Daily Papers

What is learned by sophisticated neural network agents such as AlphaZero? This question is of both scientific and practical interest. If the representations of strong neural networks bear no resemblance to human concepts, our ability to understand faithful explanations of their decisions will be restricted, ultimately limiting what we can achieve with neural network interpretability. In this work we provide evidence that human knowledge is acquired by the AlphaZero neural network as it trains on the game of chess. By probing for a broad range of human chess concepts we show when and where these concepts are represented in the AlphaZero network. We also provide a behavioural analysis focusing on opening play, including qualitative analysis from chess Grandmaster Vladimir Kramnik. Finally, we carry out a preliminary investigation looking at the low-level details of AlphaZero's representations, and make the resulting behavioural and representational analyses available online.

AI led chess systems to a superhuman level, yet these systems heavily rely on black-box algorithms. This is unsustainable in ensuring transparency to the end-user, particularly when these systems are responsible for sensitive decision-making. Recent interpretability work has shown that the inner representations of Deep Neural Networks (DNNs) were fathomable and contained human-understandable concepts. Yet, these methods are seldom contextualised and are often based on a single hidden state, which makes them unable to interpret multi-step reasoning, e.g. planning. In this respect, we propose contrastive sparse autoencoders (CSAE), a novel framework for studying pairs of game trajectories. Using CSAE, we are able to extract and interpret concepts that are meaningful to the chess-agent plans. We primarily focused on a qualitative analysis of the CSAE features before proposing an automated feature taxonomy. Furthermore, to evaluate the quality of our trained CSAE, we devise sanity checks to wave spurious correlations in our results.

Chess provides an ideal testbed for evaluating the reasoning, modeling, and abstraction capabilities of large language models (LLMs), as it has well-defined structure and objective ground truth while admitting a wide spectrum of skill levels. However, existing evaluations of LLM ability in chess are ad hoc and narrow in scope, making it difficult to accurately measure LLM chess understanding and how it varies with scale, post-training methodologies, or architecture choices. We present ChessQA, a comprehensive benchmark that assesses LLM chess understanding across five task categories (Structural, Motifs, Short Tactics, Position Judgment, and Semantic), which approximately correspond to the ascending abstractions that players master as they accumulate chess knowledge, from understanding basic rules and learning tactical motifs to correctly calculating tactics, evaluating positions, and semantically describing high-level concepts. In this way, ChessQA captures a more comprehensive picture of chess ability and understanding, going significantly beyond the simple move quality evaluations done previously, and offers a controlled, consistent setting for diagnosis and comparison. Furthermore, ChessQA is inherently dynamic, with prompts, answer keys, and construction scripts that can evolve as models improve. Evaluating a range of contemporary LLMs, we find persistent weaknesses across all five categories and provide results and error analyses by category. We will release the code, periodically refreshed datasets, and a public leaderboard to support further research.

There are an increasing number of domains in which artificial intelligence (AI) systems both surpass human ability and accurately model human behavior. This introduces the possibility of algorithmically-informed teaching in these domains through more relatable AI partners and deeper insights into human decision-making. Critical to achieving this goal, however, is coherently modeling human behavior at various skill levels. Chess is an ideal model system for conducting research into this kind of human-AI alignment, with its rich history as a pivotal testbed for AI research, mature superhuman AI systems like AlphaZero, and precise measurements of skill via chess rating systems. Previous work in modeling human decision-making in chess uses completely independent models to capture human style at different skill levels, meaning they lack coherence in their ability to adapt to the full spectrum of human improvement and are ultimately limited in their effectiveness as AI partners and teaching tools. In this work, we propose a unified modeling approach for human-AI alignment in chess that coherently captures human style across different skill levels and directly captures how people improve. Recognizing the complex, non-linear nature of human learning, we introduce a skill-aware attention mechanism to dynamically integrate players' strengths with encoded chess positions, enabling our model to be sensitive to evolving player skill. Our experimental results demonstrate that this unified framework significantly enhances the alignment between AI and human players across a diverse range of expertise levels, paving the way for deeper insights into human decision-making and AI-guided teaching tools.

Reasoning is a central capability of human intelligence. In recent years, with the advent of large-scale datasets, pretrained large language models have emerged with new capabilities, including reasoning. However, these models still struggle with long-term, complex reasoning tasks, such as playing chess. Based on the observation that expert chess players employ a dual approach combining long-term strategic play with short-term tactical play along with language explanation, we propose improving the reasoning capability of large language models in chess by integrating annotated strategy and tactic. Specifically, we collect a dataset named MATE, which consists of 1 million chess positions with candidate moves annotated by chess experts for strategy and tactics. We finetune the LLaMA-3-8B model and compare it against state-of-the-art commercial language models in the task of selecting better chess moves. Our experiments show that our models perform better than GPT, Claude, and Gemini models. We find that language explanations can enhance the reasoning capability of large language models.

Recent large language models (LLMs) have shown strong reasoning capabilities. However, a critical question remains: do these models possess genuine reasoning skills particularly complex strategic reasoning or are they primarily excelling at sophisticated pattern recognition within their training data? To address this question, this paper presents a chess testbed, ChessArena, to evaluate the strategic reasoning capabilities of LLMs. Chess requires complex strategic reasoning capabilities including long-term planning, strict rule comprehension, and multi-turn conversation memorization. Specifically, ChessArena is a competitive framework where LLMs play against each other, under four different play modes. The testbed is equipped with a ranking algorithm and a leaderboard. The testbed can also evaluate fine-grained capabilities including basic understanding, move selection, and puzzle solving. Over 13 LLMs with different modes are evaluated in ChessArena, playing over 800 games. The results reveal significant shortcomings in current LLMs: no model can beat Maia-1100 (a chess engine at human amateur level), while some even failed to defeat a random player that selects moves arbitrarily. We also present a strong baseline to the testbed: our fine-tuned Qwen3-8B substantially improved performance, approaching much larger state-of-the-art reasoning models.

We present an end-to-end learning method for chess, relying on deep neural networks. Without any a priori knowledge, in particular without any knowledge regarding the rules of chess, a deep neural network is trained using a combination of unsupervised pretraining and supervised training. The unsupervised training extracts high level features from a given position, and the supervised training learns to compare two chess positions and select the more favorable one. The training relies entirely on datasets of several million chess games, and no further domain specific knowledge is incorporated. The experiments show that the resulting neural network (referred to as DeepChess) is on a par with state-of-the-art chess playing programs, which have been developed through many years of manual feature selection and tuning. DeepChess is the first end-to-end machine learning-based method that results in a grandmaster-level chess playing performance.

Chess has long been a testbed for AI's quest to match human intelligence, and in recent years, chess AI systems have surpassed the strongest humans at the game. However, these systems are not human-aligned; they are unable to match the skill levels of all human partners or model human-like behaviors beyond piece movement. In this paper, we introduce Allie, a chess-playing AI designed to bridge the gap between artificial and human intelligence in this classic game. Allie is trained on log sequences of real chess games to model the behaviors of human chess players across the skill spectrum, including non-move behaviors such as pondering times and resignations In offline evaluations, we find that Allie exhibits humanlike behavior: it outperforms the existing state-of-the-art in human chess move prediction and "ponders" at critical positions. The model learns to reliably assign reward at each game state, which can be used at inference as a reward function in a novel time-adaptive Monte-Carlo tree search (MCTS) procedure, where the amount of search depends on how long humans would think in the same positions. Adaptive search enables remarkable skill calibration; in a large-scale online evaluation against players with ratings from 1000 to 2600 Elo, our adaptive search method leads to a skill gap of only 49 Elo on average, substantially outperforming search-free and standard MCTS baselines. Against grandmaster-level (2500 Elo) opponents, Allie with adaptive search exhibits the strength of a fellow grandmaster, all while learning exclusively from humans.

The rapid advancement of Generative AI has raised significant questions regarding its ability to produce creative and novel outputs. Our recent work investigates this question within the domain of chess puzzles and presents an AI system designed to generate puzzles characterized by aesthetic appeal, novelty, counter-intuitive and unique solutions. We briefly discuss our method below and refer the reader to the technical paper for more details. To assess our system's creativity, we presented a curated booklet of AI-generated puzzles to three world-renowned experts: International Master for chess compositions Amatzia Avni, Grandmaster Jonathan Levitt, and Grandmaster Matthew Sadler. All three are noted authors on chess aesthetics and the evolving role of computers in the game. They were asked to select their favorites and explain what made them appealing, considering qualities such as their creativity, level of challenge, or aesthetic design.

In this paper, I apply linguistic methods of analysis to non-linguistic data, chess plays, metaphorically equating one with the other and seeking analogies. Chess game notations are also a kind of text, and one can consider the records of moves or positions of pieces as words and statements in a certain language. In this article I show how word embeddings (word2vec) can work on chess game texts instead of natural language texts. I don't see how this representation of chess data can be used productively. It's unlikely that these vector models will help engines or people choose the best move. But in a purely academic sense, it's clear that such methods of information representation capture something important about the very nature of the game, which doesn't necessarily lead to a win.

This paper presents a new approach that integrates deep learning with computational chess, using both the Mixture of Experts (MoE) method and Monte-Carlo Tree Search (MCTS). Our methodology employs a suite of specialized models, each designed to respond to specific changes in the game's input data. This results in a framework with sparsely activated models, which provides significant computational benefits. Our framework combines the MoE method with MCTS, in order to align it with the strategic phases of chess, thus departing from the conventional ``one-for-all'' model. Instead, we utilize distinct game phase definitions to effectively distribute computational tasks across multiple expert neural networks. Our empirical research shows a substantial improvement in playing strength, surpassing the traditional single-model framework. This validates the efficacy of our integrated approach and highlights the potential of incorporating expert knowledge and strategic principles into neural network design. The fusion of MoE and MCTS offers a promising avenue for advancing machine learning architectures.

In this study, we present an innovative fusion of language models and query analysis techniques to unlock cognition in artificial intelligence. Our system seamlessly integrates a Chess engine with a language model, enabling it to predict moves and provide strategic explanations. Leveraging a vector database to achieve retrievable answer generation, our OpenSI AI system elucidates its decision-making process, bridging the gap between raw computation and human-like understanding. Our choice of Chess as the demonstration environment underscores the versatility of our approach. Beyond Chess, our system holds promise for diverse applications, from medical diagnostics to financial forecasting.

Modern natural language models such as the GPT-2/GPT-3 contain tremendous amounts of information about human belief in a consistently testable form. If these models could be shown to accurately reflect the underlying beliefs of the human beings that produced the data used to train these models, then such models become a powerful sociological tool in ways that are distinct from traditional methods, such as interviews and surveys. In this study, We train a version of the GPT-2 on a corpora of historical chess games, and then "launch" clusters of synthetic agents into the model, using text strings to create context and orientation. We compare the trajectories contained in the text generated by the agents/model and compare that to the known ground truth of the chess board, move legality, and historical patterns of play. We find that the percentages of moves by piece using the model are substantially similar from human patterns. We further find that the model creates an accurate latent representation of the chessboard, and that it is possible to plot trajectories of legal moves across the board using this knowledge.

As artificial intelligence becomes increasingly intelligent---in some cases, achieving superhuman performance---there is growing potential for humans to learn from and collaborate with algorithms. However, the ways in which AI systems approach problems are often different from the ways people do, and thus may be uninterpretable and hard to learn from. A crucial step in bridging this gap between human and artificial intelligence is modeling the granular actions that constitute human behavior, rather than simply matching aggregate human performance. We pursue this goal in a model system with a long history in artificial intelligence: chess. The aggregate performance of a chess player unfolds as they make decisions over the course of a game. The hundreds of millions of games played online by players at every skill level form a rich source of data in which these decisions, and their exact context, are recorded in minute detail. Applying existing chess engines to this data, including an open-source implementation of AlphaZero, we find that they do not predict human moves well. We develop and introduce Maia, a customized version of Alpha-Zero trained on human chess games, that predicts human moves at a much higher accuracy than existing engines, and can achieve maximum accuracy when predicting decisions made by players at a specific skill level in a tuneable way. For a dual task of predicting whether a human will make a large mistake on the next move, we develop a deep neural network that significantly outperforms competitive baselines. Taken together, our results suggest that there is substantial promise in designing artificial intelligence systems with human collaboration in mind by first accurately modeling granular human decision-making.

Current rating systems update ratings incrementally and may not always accurately reflect a player's true strength at all times, especially for rapidly improving players or very rusty players. To overcome this, we explore a method to estimate player ratings directly from game moves and clock times. We compiled a benchmark dataset from Lichess, encompassing various time controls and including move sequences and clock times. Our model architecture comprises a CNN to learn positional features, which are then integrated with clock-time data into a bidirectional LSTM, predicting player ratings after each move. The model achieved an MAE of 182 rating points in the test data. Additionally, we applied our model to the 2024 IEEE Big Data Cup Chess Puzzle Difficulty Competition dataset, predicted puzzle ratings and achieved competitive results. This model is the first to use no hand-crafted features to estimate chess ratings and also the first to output a rating prediction for each move. Our method highlights the potential of using move-based rating estimation for enhancing rating systems and potentially other applications such as cheating detection.

When solving decision-making tasks, humans typically depend on information from two key sources: (1) Historical policy data, which provides interaction replay from the environment, and (2) Analytical insights in natural language form, exposing the invaluable thought process or strategic considerations. Despite this, the majority of preceding research focuses on only one source: they either use historical replay exclusively to directly learn policy or value functions, or engaged in language model training utilizing mere language corpus. In this paper, we argue that a powerful autonomous agent should cover both sources. Thus, we propose ChessGPT, a GPT model bridging policy learning and language modeling by integrating data from these two sources in Chess games. Specifically, we build a large-scale game and language dataset related to chess. Leveraging the dataset, we showcase two model examples ChessCLIP and ChessGPT, integrating policy learning and language modeling. Finally, we propose a full evaluation framework for evaluating language model's chess ability. Experimental results validate our model and dataset's effectiveness. We open source our code, model, and dataset at https://github.com/waterhorse1/ChessGPT.

Chess recognition is the task of extracting the chess piece configuration from a chessboard image. Current approaches use a pipeline of separate, independent, modules such as chessboard detection, square localization, and piece classification. Instead, we follow the deep learning philosophy and explore an end-to-end approach to directly predict the configuration from the image, thus avoiding the error accumulation of the sequential approaches and eliminating the need for intermediate annotations. Furthermore, we introduce a new dataset, Chess Recognition Dataset (ChessReD), that consists of 10,800 real photographs and their corresponding annotations. In contrast to existing datasets that are synthetically rendered and have only limited angles, ChessReD has photographs captured from various angles using smartphone cameras; a sensor choice made to ensure real-world applicability. Our approach in chess recognition on the introduced challenging benchmark dataset outperforms related approaches, successfully recognizing the chess pieces' configuration in 15.26% of ChessReD's test images. This accuracy may seem low, but it is ~7x better than the current state-of-the-art and reflects the difficulty of the problem. The code and data are available through: https://github.com/ThanosM97/end-to-end-chess-recognition.

The recent breakthrough successes in machine learning are mainly attributed to scale: namely large-scale attention-based architectures and datasets of unprecedented scale. This paper investigates the impact of training at scale for chess. Unlike traditional chess engines that rely on complex heuristics, explicit search, or a combination of both, we train a 270M parameter transformer model with supervised learning on a dataset of 10 million chess games. We annotate each board in the dataset with action-values provided by the powerful Stockfish 16 engine, leading to roughly 15 billion data points. Our largest model reaches a Lichess blitz Elo of 2895 against humans, and successfully solves a series of challenging chess puzzles, without any domain-specific tweaks or explicit search algorithms. We also show that our model outperforms AlphaZero's policy and value networks (without MCTS) and GPT-3.5-turbo-instruct. A systematic investigation of model and dataset size shows that strong chess performance only arises at sufficient scale. To validate our results, we perform an extensive series of ablations of design choices and hyperparameters.

While large language models have made strides in natural language processing, their proficiency in complex reasoning tasks requiring formal language comprehension, such as chess, remains less investigated. This paper probes the performance of ChatGPT, a sophisticated language model by OpenAI in tackling such complex reasoning tasks, using chess as a case study. Through robust metrics examining both the legality and quality of moves, we assess ChatGPT's understanding of the chessboard, adherence to chess rules, and strategic decision-making abilities. Our evaluation identifies limitations within ChatGPT's attention mechanism that affect its formal language comprehension and uncovers the model's underdeveloped self-regulation abilities. Our study also reveals ChatGPT's propensity for a coherent strategy in its gameplay and a noticeable uptick in decision-making assertiveness when the model is presented with a greater volume of natural language or possesses a more lucid understanding of the state of the chessboard. These findings contribute to the growing exploration of language models' abilities beyond natural language processing, providing valuable information for future research towards models demonstrating human-like cognitive abilities.

Generative sequence models are typically trained on sample sequences from natural or formal languages. It is a crucial question whether -- or to what extent -- sample-based training is able to capture the true structure of these languages, often referred to as the ``world model''. Theoretical results indicate that we can hope for soundness at best, that is, generating valid sequences, but not necessarily all of them. However, it is still important to have practical tools that are able to verify whether a given sequence model is sound. In this study, we focus on chess, as it is a domain that provides enough complexity while having a simple rule-based world model. We propose adversarial sequence generation for verifying the soundness of the sequence model. Our adversaries generate valid sequences so as to force the sequence model to generate an invalid next move prediction. Apart from the falsification of soundness, this method is also suitable for a more fine-grained analysis of the failure modes and the effects of different choices during training. To demonstrate this, we propose a number of methods for adversarial sequence generation and evaluate the approach on a large set of chess models. We train models on random as well as high-quality chess games, using several training recipes. We find that none of the models are sound, but some training techniques and dataset choices are able to improve soundness remarkably. We also investigate the potential application of board state probes in both our training and attack methods. Our findings indicate that the extracted board states have no causal role in next token prediction in most of the models.

This work demonstrates that natural language transformers can support more generic strategic modeling, particularly for text-archived games. In addition to learning natural language skills, the abstract transformer architecture can generate meaningful moves on a chessboard. With further fine-tuning, the transformer learns complex gameplay by training on 2.8 million chess games in Portable Game Notation. After 30,000 training steps, OpenAI's Generative Pre-trained Transformer (GPT-2) optimizes weights for 774 million parameters. This fine-tuned Chess Transformer generates plausible strategies and displays game formations identifiable as classic openings, such as English or the Slav Exchange. Finally, in live play, the novel model demonstrates a human-to-transformer interface that correctly filters illegal moves and provides a novel method to challenge the transformer's chess strategies. We anticipate future work will build on this transformer's promise, particularly in other strategy games where features can capture the underlying complex rule syntax from simple but expressive player annotations.

Transformer language models have made tremendous strides in natural language understanding tasks. However, the complexity of natural language makes it challenging to ascertain how accurately these models are tracking the world state underlying the text. Motivated by this issue, we consider the task of language modeling for the game of chess. Unlike natural language, chess notations describe a simple, constrained, and deterministic domain. Moreover, we observe that the appropriate choice of chess notation allows for directly probing the world state, without requiring any additional probing-related machinery. We find that: (a) With enough training data, transformer language models can learn to track pieces and predict legal moves with high accuracy when trained solely on move sequences. (b) For small training sets providing access to board state information during training can yield significant improvements. (c) The success of transformer language models is dependent on access to the entire game history i.e. "full attention". Approximating this full attention results in a significant performance drop. We propose this testbed as a benchmark for future work on the development and analysis of transformer language models.

We develop a method of adapting the AlphaZero model to General Game Playing (GGP) that focuses on faster model generation and requires less knowledge to be extracted from the game rules. The dataset generation uses MCTS playing instead of self-play; only the value network is used, and attention layers replace the convolutional ones. This allows us to abandon any assumptions about the action space and board topology. We implement the method within the Regular Boardgames GGP system and show that we can build models outperforming the UCT baseline for most games efficiently.

We present a new dataset containing 10K human-annotated games of Go and show how these natural language annotations can be used as a tool for model interpretability. Given a board state and its associated comment, our approach uses linear probing to predict mentions of domain-specific terms (e.g., ko, atari) from the intermediate state representations of game-playing agents like AlphaGo Zero. We find these game concepts are nontrivially encoded in two distinct policy networks, one trained via imitation learning and another trained via reinforcement learning. Furthermore, mentions of domain-specific terms are most easily predicted from the later layers of both models, suggesting that these policy networks encode high-level abstractions similar to those used in the natural language annotations.

The game of chess is the most widely-studied domain in the history of artificial intelligence. The strongest programs are based on a combination of sophisticated search techniques, domain-specific adaptations, and handcrafted evaluation functions that have been refined by human experts over several decades. In contrast, the AlphaGo Zero program recently achieved superhuman performance in the game of Go, by tabula rasa reinforcement learning from games of self-play. In this paper, we generalise this approach into a single AlphaZero algorithm that can achieve, tabula rasa, superhuman performance in many challenging domains. Starting from random play, and given no domain knowledge except the game rules, AlphaZero achieved within 24 hours a superhuman level of play in the games of chess and shogi (Japanese chess) as well as Go, and convincingly defeated a world-champion program in each case.

Large Language Models (LLMs) exhibit remarkable capabilities, yet it remains unclear to what extent these reflect sophisticated recall (crystallized intelligence) or reasoning ability (fluid intelligence). We introduce chess as a controlled testbed for disentangling these faculties. Leveraging the game's structure and scalable engine evaluations, we construct a taxonomy of positions varying in training corpus proximity--ranging from common states solvable by memorization to novel ones requiring first-principles reasoning. We systematically evaluate multiple GPT generations under varying reasoning intensities. Our analysis reveals a clear gradient: performance consistently degrades as fluid intelligence demands increase. Notably, in out-of-distribution tasks, performance collapses to random levels. While newer models improve, progress slows significantly for tasks outside the training distribution. Furthermore, while reasoning-augmented inference improves performance, its marginal benefit per token decreases with distributional proximity. These results suggest current architectures remain limited in systematic generalization, highlighting the need for mechanisms beyond scale to achieve robust fluid intelligence.

While large language models perform well on a range of complex tasks (e.g., text generation, question answering, summarization), robust multi-step planning and reasoning remains a considerable challenge for them. In this paper we show that search-based planning can significantly improve LLMs' playing strength across several board games (Chess, Fischer Random / Chess960, Connect Four, and Hex). We introduce, compare and contrast two major approaches: In external search, the model guides Monte Carlo Tree Search (MCTS) rollouts and evaluations without calls to an external engine, and in internal search, the model directly generates in-context a linearized tree of potential futures and a resulting final choice. Both build on a language model pre-trained on relevant domain knowledge, capturing the transition and value functions across these games. We find that our pre-training method minimizes hallucinations, as our model is highly accurate regarding state prediction and legal moves. Additionally, both internal and external search indeed improve win-rates against state-of-the-art bots, even reaching Grandmaster-level performance in chess while operating on a similar move count search budget per decision as human Grandmasters. The way we combine search with domain knowledge is not specific to board games, suggesting direct extensions into more general language model inference and training techniques.

This paper introduces SC2LE (StarCraft II Learning Environment), a reinforcement learning environment based on the StarCraft II game. This domain poses a new grand challenge for reinforcement learning, representing a more difficult class of problems than considered in most prior work. It is a multi-agent problem with multiple players interacting; there is imperfect information due to a partially observed map; it has a large action space involving the selection and control of hundreds of units; it has a large state space that must be observed solely from raw input feature planes; and it has delayed credit assignment requiring long-term strategies over thousands of steps. We describe the observation, action, and reward specification for the StarCraft II domain and provide an open source Python-based interface for communicating with the game engine. In addition to the main game maps, we provide a suite of mini-games focusing on different elements of StarCraft II gameplay. For the main game maps, we also provide an accompanying dataset of game replay data from human expert players. We give initial baseline results for neural networks trained from this data to predict game outcomes and player actions. Finally, we present initial baseline results for canonical deep reinforcement learning agents applied to the StarCraft II domain. On the mini-games, these agents learn to achieve a level of play that is comparable to a novice player. However, when trained on the main game, these agents are unable to make significant progress. Thus, SC2LE offers a new and challenging environment for exploring deep reinforcement learning algorithms and architectures.

While Generative AI rapidly advances in various domains, generating truly creative, aesthetic, and counter-intuitive outputs remains a challenge. This paper presents an approach to tackle these difficulties in the domain of chess puzzles. We start by benchmarking Generative AI architectures, and then introduce an RL framework with novel rewards based on chess engine search statistics to overcome some of those shortcomings. The rewards are designed to enhance a puzzle's uniqueness, counter-intuitiveness, diversity, and realism. Our RL approach dramatically increases counter-intuitive puzzle generation by 10x, from 0.22\% (supervised) to 2.5\%, surpassing existing dataset rates (2.1\%) and the best Lichess-trained model (0.4\%). Our puzzles meet novelty and diversity benchmarks, retain aesthetic themes, and are rated by human experts as more creative, enjoyable, and counter-intuitive than composed book puzzles, even approaching classic compositions. Our final outcome is a curated booklet of these AI-generated puzzles, which is acknowledged for creativity by three world-renowned experts.

Otter is a 15.3M-parameter human chess AI that predicts human move selection by modeling play as a time-aware, sequential process rather than treating each position in isolation. It combines two conditioning signals: (1) a move history encoder that conditions predictions on the last 20 moves, capturing opening preferences, positional drift, and intra-game behavioral tendencies; and (2) a time control module that modulates predictions based on clock pressure. Otter is trained on 6.1 billion positions from 117 million Lichess rapid games over 30 days on a single T4 GPU. Otter achieves 55.23% top-1 and 90.95% top-5 move-prediction accuracy, surpassing the prior state-of-the-art human chess model, Maia 2, with far fewer parameters and less training data. Across 11 Elo brackets (<1100 to >=2000), accuracy peaks at 57.38% in the 1900-1999 bracket. These results show that modeling chess as a time-aware, sequential activity yields more human-accurate move prediction than position-only approaches, using a smaller model. Code, trained models, and complete training logs are publicly released.

Predicting the relative value of any given chess piece in a position remains an open challenge, as a piece's contribution depends on its spatial relationships with every other piece on the board. We demonstrate that incorporating the state of the full chess board via latent position representations derived using a CNN-based autoencoder significantly improves accuracy for MLP-based piece value prediction architectures. Using a dataset of over 12 million piece-value pairs gathered from Grandmaster-level games, with ground-truth labels generated by Stockfish 17, our enhanced piece value predictor significantly outperforms context-independent MLP-based systems, reducing validation mean absolute error by 16% and predicting relative piece value within approximately 0.65 pawns. More generally, our findings suggest that encoding the full problem state as context provides useful inductive bias for predicting the contribution of any individual component.

Modern chess language models are dense transformers trained on millions of games played by thousands of high-rated individuals. However, these monolithic networks tend to collapse into mode-averaged behavior, where stylistic boundaries are blurred, and rare but effective strategies are suppressed. To counteract homogenization, we introduce Mixture-of-Masters (MoM), the first chess mixture-of-experts model with small-sized GPT experts emulating world-class grandmasters. Each expert is trained with a combination of self-supervised learning and reinforcement learning guided by chess-specific rewards. For each move, a post-hoc learnable gating network selects the most appropriate persona to channel depending on the game state, allowing MoM to switch its style dynamically--e.g., Tal's offensive vocation or Petrosian's defensive solidity. When evaluated against Stockfish on unseen standard games, MoM outperforms both dense individual expert networks and popular GPT baselines trained on aggregated data, while ensuring generation variety, control, and interpretability.

This report presents Giraffe, a chess engine that uses self-play to discover all its domain-specific knowledge, with minimal hand-crafted knowledge given by the programmer. Unlike previous attempts using machine learning only to perform parameter-tuning on hand-crafted evaluation functions, Giraffe's learning system also performs automatic feature extraction and pattern recognition. The trained evaluation function performs comparably to the evaluation functions of state-of-the-art chess engines - all of which containing thousands of lines of carefully hand-crafted pattern recognizers, tuned over many years by both computer chess experts and human chess masters. Giraffe is the most successful attempt thus far at using end-to-end machine learning to play chess.

We demonstrate that small pretrained foundational generative language models with millions of parameters can learn the latent rules of a process from data associated with the process. Inspired by Stefan Zweig's novella "Schachnovelle," also known as "The Royal Game" in English, we show that 28M and 125M parameter pretrained foundational small language models (SLMs) can be instruction fine-tuned with 1,000-to-1,000,000 examples to learn the rules of chess, propose legal moves, and accurately solve chess problems. We also explore the impact of successive language model fine-tuning epochs on improved outcomes and demonstrate reductions in model hallucinations by increasing the number of instruction fine-tuning examples.

Game playing has long served as a fundamental benchmark for evaluating Artificial General Intelligence (AGI). While Large Language Models (LLMs) have demonstrated impressive capabilities in general reasoning, their effectiveness in spatial strategic reasoning, which is critical for complex and fully observable board games, remains insufficiently explored. In this work, we adopt Chinese Chess (Xiangqi) as a challenging and rich testbed due to its intricate rules and spatial complexity. To advance LLMs' strategic competence in such environments, we propose a training framework tailored to Xiangqi, built upon a large-scale dataset of five million board-move pairs enhanced with expert annotations and engine evaluations. Building on this foundation, we introduce Xiangqi-R1, a 7B-parameter model trained in multi-stage manner: (1) fine-tuning for legal move prediction to capture basic spatial rules, (2) incorporating strategic annotations to improve decision-making, and (3) applying reinforcement learning via Group Relative Policy Optimization (GRPO) with multi-dimensional reward signals to enhance reasoning stability. Our Experimental results indicate that, despite their size and power, general-purpose LLMs struggle to achieve satisfactory performance in these tasks. Compared to general-purpose LLMs, Xiangqi-R1 greatly advances with an 18% rise in move legality and a 22% boost in analysis accuracy. Our results point to a promising path for creating general strategic intelligence in spatially complex areas.

Recent work has fine-tuned language models on chess data and reported high benchmark scores as evidence that the resulting models can understand the rules of chess, play full chess games at a professional level, or generate human-readable explanations grounded in expert knowledge. We train KinGPT, a 25M-parameter character-level language model trained only on (position, best-move) pairs, who exceeds 3B-parameter ChessGPT on a 600-puzzle mate-in-N suite and 4B-parameter C1-4B over a 20-theme puzzle benchmark. We examine several claims made in existing literature regarding chess-trained language models and assert that their impressive benchmark performance is largely explained by pattern-matching. We also demonstrate how LLM-Modulo, a verifier-in-the-loop framework, raises RedPajama 3B's best move accuracy from 1.2% to 21.2% and move generation validity from 19.3% to 95.3% on mate-in-N chess puzzles, comparable to gains achieved from ChessGPT's fine-tuning on chess-specific web corpora at a fraction of the cost. Our results illustrate how pairing a general LLM with an external verifier offers a more flexible alternative to directly training on synthetic data for well-defined domains. We open source all training/evaluation code, datasets, puzzle samples, and KinGPT model checkpoints for reproducibility.

How can you get a language model to reason in a task it natively struggles with? We study how reasoning evolves in a language model -- from supervised fine-tuning (SFT) to reinforcement learning (RL) -- by analyzing how a set of theoretically-inspired datasets impacts language model performance in chess. We find that fine-tuning a model to directly predict the best move leads to effective RL and the strongest downstream performance -- however, the RL step elicits unfaithful reasoning (reasoning inconsistent with the chosen move). Alternatively, training on multi-move trajectories yields comparable downstream performance with faithful reasoning and more stable RL. We show that RL induces a substantial positive shift in the distribution of move quality and reduces hallucination rates as a side effect. Finally, we find several SFT-checkpoint metrics -- metrics spanning evaluation performance, hallucination rates, and reasoning quality -- to be predictive of post-RL model performance. We release checkpoints and final models as well as training data, evaluations, and code which allowed us to surpass leading open-source reasoning models in chess with a 7B-parameter model.

Large language models (LLMs) have achieved striking successes on many benchmarks, yet recent studies continue to expose fundamental weaknesses. In particular, tasks that require abstract reasoning remain challenging, often because they use representations such as grids, symbols, or visual patterns that differ from the natural language data LLMs are trained on. In this paper, we introduce Concept, a simple word-guessing board game, as a benchmark for probing abductive reasoning in a representation that is much closer to LLM pre-training data: natural language. Our results show that this game, easily solved by humans (with a success rate of over 90\%), is still very challenging for state-of-the-art LLMs (no model exceeds 40\% success rate). Specifically, we observe that LLMs struggle with interpreting other players' strategic intents, and with correcting initial hypotheses given sequential information updates. In addition, we extend the evaluation across multiple languages, and find that the LLM performance drops further in lower-resource languages (Dutch, French, and Spanish) compared to English.

As more non-AI experts use complex AI systems for daily tasks, there has been an increasing effort to develop methods that produce explanations of AI decision making that are understandable by non-AI experts. Towards this effort, leveraging higher-level concepts and producing concept-based explanations have become a popular method. Most concept-based explanations have been developed for classification techniques, and we posit that the few existing methods for sequential decision making are limited in scope. In this work, we first contribute a desiderata for defining concepts in sequential decision making settings. Additionally, inspired by the Protege Effect which states explaining knowledge often reinforces one's self-learning, we explore how concept-based explanations of an RL agent's decision making can in turn improve the agent's learning rate, as well as improve end-user understanding of the agent's decision making. To this end, we contribute a unified framework, State2Explanation (S2E), that involves learning a joint embedding model between state-action pairs and concept-based explanations, and leveraging such learned model to both (1) inform reward shaping during an agent's training, and (2) provide explanations to end-users at deployment for improved task performance. Our experimental validations, in Connect 4 and Lunar Lander, demonstrate the success of S2E in providing a dual-benefit, successfully informing reward shaping and improving agent learning rate, as well as significantly improving end user task performance at deployment time.

Reviewing the progress in artificial intelligence over the past decade, various significant advances (e.g. object detection, image generation, large language models) have enabled AI systems to produce more semantically meaningful outputs and achieve widespread adoption in internet scenarios. Nevertheless, AI systems still struggle when it comes to understanding and interacting with the physical world. This reveals an important issue: relying solely on semantic-level concepts learned from internet data (e.g. texts, images) to understand the physical world is far from sufficient -- machine intelligence currently lacks an effective way to learn about the physical world. This research introduces the idea of analytic concept -- representing the concepts related to the physical world through programs of mathematical procedures, providing machine intelligence a portal to perceive, reason about, and interact with the physical world. Except for detailing the design philosophy and providing guidelines for the application of analytic concepts, this research also introduce about the infrastructure that has been built around analytic concepts. I aim for my research to contribute to addressing these questions: What is a proper abstraction of general concepts in the physical world for machine intelligence? How to systematically integrate structured priors with neural networks to constrain AI systems to comply with physical laws?

Concepts serve as fundamental abstractions that support human reasoning and categorization. However, it remains unclear whether large language models truly capture such conceptual structures or primarily rely on surface-level pattern memorization. Existing benchmarks are largely static and fact oriented, which limits their ability to probe fine-grained semantic understanding and makes them vulnerable to data leakage and overfitting. To address this limitation, we introduce CK-Arena, a dynamic benchmark for conceptual knowledge evaluation based on a multi agent social deduction game, namely the Undercover game. In this setting, LLM based agents are assigned subtly different concept words and must describe, distinguish, and infer conceptual properties from others' statements. Model performance is evaluated through both game level outcomes and the semantic quality of generated descriptions. Furthermore, CK-Arena leverages the interaction process to automatically construct high quality question answering data for fine grained diagnostic analysis. Experimental results show that conceptual understanding varies substantially across models and categories, and is not strictly aligned with overall model capability. The data and code are available at the project homepage: https://ck-arena.site.

Reinforcement learning (RL) has become central to improving large language models (LLMs) on complex reasoning tasks, yet RL post-training is largely studied in isolation from the pretraining that precedes it. As a result, two basic questions remain open: (1) how do pretraining choices (model size, data) shape the returns to RL compute, and (2) what does RL actually do to the model? These questions are difficult to study in the standard LLM setting: pretraining corpora are vast and uncontrolled, making it hard to attribute behaviors to pretraining versus RL, and systematic compute sweeps across both stages are prohibitively expensive. To address these challenges, we use chess as a controlled testbed for studying reasoning across the full pretraining-to-post-training pipeline. We follow the standard LLM training pipeline by pretraining language models from 5M to 1B parameters on human chess games, supervised fine-tuning on synthetic reasoning traces, and running RL on chess puzzles with verifiable rewards. Using this framework, we find that the post-RL performance at given RL compute level is well-predicted from the pretraining loss, and slope of the RL reward curves improves approximately linearly with the pretraining tokens. Beyond scaling, we find that RL does not simply sharpen the SFT policy: on easy puzzles it amplifies correct moves the SFT policy already preferred, while on hard puzzles it surfaces correct moves that were nearly absent under SFT. We further test whether our findings transfer beyond chess by training a 1B language model on math-domain text, where the same predictive pattern emerges: longer-pretrained checkpoints reach higher post-RL performance and improve faster under RL. In sum, we provide a quantitative account of the pretraining-to-RL interface and a controlled testbed for studying the science of reasoning across the full pretraining-to-post-training pipeline.

Language models show a surprising range of capabilities, but the source of their apparent competence is unclear. Do these networks just memorize a collection of surface statistics, or do they rely on internal representations of the process that generates the sequences they see? We investigate this question by applying a variant of the GPT model to the task of predicting legal moves in a simple board game, Othello. Although the network has no a priori knowledge of the game or its rules, we uncover evidence of an emergent nonlinear internal representation of the board state. Interventional experiments indicate this representation can be used to control the output of the network and create "latent saliency maps" that can help explain predictions in human terms.

Large Language Models (LLMs) reasoning abilities are increasingly being applied to classical board and card games, but the dominant approach -- involving prompting for direct move generation -- has significant drawbacks. It relies on the model's implicit fragile pattern-matching capabilities, leading to frequent illegal moves and strategically shallow play. Here we introduce an alternative approach: We use the LLM to translate natural language rules and game trajectories into a formal, executable world model represented as Python code. This generated model -- comprising functions for state transition, legal move enumeration, and termination checks -- serves as a verifiable simulation engine for high-performance planning algorithms like Monte Carlo tree search (MCTS). In addition, we prompt the LLM to generate heuristic value functions (to make MCTS more efficient), and inference functions (to estimate hidden states in imperfect information games). Our method offers three distinct advantages compared to directly using the LLM as a policy: (1) Verifiability: The generated CWM serves as a formal specification of the game's rules, allowing planners to algorithmically enumerate valid actions and avoid illegal moves, contingent on the correctness of the synthesized model; (2) Strategic Depth: We combine LLM semantic understanding with the deep search power of classical planners; and (3) Generalization: We direct the LLM to focus on the meta-task of data-to-code translation, enabling it to adapt to new games more easily. We evaluate our agent on 10 different games, of which 4 are novel and created for this paper. 5 of the games are fully observed (perfect information), and 5 are partially observed (imperfect information). We find that our method outperforms or matches Gemini 2.5 Pro in 9 out of the 10 considered games.

Closed drafting or "pick and pass" is a popular game mechanic where each round players select a card or other playable element from their hand and pass the rest to the next player. Games employing closed drafting make for great studies on memory and turn order due to their explicitly calculable memory of other players' hands. In this paper, we establish first-principle benchmarks for studying model-free reinforcement learning algorithms and their comparative ability to learn memory in a popular family of closed drafting games called "Sushi Go Party!", producing state-of-the-art results on this environment along the way. Furthermore, as Sushi Go Party! can be expressed as a set of closely-related games based on the set of cards in play, we quantify the generalizability of reinforcement learning algorithms trained on various sets of cards, establishing key trends between generalized performance and the set distance between the train and evaluation game configurations. Finally, we fit decision rules to interpret the strategy of the learned models and compare them to the ranking preferences of human players, finding intuitive common rules and intriguing new moves.

Language models often lack grounded reasoning capabilities in specialized domains where training data is scarce but bespoke systems excel. We introduce a general framework for distilling expert system reasoning into natural language chain-of-thought explanations, enabling compact models to acquire domain expertise and the ability to generate faithful, grounded explanations. Rather than distilling only final outputs, we capture the full reasoning process, transforming opaque expert computations into transparent, step-by-step explanations. We demonstrate this approach in chess, a canonical reasoning domain where language models continue to underperform. Our 4B parameter model, C1, advances from a near-zero baseline to 48.1% accuracy, outperforming all open-source models and most frontier proprietary systems. Notably, C1 surpasses its distillation teacher and generates solutions in two orders of magnitude fewer tokens than baselines. Unlike prior neural chess approaches that predict only best moves, C1 generates explainable solutions revealing strategic reasoning. Our pipeline combines supervised fine-tuning and reinforcement learning with theme-balanced data sampling for comprehensive tactical coverage. Master Distillation demonstrates how to inject expert-level knowledge into compact models for under-optimized domains, offering a recipe for unlocking RLVR where LLMs lack sufficient base capabilities.

How do sequence models represent their decision-making process? Prior work suggests that Othello-playing neural network learned nonlinear models of the board state (Li et al., 2023). In this work, we provide evidence of a closely related linear representation of the board. In particular, we show that probing for "my colour" vs. "opponent's colour" may be a simple yet powerful way to interpret the model's internal state. This precise understanding of the internal representations allows us to control the model's behaviour with simple vector arithmetic. Linear representations enable significant interpretability progress, which we demonstrate with further exploration of how the world model is computed.

Recent developments in the field of model-based RL have proven successful in a range of environments, especially ones where planning is essential. However, such successes have been limited to deterministic fully-observed environments. We present a new approach that handles stochastic and partially-observable environments. Our key insight is to use discrete autoencoders to capture the multiple possible effects of an action in a stochastic environment. We use a stochastic variant of Monte Carlo tree search to plan over both the agent's actions and the discrete latent variables representing the environment's response. Our approach significantly outperforms an offline version of MuZero on a stochastic interpretation of chess where the opponent is considered part of the environment. We also show that our approach scales to DeepMind Lab, a first-person 3D environment with large visual observations and partial observability.

Constructing agents with planning capabilities has long been one of the main challenges in the pursuit of artificial intelligence. Tree-based planning methods have enjoyed huge success in challenging domains, such as chess and Go, where a perfect simulator is available. However, in real-world problems the dynamics governing the environment are often complex and unknown. In this work we present the MuZero algorithm which, by combining a tree-based search with a learned model, achieves superhuman performance in a range of challenging and visually complex domains, without any knowledge of their underlying dynamics. MuZero learns a model that, when applied iteratively, predicts the quantities most directly relevant to planning: the reward, the action-selection policy, and the value function. When evaluated on 57 different Atari games - the canonical video game environment for testing AI techniques, in which model-based planning approaches have historically struggled - our new algorithm achieved a new state of the art. When evaluated on Go, chess and shogi, without any knowledge of the game rules, MuZero matched the superhuman performance of the AlphaZero algorithm that was supplied with the game rules.

The rapid advancement of large language models (LLMs) has accelerated their application in reasoning, with strategic reasoning drawing increasing attention. To evaluate LLMs' strategic reasoning capabilities, game theory, with its concise structure, has become a preferred approach. However, current research focuses on a limited selection of games, resulting in low coverage. Classic game scenarios risk data leakage, and existing benchmarks often lack extensibility, making them inadequate for evaluating state-of-the-art models. To address these challenges, we propose TMGBench, a benchmark with comprehensive game type coverage, novel scenarios, and flexible organization. Specifically, we incorporate all 144 game types summarized by the Robinson-Goforth topology of 2x2 games, constructed as classic games. We also employ synthetic data generation to create diverse, higher-quality scenarios through topic guidance and human inspection, referred to as story-based games. Lastly, we provide a sustainable framework for increasingly powerful LLMs by treating these games as atomic units and organizing them into more complex forms via sequential, parallel, and nested structures. Our comprehensive evaluation of mainstream LLMs covers tests on rational reasoning, robustness, Theory-of-Mind (ToM), and reasoning in complex forms. Results reveal flaws in accuracy, consistency, and varying mastery of ToM. Additionally, o1-mini, OpenAI's latest reasoning model, achieved accuracy rates of 66.6%, 60.0%, and 70.0% on sequential, parallel, and nested games, highlighting TMGBench's challenges.

We developed a benchmark set to assess the generalization of state-of-the-art large language models on problems beyond linguistic tasks and evaluate it on a systematic progression of GPT models (GPT-3.5, GPT-4, GPT-4o, GPT-4o-mini). Using simple games like Tic-Tac-Toe, Connect Four, Battleship, and a Shape Recognition Game, all encoded in ASCII, we test strategic capabilities and spatial reasoning, core abilities any artificial intelligence would need to master for solving problems in chemistry. To probe generalization, we introduce two new games for spatial logic: LEGO Connect Language (LCL) and Guess-the-SMILES (GtS), a operationally simple chemistry benchmark. Our results show that GPT models provide meaningful responses for several tasks but, generally, perform poorly. A systematic performance progression with increased model capabilities (GPT-3.5, GPT-4, GPT-4o) is only observed for 4 out of the 7 benchmark tasks. All models consistently struggle with Battleship, LCL, and GtS. This suggests that while GPT models can emulate conversational proficiency and basic rule comprehension, they have limited generalization with respect to strategy and spatial reasoning. Particularly poor performance is observed for interpreting molecular graphs when encoded in ASCII. The results provided by our open-source benchmark suite (https://github.com/BlueVelvetSackOfGoldPotatoes/child-play{ChildPlay GitHub Repository}) caution against claims of emergent intelligence in GPT models, which appear more specialized than general.

Li et al. (2023) used the Othello board game as a test case for the ability of GPT-2 to induce world models, and were followed up by Nanda et al. (2023b). We briefly discuss the original experiments, expanding them to include more language models with more comprehensive probing. Specifically, we analyze sequences of Othello board states and train the model to predict the next move based on previous moves. We evaluate seven language models (GPT-2, T5, Bart, Flan-T5, Mistral, LLaMA-2, and Qwen2.5) on the Othello task and conclude that these models not only learn to play Othello, but also induce the Othello board layout. We find that all models achieve up to 99% accuracy in unsupervised grounding and exhibit high similarity in the board features they learned. This provides considerably stronger evidence for the Othello World Model Hypothesis than previous works.

Children learn though play. We introduce the analogous idea of learning programs through play. In this approach, a program induction system (the learner) is given a set of tasks and initial background knowledge. Before solving the tasks, the learner enters an unsupervised playing stage where it creates its own tasks to solve, tries to solve them, and saves any solutions (programs) to the background knowledge. After the playing stage is finished, the learner enters the supervised building stage where it tries to solve the user-supplied tasks and can reuse solutions learnt whilst playing. The idea is that playing allows the learner to discover reusable general programs on its own which can then help solve the user-supplied tasks. We claim that playing can improve learning performance. We show that playing can reduce the textual complexity of target concepts which in turn reduces the sample complexity of a learner. We implement our idea in Playgol, a new inductive logic programming system. We experimentally test our claim on two domains: robot planning and real-world string transformations. Our experimental results suggest that playing can substantially improve learning performance. We think that the idea of playing (or, more verbosely, unsupervised bootstrapping for supervised program induction) is an important contribution to the problem of developing program induction approaches that self-discover BK.

In this paper, we present a conceptual model game to examine the dynamics of asymmetric interactions in games with imperfect information. The game involves two agents with starkly contrasting capabilities: one agent can take actions but has no information of the state of the game, whereas the other agent has perfect information of the state but cannot act or observe the other agent's actions. This duality manifests an extreme form of asymmetry, and how differing abilities influence the possibility of attaining common knowledge. Using Kripke structures and epistemic logic we demonstrate that, under these conditions, common knowledge of the current game state becomes unattainable. Our findings advance the discussion on the strategic limitations of knowledge in environments where information and action are unevenly distributed.

Latent, or silent, reasoning lets language models carry out intermediate computation in continuous vector space instead of words, and is widely assumed to function as an internal scratchpad the model consults during inference. Whether that assumption survives reinforcement learning has not been tested directly: existing causal analyses of latent reasoning are confined to math and logic tasks, comparing reliance on thoughts within one checkpoint, never before and after RL. We train a chess-playing model through a staged latent-reasoning curriculum followed by reinforcement learning, and find legality climbs monotonically to 61% (from a 48% pre-RL baseline) while checkmate confabulation is eliminated entirely. To locate this gain, we run a six-condition causal intervention suite on the same model before and after RL: substituting or noising the thought vectors leaves performance unchanged, ablating them costs only mild degradation, and only exact-zero vectors cause collapse. This robustness gap is itself the finding: under exact-zero corruption, legality collapses to 1% pre-RL versus 9% post-RL, a gap that survives correction across the full battery. A 10x-larger replication of the post-RL checkpoint's own battery confirms this: removing the thoughts, with or without restoring sequence length, also reaches significance; substitution and noise remain indistinguishable from baseline. RL appears to add robustness to disruption, not reliance on thought content. These results push back against the field's default assumption that latent thoughts function as an actively consulted inference-time scratchpad, and instead indicate latent reasoning's principal effect here is shaping the model's parameters during training. We also demonstrate a working RL gain in chess, where multiple groups report the same latent-reasoning-plus-RL recipe failing to improve accuracy over SFT.

To make AI systems broadly useful for challenging real-world tasks, we need them to learn complex human goals and preferences. One approach to specifying complex goals asks humans to judge during training which agent behaviors are safe and useful, but this approach can fail if the task is too complicated for a human to directly judge. To help address this concern, we propose training agents via self play on a zero sum debate game. Given a question or proposed action, two agents take turns making short statements up to a limit, then a human judges which of the agents gave the most true, useful information. In an analogy to complexity theory, debate with optimal play can answer any question in PSPACE given polynomial time judges (direct judging answers only NP questions). In practice, whether debate works involves empirical questions about humans and the tasks we want AIs to perform, plus theoretical questions about the meaning of AI alignment. We report results on an initial MNIST experiment where agents compete to convince a sparse classifier, boosting the classifier's accuracy from 59.4% to 88.9% given 6 pixels and from 48.2% to 85.2% given 4 pixels. Finally, we discuss theoretical and practical aspects of the debate model, focusing on potential weaknesses as the model scales up, and we propose future human and computer experiments to test these properties.

We demonstrate LLM agent specification gaming by instructing models to win against a chess engine. We find reasoning models like o1 preview and DeepSeek-R1 will often hack the benchmark by default, while language models like GPT-4o and Claude 3.5 Sonnet need to be told that normal play won't work to hack. We improve upon prior work like (Hubinger et al., 2024; Meinke et al., 2024; Weij et al., 2024) by using realistic task prompts and avoiding excess nudging. Our results suggest reasoning models may resort to hacking to solve difficult problems, as observed in OpenAI (2024)'s o1 Docker escape during cyber capabilities testing.

While Large Language Models (LLMs) have demonstrated their proficiency in complex reasoning tasks, their performance in dynamic, interactive, and competitive scenarios - such as business strategy and stock market analysis - remains underexplored. To bridge this gap, we formally explore the dynamic reasoning capabilities of LLMs for decision-making in rapidly evolving environments. We introduce two game theory-based pilot challenges that mirror the complexities of real-world dynamic decision-making. These challenges are well-defined, enabling clear, controllable, and precise evaluation of LLMs' dynamic reasoning abilities. Through extensive experiments, we find that existing reasoning methods tend to falter in dynamic settings that require k-level thinking - a key concept not tackled by previous works. To address this, we propose a novel reasoning approach for LLMs, named "K-Level Reasoning". This approach adopts the perspective of rivals to recursively employ k-level thinking based on available historical information, which significantly improves the prediction accuracy of rivals' subsequent moves and informs more strategic decision-making. This research not only sets a robust quantitative benchmark for the assessment of dynamic reasoning but also markedly enhances the proficiency of LLMs in dynamic contexts.

In this work, we describe strategies and provide case-study activities that can be used to examine the properties of superposition, entanglement, tagging, complementarity, and measurement in quantum curricula geared for teacher training. Having a solid foundation in these conceptual ideas is critical for educators who will be adopting quantum ideas within the classroom. Yet they are some of the most difficult concepts to master. We show how one can systematically develop these conceptual foundations with thought experiments on light and with thought experiments that employ the Stern-Gerlach experiment. We emphasize the importance of computer animations in aiding the instruction on these concepts.

Our understanding of the visual world is centered around various concept axes, characterizing different aspects of visual entities. While different concept axes can be easily specified by language, e.g. color, the exact visual nuances along each axis often exceed the limitations of linguistic articulations, e.g. a particular style of painting. In this work, our goal is to learn a language-informed visual concept representation, by simply distilling large pre-trained vision-language models. Specifically, we train a set of concept encoders to encode the information pertinent to a set of language-informed concept axes, with an objective of reproducing the input image through a pre-trained Text-to-Image (T2I) model. To encourage better disentanglement of different concept encoders, we anchor the concept embeddings to a set of text embeddings obtained from a pre-trained Visual Question Answering (VQA) model. At inference time, the model extracts concept embeddings along various axes from new test images, which can be remixed to generate images with novel compositions of visual concepts. With a lightweight test-time finetuning procedure, it can also generalize to novel concepts unseen at training.

Recent large language models (LLMs) have shown indications of mathematical reasoning ability. However it has not been clear how they would fare on more challenging competition-level problems. And while self-generated verbalizations of intermediate reasoning steps (i.e., chain-of-thought prompting) have been shown to be helpful, whether LLMs can make use of helpful side information such as problem-specific hints has not been investigated before. In this paper, we propose a challenging benchmark dataset for enabling such analyses. The Concept and Hint-Annotated Math Problems (CHAMP) consists of high school math competition problems, annotated with concepts, or general math facts, and hints, or problem-specific tricks. These annotations allow us to explore the effects of additional information, such as relevant hints, misleading concepts, or related problems. This benchmark is difficult, with the best model only scoring 58.1% in standard settings. With concepts and hints, performance sometimes improves, indicating that some models can make use of such side information. We further annotate model-generated solutions for their correctness. Using this corpus, we find that models often arrive at the correct final answer through wrong reasoning steps. In addition, we test whether models are able to verify these solutions, and find that most models struggle. The dataset and code are available on the project website.

The deployment of ever-larger machine learning models reflects a growing consensus that the more expressive the modelx2013and the more data one has access tox2013the more one can improve performance. As models get deployed in a variety of real world scenarios, they inevitably face strategic environments. In this work, we consider the natural question of how the interplay of models and strategic interactions affects scaling laws. We find that strategic interactions can break the conventional view of scaling lawsx2013meaning that performance does not necessarily monotonically improve as models get larger and/ or more expressive (even with infinite data). We show the implications of this phenomenon in several contexts including strategic regression, strategic classification, and multi-agent reinforcement learning through examples of strategic environments in whichx2013by simply restricting the expressivity of one's model or policy classx2013one can achieve strictly better equilibrium outcomes. Motivated by these examples, we then propose a new paradigm for model-selection in games wherein an agent seeks to choose amongst different model classes to use as their action set in a game.

Recent advances in large language models have enabled natural language to serve as a flexible interface for controlling complex systems, but often at the cost of large-scale multimodal training or weakened domain-specific inductive biases. In structured decision-making domains such as chess, specialized policy networks achieve strong performance but lack semantic controllability, while prompt-conditioned language models are more flexible yet typically exhibit weaker domain grounding. We propose UniMaia, a framework for prompt-conditioned policy modulation that adapts a frozen Lc0-based chess policy network using a parameter-efficient text encoder and a ControlNet-style conditioning mechanism. UniMaia enables semantic control over gameplay, including opening selection and player strength, while preserving the pretrained policy representations. We further introduce UniMaia-Aux, which incorporates auxiliary temporal conditioning and behavioral prediction objectives. To support this work, we construct a large-scale metadata-augmented Lichess dataset, develop a semi-automated prompt-generation pipeline, and introduce benchmarks spanning both prompt-conditioned and metadata-conditioned settings. UniMaia achieves state-of-the-art expected accuracy on several prompt-conditioned benchmarks and competitive top-move accuracy on general instruction-following tasks, while remaining competitive with dedicated metadata-conditioned approaches on human move prediction benchmarks. UniMaia-Aux further improves expected accuracy and behavioral modeling across several evaluation settings, with modest trade-offs in top-move accuracy. Overall, our results demonstrate that prompt-conditioned control of domain-specific policy networks is feasible without end-to-end multimodal training, while highlighting trade-offs between controllability and predictive performance.

As Large Language Models (LLMs) are integrated into critical real-world applications, their strategic and logical reasoning abilities are increasingly crucial. This paper evaluates LLMs' reasoning abilities in competitive environments through game-theoretic tasks, e.g., board and card games that require pure logic and strategic reasoning to compete with opponents. We first propose GTBench, a language-driven environment composing 10 widely-recognized tasks, across a comprehensive game taxonomy: complete versus incomplete information, dynamic versus static, and probabilistic versus deterministic scenarios. Then, we investigate two key problems: (1) Characterizing game-theoretic reasoning of LLMs; (2) LLM-vs-LLM competitions as reasoning evaluation. We observe that (1) LLMs have distinct behaviors regarding various gaming scenarios; for example, LLMs fail in complete and deterministic games yet they are competitive in probabilistic gaming scenarios; (2) Open-source LLMs, e.g., CodeLlama-34b-Instruct, are less competitive than commercial LLMs, e.g., GPT-4, in complex games. In addition, code-pretraining greatly benefits strategic reasoning, while advanced reasoning methods such as Chain-of-Thought (CoT) and Tree-of-Thought (ToT) do not always help. Detailed error profiles are also provided for a better understanding of LLMs' behavior.

Sparse autoencoders have recently produced dictionaries of high-dimensional vectors corresponding to the universe of concepts represented by large language models. We find that this concept universe has interesting structure at three levels: 1) The "atomic" small-scale structure contains "crystals" whose faces are parallelograms or trapezoids, generalizing well-known examples such as (man-woman-king-queen). We find that the quality of such parallelograms and associated function vectors improves greatly when projecting out global distractor directions such as word length, which is efficiently done with linear discriminant analysis. 2) The "brain" intermediate-scale structure has significant spatial modularity; for example, math and code features form a "lobe" akin to functional lobes seen in neural fMRI images. We quantify the spatial locality of these lobes with multiple metrics and find that clusters of co-occurring features, at coarse enough scale, also cluster together spatially far more than one would expect if feature geometry were random. 3) The "galaxy" scale large-scale structure of the feature point cloud is not isotropic, but instead has a power law of eigenvalues with steepest slope in middle layers. We also quantify how the clustering entropy depends on the layer.

This paper introduces Alympics (Olympics for Agents), a systematic simulation framework utilizing Large Language Model (LLM) agents for game theory research. Alympics creates a versatile platform for studying complex game theory problems, bridging the gap between theoretical game theory and empirical investigations by providing a controlled environment for simulating human-like strategic interactions with LLM agents. In our pilot case study, the "Water Allocation Challenge," we explore Alympics through a challenging strategic game focused on the multi-round auction on scarce survival resources. This study demonstrates the framework's ability to qualitatively and quantitatively analyze game determinants, strategies, and outcomes. Additionally, we conduct a comprehensive human assessment and an in-depth evaluation of LLM agents in strategic decision-making scenarios. Our findings not only expand the understanding of LLM agents' proficiency in emulating human strategic behavior but also highlight their potential in advancing game theory knowledge, thereby enriching our understanding of both game theory and empowering further research into strategic decision-making domains with LLM agents. Codes, prompts, and all related resources are available at https://github.com/microsoft/Alympics.

The utilization of artificial intelligence (AI) in card games has been a well-explored subject within AI research for an extensive period. Recent advancements have propelled AI programs to showcase expertise in intricate card games such as Mahjong, DouDizhu, and Texas Hold'em. In this work, we aim to develop an AI program for an exceptionally complex and popular card game called GuanDan. This game involves four players engaging in both competitive and cooperative play throughout a long process to upgrade their level, posing great challenges for AI due to its expansive state and action space, long episode length, and complex rules. Employing reinforcement learning techniques, specifically Deep Monte Carlo (DMC), and a distributed training framework, we first put forward an AI program named DanZero for this game. Evaluation against baseline AI programs based on heuristic rules highlights the outstanding performance of our bot. Besides, in order to further enhance the AI's capabilities, we apply policy-based reinforcement learning algorithm to GuanDan. To address the challenges arising from the huge action space, which will significantly impact the performance of policy-based algorithms, we adopt the pre-trained model to facilitate the training process and the achieved AI program manages to achieve a superior performance.

Useful social science theories predict behavior across settings. However, applying a theory to make predictions in new settings is challenging: rarely can it be done without ad hoc modifications to account for setting-specific factors. We argue that AI agents put in simulations of those novel settings offer an alternative for applying theory, requiring minimal or no modifications. We present an approach for building such "general" agents that use theory-grounded natural language instructions, existing empirical data, and knowledge acquired by the underlying AI during training. To demonstrate the approach in settings where no data from that data-generating process exists--as is often the case in applied prediction problems--we design a heterogeneous population of 883,320 novel games. AI agents are constructed using human data from a small set of conceptually related but structurally distinct "seed" games. In preregistered experiments, on average, agents predict initial human play in a random sample of 1,500 games from the population better than (i) a cognitive hierarchy model, (ii) game-theoretic equilibria, and (iii) out-of-the-box agents. For a small set of separate novel games, these simulations predict responses from a new sample of human subjects better even than the most plausibly relevant published human data.

LLMs can appear cautious in risk decision-making tasks, yet cautious-looking outputs do not necessarily indicate alignment with human decision-making mechanisms. We investigate this distinction using the St. Petersburg game as a controlled testbed, a classical paradox in which the expected payoff is infinite, yet humans typically report low, finite willingness to pay. We evaluate 28 LLMs with a structured prompt suite that includes the original game; controlled decision variants that perturb truncation, repeated play, numeric endowment, and occupational identity; a human-perspective prompt that asks models to reason as human decision makers; and paired comparisons between base models and their instruction-tuned counterparts. In the original game, most models generate finite bids, creating the appearance of human-like risk behavior. However, this outcome-level resemblance masks substantial mechanism-level differences. The controlled variants reveal that rather than maintaining human-like behavior seen in the original game, models often shift to conditionally and computationally rational behavior. Human-cue prompting and instruction tuning often lower bids and reduce some visible pathologies, but most mechanism-level response patterns remain largely unchanged. These findings show that behavioral alignment in risk decision-making can be surface-level: LLMs may produce human-like risk decisions without exhibiting human-consistent mechanisms. High-stakes evaluations of LLM decision-making should therefore move beyond outcome similarity and examine whether the alignment is supported by mechanism-level consistency.

Integer sequences are of central importance to the modeling of concepts admitting complete finitary descriptions. We introduce a novel view on the learning of such concepts and lay down a set of benchmarking tasks aimed at conceptual understanding by machine learning models. These tasks indirectly assess model ability to abstract, and challenge them to reason both interpolatively and extrapolatively from the knowledge gained by observing representative examples. To further aid research in knowledge representation and reasoning, we present FACT, the Finitary Abstraction Comprehension Toolkit. The toolkit surrounds a large dataset of integer sequences comprising both organic and synthetic entries, a library for data pre-processing and generation, a set of model performance evaluation tools, and a collection of baseline model implementations, enabling the making of the future advancements with ease.

In the post-AlphaGo era, there has been a renewed interest in search techniques such as Monte Carlo Tree Search (MCTS), particularly in their application to Large Language Models (LLMs). This renewed attention is driven by the recognition that current next-token prediction models often lack the ability for long-term planning. Is it possible to instill search-like abilities within the models to enhance their planning abilities without relying on explicit search? We propose DiffuSearch , a model that does implicit search by looking into the future world via discrete diffusion modeling. We instantiate DiffuSearch on a classical board game, Chess, where explicit search is known to be essential. Through extensive controlled experiments, we show DiffuSearch outperforms both the searchless and explicit search-enhanced policies. Specifically, DiffuSearch outperforms the one-step policy by 19.2% and the MCTS-enhanced policy by 14% on action accuracy. Furthermore, DiffuSearch demonstrates a notable 30% enhancement in puzzle-solving abilities compared to explicit search-based policies, along with a significant 540 Elo increase in game-playing strength assessment. These results indicate that implicit search via discrete diffusion is a viable alternative to explicit search over a one-step policy. All codes are publicly available at https://github.com/HKUNLP/DiffuSearch{https://github.com/HKUNLP/DiffuSearch}.

The pursuit of artificial agents that can learn to master complex environments has led to remarkable successes, yet prevailing deep reinforcement learning methods often rely on immense experience, encoding their knowledge opaquely within neural network weights. We propose a different paradigm, one in which an agent learns to play by reasoning and planning. We introduce Cogito, ergo ludo (CEL), a novel agent architecture that leverages a Large Language Model (LLM) to build an explicit, language-based understanding of its environment's mechanics and its own strategy. Starting from a tabula rasa state with no prior knowledge (except action set), CEL operates on a cycle of interaction and reflection. After each episode, the agent analyzes its complete trajectory to perform two concurrent learning processes: Rule Induction, where it refines its explicit model of the environment's dynamics, and Strategy and Playbook Summarization, where it distills experiences into an actionable strategic playbook. We evaluate CEL on diverse grid-world tasks (i.e., Minesweeper, Frozen Lake, and Sokoban), and show that the CEL agent successfully learns to master these games by autonomously discovering their rules and developing effective policies from sparse rewards. Ablation studies confirm that the iterative process is critical for sustained learning. Our work demonstrates a path toward more general and interpretable agents that not only act effectively but also build a transparent and improving model of their world through explicit reasoning on raw experience.

We study the Levine hat problem, a classic combinatorial puzzle introduced by Lionel Levine in 2010. This problem involves a game in which n geq 2 players, each seeing an infinite stack of hats on each of their teammates' heads but not on their own, must simultaneously guess the index of a black hat on their own stack. If one of the players fails to do so, the team loses collectively. The players must therefore come up with a good strategy before the game starts. While the optimal winning probability V_{n} remains unknown even for n=2, we make three key advances. First, we develop a novel geometric framework for representing strategies through measurable functions, providing a new expression of V_{n} and a unified treatment of the game for finite and for infinite stacks via integral formulations. Secondly, we construct a new strategy K_{5} that reaches the conjectured optimal probability of victory : 0.35. We also show that K_{5} is part of a larger class of strategies that allow us to improve current bounds and resolve conjectured inequalities. Finally, we introduce and entirely solve a continuous generalization of the problem, demonstrating that extending to uncountable hat stacks increases the optimal winning probability to exactly 1/2. This generalization naturally leads to a broader and smoother strategic framework, within which we also describe how to compute optimal responses to a range of strategies.

We introduce a new language learning setting relevant to building adaptive natural language interfaces. It is inspired by Wittgenstein's language games: a human wishes to accomplish some task (e.g., achieving a certain configuration of blocks), but can only communicate with a computer, who performs the actual actions (e.g., removing all red blocks). The computer initially knows nothing about language and therefore must learn it from scratch through interaction, while the human adapts to the computer's capabilities. We created a game in a blocks world and collected interactions from 100 people playing it. First, we analyze the humans' strategies, showing that using compositionality and avoiding synonyms correlates positively with task performance. Second, we compare computer strategies, showing how to quickly learn a semantic parsing model from scratch, and that modeling pragmatics further accelerates learning for successful players.

Human explanations of high-level decisions are often expressed in terms of key concepts the decisions are based on. In this paper, we study such concept-based explainability for Deep Neural Networks (DNNs). First, we define the notion of completeness, which quantifies how sufficient a particular set of concepts is in explaining a model's prediction behavior based on the assumption that complete concept scores are sufficient statistics of the model prediction. Next, we propose a concept discovery method that aims to infer a complete set of concepts that are additionally encouraged to be interpretable, which addresses the limitations of existing methods on concept explanations. To define an importance score for each discovered concept, we adapt game-theoretic notions to aggregate over sets and propose ConceptSHAP. Via proposed metrics and user studies, on a synthetic dataset with apriori-known concept explanations, as well as on real-world image and language datasets, we validate the effectiveness of our method in finding concepts that are both complete in explaining the decisions and interpretable. (The code is released at https://github.com/chihkuanyeh/concept_exp)

Reasoning is not just about solving problems -- it is also about evaluating which problems are worth solving at all. Evaluations of artificial intelligence (AI) systems primarily focused on problem solving, historically by studying how models play games such as chess and Go. In this paper, we advocate for a new paradigm that assesses AI systems' evaluation of games. First, we introduce a formalism for evaluating such evaluations. We then leverage a large-scale dataset of over 100 novel board games and over 450 human judgments to compare evaluations produced by modern language and reasoning models against those of people and symbolic computational agents. We consider two kinds of evaluative queries: assessing the payoff (or fairness) and the funness of games. These queries span two dimensions relevant to the design of evaluations of AI evaluations: how complex a query is to compute and how difficult a query is to quantify. Our results show that reasoning models are generally more aligned to people in their evaluations of games than non-reasoning language models. However, we observe a non-monotonic relationship: as models get closer to game-theoretic optimal, their fit to human data weakens. We also observe more "jaggedness" across models for assessing funness, in line with the greater difficulty of quantifying this query. Across queries and games, reasoning models show highly variable and unpredictable resource usage when assessing queries, pointing to the importance of imbuing more resource-rational meta-reasoning in language and reasoning models.

While researchers are finding concepts represented as linear directions in language models, a bag of linear directions fails to capture relational structure. To better understand this dichotomy, we study a model with known linear representations, but trained in a highly structured domain -- the board game Othello. While the model's internal board-state representation is linearly decodable, we find additional structure in the form of tensor product representations (TPRs). We train TPR probes to recover shared structure amongst the linear probes, yielding a factorization into square-embeddings, color-embeddings, and a binding matrix that composes them to construct the model's board-state representation. We find geometric signatures within the weights of our TPR probe that align with the structure of the board, but perhaps more importantly, that the linear probes can be recovered directly from the parameters of our TPR probe. Our findings suggest that directional representations may be projections of more structured underlying representations.

In this paper, we propose the use of the popular word-based board game Codenames as a suitable benchmark for evaluating the reasoning capabilities of Large Language Models (LLMs). Codenames presents a highly interesting challenge for achieving successful AI performance, requiring both a sophisticated understanding of language, theory of mind, and epistemic reasoning capabilities. Prior attempts to develop agents for Codenames have largely relied on word embedding techniques, which have a limited vocabulary range and perform poorly when paired with differing approaches. LLMs have demonstrated enhanced reasoning and comprehension capabilities for language-based tasks, but can still suffer in lateral thinking challenges. We evaluate the capabilities of several state-of-the-art LLMs, including GPT-4o, Gemini 1.5, Claude 3.5 Sonnet, and Llama 3.1, across a variety of board setups. Our results indicate that while certain LLMs perform better than others overall, different models exhibit varying emergent behaviours during gameplay and excel at specific roles. We also evaluate the performance of different combinations of LLMs when playing cooperatively together, demonstrating that LLM agents are more generalisable to a wider range of teammates than prior techniques.

This paper argues that the finite horizon paradox, where game theory contradicts intuition, stems from the limitations of standard number systems in modelling the cognitive perception of infinity. To address this issue, we propose a new framework based on Alternative Set Theory (AST). This framework represents different cognitive perspectives on a long history of events using distinct topologies. These topologies define an indiscernibility equivalence that formally treats huge, indistinguishable quantities as equivalent. This offers criterion-dependent resolutions to long-standing paradoxes, such as Selten's chain store paradox and Rosenthal's centipede game. Our framework reveals new intuitive subgame perfect equilibria, the characteristics of which depend on the chosen temporal perspective and payoff evaluation. Ultimately, by grounding its mathematical foundation in different modes of human cognition, our work expands the explanatory power of game theory for long-horizon scenarios.

Research has a long history of discussing what is superior in predicting certain outcomes: statistical methods or the human brain. This debate has repeatedly been sparked off by the remarkable technological advances in the field of artificial intelligence (AI), such as solving tasks like object and speech recognition, achieving significant improvements in accuracy through deep-learning algorithms (Goodfellow et al. 2016), or combining various methods of computational intelligence, such as fuzzy logic, genetic algorithms, and case-based reasoning (Medsker 2012). One of the implicit promises that underlie these advancements is that machines will 1 day be capable of performing complex tasks or may even supersede humans in performing these tasks. This triggers new heated debates of when machines will ultimately replace humans (McAfee and Brynjolfsson 2017). While previous research has proved that AI performs well in some clearly defined tasks such as playing chess, playing Go or identifying objects on images, it is doubted that the development of an artificial general intelligence (AGI) which is able to solve multiple tasks at the same time can be achieved in the near future (e.g., Russell and Norvig 2016). Moreover, the use of AI to solve complex business problems in organizational contexts occurs scarcely, and applications for AI that solve complex problems remain mainly in laboratory settings instead of being implemented in practice. Since the road to AGI is still a long one, we argue that the most likely paradigm for the division of labor between humans and machines in the next decades is Hybrid Intelligence. This concept aims at using the complementary strengths of human intelligence and AI, so that they can perform better than each of the two could separately (e.g., Kamar 2016).

In recent years, concept-based approaches have emerged as some of the most promising explainability methods to help us interpret the decisions of Artificial Neural Networks (ANNs). These methods seek to discover intelligible visual 'concepts' buried within the complex patterns of ANN activations in two key steps: (1) concept extraction followed by (2) importance estimation. While these two steps are shared across methods, they all differ in their specific implementations. Here, we introduce a unifying theoretical framework that comprehensively defines and clarifies these two steps. This framework offers several advantages as it allows us: (i) to propose new evaluation metrics for comparing different concept extraction approaches; (ii) to leverage modern attribution methods and evaluation metrics to extend and systematically evaluate state-of-the-art concept-based approaches and importance estimation techniques; (iii) to derive theoretical guarantees regarding the optimality of such methods. We further leverage our framework to try to tackle a crucial question in explainability: how to efficiently identify clusters of data points that are classified based on a similar shared strategy. To illustrate these findings and to highlight the main strategies of a model, we introduce a visual representation called the strategic cluster graph. Finally, we present https://serre-lab.github.io/Lens, a dedicated website that offers a complete compilation of these visualizations for all classes of the ImageNet dataset.

Let L be a finite lattice. An Ungar move sends an element xin L to the meet of {x}cup T, where T is a subset of the set of elements covered by x. We introduce the following Ungar game. Starting at the top element of L, two players -- Atniss and Eeta -- take turns making nontrivial Ungar moves; the first player who cannot do so loses the game. Atniss plays first. We say L is an Atniss win (respectively, Eeta win) if Atniss (respectively, Eeta) has a winning strategy in the Ungar game on L. We first prove that the number of principal order ideals in the weak order on S_n that are Eeta wins is O(0.95586^nn!). We then consider a broad class of intervals in Young's lattice that includes all principal order ideals, and we characterize the Eeta wins in this class; we deduce precise enumerative results concerning order ideals in rectangles and type-A root posets. We also characterize and enumerate principal order ideals in Tamari lattices that are Eeta wins. Finally, we conclude with some open problems and a short discussion of the computational complexity of Ungar games.

Recent advances in artificial intelligence have been driven by the presence of increasingly realistic and complex simulated environments. However, many of the existing environments provide either unrealistic visuals, inaccurate physics, low task complexity, restricted agent perspective, or a limited capacity for interaction among artificial agents. Furthermore, many platforms lack the ability to flexibly configure the simulation, making the simulated environment a black-box from the perspective of the learning system. In this work, we propose a novel taxonomy of existing simulation platforms and discuss the highest level class of general platforms which enable the development of learning environments that are rich in visual, physical, task, and social complexity. We argue that modern game engines are uniquely suited to act as general platforms and as a case study examine the Unity engine and open source Unity ML-Agents Toolkit. We then survey the research enabled by Unity and the Unity ML-Agents Toolkit, discussing the kinds of research a flexible, interactive and easily configurable general platform can facilitate.

AI has achieved remarkable mastery over games such as Chess, Go, and Poker, and even Jeopardy, but the rich variety of standardized exams has remained a landmark challenge. Even in 2016, the best AI system achieved merely 59.3% on an 8th Grade science exam challenge. This paper reports unprecedented success on the Grade 8 New York Regents Science Exam, where for the first time a system scores more than 90% on the exam's non-diagram, multiple choice (NDMC) questions. In addition, our Aristo system, building upon the success of recent language models, exceeded 83% on the corresponding Grade 12 Science Exam NDMC questions. The results, on unseen test questions, are robust across different test years and different variations of this kind of test. They demonstrate that modern NLP methods can result in mastery on this task. While not a full solution to general question-answering (the questions are multiple choice, and the domain is restricted to 8th Grade science), it represents a significant milestone for the field.

The categories of open learners (due to Fong, Spivak and Tuy\'eras) and open games (due to the present author, Ghani, Winschel and Zahn) bear a very striking and unexpected similarity. The purpose of this short note is to prove that there is a faithful symmetric monoidal functor from the former to the latter, which means that any supervised neural network (without feedback or other complicating features) can be seen as an open game in a canonical way. Roughly, each parameter is controlled by a different player, and the game's best response relation encodes the dynamics of gradient descent. We suggest paths for further work exploiting the link.

Large language models handle single-turn generation well, but multi-turn interactions still require the model to reconstruct user intent and task state from an expanding token history because internal representations do not persist across turns. This token-first paradigm leads to drift, inconsistent reasoning modes, and growing prompts as conversations deepen. We propose CORE, a concept-first interaction layer that improves multi-turn stability without modifying model weights. CORE combines a small library of universal cognitive operators with a persistent Local Concept - a compact semantic state capturing the task, constraints, preferences, and intermediate results. Each model call receives only this concept state, the user's latest instruction, and the selected operator, eliminating the need to replay full history. A preliminary prototype simulating CORE's behavior shows about 42% reduction in cumulative prompt tokens, though this number reflects prototype conditions and should not be interpreted as a real-world performance estimate. CORE offers a model-agnostic mechanism that separates conceptual reasoning from language generation, suggesting a scalable direction for more stable multi-turn systems.

Competitive Pok\'emon Singles (CPS) is a popular strategy game where players learn to exploit their opponent based on imperfect information in battles that can last more than one hundred stochastic turns. AI research in CPS has been led by heuristic tree search and online self-play, but the game may also create a platform to study adaptive policies trained offline on large datasets. We develop a pipeline to reconstruct the first-person perspective of an agent from logs saved from the third-person perspective of a spectator, thereby unlocking a dataset of real human battles spanning more than a decade that grows larger every day. This dataset enables a black-box approach where we train large sequence models to adapt to their opponent based solely on their input trajectory while selecting moves without explicit search of any kind. We study a progression from imitation learning to offline RL and offline fine-tuning on self-play data in the hardcore competitive setting of Pok\'emon's four oldest (and most partially observed) game generations. The resulting agents outperform a recent LLM Agent approach and a strong heuristic search engine. While playing anonymously in online battles against humans, our best agents climb to rankings inside the top 10% of active players.

Starting with early successes in computer vision tasks, deep learning based techniques have since overtaken state of the art approaches in a multitude of domains. However, it has been demonstrated time and again that these techniques fail to capture semantic context and logical constraints, instead often relying on spurious correlations to arrive at the answer. Since application of deep learning techniques to critical scenarios are dependent on adherence to domain specific constraints, several attempts have been made to address this issue. One limitation holding back a thorough exploration of this area, is a lack of suitable datasets which feature a rich set of rules. In order to address this, we present the VALUE (Vision And Logical Understanding Evaluation) Dataset, consisting of 200,000+ annotated images and an associated rule set, based on the popular board game - chess. The curated rule set considerably constrains the set of allowable predictions, and are designed to probe key semantic abilities like localization and enumeration. Alongside standard metrics, additional metrics to measure performance with regards to logical consistency is presented. We analyze several popular and state of the art vision models on this task, and show that, although their performance on standard metrics are laudable, they produce a plethora of incoherent results, indicating that this dataset presents a significant challenge for future works.

Modern generative models demonstrate impressive capabilities, likely stemming from an ability to identify and manipulate abstract concepts underlying their training data. However, fundamental questions remain: what determines the concepts a model learns, the order in which it learns them, and its ability to manipulate those concepts? To address these questions, we propose analyzing a model's learning dynamics via a framework we call the concept space, where each axis represents an independent concept underlying the data generating process. By characterizing learning dynamics in this space, we identify how the speed at which a concept is learned, and hence the order of concept learning, is controlled by properties of the data we term concept signal. Further, we observe moments of sudden turns in the direction of a model's learning dynamics in concept space. Surprisingly, these points precisely correspond to the emergence of hidden capabilities, i.e., where latent interventions show the model possesses the capability to manipulate a concept, but these capabilities cannot yet be elicited via naive input prompting. While our results focus on synthetically defined toy datasets, we hypothesize a general claim on emergence of hidden capabilities may hold: generative models possess latent capabilities that emerge suddenly and consistently during training, though a model might not exhibit these capabilities under naive input prompting.

Mahjong is a popular multi-player imperfect-information game developed in China in the late 19th-century, with some very challenging features for AI research. Sanma, being a 3-player variant of the Japanese Riichi Mahjong, possesses unique characteristics including fewer tiles and, consequently, a more aggressive playing style. It is thus challenging and of great research interest in its own right, but has not yet been explored. In this paper, we present Meowjong, an AI for Sanma using deep reinforcement learning. We define an informative and compact 2-dimensional data structure for encoding the observable information in a Sanma game. We pre-train 5 convolutional neural networks (CNNs) for Sanma's 5 actions -- discard, Pon, Kan, Kita and Riichi, and enhance the major action's model, namely the discard model, via self-play reinforcement learning using the Monte Carlo policy gradient method. Meowjong's models achieve test accuracies comparable with AIs for 4-player Mahjong through supervised learning, and gain a significant further enhancement from reinforcement learning. Being the first ever AI in Sanma, we claim that Meowjong stands as a state-of-the-art in this game.

Complex games have long been an important benchmark for testing the progress of artificial intelligence algorithms. AlphaGo, AlphaZero, and MuZero have defeated top human players in Go and Chess, garnering widespread societal attention towards artificial intelligence. Concurrently, large language models (LLMs) have exhibited remarkable capabilities across various tasks, raising the question of whether LLMs can achieve similar success in complex games. In this paper, we explore the potential of LLMs in mastering complex card games. We systematically assess the learning capabilities of LLMs across eight diverse card games, evaluating the impact of fine-tuning on high-quality gameplay data, and examining the models' ability to retain general capabilities while mastering these games. Our findings indicate that: (1) LLMs can approach the performance of strong game AIs through supervised fine-tuning on high-quality data, (2) LLMs can achieve a certain level of proficiency in multiple complex card games simultaneously, with performance augmentation for games with similar rules and conflicts for dissimilar ones, and (3) LLMs experience a decline in general capabilities when mastering complex games, but this decline can be mitigated by integrating a certain amount of general instruction data. The evaluation results demonstrate strong learning ability and versatility of LLMs. The code is available at https://github.com/THUDM/LLM4CardGame

Humans invent new words when there is a rising demand for a new useful concept (e.g., doomscrolling). We explore and validate a similar idea in our communication with LLMs: introducing new words to better understand and control the models, expanding on the recently introduced neologism learning. This method introduces a new word by adding a new word embedding and training with examples that exhibit the concept with no other changes in model parameters. We show that adding a new word allows for control of concepts such as flattery, incorrect answers, text length, as well as more complex concepts in AxBench. We discover that neologisms can also further our understanding of the model via self-verbalization: models can describe what each new word means to them in natural language, like explaining that a word that represents a concept of incorrect answers means ``a lack of complete, coherent, or meaningful answers...'' To validate self-verbalizations, we introduce plug-in evaluation: we insert the verbalization into the context of a model and measure whether it controls the target concept. In some self-verbalizations, we find machine-only synonyms: words that seem unrelated to humans but cause similar behavior in machines. Finally, we show how neologism learning can jointly learn multiple concepts in multiple words.

As part of a series on first-principles modeling of cognitive functions, this paper attempts to provide a mathematical formulation of thinking and perception. It formally derives slow thinking or more generally, active perception, and encompasses the design, training and inference of slow thinking large language models. Our starting point is the lifting and projection of probability distributions on the observable and latent spaces, with the objective of representing complex data distributions by simple function families such as neural networks. A theory called "active lifting" is proposed, based on the sampling of latent sequences and an intrinsic drive to reduce uncertainty with maximum rate. It derives a large design space, containing the slow thinking models in a subspace that we call the static theory. These models are positioned on the representation hierarchy and sampler hierarchy induced by the static theory, and can be upgraded by climbing the two hierarchies. Active lifting further derives an inference process with an internal time axis, and a training objective that resembles minimum-length coding as well as the invention of languages. Thus, it characterizes the agency of perception, including the emergence of the slow thinking formats. Technical by-products of this theory include a three-stage pathway for improving slow thinking models, a unified approach to constructing encoders and generative models for all data modalities, a priori formation of human-like visual representations, and a possible solution to policy collapse.

Generative AI technologies are growing in power, utility, and use. As generative technologies are being incorporated into mainstream applications, there is a need for guidance on how to design those applications to foster productive and safe use. Based on recent research on human-AI co-creation within the HCI and AI communities, we present a set of seven principles for the design of generative AI applications. These principles are grounded in an environment of generative variability. Six principles are focused on designing for characteristics of generative AI: multiple outcomes & imperfection; exploration & control; and mental models & explanations. In addition, we urge designers to design against potential harms that may be caused by a generative model's hazardous output, misuse, or potential for human displacement. We anticipate these principles to usefully inform design decisions made in the creation of novel human-AI applications, and we invite the community to apply, revise, and extend these principles to their own work.

We introduce a real-time strategy game environment based on Generals.io, a game with thousands of weekly active players. Our environment is fully compatible with Gymnasium and PettingZoo and is capable of running thousands of frames per second on commodity hardware. We also present a reference agent, trained with supervised pre-training and self-play, which reached the top 0.003% of the 1v1 human leaderboard after only 36 hours on a single H100 GPU. To accelerate learning, we incorporate potential-based reward shaping and memory features. Our contributions of a modular RTS benchmark and a competitive baseline agent provide an accessible yet challenging platform for advancing multi-agent reinforcement learning research. The documented code, together with examples and tutorials, is available at https://github.com/strakam/generals-bots.

Automatically generating novel and interesting games is a complex task. Challenges include representing game rules in a computationally workable form, searching through the large space of potential games under most such representations, and accurately evaluating the originality and quality of previously unseen games. Prior work in automated game generation has largely focused on relatively restricted rule representations and relied on domain-specific heuristics. In this work, we explore the generation of novel games in the comparatively expansive Ludii game description language, which encodes the rules of over 1000 board games in a variety of styles and modes of play. We draw inspiration from recent advances in large language models and evolutionary computation in order to train a model that intelligently mutates and recombines games and mechanics expressed as code. We demonstrate both quantitatively and qualitatively that our approach is capable of generating new and interesting games, including in regions of the potential rules space not covered by existing games in the Ludii dataset. A sample of the generated games are available to play online through the Ludii portal.

AI algorithms for imperfect-information games are typically compared using performance metrics on individual games, making it difficult to assess robustness across game choices. Card games are a natural domain for imperfect information due to hidden hands and stochastic draws. To facilitate comparative research on imperfect-information game-playing algorithms and game systems, we introduce Valet, a diverse and comprehensive testbed of 21 traditional imperfect-information card games. These games span multiple genres, cultures, player counts, deck structures, mechanics, winning conditions, and methods of hiding and revealing information. To standardize implementations across systems, we encode the rules of each game in RECYCLE, a card game description language. We empirically characterize each game's branching factor and duration using random simulations, reporting baseline score distributions for a Monte Carlo Tree Search player against random opponents to demonstrate the suitability of Valet as a benchmarking suite.

Multi-party dialogue is a critical setting for studying collaborative reasoning and decision-making, yet existing datasets rarely focus on structured, in-depth complex reasoning tasks. We introduce DeliChess, a novel dataset of group deliberation dialogues in which participants collaboratively solve multiple-choice chess puzzles. Each group first completes the puzzle individually, then engages in a multi-party discussion before submitting a revised collective answer. The dataset includes 107 dialogues with full transcripts, pre- and post-discussion choices, and metadata on puzzle difficulty and move quality. We evaluate performance using three metrics based on chess engine evaluations, and find that deliberation significantly improves group accuracy. We further analyse the role of probing utterances (i.e., messages that elicit proposals, justifications, or strategic reflection) using a classifier trained on prior deliberation data. While probing makes group performance more variable after discussion, it does not consistently lead to better performance. Our dataset offers a rich testbed for modelling group reasoning, dialogue dynamics, and the resolution of differing perspectives and opinions in a well-defined strategic domain.

In this article, we introduce a neuro-symbolic approach that combines a low-level perception task performed by a neural network with a high-level reasoning task performed by a possibilistic rule-based system. The goal is to be able to derive for each input instance the degree of possibility that it belongs to a target (meta-)concept. This (meta-)concept is connected to intermediate concepts by a possibilistic rule-based system. The probability of each intermediate concept for the input instance is inferred using a neural network. The connection between the low-level perception task and the high-level reasoning task lies in the transformation of neural network outputs modeled by probability distributions (through softmax activation) into possibility distributions. The use of intermediate concepts is valuable for the explanation purpose: using the rule-based system, the classification of an input instance as an element of the (meta-)concept can be justified by the fact that intermediate concepts have been recognized. From the technical side, our contribution consists of the design of efficient methods for defining the matrix relation and the equation system associated with a possibilistic rule-based system. The corresponding matrix and equation are key data structures used to perform inferences from a possibilistic rule-based system and to learn the values of the rule parameters in such a system according to a training data sample. Furthermore, leveraging recent results on the handling of inconsistent systems of fuzzy relational equations, an approach for learning rule parameters according to multiple training data samples is presented. Experiments carried out on the MNIST addition problems and the MNIST Sudoku puzzles problems highlight the effectiveness of our approach compared with state-of-the-art neuro-symbolic ones.

Large reasoning models (LRMs) have demonstrated impressive reasoning capabilities across a broad range of tasks including Olympiad-level mathematical problems, indicating evidence of their complex reasoning abilities. While many reasoning benchmarks focus on the STEM domain, the ability of LRMs to reason correctly in broader task domains remains underexplored. In this work, we introduce TTT-Bench, a new benchmark that is designed to evaluate basic strategic, spatial, and logical reasoning abilities in LRMs through a suite of four two-player Tic-Tac-Toe-style games that humans can effortlessly solve from a young age. We propose a simple yet scalable programmatic approach for generating verifiable two-player game problems for TTT-Bench. Although these games are trivial for humans, they require reasoning about the intentions of the opponent, as well as the game board's spatial configurations, to ensure a win. We evaluate a diverse set of state-of-the-art LRMs, and discover that the models that excel at hard math problems frequently fail at these simple reasoning games. Further testing reveals that our evaluated reasoning models score on average downarrow 41\% \& downarrow 5\% lower on TTT-Bench compared to MATH 500 \& AIME 2024 respectively, with larger models achieving higher performance using shorter reasoning traces, where most of the models struggle on long-term strategic reasoning situations on simple and new TTT-Bench tasks.

Rigorously evaluating machine intelligence against the broad spectrum of human general intelligence has become increasingly important and challenging in this era of rapid technological advance. Conventional AI benchmarks typically assess only narrow capabilities in a limited range of human activity. Most are also static, quickly saturating as developers explicitly or implicitly optimize for them. We propose that a more promising way to evaluate human-like general intelligence in AI systems is through a particularly strong form of general game playing: studying how and how well they play and learn to play all conceivable human games, in comparison to human players with the same level of experience, time, or other resources. We define a "human game" to be a game designed by humans for humans, and argue for the evaluative suitability of this space of all such games people can imagine and enjoy -- the "Multiverse of Human Games". Taking a first step towards this vision, we introduce the AI GameStore, a scalable and open-ended platform that uses LLMs with humans-in-the-loop to synthesize new representative human games, by automatically sourcing and adapting standardized and containerized variants of game environments from popular human digital gaming platforms. As a proof of concept, we generated 100 such games based on the top charts of Apple App Store and Steam, and evaluated seven frontier vision-language models (VLMs) on short episodes of play. The best models achieved less than 10\% of the human average score on the majority of the games, and especially struggled with games that challenge world-model learning, memory and planning. We conclude with a set of next steps for building out the AI GameStore as a practical way to measure and drive progress toward human-like general intelligence in machines.

When one agent interacts with a multi-agent environment, it is challenging to deal with various opponents unseen before. Modeling the behaviors, goals, or beliefs of opponents could help the agent adjust its policy to adapt to different opponents. In addition, it is also important to consider opponents who are learning simultaneously or capable of reasoning. However, existing work usually tackles only one of the aforementioned types of opponents. In this paper, we propose model-based opponent modeling (MBOM), which employs the environment model to adapt to all kinds of opponents. MBOM simulates the recursive reasoning process in the environment model and imagines a set of improving opponent policies. To effectively and accurately represent the opponent policy, MBOM further mixes the imagined opponent policies according to the similarity with the real behaviors of opponents. Empirically, we show that MBOM achieves more effective adaptation than existing methods in a variety of tasks, respectively with different types of opponents, i.e., fixed policy, na\"ive learner, and reasoning learner.

Games are abstractions of the real world, where artificial agents learn to compete and cooperate with other agents. While significant achievements have been made in various perfect- and imperfect-information games, DouDizhu (a.k.a. Fighting the Landlord), a three-player card game, is still unsolved. DouDizhu is a very challenging domain with competition, collaboration, imperfect information, large state space, and particularly a massive set of possible actions where the legal actions vary significantly from turn to turn. Unfortunately, modern reinforcement learning algorithms mainly focus on simple and small action spaces, and not surprisingly, are shown not to make satisfactory progress in DouDizhu. In this work, we propose a conceptually simple yet effective DouDizhu AI system, namely DouZero, which enhances traditional Monte-Carlo methods with deep neural networks, action encoding, and parallel actors. Starting from scratch in a single server with four GPUs, DouZero outperformed all the existing DouDizhu AI programs in days of training and was ranked the first in the Botzone leaderboard among 344 AI agents. Through building DouZero, we show that classic Monte-Carlo methods can be made to deliver strong results in a hard domain with a complex action space. The code and an online demo are released at https://github.com/kwai/DouZero with the hope that this insight could motivate future work.

Competitive sports require sophisticated tactical analysis, yet combat disciplines like boxing remain underdeveloped in AI-driven analytics due to the complexity of action dynamics and the lack of structured tactical representations. To address this, we present BoxMind, a closed-loop AI expert system validated in elite boxing competition. By defining atomic punch events with precise temporal boundaries and spatial and technical attributes, we parse match footage into 18 hierarchical technical-tactical indicators. We then propose a graph-based predictive model that fuses these explicit technical-tactical profiles with learnable, time-variant latent embeddings to capture the dynamics of boxer matchups. Modeling match outcome as a differentiable function of technical-tactical indicators, we turn winning probability gradients into executable tactical adjustments. Experiments show that the outcome prediction model achieves state-of-the-art performance, with 69.8% accuracy on BoxerGraph test set and 87.5% on Olympic matches. Using this predictive model as a foundation, the system generates strategic recommendations that demonstrate proficiency comparable to human experts. BoxMind is validated through a closed-loop deployment during the 2024 Paris Olympics, directly contributing to the Chinese National Team's historic achievement of three gold and two silver medals. BoxMind establishes a replicable paradigm for transforming unstructured video data into strategic intelligence, bridging the gap between computer vision and decision support in competitive sports.

StarCraft II is one of the most challenging simulated reinforcement learning environments; it is partially observable, stochastic, multi-agent, and mastering StarCraft II requires strategic planning over long time horizons with real-time low-level execution. It also has an active professional competitive scene. StarCraft II is uniquely suited for advancing offline RL algorithms, both because of its challenging nature and because Blizzard has released a massive dataset of millions of StarCraft II games played by human players. This paper leverages that and establishes a benchmark, called AlphaStar Unplugged, introducing unprecedented challenges for offline reinforcement learning. We define a dataset (a subset of Blizzard's release), tools standardizing an API for machine learning methods, and an evaluation protocol. We also present baseline agents, including behavior cloning, offline variants of actor-critic and MuZero. We improve the state of the art of agents using only offline data, and we achieve 90% win rate against previously published AlphaStar behavior cloning agent.

During 2023, two interesting results were proven about the limit behavior of game dynamics: First, it was shown that there is a game for which no dynamics converges to the Nash equilibria. Second, it was shown that the sink equilibria of a game adequately capture the limit behavior of natural game dynamics. These two results have created a need and opportunity to articulate a principled computational theory of the meaning of the game that is based on game dynamics. Given any game in normal form, and any prior distribution of play, we study the problem of computing the asymptotic behavior of a class of natural dynamics called the noisy replicator dynamics as a limit distribution over the sink equilibria of the game. When the prior distribution has pure strategy support, we prove this distribution can be computed efficiently, in near-linear time to the size of the best-response graph. When the distribution can be sampled -- for example, if it is the uniform distribution over all mixed strategy profiles -- we show through experiments that the limit distribution of reasonably large games can be estimated quite accurately through sampling and simulation.

The 13-card variant of Classic Indian Rummy is a sequential game of incomplete information that requires probabilistic reasoning and combinatorial decision-making. This paper proposes a rule-based framework for strategic play, driven by a new hand-evaluation metric termed MinDist. The metric modifies the MinScore metric by quantifying the edit distance between a hand and the nearest valid configuration, thereby capturing structural proximity to completion. We design a computationally efficient algorithm derived from the MinScore algorithm, leveraging dynamic pruning and pattern caching to exactly calculate this metric during play. Opponent hand-modeling is also incorporated within a two-player zero-sum simulation framework, and the resulting strategies are evaluated using statistical hypothesis testing. Empirical results show significant improvement in win rates for MinDist-based agents over traditional heuristics, providing a formal and interpretable step toward algorithmic Rummy strategy design.

After the recent groundbreaking results of AlphaGo and AlphaZero, we have seen strong interests in deep reinforcement learning and artificial general intelligence (AGI) in game playing. However, deep learning is resource-intensive and the theory is not yet well developed. For small games, simple classical table-based Q-learning might still be the algorithm of choice. General Game Playing (GGP) provides a good testbed for reinforcement learning to research AGI. Q-learning is one of the canonical reinforcement learning methods, and has been used by (Banerjee & Stone, IJCAI 2007) in GGP. In this paper we implement Q-learning in GGP for three small-board games (Tic-Tac-Toe, Connect Four, Hex)source code: https://github.com/wh1992v/ggp-rl, to allow comparison to Banerjee et al.. We find that Q-learning converges to a high win rate in GGP. For the ε-greedy strategy, we propose a first enhancement, the dynamic ε algorithm. In addition, inspired by (Gelly & Silver, ICML 2007) we combine online search (Monte Carlo Search) to enhance offline learning, and propose QM-learning for GGP. Both enhancements improve the performance of classical Q-learning. In this work, GGP allows us to show, if augmented by appropriate enhancements, that classical table-based Q-learning can perform well in small games.

The prototyping of computer games, particularly card games, requires extensive human effort in creative ideation and gameplay evaluation. Recent advances in Large Language Models (LLMs) offer opportunities to automate and streamline these processes. However, it remains challenging for LLMs to design novel game mechanics beyond existing databases, generate consistent gameplay environments, and develop scalable gameplay AI for large-scale evaluations. This paper addresses these challenges by introducing a comprehensive automated card game prototyping framework. The approach highlights a graph-based indexing method for generating novel game designs, an LLM-driven system for consistent game code generation validated by gameplay records, and a gameplay AI constructing method that uses an ensemble of LLM-generated action-value functions optimized through self-play. These contributions aim to accelerate card game prototyping, reduce human labor, and lower barriers to entry for game developers.

We introduce "Method Actors" as a mental model for guiding LLM prompt engineering and prompt architecture. Under this mental model, LLMs should be thought of as actors; prompts as scripts and cues; and LLM responses as performances. We apply this mental model to the task of improving LLM performance at playing Connections, a New York Times word puzzle game that prior research identified as a challenging benchmark for evaluating LLM reasoning. Our experiments with GPT-4o show that a "Method Actors" approach can significantly improve LLM performance over both a vanilla and "Chain of Thoughts" approach. A vanilla approach solves 27% of Connections puzzles in our dataset and a "Chain of Thoughts" approach solves 41% of puzzles, whereas our strongest "Method Actor" approach solves 86% of puzzles. We also test OpenAI's newest model designed specifically for complex reasoning tasks, o1-preview. When asked to solve a puzzle all at once, o1-preview solves 79% of Connections puzzles in our dataset, and when allowed to build puzzle solutions one guess at a time over multiple API calls, o1-preview solves 100% of the puzzles. Incorporating a "Method Actor" prompt architecture increases the percentage of puzzles that o1-preview solves perfectly from 76% to 87%.

To make deliberate progress towards more intelligent and more human-like artificial systems, we need to be following an appropriate feedback signal: we need to be able to define and evaluate intelligence in a way that enables comparisons between two systems, as well as comparisons with humans. Over the past hundred years, there has been an abundance of attempts to define and measure intelligence, across both the fields of psychology and AI. We summarize and critically assess these definitions and evaluation approaches, while making apparent the two historical conceptions of intelligence that have implicitly guided them. We note that in practice, the contemporary AI community still gravitates towards benchmarking intelligence by comparing the skill exhibited by AIs and humans at specific tasks such as board games and video games. We argue that solely measuring skill at any given task falls short of measuring intelligence, because skill is heavily modulated by prior knowledge and experience: unlimited priors or unlimited training data allow experimenters to "buy" arbitrary levels of skills for a system, in a way that masks the system's own generalization power. We then articulate a new formal definition of intelligence based on Algorithmic Information Theory, describing intelligence as skill-acquisition efficiency and highlighting the concepts of scope, generalization difficulty, priors, and experience. Using this definition, we propose a set of guidelines for what a general AI benchmark should look like. Finally, we present a benchmark closely following these guidelines, the Abstraction and Reasoning Corpus (ARC), built upon an explicit set of priors designed to be as close as possible to innate human priors. We argue that ARC can be used to measure a human-like form of general fluid intelligence and that it enables fair general intelligence comparisons between AI systems and humans.

RLCard is an open-source toolkit for reinforcement learning research in card games. It supports various card environments with easy-to-use interfaces, including Blackjack, Leduc Hold'em, Texas Hold'em, UNO, Dou Dizhu and Mahjong. The goal of RLCard is to bridge reinforcement learning and imperfect information games, and push forward the research of reinforcement learning in domains with multiple agents, large state and action space, and sparse reward. In this paper, we provide an overview of the key components in RLCard, a discussion of the design principles, a brief introduction of the interfaces, and comprehensive evaluations of the environments. The codes and documents are available at https://github.com/datamllab/rlcard

Adversarial board games, as a paradigmatic domain of strategic reasoning and intelligence, have long served as both a popular competitive activity and a benchmark for evaluating artificial intelligence (AI) systems. Building on this foundation, we propose an adversarial benchmarking framework to assess the comprehensive performance of Large Language Models (LLMs) through board games competition, compensating the limitation of data dependency of the mainstream Question-and-Answer (Q&A) based benchmark method. We introduce Qi Town, a specialized evaluation platform that supports 5 widely played games and involves 20 LLM-driven players. The platform employs both the Elo rating system and a novel Performance Loop Graph (PLG) to quantitatively evaluate the technical capabilities of LLMs, while also capturing Positive Sentiment Score (PSS) throughout gameplay to assess mental fitness. The evaluation is structured as a round-robin tournament, enabling systematic comparison across players. Experimental results indicate that, despite technical differences, most LLMs remain optimistic about winning and losing, demonstrating greater adaptability to high-stress adversarial environments than humans. On the other hand, the complex relationship between cyclic wins and losses in PLGs exposes the instability of LLMs' skill play during games, warranting further explanation and exploration.

Recent advances in reinforcement learning have shown that language models can develop sophisticated reasoning through training on tasks with verifiable rewards, but these approaches depend on human-curated problem-answer pairs and domain-specific reward engineering. We introduce SPIRAL, a self-play framework where models learn by playing multi-turn, zero-sum games against continuously improving versions of themselves, eliminating the need for human supervision. Through self-play, SPIRAL generates an infinite curriculum of progressively challenging problems as models must constantly adapt to stronger opponents. To enable this self-play training at scale, We implement a fully online, multi-turn, multi-agent reinforcement learning system for LLMs and propose role-conditioned advantage estimation (RAE) to stabilize multi-agent training. Using SPIRAL, self-play on zero-sum games produces reasoning capabilities that transfer broadly. Training Qwen3-4B-Base on Kuhn Poker alone achieves 8.6% improvement on math and 8.4% on general reasoning, outperforming SFT on 25,000 expert game trajectories. Analysis reveals that this transfer occurs through three cognitive patterns: systematic decomposition, expected value calculation, and case-by-case analysis. Multi-game training (TicTacToe, Kuhn Poker, Simple Negotiation) further enhances performance as each game develops distinct reasoning strengths. Applying SPIRAL to a strong reasoning model (DeepSeek-R1-Distill-Qwen-7B) can still lead to 2.0% average improvement. These results demonstrate that zero-sum games naturally develop transferable reasoning capabilities, highlighting a promising direction for autonomous reasoning development.

Game theory is a powerful framework for reasoning about strategic interactions, with applications in domains ranging from day-to-day life to international politics. However, applying formal reasoning tools in such contexts is challenging, as these scenarios are often expressed in natural language. To address this, we introduce a framework for the autoformalization of game-theoretic scenarios, which translates natural language descriptions into formal logic representations suitable for formal solvers. Our approach utilizes one-shot prompting and a solver that provides feedback on syntactic correctness to allow LLMs to refine the code. We evaluate the framework using GPT-4o and a dataset of natural language problem descriptions, achieving 98% syntactic correctness and 88% semantic correctness. These results show the potential of LLMs to bridge the gap between real-life strategic interactions and formal reasoning.

A new prior is proposed for learning representations of high-level concepts of the kind we manipulate with language. This prior can be combined with other priors in order to help disentangling abstract factors from each other. It is inspired by cognitive neuroscience theories of consciousness, seen as a bottleneck through which just a few elements, after having been selected by attention from a broader pool, are then broadcast and condition further processing, both in perception and decision-making. The set of recently selected elements one becomes aware of is seen as forming a low-dimensional conscious state. This conscious state is combining the few concepts constituting a conscious thought, i.e., what one is immediately conscious of at a particular moment. We claim that this architectural and information-processing constraint corresponds to assumptions about the joint distribution between high-level concepts. To the extent that these assumptions are generally true (and the form of natural language seems consistent with them), they can form a useful prior for representation learning. A low-dimensional thought or conscious state is analogous to a sentence: it involves only a few variables and yet can make a statement with very high probability of being true. This is consistent with a joint distribution (over high-level concepts) which has the form of a sparse factor graph, i.e., where the dependencies captured by each factor of the factor graph involve only very few variables while creating a strong dip in the overall energy function. The consciousness prior also makes it natural to map conscious states to natural language utterances or to express classical AI knowledge in a form similar to facts and rules, albeit capturing uncertainty as well as efficient search mechanisms implemented by attention mechanisms.

While vision-and-language models perform well on tasks such as visual question answering, they struggle when it comes to basic human commonsense reasoning skills. In this work, we introduce WinoGAViL: an online game of vision-and-language associations (e.g., between werewolves and a full moon), used as a dynamic evaluation benchmark. Inspired by the popular card game Codenames, a spymaster gives a textual cue related to several visual candidates, and another player tries to identify them. Human players are rewarded for creating associations that are challenging for a rival AI model but still solvable by other human players. We use the game to collect 3.5K instances, finding that they are intuitive for humans (>90% Jaccard index) but challenging for state-of-the-art AI models, where the best model (ViLT) achieves a score of 52%, succeeding mostly where the cue is visually salient. Our analysis as well as the feedback we collect from players indicate that the collected associations require diverse reasoning skills, including general knowledge, common sense, abstraction, and more. We release the dataset, the code and the interactive game, allowing future data collection that can be used to develop models with better association abilities.

Game theory is a foundational framework for analyzing strategic interactions, and its intersection with large language models (LLMs) is a rapidly growing field. However, existing surveys mainly focus narrowly on using game theory to evaluate LLM behavior. This paper provides the first comprehensive survey of the bidirectional relationship between Game Theory and LLMs. We propose a novel taxonomy that categorizes the research in this intersection into four distinct perspectives: (1) evaluating LLMs in game-based scenarios; (2) improving LLMs using game-theoretic concepts for better interpretability and alignment; (3) modeling the competitive landscape of LLM development and its societal impact; and (4) leveraging LLMs to advance game models and to solve corresponding game theory problems. Furthermore, we identify key challenges and outline future research directions. By systematically investigating this interdisciplinary landscape, our survey highlights the mutual influence of game theory and LLMs, fostering progress at the intersection of these fields.

Wordle is an online word puzzle game that gained viral popularity in January 2022. The goal is to guess a hidden five letter word. After each guess, the player gains information about whether the letters they guessed are present in the word, and whether they are in the correct position. Numerous blogs have suggested guessing strategies and starting word lists that improve the chance of winning. Optimized algorithms can win 100% of games within five of the six allowed trials. However, it is infeasible for human players to use these algorithms due to an inability to perfectly recall all known 5-letter words and perform complex calculations that optimize information gain. Here, we present two different methods for choosing starting words along with a framework for discovering the optimal human strategy based on reinforcement learning. Human Wordle players can use the rules we discover to optimize their chance of winning.

This position paper argues that, in order to understand AI, we cannot rely on our existing vocabulary of human words. Instead, we should strive to develop neologisms: new words that represent precise human concepts that we want to teach machines, or machine concepts that we need to learn. We start from the premise that humans and machines have differing concepts. This means interpretability can be framed as a communication problem: humans must be able to reference and control machine concepts, and communicate human concepts to machines. Creating a shared human-machine language through developing neologisms, we believe, could solve this communication problem. Successful neologisms achieve a useful amount of abstraction: not too detailed, so they're reusable in many contexts, and not too high-level, so they convey precise information. As a proof of concept, we demonstrate how a "length neologism" enables controlling LLM response length, while a "diversity neologism" allows sampling more variable responses. Taken together, we argue that we cannot understand AI using our existing vocabulary, and expanding it through neologisms creates opportunities for both controlling and understanding machines better.