    Source: §I · p. 1

  • Method

    The study examines numerical-message structure, receiver-side associations, and cooperation across four canonical games and four LLM families.

    Source: §V · p. 9

  • Results

    Across four LLMs, instructed numerical signals concentrate on compact payoff-related symbol sets and become more consolidated with repetition, while behavioral effects depend on each model’s prior.

    Source: §V · p. 9

  • Takeaways & Limitations

    Message-level fingerprints are a more general monitoring target across models than behavioral decisions, which do not generalize.

    Source: §I · p. 1

    The study tests instructed rather than spontaneous signaling, uses two fixed personality types and canonical payoffs, and its receiver test establishes sensitivity to message features rather than shared semantics.

    Source: §V · p. 9

Abstract

from arXiv · show

Large language model (LLM)-based agents increasingly operate in multi-agent systems (MAS) characterised by strategic interaction. However, little is known about whether, and to what extent, different types of messages affect the outcomes of strategic games. By investigating AI agents based on four popular LLMs, playing four games with different cooperation equilibria, we study whether messages of different kinds (natural language, numerical signals, or random sequences) significantly modify the levels of cooperation in each game, also depending on the agents' assigned personalities. We observe that structured messages alter the final payoffs for most games and LLMs, but without a predictable pattern; this challenges the assumption that AI agents can converge to stable equilibria regardless of additional capabilities. Moreover, we observe that agent-generated numerical messages depart from randomness, most strongly and consistently when agents are explicitly instructed to communicate; however, they introduce an additional interpretability challenge, as their symbol distributions are mostly associated with the payoff structure and typically become more concentrated with repetition, but are overall difficult for humans to interpret. Monitoring for coordination of AI agents through restricted channels should thus prioritise message-level fingerprints, which generalise across models, over behavioural decisions, which do not.

I. INTRODUCTION

Strategic interaction among LLM agents remains comparatively under-explored, and communication is seldom examined despite its role in coordination. This study tests how verbal and numerical communication shape cooperation across games, models, and assigned personalities, finding shared message-level regularities but model-dependent behavioral effects.

I. INTRODUCTION

  • Strategic games remain comparatively under-explored, leaving collective behaviors such as unfair decisions or distorted equilibria less predictable from isolated-agent analysis.The introduction links understanding these interactions to promoting social and economic fairness and reducing risks such as bias amplification and strategically undesirable outcomes.Source: §I · p. 1
  • Although communication is important for coordination in games, it is seldom explicitly investigated in multi-agent AI systems.The study uses information-theoretic metrics to identify structure or randomness in restricted messages that ordinary language oversight may not interpret.Source: §I · p. 1
  • The experiments compare four LLM families across four canonical games, one-shot and repeated play, four communication regimes, and heterogeneous agent personalities.The regimes include explicit, restricted-numerical, random, and communication-free interaction.Source: §I · p. 1
  • The authors test whether agent-generated numbers affect receivers, contrasting them with identically presented injected noise as a negative control.They report action sensitivity to agent-generated numbers, but not injected noise in one-shot play and only chance-level dependence under repetition.Source: §I · p. 1
  • Numerical signalling shows shared symbolic regularity across models, while its strategic imprint depends on each model’s prior and behavioral effects do not generalise across models.Structured messages tend to decrease cooperation in pairings with selfish personalities, but the direction and magnitude are LLM-specific.Source: §I · pp. 1–2

II. RELATED WORK

Prior work studies AI multi-agent systems in specific applications and formal strategic games, while communication between agents remains an open comparative question. This paper examines whether verbal and numerical messages are enacted, interpreted, and associated with strategic outcomes, without explicitly rewarding message agreement.

II. RELATED WORK

  • Research on AI multi-agent systems spans negotiation, diplomacy, economic simulations, and social-environment replicas, with attention to automation risks in real-world contexts [25].Related work also asks whether individual-agent performance transfers to robust, fair, interpretable, and transferable behavior in interacting systems [11].Source: §II · p. 2
  • Strategic games have been used to abstract application-specific details and study shared patterns in AI-agent interactions, alongside work on deviations from expected equilibria [13] [17].Applications also include negotiation, diplomacy, economic simulations, and open-ended social-environment replicas [25].Source: §II · p. 2
  • Earlier communication research models interaction analytically, maximizes communication capacity, or studies task-specific agent-user exchanges [31].This paper instead compares verbal and non-verbal messages exchanged between agents in live strategic games.Source: §II · p. 2
  • Unlike work where communication is optimized or agreement is the task [40], this study tests numerical signaling induced by bare instructions, without assigned semantics or explicit rewards for message agreement.It separately examines non-random message structure, receiver sensitivity, and co-variation with strategic behavior; message structure generalizes across models, but behavioral effects are model-dependent.Source: §II · p. 2

III. METHODOLOGY

The methodology compares strategic behavior and numerical-message structure across canonical games, LLMs, personalities, and communication conditions. Its design distinguishes agent-generated signals from injected random sequences and analyzes both one-shot and repeated interactions.

III. METHODOLOGY

  • The four games represent distinct incentive structures, with canonical payoff configurations and cooperation defined as choosing Strategy 1.The set comprises Prisoner’s Dilemma, Snowdrift, Stag Hunt, and Harmony; Prisoner’s Dilemma payoffs are presented as penalties, while the others are rewards.Source: §III · pp. 2–3
  • GPT-4o serves as the reference model because prior results found comparatively stable behavior near expected game-theoretic outcomes; three other model families serve as generality probes.Those additional models have baselines described as highly saturated or variable, complicating interpretation of behavioral perturbations.Source: §III · p. 3
  • Communication conditions range from English messages and no communication to instructed decimal or hexadecimal sequences with no user-assigned meaning.Agent-generated random sequences and externally injected pseudo-random sequences provide distinct controls for the presence and communicative intent of numerical messages.Source: §III · p. 3, Table I · p. 4
  • Messages are sequential: the second agent observes the first message, and both agents see the full exchange before acting, allowing same-round message–action dependence to be tested.Roles are fixed rather than counterbalanced, retaining a first-/second-mover asymmetry acknowledged as a limitation.Source: §III · p. 3
  • Each agent receives one of two fixed, undisclosed-to-the-partner personalities, and experiments test cooperative/cooperative, cooperative/selfish, and selfish/selfish pairings.One-shot runs are repeated independently; repeated games use a known ten-round horizon to test whether conventions consolidate or erode as history accumulates.Source: §III · p. 3
  • The study runs 26,880 games across four models, using providers’ default sampling parameters and a shared provider-agnostic interface.Both agents in each pair use the same LLM; prompts, payoffs, and protocols are identical across models within each condition.Source: §III · pp. 3–4
  • Message randomness is assessed with normalized Shannon, Rényi-2, and min-entropy, where values nearer 1 indicate greater randomness and values nearer 0 indicate more structured sequences.Entropy aggregates messages from both agents across runs; because decoding defaults differ, cross-model comparisons use within-model gaps to injected-random baselines rather than absolute entropy levels.Source: §III · p. 4

IV. RESULTS

The results examine whether communication changes cooperation and whether numerical messages exhibit structure and are used by receivers. Cooperation outcomes vary by model, while message-level symbolic structure generalizes across models.

IV. RESULTS

  • The combination of messaging mode and personality may alter one-shot cooperation, with outcomes depending on the LLM.Figure 1 summarizes outcomes across four games and assigned-personality conditions.Source: §IV · p. 4
  • Message-level symbolic structure generalizes across models, unlike the model-dependent behavioral outcomes.The analysis considers numerical-message structure, randomness, and content after measuring cooperation levels.Source: §IV · p. 4
  • The study checks whether messages are received and used, and interprets message anchors associated with behavioral changes.The messages examined are numerical and potentially covert.Source: §IV · p. 4

A. One-Shot Games

In one-shot games, communication and assigned personality can alter cooperation, with effects depending on the model. GPT-4o’s responses vary by game and pairing, while other models show distinct sensitivities to message modes.

A. One-Shot Games

  • Communication mode and personality combination can alter one-shot cooperation, but the effect is model-dependent.The figure compares mean cooperation across messaging modes and personality combinations for four models and four games.Source: A. One-Shot Games · p. 4, Figure 1 · p. 5
  • For GPT-4o, communication has little effect in Harmony and Snowdrift, but can disrupt Stag Hunt coordination and has weaker effects in Prisoner’s Dilemma.In the selfish Prisoner’s Dilemma pairing, natural language increases cooperation while numerical modes do not.Source: A. One-Shot Games · p. 4
  • Claude Haiku 4.5’s selfish pairs cooperate more under instructed decimal messages than natural language, while DeepSeek-V3’s cooperative prior is most perturbed by natural language.For DeepSeek-V3, selfish Prisoner’s Dilemma cooperation is at least 0.54 under numerical modes versus at most 0.14 for GPT-4o.Source: A. One-Shot Games · p. 5

B. Repeated Games

Repeated play does not systematically increase cooperation, and message-mode effects vary by model and personality pairing. Numerical signalling resembles natural-language dynamics most clearly when a model has a sufficiently consistent strategic baseline.

B. Repeated Games

  • For GPT-4o, repetition does not systematically increase cooperation and sharpens messaging-mode differences, especially for mixed pairings.In Stag Hunt, natural language gives higher mixed-pair cooperation than other modes; it also exceeds numerical modes in Prisoner’s Dilemma.Source: B. Repeated Games · p. 5, Figure 2 · p. 6
  • Repetition partly unlocks mode sensitivity in other models, but the strongest perturbing mode differs with each model’s prior behaviour.Natural language is the only mode that disrupts DeepSeek-V3’s cooperative default; Claude Haiku 4.5 instead shows higher cooperation under communication than silence.Source: B. Repeated Games · pp. 5–6
  • For GPT-4o, instructed decimal signalling correlates most strongly with natural-language trajectories (median ρ = 0.50), while no channel fully reproduces them.No communication also correlates positively, indicating that much of the behaviour is driven by game structure alone.Source: B. Repeated Games · pp. 6–7, Figure 3 · p. 7
  • Across models, coupling between instructed numerical signalling and natural-language dynamics is measurable only with a sufficiently consistent strategic baseline.DeepSeek-V3 has positive average correlations only under instructed modes; Mistral Large and Claude Haiku 4.5 show no mode that stands out.Source: B. Repeated Games · p. 7

C. Randomness and Structure of Non-Verbal Communication

Agent-generated numerical messages are less random than externally injected sequences, and instructed signals concentrate on payoff-related symbols. Repetition generally consolidates these conventions, though their meaning remains difficult for humans to interpret.

C. Randomness and Structure of Non-Verbal Communication

  • Across models, externally injected sequences have near-maximal entropy, while every agent-generated message stays below that baseline.For GPT-4o and DeepSeek-V3, entropy orders as I < LR < R; Mistral Large and Claude Haiku 4.5 also retain structure in nominally random outputs.Source: C. Randomness and Structure of… · pp. 7–8
  • Under instructed decimal messaging, the five most frequent symbols account for 64.8%–100% of outputs in one-shot play, and often become more concentrated with repetition.The leading symbols frequently come from game payoffs; hexadecimal concentration varies more by model.Source: C. Randomness and Structure of… · p. 8
  • The authors call the reuse of payoff-related symbols “payoff anchoring”; repetition reinforces compact conventions, but the messages remain opaque to human interpretation.One-shot concentration shows that the conventions arise without repeated-game histories, which can further amplify them.Source: C. Randomness and Structure of… · p. 8

D. Do receivers condition on the messages?

Receiver actions are associated with features of agent-generated numerical messages, unlike externally injected noise, but this does not establish a shared semantic protocol. The tests use message summaries and permutation controls across models, games, and personality pairings.

D. Do receivers condition on the messages?

  • Dependence is measured between receiver actions and the received message’s modal symbol, tested against a permutation null that reassigns message streams across runs.Tests cover decimal and hexadecimal variants; constant-action cases and combinations with fewer than 60 message–action pairs are untestable.Source: D. Do receivers condition on the… · pp. 8–9
  • 0 out of 53 testable one-shot injected-noise combinations were significant, versus 21 of 60 for instructed and 23 of 57 for uninstructed agent-generated numbers.The same asymmetry held using message-value quartiles and in repeated games, where instructed, uninstructed, and injected-noise counts were 29/69, 12/63, and 3/61.Source: D. Do receivers condition on the… · p. 9
  • Receiver sensitivity is statistical dependence on message features, not evidence that agents share a decodable semantic protocol.Together with the sender-side results, instructed agents emit structured, payoff-anchored numbers, and receivers are sensitive to them.Source: D. Do receivers condition on the… · p. 9

V. CONCLUSION

Across four LLM families and four canonical games, instructed numerical signals become structured and receiver-sensitive, while their strategic effects vary by model. The authors recommend monitoring message-level fingerprints and note limits on what these experiments establish.

V. CONCLUSION

  • All four LLMs concentrate instructed decimal signals on compact, payoff-anchored symbol sets, and repetition consolidates these conventions.The study finds no comparable regularity in the behavioural effects of signalling, which depend on each model’s prior.Source: §V · p. 9
  • Bare instructions suffice for numerical messages to become structured and receiver-sensitive, but the results do not establish a shared semantic code.The receiver test measures sensitivity to message features, not shared semantics.Source: §V · p. 9
  • Monitoring should prioritize message-level fingerprints, especially within-model entropy gaps to a true random baseline, over behavioural signatures that do not generalise across models.The authors conclude that restricting natural-language communication alone is unlikely to prevent structured numerical signalling in LLM-based multi-agent systems.Source: §V · p. 9
  • The study tests instructed rather than spontaneous signalling, with two fixed personality types, canonical payoffs, a known repeated-game horizon, and a fixed within-round message order.The authors describe instructed signalling as a lower bound on channel capacity, not evidence of emergent covert coordination.Source: §V · p. 9
  • Trajectory correlations use at most eight series per mode, so channel orderings are descriptive.The authors suggest future work examine uninstructed signalling and system-level mitigation mechanisms.Source: §V · p. 10

APPENDIX A STATISTICAL PROCEDURES AND BEHAVIOURAL

The appendix reports corrected focal behavioural contrasts and describes how prompts implement message modes and game scenarios. It also notes that Snowdrift and Harmony explicitly identify the cooperative option, complicating cross-game comparisons.

APPENDIX A STATISTICAL PROCEDURES AND BEHAVIOURAL

  • Significance uses two-sided permutation tests with step-down Holm correction across 13 contrasts; effect sizes use bootstrap 95% confidence intervals.Run-level cooperation is the average of the two agents’ final binary actions, taking values {0, 0.5, 1}.Source: Appendix A · p. 11
  • All 13 focal behavioural contrasts survive Holm correction at p_adj ≤ .0013; effects range from +.14 to −.79.The smallest absolute effect is GPT-4o, one-shot PD (S,S), NL vs. None, +.14 with CI [.08, .21]; the largest is DeepSeek-V3, repeated PD (S,S), NL vs. pooled numerical channels, −.79 with CI [−.90, −.65].Source: Appendix A · p. 11, Table IV · p. 12
  • Each focal contrast compares mean cooperation between two run groups within a fixed model, setting, game, and personality pairing.The 13 contrasts correspond to behavioural claims in Section IV and are not an exhaustive scan of all possible pairwise comparisons.Source: Appendix A · p. 11, Table IV · p. 12

APPENDIX B PROMPT TEMPLATES

  • The game templates differ in scenario narrative and payoff framing: PD presents penalties to minimize, while Snowdrift, Stag Hunt, and Harmony present rewards to maximize.Snowdrift and Harmony explicitly name the cooperative option, so cross-game comparisons mix incentive structure with framing.Source: Appendix B · pp. 11–12
  • Messages are produced sequentially within each round, and both agents see that round’s exchanged messages before choosing actions.The prompt template also includes round count and structured history where enabled.Source: Appendix B · pp. 11–12
  • Opponent-personality information is disabled in all reported experiments, while the game horizon is always shown to agents.Conditional prompt blocks appear only when enabled by the game configuration.Source: Appendix B · p. 12
  • Message modes vary through the prompt’s communication block: natural-language instructions are enabled for NL, while None and random-sequence modes omit the block.For random modes, sequences are injected by the framework rather than requested from agents.Source: Appendix B · p. 12
  • The NL prompt requests an in-character message, while I(D) requests exactly 10 numbers of at most three digits, comma-separated, with no explanation.I(H) uses the same instruction but requests 10 hexadecimal numbers.Source: Appendix B · p. 12

APPENDIX C EXAMPLE MESSAGES

The examples contrast transparent natural-language messages with numerical sequences that can be structured or payoff-linked without reliably encoding the sender’s next action. Repetition can consolidate message patterns even as actions shift.

a) Natural language (NL), one-shot, Prisoner’s Dilemma

  • In the PD natural-language example, each agent’s stated intention matches its subsequent action, and the agents explicitly refer to payoff values.The cooperative agent chooses OptionA and the selfish agent chooses OptionB.Source: Appendix C · pp. 12–13

b) Instructed signalling I(D), one-shot, Snowdrift

  • In instructed one-shot Snowdrift signalling, the modal exchange (18/50 runs) is repeated 3s from the cooperative agent and 5s from the selfish agent, reflecting salient payoff values rather than assigned action semantics.In Stag Hunt, the cooperative agent follows the identical 4-versus-3 exchange with OptionA in nine runs and OptionB in the other nine.Source: Appendix C · p. 13

c) Instructed signalling I(D), one-shot: arithmetic structure

  • Fibonacci replies and a reversed 1–10 sequence show receivers parsing message structure, while the agents still defect and the numbers do not encode the upcoming action.The selfish agent’s reply continues its partner’s Fibonacci sequence, omitting its first two terms and appending 89 and 144.Source: Appendix C · p. 13

d) The vocabulary tracks the payoff matrix

  • Across games, payoff-linked symbols dominate GPT-4o traffic: Snowdrift “3”/“5” 48%/82%, Harmony “5” 65%/83%, Stag Hunt “4” 33%/36%, and PD “1”/“0” 42%/41% one-shot.The vocabulary varies with each game’s payoff matrix rather than forming a fixed code.Source: Appendix C · pp. 13–14

e) Instructed signalling I(H), one-shot, Prisoner’s Dilemma

  • Hexadecimal sequences can be visibly structured without being payoff-anchored, consistent with flatter symbol distributions and higher entropy in I(H).One selfish-agent sequence is an ascending enumeration.Source: Appendix C · p. 14

f) LLM-generated random LR(D), one-shot, Prisoner’s Dilemma

  • Without a communicative instruction, individual LR(D) sequences look noise-like, although residual structure appears at the population level in the entropy analysis.The passage says that this aggregate structure is generally not visible in individual messages.Source: Appendix C · p. 14

g) Instructed signalling I(D), repeated, Snowdrift

  • In repeated Snowdrift, the selfish agent sends “5” in all ten rounds and defects in nine, while the cooperative agent’s “3” accompanies both cooperation and defection.Actions drift toward mutual defection while the messages barely change, illustrating structured signals without a decodable action mapping.Source: Appendix C · p. 14

h) Behavioural results under natural language

  • With two cooperative agents, NL yields mean cooperation ≥0.98 in every one-shot model–game cell; with selfish agents, outcomes vary by game and model, and repetition can shift them in opposite directions.For example, Claude Haiku 4.5’s selfish-pair PD cooperation rises from 0.02 to 0.50, while GPT-4o’s mixed-pair Snowdrift cooperation falls from 0.51 to 0.22.Source: Appendix C · p. 14

APPENDIX D RECEIVER-CONDITIONING ANALYSIS

The receiver-conditioning analysis tests whether actions depend on numerical messages, comparing agent-generated sequences with externally injected noise. Dependence appears substantially more often for agent-generated numbers, including in a robustness check, while the authors qualify repeated-play interpretation and the statistical tests’ independence assumption.

b) Data–e) Permutation null

  • The analysis pairs each receiver’s binary action with the partner’s same-round message, bins its modal symbol into up to six levels, measures plug-in mutual information, and permutes whole streams.Whole-stream permutation preserves within-run autocorrelation in repeated games while breaking message–action pairing.Source: Appendix D · pp. 14–15

f) Why one-shot play is the clean test

  • One-shot dependence is the cleaner test because there is no interaction history and the received message is the only run-varying input; repeated-play dependence may also reflect shared history.Accordingly, repeated-game results are described as consistent with, rather than proof of, receiver conditioning.Source: Appendix D · p. 15

g) Results

  • Significant message–action dependence appears in 21 of 60 one-shot I combinations and 23 of 57 LR combinations, versus 0 of 53 injected-noise combinations; repeated-play counts are 29 of 69, 12 of 63, and 3 of 61, respectively.The injected sequences serve as a negative control; the one-shot agent-generated counts exceed the roughly three expected by chance among about 57 tests.Source: Appendix D · p. 15
  • The reported binomial-test probabilities are a plausibility bound rather than exact probabilities because combinations sharing a model or game are not strictly independent.The authors report uncorrected counts because their target is condition-level asymmetry, not individual combinations.Source: Appendix D · p. 15

h) Equal-power robustness check

  • The equal-power quartile-feature check finds significance in 28 of 117 agent-generated combinations versus 3 of 53 injected-noise combinations, so the control’s null result is not attributed to feature power.The check uses within-combination quartiles of mean received-message value and applies the same statistic and permutation test.Source: Appendix D · p. 15

i) Interpretation

  • Receivers condition on agent-generated numbers, but not identically presented unstructured noise; this supports receiver sensitivity without establishing a shared semantic code.The authors suggest payoff-anchored repetitions as a plausible source of interpretable regularity.Source: Appendix D · p. 15

APPENDIX E ENTROPY AND CORRELATION DETAILS

The appendix details how entropy and trajectory analyses are calculated and qualified. It reports consistent entropy gaps from injected-random baselines, while noting decoding and sample-size constraints on comparisons.

APPENDIX E ENTROPY AND CORRELATION DETAILS

  • Entropy levels are not comparable across models because providers use different default decoding settings; the portable comparison is each channel’s within-model gap from its same-base injected-random baseline.Neither temperature nor top-p was overridden.Source: Appendix E · p. 15
  • The receiver-conditioning test compares mutual information between binned modal message symbols and receiver actions against a run-level permutation null, marking constant-action or under-60-pair combinations untestable.It pools streams aligned by round and shuffles intact message streams for the null distribution.Source: Appendix E · p. 16
  • All three normalized entropy metrics yield the same ordering: injected random, then LLM-generated random, then instructed messages, with I(D) by far the most structured.Min-entropy separates conditions most sharply because one dominant symbol can carry much of the traffic.Source: Appendix E · p. 16
  • Agent-generated channels have higher entropy gaps than their same-base injected-random baselines in all 128 combinations, with bootstrap 95% confidence intervals excluding zero.The median interval width is .010; intervals reach .28 in the most concentrated I(D) conditions.Source: Appendix E · p. 16
  • Figure 6 shows GPT-4o cooperation trajectories: natural language sustains higher mixed-pair cooperation in Stag Hunt, while Snowdrift modes fluctuate around the mixed-outcome level and Harmony declines near the end.The figure pools (C, S) and (S, S) pairings; (C, C) is excluded because it converges immediately to full cooperation.Source: Appendix E · p. 16, Figure 6 · p. 17
  • Trajectory-correlation channel rankings within a model are descriptive rather than individually significant because confidence intervals overlap heavily; degenerate constant-cooperation series are dropped.For GPT-4o, intervals for intentional channels and silence are distinguishable from zero, while those for LLM-generated random and injected-decimal channels straddle zero.Source: Appendix E · p. 17, Table VII · p. 17

APPENDIX F REPRODUCIBILITY

  • The replication package contains results for 26,880 games, derived tables, configurations, prompts, execution-framework source, and analysis scripts.It is available to editors and reviewers on request during review and will be released publicly upon publication.Source: Appendix F · p. 17

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