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Cooperation Without Adaptation: LLM Agents in a Collective-Risk Dilemma

Trung-Kiet Huynh, Dao-Sy Duy-Minh, Chi-Nguyen Tran, Phu-Hoa Pham, Phu-Quy Nguyen-Lam, Le Hong Trang, The Anh Han

arXiv:2610.22882v1cs.GTcs.AIcs.CLcs.MA

TL;DR

The paper asks whether language-model agents adapt cooperation to changing risks and partner behaviour, rather than relying on fixed defaults. It evaluates five production models in a collective-risk game using risk variation, payoff-dominated-action tests, scripted partners, mixed groups, and population selection. Contributions often remained close to the fair share despite changing incentives, and selection outcomes depended on population structure.

  • Problem

    The paper asks whether language-model agents condition cooperation on collective risk, other agents’ behaviour, and selection pressures, rather than relying on relatively fixed defaults.

    Source: §1 · p. 1

  • Method

    The authors evaluate five production models in a collective-risk game using risk variation, dominated-action tests, fixed-partner best responses, and selection in mixed populations.

    Source: §1 · p. 2

  • Results

    Contributions often stayed near the fair share across positive risks, while selection favoured the smallest contributor in a one-table population and the fair-share model in larger mixed populations at moderate and high risk.

    Source: §5 · p. 8

  • Takeaways & Limitations

    Cooperative self-play alone is not evidence of strategic competence or reliable adaptation; evaluations should test agents across incentives, partner actions, and population compositions.

    Source: §5 · p. 8

    The evidence covers five models in English and one game, with ten games per condition limiting statistical power.

    Source: §5 · p. 9

Abstract

from arXiv · show

Language-model agents are increasingly deployed to act on behalf of people in settings where individual decisions shape shared outcomes. We examine whether five production models adapt their contributions to changing risks and partner behaviour, or instead follow a relatively fixed strategy. In a collective-risk dilemma game (CRG), six players contribute to a shared pool over ten rounds; if the pool falls short of its target, a catastrophe may destroy their remaining resources. Four models contributed approximately an equal share at every positive risk level, and four of the five also contributed when risk was zero, even though contributing could not improve their payoff. Explicitly stating that only the agent's own payoff mattered reduced zero-risk contributions for most models, while a more neutral prompt did not. Several models correctly identified which action offered the higher expected payoff at different risk levels, yet their contributions changed little. Against fixed partners, no model consistently chose the payoff-maximising response, and in mixed-model groups, a single agent's shortfall in the final round could cause an otherwise successful group to fail. Payoff-based selection also depended on population composition: in a single six-agent group, it favoured the model that contributed least and reduced group success, whereas in larger, randomly mixed populations, it favoured an exact fair-share model at moderate and high risk. These findings show that cooperative self-play alone is not evidence of strategic adaptability. Consistent with a machine-behaviour approach, evaluations of agents in multi-agent systems should test how they respond to incentives, partner actions, and population composition.

1 Introduction

This paper tests whether five production language models adjust cooperation to changing risk, partner behaviour, and payoff-based selection, using a collective-risk game and game-theoretic benchmarks. Across these tests, cooperative-looking self-play often reflects a relatively fixed contribution pattern rather than reliable adaptation.

2 Related work

  • Prior work motivates testing varied environments: identical-model self-play can make stake-sensitive and fixed policies look alike, while target-focused prompts may make equal splits appear cooperative.The paper situates its tests among work on collective-risk games, strategic language-model behaviour, and multi-agent evaluation.Source: §1 · p. 1, §2 · pp. 2–3

3.1 Collective-risk game and benchmarks

  • For risk-neutral players, the welfare optimum is no contribution below p = 1/2 and an exact target-reaching contribution above it; positive contributions are payoff-dominated at zero risk and after the outcome is settled.The fair share is an equal split of the target, but the benchmark is not a unique equilibrium or a prediction of individual play.Source: §1 · p. 2, §3.1 · pp. 3–4

1 Introduction

  • Most models contributed roughly the fair share across positive risk levels, while zero-risk contributions often persisted; partner responses and selection also varied with population structure.Stating that only the agent’s own payoff mattered reduced zero-risk contributions for most models, but contributions otherwise changed little; one-table selection favoured the smallest contributor, whereas larger randomly mixed populations favoured the fair-share model at moderate and high risk.Source: §1 · p. 2

3.2 Models, protocol and measures

  • The protocol tests one production model per provider in English, supplies the full contribution history each round, and compares expected payoffs with a welfare benchmark.Games, rather than seats, are the unit of analysis; uncertainty intervals use bootstraps over games.Source: §3.2 · p. 4

3.3 Evaluation designs

  • Four designs test risk-grid self-play and prompt controls, best responses to scripted partners, mixed-model tables, and payoff-based selection across population sizes.Fixed-partner best responses are computed by enumerating the model’s possible ten-move sequences; selection treats each model as a strategy in an empirical game.Source: §3.3 · p. 4

4 Results

Across the five models, contributions were often weakly responsive to risk and partner behavior. Tests with fixed partners, mixed groups, and payoff-based selection revealed different ways that cooperative outcomes diverged from payoff-sensitive behavior.

4.1.1 Self-play across risk.

  • Self-play losses differed: three models paid toward an unjustified target at low risk, Grok overpaid after the pool was safe, and Qwen often missed through last-round zeros.Source: Table 3 · p. 5, §4.1.1 · p. 5
  • Four of five models contributed approximately the fair share at every positive risk level, and four also paid at zero risk, where contributions only reduced their own payoff.Source: §4.1.1 · p. 5, Figure 2 · p. 6

4.1.2 Prompt and objective controls.

  • Stating the goal reduced zero-risk contributions for Claude, Gemini, and Qwen, but did not make most models responsive across positive risks; neutral wording did not reduce contributions.Source: §4.1.2 · pp. 5–6

4.2.1 Fixed partners.

  • Only 186 of 1,000 scripted games were exactly optimal, and all were cases where the fair share coincided with the optimum; contributions after settlement were also wasted.Source: §4.2.1 · p. 6, Table 5 · p. 7
  • Partner type explained 69%–92% of contribution variation for four models, while risk and its interaction explained at most 5%; responses still diverged from best responses.Source: §4.2.1 · pp. 6–7

4.2.2 Mixed tables.

  • Five Gemini 3.5 Flash-Lite seats with one Qwen3-235B seat succeeded in only 11 of 30 games, as Qwen's final-round zero could leave the target unmet.Source: Figure 4 · p. 7, §4.2.2 · p. 7
  • As Grok 4.20 seats increased from one to five, most focal models contributed less; alongside five Grok seats, GPT-5.6 Luna contributed 10.5 units.Source: Figure 4 · p. 7, §4.2.2 · p. 8

4.3 Selection: population-size dependent

  • At p=0.9, one-table payoff selection reached the target in 47% of selected games versus 76% under random choice, while larger populations favoured Gemini at moderate and high risk.Winners were stable under bootstrap resampling, stronger selection, larger N, and grid-only payoffs, but not weak selection or candidate removal.Source: §4.3 · p. 8

5 Conclusion and discussion

Cooperative-looking self-play does not establish incentive sensitivity or partner adaptation: contributions often stay near fair share as risk changes, and one late shortfall can undo a group’s success. Selection outcomes also depend on population structure, while the mechanisms behind the observed choices remain uncertain.

5 Conclusion and discussion

  • In mixed tables, one late shortfall can make an exact fair-share target fail; cooperative-looking self-play alone is not sufficient evidence that an agent responds appropriately to incentives or adapts to partners.Source: §5 · p. 8
  • Cooperative-looking self-play alone is not evidence that agents respond appropriately to incentives or adapt to partners.Evaluations should vary incentives and partner behaviour, test mixed populations, and compare individual payoffs with group welfare.Source: §5 · p. 8
  • In a one-table population, selection can favour the lowest contributor at the group’s expense, while larger randomly mixed populations favour fair share at moderate and high risk.Figure 5 compares stationary mass at N=6 and N=30 and shows payoffs by table composition.Source: §5 · p. 8, Figure 5 · p. 8
  • The mechanisms remain uncertain because the data show choices, not reasoning; focal-point and prompt-response accounts are hypotheses, not established explanations.The authors distinguish models’ stated payoff rankings from their actions in play.Source: §5 · pp. 8–9
  • The goal sentence is tested in one phrasing and in self-play; without a group-payoff goal, objective explicitness and instruction direction cannot be separated.The running pool is not displayed, so arithmetic errors may explain some contributions after the target has been met.Source: §5 · p. 9
  • The study tests five models from five providers in English and one game, so its claims concern this panel and setting, not language models generally.Ten games per condition limit statistical power; prompt rewordings cover only a small portion of the prompt space, and selection assumes rare entry.Source: §5 · p. 9

Supplementary Material

The supplementary document supports the paper with underlying values, proofs, prompts, questions, and methods. Its section-number convention distinguishes supplement sections from sections of the paper.

Supplementary Material

  • Trung-Kiet Huynh, Dao-Sy Duy-Minh, and Chi-Nguyen Tran are marked as equal contributors.The author list also names Phu-Hoa Pham, Phu-Quy Nguyen-Lam, Le Hong Trang, and The Anh Han.Source: Supplementary Material · p. 11
  • The supplement provides the values behind the paper’s verbal results, full proofs, prompts and questions word for word, and methods.Source: Supplementary Material · p. 11
  • Supplementary section numbers begin with S; an unprefixed reference such as “Section 4” refers to the paper.Source: Supplementary Material · p. 11

S1 How to read this supplement

This supplement maps paper sections to supporting material and explains its conventions for results, measures, and uncertainty intervals.

S1 How to read this supplement

  • Table S1 maps parts of the paper to the corresponding supplementary sections.Source: S1 How to read this supplement · p. 13, Table S1 · p. 13
  • Results sections S6–S11 begin with tables listing each main-text result statement beside its precise value, followed by breakdowns by model, risk, and condition.Source: S1 How to read this supplement · p. 13
  • Games are the unit of analysis; payoffs are expected over the catastrophe lottery, seat totals are out of 40, and intervals are 95% percentile bootstraps over games.Models are listed in a fixed order: Claude Haiku 4.5, Gemini 3.5 Flash-Lite, GPT-5.6 Luna, Qwen3-235B, and Grok 4.20.Source: S1 How to read this supplement · p. 13

S2 The game and its benchmarks

The supplement formalizes the collective-risk game’s equilibrium and welfare benchmarks, then extends them to repeated play and payoff comparisons. It also derives loss measures and best responses to scripted partners.

S2.1 Rules and notation

  • Six players receive 40 each, contribute to a shared pool over ten rounds, and face a catastrophe lottery if their combined total misses the target.The fair share is 20 per player, or 2 per round; the pivot risk is p*=1/2.Source: S2.1 Rules and notation · p. 13

S2.2 Proposition 1 and its proof

  • A Nash equilibrium is either zero contributions or an exact-target profile with each player contributing at most pe; the welfare optimum switches at p*=1/2.At p<p*, zero contributions maximize welfare; at p>p*, meeting the target exactly does, and both outcomes are optimal at the pivot.Source: S2.2 Proposition 1 and its proof · pp. 13–14

S2.3 The ten-round game

  • The ten-round game reaches exactly the same on-path pure-strategy Nash-equilibrium totals as the one-shot game, although the paper makes no claim about subgame perfection.Each such outcome can be implemented by fixed schedules that ignore history.Source: S2.3 The ten-round game · p. 14

S2.4 Proposition 2 and its proof

  • When two seats face the same pool outcome, their payoff difference equals the other seat’s contribution minus their own, scaled by 1−p if the target is missed.The identity also holds for realized payoffs under the shared table lottery.Source: S2.4 Proposition 2 and its proof · p. 15

S2.5 The optimum and the three ways to lose

  • The paper decomposes each game’s loss from the attainable optimum into three non-negative components whose sum equals total loss.The components distinguish costly target-reaching below the pivot, missing the target below the pivot, and loss from missing it at or above the pivot.Source: S2.5 The optimum and the three ways to… · p. 15

S2.6 How much risk aversion would explain paying at low risk

  • At p=0.1, preferring a certain fair-share payoff of 20 over a lottery paying 40 with probability 0.9 requires CRRA risk aversion γ≥0.85.At zero risk, no increasing utility prefers the certain 20 to the certain 40.Source: S2.6 How much risk aversion would… · p. 15

S2.7 Scripted partners and their best responses

  • Against partners who always contribute zero or four, paying nothing is optimal; against partners paying two or copying the group, the best response switches from zero below p* to 20 above it.At p*, both zero and 20 are best responses against the latter partner types.Source: S2.7 Scripted partners and their best… · p. 16, Table S3 · p. 16

S3 Models and protocol

The protocol evaluates five models under shared serving settings, separate simultaneous calls, and a defined procedure for extracting each move from model replies.

S3.1 Models

S3.2 How a round is played

  • Each model-seat receives a separate call with the full prior contribution history, but no memory of earlier calls; all seats see the same history and decide simultaneously.The running pool is not printed, and prompts are in English without personas.Source: S3.2 How a round is played · p. 17

S3.3 How replies are read

  • Moves are extracted from the final CONTRIBUTION line, with retries for unusable or truncated replies; no reply was unreadable across 187,000 decisions.Two Grok replies lacked the required line, and their moves were read from the rest of the reply.Source: S3.3 How replies are read · p. 17

S4 Prompts, word for word

The appendix records the prompts used to elicit each contribution, including the base wording, a no-hint version, neutral alternatives, and two rewordings.

S4.1 Base prompt

  • The base prompt describes a six-player, ten-round game with private simultaneous contributions, a target of 120, and a lottery if the group falls short.It reveals contributions after each round and prohibits communication among players.Source: S4.1 Base prompt · p. 18
  • The prompts ask the model to choose 0, 2, or 4 for the current round and return the choice on a final CONTRIBUTION line.The base prompt supplies the round number, remaining money, and prior contributions by fixed seat.Source: S4.1 Base prompt · p. 18

S4.2 No hint

  • The no-hint prompt removes the equal-split average and example while leaving the rest of the prompt unchanged.The removed text names an average contribution of 2 per player per round and an example of contributing 2 each round.Source: S4.2 No hint · p. 18

S4.3 Neutral wording

  • Neutral wording replaces normative or climate-related labels while keeping the equal-split phrases; its pool line is also reworded.The paper notes that no experimental condition displays the pool line.Source: S4.3 Neutral wording · p. 18

S4.4 Neutral wording, goal stated

  • A separate neutral prompt adds that only the participant’s own final cash payoff matters.The opening is added to the neutral-wording prompt while retaining the game rules and equal-split phrases.Source: S4.4 Neutral wording, goal stated · p. 19

S4.5 Two rewordings

  • The two rewordings preserve the rules, equal-split phrases, and final instruction while changing only phrasing.They provide alternative formulations of the role, endowment, contributions, target, lottery, and information rules.Source: S4.5 Two rewordings · p. 19

S5 Designs, data and statistics

The supplement specifies game counts and seat layouts, explains why analyses use expected rather than realized payoffs, and documents statistical and reproducibility procedures.

S5.1 Games in each design

  • Each design cell contains ten games, and the two-model design covers all unordered model pairs with fixed contiguous seat blocks whose order alternates across cells.Every model plays the same number of games in each design.Source: S5.1 Games in each design · p. 21

S5.2 The lottery and why the paper uses expected payoffs

  • All ten shared lottery draws fall below 0.47, so every missed game at p≥0.5 became a catastrophe and low-risk catastrophes occurred more often than their risks imply.The self-play grid had 96 catastrophes among 143 missed games, versus 69 expected; the paper therefore reports expected payoffs, and agents never see the draw.Source: S5.2 The lottery and why the paper… · p. 21, Table S6 · p. 22

S5.3 Statistical methods

  • Statistical units are games, with 95% percentile bootstrap intervals and two-sided permutation tests used for differences in means.Pooled-risk intervals are stratified for self-play and prompt controls but unstratified for scripted and two-model designs.Source: S5.3 Statistical methods · p. 22
  • Additional analyses use exact sign-flip tests for Qwen’s tablemate comparisons, balanced sums of squares for scripted-design variance shares, and bootstrapping for selection uncertainty.Selection resampling recomputes the full analysis after resampling mixed and self-play cells.Source: S5.3 Statistical methods · p. 22

S5.4 Reproducibility

  • The design is reproducible, but model text is not: the service does not return identical text for a seed, even at temperature zero.A replay is a new sample of the same design rather than a copy of the earlier game records.Source: S5.4 Reproducibility · pp. 22–23

S6 Self-play across risk

In self-play, four models stayed near the fair-share contribution across positive risk levels, while Grok 4.20 paid more and did not increase contributions steadily with risk. Group outcomes and losses varied substantially, with some failures arising from last-round contributions.

S6.1 Numbers behind the main text

  • All 121 missed positive-risk games were at or ahead of the fair pace through round 9, and GPT-5.6 Luna missed the target in 23 of 100 games at positive risk.The section’s numerical summary also reports that the pooled panel lost 42% of attainable payoff, with interval [41, 43].Source: S6.1 Numbers behind the main text · p. 23

S6.2 What a seat pays at each risk level

  • At positive risk, four models’ mean seat totals stayed close to the fair share of 20, while Grok 4.20 paid more and did not rise steadily with risk.The four models remained near 20 whether the optimum was 0 or 20; Grok 4.20’s highest payments occurred at zero and the highest risk levels.Source: S6.2 What a seat pays at each risk level · p. 23
  • Gemini 3.5 Flash-Lite put almost every seat exactly at 20, while Qwen3-235B usually skipped its last payment and Grok 4.20 ranged from 20 to 40.Figure S1 shows the distribution of seat totals across risk levels, with outlined cells marking the optimal total.Source: S6.2 What a seat pays at each risk level · p. 23, Figure S1 · p. 24

S6.3 Outcomes at each risk level

  • Gemini 3.5 Flash-Lite and Grok 4.20 reached the target at every risk level, Claude Haiku 4.5 did so at every positive level, and Qwen3-235B almost never did.At low risk (p=0.1), the three fair-share models lost close to half of attainable payoff; above the pivot, Claude and Gemini lost little, while GPT-5.6 Luna still lost through missed targets.Source: S6.3 Outcomes at each risk level · p. 24, Table S8 · p. 25

S6.4 Where the payoff goes

  • Claude Haiku 4.5, Gemini 3.5 Flash-Lite and GPT-5.6 Luna lost mostly by paying below the pivot; Grok 4.20 lost mostly by paying past 120.Qwen3-235B’s losses were nearly evenly split between paying below the pivot and missing the target above it.Source: S6.4 Where the payoff goes · p. 25, Table S9 · p. 26

S6.5 The end of the game

  • Qwen3-235B usually paid 2 through round 9 and 0 in round 10, leaving its groups short; repeating round 9’s move would have brought every such game to the target.GPT-5.6 Luna’s missed positive-risk games had been at or ahead of the fair pace until the last round.Source: S6.5 The end of the game · p. 26, Table S10 · p. 26

S7 Prompt and decoding controls

Prompt and decoding controls tested whether the self-play patterns persisted across reruns, rewordings, temperature settings, and stated goals. Most models’ average contributions changed little under routine controls, while some prompt changes shifted contributions or model-specific move patterns.

S7 Prompt and decoding controls

  • The control design compared self-play under a base prompt, a rerun, prompt rewordings, and temperature zero, with in-game questions kept outside the game.The rerun repeated the base prompt; the separate-call questions never entered play.Source: S7 Prompt and decoding controls · p. 26

S7.1 Numbers behind the main text

  • Under neutral wording at p=0, four models still had mean totals above half the fair share, whereas stating the goal left 132 of 300 seats paying something.These comparisons are among the reported prompt-control quantities.Source: S7.1 Numbers behind the main text · p. 27

S7.2 Every change against the base grid

S7.3 The goal test at each risk level

  • At zero risk, stating that only the agent’s payoff mattered removed almost all contributions for Claude Haiku 4.5 and Gemini 3.5 Flash-Lite and most for Qwen3-235B, while Grok 4.20 kept paying.Neutral wording left most zero-risk payment in place.Source: S7.3 The goal test at each risk level · p. 29, Table S12 · p. 29
  • With the stated goal, Claude Haiku 4.5 and Gemini 3.5 Flash-Lite stayed at the fair share at p=0.1, but only GPT-5.6 Luna returned to it at p=0.9.GPT-5.6 Luna and Qwen3-235B contributed far below the fair share at p=0.1; Qwen3-235B never reached the target under the goal prompt.Source: S7.3 The goal test at each risk level · p. 29, Table S12 · p. 29

S7.4 How each model’s signature changes

S7.5 Removing the hint

  • Removing the hint did not make any model pay significantly less on average, and two paid more; Qwen3-235B reached the target but earned less because it overpaid.Without the hint, fewer seats played the exact split, and Grok 4.20 moved almost entirely to paying 4 every round.Source: S7.5 Removing the hint · p. 30, Table S14 · p. 31

S8 In-game questions

The section uses separate-call questions to test rule recall and compare expected payoffs without allowing answers to affect play. Several models changed their stated payoff preference with risk while their contributions barely changed, so correct answers did not establish incentive-sensitive play.

S8 In-game questions

S8.4 Rule questions

  • Every model answered every rule question correctly, showing that agents read the rules but not that they could use them to choose actions.The questions asked about information printed in the prompt.Source: S8.4 Rule questions · p. 34, Table S15 · p. 34

S8.5 The value comparison

  • The value comparison contrasted paying 2 each round and reaching the target with paying nothing and missing it; the correct preference switches at p*.The question was asked in separate calls and did not affect the game.Source: S8.5 The value comparison · p. 34
  • Claude Haiku 4.5 and Gemini 3.5 Flash-Lite answered the value comparison almost always correctly, and GPT-5.6 Luna did so most of the time; Qwen3-235B always favored paying nothing.Qwen3-235B was correct only at p=0.1; Grok 4.20 also leaned toward that answer.Source: S8.5 The value comparison · p. 34, Table S16 · p. 34

S8.6 Answers against play

  • Claude Haiku 4.5, Gemini 3.5 Flash-Lite, and GPT-5.6 Luna changed their answers with risk while their contributions barely changed.Figure S3 compares answers and play at p=0.1 and p=0.9; Table S18 reports the shifts and intervals.Source: S8.6 Answers against play · p. 35, Figure S3 · p. 36, Table S18 · p. 36
  • Correct value answers do not establish knowledge of a best response because the question supplies hypothetical outcomes rather than partners’ responses to a deviation.The paper treats the answer-play difference as evidence about reported answers and actions, not what an agent computes.Source: S8.6 Answers against play · p. 35

S8.7 The second value question

  • The second value question was excluded because answers could reflect either whole-game cash or cash remaining when asked, making a wrong answer ambiguous.Claude Haiku 4.5 often gave the latter reading in round 5, when it had already spent part of its endowment.Source: S8.7 The second value question · p. 37, Table S19 · p. 37

S9 One model beside scripted partners

Beside scripted partners, models generally failed to adjust contributions to the payoff-maximising response, whether that meant paying nothing or changing contributions with risk. Contributions also continued after some outcomes were settled, and Qwen3-235B’s characteristic last-round shortfall often left fair-share groups below target.

S9 One model beside scripted partners

S9.2 What the models pay

  • Against partners who always paid 0 or 4, every model contributed more than the best response of 0 in every game; no model paid nothing in any of 1,000 scripted games.Only 186 of 1,000 scripted games were exactly optimal, and every such game had total payment 20.Source: S9.2 What the models pay · p. 38, S9.1 Numbers behind the main text · p. 38, Table S20 · p. 39
  • Beside fair-share partners, Gemini 3.5 Flash-Lite and GPT-5.6 Luna paid almost exactly 2 each round at every risk, while Qwen3-235B skipped the final payment and Grok 4.20 paid more than 2.Figure S5 compares round-by-round payments at low and high risk; none of the rows changed between those panels.Source: S9.2 What the models pay · p. 38, Figure S5 · p. 39

S9.3 Outcomes and distance from the best response

  • Beside partners who always paid 4, every model reached the target in every game, making its gap to the best response money spent for nothing; beside fair-share partners, Qwen3-235B almost never reached the target.Claude Haiku 4.5, Gemini 3.5 Flash-Lite, and GPT-5.6 Luna reached it in almost every fair-share-partner game.Source: S9.3 Outcomes and distance from the… · p. 40, Table S21 · p. 40
  • Beside partners who always paid 0, the target could not be reached, so every unit the model paid was kept by a partner who paid nothing.The cost of those payments shrank as risk rose because the lottery was more likely to destroy the partner’s retained money.Source: S9.3 Outcomes and distance from the… · p. 40, Table S22 · p. 41

S9.4 Partner versus risk

  • For four models, partner type explained most contribution variation and risk explained little; Grok 4.20’s variation was mostly residual and traced to its opening move.Four models opened with 2 in all 800 games, while Grok 4.20 opened with 4 in about half its games and mostly repeated its opening.Source: S9.4 Partner versus risk · p. 41, Table S23 · p. 41, Table S24 · p. 42

S9.5 Paying after the outcome is settled

S9.6 Fair-share partners

  • Beside fair-share partners, Gemini 3.5 Flash-Lite played the exact split in every game, while Qwen3-235B’s most common sequence left the group exactly 2 short.Grok 4.20 split between the exact split and paying 4 in every round.Source: S9.6 Fair-share partners · p. 42, Table S26 · p. 43

S10 Two models at one table

Mixed-model outcomes varied with partner composition, and Qwen3-235B’s final-round zeros often coincided with failed targets. Grok 4.20’s generous contributions also changed how other models paid.

S10 Two models at one table

S10.2 Who earns more at a shared table

S10.3 Which tables reach the target

  • Tables mixing Qwen3-235B with Gemini 3.5 Flash-Lite or GPT-5.6 Luna failed at almost every split, while tables with Grok 4.20 usually succeeded.Exceptions included five GPT-5.6 Luna seats beside one Grok 4.20 seat and four or five Qwen3-235B seats beside Grok 4.20.Source: S10.3 Which tables reach the target · p. 44, Table S29 · p. 45

S10.4 The last round

S10.5 Slack and generous partners

S11 Selection among models

Selection outcomes differed between a single six-agent table and larger randomly mixed populations. The analysis links those outcomes to model payoffs across table compositions and reports checks on selection settings and candidate sets.

S11.1 Method

  • The selection analysis computes each model’s mean payoff over randomly sampled six-agent tables, then uses pairwise comparison and rare-mutation fixation to derive stationary masses.It reports finite-intensity, frequency-dependent selection; changing population size changes both sampling probabilities and the fixation path.Source: S11.1 Method · p. 48

S11.3 Payoff functions

  • Models’ table-composition payoffs helped drive selection: Gemini 3.5 Flash-Lite and GPT-5.6 Luna earned little beside Qwen3-235B when targets were missed, while Grok 4.20 earned less than its tablemates because it paid the most.The payoff functions were computed for ordered model pairs at each risk level.Source: S11.3 Payoff functions · p. 49, Table S33 · p. 50, Table S34 · p. 50

S11.4 Every selection setting

  • The selection winners remained on top in at least 93% of bootstrap samples and were unchanged at β=10, for N=60 and 120, and with grid-only self-play payoffs.Under weak selection (β=0.1), the largest one-table mass was 0.34, and GPT-5.6 Luna narrowly led at moderate and high risk.Source: S11.4 Every selection setting · p. 52, Table S37 · p. 54
  • In a single-table population, stronger selection concentrated mass on Qwen3-235B at every risk level and group success fell almost to zero; in larger populations, Gemini 3.5 Flash-Lite took all mass from β=1 at moderate and high risk.At low risk in the larger population, Qwen3-235B took the mass; Grok 4.20 lost mass under both population sizes once selection was more than very weak.Source: S11.4 Every selection setting · p. 52, Table S36 · p. 52, Figure S6 · p. 53

S11.5 Facts about single tables

  • A single seat of any model beside five Grok 4.20 seats earned more than those Grok seats in every cell; Qwen3-235B was significantly ahead of tablemates in most cells and behind in one.A population only slightly larger than one table was enough for selection to favour Gemini 3.5 Flash-Lite.Source: S11.5 Facts about single tables · p. 55, Table S38 · p. 55

S11.6 Single entrants, candidate sets and welfare

  • Removing Qwen3-235B made GPT-5.6 Luna the winner in the one-table population at all tested risks; removing Gemini 3.5 Flash-Lite made Luna win the larger population at moderate and high risk.These post-hoc checks reused observed payoffs and rebuilt the mutation chain over the remaining models.Source: S11.6 Single entrants, candidate sets… · p. 55, Table S42 · p. 57
  • The analysis also reports stationary-mass-weighted expected payoff at p=0.5, where zero contribution and reaching the target both earn 20 per seat; target rate alone therefore need not indicate payoff loss.These are rare-mutation one-model outcomes, not success rates during invasion.Source: S11.6 Single entrants, candidate sets… · p. 55, Table S43 · p. 58

S12 Strategies, the last round and equilibrium benchmarks

Across last-round choices, equilibrium comparisons, rule-based descriptions, and partner responses, the models’ play showed different patterns by model and game state. The reported association between Grok 4.20’s opening move and partner payments is observational.

S12.1 The last round

  • Qwen3-235B skipped its final contribution in 93% of seats both when it was needed and when others’ fair shares would suffice, while Claude Haiku 4.5 mostly stopped after the target was reached.GPT-5.6 Luna skipped in 2% of cases when needed and 53% when others’ fair shares sufficed; ahead-of-pace self-play tables often missed the target.Source: S12.1 The last round · p. 58, Table S44 · p. 59

S12.2 Play against the equilibria of Proposition 1

  • Above the risk pivot, most Qwen3-235B and Grok 4.20 self-play games fell below the worst Nash-equilibrium payoff, while this never occurred for Gemini 3.5 Flash-Lite or Claude Haiku 4.5.Qwen3-235B missed the target after paying, whereas Grok 4.20 overpaid.Source: S12.2 Play against the equilibria of… · p. 59, Table S45 · p. 59

S12.3 Simple rules describe the moves

  • For every model, one simple rule predicted most moves, and a fitted tree added little; removing risk level from the tree inputs did not lower accuracy for any model.A rule close to the risk-neutral optimum predicted far fewer moves than the best rule for every model.Source: S12.3 Simple rules describe the moves · p. 59, Table S46 · p. 60

S12.4 Responses to Grok 4.20’s opening move

  • Higher Grok 4.20 opening moves were associated with lower GPT-5.6 Luna payments while the pool was short, but with higher Qwen3-235B payments.The relationship was observational, not a randomized opening intervention or proof of an internal reasoning mechanism; Claude Haiku 4.5’s payments barely changed and Gemini 3.5 Flash-Lite’s changed only slightly.Source: S12.4 Responses to Grok 4.20’s opening… · p. 60, Table S47 · p. 60
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