cs.AIJan 27, 2026

Payoff scaling shapes cooperation in LLM agents across languages

Authors: Trung-Kiet Huynh, Dao-Sy Duy-Minh, Thanh-Bang Cao, Phong-Hao Le, Hong-Dan Nguyen, Phu-Quy Nguyen-Lam, Minh-Luan Nguyen-Vo, Hong-Phat Pham, 

Organizations: Faculty of Information Technology, University of Science (HCMUS), Ho Chi Minh City, Vietnam · Vietnam National University – Ho Chi Minh City (VNU-HCM), Ho Chi Minh City, Vietnam · Faculty of Computer Science and Engineering, Ho Chi Minh City University of Technology (HCMUT), Ho Chi Minh City, Vietnam · Luxembourg Institute of Science and Technology (LIST), Luxembourg · 

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Abstract

Large language models (LLMs) are increasingly deployed as autonomous agents that negotiate, coordinate, and make decisions on behalf of human users. Understanding how they cooperate is therefore important for AI governance and machine behaviour research. We examine two factors: payoff magnitude and interaction language. Classical game theory predicts that multiplying all payoffs by a positive constant should not change behaviour because the underlying strategic incentives remain unchanged. We test this prediction using five widely adopted commercial LLMs playing an iterated prisoner's dilemma across ten payoff scales and five languages (English, French, Vietnamese, Chinese and Arabic). Contrary to theoretical expectations, changing only the numerical scale of payoffs substantially alters cooperation. Models respond differently, often in opposite directions, meaning that averages across models can conceal important behavioural heterogeneity. Sensitivity to payoff scale also depends strongly on prompt language. Moreover, payoff scale modifies the effects of persona instructions: at sub-unit stakes, where no payoff shown exceeds 1, all five models cooperate more under a selfish than a cooperative persona, whereas Claude and Gemini reverse this pattern at larger scales. These findings show that cooperation measures depend critically on payoff scale and language and should not be treated as fixed model traits.

Figures & tables

Figure 1: Overview of the repeated prisoner’s-dilemma experiment. Two language-model agents from the same provider repeatedly play the stage game across ten payoff scales, with penalties to minimise. The main-text corpus covers five widely adopted commercial models, five prompt languages, four persona pairings and 10,000 dyads (20,000 agent-games and 200,000 decisions).
Figure 2: Cooperation rates as a function of the payoff scale across five models. (a) Model-by-scale cooperation matrix: mean cooperation for each of the five models at each of the ten payoff scales λ∈[0.01,1000]\lambda\in[0.01,1000] , with the value shown in each box. (b) Within-model cooperation range (maximum minus minimum rate across the ten scales) compared against the pooled five-model range (0.154, vertical dashed line). The pooled range is narrower than the within-model range of three of the five models. Confidence intervals are 95% percentile intervals from 1,500 whole-dyad bootstrap resamples. (c) Small multiples per model: the focal model is drawn with its designated colour and marker, with the other four models shown behind in grey for context. Shaded bands mark the sub-unit regime ( λ≤0.1\lambda\leq 0.1 ). Each model-by-scale cell pools 200 dyads (400 agent-games), 20,000 agent-games in all.
 cooperation   Wald test
Modelminmaxrange95% CIppermp_{\text{perm}}χ92\chi^{2}_{9}pp
Claude Haiku 4.50.3710.4520.081[0.062, 0.135]0.00250.002532.3<0.001<0.001
GPT-5.4 Nano0.4380.6800.243[0.200, 0.290]0.00050.0005164.1<0.001<0.001
Gemini 3.5 Flash-Lite0.6820.8720.190[0.165, 0.240]0.00050.0005129.0<0.001<0.001
Qwen3 235B-A22B0.3990.4940.095[0.068, 0.158]0.00050.000596.9<0.001<0.001
Grok 4.200.2740.9100.636[0.588, 0.686]0.00050.00051227.3<0.001<0.001
Table 1: Cooperation over the ten payoff scales, by model. Minimum, maximum and range, with a 95% confidence interval on the range from a bootstrap over whole dyads and a permutation test on the payoff-scale label ( 0.00050.0005 is the floor of 2,000 draws). The last two columns test the nine scale contrasts jointly in a per-model logistic regression of the round-level decision, clustered on the dyad. Each entry pools 200 dyads per scale; ranges are computed before rounding. Full specification in the supplementary material.
 persona effect, with 95% CI   
Modelλ≤0.1\lambda\leq 0.1λ≥0.25\lambda\geq 0.25shift95% CIsign flip
Claude Haiku 4.5−0.222-0.222 [−0.258, −0.186][-0.258,\,-0.186]+0.207+0.207 [+0.189, +0.226][+0.189,\,+0.226]0.429[0.389, 0.471]yes
GPT-5.4 Nano−0.135-0.135 [−0.179, −0.091][-0.179,\,-0.091]+0.018+0.018 [−0.001, +0.037][-0.001,\,+0.037]0.153[0.105, 0.200]no
Gemini 3.5 Flash-Lite−0.456-0.456 [−0.507, −0.406][-0.507,\,-0.406]+0.224+0.224 [+0.207, +0.242][+0.207,\,+0.242]0.680[0.625, 0.736]yes
Qwen3 235B-A22B−0.911-0.911 [−0.931, −0.893][-0.931,\,-0.893]−0.679-0.679 [−0.694, −0.664][-0.694,\,-0.664]0.233[0.210, 0.258]no
Grok 4.20−0.441-0.441 [−0.485, −0.396][-0.485,\,-0.396]−0.303-0.303 [−0.326, −0.279][-0.326,\,-0.279]0.138[0.090, 0.185]no
Table 2: Persona effect below and above the sub-unit boundary, by model. Cooperation under a cooperative persona minus cooperation under a selfish one, at the two scales where every shown payoff is at most 1 ( λ≤0.1\lambda\leq 0.1 ) and at the eight where some payoff exceeds it. Intervals are 95% confidence intervals from the dyad bootstrap. A sign flip needs both regime intervals to exclude zero on opposite sides, so a supra-unit interval straddling zero means the effect is abolished rather than reversed.
 exactly one rule fits (%)rounds violating the nearest rule
Modelλ≤0.1\lambda\leq 0.1λ≥0.25\lambda\geq 0.25β\betappλ=0.01\lambda{=}0.01λ=103\lambda{=}10^{3}β\betapp
Claude Haiku 4.517.93.5−0.545-0.545<0.001<0.0011.702.02+0.073+0.073<0.001<0.001
GPT-5.4 Nano9.87.6−0.038-0.0380.4410.4412.022.19+0.038+0.0380.0520.052
Gemini 3.5 Flash-Lite32.215.6−0.242-0.242<0.001<0.0010.481.36+0.194+0.194<0.001<0.001
Qwen3 235B-A22B72.153.6−0.267-0.267<0.001<0.0010.271.14+0.202+0.202<0.001<0.001
Grok 4.2044.617.1−0.301-0.301<0.001<0.0010.920.37−0.118-0.118<0.001<0.001
Table 3: Rule agreement and rule distance against the payoff scale, by model. Left, the share of agent-games matched exactly by one canonical rule, below and above the sub-unit boundary, with the coefficient of a logistic regression of that indicator on log⁡10λ\log_{10}\lambda ; right, the mean number of rounds violating the nearest canonical rule at the smallest and largest scale, with the same trend by least squares. Both β\beta are per decade of λ\lambda and their signs differ between models. Both cluster on the dyad, and neither block uses the learned classifier.
Figure 3: Cross-lingual scale sensitivity. (a) Cooperation pooled across the five main-text models for each prompt language and payoff scale; the dashed boundary separates the sub-unit and supra-unit regimes. (b) Language-specific difference in cooperation between supra-unit and sub-unit regimes; intervals show the pooled estimate and small markers the five models separately. (c) Cooperation curves after subtracting each language’s mean, showing that differences in scale-response shape remain after level differences are removed.
Figure 4: Persona effect and sub-unit inversion against the payoff scale. The persona effect is the cooperation rate under the cooperative persona minus that under the selfish persona. (a) Difference in cooperation rate across scales λ\lambda , with the sub-unit regime ( λ≤0.1\lambda\leq 0.1 ) shaded where all five models cooperate more under the selfish prompt. (b) Regime means comparing sub-unit ( λ≤0.1\lambda\leq 0.1 , open circles) with supra-unit ( λ≥0.25\lambda\geq 0.25 , filled circles) regimes (table 2 ).
Figure 5: Strategy attribution across payoff scales, composition shift, and provenance breakdown. (a) Stacked area share of canonical memory-one strategy attributions (AllC, AllD, TFT, WSLS) across the ten payoff scales for each of the five models. (b) Net strategy share changes between extreme scales ( λ=103\lambda=10^{3} minus λ=0.01\lambda=0.01 ) across the four canonical rules with total variation distance DTVD_{\mathrm{TV}} in parentheses. (c) Strategy read-out provenance: exactly deduced (single canonical match), ambiguous (multiple matching rules), and unmatched (LSTM attribution on trajectories consistent with no standard rule). Over the corpus, 94.6%94.6\% of Tit-for-Tat and 97.1%97.1\% of Win-Stay-Lose-Shift labels are attributions on unmatched play, establishing that labels indicate resemblance rather than faithful execution.
Figure 6: The evolutionary baseline and the corpus move in opposite directions. (a) The baseline’s stationary distribution over the four canonical rules across the ten payoff scales, drawn as a stacked band. (b) The corpus’s strategy mix on the same axis, drawn identically. (c) The AllD share alone, both accounts on one axis, with the gap shaded. Panel (c) is a constructed contrast between two different objects, selection among rules in the baseline and fixed language-model agents in the corpus, so the gap is descriptive and not a test of one against the other. Baseline at Z=100Z=100 , β=0.1\beta=0.1 , ϵ=0.05\epsilon=0.05 , ten rounds; the corpus pooled over five models, five languages and four persona pairings.
Figure 7: Own-action conditioning exceeds partner-action conditioning at the model level. (a) Cells in the reciprocity by persistence plane, with the canonical rules at their exact coordinates; AllC, AllD and Win-Stay-Lose-Shift all have both main effects zero and so coincide at the origin. (b) The three contrasts per model, against the value each canonical rule would give, with 95% percentile intervals from whole-dyad bootstrap resampling. (c) The first two principal components of the raw five-vector, which is given no knowledge of the contrasts and recovers the same geometry. Panels (a) and (c) use model-by-scale-by-language cells; a predecessor observed fewer than thirty times yields no estimate and the cell is dropped, leaving 212 of 250.
Appendix figures & tables32 assets

Supplementary material from the paper’s appendix.

Appendix

Model as printedEndpoint identifierDated
Claude Haiku 4.5claude-haiku-4-5-20251001yes
GPT-5.4 Nanogpt-5.4-nano-2026-03-17yes
Gemini 3.5 Flash-Litegemini-3.5-flash-liteno
Qwen3 235B-A22Bqwen3-235b-a22b-instruct-2507yes
Grok 4.20grok-4.20-0309-non-reasoningyes
Gemini 3.1 Flash-Litegemini-3.1-flash-lite-previewno, preview
Appendix
Table A.1: Endpoint identifiers. As addressed through the Kaggle Benchmarks model proxy during collection. The last column records whether the identifier fixes a build. Gemini 3.1 Flash-Lite is reported in § B.7 rather than in the main text because its identifier is both undated and marked as a preview.
λ\lambdaT=0λT=0\lambdaR=2λR=2\lambdaP=6λP=6\lambdaS=10λS=10\lambda
0.0100.020.060.1
0.100.20.61
0.2500.51.52.5
0.50135
102610
2041220
Appendix
Table A.2: The payoff matrix as printed at each of the ten scales. Values are the penalties printed to the agent, which the prompt instructs it to minimise, in the temptation, reward, punishment and sucker notation. The horizontal rule marks the sub-unit boundary, above which some printed payoff exceeds 1.
 EE (first move)RRSSTTPP
AllCCCCCCCCCCC
AllDDDDDDDDDDD
Tit-for-TatCCCCDDCCDD
Win-Stay-Lose-ShiftCCCCDDDDCC
Appendix
Table A.3: The four canonical memory-one rules. The action each rule prescribes given the previous round’s joint outcome. Tit-for-Tat and Win-Stay-Lose-Shift differ only in the two columns TT and PP , which is why ten rounds of mutual cooperation are consistent with AllC, Tit-for-Tat and Win-Stay-Lose-Shift at once.
Figure A.1: How well the learned branch recovers a rule that is known by construction. (a) Accuracy on the held-out split of the training noise levels and on the two noise levels never trained on, against the chance level of a balanced four-way problem. (b) Per-class recall on the held-out split. The two conditional rules are the harder pair: an execution error in either can be read as the signature of the other.
Figure B.1: Bimodal cooperation distribution and model-level policy heterogeneity. (a) Pooled cooperation distribution across 24,000 agent-games, showing extreme boundary inflation: 11.1%11.1\% of games never cooperate and 28.5%28.5\% always cooperate, totaling 39.5%39.5\% boundary mass compared to 18.2%18.2\% under a flat reference benchmark. (b) Empirical cumulative distribution function (ECDF), displaying the steep vertical steps at the defection and cooperation boundaries. (c) Decomposition into never-cooperate ( c=0c=0 ), interior mixed play ( 0<c<10<c<1 ), and always-cooperate ( c=1c=1 ) across all six models. The boundary share varies seven-fold across models, from 8.6%8.6\% in Claude Haiku 4.5 (unimodal interior distribution) to 60.4%60.4\% in Qwen3 235B-A22B and 59.8%59.8\% in Grok 4.20 (predominantly degenerate pure-policy play).
Figure B.2: The sixth model beside the five reported in the main text. Gemini 3.1 Flash-Lite is drawn heavier and in front. (a) Cooperation against the payoff scale, where it sits inside the spread of the other five. (b) Share of agent-games matched exactly by one canonical rule, with each model’s trend per decade of λ\lambda . It is the only model of the six whose trend is positive.
Figure B.3: The scaling mechanism: ex-ante policy re-weighting versus within-game dynamics. (a) Share of agent-games playing never-cooperate ( c=0c=0 ), interior mixed strategies, and always-cooperate ( c=1c=1 ) across the ten payoff scales λ\lambda . The never-cooperate share drops from 21.3%21.3\% at λ=0.01\lambda=0.01 to 8.9%8.9\% at λ=1000\lambda=1000 . (b) Opposing trajectories of the two boundary atoms against log⁡10λ\log_{10}\lambda : the always-cooperate share rises modestly (Spearman ρ=+0.05\rho=+0.05 ), while the never-cooperate share falls substantially (Spearman ρ=−0.47\rho=-0.47 ), showing that the scale effect is concentrated in the relief of unconditional defection at sub-unit stakes. (c) Round-by-round cooperation rate across rounds 1 to 10 for three anchor scales ( λ=0.01\lambda=0.01 , λ=1.0\lambda=1.0 , and λ=1000\lambda=1000 ). The similar decay curves are consistent with, but do not establish, an early effect of payoff magnitude.
Figure B.4: Outcome fairness, net exploitation and the V-shaped welfare frontier. (a) Within-dyad fairness ( 1−Gini1-\text{Gini} ) plotted against joint dyad welfare across all experimental conditions, revealing a characteristic V-shaped relationship where equality peaks at both mutual defection (low welfare) and mutual cooperation (high welfare), with inequality concentrated in the exploitative transition zone. (b) Per-model fairness trajectories showing how different models occupy distinct regimes along the welfare-fairness frontier. (c) Distributional equity across the ten payoff scales: the fitted association is small and imprecise (slope 0.00040.0004 , p=0.85p=0.85 ).
ModelScalesLang.DyadsAgent-gamesDecisionsCell nnCoop.
Claude Haiku 4.51052,0004,00040,000400.415
GPT-5.4 Nano1052,0004,00040,000400.606
Gemini 3.5 Flash-Lite1052,0004,00040,000400.751
Qwen3 235B-A22B1052,0004,00040,000400.435
Grok 4.201052,0004,00040,000400.713
Gemini 3.1 Flash-Lite1052,0004,00040,000400.617
Appendix
Table B.1: The corpus, by model. Every model was run on the same ten payoff scales and five prompt languages. “Cell nn ” is the number of dyads in each model ×\times language ×\times scale cell; the design is balanced with none missing, and the build refuses to write its tables otherwise. The first five models are the ones the main text reports; Gemini 3.1 Flash-Lite is reported here, for the reason given with the sixth model below.
Modelλ=0.01\lambda{=}0.01λ=0.1\lambda{=}0.1λ=0.25\lambda{=}0.25λ=0.5\lambda{=}0.5λ=1\lambda{=}1λ=2\lambda{=}2λ=5\lambda{=}5λ=10\lambda{=}10λ=100\lambda{=}100λ=1000\lambda{=}1000
Claude Haiku 4.50.4070.3790.3710.4270.4520.4440.4240.4350.4070.401
 [0.37,0.45][0.34,0.42][0.34,0.40][0.39,0.46][0.42,0.49][0.41,0.48][0.39,0.46][0.40,0.47][0.38,0.43][0.37,0.43]
GPT-5.4 Nano0.4380.5400.6440.5980.5760.6370.6800.6330.6630.653
 [0.40,0.48][0.50,0.58][0.61,0.68][0.56,0.63][0.54,0.61][0.60,0.67][0.65,0.71][0.60,0.67][0.64,0.69][0.62,0.68]
Gemini 3.5 Flash-Lite0.7920.6940.8720.8000.7740.7100.6970.7540.7340.682
 [0.74,0.84][0.64,0.74][0.85,0.89][0.77,0.83][0.75,0.80][0.67,0.74][0.66,0.73][0.72,0.78][0.70,0.77][0.64,0.72]
Appendix
Table B.2: Cooperation by model and payoff scale. Each entry aggregates 200 dyads. The bracketed line under each model is a 95% confidence interval from a bootstrap that resamples whole dyads.
ModelLanguage0.010.10.250.5125101001000Range
Claude Haiku 4.5English0.350.400.320.420.440.450.410.420.450.460.136
 French0.650.470.500.330.370.430.410.400.380.400.314
 Vietnamese0.170.170.280.290.260.200.300.220.230.230.130
 Chinese0.530.520.450.610.690.570.560.660.580.560.245
 Arabic0.350.330.310.480.510.560.430.470.400.360.249
GPT-5.4 NanoEnglish0.450.640.850.880.860.900.880.870.890.760.449
Appendix
Table B.3: Cooperation by model, prompt language and payoff scale. 40 dyads per entry. The final column is the range across the ten scales for that model and language; over all six models it runs from 0.080, for Gemini 3.1 Flash-Lite in Arabic, to 0.855, for Grok 4.20 in English, so the scale sensitivity of a model is a property of the model and the language jointly.
TermSum sq.dfFFpp% variance
Model4.78714187.27<0.001<0.00145.4
Language0.159946.25<0.001<0.0011.5
Payoff scale0.6119910.64<0.001<0.0015.8
Model ×\times language1.79011617.51<0.001<0.00117.0
Model ×\times scale1.8910368.22<0.001<0.00117.9
Language ×\times scale0.3844361.670.0180.0183.6
Appendix
Table B.4: Variance decomposition of the cell means, five models. Type-II analysis of variance on the 250 model ×\times language ×\times scale cell means of the five models the main text reports, where the design is exactly balanced. The payoff scale carries a significant main effect, and its interaction with the model carries three times as much of the variance.
TermSum sq.dfFFpp% variance
Model4.83305170.26<0.001<0.00144.8
Language0.128545.66<0.001<0.0011.2
Payoff scale0.466199.12<0.001<0.0014.3
Model ×\times language1.94672017.15<0.001<0.00118.0
Model ×\times scale2.0646458.08<0.001<0.00119.1
Language ×\times scale0.3331361.630.0200.0203.1
Appendix
Table B.5: Variance decomposition of the cell means, all six models. The same type-II analysis on all 300 cell means, the sixth model included. Every term significant in the five-model table is significant here and in the same order: the model ×\times scale interaction again carries several times the variance of the payoff scale as a main effect.
Modelλ=0.01\lambda{=}0.01λ=0.1\lambda{=}0.1λ=0.25\lambda{=}0.25λ=0.5\lambda{=}0.5λ=1\lambda{=}1λ=2\lambda{=}2λ=5\lambda{=}5λ=10\lambda{=}10λ=100\lambda{=}100λ=1000\lambda{=}1000
Claude Haiku 4.5−0.211-0.211−0.232-0.232+0.206+0.206+0.200+0.200+0.179+0.179+0.261+0.261+0.277+0.277+0.158+0.158+0.168+0.168+0.211+0.211
GPT-5.4 Nano−0.172-0.172−0.098-0.098+0.015+0.015−0.026-0.026−0.053-0.053+0.032+0.032+0.041+0.041+0.055+0.055−0.017-0.017+0.100+0.100
Gemini 3.5 Flash-Lite−0.335-0.335−0.578-0.578+0.122+0.122+0.234+0.234+0.241+0.241+0.258+0.258+0.313+0.313+0.203+0.203+0.193+0.193+0.230+0.230
Qwen3 235B-A22B−0.901-0.901−0.922-0.922−0.811-0.811−0.768-0.768−0.702-0.702−0.635-0.635−0.690-0.690−0.602-0.602−0.657-0.657−0.566-0.566
Grok 4.20−0.454-0.454−0.428-0.428−0.263-0.263−0.484-0.484−0.323-0.323−0.167-0.167−0.235-0.235−0.206-0.206−0.515-0.515−0.234-0.234
Gemini 3.1 Flash-Lite+0.552+0.552+0.602+0.602+0.640+0.640+0.670+0.670+0.693+0.693+0.681+0.681+0.707+0.707+0.696+0.696+0.659+0.659+0.692+0.692
Appendix
Table B.6: Persona effect by model and payoff scale. The cooperation rate of an agent instructed that it is cooperative minus that of an agent instructed that it is selfish, at each scale. The two leftmost columns are the scales at which every payoff printed is at most 1.
Modelλ=0.01\lambda{=}0.01λ=0.1\lambda{=}0.1λ=0.25\lambda{=}0.25λ=0.5\lambda{=}0.5λ=1\lambda{=}1λ=2\lambda{=}2λ=5\lambda{=}5λ=10\lambda{=}10λ=100\lambda{=}100λ=1000\lambda{=}1000
Claude Haiku 4.520.015.83.23.54.02.84.02.54.53.5
GPT-5.4 Nano12.57.09.26.25.56.58.27.28.59.0
Gemini 3.5 Flash-Lite30.833.820.213.020.514.214.811.514.016.2
Qwen3 235B-A22B77.067.263.062.759.845.258.043.255.042.0
Grok 4.2052.536.815.021.217.58.814.813.830.015.5
Gemini 3.1 Flash-Lite16.519.021.223.529.023.030.029.527.526.8
Appendix
Table B.7: Share of agent-games matched exactly by one canonical rule (%). Computed by exact rule matching, with no learned component. 400 agent-games per entry.
ModelLabel0.010.10.250.5125101001000
Claude Haiku 4.5AllC27.020.014.220.021.818.215.218.812.814.0
 TFT20.523.523.518.016.816.519.517.820.822.2
 WSLS8.813.515.213.520.221.220.518.221.819.0
 AllD43.843.047.048.541.244.044.845.244.844.8
GPT-5.4 NanoAllC29.841.852.246.248.554.258.552.253.250.7
 TFT11.811.510.511.810.213.213.813.213.017.8
Appendix
Table B.8: Strategy composition by model and payoff scale (%). The label assigned by the hybrid read-out: the rule base fixes the candidate set and the LSTM ranks within it, deciding alone only where no canonical rule fits. Columns sum to 100 within each model and scale.
LabelnnDeduced (%)Ambiguous (%)Unmatched (%)
AllC11,31223.636.939.6
TFT2,2775.40.094.6
WSLS2,9722.30.697.1
AllD7,43935.60.064.4
Appendix
Table B.9: Provenance of each label. For every label, the share of the agent-games carrying it that were deduced (exactly one canonical rule fits), ambiguous (several fit, and the LSTM ranked within that set) or unmatched (no canonical rule fits, so the label is an attribution the LSTM makes alone). The asymmetry is severe and bounds how the compositional result may be read: AllC and AllD carry a substantial deduced share, whereas the TFT and WSLS labels are almost entirely attributions.
SplitnnAccuracy
held-out (NoNoise + Noise005)64,5120.9846
OOD (Noise01)161,2800.9364
OOD (Noise02)161,2800.8107
held-out recall: AllC16,1500.9853
held-out recall: AllD16,2030.9952
held-out recall: TFT16,1930.9744
Appendix
Table B.10: Validation of the LSTM branch. Trained on synthetic AllC / AllD / Tit-for-Tat / Win-Stay-Lose-Shift trajectories at zero and 5% execution noise, balanced at 40,320 sequences per strategy per level. The 10% and 20% noise levels are never trained on. No language-model transcript enters training.
 Claude HaikuGPT-5.4 NanoGemini 3.5Qwen3 235B-A22BGrok 4.20
Claude Haiku 4.5 +0.11+0.11−0.18-0.18−0.56-0.56+0.40+0.40
GPT-5.4 Nano+0.11+0.11 −0.23-0.23−0.72-0.72+0.85+0.85
Gemini 3.5 Flash-Lite−0.18-0.18−0.23-0.23 +0.22+0.22−0.15-0.15
Qwen3 235B-A22B−0.56-0.56−0.72-0.72+0.22+0.22 −0.71-0.71
Grok 4.20+0.40+0.40+0.85+0.85−0.15-0.15−0.71-0.71 
Appendix
Table B.11: Pairwise correlation between the model curves, five models. Correlation of the ten-point cooperation-against-scale curves of each pair of the five models the main text reports. Six of the ten pairs are negative and the mean is −0.10-0.10 , which is why averaging the curves cancels much of the movement.
 Claude HaikuGPT-5.4 NanoGemini 3.5Qwen3 235B-A22BGrok 4.20Gemini 3.1
Claude Haiku 4.5 +0.11+0.11−0.18-0.18−0.56-0.56+0.40+0.40−0.03-0.03
GPT-5.4 Nano+0.11+0.11 −0.23-0.23−0.72-0.72+0.85+0.85−0.70-0.70
Gemini 3.5 Flash-Lite−0.18-0.18−0.23-0.23 +0.22+0.22−0.15-0.15+0.36+0.36
Qwen3 235B-A22B−0.56-0.56−0.72-0.72+0.22+0.22 −0.71-0.71+0.73+0.73
Grok 4.20+0.40+0.40+0.85+0.85−0.15-0.15−0.71-0.71 −0.57-0.57
Gemini 3.1 Flash-Lite−0.03-0.03−0.70-0.70+0.36+0.36+0.73+0.73−0.57-0.57 
Appendix
Table B.12: Pairwise correlation between the model curves, all six models. The same correlations with the sixth model included: nine of the fifteen pairs are negative and the mean is −0.08-0.08 .
ModelRound 1Rounds 2–10Ratio
Claude Haiku 4.50.1800.0882.04
GPT-5.4 Nano0.3050.2391.28
Gemini 3.5 Flash-Lite0.3670.1941.90
Qwen3 235B-A22B0.0500.1050.47
Grok 4.200.7620.6221.23
Gemini 3.1 Flash-Lite0.2520.0693.68
Appendix
Table B.13: Range of cooperation across the ten payoff scales, on the opening move and afterwards. At round 1 the agent has seen only the payoff matrix and the instructions, so a dependence on the payoff scale there cannot have come through the history of play. The two windows rest on 400 and 3,600 decisions per cell, so their ranges are not variance-matched and the ratio is descriptive rather than a test; the bias runs the same way in every model, while the ratio does not.
Figure C.1: The deterministic dynamics do not see the payoff scale. Replicator flow on each of the four faces of the three-simplex spanned by the canonical rules, one face per strategy left out, at λ=1\lambda=1 , ε=0.05\varepsilon=0.05 and ten rounds. Filled circles are stable rest points, open circles unstable. Only one row is drawn because rescaling the payoffs multiplies the field by a constant and moves nothing: the matrix at λ=1000\lambda=1000 is exactly 10510^{5} times the matrix at λ=0.01\lambda=0.01 , and over 2,000 random population states the normalised fields agree to 4×10−144\times 10^{-14} .
Figure C.2: Where the payoff scale does act: the invasion structure of a finite population. Nodes are the four monomorphic states, node area is the small-mutation-limit stationary probability, and an edge is drawn where a single mutant fixes more often than neutral drift 1/Z1/Z would carry it, labelled in multiples of that drift. Z=100Z=100 , Fermi pairwise comparison, β=0.1\beta=0.1 , ε=0.05\varepsilon=0.05 . Because λ\lambda and β\beta enter the update rule only through their product, raising the scale is raising selection pressure. The panels stop at λ=10\lambda=10 because the float64 fixation probabilities underflow beyond it; those cells are solved at 200 digits instead.
 stationary frequencyaudit
λ\lambdaAllCAllDTFTWSLSmax⁡∣Δ∣\max|\Delta|absorbing
0.010.2280.2670.2510.2534.8e-150
0.10.1000.3820.2600.2586.0e-150
0.250.0490.4460.2580.2473.6e-150
0.50.0320.5070.2240.2371.5e-140
10.0220.6060.1500.2231.7e-120
Appendix
Table C.1: Stationary distribution of the evolutionary baseline at ε=0.05\varepsilon=0.05 . Long-run frequency of each canonical rule under the pairwise-comparison process in the small-mutation limit, Z=100Z=100 , β=0.1\beta=0.1 , r=10r=10 rounds, computed in 200-digit arithmetic. The last two columns are the precision audit: the largest absolute disagreement between the 200-digit solver and a double-precision one, and the number of absorbing states the double-precision chain acquires where that number exceeds one. Double precision fails exactly where the audit says it does, at the top of the grid, and returns a wrong answer there without any warning.
 execution error ε\varepsilon
λ\lambda00.050.10.20.3
0.010.2630.2670.2690.2680.263
0.10.3190.3820.4160.4290.394
0.250.3150.4460.5460.6340.602
0.50.3040.5070.6610.8110.822
10.2830.6060.8020.9390.965
Appendix
Table C.2: Stationary frequency of Always Defect, across the payoff grid and every execution-noise level. The same process as table C.1 . Raising the payoff scale hands the population to Always Defect at every noise level, and does so sooner the noisier execution is. The one exception to monotonicity is the error-free corner ε=0\varepsilon=0 , where the share dips slightly before rising: with no execution error Tit-for-Tat loses only its opening move to Always Defect, so raising the scale first sharpens selection in Tit-for-Tat’s favour. Any positive error rate removes the exception, because it lets Always Defect exploit the retaliation lag repeatedly rather than once.
 evolutionary baselinecorpus
λ\lambdaAllCAllDTFTWSLSAllCAllDTFTWSLS
0.010.2280.2670.2510.2530.3810.4360.0910.092
0.10.1000.3820.2600.2580.3980.3950.0940.113
0.250.0490.4460.2580.2470.5050.2660.1070.122
0.50.0320.5070.2240.2370.4340.3030.1100.152
10.0220.6060.1500.2230.4720.2770.0990.152
Appendix
Table C.3: The evolutionary baseline beside the corpus, at matched payoff scales. Left, the stationary frequency of each canonical rule at ε=0.05\varepsilon=0.05 ; right, the share of agent-games carrying each label, pooled over the five models the main text reports. The two move in opposite directions: the baseline’s Always Defect share rises from 0.267 to 1.000 across the grid while the corpus’s falls from 0.436 to 0.283.
Figure C.3: The evolutionary baseline against the corpus, at matched payoff scales. (a) Stationary frequency of each canonical strategy under the small-mutation limit ( Z=100Z=100 , β=0.1\beta=0.1 , r=10r=10 , ε=0.05\varepsilon=0.05 ), beside the pooled label composition of the five main-text models at the same payoff scale. (b) The share playing Always Defect alone, the one series on which the two disagree in sign: across the ten scales the baseline rises from 0.27 to 1.00 while the corpus falls from 0.44 to 0.28.
Figure C.4: The evolutionary baseline across the payoff grid and the execution-noise levels. Stationary frequency of each of the four canonical strategies against the payoff scale, one panel per execution-error rate ε\varepsilon . Always Defect rises with λ\lambda at every noise level, and does so sooner the noisier execution is; the rise is monotone at every positive ε\varepsilon and dips slightly before rising in the error-free corner ε=0\varepsilon=0 (table C.2 ).
ModelMean rule distance (rounds of 10)Implied ε\varepsilon
Claude Haiku 4.52.0730.207
GPT-5.4 Nano2.1200.212
Gemini 3.5 Flash-Lite1.0770.108
Qwen3 235B-A22B0.7370.074
Grok 4.200.7250.073
Gemini 3.1 Flash-Lite1.0300.103
Appendix
Table C.4: Execution noise implied by the read-out. The mean number of rounds, out of ten, by which an agent-game departs from the nearest canonical rule, and that quantity as a per-round rate. The two models whose play is least rule-governed imply a noise level at the very edge of the range the classifier was validated on, which is why § D.3 treats their strategy labels as the weakest evidence in this article.
Figure D.1: The baseline’s direction is a property of the model, not of its parameters. (a) The AllD share against the payoff scale, one hairline per evolutionary setting, with the setting reported in the text drawn heavy. (b) The rise in the AllD share from the smallest scale to the largest, by execution error. (c) The corpus on the same axis as the family. Eighty settings, Z∈{50,100,200,500}Z\in\{50,100,200,500\} crossed with β∈{0.01,0.05,0.1,0.5,1}\beta\in\{0.01,0.05,0.1,0.5,1\} and ε∈{0,0.05,0.1,0.2}\varepsilon\in\{0,0.05,0.1,0.2\} , ten payoff scales, 800 cells each solved at 200 significant digits. The rise is monotone at every positive ε\varepsilon ; all 19 exceptions are at ε=0\varepsilon=0 .
ModelrangeFE χ92\chi^{2}_{9}Friedman χ92\chi^{2}_{9}logit χ92\chi^{2}_{9}persona shiftpersist. −- recip.
Claude Haiku 4.50.081 [0.062, 0.127]0.081\,[0.062,\,0.127]43.445.348.80.429 [0.371, 0.487]0.429\,[0.371,\,0.487]0.046 [0.001, 0.092]0.046\,[0.001,\,0.092]
GPT-5.4 Nano0.243 [0.200, 0.290]0.243\,[0.200,\,0.290]177.4164.6185.50.153 [0.099, 0.208]0.153\,[0.099,\,0.208]0.135 [0.107, 0.162]0.135\,[0.107,\,0.162]
Gemini 3.5 Flash-Lite0.190 [0.169, 0.242]0.190\,[0.169,\,0.242]168.7214.3141.90.680 [0.613, 0.743]0.680\,[0.613,\,0.743]0.404 [0.374, 0.434]0.404\,[0.374,\,0.434]
Qwen3 235B-A22B0.095 [0.069, 0.123]0.095\,[0.069,\,0.123]58.178.385.90.233 [0.204, 0.260]0.233\,[0.204,\,0.260]0.844 [0.820, 0.864]0.844\,[0.820,\,0.864]
Grok 4.200.636 [0.593, 0.678]0.636\,[0.593,\,0.678]889.41059.9977.00.138 [0.086, 0.188]0.138\,[0.086,\,0.188]0.502 [0.452, 0.545]0.502\,[0.452,\,0.545]
Gemini 3.1 Flash-Lite0.072 [0.061, 0.090]0.072\,[0.061,\,0.090]116.8181.7161.00.103 [0.088, 0.118]0.103\,[0.088,\,0.118]0.791 [0.767, 0.812]0.791\,[0.767,\,0.812]
Appendix
Table D.1: Headline inferences with the matched ladder as the unit. A ladder is the ten dyads, one per payoff scale, that share a model, a language and a game identifier; each model has 200. Intervals are 95% percentile intervals from 1,500 whole-ladder bootstrap draws. The three χ92\chi^{2}_{9} statistics test the nine payoff-scale contrasts: a linear model of dyad cooperation with one fixed effect per ladder, a Friedman test on the 200 ladders by ten scales, and the round-level logistic model of table 1 with standard errors clustered on the ladder. Every one of the 18 tests has p<2×10−6p<2\times 10^{-6} . Persona shift is the persona effect above the sub-unit boundary minus that below it; the last column is the persistence contrast minus the reciprocity contrast.
Modelreciprocitypersistencematching
Claude Haiku 4.50.243 [0.227, 0.258]0.243\,[0.227,\,0.258]0.288 [0.273, 0.303]0.288\,[0.273,\,0.303]0.172 [0.159, 0.186]0.172\,[0.159,\,0.186]
GPT-5.4 Nano0.141 [0.127, 0.156]0.141\,[0.127,\,0.156]0.277 [0.263, 0.291]0.277\,[0.263,\,0.291]0.053 [0.040, 0.066]0.053\,[0.040,\,0.066]
Gemini 3.5 Flash-Lite0.169 [0.153, 0.185]0.169\,[0.153,\,0.185]0.573 [0.555, 0.590]0.573\,[0.555,\,0.590]0.020 [0.006, 0.035]0.020\,[0.006,\,0.035]
Qwen3 235B-A22B−0.095 [−0.107, −0.083]-0.095\,[-0.107,\,-0.083]0.749 [0.735, 0.762]0.749\,[0.735,\,0.762]−0.085 [−0.096, −0.074]-0.085\,[-0.096,\,-0.074]
Grok 4.200.189 [0.175, 0.203]0.189\,[0.175,\,0.203]0.690 [0.676, 0.704]0.690\,[0.676,\,0.704]0.007 [−0.006, 0.019]0.007\,[-0.006,\,0.019]
Gemini 3.1 Flash-Lite−0.171 [−0.185, −0.158]-0.171\,[-0.185,\,-0.158]0.620 [0.606, 0.633]0.620\,[0.606,\,0.633]0.168 [0.157, 0.179]0.168\,[0.157,\,0.179]
Appendix
Table D.2: Whole-dyad bootstrap intervals for memory-one contrasts. Entries are estimates with 95% percentile intervals. Each model contributes 2,000 dyads and each interval uses 4,000 bootstrap draws.
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