GTBench: Uncovering the Strategic Reasoning Limitations of LLMs via Game-Theoretic Evaluations

Motivation

• Existing role-playing evaluations can obscure pure reasoning because narrative complexity and fluent language may mask concrete reasoning deficiencies.Game-theoretic tasks instead emphasize pure logic, strategic reasoning, and well-defined action and state spaces.Source: §1 · p. 2

GTBENCH

• GTBENCH comprises 10 recognized games spanning complete versus incomplete information, deterministic versus probabilistic settings, and static versus dynamic interaction.The tasks cover abilities including board strategy, collaboration, auction, and bidding.Source: §1 · pp. 2–3

Research questions

• The study characterizes LLM strategic reasoning across scenarios, compares LLMs with conventional solvers, and examines how model and reasoning factors affect performance.It also uses LLM-versus-LLM competition as an automated reasoning evaluation.Source: §1 · pp. 2–3

Main findings

• LLMs generally fail against simple MCTS opponents in complete deterministic games but remain competitive in incomplete probabilistic scenarios.The paper also reports benefits from code pretraining, inconsistent gains from CoT, SC-CoT, and ToT, and a recovery by Llama-3-70b-Instruct relative to other open-source models.Source: §1 · p. 3

2.1 Background and Related Work

• Existing gaming-agent studies cover negotiation, deception, coordination, diplomacy, and related abilities, but often evaluate only one or two games.Such narrow coverage may provide less robust assurance of overall LLM abilities.Source: §2.1 · pp. 3–4
• GTBENCH differs from prior unified suites by combining broad strategic coverage, reasoning-method comparisons, game-theoretic properties, and competition against conventional solvers.It also enables direct LLM-agent-versus-LLM-agent comparisons.Source: §2.1 · p. 4

2.2 Problem Definition

• Gameplay is formulated as an alternating two-participant Markov Decision Process with states, actions, observations, policies, transitions, and terminal outcomes.Participants may be optimization-based solvers or LLM-driven agents.Source: §2.2 · p. 4
• Normalized Relative Advantage measures one participant’s relative performance against an opponent and is interpretable on a [−1, 1] scale.Positive, negative, and zero values indicate being better, worse, or equally competitive, respectively.Source: §2.2 · pp. 4–5
• The score function uses win, loss, and draw values for zero-sum games and earned rewards for non-zero-sum games.Elo Rating is additionally used to estimate relative skill levels in zero-sum games.Source: §2.2 · pp. 4–5

3 GTBENCH: Game-Theoretic Evaluation of LLMs

• GTBENCH provides unified interfaces for participants and games through Environment, Prompt Adapter, and Participant components.The environment manages gameplay, the adapter converts observations and extracts actions, and participants generate reasoning and actions.Source: §3 · p. 5

3.1 Taxonomy of Game-Theoretic Tasks

• The benchmark covers 10 games across complete or incomplete information, static or dynamic interaction, and probabilistic or deterministic settings.The games probe board strategy, bidding, collaboration, bluffing, and mathematical reasoning.Source: §3.1 · p. 5

3.2 Participants and Protocols

• Conventional comparisons use 1,000-simulation MCTS for most games, Tit-for-Tat for Iterated Prisoner’s Dilemma, and Random Agents as baselines.Tit-for-Tat is selected because the dynamic task has a very limited action space.Source: §3.2 · p. 5
• The LLM agents are evaluated with direct prompting, CoT, five-trajectory self-consistent CoT, and three-sequence ToT reasoning schemes.The prompt templates separate system, head, observation, and reasoning components and can adapt to new games.Source: §3.2 · p. 5
• All LLM agents achieve at least 90% task completion, indicating that the prompts support instruction following sufficient to finish games.This sanity check concerns completion rather than strategic quality.Source: §3.2 · p. 5
• Figure 2 reports NRA for LLM-driven reasoning agents against MCTS and Random Agents in complete deterministic scenarios, while Figure 3 reports game-wise NRA against MCTS/TfT and Random Agents in incomplete probabilistic scenarios.Figure 2 uses red and gray lines for the maximum NRA achieved by LLM agents; Figure 3 uses green and gray lines and error bars over different reasoning methods.Source: Figure 2 · p. 6, Figure 3 · p. 6

4.1 Complete and Deterministic Gaming

• NRA reaches −1 against MCTS across complete deterministic games, indicating LLM agents barely win even one match against near-optimal conventional solvers.The evaluated tasks are Tic-Tac-Toe, Connect-4, Breakthrough, and Nim; sufficient MCTS simulations produce near-optimal strategies in these games.Source: §4.1 · p. 6

4.2 Probabilistic and Dynamic Gaming

• NRA stays near 0 across most probabilistic and dynamic tasks, with GPT-4 outperforming MCTS in Kuhn Poker, although limited-stochasticity Liar’s Dice remains difficult.The tasks include Kuhn Poker, Liar’s Dice, Blind Auction, Negotiation, Pig, and Iterated Prisoner’s Dilemma; Random Agent is omitted for Negotiation and Iterated Prisoner’s Dilemma because it rarely collaborates.Source: §4.2 · p. 7

4.3 LLM-vs.-LLM Competition

• GPT-4 is the strongest examined model overall, while Llama-3-70b-Instruct performs comparably to GPT-4 and outperforms GPT-3.5-turbo in LLM-vs.-LLM competition.The comparison uses GPT-3.5-turbo with Prompt Agent as the common opponent and reports both NRA and Elo results.Source: §4.3 · p. 7, Table 6 · p. 8
• Code-pretrained models significantly outperform larger chat models with fewer than half the parameters, supporting a benefit of code-pretraining for game-theoretic reasoning.Examples include CodeLlama-34b-Instruct and Deepseek-Coder-6.7b-Instruct versus Llama-2-70b-chat and Deepseek-LLM-67b-chat.Source: §4.3 · p. 7, Table 2 · p. 7
• Advanced reasoning is inconsistent: it substantially improves Mistral-7b-Orca and benefits GPT-3.5-turbo, but worsens results for other models and all tested CoT prompts underperform the naive Prompt Agent.The authors characterize advanced reasoning as a double-edged sword because powerful models can leverage it while ordinary models may incur additional reasoning errors.Source: §4.3 · pp. 7–8, Figure 4 · p. 7, Table 3 · p. 7
• Complex board states expose an open-source gap for Llama-2-70b-chat and CodeLlama-34b-Instruct, whereas Llama-3-70b-Instruct performs significantly better in Breakthrough and reaches comparable capabilities to commercial models.Table 4 compares average NRA with Breakthrough included and excluded; Breakthrough has a larger action/state space than other complete-information games.Source: §4.3 · p. 8, Table 4 · p. 8, §4.4 · p. 8

4.4 Error Profiles

• Error analysis finds that LLMs often form reasonable strategies but fail to execute matching actions and miss immediate winning or losing moves.The error profile covers misinterpretation, factual inaccuracies, overconfidence, calculation mistakes, and endgame misdetection, based on 157 manually examined turns.Source: §4.4 · p. 8, Table 5 · p. 8

Nash Equilibrium with Regret

• GPT-4 has lower regret in Blind Auction, whereas CodeLlama-34b-Instruct has lower regret than GPT-4 in Iterated Prisoner’s Dilemma.The difference is associated with GPT-4 choosing <Silent> more often and CodeLlama choosing <Testify> more often.Source: §5 · p. 9

Pareto Efficiency

• Most agreements provide substantial value to both negotiators, but Llama-2-70b-chat and CodeLlama-34b-Instruct sometimes accept unfair divisions.GPT-4 and Mistral instead struggle to reach agreements and tend to negotiate for Pareto improvements.Source: §5 · p. 9
• Repeated Prisoner’s Dilemma produces Pareto improvements across rounds, indicating that LLMs can use play history to adjust their strategies.Source: §5 · p. 9

6 Conclusion

• Evaluations across 10 representative games reveal failures in complete and deterministic games, stronger reasoning in incomplete and probabilistic games, and benefits from code-generation pretraining and suitable prompts.Source: §6 · p. 9
• The benchmark evaluates LLM strategic and logical reasoning using varied classic and LLM-based gaming agents across a broad game scope.Source: §6 · p. 9

Limitations

• Results may vary because the evaluation relies on predefined prompt templates and is sensitive to prompt design.Source: §6 · p. 9
• The selected games have limited action and state spaces, which may not distinguish LLMs at similar skill levels.Source: §6 · pp. 9–10
• Generated actions may be illegal because of instruction-following incapabilities.Source: §6 · p. 10

Impact Statements

• The study measures current model capabilities rather than training models to become competent at game theory, bluffing, or defection.Source: §6 · p. 10

A1 Overall Architecture

• GTBENCH comprises an Environment, Prompt Adapter, and Participant that jointly manage game execution and language-model interaction.Source: A1 Overall Architecture · p. 15
• The Environment builds observations, manages gameplay, applies participant actions, and is implemented on OpenSpiel.Source: A1 Overall Architecture · p. 15
• The Prompt Adapter converts environment observations into unified prompts and transforms participant responses into legal actions.Source: A1 Overall Architecture · p. 15
• Participants generate intended actions from the observation prompts for each turn.Source: A1 Overall Architecture · p. 15

Other Games

• Tic-Tac-Toe, Connect Four, and Liar’s Dice provide additional board and deception settings with turn-based grid or challenge actions.Source: A2.1 Games Introduction · pp. 15–16

Prisoner’s Dilemma

• Prisoner’s Dilemma models cooperation or defection between two rational agents, with simultaneous <Silent> or <Testify> actions.Source: A2.1 Games Introduction · p. 15

Breakthrough

• Breakthrough uses a 3*8 board where players move pieces toward the opponent’s home row, observing board status and move histories.Source: A2.1 Games Introduction · pp. 15–16

Blind Auction

• Blind Auction has simultaneous sealed bids, with the highest bidder paying the submitted price; observations provide item valuations and actions specify bid amounts.Source: A2.1 Games Introduction · p. 16

Kuhn Poker

• Kuhn Poker is a three-card, imperfect-information, zero-sum game where players choose <Pass> or <Bet> using private cards and move history.Source: A2.1 Games Introduction · p. 16

Pig

• Pig is a probabilistic turn-taking game in which players repeatedly roll, risk losing turn points on a 1, or stop and bank their total.Source: A2.1 Games Introduction · pp. 16–17

Nim

• Nim requires players to remove any positive number of objects from one heap, using pile counts as observations and pile-and-quantity actions.Source: A2.1 Games Introduction · p. 17

Negotiation

• Negotiation exposes item quantities, proposals, utterances, and player valuations, with actions consisting of agreement or requested item quantities.Source: A2.1 Games Introduction · p. 17

• Games differ by information structure: complete-information games reveal strategies, payoffs, and actions, whereas incomplete-information games hide some game parameters or opponents’ actions.Source: A2.2 Gaming-Theoretic Taxonomy · p. 17
• Games are static when players decide simultaneously and dynamic when they observe previous moves before acting.Source: A2.2 Gaming-Theoretic Taxonomy · p. 17
• Deterministic games produce fully rule- and strategy-determined outcomes, while probabilistic games include randomness such as dice rolls or card draws.Source: A2.2 Gaming-Theoretic Taxonomy · p. 17

A3.1 Conventional Agent

• GTBench includes conventional MCTS and Tit-for-Tat agents, respectively representing heuristic search through simulated future outcomes and reciprocal repeated-game behavior.MCTS focuses computation on promising branches, while Tit-for-Tat begins cooperatively and then copies the opponent’s previous action.Source: A3.1 Conventional Agent · p. 18

A4 LLM-vs-LLM Results

• The appendix presents an NRA confusion matrix comparing LLM agents against GPT-3.5-turbo and GPT-4 across ten games.GPT-3.5-turbo with Prompt Agent serves as the common opponent for multiple LLM-agent combinations.Source: A4 LLM-vs-LLM Results · p. 18, Figure A6 · p. 18

A5.1 Modular Prompt Structure

• The modular prompt gives participants shared system guidance, game rules, current-state observations, and reasoning instructions, with variables populated from a unified observation object.This structure ensures participants access the same observations and information while supporting game-specific legal actions and reasoning formats.Source: A5.1 Modular Prompt Structure · p. 19

A5.2 System Prompt / A5.3 Head Prompt

• System prompts use a common gaming-agent instruction requiring legal moves, whereas head prompts provide game-specific rules for the ten environments.The appendix gives examples covering board games, repeated dilemmas, auctions, imperfect-information games, dice games, Nim, and negotiation.Source: A5.2 System Prompt · p. 19, A5.3 Head Prompt · pp. 19–20

A5.4 Observations

• Observation prompts encode game-specific state, histories, scores, private information, bids, dice, piles, and negotiation proposals or utterances.These templates expose the current environment state and relevant prior actions for decision-making in each task.Source: A5.4 Observations · pp. 21–22

A5.5 Reasoning Prompt / A8 Chain-of-Thought Sensitivity / A9 Error Pattern Demonstrations

• The appendix compares Prompt, CoT, SC-CoT, and ToT agents, including multi-trajectory action selection for SC-CoT and candidate generation and voting for ToT.It also documents illegal outcomes produced by some CoT strategies and five recurring GTBENCH error patterns: misinterpretation, factual error, math calculation, over-confidence, and another demonstrated category.Source: A5.5 Reasoning Prompt · pp. 22–23, A8 Chain-of-Thought Sensitivity · p. 25, Table A10 · p. 25

A5.6 Sanity Check

• All LLM agents achieve ≥90% completion rates across games, indicating that the prompts are properly configured and agents can follow instructions to finish matches.A valid match requires every participant to generate legal moves at each turn; completion rates are based on 50 valid matches.Source: A5.6 Sanity Check · p. 23, Table A7 · p. 23

A6 How Temperature Affects LLM Performance

• Higher generation temperature worsens performance in deterministic games, while its effect on probabilistic games depends on the model.Experiments compare temperature 0.2 against 0.4, 0.6, and 0.8 for CodeLlama-34b-Instruct and GPT-3.5-turbo-1106 across 20 matches per setting using PromptAgent.Source: A6 How Temperature Affects LLM… · p. 23, Table A8 · p. 24

A10.1 Regret Value for Blind Auction

• Blind-auction regret is defined conditionally from the player’s and opponent’s last bids and the player’s valuation.When b_1 exceeds b_2 + 1, regret is b_1 - (b_2 + 1); otherwise, when b_2 + 1 is below v, regret is v - (b_2 + 1).Source: A10.1 Regret Value for Blind Auction · p. 27

A10.2 Regret Value for Iterated Prisoner’s Dilemma

• Iterated Prisoner’s Dilemma regret is calculated by accumulating the regret value from each individual turn.The supplied rule includes a per-turn regret value of 1.Source: A10.2 Regret Value for Iterated… · p. 27

A11 User Interfaces of GTBench Leaderboard

• The GTBench leaderboard user interfaces are presented in Figures A8 and A9, including selection of LLMs or agents and opponents.Figure A8 shows the leaderboard interface, while Figure A9 shows the interface after various LLMs/agents and opponents are selected.Source: A11 User Interfaces of GTBench… · p. 27, Figure A8 · p. 28, Figure A9 · p. 29