This paper investigates the problem of computing the equilibrium of competitive games, which is often modeled as a constrained saddle-point optimization problem with probability simplex constraints. Despite recent efforts in understanding the last-iterate convergence of extragradient methods in the unconstrained setting, the theoretical underpinnings of these methods in the constrained settings, especially those using multiplicative updates, remain highly inadequate, even when the objective function is bilinear. Motivated by the algorithmic role of entropy regularization in single-agent reinforcement learning and game theory, we develop provably efficient extragradient methods to find the quantal response equilibrium (QRE)---which are solutions to zero-sum two-player matrix games with entropy regularization---at a linear rate. The proposed algorithms can be implemented in a decentralized manner, where each player executes symmetric and multiplicative updates iteratively using its own payoff without observing the opponent's actions directly. In addition, by controlling the knob of entropy regularization, the proposed algorithms can locate an approximate Nash equilibrium of the unregularized matrix game at a sublinear rate without assuming the Nash equilibrium to be unique. Our methods also lead to efficient policy extragradient algorithms for solving entropy-regularized zero-sum Markov games at a linear rate. All of our convergence rates are nearly dimension-free, which are independent of the size of the state and action spaces up to logarithm factors, highlighting the positive role of entropy regularization for accelerating convergence.

Developing large language models (LLMs) to cooperate and compete effectively within multi-agent systems (MASs) is a critical step towards more advanced intelligence. While reinforcement learning (RL) has proven effective for enhancing reasoning in single-agent tasks, its extension to multi-turn, multi-agent scenarios remains underexplored due to the challenges of long-horizon credit assignment and agent-specific advantage estimation. To address these challenges, we introduce MARSHAL, an end-to-end RL framework that incentivizes Multi-Agent Reasoning through Self-play witH strAtegic LLMs in both cooperative and competitive games. MARSHAL features a turn-level advantage estimator that aligns learning signals with each interaction for credit assignment, and an agent-specific advantage normalization to stabilize multi-agent training. By learning with self-play across cooperative and competitive games, MARSHAL agent trained from Qwen3-4B develops strong strategic abilities that generalize to held-out games with up to 28.7% performance improvements. More importantly, the capability acquired through self-play generalizes beyond games, yielding consistent performance gains of MASs in reasoning benchmarks. When integrated into leading MASs, our MARSHAL agent achieves significant performance gains of up to 10.0% on AIME, 6.6% on GPQA-Diamond, and 3.5% on average across all benchmarks. These results establish end-to-end RL training with self-play in strategic games as a powerful approach for developing generalizable multi-agent reasoning capabilities in LLMs.

This paper investigates the problem of computing the equilibrium of competitive games, which is often modeled as a constrained saddle-point optimization problem with probability simplex constraints. Despite recent efforts in understanding the last-iterate convergence of extragradient methods in the unconstrained setting, the theoretical underpinnings of these methods in the constrained settings, especially those using multiplicative updates, remain highly inadequate, even when the objective function is bilinear. Motivated by the algorithmic role of entropy regularization in single-agent reinforcement learning and game theory, we develop provably efficient extragradient methods to find the quantal response equilibrium (QRE)---which are solutions to zero-sum two-player matrix games with entropy regularization---at a linear rate. The proposed algorithms can be implemented in a decentralized manner, where each player executes symmetric and multiplicative updates iteratively using its own payoff without observing the opponent's actions directly. In addition, by controlling the knob of entropy regularization, the proposed algorithms can locate an approximate Nash equilibrium of the unregularized matrix game at a sublinear rate without assuming the Nash equilibrium to be unique.

Recent advances in reinforcement learning (RL) heavily rely on a variety of well-designed benchmarks, which provide environmental platforms and consistent criteria to evaluate existing and novel algorithms. Specifically, in multi-agent RL (MARL), a plethora of benchmarks based on cooperative games have spurred the development of algorithms that improve the scalability of cooperative multi-agent systems. However, for the competitive setting, a lightweight and open-sourced benchmark with challenging gaming dynamics and visual inputs has not yet been established. In this work, we present FightLadder, a real-time fighting game platform, to empower competitive MARL research. Along with the platform, we provide implementations of state-of-the-art MARL algorithms for competitive games, as well as a set of evaluation metrics to characterize the performance and exploitability of agents. We demonstrate the feasibility of this platform by training a general agent that consistently defeats 12 built-in characters in single-player mode, and expose the difficulty of training a non-exploitable agent without human knowledge and demonstrations in two-player mode. FightLadder provides meticulously designed environments to address critical challenges in competitive MARL research, aiming to catalyze a new era of discovery and advancement in the field. Videos and code at https://sites.google.com/view/fightladder/home.

Offline Reinforcement Learning (RL) has received significant interest due to its ability to improve policies in previously collected datasets without online interactions. Despite its success in the single-agent setting, offline multi-agent RL remains a challenge, especially in competitive games. Firstly, unaware of the game structure, it is impossible to interact with the opponents and conduct a major learning paradigm, self-play, for competitive games. Secondly, real-world datasets cannot cover all the state and action space in the game, resulting in barriers to identifying Nash equilibrium (NE). To address these issues, this paper introduces Off-FSP, the first practical model-free offline RL algorithm for competitive games. We start by simulating interactions with various opponents by adjusting the weights of the fixed dataset with importance sampling. This technique allows us to learn best responses to different opponents and employ the Offline Self-Play learning framework. In this framework, we further implement Fictitious Self-Play (FSP) to approximate NE. In partially covered real-world datasets, our methods show the potential to approach NE by incorporating any single-agent offline RL method. Experimental results in Leduc Hold'em Poker show that our method significantly improves performances compared with state-of-the-art baselines.

This paper addresses policy learning in non-stationary environments and games with continuous actions. Rather than the classical reward maximization mechanism, inspired by the ideas of follow-the-regularized-leader (FTRL) and mirror descent (MD) update, we propose a no-regret style reinforcement learning algorithm PORL for continuous action tasks. We prove that PORL has a last-iterate convergence guarantee, which is important for adversarial and cooperative games. Empirical studies show that, in stationary environments such as MuJoCo locomotion controlling tasks, PORL performs equally well as, if not better than, the soft actor-critic (SAC) algorithm; in non-stationary environments including dynamical environments, adversarial training, and competitive games, PORL is superior to SAC in both a better final policy performance and a more stable training process.

We study convergence properties of the mixed strategies that result from a general class of optimal no regret learning strategies in a repeated game setting where the stage game is any 2 by 2 competitive game (i.e. game for which all the Nash equilibria (NE) of the game are completely mixed). We consider the class of strategies whose information set at each step is the empirical average of the opponent's realized play (and the step number), that we call mean based strategies. We first show that there does not exist any optimal no regret, mean based strategy for player 1 that would result in the convergence of her mixed strategies (in probability) against an opponent that plays his Nash equilibrium mixed strategy at each step. Next, we show that this last iterate divergence necessarily occurs if player 2 uses any adaptive strategy with a minimal randomness property. This property is satisfied, for example, by any fixed sequence of mixed strategies for player 2 that converges to NE. We conjecture that this property holds when both players use optimal no regret learning strategies against each other, leading to the divergence of the mixed strategies with a positive probability. Finally, we show that variants of mean based strategies using recency bias, which have yielded last iterate convergence in deterministic min max optimization, continue to lead to this last iterate divergence. This demonstrates a crucial difference in outcomes between using the opponent's mixtures and realizations to make strategy updates.

Just as the kaleidoscopic dramas of Leo Tolstoy's novel Anna Karenina, the pseudo-non-fiction murk of Alan Moore's comic From Hell and the domestic pragmatism of Jamie Oliver's 15 Minute Meals meet under the fat banner of prose, so the body of video games becomes an ever broader church. It is impossible to enforce orthodoxy in a medium where shifting technology defines the canvas. The artform now embraces work from a dizzying spectrum. A challenging time, then, for the Victoria and Albert Museum to stage its first major video game exhibition. Rather than reach into the primordial digital soup of the 1950s, or the gambling-adjacent squalor of the Pac-Man and Space Invaders arcade era, the V&A's exhibition, titled Videogames: Design/Play/Disrupt, begins in the mid-2000s. This was the moment at which technological advances began to alter dramatically the way in which games were designed, made and played.