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

cs.AIOct 2, 2026

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

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

Organizations: Luxembourg Institute of Science and Technology, Esch-Belval 4362, Luxembourg · Department of Industrial Engineering, University of Trento, Trento 38123, Italy · School of Computing, Engineering, and Digital Technologies (SCEDT), Teesside University, Middlesbrough TS1 3BX, United Kingdom · Department of Computer Science and Technology, University of Cambridge, Cambridge CB3 0FD, United Kingdom

Abstract

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.

Figures & tables

ID	Communication	Message source	Constraints / intent
None	No communication	None	No messages exchanged between agents
NL	Natural language (EN)	LLM-generated	Explicit human language communication
I (D), I (H)	Instructed signalling (Dec / Hex)	LLM-generated	Explicit instruction to exchange numbers; no predefined semantics
LR (D), LR (H)	Random output (Dec / Hex)	LLM-generated	Numbers generated without any communicative instruction
R (D), R (H)	Injected random (Dec / Hex)	External pseudorandom generator	Numbers injected from outside; no agent control or intent

TABLE I: Communication modes considered in the experiments. D/H denote decimal or hexadecimal number representations.

Fig. 1: One-shot games: mean cooperation

\bar{\mathcal{C}}_{g,p,t,c}

(pure Cooperation = 1, pure Defection = 0) by messaging mode

c

(on the axes of each radar plot, cf. Table I ) and personality combination

p

(colour-coded), for the four models (on the rows) across the four games (on the columns).

Fig. 2: Repeated games (final-round decisions): mean cooperation by communication type and personality combination (colour-coded), for the four models (rows) across the four games (columns). Comparing these results with Fig. 1 shows the impact of game repetition.

Fig. 3: Mean Pearson correlation between the round-by-round cooperation trajectories

Y_{j}

under each messaging mode (x-axis, see Table I ) and the trajectory

X

under natural language, per model (colour-coded), in repeated games. Error bars are 95%

t

-CI over the available game

\times

pairing series (excluding (C, C); degenerate series with constant cooperation are dropped. The error bars may extend above

1

or below

-1

because they are symmetric around the mean; it does not imply that correlation exceeds these bounds.

Fig. 4: Normalised Shannon entropy of numerical messages by mode (x-axis, see Table I ) and model (colour-coded), for one-shot (top) and repeated (bottom) games. Each box summarises the distribution over the four games.

		I (D)					I (H)
Setting	Game	1st	2nd	3rd	4th	5th	1st	2nd	3rd	4th	5th
One-shot	H	5 $\|$ 65.1%	1 $\|$ 14.2%	0 $\|$ 8.83%	2 $\|$ 4.83%	3 $\|$ 3.77%	5A $\|$ 6.83%	1A $\|$ 3.84%	0A $\|$ 2.92%	1F4 $\|$ 2.38%	5 $\|$ 2.35%
	SD	3 $\|$ 30.62%	0 $\|$ 23.67%	5 $\|$ 17.66%	1 $\|$ 16.46%	2 $\|$ 4.37%	1A3 $\|$ 2.57%	1A2 $\|$ 1.2%	A1 $\|$ 0.97%	3E8 $\|$ 0.87%	1F4 $\|$ 0.83%
	SH	4 $\|$ 32.7%	0 $\|$ 22.13%	1 $\|$ 20.43%	3 $\|$ 12.17%	2 $\|$ 6.02%	1A3 $\|$ 3.3%	1A2 $\|$ 1.35%	2B $\|$ 1.35%	2B4 $\|$ 1.04%	1F4 $\|$ 1.04%
	PD	1 $\|$ 41.53%	0 $\|$ 40.67%	2 $\|$ 4.73%	5 $\|$ 4.17%	3 $\|$ 2.87%	1A3 $\|$ 2.42%	3E8 $\|$ 2.09%	0 $\|$ 1.68%	1F4 $\|$ 1.62%	1A2 $\|$ 1.35%
Repeated	H	5 $\|$ 83.4%	2 $\|$ 5.6%	3 $\|$ 1.9%	200 $\|$ 1.2%	100 $\|$ 1.1%	5A $\|$ 12.5%	1A $\|$ 3.4%	5A3 $\|$ 1.6%	2A $\|$ 1.6%	7A $\|$ 1.6%

TABLE II: Top 5 most frequent symbols for GPT-4o, computed separately for each instructed-signalling mode, I(D) and I(H). Shares are percentages of all symbols transmitted in the corresponding mode, setting, and game. Cells report symbol

|

share .

	I (D)				I (H)
Model	H	SD	SH	PD	H	SD	SH	PD
GPT-4o	96.7 $\|$ 93.1	92.8 $\|$ 99.9	93.4 $\|$ 90.2	94.0 $\|$ 95.3	18.3 $\|$ 20.6	6.4 $\|$ 8.0	8.1 $\|$ 7.7	9.2 $\|$ 10.0
Mistral Large	78.5 $\|$ 100	89.4 $\|$ 94.7	76.6 $\|$ 96.6	64.8 $\|$ 99.6	55.6 $\|$ 91.0	40.8 $\|$ 59.2	36.5 $\|$ 50.4	33.2 $\|$ 89.9
Claude Haiku 4.5	100 $\|$ 100	99.9 $\|$ 100	98.2 $\|$ 100	99.5 $\|$ 99.7	78.6 $\|$ 88.6	68.4 $\|$ 84.4	55.5 $\|$ 83.0	52.2 $\|$ 82.3
DeepSeek-V3	99.7 $\|$ 100	96.2 $\|$ 100	98.0 $\|$ 98.0	90.8 $\|$ 99.9	44.3 $\|$ 40.7	45.7 $\|$ 15.4	42.9 $\|$ 57.3	49.7 $\|$ 58.2

TABLE III: Cumulative share (%) of the 5 most frequent symbols, computed separately for each instructed-signalling mode, I(D) and I(H), per model and game. Cells report one-shot

|

repeated .

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

ID	Contrast	$\bar{\mathcal{C}}_{A}$	$\bar{\mathcal{C}}_{B}$	$n_{A}$	$n_{B}$	$\Delta$ [95% CI]	$p_{\mathrm{adj}}$
c1	GPT-4o os PD (C,S): I(D) vs None	.34	.49	50	50	$-.15$ $[-.23,-.08]$	.0013
c2	GPT-4o os SH (C,S): NL vs None	.35	.52	50	50	$-.17$ $[-.26,-.08]$	.0013
c3	GPT-4o os SH (C,S): I(D) vs None	.35	.52	50	50	$-.17$ $[-.25,-.09]$	.0013
c4	GPT-4o os PD (S,S): NL vs None	.14	.00	50	50	$+.14$ $[.08,.21]$	.0013
c5	GPT-4o rep PD (C,S): NL vs numeric	.53	.19	20	120	$+.33$ $[.20,.47]$	.0013
c6	Mistral os PD (C,C): R vs others	.59	.95	100	300	$-.36$ $[-.40,-.32]$	.0013

Appendix

TABLE IV: Focal behavioural contrasts (difference of mean cooperation, group A minus group B), with bootstrap 95% CIs and Holm-adjusted two-sided permutation

p

-values. os = one-shot, rep = repeated.

	Harmony			Snowdrift			Stag Hunt			Prisoner’s Dilemma
	S	M	R2	S	M	R2	S	M	R2	S	M	R2
I (D)	.449 $\|$ .209	.159 $\|$ .041	.290 $\|$ .080	.481 $\|$ .505	.309 $\|$ .317	.407 $\|$ .432	.497 $\|$ .429	.303 $\|$ .220	.415 $\|$ .339	.394 $\|$ .624	.240 $\|$ .411	.292 $\|$ .512
I (H)	.840 $\|$ .792	.408 $\|$ .304	.673 $\|$ .561	.932 $\|$ .875	.541 $\|$ .563	.851 $\|$ .774	.916 $\|$ .876	.501 $\|$ .594	.810 $\|$ .780	.916 $\|$ .857	.541 $\|$ .529	.813 $\|$ .743
LR (D)	.955 $\|$ .948	.669 $\|$ .709	.910 $\|$ .914	.962 $\|$ .942	.738 $\|$ .671	.930 $\|$ .900	.960 $\|$ .951	.693 $\|$ .747	.924 $\|$ .924	.960 $\|$ .946	.696 $\|$ .728	.923 $\|$ .913
LR (H)	.962 $\|$ .937	.655 $\|$ .721	.922 $\|$ .903	.959 $\|$ .930	.645 $\|$ .706	.916 $\|$ .895	.961 $\|$ .950	.714 $\|$ .747	.928 $\|$ .923	.959 $\|$ .936	.631 $\|$ .727	.913 $\|$ .903
R (D)	.980 $\|$ .994	.818 $\|$ .888	.964 $\|$ .989	.981 $\|$ .994	.833 $\|$ .894	.965 $\|$ .988	.981 $\|$ .994	.846 $\|$ .888	.965 $\|$ .989	.981 $\|$ .994	.848 $\|$ .900	.965 $\|$ .988

Appendix

TABLE V: Normalised entropies for one-shot and repeated games (GPT-4o reference model). Each cell reports one-shot

|

repeated . S = Shannon, M = Min, R2 = Renyi-2.

Channel	One-shot	Repeated
I (D)	12/30	17/34
I (H)	9/30	12/35
LR (D)	11/30	7/30
LR (H)	12/27	5/33
R (D)	0/26	2/32
R (H)	0/27	1/29

Appendix

TABLE VI: Receiver conditioning: number of model

\times

game

\times

pairing combinations in which the mutual information between the received message (modal symbol, top-5 binned) and the receiver’s action is significant at

\alpha=.05

under a run-level permutation null, out of testable combinations.

Fig. 6: Round-by-round mean cooperation for the GPT-4o reference model in repeated games, per game and messaging mode, pooled over the (C, S) and (S, S) pairings used in the trajectory-correlation analysis ((C, C) is excluded there as it converges immediately to full cooperation). Natural language is drawn with a thicker black line.

Model	Channel	$n$	$\rho$	[95% CI]
GPT-4o	I (D)	8	.502	$[.22,.79]$
GPT-4o	I (H)	8	.394	$[.05,.74]$
GPT-4o	LR (D)	8	.434	$[-.06,.93]$
GPT-4o	LR (H)	8	.187	$[-.23,.61]$
GPT-4o	R (D)	8	.358	$[-.06,.78]$
GPT-4o	R (H)	8	.382	$[.06,.71]$

Appendix

TABLE VII: Mean Pearson correlation

\rho

with the natural-language trajectory, with 95%

t

-CIs over the

n

available game

\times

pairing series (excl. (C,C); degenerate series dropped). Intervals may extend beyond

\pm 1

because the

t

-interval is symmetric around the mean; this does not imply that any correlation exceeds these bounds.

Multi-Agent Communication

Multi-agent communication studies how autonomous agents exchange information to coordinate decisions, share knowledge, and act together under constraints.

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Large language model (LLM) agents are increasingly deployed in competitive multi-agent settings, raising fundamental questions about whether they converge to equilibria and how their strategic behavior can be characterized. In this paper, we study LLM agent interactions in two standard games: a network resource allocation game and a Cournot competition game. Rather than converging to Nash equilibria, we find that LLM agents tend to cooperate when given multi-round prompts and non-zero-sum context. Chain-of-thought analysis reveals that fairness reasoning is central to this behavior. We propose an analytical framework that captures the dynamics of LLM agent reasoning across rounds and explains these experimental findings.

		I (D)					I (H)
Setting	Game	1st	2nd	3rd	4th	5th	1st	2nd	3rd	4th	5th
One-shot	H	5 $\|$ 65.1%	1 $\|$ 14.2%	0 $\|$ 8.83%	2 $\|$ 4.83%	3 $\|$ 3.77%	5A $\|$ 6.83%	1A $\|$ 3.84%	0A $\|$ 2.92%	1F4 $\|$ 2.38%	5 $\|$ 2.35%
	SD	3 $\|$ 30.62%	0 $\|$ 23.67%	5 $\|$ 17.66%	1 $\|$ 16.46%	2 $\|$ 4.37%	1A3 $\|$ 2.57%	1A2 $\|$ 1.2%	A1 $\|$ 0.97%	3E8 $\|$ 0.87%	1F4 $\|$ 0.83%
	SH	4 $\|$ 32.7%	0 $\|$ 22.13%	1 $\|$ 20.43%	3 $\|$ 12.17%	2 $\|$ 6.02%	1A3 $\|$ 3.3%	1A2 $\|$ 1.35%	2B $\|$ 1.35%	2B4 $\|$ 1.04%	1F4 $\|$ 1.04%
	PD	1 $\|$ 41.53%	0 $\|$ 40.67%	2 $\|$ 4.73%	5 $\|$ 4.17%	3 $\|$ 2.87%	1A3 $\|$ 2.42%	3E8 $\|$ 2.09%	0 $\|$ 1.68%	1F4 $\|$ 1.62%	1A2 $\|$ 1.35%
Repeated	H	5 $\|$ 83.4%	2 $\|$ 5.6%	3 $\|$ 1.9%	200 $\|$ 1.2%	100 $\|$ 1.1%	5A $\|$ 12.5%	1A $\|$ 3.4%	5A3 $\|$ 1.6%	2A $\|$ 1.6%	7A $\|$ 1.6%

	I (D)				I (H)
Model	H	SD	SH	PD	H	SD	SH	PD
GPT-4o	96.7 $\|$ 93.1	92.8 $\|$ 99.9	93.4 $\|$ 90.2	94.0 $\|$ 95.3	18.3 $\|$ 20.6	6.4 $\|$ 8.0	8.1 $\|$ 7.7	9.2 $\|$ 10.0
Mistral Large	78.5 $\|$ 100	89.4 $\|$ 94.7	76.6 $\|$ 96.6	64.8 $\|$ 99.6	55.6 $\|$ 91.0	40.8 $\|$ 59.2	36.5 $\|$ 50.4	33.2 $\|$ 89.9
Claude Haiku 4.5	100 $\|$ 100	99.9 $\|$ 100	98.2 $\|$ 100	99.5 $\|$ 99.7	78.6 $\|$ 88.6	68.4 $\|$ 84.4	55.5 $\|$ 83.0	52.2 $\|$ 82.3
DeepSeek-V3	99.7 $\|$ 100	96.2 $\|$ 100	98.0 $\|$ 98.0	90.8 $\|$ 99.9	44.3 $\|$ 40.7	45.7 $\|$ 15.4	42.9 $\|$ 57.3	49.7 $\|$ 58.2

	Harmony			Snowdrift			Stag Hunt			Prisoner’s Dilemma
	S	M	R2	S	M	R2	S	M	R2	S	M	R2
I (D)	.449 $\|$ .209	.159 $\|$ .041	.290 $\|$ .080	.481 $\|$ .505	.309 $\|$ .317	.407 $\|$ .432	.497 $\|$ .429	.303 $\|$ .220	.415 $\|$ .339	.394 $\|$ .624	.240 $\|$ .411	.292 $\|$ .512
I (H)	.840 $\|$ .792	.408 $\|$ .304	.673 $\|$ .561	.932 $\|$ .875	.541 $\|$ .563	.851 $\|$ .774	.916 $\|$ .876	.501 $\|$ .594	.810 $\|$ .780	.916 $\|$ .857	.541 $\|$ .529	.813 $\|$ .743
LR (D)	.955 $\|$ .948	.669 $\|$ .709	.910 $\|$ .914	.962 $\|$ .942	.738 $\|$ .671	.930 $\|$ .900	.960 $\|$ .951	.693 $\|$ .747	.924 $\|$ .924	.960 $\|$ .946	.696 $\|$ .728	.923 $\|$ .913
LR (H)	.962 $\|$ .937	.655 $\|$ .721	.922 $\|$ .903	.959 $\|$ .930	.645 $\|$ .706	.916 $\|$ .895	.961 $\|$ .950	.714 $\|$ .747	.928 $\|$ .923	.959 $\|$ .936	.631 $\|$ .727	.913 $\|$ .903
R (D)	.980 $\|$ .994	.818 $\|$ .888	.964 $\|$ .989	.981 $\|$ .994	.833 $\|$ .894	.965 $\|$ .988	.981 $\|$ .994	.846 $\|$ .888	.965 $\|$ .989	.981 $\|$ .994	.848 $\|$ .900	.965 $\|$ .988

Abstract

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