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Evolutionary Dynamics of Cooperation in Next-Generation LLM Agent Systems: A Cross-Provider Empirical Extension

Francisco León Zúñiga Bolívar

Year: 2026Venue: arXiv preprintArea: cs.MAType: PreprintEmbeddings: 55

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 97%

Last extracted: 7/8/2026, 10:24:33 PM

Summary

This paper extends a prior benchmark to evaluate four next-generation LLMs (Claude Sonnet 4.6, Gemini 2.5 Flash, Gemini 3.1 Pro, GPT-5.4 Mini) across three providers using evolutionary game theory in the Iterated Prisoner's Dilemma. It tests four hypotheses regarding cooperative bias, aggressive capability parity, cross-provider divergence, and noise robustness. Results confirm a persistent cooperative bias across most model-prompt combinations, substantial cross-provider divergence (Gemini models lean aggressive, OpenAI leans cooperative), partial support for aggressive capability parity via Self-Refine prompting, and directionally positive but unconfirmed noise robustness. Provider identity emerges as the strongest correlate of equilibrium outcomes.

Entities (13)

Anthropic · provider · 99%Google · provider · 99%OpenAI · provider · 99%Claude Sonnet 4.6 · llm-model · 98%Gemini-2.5-flash · llm-model · 98%Gemini 3.1 Pro · llm-model · 98%gpt-5.4-mini · llm-model · 98%Hypothesis H1 · research-hypothesis · 98%Hypothesis H3 · research-hypothesis · 98%Iterated Prisoner's Dilemma · game-framework · 97%Moran Process · evolutionary-algorithm · 97%Self-Refine · prompting-technique · 96%Index of Differential Capabilities · metric · 95%

Relation Signals (10)

Claude Sonnet 4.6 → developedby → Anthropic

confidence 99% · Claude Sonnet 4.6 (Anthropic) — successor to Claude 3.5 Sonnet

Gemini-2.5-flash → developedby → Google

confidence 99% · Gemini 2.5 Flash (Google) — high-throughput frontier model

gpt-5.4-mini → developedby → OpenAI

confidence 99% · GPT-5.4 Mini (OpenAI) — successor to ChatGPT-4o

Hypothesis H3 → supportedby → Cross-Provider Divergence

confidence 98% · Cross-provider divergence is substantial... Provider identity, rather than model generation, is the strongest correlate of equilibrium outcomes

Hypothesis H1 → supportedby → Cooperative Bias

confidence 98% · Ten of twelve model–prompt combinations favour cooperative equilibria in balanced noiseless conditions.

Hypothesis H1 → testedon → LLM Models

confidence 98% · H1: Cooperative bias persists. Next-generation models will maintain or increase the cooperative bias

Claude Sonnet 4.6 → evaluatedin → Iterated Prisoner's Dilemma

confidence 97% · We apply the identical experimental protocol... iterated prisoner's dilemma

Iterated Prisoner's Dilemma → simulatedvia → Moran Process

confidence 97% · simulating populations of agents through a Moran evolutionary process

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Abstract

Abstract:Do next-generation LLM agents inherit the cooperative biases documented in their predecessors, or does scale and provider diversity reshape equilibrium behaviour in competitive multi-agent settings? Willis et al. established a benchmark for this question using evolutionary game theory and the Iterated Prisoner's Dilemma (IPD), finding consistent cooperative biases in ChatGPT-4o and Claude 3.5 Sonnet. We extend this benchmark to four frontier models released in 2025-2026 - Claude Sonnet 4.6, Gemini 2.5 Flash, Gemini 3.1 Pro, and GPT-5.4 Mini - applying the identical protocol across three prompting styles (Default, Prose, Self-Refine) and four population compositions (balanced and biased, with and without noise). Cooperative bias persists across providers (H1): ten of twelve model-prompt combinations favour cooperative equilibria in balanced noiseless conditions. Cross-provider divergence is substantial (H3): Gemini 2.5 Flash reaches up to 77% aggressive equilibria under biased conditions, while GPT-5.4 Mini reaches 70% cooperative equilibria under Self-Refine. Support for aggressive capability parity is partial (H2): Self-Refine raises ICD in all models and Gemini 3.1 Pro Refine achieves the highest ICD in the dataset (0.925), but Default and Prose prompts show no systematic narrowing. Evidence on noise robustness is directionally positive but not robustly confirmed (H4): with n=500 Moran iterations per condition, average noise sensitivity is about 6 percentage points for Claude Sonnet 4.6 versus 13 pp for Claude 3.5 Sonnet, but this cross-study gap is not statistically significant once the predecessor's unreported sampling error is propagated. Provider identity, rather than model generation, is the strongest correlate of equilibrium outcomes; noise remains a universal challenge regardless of model size or vintage.

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Evolutionary Dynamics of Cooperation in Next-Generation LLM Agent Systems: A Cross-Provider Empirical Extension Francisco León Zúñiga Bolívar Institución Universitaria Colegio Mayor del CaucaPopayánColombia franciscoleon@unimayor.edu.co Abstract. Do next-generation llmagents inherit the cooperative biases documented in their predecessors, or does scale and provider diversity reshape equilibrium behaviour in competitive multi-agent settings? Willis et al. (Willis et al., 2025) established a benchmark for this question using evolutionary game theory and the Iterated Prisoner’s Dilemma (ipd), finding consistent cooperative biases in ChatGPT-4o and Claude 3.5 Sonnet. We extend this benchmark to four frontier models released in 2025–2026—Claude Sonnet 4.6, Gemini 2.5 Flash, Gemini 3.1 Pro, and GPT-5.4 Mini—applying the identical protocol across three prompting styles (Default, Prose, Self-Refine) and four population compositions (balanced and biased, with and without noise). Cooperative bias persists across providers (H1): ten of twelve model–prompt combinations favour cooperative equilibria in balanced noiseless conditions. Cross-provider divergence is substantial (H3): Gemini 2.5 Flash reaches up to 77% aggressive equilibria under biased conditions, while GPT-5.4 Mini reaches 70% cooperative equilibria under Self-Refine. Support for aggressive capability parity is partial (H2): Self-Refine raises ICDICDin all models and Gemini 3.1 Pro Refine achieves the highest ICDICDin the dataset (0.925), but Default and Prose prompts show no systematic narrowing. Evidence on noise robustness is directionally positive but not robustly confirmed (H4): with n=500n=500 Moran iterations per condition, average Δnoise _noise is ≈6≈ 6 percentage points for Claude Sonnet 4.6 versus 13 p for Claude 3.5 Sonnet, but this cross-study gap is not statistically significant once the predecessor’s unreported sampling error is propagated. Provider identity, rather than model generation, is the strongest correlate of equilibrium outcomes; noise remains a universal challenge regardless of model size or vintage. Large Language Models, Iterated Prisoner’s Dilemma, Multi-Agent Systems, Evolutionary Game Theory, Moran Process, Cooperative AI 1. Introduction When llm-powered agents interact repeatedly in competitive settings, do they cooperate or defect? The question is not merely theoretical: autonomous llmagents are already deployed to negotiate contracts, allocate computational resources, and bid in markets (Wang et al., 2024). In each of these settings, the long-run social welfare of the system hinges on whether evolutionary pressure selects for cooperative or aggressive behaviour. Willis et al. (2025) provided the first systematic treatment of this question, using the Iterated Prisoner’s Dilemma (ipd) as a formal testbed. Rather than prompting llms to output individual actions—an approach that prior work found unreliable (Fan et al., 2024)—they prompt models to generate complete strategies in natural language, which are subsequently implemented as Python algorithms. By simulating populations of agents through a Moran evolutionary process, they characterised the equilibrium tendencies of ChatGPT-4o and Claude 3.5 Sonnet. Their central finding is a persistent cooperative bias: in balanced populations and under most prompting conditions, cooperative strategies dominate, with aggressive equilibria arising at substantially below prior probability. Two gaps limit the reach of that benchmark. First, the llmlandscape has shifted substantially since that study. A new generation of frontier models, released throughout 2025–2026, incorporates extended context, revised alignment training, and improved reasoning capabilities. Whether their cooperative tendencies persist, deepen, or erode under evolutionary pressure is unknown. Second, the original study covered only two models from two providers, leaving it unclear whether cooperative bias is a universal property of frontier llms or an artefact of specific training choices. No systematic cross-provider comparison has been conducted. We address both gaps by applying the identical experimental protocol to four next-generation models spanning three providers: • Claude Sonnet 4.6 (Anthropic) — successor to Claude 3.5 Sonnet; • Gemini 2.5 Flash (Google) — high-throughput frontier model; • Gemini 3.1 Pro (Google) — Google’s premium-tier model; • GPT-5.4 Mini (OpenAI) — successor to ChatGPT-4o. We evaluate four hypotheses, stated in advance of analysis: H1: Cooperative bias persists. Next-generation models will maintain or increase the cooperative bias documented by Willis et al., reflecting continued improvements in value alignment. H2: Aggressive capability parity. Better reasoning will enable more effective aggressive strategies, reducing the payoff advantage that cooperative agents currently hold over aggressive ones. H3: Cross-provider divergence. Differences in cooperative tendencies across providers will be large enough to attribute to distinct training and alignment choices rather than sampling noise. H4: Noise robustness improves. Newer models will be more robust to action noise, particularly Claude, which exhibited marked noise sensitivity in the original benchmark. Three contributions follow from this design. First, we extend the Willis et al. (2025) benchmark—its specific Moran/attitude-agent framework—to four new frontier models and three providers, providing the first within-lineage longitudinal comparison under that framework (prior cross-provider LLM-IPD work exists but does not track generational successors within a provider); H1 is confirmed in 10 of 12 model–prompt combinations, establishing cooperative bias as a broadly shared property of current frontier llms. Second, we introduce the Index of Differential Capabilities (ICDICD), a scalar summary of the aggressive–cooperative payoff gap; across conditions, ICDICDranges from 0.454 to 0.925, with Gemini 3.1 Pro under Self-Refine prompting reaching the highest value in the dataset. Third, we provide a longitudinal comparison within the Anthropic and OpenAI lineages; provider identity is the strongest correlate of equilibrium outcome—Gemini 2.5 Flash produces aggressive equilibria in up to 77% of biased-population runs, while GPT-5.4 Mini reaches 70% cooperative equilibria under Refine prompting, confirming H3. All Moran results use n=500n=500 iterations per condition; this resolves the n=100n=100 underpowering of H4, though the remaining cross-study comparison to the predecessor leaves H4 directionally suggestive rather than robustly confirmed. The remainder of this paper is organised as follows. Section 2 reviews related work. Section 3 summarises the experimental protocol. Section 4 presents our findings. Section 5 evaluates the four hypotheses and discusses implications for masdesign. Section 6 concludes. 2. Related Work LLMs in game-theoretic settings. The intersection of llms and game theory has developed rapidly into a coherent subfield (Sun et al., 2025). Aher et al. (2023) used llms to replicate human subject behaviour in behavioural economics experiments. Brookins and DeBacker (2023) and Guo (2023) investigated whether llms approximate Nash-rational play, finding mixed evidence across game types. Akata et al. (2025) studied repeated normal-form games, while Fan et al. (2024) demonstrated systematic failures when llms are prompted to output individual game actions. These limitations motivate our strategy-generation approach, which operates at a higher level of abstraction and separates policy specification from action execution. LLMs in social dilemmas. Yocum et al. (2023) and Piatti et al. (2024) studied llmagent behaviour in Markov social dilemmas. Park et al. (2023) introduced generative agents for simulating social behaviour. De Zarzà et al. (2023) explored emergent cooperation in llm-extended coevolutionary theory. Leibo et al. (2017) grounded such studies in multi-agent reinforcement learning. Our contribution to this strand is the analysis of evolutionary population dynamics—how strategy types rise or collapse across generations—rather than individual agent decisions at a single time step. The Willis et al. (2025) benchmark. Willis et al. (2025) introduced the framework we extend. They prompt llms to generate natural-language ipdstrategies (converted to Python), then simulate Moran evolutionary processes over populations of attitude-agents (Aggressive, Cooperative, Neutral). Key findings include a cooperative bias in both ChatGPT-4o and Claude 3.5 Sonnet, with Claude exhibiting greater difficulty generating effective aggressive strategies and notable fragility under action noise. Applying the Self-Refine framework (Madaan et al., 2023) to strategy generation narrowed this gap (Willis et al., 2025), demonstrating that iterative self-critique can inadvertently enable aggressive strategies, a result that complicates alignment-by-design assumptions. Concurrent and subsequent work. Three papers have since tested related questions with different frameworks. Payne and Alloui-Cros (2025) run evolutionary ipdtournaments across GPT, Gemini, and Claude frontier models using variable termination probabilities and 32,000 prose rationales, without a Moran process or attitude-agent populations; they find the same provider fingerprint we document (Gemini aggressive, OpenAI cooperative), providing independent corroboration for our cross-provider divergence finding (H3, Section 5) via a mechanistically distinct design. Vallinder and Hughes (2024) apply cross-provider evolutionary selection to the Donor Game under indirect reciprocity with older models (Claude 3.5 Sonnet, Gemini 1.5 Flash, GPT-4o) and find the ranking Claude >> Gemini >> GPT; our GPT-5.4 Mini emerges as the most cooperative of the four models we test, inverting that OpenAI position and suggesting that model generation, not just provider, shapes cooperative disposition. Willis et al. (2026) scale the Willis et al. framework to hundreds of agents under cultural evolutionary dynamics and find that newer llms worsen societal outcomes in their setting—an apparent tension with our result that cooperative bias persists under Moran selection at n=12n=12. Whether the selection mechanism (pairwise Moran versus cultural evolution) or population scale reconciles the divergence is an open question we flag for future work. Cross-generational and cross-provider comparisons. Prior capability comparisons across llmproviders focused on general reasoning benchmarks—MMLU (Hendrycks et al., 2021), HumanEval (Chen et al., 2021), and similar—where cooperative dispositions are absent by design. The work above establishes that provider-level differences emerge in evolutionary social dilemmas, but each study either uses older model generations or a single dilemma type. Within-lineage longitudinal analysis—whether successive generations from the same provider preserve or reverse established cooperation patterns—remains absent from the literature. Our study targets that gap by pairing current frontier models against their immediate predecessors under the same Moran ipdframework. Evolutionary game theory foundations. Our simulations rest on the Moran process (Moran, 1958), the classical model of evolutionary selection in finite populations; Nowak et al. (2004) established its connection to cooperation in ipdgames in finite populations, and Traulsen et al. (2006) derived fixation probabilities under this process. The ipdframework follows Axelrod (1984); Axelrod and Hamilton (1981); Nowak (2006) provides the backbone for convergence analysis in finite evolutionary games. The noise mechanism follows Wu and Axelrod (1995) and Wahl and Nowak (1999). These threads—llmstrategic behaviour, multi-agent social dilemmas, and evolutionary population dynamics—jointly define the space this paper occupies: a cross-provider benchmark for next-generation llms, including within-lineage longitudinal comparisons, evaluated under Darwinian selection over ipdstrategy populations. 3. Method We apply the identical experimental protocol of Willis et al. (Willis et al., 2025) to four new models, adding the first Google provider entry and enabling cross-provider comparison. We summarise the key components below. 3.1. Strategy Generation For each combination of model and prompt style, we generate 25 natural-language ipdstrategies per attitude (Aggressive, Cooperative, Neutral), yielding 75 strategies per model–prompt pair. Three prompt styles are used (Table 1): Default (direct elicitation with game-theoretic language), Refine (Self-Refine (Madaan et al., 2023) applied to the default output), and Prose (obfuscated scenario without game-theoretic terminology). Table 1. Prompt styles (following Willis et al. (Willis et al., 2025)). Style Description Default Direct prompt with game-theoretic language; strategy generated in natural language. Refine Default output refined via Self-Refine (Madaan et al., 2023): the model critiques and rewrites its own strategy. Prose Game-theoretic framing obfuscated as a real-world scenario (e.g., trade negotiation). Strategy generated, then translated to the ipdcontext. Strategies are then converted to Python algorithms using the same llmthat generated them (one per provider), preserving consistency within each provider. Strategies that cannot be executed without error are replaced by regeneration until 25 valid strategies per attitude are obtained. 3.2. Models We apply the protocol to the four models listed in Table 2. These represent the direct successors of the models studied by Willis et al. (Claude lineage, OpenAI lineage) and a new provider (Google). Table 2. Models evaluated in this study. Model Provider Predecessor Released Claude Sonnet 4.6 Anthropic Claude 3.5 Sonnet 2025 Gemini 2.5 Flash Google (new provider) 2025 Gemini 3.1 Pro Google (new provider) 2025 GPT-5.4 Mini OpenAI ChatGPT-4o 2025 3.3. IPD Tournament All 75 strategies compete in an all-play-all tournament using the Axelrod Python library (Knight et al., 2016). Each match consists of 1,000 rounds of the standard ipd(payoff matrix: R=3R=3, S=0S=0, T=5T=5, P=1P=1). Noise conditions introduce a 10% probability of action-flip per player per round. Tournaments are repeated 20 times. 3.4. Attitude-Agents Following Willis et al. (Willis et al., 2025), we define three attitude-agents, each uniformly sampling from its corresponding strategy set for each match. This captures populations of agents with distinct strategic dispositions rather than fixed individual strategies. 3.5. Moran Process We simulate Moran evolutionary processes with population size n=12n=12 and 500 iterations per condition. Four population compositions are evaluated: (1) Balanced, clean (4:4:4) — equal priors; (2) Biased, clean (8:2:2) — aggressive majority; (3) Balanced, noise (4:4:4 with noise) — equal priors with action noise; (4) Biased, noise (8:2:2 with noise) — aggressive majority with action noise. Convergence is assessed by the proportion of runs reaching each monoculture equilibrium (all-Aggressive, all-Cooperative, or all-Neutral). Figure 1 illustrates a single trajectory. Line chart showing population fractions over evolutionary generations for three attitude groups (Aggressive, Cooperative, Neutral). Cooperative fraction rises to fixation. Figure 1. Example Moran process trajectory for Claude 4.6 Default (balanced 4:4:4 population, single seed). Shading represents the fraction of agents with each attitude at each generation; the process terminates when one attitude reaches fixation. 3.6. Derived Metrics We introduce the Index of Differential Capabilities (ICDICD) to summarise the head-to-head payoff gap between aggressive and cooperative agents: (1) ICD=u¯(A)u¯(C),u¯(k)=13∑j∈A,C,Nu(k,j),ICD= u(A) u(C), u(k)= 13 _j∈\A,C,N\u(k,j), where u(k,j)u(k,j) is the normalised payoff of attitude k against attitude j. An ICDICDof 1.0 implies equal capability; values below 1.0 indicate a cooperative advantage. We also define the noise sensitivity Δnoise _noise: (2) Δnoise=PCclean−PCnoise, _noise=P_C^clean-P_C^noise, measuring the drop in cooperative equilibrium probability under action noise. 4. Results 4.1. Strategy Validation: Cooperation Propensity Table 3 reports the normalised propensity to cooperate for the Default prompt without noise, analogous to Table 3 in Willis et al. Table 3. Normalised cooperation propensity (Default prompt, no noise). Rows: row-player attitude; columns: opponent attitude. Model Att. vs A vs C vs N Claude 4.6 A 0.000 0.297 0.286 C 0.298 0.991 0.992 N 0.287 0.993 1.000 Gemini 2.5 Flash A 0.082 0.087 0.087 C 0.088 1.000 1.000 N 0.088 1.000 1.000 Gemini 3.1 Pro A 0.413 0.629 0.629 C 0.630 0.999 0.999 N 0.630 0.999 0.999 GPT-5.4 Mini A 0.006 0.025 0.023 C 0.026 1.000 0.940 N 0.024 0.940 0.883 All four models exhibit the expected attitude separation: aggressive strategies cooperate markedly less than cooperative or neutral strategies. Gemini 3.1 Pro is the notable exception, with aggressive strategies cooperating at 41% even against other aggressive strategies—substantially higher than any model tested by Willis et al. By contrast, Gemini 2.5 Flash and GPT-5.4 Mini generate highly committed aggressive strategies (cooperation rates near zero against all opponents), while Claude 4.6 closely mirrors Claude 3.5 Sonnet’s pattern: aggressive strategies cooperate at ≈0≈ 0% against other aggressors but rise to ≈29≈ 29% against cooperative opponents. 4.2. Head-to-Head Payoffs and Differential Capabilities Table 4 presents the normalised mean payoffs across all 25 matches for each attitude pairing (Default prompt, no noise), together with the ICDICD(Eq. 1). Table 4. Normalised head-to-head payoffs (Default prompt, no noise) and Index of Differential Capabilities (ICDICD). Lower ICDICDindicates a larger cooperative advantage. Model Prompt Aggressive payoff vs. Cooperative payoff vs. ICDICD A C N A C N Claude 4.6 Default 1.000 1.893 1.861 1.890 2.982 2.988 0.605 Prose 1.242 1.917 1.784 1.507 3.000 2.836 0.673 Refine 2.432 2.820 2.683 2.772 2.997 2.924 0.913 Gemini 2.5 Flash Default 1.232 1.244 1.244 1.239 3.000 3.000 0.514 Prose 1.290 1.796 1.551 0.985 2.574 2.086 0.822 Refine 1.486 2.288 2.022 1.470 2.998 2.635 0.816 Gemini 3.1 Pro Default 1.992 2.400 2.400 2.397 2.998 2.998 0.809 Prose 2.009 1.971 1.972 1.702 2.979 2.858 0.789 Refine 2.159 2.405 2.433 2.259 2.685 2.617 0.925 GPT-5.4 Mini Default 1.017 1.079 1.073 1.074 3.000 2.900 0.454 Prose 1.982 2.395 2.423 2.394 2.914 2.937 0.825 Refine 1.066 1.427 1.252 1.343 2.846 2.305 0.577 Grouped bar chart showing ICD values for four LLM models (Claude 4.6, Gemini 2.5 Flash, Gemini 3.1 Pro, GPT-5.4 Mini) across three prompt styles (Default, Prose, Refine). Values range from 0.454 to 0.925. Figure 2. Index of Differential Capabilities (ICDICD) per model and prompt style. Lower values indicate a larger cooperative payoff advantage. Shaded bands show the range of ICDICDvalues reported by Willis et al. for ChatGPT-4o (purple) and Claude 3.5 Sonnet (brown); markers indicate per-model averages. ICDICDvalues range from 0.454 (GPT-5.4 Mini Default) to 0.925 (Gemini 3.1 Pro Refine) and vary substantially across prompts. Cooperative and neutral attitudes achieve near-mutual-cooperation payoffs (≈3.0≈ 3.0) in most model–prompt combinations, consistent with the original findings. GPT-5.4 Mini Default’s ICDICDof 0.454 means aggressive strategies achieve less than half the payoff of cooperative ones; at the other extreme, Gemini 3.1 Pro Refine (0.925) and Claude 4.6 Refine (0.913) approach payoff parity. The Self-Refine prompt increases ICDICDrelative to Default in all four models, replicating the original finding that Self-Refine narrows the aggressive–cooperative gap. For Claude 4.6 Refine, this effect is particularly pronounced: aggressive strategies achieve payoffs of 2.43–2.82, approaching cooperative strategies at 2.77–3.00. One notable exception to the monotonic Default→ → trend is GPT-5.4 Mini: its Refine ICDICD(0.577) is substantially lower than its Prose ICDICD(0.825), the only reversal in the dataset. This suggests that GPT-5.4 Mini’s Self-Refine mechanism improves strategy coherence at the expense of aggressive payoff, producing more cooperative-dominated refinements rather than capability-balanced ones. Compared to Willis et al.’s Claude 3.5 Sonnet (Table 4 therein), Claude 4.6 shows systematically higher payoffs for aggressive strategies under Refine, suggesting improved aggressive strategy generation in the newer version. GPT-5.4 Mini Default, however, shows weaker aggressive performance than ChatGPT-4o Default (1.05 vs. 1.81 average), indicating that model scaling does not uniformly improve aggressive capability. 4.3. Evolutionary Equilibria Table 5 reports the main Moran process results: the proportion of 500 runs converging to each equilibrium attitude, for all four population conditions. The Willis et al. reference values are shown for comparison. Table 5. Moran process equilibrium proportions (%A / %C / %N) across four population conditions. Prior probability indicates the baseline for each initial ratio. Reference values from Willis et al. (Willis et al., 2025) (Table 6) are shown in the bottom rows. Bold: values highlighted in the text discussion. Model Prompt 4:4:4 clean 4:4:4 noise 8:2:2 clean 8:2:2 noise (prior: 33/33/33) (prior: 33/33/33) (prior: 67/17/17) (prior: 67/17/17) Claude 4.6 Default 2/49/49 24/40/36 12/47/41 60/23/17 Prose 5/48/47 28/37/36 32/38/30 64/17/19 Refine 23/39/38 29/41/30 59/22/20 62/23/15 Gemini 2.5 Flash Default 2/49/48 20/39/41 24/38/38 61/23/16 Prose 24/47/30 40/28/32 65/19/16 77/9/14 Refine 21/47/32 36/32/32 56/22/22 73/13/13 Gemini 3.1 Pro Default 14/42/45 31/35/34 36/32/32 63/20/17 Prose 19/43/38 36/29/34 60/21/20 70/12/17 Refine 27/40/33 32/37/31 62/18/20 66/18/16 GPT-5.4 Mini Default 2/53/45 14/47/39 22/38/40 52/25/23 Prose 12/45/43 31/34/35 46/28/26 66/16/18 Refine 4/70/26 25/42/33 28/46/26 60/19/20 ChatGPT-4o† Default 14/53/33 16/42/42 66/19/17 59/20/21 Prose 13/38/49 23/41/36 35/27/38 60/18/22 Refine 19/48/33 28/38/34 49/30/21 63/19/18 Claude 3.5 Sonnet† Default 4/49/47 15/37/48 36/24/40 41/20/39 Prose 14/42/44 17/33/50 41/30/29 61/26/13 Refine 16/51/33 37/34/29 50/22/28 60/18/22 †From Willis et al. (Willis et al., 2025), Table 6. Four-panel stacked bar chart (balanced clean, balanced noisy, biased clean, biased noisy) showing Aggressive, Cooperative, and Neutral equilibrium proportions for all 12 model-prompt combinations. Cooperative dominates balanced conditions; Aggressive rises under noise and bias. Figure 3. Moran process equilibrium distributions across all 48 conditions (12 model–prompt combinations × 4 population regimes). Stacked bars show the proportion of 500 runs converging to Aggressive (red), Cooperative (blue), or Neutral (grey) equilibrium. Dashed line indicates the theoretical prior (33% for balanced, 67% for biased conditions). Vertical grey lines separate the four LLM families. Balanced, noiseless condition (4:4:4 clean). This condition provides the most direct comparison with the original benchmark. Ten of twelve model–prompt combinations favour cooperative equilibria as the plurality outcome (PC>PAP_C>P_A and PC>PNP_C>P_N), confirming a widespread cooperative bias. The strongest cooperative equilibria are achieved by GPT-5.4 Mini Refine (70%), GPT-5.4 Mini Default (53%), and Gemini 2.5 Flash Default (49%). The two exceptions are both cases where the Neutral attitude achieves the plurality: Claude 4.6 Default (2%A/49%C/49%N, a near-tie resolved in favour of Neutral by 4 of 500 runs) and Gemini 3.1 Pro Default (14%A/42%C/45%N). Notably, neither exception is driven by aggressive dominance; in both cases aggressive equilibria remain well below the 33% prior. The Neutral plurality for Gemini 3.1 Pro Default likely reflects that its aggressive strategies cooperate at unusually high rates (Table 3, 41% vs. own attitude), blurring the boundary between aggressive and neutral behaviour and allowing neutral strategies to benefit from mutual cooperation without defection penalties. The larger n=500n=500 sample, together with the corrected complete strategy libraries (see the erratum note), resolves several earlier ambiguities: GPT-5.4 Mini Default, Gemini 3.1 Pro Prose, and Gemini 3.1 Pro Refine all favour a clear cooperative plurality (53%C, 43%C, and 40%C respectively). Comparing within-lineage successors: Claude 4.6 Default achieves 2%A/49%C/49%N, virtually identical to Claude 3.5 Sonnet Default at 4%A/49%C/47%N. This is the closest generational replication in the dataset. GPT-5.4 Mini Default achieves 2%A/53%C/45%N, far lower aggressive than ChatGPT-4o Default (14%A), confirming improved cooperative bias in the OpenAI lineage for this condition. Balanced, noisy condition (4:4:4 noise). Noise consistently shifts equilibria toward aggression across all twelve model–prompt combinations; all twelve pairs show PAnoise>PAcleanP_A^noise>P_A^clean. With n=500n=500, the apparent n=100n=100 anomaly in which Gemini 3.1 Pro Default’s cooperative equilibria rose under noise disappears: its Δnoise _noise is now +6+6p (normal degradation), confirming that the earlier negative value was sampling noise. The most noise-resilient combination is Claude 4.6 Refine, whose cooperative equilibrium is essentially flat under noise (Δnoise=−1 _noise=-1p; see Section 4.6 and Table 8). Gemini 2.5 Flash Prose shows the largest cooperative degradation (47% to 28%, Δnoise=19 _noise=19p), and GPT-5.4 Mini Refine the largest overall (Δnoise=28 _noise=28p). Biased, noiseless condition (8:2:2 clean). With aggressive agents as the initial majority, the prior for aggressive equilibria is 67%. Below this threshold indicates resistance to aggressive takeover; above it indicates susceptibility. At n=500n=500, all twelve model–prompt combinations finish below the 67% prior, indicating that cooperative/neutral attitudes resist aggressive takeover even when seeded as the minority. The most resistant are Claude 4.6 Default (12%A) and GPT-5.4 Mini Default (22%A). The closest to the prior is Gemini 2.5 Flash Prose at 65%A—just below the seeding proportion, suggesting aggressive agents enjoy roughly no disadvantage in that condition but do not amplify beyond it. Biased, noisy condition (8:2:2 noise). In the most adverse condition (aggressive majority with noise), all twelve model–prompt combinations converge to aggressive equilibria at rates of 52–77%. Contrary to a naive reading of the aggressive-majority prior (67%), only 2 of 12 combinations exceed this threshold—Gemini 2.5 Flash Prose (77%) and Gemini 2.5 Flash Refine (73%). The remaining 10 of 12 fall at or below the prior, indicating that the Moran process does not amplify aggressive dominance beyond the initial seeding for most model–prompt combinations, even in this worst-case scenario. The most resistant are GPT-5.4 Mini Default (52%) and Claude 4.6 Default (60%), which finish substantially below the prior. 4.4. Cross-Provider Significance Tests Table 6 reports the pairwise two-sample z-tests for PAP_A in the balanced noiseless condition (Default prompt), with Holm-Bonferroni correction for the six simultaneous comparisons. Table 6. Pairwise z-tests for aggressive equilibrium proportion PAP_A (balanced 4:4:4, noiseless, Default prompt; n=500n=500 per condition). Holm-Bonferroni adjusted thresholds αadj _adj shown; ∗: p<αadjp< _adj, ns: not significant. Model A Model B z Sig. G3.1 Pro (14%) GPT (2%) +7.24+7.24 ∗ Claude (2%) G3.1 Pro (14%) −6.92-6.92 ∗ G2.5 Flash (2%) G3.1 Pro (14%) −6.61-6.61 ∗ G2.5 Flash (2%) GPT (2%) +0.90+0.90 ns Claude (2%) GPT (2%) +0.48+0.48 ns Claude (2%) G2.5 Flash (2%) −0.43-0.43 ns 4.5. Strategy Diversity To characterise how behaviourally varied the 25 strategies per attitude are, we compute the Shannon entropy H of the per-strategy cooperation rate distribution within each attitude group, per prompt (no noise): (3) H=−∑ipilog⁡pi,H=- _ip_i p_i, where pip_i is the fraction of strategies falling in cooperation-rate bin i (10 equal bins on [0,1][0,1]). We also report the attitude separation, defined as the difference in mean cooperation rate between the Cooperative and Aggressive attitude agents. Table 7 summarises the results. Table 7. Strategy diversity: mean Shannon entropy H (nats) across attitudes, and attitude separation (Cooperative minus Aggressive mean cooperation rate). Higher entropy indicates more varied within-attitude behaviour; higher separation indicates clearer attitude distinction. Model Prompt H¯ H Sep. Claude 4.6 Default 0.28 0.56 Prose 1.35 0.56 Refine 0.68 0.16 Gemini 2.5 Flash Default 0.20 0.61 Prose 1.07 0.41 Refine 1.48 0.52 Gemini 3.1 Pro Default 0.65 0.32 Prose 0.87 0.42 Refine 1.69 0.17 GPT-5.4 Mini Default 0.22 0.63 Prose 0.57 0.28 Refine 0.62 0.47 Three patterns emerge. First, attitude separation for Gemini 3.1 Pro is consistently the lowest of the four models (avg. 0.30 vs. 0.47 for the others), consistent with its unusually high aggressive cooperation rate (Table 3) and the consequent neutral-plurality outcomes in the Moran process. When aggressive strategies cooperate almost as often as neutral ones, the fitness differential that normally drives cooperative equilibria is eroded. Second, the Refine prompt reduces separation in Claude 4.6 (0.16) and Gemini 3.1 Pro (0.17), corroborating the ICDICDanalysis: self-refinement narrows the aggressive–cooperative gap at the individual strategy level. Third, Gemini 3.1 Pro Refine attains the highest within-attitude entropy in the dataset (H¯=1.69 H=1.69): its strategies are behaviourally varied yet weakly separated by attitude, so high diversity within attitudes coexists with the low distinction between them noted above—a combination that further blurs the fitness signal driving the Moran dynamics. 4.6. Noise Sensitivity (Δnoise _noise) Table 8 summarises Δnoise _noise (Eq. 2) for the balanced condition, providing a direct measure of noise robustness. Table 8. Noise sensitivity Δnoise _noise (balanced population, 4:4:4). Positive values indicate degradation of cooperative equilibria under noise. Model Default Prose Refine Avg. Claude 4.6 9 12 −1-1 6 Gemini 2.5 Flash 10 19 14 15 Gemini 3.1 Pro 6 14 3 8 GPT-5.4 Mini 7 11 28 15 Claude 3.5 Sonnet† 12 9 17 13 ChatGPT-4o† 11 −3-3 10 6 †Derived from Willis et al. (Willis et al., 2025), Table 6. With n=500n=500, Claude 4.6 stands out as the most noise-robust on average (Δnoise≈6 _noise≈ 6p), with its Refine condition essentially flat under noise (Δnoise=−1 _noise=-1p). GPT-5.4 Mini and Gemini 2.5 Flash are the most noise-sensitive (≈15≈ 15p each); GPT-5.4 Mini’s figure is driven almost entirely by its Refine prompt (Δnoise=28 _noise=28p). Claude 4.6’s average (≈6≈ 6p) is numerically lower than Claude 3.5 Sonnet’s 13p. While our own estimate is now precise (SE ≈2.2≈ 2.2p at n=500n=500), this is a cross-study comparison: the predecessor’s value carries its own (unreported, likely n=100n=100) sampling error, and propagating it leaves the ≈7≈ 7p gap short of conventional significance. We therefore treat this as directionally suggestive evidence of improved noise robustness rather than a confirmed effect; the full inferential argument is given under H4 (Section 5). 5. Discussion 5.1. Hypothesis Evaluation H1 — Cooperative bias persists (Confirmed within the tested strategy library). The cooperative bias documented by Willis et al. is consistent across model generations and providers. All verdicts in this section are conditional on the single 75-strategy-per-model library we generated; we did not resample alternative strategy libraries, so the verdicts speak to these libraries’ evolutionary dynamics rather than to a population-level claim over all strategies a model could produce. In the balanced noiseless condition—the most direct comparison to the original benchmark—10 of 12 model–prompt combinations favour cooperative equilibria as the plurality outcome (PC>PAP_C>P_A and PC>PNP_C>P_N), and aggressive equilibrium rates remain well below the 33% prior for the majority of configurations. The successor models (Claude 4.6, GPT-5.4 Mini) show cooperative bias comparable to or stronger than their predecessors. The within-lineage consistency is striking: Claude 4.6 Default (2%A/49%C) and Claude 3.5 Sonnet Default (4%A/49%C) are indistinguishable, suggesting that the fundamental strategic disposition of Anthropic models is stable across generations. The n=500n=500 sample, together with the corrected complete libraries, sharpens this picture: GPT-5.4 Mini Default, Gemini 3.1 Pro Prose and Gemini 3.1 Pro Refine all favour an unambiguous Cooperative-plurality, bringing the count of cooperative-plurality combinations to 10 of 12. We hypothesise that this stability reflects convergence in alignment training objectives: both models are trained with RLHF and Constitutional AI techniques that strongly penalise defection-like behaviours. The persistence of cooperative bias is consistent with this interpretation but does not rule out alternative explanations (e.g., convergence in the training data distribution for game-theoretic scenarios). H2 — Aggressive capability parity (Partially Supported). Evidence for H2 is mixed and prompt-dependent. The Self-Refine prompt increases ICDICDin all four models, replicating the finding from the original paper that iterative self-critique improves aggressive strategy quality. Gemini 3.1 Pro Refine achieves the highest ICDICDin the dataset (0.925), approaching payoff parity between aggressive and cooperative agents; within the Anthropic lineage, Claude 4.6 Refine (0.913) shows a larger narrowing than Claude 3.5 Sonnet Refine. This suggests that more capable models can use self-refinement more effectively to generate strategic diversity. However, under Default and Prose prompts, most models show ICDICDvalues in the range 0.45–0.82, not systematically higher than the original models. Moreover, the models with the highest ICDICDunder Refine (Gemini 3.1 Pro at 0.925 and Claude 4.6 at 0.913) are also those where the biased condition shows high aggressive equilibria (62% and 59%A respectively), suggesting that capability parity may genuinely increase the viability of aggressive strategies. This finding reinforces the safety implication identified by Willis et al.: prompt engineering techniques that improve aggressive capabilities may inadvertently increase the risk of aggressive equilibria in deployed systems. H3 — Cross-provider divergence (Confirmed within the tested strategy library). The four models exhibit substantially different evolutionary profiles. Two-sample z-tests for proportions confirm that the cross-model differences are not sampling artefacts. At n=500n=500 the tests gain substantial power: in the balanced noiseless condition (Default prompt), 3 of 6 pairwise comparisons of PAP_A are significant, all involving Gemini 3.1 Pro (14%A), which differs from GPT-5.4 Mini (z=7.24z=7.24), Claude 4.6 (z=6.92z=6.92), and Gemini 2.5 Flash (z=6.61z=6.61), each with p<10−10p<10^-10. All three survive Holm-Bonferroni correction for six simultaneous tests trivially (Table 6). Under biased conditions the effect sizes are even larger: Gemini 2.5 Flash Prose (65%A) versus Claude 4.6 Default (12%A) yields z=17.2z=17.2, and versus GPT-5.4 Mini Default (22%A) yields z=13.7z=13.7, both with p below machine precision. Gemini 2.5 Flash is systematically the most aggressive: its Prose and Refine prompts produce aggressive equilibrium rates of 21–24% in balanced conditions and 56–77% in biased conditions. GPT-5.4 Mini is the most cooperative: its Default and Refine prompts produce aggressive equilibria of just 2–4% in balanced conditions, and it achieves the highest cooperative equilibrium rate in the dataset (70% under Refine). Claude 4.6 and Gemini 3.1 Pro occupy an intermediate position. These cross-provider differences are associations, not isolated causal effects of provider identity. Our design confounds three factors that cannot be separated: (i) the model’s strategy-generation behaviour, (i) provider-level alignment choices, and (i) the fact that each provider’s natural-language strategies are code-converted by that same provider’s model (Section 3), so coding fidelity is entangled with provider identity. Indeed, the two Google models (Flash and Pro) differ substantially from each other—Flash being more aggressive, Pro more cooperative—a within-provider gap that, in some comparisons, exceeds cross-provider gaps and directly cautions against reading “provider” as a clean causal lever. What we can defensibly claim is that the provider label is the strongest observable correlate of equilibrium outcome in our data, and that this correlation is too large to attribute to sampling variation (Section 4.4); disentangling its causal components—training corpus, reward model, safety fine-tuning, and code-conversion ability—requires a factorial design that holds the code-conversion model fixed across providers, which we leave to future work. H4 — Noise robustness improves (Suggestive — not robustly confirmed). With the full n=500n=500 Moran sample, our own per-proportion standard error falls to ≈2.2≈ 2.2p, so Claude 4.6’s average noise sensitivity (Δnoise≈6 _noise≈ 6p) is now precisely estimated. The corresponding figure for Claude 3.5 Sonnet in the original benchmark is 1313p—a ≈7≈ 7p numerical improvement in the hypothesised direction. We deliberately stop short of declaring H4 confirmed, for two reasons. First, the comparison is cross-study: the Willis et al. baseline does not report a sampling error, and their protocol used n=100n=100 iterations, implying a per-proportion SE of ≈5≈ 5p on each constituent of their Δnoise _noise. Propagating that uncertainty conservatively (treating prompt as a fixed factor, no #prompts \#prompts shrinkage), the SE of the difference is ≈7.8≈ 7.8p, yielding z≈0.9z≈ 0.9 (p≈0.4p≈ 0.4)—not significant at conventional thresholds; even the most favourable shrinkage assumption keeps p>0.05p>0.05. Second, the only clean test would re-run Claude 3.5 Sonnet through our identical n=500n=500 harness; we choose not to, because selecting that single additional experiment specifically to resolve H4 would be a form of result-driven analysis inconsistent with our pre-registered design. We therefore report H4 as directionally suggestive but not robustly established: the point estimate moves as hypothesised, yet the available evidence cannot rule out that the apparent gain reflects between-study variation rather than a genuine generational improvement. The within-Claude 4.6 pattern is itself informative: the gain concentrates in the Refine condition (Δnoise=−1 _noise=-1p, essentially flat) and Default (Δnoise=9 _noise=9p), while Prose remains sensitive (Δnoise=12 _noise=12p)—a heterogeneity that further cautions against collapsing the three prompts into a single averaged test. Among the new models, Claude 4.6 is the most noise-robust on average (≈6≈ 6p), followed by Gemini 3.1 Pro (≈9≈ 9p); GPT-5.4 Mini and Gemini 2.5 Flash are the most noise-sensitive (≈15≈ 15p each), with GPT-5.4 Mini’s figure driven almost entirely by its Refine prompt (Δnoise=28 _noise=28p). A structural caveat remains: if cooperative strategies rely on clean mutual cooperation signals (as Tit-For-Tat-like strategies do), residual noise sensitivity is expected regardless of model capability. The Claude 4.6 point estimate is consistent with—but does not establish—the possibility that this floor can be lowered by improved alignment training. 5.2. Implications for MAS Design The findings carry direct implications for the deployment of llm-based multi-agent systems. Provider choice is strongly predictive. The choice of llmprovider is strongly associated with the emergent equilibrium dynamics of a deployed system: empirically, a population of Gemini 2.5 Flash agents exhibits substantially higher aggressive-equilibrium rates than one of GPT-5.4 Mini or Claude 4.6 agents. Because provider identity is confounded with code-conversion fidelity and model scale in our design, this is a deployment-relevant predictive regularity rather than evidence that provider-level alignment alone causes the difference. Noise is a universal threat. Across all models and prompts, action noise consistently shifts equilibria toward aggression in biased populations. Systems deployed in noisy environments (e.g., with communication errors or stochastic action execution) should be initialised with non-aggressive majorities, or designed with noise-robust cooperation mechanisms. Self-Refine as a double-edged tool. The Refine prompt generally improves strategy quality but also narrows the gap between aggressive and cooperative capabilities. Designers should be aware that using self-refinement in strategy generation may inadvertently increase the viability of aggressive equilibria. Intra-generation stability of the cooperative baseline. The near-identical balanced-condition results between Claude 3.5 Sonnet and Claude 4.6 Default suggest that generational model updates do not substantially alter the baseline cooperative bias. Whether other dimensions—such as resilience to action noise—shift across generations remains an open question: our point estimates hint at improved noise robustness (H4), but the cross-study comparison is not conclusive. Designers should not assume that all strategic properties transfer unchanged between model versions, nor that they necessarily change. 5.3. Limitations With n=500n=500 Moran iterations per condition the per-proportion standard error is ≈2.2≈ 2.2p, adequate for the hypothesis tests reported here; residual estimation variance nonetheless remains for the smallest effect sizes (e.g., sub-2p differences in balanced aggressive rates). Second, the ipdis a stylised social dilemma; results may not generalise to more complex multi-player or asymmetric games. Third, we use the same model for strategy generation and code conversion within each provider, which may conflate generation capability with coding capability. Fourth, model API versions may change over time, affecting reproducibility; we report exact version identifiers in our codebase (the simulation code and replication package, available at https://github.com/arqFranciscoLeon/evollm and permanently archived at https://doi.org/10.5281/zenodo.20248615). 6. Conclusion We extended the Willis et al. (2025) llmevolutionary game-theory benchmark to four next-generation frontier models spanning three providers—48 Moran process conditions, 4,800 simulation runs, identical experimental protocol. The headline result is that cooperative bias survives the transition to a new model generation: ten of twelve model–prompt combinations favour cooperative equilibria in balanced noiseless conditions, and within the Anthropic lineage Claude 4.6 is virtually indistinguishable from its predecessor. What most strongly differentiates outcomes is not which generation a model belongs to but which provider trained it. Gemini 2.5 Flash reaches up to 77% aggressive equilibria; GPT-5.4 Mini reaches up to 70% cooperative under Self-Refine. That cross-provider gap is larger than any intra-lineage shift we observe, pointing to provider-level factors—of which alignment objectives are one, alongside the confounded code-conversion pathway—as the strongest observed correlate of equilibrium behaviour. The Self-Refine prompt sharpens this picture in a troubling direction. It narrows the aggressive–cooperative payoff gap (ICDICDrises toward 1.0) consistently across all four models, with Gemini 3.1 Pro Refine reaching the dataset maximum (ICDICD= 0.925). Prompt engineering designed to improve strategy quality may therefore inadvertently reduce the cooperative advantage, raising the equilibrium aggression rate in deployed multi-agent populations. Noise robustness shows a directionally encouraging but unconfirmed shift: at n=500n=500 Claude 4.6 shows Δnoise≈6 _noise≈ 6 p versus the 13 p reported for Claude 3.5 Sonnet, a gap that does not reach significance once the predecessor’s cross-study sampling error is propagated. Meanwhile biased noisy conditions converge to aggressive equilibria in the 52–77% range, with only 2 of 12 configurations exceeding the 67% prior. These findings raise questions the current design cannot answer. Do the cross-provider differences reflect divergent RLHF reward signals, differences in training corpora, or systematic variation in post-training alignment objectives? Do cooperative equilibria persist in n-player or asymmetric games, or are they an artefact of the symmetric two-player IPD structure? And would mixed-provider populations—where a Gemini agent competes against a GPT agent—generate equilibrium dynamics that neither provider’s models produce in isolation? And is the apparent reduction in noise sensitivity across two Anthropic generations a genuine effect or between-study variation—a question only a matched re-run of the predecessor under an identical harness can settle? Having this baseline makes those questions tractable. The benchmark protocol and complete strategy datasets are reproducible and extensible; we release the simulation code, strategy libraries, and the n=500 results as a replication package at https://github.com/arqFranciscoLeon/evollm (preserving the upstream Willis et al. structure), permanently archived with DOI 10.5281/zenodo.20248615, so that each new frontier model can be slotted into the same 48-condition design, turning this cross-provider snapshot into a running longitudinal record. Acknowledgements The author thanks the open-source community behind the evollm simulation framework originally developed by Willis et al., on which this study is directly built. AI Use Disclosure AI assistance (Claude, by Anthropic, via Claude Code) was used in the development of the simulation code, the cloud execution harness, the analysis and reproduction scripts, manuscript drafting and translation assistance, and an internal peer-review simulation. The author reviewed, verified, and takes full responsibility for the experimental design, results, claims, and conclusions of this work. Note on Version 2 (Erratum) This version corrects an error in the version 1 preprint. The error leaves the paper’s central conclusions intact and, as we set out below, marginally reinforces them—the count of cooperative-plurality outcomes rises rather than falls. We pre-registered the correction before regenerating any data, and the regenerated artifacts are available in the public replication repository. Incomplete strategy libraries. The Gemini 3.1 Pro Prose and Self-Refine strategy libraries used in v1 were truncated during generation, holding 21 and 7 strategies respectively where the protocol calls for 75; the corresponding noise-aware libraries were affected in the same way. We regenerated all four libraries in full under an identical protocol—the same model (gemini-3.1-pro-preview), temperature, prompts, and per-provider code conversion—and re-ran them through the head-to-head tournaments and the Moran process at n=500n=500. The Default libraries were complete in v1 and remain unchanged, as do all results for the other three models. Effect on results. With the corrected libraries, the balanced noiseless condition now yields a cooperative plurality in 10 of 12 model–prompt combinations rather than 9: Gemini 3.1 Pro Refine shifts from a Neutral to a cooperative plurality (40%C). The highest ICDICDin the dataset is likewise reassigned—now Gemini 3.1 Pro Refine at 0.925, where v1 reported Claude 4.6 Refine at 0.913. We have updated the affected entries in Tables 5, 4, 8 and 7, together with Figures 3 and 2. None of the hypothesis-level verdicts (H1–H4) changes. References (1) Aher et al. (2023) Gati Aher, Rosa I. Arriaga, and Adam Tauman Kalai. 2023. 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