Pgx is available at https://github.com/sotetsuk/pgx. Figure 1: Example games included in Pgx.

The competitive and cooperative forces of natural selection have driven the evolution of intelligence for millions of years, culminating in nature's vast biodiversity and the complexity of human minds. Inspired by this process, we propose a novel multi-agent reinforcement learning framework where each agent is assigned a genotype and where reward functions are modelled after the concept of inclusive fitness. An agent's genetic material may be shared with other agents, and our inclusive reward function naturally accounts for this. We study the resulting social dynamics in two types of network games with prisoner's dilemmas and find that our results align with well-established principles from biology, such as Hamilton's rule. Furthermore, we outline how this framework can extend to more open-ended environments with spatial and temporal structure, finite resources, and evolving populations. We hypothesize the emergence of an arms race of strategies, where each new strategy is a gradual improvement over earlier adaptations of other agents, effectively producing a multi-agent autocurriculum analogous to biological evolution. In contrast to the binary team-based structures prevalent in earlier research, our gene-based reward structure introduces a spectrum of cooperation ranging from full adversity to full cooperativeness based on genetic similarity, enabling unique non team-based social dynamics. For example, one agent having a mutual cooperative relationship with two other agents, while the two other agents behave adversarially towards each other. We argue that incorporating inclusive fitness in agents provides a foundation for the emergence of more strategically advanced and socially intelligent agents.

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.

We propose Pgx, a suite of board game reinforcement learning (RL) environments written in JAX and optimized for GPU/TPU accelerators. By leveraging JAX's auto-vectorization and parallelization over accelerators, Pgx can efficiently scale to thousands of simultaneous simulations over accelerators. In our experiments on a DGX-A100 workstation, we discovered that Pgx can simulate RL environments 10-100x faster than existing implementations available in Python. Pgx includes RL environments commonly used as benchmarks in RL research, such as backgammon, chess, shogi, and Go. Additionally, Pgx offers miniature game sets and baseline models to facilitate rapid research cycles.

Case-based reasoning (CBR) is an experience-based approach to problem solving, where a repository of solved cases is adapted to solve new cases. Recent research shows that Large Language Models (LLMs) with Retrieval-Augmented Generation (RAG) can support the Retrieve and Reuse stages of the CBR pipeline by retrieving similar cases and using them as additional context to an LLM query. Most studies have focused on text-only applications, however, in many real-world problems the components of a case are multimodal. In this paper we present MCBR-RAG, a general RAG framework for multimodal CBR applications. The MCBR-RAG framework converts non-text case components into text-based representations, allowing it to: 1) learn application-specific latent representations that can be indexed for retrieval, and 2) enrich the query provided to the LLM by incorporating all case components for better context. We demonstrate MCBR-RAG's effectiveness through experiments conducted on a simplified Math-24 application and a more complex Backgammon application. Our empirical results show that MCBR-RAG improves generation quality compared to a baseline LLM with no contextual information provided.

We present a Monte-Carlo simulation algorithm for real-time policy improvement of an adaptive controller. In the Monte-Carlo simulation, the long-term expected reward of each possible action is statistically measured, using the initial policy to make decisions in each step of the simulation. The action maximizing the measured expected reward is then taken, resulting in an improved policy. Our algorithm is easily parallelizable and has been implemented on the IBM SP1 and SP2 parallel-RISC supercomputers. We have obtained promising initial results in applying this algorithm to the domain of backgammon. Results are reported for a wide variety of initial policies, ranging from a random policy to TD-Gammon, an extremely strong multi-layer neural network. In each case, the Monte-Carlo algorithm gives a substantial reduction, by as much as a factor of 5 or more, in the error rate of the base players. The algorithm is also potentially useful in many other adaptive control applications in which it is possible to simulate the environment.

We propose Pgx, a suite of board game reinforcement learning (RL) environments written in JAX and optimized for GPU/TPU accelerators. By leveraging JAX's auto-vectorization and parallelization over accelerators, Pgx can efficiently scale to thousands of simultaneous simulations over accelerators. In our experiments on a DGX-A100 workstation, we discovered that Pgx can simulate RL environments 10-100x faster than existing implementations available in Python. Pgx includes RL environments commonly used as benchmarks in RL research, such as backgammon, chess, shogi, and Go. Additionally, Pgx offers miniature game sets and baseline models to facilitate rapid research cycles. We demonstrate the efficient training of the Gumbel AlphaZero algorithm with Pgx environments. Overall, Pgx provides high-performance environment simulators for researchers to accelerate their RL experiments. Pgx is available at https://github.com/sotetsuk/pgx.

TIle networks were trained by a supervised learning procedure on a large set of sample positions evaluated by a human expert. In actual match play against humans and conventional computer programs, the networks demonstrate substantial ability to generalize on the basis of expert knowledge. Our study touches on some of the most important issues in net(cid:173) work learning theory, including the development of efficient coding schemes and training procedures, scaling, generalization, the use of real-valued inputs and out(cid:173) puts, and techniques for escaping from local minima. Practical applications in games and other domains are also discussed.

This paper examines whether temporal difference methods for training connectionist networks, such as Suttons's TO(') algorithm, can be suc(cid:173) cessfully applied to complex real-world problems. A number of important practical issues are identified and discussed from a general theoretical per(cid:173) spective. These practical issues are then examined in the context of a case study in which TO(') is applied to learning the game of backgammon from the outcome of self-play. This is apparently the first application of this algorithm to a complex nontrivial task. It is found that, with zero knowledge built in, the network is able to learn from scratch to play the entire game at a fairly strong intermediate level of performance, which is clearly better than conventional commercial programs, and which in fact surpasses comparable networks trained on a massive human expert data set.