GIFT: Games as Informal Training for Generalizable LLMs

Bingbing Xu; Huawei Shen; Nuoyan Lyu; Tat-Seng Chua; Weihao Meng; Xueyun Tian; Yang Zhang; Yige Yuan; Zhiyong Huang

REVIEW 4 major objections 5 minor 14 references

The paper claims that adding game-based informal learning to formal math training improves general LLM abilities, and that a nested training framework avoids the interference of naive task mixing.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 11:34 UTC pith:WQHS7A5J

load-bearing objection The paper's central AND-objective claim is contradicted by its own appendix, which implements an average reward; the empirical study is substantial enough to warrant referee time, but not near acceptance. the 4 major comments →

arxiv 2601.05633 v2 pith:WQHS7A5J submitted 2026-01-09 cs.CL

GIFT: Games as Informal Training for Generalizable LLMs

Nuoyan Lyu , Bingbing Xu , Xueyun Tian , Weihao Meng , Yige Yuan , Yang Zhang , Zhiyong Huang , Tat-Seng Chua

show 1 more author

Huawei Shen

This is my paper

classification cs.CL

keywords informal learninggamesreinforcement learningmulti-task RLLLM generalizationnested trainingreasoning abilitiessocial reasoning

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tries to establish that playing games can serve as informal learning for LLMs, complementing formal math training to improve broad abilities such as reasoning, creativity, and social inference. It reports that adding game reinforcement learning to math raises average general-benchmark performance from 38.34% to 42.43% for 1.5B models and from 42.00% to 55.84% for 7B models. It also proposes a nested training framework that concatenates math and game sub-tasks into a single trajectory, claiming this changes the optimization from an OR-type objective to an AND-type objective and prevents task interference. A sympathetic reader would care because this points toward an annotation-free, scalable route to broader LLM competence.

Core claim

The central discovery is that game-based informal learning transfers to general abilities beyond formal training alone, and that naive multi-task mixing undermines this transfer. The paper's proposed fix is nested training: execute all sub-tasks sequentially in one rollout and reward the average success across them. The authors argue this substitutes an implicit OR objective, where one task can dominate, with an explicit AND objective requiring joint success, which stabilizes gradients, preserves in-domain performance, and improves generalization. The empirical support is the consistent advantage of nested over mixed training across task combinations and model scales.

What carries the argument

The Nested Training Framework: a composite task is built by sequentially composing sub-task trajectories (e.g., math then a game) within a single episode, and the trajectory reward is the average success across sub-tasks. The paper claims this enforces joint success—an explicit AND-type objective—whereas mixed training optimizes an additive OR-type objective. This mechanism carries the argument because it is what purportedly stabilizes gradients and prevents any single task from dominating.

Load-bearing premise

The paper assumes that averaging per-sub-task successes across concatenated trajectories instantiates an explicit AND-type objective that coordinates subtask gradients, but the appendix specifies the reward as an average, which under uniform task sampling yields the same expected objective as naive mixed training.

What would settle it

Train two models on identical math and game tasks: one with nested concatenated trajectories and one with independently sampled sub-task trajectories but the same average reward. If their general-benchmark performance converges, the nested structure is not the driver. Alternatively, compute the expected gradient of the nested average-reward objective and show it equals the mixed-training gradient.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

Game-based informal learning plus formal math raises average general-benchmark performance by about 4 percentage points for 1.5B models and about 14 points for 7B models.
Nested training beats naive mixed training on general abilities in every task combination and at both model scales tested.
Nested training also helps in an informal-only setting, improving MMLU from 39.57% to 43.80% and average general ability from 33.17% to 34.46%.
The nested objective is order-invariant: reversing the order of math and Matrix Games yields nearly identical average general performance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The appendix's average-success reward, under uniform task sampling, has the same expected value as the mixed-training additive objective; so the claimed OR-to-AND shift is not a change in the objective itself. The reported gains would have to come from trajectory-level credit assignment or rollout composition, which the paper does not isolate.
A direct test would compare nested training against mixed training with matched expected rewards per trajectory, or against averaging rewards from independently sampled sub-tasks; if performance equalizes, concatenation is not the active ingredient.
Because opponent strength changes game success and downstream transfer, self-play or adaptive opponents could amplify or alter the effects reported here.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

The paper's central AND-objective claim is contradicted by its own appendix, which implements an average reward; the empirical study is substantial enough to warrant referee time, but not near acceptance.

read the letter

What you should know up front: the paper's headline methodological claim does not hold. Section 3.4 says nested training enforces an explicit AND-type objective, distinct from naive mixed training's OR objective. Appendix B.2, however, defines the nested reward as "the average success across all sub-tasks, rather than a strict conjunction." For uniformly sampled sub-tasks, that average is the same additive objective as Eq. (1). So the claimed OR-to-AND transformation is not implemented. This is not a nitpick; it is the load-bearing distinction of the paper.

That said, there is real substance here. The empirical study is extensive: two model scales, three game genres, formal math, and a consistent set of in-domain and general benchmarks. The tables are detailed, and the ablations on opponent choice and task ordering are useful. The specific configuration — sequential concatenation of heterogeneous task trajectories in one rollout — is new, and the observed pattern that nested training often beats naive mixing on average general ability is directionally positive. The framing of games as informal learning is a reasonable lens, even if the educational theory is doing more rhetorical than analytical work.

The soft spots are significant. No error bars, no multiple seeds, single fixed seed. In-domain performance often degrades badly in nested training (1.5B F+I2: MATH 16.2 vs 28.0 mixed; TicTacToe 16 vs 50). The 7B formal-math-only baseline drops on general benchmarks relative to the base model, which muddies the "informal learning helps" story — it could be forgetting avoidance rather than transfer. The abstract claims code is public, but the GitHub link is a placeholder. And the F+I3 warm-up for 1.5B is a hand-placed crutch that deserves more scrutiny.

On the math: the stress-test note is correct. The average reward does not couple sub-task gradients through the objective; partial success gets linear credit. The only structural differences left are trajectory-level composition, longer max_turn, and balanced per-rollout task presence — none of which are isolated from the objective framing. The paper's own B.2 note is an admission that the implementation does not match Sec. 3.4.

Bottom line: the empirical question — can game-based RL improve general abilities beyond math-only training? — is worth asking, and the data are detailed enough to merit referee time. But the central methodological claim is a mischaracterization, and the current draft would not survive review as written. I'd recommend sending it to peer review with heavy revision expected: fix the objective framing or drop the AND language, add seeds and error bars, release code, and isolate the trajectory-composition effect. This is a useful paper to engage with, but it needs major rework before it can be believed.

Referee Report

4 major / 5 minor

Summary. The paper proposes treating games as informal learning environments for LLMs, trained with GRPO alongside formal math tasks. It introduces a 'nested training' framework that concatenates sub-task trajectories and rewards their average success, claiming this converts an implicit 'OR' objective into an explicit 'AND' objective. Experiments with Qwen2.5-1.5B and 7B on Matrix Games, TicTacToe, and Who's the Spy report improvements on general ability benchmarks (MMLU, MMLU-Pro, CommonGen, SocialIQA) over math-only and mixed training, alongside ablations on opponent choice and task order.

Significance. If the central claims held, this would be a useful contribution to multi-task RL for LLMs, with practical implications for scalable, annotation-free training. The paper has strengths: it uses external benchmarks, reports two model scales, provides detailed environment descriptions, includes several ablations, and publicly releases code (though the link is a placeholder). However, the main methodological claim is contradicted by the paper's own appendix: the implemented nested reward is an average, i.e., an additive objective, not the AND-type objective claimed in Sec. 3.4. A numerical error in Table 3 inflates the headline 7B gain. In-domain performance under nested training often degrades substantially, contradicting the abstract's claim of preserving in-domain subtask performance. These are load-bearing issues for the paper's contribution as stated.

major comments (4)

[Sec. 3.4 and Appendix B.2, Eq. (1)-(2)] The central claim that nested training replaces an OR-type objective with an explicit AND-type objective is not supported by the implementation. Appendix B.2 states the nested reward is 'the average success across all sub-tasks, rather than a strict conjunction'; for K sub-tasks this is R_nested = (1/K) Σ_k R_k(τ_k). Under uniform task sampling, mixed training in Eq. (1) has the same expected value as this sum. An average reward is linear: partial success receives partial credit, and the gradient is a sum of per-task gradients, so it does not force all sub-tasks to succeed and does not couple sub-task gradients in the manner claimed. The paper explicitly denies using a stricter or multiplicative objective, but a nonlinear/AND coupling is exactly what would be needed for the claimed distinction. The only remaining structural differences—concatenated rollouts, max_turn as the sum rather th
[Table 3 and Abstract] The reported 7B formal-only average is 42.00%, but the four general-ability columns in that row (MMLU 66.51, MMLU-Pro 32.45, Common 35.18, SocialIQA 75.90) average to 52.51%. The abstract therefore overstates the 7B formal-to-nested gain: the nested F+I3 average of 55.84% is only 3.33 points above the corrected formal-only baseline, not 13.84 points. Since the abstract and Sec. 4.2 emphasize this large gain, the arithmetic error changes the quantitative support for the paper's headline result. Please correct the table and re-evaluate all conclusions that depend on this baseline.
[Tables 2 and 3; Abstract] The abstract claims nested training preserves 'in-domain subtask performance,' but the tables show substantial in-domain degradation in several settings. For 1.5B F+I2, nested MATH is 16.2 vs 43.2 in math-only training and TicTacToe is 16 vs 75 in TicTacToe-only training; nested also underperforms mixed training on all three in-domain tasks in this row (MATH 16.2 vs 28.0, Matrix 20 vs 30, TicTacToe 16 vs 50). For 7B F+I2, nested Matrix is 40 vs mixed 63 and TicTacToe is 33 vs mixed 45. These are not minor fluctuations; they contradict the stated claim that nested training preserves in-domain performance. The paper should either revise the claim or provide a definition of 'preserve' that accommodates these drops.
[Sec. 4.1 and Appendix B.3] All reported training runs appear to use a single fixed seed and no error bars or multiple-seed statistics. Several key comparisons are small (e.g., Table 2 F+I3 Avg 42.43 vs 41.68; Table 6 Avg 54.95 vs 54.59), and game evaluations involve stochastic opponents and only 100 rounds. Without multiple seeds or significance testing, the reader cannot determine whether nested training's advantages over mixed training are robust or due to noise. The authors should report means and standard deviations over at least 3-5 seeds for the main comparisons, or justify why the deterministic components make this unnecessary.

minor comments (5)

[Appendix B.3] Typo: 'rollout-upodate' should be 'rollout-update'.
[Table 2] In the F+I3 nested row, '21.1924.37' appears to be missing a separator; it should likely be '21.19 24.37' or similar.
[Abstract and footnote] The GitHub URL is a placeholder ('XXX/XXXX'), despite the text claiming code and data are publicly available. A working link or repository snapshot should be provided.
[Sec. 4.4.4] The statement that the nested objective is 'theoretically invariant' to sub-task order is stronger than what the average-reward definition supports: while the summed reward is order-invariant at a symbolic level, the concatenated trajectory and credit assignment can make the training dynamics order-dependent. Only two orders are tested, so the claim should be phrased empirically.
[Figures 3-4] The text refers to 'gradient magnitudes' in Fig. 4, but the figure's axes are not described. Please clarify what is plotted and how gradients are measured.

Circularity Check

0 steps flagged

No significant circularity: empirical claims rest on external benchmarks; B.2's average-reward note is a non-circular consistency gap.

full rationale

The paper's central empirical claims—that game-based informal RL improves held-out general benchmarks and that nested training improves over mixed training—are evaluated on externally defined benchmarks (MMLU, MMLU-Pro, CommonGen, SocialIQA, MATH500, and fixed-opponent game success) that are not used to set training constants or fitted parameters. Eq. (1) and Eq. (2) are objectives, not predictions derived from those benchmarks. No fitted target is renamed as a prediction, and no uniqueness theorem or ansatz is imported through self-citation; the cited RAGEN/StarPO* backbone is an implementation reference rather than the source of the generalization claim. The one in-text tension worth flagging is methodological, not circular: Sec. 3.4 claims an explicit AND-type objective, while Appendix B.2 states, 'The reward for a nested trajectory is defined as the average success across all sub-tasks, rather than a strict conjunction.' Under uniform task sampling, that average has the same expected value as the additive mixed objective in Eq. (1), so the OR-to-AND transformation is not guaranteed by the implemented reward. This is a mathematical consistency issue and a potential confound with trajectory-level credit assignment, but it does not make the reported benchmark improvements equivalent to the paper's inputs. The empirical results remain externally falsifiable, so no circular reduction is present.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The ledger counts hand-chosen reward values, warm-up, and opponent selection as free parameters. The core theoretical premise—that average reward implements an AND objective—is the most important ad hoc assumption and is contradicted by the paper's own implementation.

free parameters (4)

Draw reward in TicTacToe = 0.5
Hand-chosen reward for draws; used in all TicTacToe training, unablated.
Format penalty = -0.1
Hand-chosen penalty for output-format violations; affects all RL training.
Warm-up mixed-task phase for F+I3 1.5B = unspecified
Ad hoc cold-start mitigation introduced because direct nested training from base initialization failed; necessary for the headline F+I3 1.5B result but its duration and data are not specified.
Opponent model for multi-turn games = Qwen3-14B
Opponent choice was selected after comparing to Gemini-2.5-Flash (Table 5) because it gave better training outcomes; results are tied to this choice.

axioms (5)

standard math Random affine transformations of payoff matrices preserve Nash equilibria
Used in Matrix Games training to augment payoff tables; true for affine transformations, so a safe standard assumption.
domain assumption GRPO/StarPO* with fixed Qwen3-14B opponents gives a reliable training signal
The entire method assumes trajectory-level GRPO on fixed opponents is sufficient to learn transferable abilities; no comparison to other RL algorithms.
domain assumption The three chosen games are representative informal-learning environments that transfer to general abilities
Core framing assumption; only three hand-picked games are used, with no criterion for representativeness.
ad hoc to paper Average reward over nested sub-task successes enforces an AND-type joint objective
Central to the nested-training contribution, but contradicted by B.2 where reward is an average, not a conjunction; in expectation it equals the mixed objective.
domain assumption MMLU, MMLU-Pro, CommonGen and SocialIQA measure broad general abilities relevant to informal learning
Standard benchmarks, but whether gains on them reflect informal-learning transfer is exactly what is being claimed.

pith-pipeline@v1.3.0-alltime-deepseek · 21597 in / 14642 out tokens · 145598 ms · 2026-08-03T11:34:50.898851+00:00 · methodology

0 comments

read the original abstract

Recent LLMs excel at formal tasks such as mathematical reasoning and code generation, but still struggle with broader abilities such as planning, creativity, and social intelligence. Inspired by human learning, where formal instruction and informal experience jointly shape intelligence, we introduce informal learning into LLM training and use games as annotation-free, feedback-driven environments. To cover diverse abilities including abstract reasoning, planning, creativity, and social interaction, we combine formal math tasks with three representative game tasks, including Matrix Games, TicTacToe, and Who's the Spy. However, directly mixing these tasks under a unified RL objective can blur task-specific learning signals and provides no explicit guidance for coordinating task-gradient directions. To combat these, we propose Coordinated Subtask Training (CST), which replaces a single mixed update with sequential subtask-specific updates, separating heterogeneous RL signals while implicitly promoting coordination among subtasks. Experiments on ability-oriented benchmarks show that game-based informal learning improves generalization beyond formal training alone, while CST further enhances multi-task RL by preserving in-domain subtask performance and improving broader general abilities. Code and data are publicly available.

Figures

Figures reproduced from arXiv: 2601.05633 by Bingbing Xu, Huawei Shen, Nuoyan Lyu, Tat-Seng Chua, Weihao Meng, Xueyun Tian, Yang Zhang, Yige Yuan, Zhiyong Huang.

**Figure 2.** Figure 2: Overview of the proposed nested training framework with formal and informal learning tasks. The middle [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** Figure 4: Comparison between mixed and nested train [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 3.** Figure 3: Comparison between mixed and nested train [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 5.** Figure 5: Case Studies on MATH500 and CommonGen. and 8). On the MATH500 benchmark, the base 7B model follows the correct high-level reasoning but makes a subtle arithmetic error, leading to an incorrect result, whereas the Matrix Games-trained model maintains a more explicit and verifiable derivation. Moreover, it chooses a = 1 creatively to simplify the calculation. On the CommonGen benchmark, while mixed training … view at source ↗

**Figure 6.** Figure 6: Detailed payoff matrices used in matrix game environments. [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: An example of the effectiveness of informal learning on the MATH500 benchmark, with Qwen2.5-7B [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

**Figure 8.** Figure 8: An example of the effectiveness of nested training on the CommonGen benchmark, comparing mixed [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 6 linked inside Pith

[1]

The game is played on a 3 x3 vertical grid
[2]

** Winning Conditions **: The game ends when a player forms a line of 3 of their own pieces

Players take turns setting one of their pieces into any available slot . ** Winning Conditions **: The game ends when a player forms a line of 3 of their own pieces . The line can be :
[3]

process

** Diagonal ** ( connected at an angle ) * Example of a diagonal win ( bottom - left to top - right ) for Player 1:* ``` . . O . O X <-- 3'O's in a diagonal line O X . ``` * Example of another diagonal win ( top - left to bottom - right ) for Player 2:* ``` X . O . X O <-- 3'X's in a diagonal line . O X ``` ** Draw Condition **: If the entire grid is fill...

2024
[4]

Aixin Liu, Aoxue Mei, Bangcai Lin, Bing Xue, Bingx- uan Wang, Bingzheng Xu, Bochao Wu, Bowei Zhang, Chaofan Lin, Chen Dong, and 1 others

Deepseek-v3 technical report.arXiv preprint arXiv:2412.19437. Aixin Liu, Aoxue Mei, Bangcai Lin, Bing Xue, Bingx- uan Wang, Bingzheng Xu, Bochao Wu, Bowei Zhang, Chaofan Lin, Chen Dong, and 1 others. 2025a. Deepseek-v3. 2: Pushing the frontier of open large language models.arXiv preprint arXiv:2512.02556. Bo Liu, Leon Guertler, Simon Yu, Zichen Liu, Pengh...

Pith/arXiv arXiv 1981
[6]

Maarten Sap, Hannah Rashkin, Derek Chen, Ronan LeBras, and Yejin Choi

The game between game theory and gaming simulations: design choices.Simulation & Gaming, 50(2):180–201. Maarten Sap, Hannah Rashkin, Derek Chen, Ronan LeBras, and Yejin Choi. 2019. Social iqa: Com- monsense reasoning about social interactions. In EMNLP. Sylvia Scribner and Michael Cole. 1973. Cognitive consequences of formal and informal education: New ac...

Pith/arXiv arXiv 2019
[7]

Qwen Team

Game-theoretic regularized self-play align- ment of large language models.arXiv preprint arXiv:2503.00030. Qwen Team. 2024. Qwen2.5: A party of foundation models. Qwen Team. 2025. Qwen3 technical report.Preprint, arXiv:2505.09388. Cecilia Thorsen, Jan-Eric Gustafsson, and Christina Cliffordson. 2014. The influence of fluid and crystal- lized intelligence ...

Pith/arXiv arXiv 2024
[8]

Yunfei Xie, Yinsong Ma, Shiyi Lan, Alan Yuille, Jun- fei Xiao, and Chen Wei

Imbalanced gradients in rl post-training of multi-task llms.arXiv preprint arXiv:2510.19178. Yunfei Xie, Yinsong Ma, Shiyi Lan, Alan Yuille, Jun- fei Xiao, and Chen Wei. 2025. Play to generalize: Learning to reason through game play.arXiv preprint arXiv:2506.08011. Fengli Xu, Qianyue Hao, Zefang Zong, Jingwei Wang, Yunke Zhang, Jingyi Wang, Xiaochong Lan,...

arXiv 2025
[9]

arXiv preprint arXiv:2501.09686

Towards large reasoning models: A survey of reinforced reasoning with large language models. arXiv preprint arXiv:2501.09686. An Yang, Baosong Yang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Zhou, Chengpeng Li, Chengyuan Li, Dayiheng Liu, Fei Huang, Guanting Dong, Hao- ran Wei, Huan Lin, Jialong Tang, Jialin Wang, Jian Yang, Jianhong Tu, Jianwei Zhang, Jianx...

Pith/arXiv arXiv 2024
[12]

O O O <-- 3'O's in row 2

** Horizontal ** ( side - by - side in a row ) * Example of a horizontal win for Player 1 ('O') :* ``` X . . O O O <-- 3'O's in row 2 . X . ```
[13]

X O O X O <-- 3'X's in column 2

** Vertical ** ( stacked on top of each other in a column ) * Example of a vertical win for Player 2 ('X') :* ``` . X O O X O <-- 3'X's in column 2 . X . ```
[2019]

Henrik Nygren, Kari Nissinen, Raija Hämäläinen, and Bram De Wever

The effectiveness of intervention with board games: a systematic review.BioPsychoSocial medicine, 13(1):22. Henrik Nygren, Kari Nissinen, Raija Hämäläinen, and Bram De Wever. 2019. Lifelong learning: Formal, non-formal and informal learning in the context of the use of problem-solving skills in technology-rich environments.British Journal of Educational T...

2019
[2023]

Bill Yuchen Lin, Wangchunshu Zhou, Ming Shen, Pei Zhou, Chandra Bhagavatula, Yejin Choi, and Xiang Ren

Let’s verify step by step.arXiv preprint arXiv:2305.20050. Bill Yuchen Lin, Wangchunshu Zhou, Ming Shen, Pei Zhou, Chandra Bhagavatula, Yejin Choi, and Xiang Ren. 2020. CommonGen: A constrained text gen- eration challenge for generative commonsense rea- soning. InFindings of the Association for Computa- tional Linguistics: EMNLP 2020, pages 1823–1840, Onl...

Pith/arXiv arXiv 2020
[2024]

In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 17960–17967

Can large language models serve as rational players in game theory? a systematic analysis. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 17960–17967. Ralph Fevre, S Gorad, and Gareth Rees. 2001. Nec- essary and unnecessary learning: the acquisition of knowledge and “skills” in and outside employment in south wales in t...

Pith/arXiv arXiv 2001
[2025]

Hunter Lightman, Vineet Kosaraju, Yura Burda, Harri Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe

Multi-agent undercover gaming: Hallucina- tion removal via counterfactual test for multimodal reasoning.arXiv preprint arXiv:2511.11182. Hunter Lightman, Vineet Kosaraju, Yura Burda, Harri Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe

arXiv

[1] [1]

The game is played on a 3 x3 vertical grid

[2] [2]

** Winning Conditions **: The game ends when a player forms a line of 3 of their own pieces

Players take turns setting one of their pieces into any available slot . ** Winning Conditions **: The game ends when a player forms a line of 3 of their own pieces . The line can be :

[3] [3]

process

** Diagonal ** ( connected at an angle ) * Example of a diagonal win ( bottom - left to top - right ) for Player 1:* ``` . . O . O X <-- 3'O's in a diagonal line O X . ``` * Example of another diagonal win ( top - left to bottom - right ) for Player 2:* ``` X . O . X O <-- 3'X's in a diagonal line . O X ``` ** Draw Condition **: If the entire grid is fill...

2024

[4] [4]

Aixin Liu, Aoxue Mei, Bangcai Lin, Bing Xue, Bingx- uan Wang, Bingzheng Xu, Bochao Wu, Bowei Zhang, Chaofan Lin, Chen Dong, and 1 others

Deepseek-v3 technical report.arXiv preprint arXiv:2412.19437. Aixin Liu, Aoxue Mei, Bangcai Lin, Bing Xue, Bingx- uan Wang, Bingzheng Xu, Bochao Wu, Bowei Zhang, Chaofan Lin, Chen Dong, and 1 others. 2025a. Deepseek-v3. 2: Pushing the frontier of open large language models.arXiv preprint arXiv:2512.02556. Bo Liu, Leon Guertler, Simon Yu, Zichen Liu, Pengh...

Pith/arXiv arXiv 1981

[5] [6]

Maarten Sap, Hannah Rashkin, Derek Chen, Ronan LeBras, and Yejin Choi

The game between game theory and gaming simulations: design choices.Simulation & Gaming, 50(2):180–201. Maarten Sap, Hannah Rashkin, Derek Chen, Ronan LeBras, and Yejin Choi. 2019. Social iqa: Com- monsense reasoning about social interactions. In EMNLP. Sylvia Scribner and Michael Cole. 1973. Cognitive consequences of formal and informal education: New ac...

Pith/arXiv arXiv 2019

[6] [7]

Qwen Team

Game-theoretic regularized self-play align- ment of large language models.arXiv preprint arXiv:2503.00030. Qwen Team. 2024. Qwen2.5: A party of foundation models. Qwen Team. 2025. Qwen3 technical report.Preprint, arXiv:2505.09388. Cecilia Thorsen, Jan-Eric Gustafsson, and Christina Cliffordson. 2014. The influence of fluid and crystal- lized intelligence ...

Pith/arXiv arXiv 2024

[7] [8]

Yunfei Xie, Yinsong Ma, Shiyi Lan, Alan Yuille, Jun- fei Xiao, and Chen Wei

Imbalanced gradients in rl post-training of multi-task llms.arXiv preprint arXiv:2510.19178. Yunfei Xie, Yinsong Ma, Shiyi Lan, Alan Yuille, Jun- fei Xiao, and Chen Wei. 2025. Play to generalize: Learning to reason through game play.arXiv preprint arXiv:2506.08011. Fengli Xu, Qianyue Hao, Zefang Zong, Jingwei Wang, Yunke Zhang, Jingyi Wang, Xiaochong Lan,...

arXiv 2025

[8] [9]

arXiv preprint arXiv:2501.09686

Towards large reasoning models: A survey of reinforced reasoning with large language models. arXiv preprint arXiv:2501.09686. An Yang, Baosong Yang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Zhou, Chengpeng Li, Chengyuan Li, Dayiheng Liu, Fei Huang, Guanting Dong, Hao- ran Wei, Huan Lin, Jialong Tang, Jialin Wang, Jian Yang, Jianhong Tu, Jianwei Zhang, Jianx...

Pith/arXiv arXiv 2024

[9] [12]

O O O <-- 3'O's in row 2

** Horizontal ** ( side - by - side in a row ) * Example of a horizontal win for Player 1 ('O') :* ``` X . . O O O <-- 3'O's in row 2 . X . ```

[10] [13]

X O O X O <-- 3'X's in column 2

** Vertical ** ( stacked on top of each other in a column ) * Example of a vertical win for Player 2 ('X') :* ``` . X O O X O <-- 3'X's in column 2 . X . ```

[11] [2019]

Henrik Nygren, Kari Nissinen, Raija Hämäläinen, and Bram De Wever

The effectiveness of intervention with board games: a systematic review.BioPsychoSocial medicine, 13(1):22. Henrik Nygren, Kari Nissinen, Raija Hämäläinen, and Bram De Wever. 2019. Lifelong learning: Formal, non-formal and informal learning in the context of the use of problem-solving skills in technology-rich environments.British Journal of Educational T...

2019

[12] [2023]

Bill Yuchen Lin, Wangchunshu Zhou, Ming Shen, Pei Zhou, Chandra Bhagavatula, Yejin Choi, and Xiang Ren

Let’s verify step by step.arXiv preprint arXiv:2305.20050. Bill Yuchen Lin, Wangchunshu Zhou, Ming Shen, Pei Zhou, Chandra Bhagavatula, Yejin Choi, and Xiang Ren. 2020. CommonGen: A constrained text gen- eration challenge for generative commonsense rea- soning. InFindings of the Association for Computa- tional Linguistics: EMNLP 2020, pages 1823–1840, Onl...

Pith/arXiv arXiv 2020

[13] [2024]

In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 17960–17967

Can large language models serve as rational players in game theory? a systematic analysis. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 17960–17967. Ralph Fevre, S Gorad, and Gareth Rees. 2001. Nec- essary and unnecessary learning: the acquisition of knowledge and “skills” in and outside employment in south wales in t...

Pith/arXiv arXiv 2001

[14] [2025]

Hunter Lightman, Vineet Kosaraju, Yura Burda, Harri Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe

Multi-agent undercover gaming: Hallucina- tion removal via counterfactual test for multimodal reasoning.arXiv preprint arXiv:2511.11182. Hunter Lightman, Vineet Kosaraju, Yura Burda, Harri Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe

arXiv