Daily Papers

Generative flow networks (GFlowNets) are sequential sampling models trained to match a given distribution. GFlowNets have been successfully applied to various structured object generation tasks, sampling a diverse set of high-reward objects quickly. We propose expected flow networks (EFlowNets), which extend GFlowNets to stochastic environments. We show that EFlowNets outperform other GFlowNet formulations in stochastic tasks such as protein design. We then extend the concept of EFlowNets to adversarial environments, proposing adversarial flow networks (AFlowNets) for two-player zero-sum games. We show that AFlowNets learn to find above 80% of optimal moves in Connect-4 via self-play and outperform AlphaZero in tournaments.

In their seminal work, Nayyar et al. (2013) showed that imperfect information can be abstracted away from common-payoff games by having players publicly announce their policies as they play. This insight underpins sound solvers and decision-time planning algorithms for common-payoff games. Unfortunately, a naive application of the same insight to two-player zero-sum games fails because Nash equilibria of the game with public policy announcements may not correspond to Nash equilibria of the original game. As a consequence, existing sound decision-time planning algorithms require complicated additional mechanisms that have unappealing properties. The main contribution of this work is showing that certain regularized equilibria do not possess the aforementioned non-correspondence problem -- thus, computing them can be treated as perfect-information problems. Because these regularized equilibria can be made arbitrarily close to Nash equilibria, our result opens the door to a new perspective to solving two-player zero-sum games and yields a simplified framework for decision-time planning in two-player zero-sum games, void of the unappealing properties that plague existing decision-time planning approaches.

One prominent approach toward resolving the adversarial vulnerability of deep neural networks is the two-player zero-sum paradigm of adversarial training, in which predictors are trained against adversarially chosen perturbations of data. Despite the promise of this approach, algorithms based on this paradigm have not engendered sufficient levels of robustness and suffer from pathological behavior like robust overfitting. To understand this shortcoming, we first show that the commonly used surrogate-based relaxation used in adversarial training algorithms voids all guarantees on the robustness of trained classifiers. The identification of this pitfall informs a novel non-zero-sum bilevel formulation of adversarial training, wherein each player optimizes a different objective function. Our formulation yields a simple algorithmic framework that matches and in some cases outperforms state-of-the-art attacks, attains comparable levels of robustness to standard adversarial training algorithms, and does not suffer from robust overfitting.

Finding the mixed Nash equilibria (MNE) of a two-player zero sum continuous game is an important and challenging problem in machine learning. A canonical algorithm to finding the MNE is the noisy gradient descent ascent method which in the infinite particle limit gives rise to the {\em Mean-Field Gradient Descent Ascent} (GDA) dynamics on the space of probability measures. In this paper, we first study the convergence of a two-scale Mean-Field GDA dynamics for finding the MNE of the entropy-regularized objective. More precisely we show that for each finite temperature (or regularization parameter), the two-scale Mean-Field GDA with a suitable {\em finite} scale ratio converges exponentially to the unique MNE without assuming the convexity or concavity of the interaction potential. The key ingredient of our proof lies in the construction of new Lyapunov functions that dissipate exponentially along the Mean-Field GDA. We further study the simulated annealing of the Mean-Field GDA dynamics. We show that with a temperature schedule that decays logarithmically in time the annealed Mean-Field GDA converges to the MNE of the original unregularized objective.

Conventional language model (LM) safety alignment relies on a reactive, disjoint procedure: attackers exploit a static model, followed by defensive fine-tuning to patch exposed vulnerabilities. This sequential approach creates a mismatch -- attackers overfit to obsolete defenses, while defenders perpetually lag behind emerging threats. To address this, we propose Self-RedTeam, an online self-play reinforcement learning algorithm where an attacker and defender agent co-evolve through continuous interaction. We cast safety alignment as a two-player zero-sum game, where a single model alternates between attacker and defender roles -- generating adversarial prompts and safeguarding against them -- while a reward LM adjudicates outcomes. This enables dynamic co-adaptation. Grounded in the game-theoretic framework of zero-sum games, we establish a theoretical safety guarantee which motivates the design of our method: if self-play converges to a Nash Equilibrium, the defender will reliably produce safe responses to any adversarial input. Empirically, Self-RedTeam uncovers more diverse attacks (+21.8% SBERT) compared to attackers trained against static defenders and achieves higher robustness on safety benchmarks (e.g., +65.5% on WildJailBreak) than defenders trained against static attackers. We further propose hidden Chain-of-Thought, allowing agents to plan privately, which boosts adversarial diversity and reduces over-refusals. Our results motivate a shift from reactive patching to proactive co-evolution in LM safety training, enabling scalable, autonomous, and robust self-improvement of LMs via multi-agent reinforcement learning (MARL).

Despite recent advancements in AI and NLP, negotiation remains a difficult domain for AI agents. Traditional game theoretic approaches that have worked well for two-player zero-sum games struggle in the context of negotiation due to their inability to learn human-compatible strategies. On the other hand, approaches that only use human data tend to be domain-specific and lack the theoretical guarantees provided by strategies grounded in game theory. Motivated by the notion of fairness as a criterion for optimality in general sum games, we propose a negotiation framework called FDHC which incorporates fairness into both the reward design and search to learn human-compatible negotiation strategies. Our method includes a novel, RL+search technique called LGM-Zero which leverages a pre-trained language model to retrieve human-compatible offers from large action spaces. Our results show that our method is able to achieve more egalitarian negotiation outcomes and improve negotiation quality.

In online reinforcement learning (online RL), balancing exploration and exploitation is crucial for finding an optimal policy in a sample-efficient way. To achieve this, existing sample-efficient online RL algorithms typically consist of three components: estimation, planning, and exploration. However, in order to cope with general function approximators, most of them involve impractical algorithmic components to incentivize exploration, such as optimization within data-dependent level-sets or complicated sampling procedures. To address this challenge, we propose an easy-to-implement RL framework called Maximize to Explore (MEX), which only needs to optimize unconstrainedly a single objective that integrates the estimation and planning components while balancing exploration and exploitation automatically. Theoretically, we prove that MEX achieves a sublinear regret with general function approximations for Markov decision processes (MDP) and is further extendable to two-player zero-sum Markov games (MG). Meanwhile, we adapt deep RL baselines to design practical versions of MEX, in both model-free and model-based manners, which can outperform baselines by a stable margin in various MuJoCo environments with sparse rewards. Compared with existing sample-efficient online RL algorithms with general function approximations, MEX achieves similar sample efficiency while enjoying a lower computational cost and is more compatible with modern deep RL methods.

Robust reinforcement learning (RL) seeks to train policies that can perform well under environment perturbations or adversarial attacks. Existing approaches typically assume that the space of possible perturbations remains the same across timesteps. However, in many settings, the space of possible perturbations at a given timestep depends on past perturbations. We formally introduce temporally-coupled perturbations, presenting a novel challenge for existing robust RL methods. To tackle this challenge, we propose GRAD, a novel game-theoretic approach that treats the temporally-coupled robust RL problem as a partially-observable two-player zero-sum game. By finding an approximate equilibrium in this game, GRAD ensures the agent's robustness against temporally-coupled perturbations. Empirical experiments on a variety of continuous control tasks demonstrate that our proposed approach exhibits significant robustness advantages compared to baselines against both standard and temporally-coupled attacks, in both state and action spaces.

We study offline multi-agent reinforcement learning (RL) in Markov games, where the goal is to learn an approximate equilibrium -- such as Nash equilibrium and (Coarse) Correlated Equilibrium -- from an offline dataset pre-collected from the game. Existing works consider relatively restricted tabular or linear models and handle each equilibria separately. In this work, we provide the first framework for sample-efficient offline learning in Markov games under general function approximation, handling all 3 equilibria in a unified manner. By using Bellman-consistent pessimism, we obtain interval estimation for policies' returns, and use both the upper and the lower bounds to obtain a relaxation on the gap of a candidate policy, which becomes our optimization objective. Our results generalize prior works and provide several additional insights. Importantly, we require a data coverage condition that improves over the recently proposed "unilateral concentrability". Our condition allows selective coverage of deviation policies that optimally trade-off between their greediness (as approximate best responses) and coverage, and we show scenarios where this leads to significantly better guarantees. As a new connection, we also show how our algorithmic framework can subsume seemingly different solution concepts designed for the special case of two-player zero-sum games.

We study the problem of convergence to a stationary point in zero-sum games. We propose competitive gradient optimization (CGO ), a gradient-based method that incorporates the interactions between the two players in zero-sum games for optimization updates. We provide continuous-time analysis of CGO and its convergence properties while showing that in the continuous limit, CGO predecessors degenerate to their gradient descent ascent (GDA) variants. We provide a rate of convergence to stationary points and further propose a generalized class of alpha-coherent function for which we provide convergence analysis. We show that for strictly alpha-coherent functions, our algorithm convergences to a saddle point. Moreover, we propose optimistic CGO (OCGO), an optimistic variant, for which we show convergence rate to saddle points in alpha-coherent class of functions.

Data-free quantization (DFQ) recovers the performance of quantized network (Q) without the original data, but generates the fake sample via a generator (G) by learning from full-precision network (P), which, however, is totally independent of Q, overlooking the adaptability of the knowledge from generated samples, i.e., informative or not to the learning process of Q, resulting into the overflow of generalization error. Building on this, several critical questions -- how to measure the sample adaptability to Q under varied bit-width scenarios? whether the largest adaptability is the best? how to generate the samples with adaptive adaptability to improve Q's generalization? To answer the above questions, in this paper, we propose an Adaptive Data-Free Quantization (AdaDFQ) method, which revisits DFQ from a zero-sum game perspective upon the sample adaptability between two players -- a generator and a quantized network. Following this viewpoint, we further define the disagreement and agreement samples to form two boundaries, where the margin is optimized to adaptively regulate the adaptability of generated samples to Q, so as to address the over-and-under fitting issues. Our AdaDFQ reveals: 1) the largest adaptability is NOT the best for sample generation to benefit Q's generalization; 2) the knowledge of the generated sample should not be informative to Q only, but also related to the category and distribution information of the training data for P. The theoretical and empirical analysis validate the advantages of AdaDFQ over the state-of-the-arts. Our code is available at https://github.com/hfutqian/AdaDFQ.

Compositional reinforcement learning is a promising approach for training policies to perform complex long-horizon tasks. Typically, a high-level task is decomposed into a sequence of subtasks and a separate policy is trained to perform each subtask. In this paper, we focus on the problem of training subtask policies in a way that they can be used to perform any task; here, a task is given by a sequence of subtasks. We aim to maximize the worst-case performance over all tasks as opposed to the average-case performance. We formulate the problem as a two agent zero-sum game in which the adversary picks the sequence of subtasks. We propose two RL algorithms to solve this game: one is an adaptation of existing multi-agent RL algorithms to our setting and the other is an asynchronous version which enables parallel training of subtask policies. We evaluate our approach on two multi-task environments with continuous states and actions and demonstrate that our algorithms outperform state-of-the-art baselines.

The growing capabilities and increasingly widespread deployment of AI systems necessitate robust benchmarks for measuring their cooperative capabilities. Unfortunately, most multi-agent benchmarks are either zero-sum or purely cooperative, providing limited opportunities for such measurements. We introduce a general-sum variant of the zero-sum board game Diplomacy -- called Welfare Diplomacy -- in which players must balance investing in military conquest and domestic welfare. We argue that Welfare Diplomacy facilitates both a clearer assessment of and stronger training incentives for cooperative capabilities. Our contributions are: (1) proposing the Welfare Diplomacy rules and implementing them via an open-source Diplomacy engine; (2) constructing baseline agents using zero-shot prompted language models; and (3) conducting experiments where we find that baselines using state-of-the-art models attain high social welfare but are exploitable. Our work aims to promote societal safety by aiding researchers in developing and assessing multi-agent AI systems. Code to evaluate Welfare Diplomacy and reproduce our experiments is available at https://github.com/mukobi/welfare-diplomacy.

We consider the problem of decentralized multi-agent reinforcement learning in Markov games. A fundamental question is whether there exist algorithms that, when adopted by all agents and run independently in a decentralized fashion, lead to no-regret for each player, analogous to celebrated convergence results in normal-form games. While recent work has shown that such algorithms exist for restricted settings (notably, when regret is defined with respect to deviations to Markovian policies), the question of whether independent no-regret learning can be achieved in the standard Markov game framework was open. We provide a decisive negative resolution this problem, both from a computational and statistical perspective. We show that: - Under the widely-believed assumption that PPAD-hard problems cannot be solved in polynomial time, there is no polynomial-time algorithm that attains no-regret in general-sum Markov games when executed independently by all players, even when the game is known to the algorithm designer and the number of players is a small constant. - When the game is unknown, no algorithm, regardless of computational efficiency, can achieve no-regret without observing a number of episodes that is exponential in the number of players. Perhaps surprisingly, our lower bounds hold even for seemingly easier setting in which all agents are controlled by a a centralized algorithm. They are proven via lower bounds for a simpler problem we refer to as SparseCCE, in which the goal is to compute a coarse correlated equilibrium that is sparse in the sense that it can be represented as a mixture of a small number of product policies. The crux of our approach is a novel application of aggregation techniques from online learning, whereby we show that any algorithm for the SparseCCE problem can be used to compute approximate Nash equilibria for non-zero sum normal-form games.

Evaluating the capabilities of Large Language Models (LLMs) has traditionally relied on static benchmark datasets, human assessments, or model-based evaluations - methods that often suffer from overfitting, high costs, and biases. ZeroSumEval is a novel competition-based evaluation protocol that leverages zero-sum games to assess LLMs with dynamic benchmarks that resist saturation. ZeroSumEval encompasses a diverse suite of games, including security challenges (PyJail), classic games (Chess, Liar's Dice, Poker), knowledge tests (MathQuiz), and persuasion challenges (Gandalf, Debate). These games are designed to evaluate a range of AI capabilities such as strategic reasoning, planning, knowledge application, and creativity. Building upon recent studies that highlight the effectiveness of game-based evaluations for LLMs, ZeroSumEval enhances these approaches by providing a standardized and extensible framework. To demonstrate this, we conduct extensive experiments with >7000 simulations across 7 games and 13 models. Our results show that while frontier models from the GPT and Claude families can play common games and answer questions, they struggle to play games that require creating novel and challenging questions. We also observe that models cannot reliably jailbreak each other and fail generally at tasks requiring creativity. We release our code at https://github.com/facebookresearch/ZeroSumEval.

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.

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.

Imperfect information games (IIG) are games in which each player only partially observes the current game state. We study how to learn epsilon-optimal strategies in a zero-sum IIG through self-play with trajectory feedback. We give a problem-independent lower bound mathcal{O}(H(A_{X}+B_{Y})/epsilon^2) on the required number of realizations to learn these strategies with high probability, where H is the length of the game, A_{X} and B_{Y} are the total number of actions for the two players. We also propose two Follow the Regularized leader (FTRL) algorithms for this setting: Balanced FTRL which matches this lower bound, but requires the knowledge of the information set structure beforehand to define the regularization; and Adaptive FTRL which needs mathcal{O}(H^2(A_{X}+B_{Y})/epsilon^2) realizations without this requirement by progressively adapting the regularization to the observations.

Despite the great empirical success of deep reinforcement learning, its theoretical foundation is less well understood. In this work, we make the first attempt to theoretically understand the deep Q-network (DQN) algorithm (Mnih et al., 2015) from both algorithmic and statistical perspectives. In specific, we focus on a slight simplification of DQN that fully captures its key features. Under mild assumptions, we establish the algorithmic and statistical rates of convergence for the action-value functions of the iterative policy sequence obtained by DQN. In particular, the statistical error characterizes the bias and variance that arise from approximating the action-value function using deep neural network, while the algorithmic error converges to zero at a geometric rate. As a byproduct, our analysis provides justifications for the techniques of experience replay and target network, which are crucial to the empirical success of DQN. Furthermore, as a simple extension of DQN, we propose the Minimax-DQN algorithm for zero-sum Markov game with two players. Borrowing the analysis of DQN, we also quantify the difference between the policies obtained by Minimax-DQN and the Nash equilibrium of the Markov game in terms of both the algorithmic and statistical rates of convergence.

We initiate the study of Multi-Agent Reinforcement Learning from Human Feedback (MARLHF), exploring both theoretical foundations and empirical validations. We define the task as identifying Nash equilibrium from a preference-only offline dataset in general-sum games, a problem marked by the challenge of sparse feedback signals. Our theory establishes the upper complexity bounds for Nash Equilibrium in effective MARLHF, demonstrating that single-policy coverage is inadequate and highlighting the importance of unilateral dataset coverage. These theoretical insights are verified through comprehensive experiments. To enhance the practical performance, we further introduce two algorithmic techniques. (1) We propose a Mean Squared Error (MSE) regularization along the time axis to achieve a more uniform reward distribution and improve reward learning outcomes. (2) We utilize imitation learning to approximate the reference policy, ensuring stability and effectiveness in training. Our findings underscore the multifaceted approach required for MARLHF, paving the way for effective preference-based multi-agent systems.

Card games are widely used to study sequential decision-making under uncertainty, with real-world analogues in negotiation, finance, and cybersecurity. These games typically fall into three categories based on the flow of control: strictly sequential (players alternate single actions), deterministic response (some actions trigger a fixed outcome), and unbounded reciprocal response (alternating counterplays are permitted). A less-explored but strategically rich structure is the bounded one-sided response, where a player's action briefly transfers control to the opponent, who must satisfy a fixed condition through one or more moves before the turn resolves. We term games featuring this mechanism Bounded One-Sided Response Games (BORGs). We introduce a modified version of Monopoly Deal as a benchmark environment that isolates this dynamic, where a Rent action forces the opponent to choose payment assets. The gold-standard algorithm, Counterfactual Regret Minimization (CFR), converges on effective strategies without novel algorithmic extensions. A lightweight full-stack research platform unifies the environment, a parallelized CFR runtime, and a human-playable web interface. The trained CFR agent and source code are available at https://monopolydeal.ai.

OpenSpiel is a collection of environments and algorithms for research in general reinforcement learning and search/planning in games. OpenSpiel supports n-player (single- and multi- agent) zero-sum, cooperative and general-sum, one-shot and sequential, strictly turn-taking and simultaneous-move, perfect and imperfect information games, as well as traditional multiagent environments such as (partially- and fully- observable) grid worlds and social dilemmas. OpenSpiel also includes tools to analyze learning dynamics and other common evaluation metrics. This document serves both as an overview of the code base and an introduction to the terminology, core concepts, and algorithms across the fields of reinforcement learning, computational game theory, and search.

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.

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.

As Large Language Models (LLMs) integrate into our social and economic interactions, we need to deepen our understanding of how humans respond to LLMs opponents in strategic settings. We present the results of the first controlled monetarily-incentivised laboratory experiment looking at differences in human behaviour in a multi-player p-beauty contest against other humans and LLMs. We use a within-subject design in order to compare behaviour at the individual level. We show that, in this environment, human subjects choose significantly lower numbers when playing against LLMs than humans, which is mainly driven by the increased prevalence of `zero' Nash-equilibrium choices. This shift is mainly driven by subjects with high strategic reasoning ability. Subjects who play the zero Nash-equilibrium choice motivate their strategy by appealing to perceived LLM's reasoning ability and, unexpectedly, propensity towards cooperation. Our findings provide foundational insights into the multi-player human-LLM interaction in simultaneous choice games, uncover heterogeneities in both subjects' behaviour and beliefs about LLM's play when playing against them, and suggest important implications for mechanism design in mixed human-LLM systems.

We generalize gradient descent with momentum for optimization in differentiable games to have complex-valued momentum. We give theoretical motivation for our method by proving convergence on bilinear zero-sum games for simultaneous and alternating updates. Our method gives real-valued parameter updates, making it a drop-in replacement for standard optimizers. We empirically demonstrate that complex-valued momentum can improve convergence in realistic adversarial games - like generative adversarial networks - by showing we can find better solutions with an almost identical computational cost. We also show a practical generalization to a complex-valued Adam variant, which we use to train BigGAN to better inception scores on CIFAR-10.

The computational study of equilibria involving constraints on players' strategies has been largely neglected. However, in real-world applications, players are usually subject to constraints ruling out the feasibility of some of their strategies, such as, e.g., safety requirements and budget caps. Computational studies on constrained versions of the Nash equilibrium have lead to some results under very stringent assumptions, while finding constrained versions of the correlated equilibrium (CE) is still unexplored. In this paper, we introduce and computationally characterize constrained Phi-equilibria -- a more general notion than constrained CEs -- in normal-form games. We show that computing such equilibria is in general computationally intractable, and also that the set of the equilibria may not be convex, providing a sharp divide with unconstrained CEs. Nevertheless, we provide a polynomial-time algorithm for computing a constrained (approximate) Phi-equilibrium maximizing a given linear function, when either the number of constraints or that of players' actions is fixed. Moreover, in the special case in which a player's constraints do not depend on other players' strategies, we show that an exact, function-maximizing equilibrium can be computed in polynomial time, while one (approximate) equilibrium can be found with an efficient decentralized no-regret learning algorithm.

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.

By incorporating regret minimization, double oracle methods have demonstrated rapid convergence to Nash Equilibrium (NE) in normal-form games and extensive-form games, through algorithms such as online double oracle (ODO) and extensive-form double oracle (XDO), respectively. In this study, we further examine the theoretical convergence rate and sample complexity of such regret minimization-based double oracle methods, utilizing a unified framework called Regret-Minimizing Double Oracle. Based on this framework, we extend ODO to extensive-form games and determine its sample complexity. Moreover, we demonstrate that the sample complexity of XDO can be exponential in the number of information sets |S|, owing to the exponentially decaying stopping threshold of restricted games. To solve this problem, we propose the Periodic Double Oracle (PDO) method, which has the lowest sample complexity among all existing double oracle methods, being only polynomial in |S|. Empirical evaluations on multiple poker and board games show that PDO achieves significantly faster convergence than previous double oracle algorithms and reaches a competitive level with state-of-the-art regret minimization methods.

We present Self-Play Preference Optimization (SPO), an algorithm for reinforcement learning from human feedback. Our approach is minimalist in that it does not require training a reward model nor unstable adversarial training and is therefore rather simple to implement. Our approach is maximalist in that it provably handles non-Markovian, intransitive, and stochastic preferences while being robust to the compounding errors that plague offline approaches to sequential prediction. To achieve the preceding qualities, we build upon the concept of a Minimax Winner (MW), a notion of preference aggregation from the social choice theory literature that frames learning from preferences as a zero-sum game between two policies. By leveraging the symmetry of this game, we prove that rather than using the traditional technique of dueling two policies to compute the MW, we can simply have a single agent play against itself while maintaining strong convergence guarantees. Practically, this corresponds to sampling multiple trajectories from a policy, asking a rater or preference model to compare them, and then using the proportion of wins as the reward for a particular trajectory. We demonstrate that on a suite of continuous control tasks, we are able to learn significantly more efficiently than reward-model based approaches while maintaining robustness to the intransitive and stochastic preferences that frequently occur in practice when aggregating human judgments.

When reward functions are hand-designed, deep reinforcement learning algorithms often suffer from reward misspecification, causing them to learn suboptimal policies in terms of the intended task objectives. In the single-agent case, inverse reinforcement learning (IRL) techniques attempt to address this issue by inferring the reward function from expert demonstrations. However, in multi-agent problems, misalignment between the learned and true objectives is exacerbated due to increased environment non-stationarity and variance that scales with multiple agents. As such, in multi-agent general-sum games, multi-agent IRL algorithms have difficulty balancing cooperative and competitive objectives. To address these issues, we propose Multi-Agent Marginal Q-Learning from Demonstrations (MAMQL), a novel sample-efficient framework for multi-agent IRL. For each agent, MAMQL learns a critic marginalized over the other agents' policies, allowing for a well-motivated use of Boltzmann policies in the multi-agent context. We identify a connection between optimal marginalized critics and single-agent soft-Q IRL, allowing us to apply a direct, simple optimization criterion from the single-agent domain. Across our experiments on three different simulated domains, MAMQL significantly outperforms previous multi-agent methods in average reward, sample efficiency, and reward recovery by often more than 2-5x. We make our code available at https://sites.google.com/view/mamql .

We investigate learning the equilibria in non-stationary multi-agent systems and address the challenges that differentiate multi-agent learning from single-agent learning. Specifically, we focus on games with bandit feedback, where testing an equilibrium can result in substantial regret even when the gap to be tested is small, and the existence of multiple optimal solutions (equilibria) in stationary games poses extra challenges. To overcome these obstacles, we propose a versatile black-box approach applicable to a broad spectrum of problems, such as general-sum games, potential games, and Markov games, when equipped with appropriate learning and testing oracles for stationary environments. Our algorithms can achieve Oleft(Delta^{1/4}T^{3/4}right) regret when the degree of nonstationarity, as measured by total variation Delta, is known, and Oleft(Delta^{1/5}T^{4/5}right) regret when Delta is unknown, where T is the number of rounds. Meanwhile, our algorithm inherits the favorable dependence on number of agents from the oracles. As a side contribution that may be independent of interest, we show how to test for various types of equilibria by a black-box reduction to single-agent learning, which includes Nash equilibria, correlated equilibria, and coarse correlated equilibria.

Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting finite-sum structure, which generically arises in empirical variants of learning problems in these contexts. Further, methods with computable approximation errors are highly desirable, as they provide verifiable exit criteria. Motivated by these applications, we study finite-sum monotone inclusion problems, which model broad classes of equilibrium problems. Our main contributions are variants of the classical Halpern iteration that employ variance reduction to obtain improved complexity guarantees in which n component operators in the finite sum are ``on average'' either cocoercive or Lipschitz continuous and monotone, with parameter L. The resulting oracle complexity of our methods, which provide guarantees for the last iterate and for a (computable) operator norm residual, is mathcal{O}( n + nLvarepsilon^{-1}), which improves upon existing methods by a factor up to n. This constitutes the first variance reduction-type result for general finite-sum monotone inclusions and for more specific problems such as convex-concave optimization when operator norm residual is the optimality measure. We further argue that, up to poly-logarithmic factors, this complexity is unimprovable in the monotone Lipschitz setting; i.e., the provided result is near-optimal.

We improve the framework of open games with agency by showing how the players' counterfactual analysis giving rise to Nash equilibria can be described in the dynamics of the game itself (hence diegetically), getting rid of devices such as equilibrium predicates. This new approach overlaps almost completely with the way gradient-based learners are specified and trained. Indeed, we show feedback propagation in games can be seen as a form of backpropagation, with a crucial difference explaining the distinctive character of the phenomenology of non-cooperative games. We outline a functorial construction of arena of games, show players form a subsystem over it, and prove that their 'fixpoint behaviours' are Nash equilibria.

We give a quantum algorithm for computing an epsilon-approximate Nash equilibrium of a zero-sum game in a m times n payoff matrix with bounded entries. Given a standard quantum oracle for accessing the payoff matrix our algorithm runs in time O(m + ncdot epsilon^{-2.5} + epsilon^{-3}) and outputs a classical representation of the epsilon-approximate Nash equilibrium. This improves upon the best prior quantum runtime of O(m + n cdot epsilon^{-3}) obtained by [vAG19] and the classic O((m + n) cdot epsilon^{-2}) runtime due to [GK95] whenever epsilon = Omega((m +n)^{-1}). We obtain this result by designing new quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.

This paper presents novel game-playing AI agents based on the AlphaZero framework, enhanced with Vision Transformer (ViT): AlphaViT, AlphaViD, and AlphaVDA. These agents are designed to play multiple board games of various sizes using a single network with shared weights, thereby overcoming AlphaZero's limitation of fixed-board-size constraints. AlphaViT employs only a transformer encoder, whereas AlphaViD and AlphaVDA incorporate both transformer encoders and decoders. In AlphaViD, the decoder processes outputs from the encoder, whereas AlphaVDA uses a learnable embeddings as the decoder input. The additional decoder layers in AlphaViD and AlphaVDA provide flexibility to adapt to various action spaces and board sizes. Experimental results show that the proposed agents, trained on either individual games or multiple games simultaneously, consistently outperform traditional algorithms such as Minimax and Monte Carlo Tree Search and approach the performance of AlphaZero, despite using a single deep neural network (DNN) with shared weights. In particular, AlphaViT shows strong performance across all tested games. Furthermore, fine-tuning the DNN using pre-trained weights from small-board games accelerates convergence and improves performance, particularly in Gomoku. Interestingly, simultaneous training on multiple games yields performance comparable to, or even surpassing, single-game training. These results indicate the potential of transformer-based architectures to develop more flexible and robust game-playing AI agents that excel in multiple games and dynamic environments.

Conventional federated learning frameworks suffer from several challenges including performance bottlenecks at the central aggregation server, data bias, poor model convergence, and exposure to model poisoning attacks, and limited trust in the centralized infrastructure. In the current paper, a novel game theory-based approach called pFedGame is proposed for decentralized federated learning, best suitable for temporally dynamic networks. The proposed algorithm works without any centralized server for aggregation and incorporates the problem of vanishing gradients and poor convergence over temporally dynamic topology among federated learning participants. The solution comprises two sequential steps in every federated learning round, for every participant. First, it selects suitable peers for collaboration in federated learning. Secondly, it executes a two-player constant sum cooperative game to reach convergence by applying an optimal federated learning aggregation strategy. Experiments performed to assess the performance of pFedGame in comparison to existing methods in decentralized federated learning have shown promising results with accuracy higher than 70% for heterogeneous data.

Assistance games are a promising alternative to reinforcement learning from human feedback (RLHF) for training AI assistants. Assistance games resolve key drawbacks of RLHF, such as incentives for deceptive behavior, by explicitly modeling the interaction between assistant and user as a two-player game where the assistant cannot observe their shared goal. Despite their potential, assistance games have only been explored in simple settings. Scaling them to more complex environments is difficult because it requires both solving intractable decision-making problems under uncertainty and accurately modeling human users' behavior. We present the first scalable approach to solving assistance games and apply it to a new, challenging Minecraft-based assistance game with over 10^{400} possible goals. Our approach, AssistanceZero, extends AlphaZero with a neural network that predicts human actions and rewards, enabling it to plan under uncertainty. We show that AssistanceZero outperforms model-free RL algorithms and imitation learning in the Minecraft-based assistance game. In a human study, our AssistanceZero-trained assistant significantly reduces the number of actions participants take to complete building tasks in Minecraft. Our results suggest that assistance games are a tractable framework for training effective AI assistants in complex environments. Our code and models are available at https://github.com/cassidylaidlaw/minecraft-building-assistance-game.

We study an online learning problem in general-sum Stackelberg games, where players act in a decentralized and strategic manner. We study two settings depending on the type of information for the follower: (1) the limited information setting where the follower only observes its own reward, and (2) the side information setting where the follower has extra side information about the leader's reward. We show that for the follower, myopically best responding to the leader's action is the best strategy for the limited information setting, but not necessarily so for the side information setting -- the follower can manipulate the leader's reward signals with strategic actions, and hence induce the leader's strategy to converge to an equilibrium that is better off for itself. Based on these insights, we study decentralized online learning for both players in the two settings. Our main contribution is to derive last-iterate convergence and sample complexity results in both settings. Notably, we design a new manipulation strategy for the follower in the latter setting, and show that it has an intrinsic advantage against the best response strategy. Our theories are also supported by empirical results.

Real-world reinforcement learning (RL) is often severely limited since typical RL algorithms heavily rely on the reset mechanism to sample proper initial states. In practice, the reset mechanism is expensive to implement due to the need for human intervention or heavily engineered environments. To make learning more practical, we propose a generic no-regret reduction to systematically design reset-free RL algorithms. Our reduction turns reset-free RL into a two-player game. We show that achieving sublinear regret in this two-player game would imply learning a policy that has both sublinear performance regret and sublinear total number of resets in the original RL problem. This means that the agent eventually learns to perform optimally and avoid resets. By this reduction, we design an instantiation for linear Markov decision processes, which is the first provably correct reset-free RL algorithm to our knowledge.

For a two-player imperfect-information extensive-form game (IIEFG) with K time steps and a player action space of size U, the game tree complexity is U^{2K}, causing existing IIEFG solvers to struggle with large or infinite (U,K), e.g., differential games with continuous action spaces. To partially address this scalability challenge, we focus on an important class of 2p0s games where the informed player (P1) knows the payoff while the uninformed player (P2) only has a belief over the set of I possible payoffs. Such games encompass a wide range of scenarios in sports, defense, cybersecurity, and finance. We prove that under mild conditions, P1's (resp. P2's) equilibrium strategy at any infostate concentrates on at most I (resp. I+1) action prototypes. When Ill U, this equilibrium structure causes the game tree complexity to collapse to I^K for P1 when P2 plays pure best responses, and (I+1)^K for P2 in a dual game where P1 plays pure best responses. We then show that exploiting this structure in standard learning modes, i.e., model-free multiagent reinforcement learning and model predictive control, is straightforward, leading to significant improvements in learning accuracy and efficiency from SOTA IIEFG solvers. Our demonstration solves a 22-player football game (K=10, U=infty) where the attacking team has to strategically conceal their intention until a critical moment in order to exploit information advantage. Code is available at https://github.com/ghimiremukesh/cams/tree/iclr

Randomized mechanisms can have good normative properties compared to their deterministic counterparts. However, randomized mechanisms are problematic in several ways such as in their verifiability. We propose here to derandomize such mechanisms by having agents play a game instead of tossing a coin. The game is designed so an agent's best action is to play randomly, and this play then injects ``randomness'' into the mechanism. This derandomization retains many of the good normative properties of the original randomized mechanism but gives a mechanism that is deterministic and easy, for instance, to audit. We consider three related methods to derandomize randomized mechanism in six different domains: voting, facility location, task allocation, school choice, peer selection, and resource allocation. We propose a number of novel derandomized mechanisms for these six domains with good normative properties. Each mechanism has a mixed Nash equilibrium in which agents play a modular arithmetic game with an uniform mixed strategy. In all but one mixed Nash equilibrium, agents report their preferences over the original problem sincerely. The derandomized methods are thus ``quasi-strategy proof''. In one domain, we additionally show that a new and desirable normative property emerges as a result of derandomization.

Reinforcement learning from human feedback (RLHF) has emerged as the standard paradigm for aligning large language models (LLMs) with human preferences. However, reward-based methods built on the Bradley-Terry assumption struggle to capture the non-transitive and heterogeneous nature of real-world preferences. To address this, recent studies have reframed alignment as a two-player Nash game, giving rise to Nash learning from human feedback (NLHF). While this perspective has inspired algorithms such as INPO, ONPO, and EGPO with strong theoretical and empirical guarantees, they remain fundamentally restricted to two-player interactions, creating a single-opponent bias that fails to capture the full complexity of realistic preference structures. In this work, we introduce Multiplayer Nash Preference Optimization (MNPO), a novel framework that generalizes NLHF to the multiplayer regime. It formulates alignment as an n-player game, where each policy competes against a population of opponents while being regularized toward a reference model. Our framework establishes well-defined Nash equilibria in multiplayer settings and extends the concept of duality gap to quantify approximation quality. We demonstrate that MNPO inherits the equilibrium guarantees of two-player methods while enabling richer competitive dynamics and improved coverage of diverse preference structures. Through comprehensive empirical evaluation, we show that MNPO consistently outperforms existing NLHF baselines on instruction-following benchmarks, achieving superior alignment quality under heterogeneous annotator conditions and mixed-policy evaluation scenarios. Together, these results establish MNPO as a principled and scalable framework for aligning LLMs with complex, non-transitive human preferences. Code is available at https://github.com/smiles724/MNPO.

We examine the problem of regret minimization when the learner is involved in a continuous game with other optimizing agents: in this case, if all players follow a no-regret algorithm, it is possible to achieve significantly lower regret relative to fully adversarial environments. We study this problem in the context of variationally stable games (a class of continuous games which includes all convex-concave and monotone games), and when the players only have access to noisy estimates of their individual payoff gradients. If the noise is additive, the game-theoretic and purely adversarial settings enjoy similar regret guarantees; however, if the noise is multiplicative, we show that the learners can, in fact, achieve constant regret. We achieve this faster rate via an optimistic gradient scheme with learning rate separation -- that is, the method's extrapolation and update steps are tuned to different schedules, depending on the noise profile. Subsequently, to eliminate the need for delicate hyperparameter tuning, we propose a fully adaptive method that attains nearly the same guarantees as its non-adapted counterpart, while operating without knowledge of either the game or of the noise profile.

Zero-shot coordination in cooperative artificial intelligence (AI) remains a significant challenge, which means effectively coordinating with a wide range of unseen partners. Previous algorithms have attempted to address this challenge by optimizing fixed objectives within a population to improve strategy or behaviour diversity. However, these approaches can result in a loss of learning and an inability to cooperate with certain strategies within the population, known as cooperative incompatibility. To address this issue, we propose the Cooperative Open-ended LEarning (COLE) framework, which constructs open-ended objectives in cooperative games with two players from the perspective of graph theory to assess and identify the cooperative ability of each strategy. We further specify the framework and propose a practical algorithm that leverages knowledge from game theory and graph theory. Furthermore, an analysis of the learning process of the algorithm shows that it can efficiently overcome cooperative incompatibility. The experimental results in the Overcooked game environment demonstrate that our method outperforms current state-of-the-art methods when coordinating with different-level partners. Our demo is available at https://sites.google.com/view/cole-2023.

We study multi-agent reinforcement learning (MARL) for the general-sum Markov Games (MGs) under the general function approximation. In order to find the minimum assumption for sample-efficient learning, we introduce a novel complexity measure called the Multi-Agent Decoupling Coefficient (MADC) for general-sum MGs. Using this measure, we propose the first unified algorithmic framework that ensures sample efficiency in learning Nash Equilibrium, Coarse Correlated Equilibrium, and Correlated Equilibrium for both model-based and model-free MARL problems with low MADC. We also show that our algorithm provides comparable sublinear regret to the existing works. Moreover, our algorithm combines an equilibrium-solving oracle with a single objective optimization subprocedure that solves for the regularized payoff of each deterministic joint policy, which avoids solving constrained optimization problems within data-dependent constraints (Jin et al. 2020; Wang et al. 2023) or executing sampling procedures with complex multi-objective optimization problems (Foster et al. 2023), thus being more amenable to empirical implementation.

Poker, also known as Texas Hold'em, has always been a typical research target within imperfect information games (IIGs). IIGs have long served as a measure of artificial intelligence (AI) development. Representative prior works, such as DeepStack and Libratus heavily rely on counterfactual regret minimization (CFR) to tackle heads-up no-limit Poker. However, it is challenging for subsequent researchers to learn CFR from previous models and apply it to other real-world applications due to the expensive computational cost of CFR iterations. Additionally, CFR is difficult to apply to multi-player games due to the exponential growth of the game tree size. In this work, we introduce PokerGPT, an end-to-end solver for playing Texas Hold'em with arbitrary number of players and gaining high win rates, established on a lightweight large language model (LLM). PokerGPT only requires simple textual information of Poker games for generating decision-making advice, thus guaranteeing the convenient interaction between AI and humans. We mainly transform a set of textual records acquired from real games into prompts, and use them to fine-tune a lightweight pre-trained LLM using reinforcement learning human feedback technique. To improve fine-tuning performance, we conduct prompt engineering on raw data, including filtering useful information, selecting behaviors of players with high win rates, and further processing them into textual instruction using multiple prompt engineering techniques. Through the experiments, we demonstrate that PokerGPT outperforms previous approaches in terms of win rate, model size, training time, and response speed, indicating the great potential of LLMs in solving IIGs.

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.

Modern reinforcement learning (RL) systems capture deep truths about general, human problem-solving. In domains where new data can be simulated cheaply, these systems uncover sequential decision-making policies that far exceed the ability of any human. Society faces many problems whose solutions require this skill, but they are often in domains where new data cannot be cheaply simulated. In such scenarios, we can learn simulators from existing data, but these will only ever be approximately correct, and can be pathologically incorrect when queried outside of their training distribution. As a result, a misalignment between the environments in which we train our agents and the real-world in which we wish to deploy our agents is inevitable. Dealing with this misalignment is the primary concern of zero-shot reinforcement learning, a problem setting where the agent must generalise to a new task or domain with zero practice shots. Whilst impressive progress has been made on methods that perform zero-shot RL in idealised settings, new work is needed if these results are to be replicated in real-world settings. In this thesis, we argue that doing so requires us to navigate (at least) three constraints. First, the data quality constraint: real-world datasets are small and homogeneous. Second, the observability constraint: states, dynamics and rewards in the real-world are often only partially observed. And third, the data availability constraint: a priori access to data cannot always be assumed. This work proposes a suite of methods that perform zero-shot RL subject to these constraints. In a series of empirical studies we expose the failings of existing methods, and justify our techniques for remedying them. We believe these designs take us a step closer to RL methods that can be deployed to solve real-world problems.

In various real-world scenarios, interactions among agents often resemble the dynamics of general-sum games, where each agent strives to optimize its own utility. Despite the ubiquitous relevance of such settings, decentralized machine learning algorithms have struggled to find equilibria that maximize individual utility while preserving social welfare. In this paper we introduce Learning with Opponent Q-Learning Awareness (LOQA), a novel, decentralized reinforcement learning algorithm tailored to optimizing an agent's individual utility while fostering cooperation among adversaries in partially competitive environments. LOQA assumes the opponent samples actions proportionally to their action-value function Q. Experimental results demonstrate the effectiveness of LOQA at achieving state-of-the-art performance in benchmark scenarios such as the Iterated Prisoner's Dilemma and the Coin Game. LOQA achieves these outcomes with a significantly reduced computational footprint, making it a promising approach for practical multi-agent applications.

Multi-agent football poses an unsolved challenge in AI research. Existing work has focused on tackling simplified scenarios of the game, or else leveraging expert demonstrations. In this paper, we develop a multi-agent system to play the full 11 vs. 11 game mode, without demonstrations. This game mode contains aspects that present major challenges to modern reinforcement learning algorithms; multi-agent coordination, long-term planning, and non-transitivity. To address these challenges, we present TiZero; a self-evolving, multi-agent system that learns from scratch. TiZero introduces several innovations, including adaptive curriculum learning, a novel self-play strategy, and an objective that optimizes the policies of multiple agents jointly. Experimentally, it outperforms previous systems by a large margin on the Google Research Football environment, increasing win rates by over 30%. To demonstrate the generality of TiZero's innovations, they are assessed on several environments beyond football; Overcooked, Multi-agent Particle-Environment, Tic-Tac-Toe and Connect-Four.

Recent advances in Competitive Self-Play (CSP) have achieved, or even surpassed, human level performance in complex game environments such as Dota 2 and StarCraft II using Distributed Multi-Agent Reinforcement Learning (MARL). One core component of these methods relies on creating a pool of learning agents -- consisting of the Main Agent, past versions of this agent, and Exploiter Agents -- where Exploiter Agents learn counter-strategies to the Main Agents. A key drawback of these approaches is the large computational cost and physical time that is required to train the system, making them impractical to deploy in highly iterative real-life settings such as video game productions. In this paper, we propose the Minimax Exploiter, a game theoretic approach to exploiting Main Agents that leverages knowledge of its opponents, leading to significant increases in data efficiency. We validate our approach in a diversity of settings, including simple turn based games, the arcade learning environment, and For Honor, a modern video game. The Minimax Exploiter consistently outperforms strong baselines, demonstrating improved stability and data efficiency, leading to a robust CSP-MARL method that is both flexible and easy to deploy.

AI agents will be predictable in certain ways that traditional agents are not. Where and how can we leverage this predictability in order to improve social welfare? We study this question in a game-theoretic setting where one agent can pay a fixed cost to simulate the other in order to learn its mixed strategy. As a negative result, we prove that, in contrast to prior work on pure-strategy simulation, enabling mixed-strategy simulation may no longer lead to improved outcomes for both players in all so-called "generalised trust games". In fact, mixed-strategy simulation does not help in any game where the simulatee's action can depend on that of the simulator. We also show that, in general, deciding whether simulation introduces Pareto-improving Nash equilibria in a given game is NP-hard. As positive results, we establish that mixed-strategy simulation can improve social welfare if the simulator has the option to scale their level of trust, if the players face challenges with both trust and coordination, or if maintaining some level of privacy is essential for enabling cooperation.

Competitions for shareable and limited resources have long been studied with strategic agents. In reality, agents often have to learn and maximize the rewards of the resources at the same time. To design an individualized competing policy, we model the competition between agents in a novel multi-player multi-armed bandit (MPMAB) setting where players are selfish and aim to maximize their own rewards. In addition, when several players pull the same arm, we assume that these players averagely share the arms' rewards by expectation. Under this setting, we first analyze the Nash equilibrium when arms' rewards are known. Subsequently, we propose a novel SelfishMPMAB with Averaging Allocation (SMAA) approach based on the equilibrium. We theoretically demonstrate that SMAA could achieve a good regret guarantee for each player when all players follow the algorithm. Additionally, we establish that no single selfish player can significantly increase their rewards through deviation, nor can they detrimentally affect other players' rewards without incurring substantial losses for themselves. We finally validate the effectiveness of the method in extensive synthetic experiments.

An abundance of recent impossibility results establish that regret minimization in Markov games with adversarial opponents is both statistically and computationally intractable. Nevertheless, none of these results preclude the possibility of regret minimization under the assumption that all parties adopt the same learning procedure. In this work, we present the first (to our knowledge) algorithm for learning in general-sum Markov games that provides sublinear regret guarantees when executed by all agents. The bounds we obtain are for swap regret, and thus, along the way, imply convergence to a correlated equilibrium. Our algorithm is decentralized, computationally efficient, and does not require any communication between agents. Our key observation is that online learning via policy optimization in Markov games essentially reduces to a form of weighted regret minimization, with unknown weights determined by the path length of the agents' policy sequence. Consequently, controlling the path length leads to weighted regret objectives for which sufficiently adaptive algorithms provide sublinear regret guarantees.

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.

We consider online learning in multi-player smooth monotone games. Existing algorithms have limitations such as (1) being only applicable to strongly monotone games; (2) lacking the no-regret guarantee; (3) having only asymptotic or slow O(1{T}) last-iterate convergence rate to a Nash equilibrium. While the O(1{T}) rate is tight for a large class of algorithms including the well-studied extragradient algorithm and optimistic gradient algorithm, it is not optimal for all gradient-based algorithms. We propose the accelerated optimistic gradient (AOG) algorithm, the first doubly optimal no-regret learning algorithm for smooth monotone games. Namely, our algorithm achieves both (i) the optimal O(T) regret in the adversarial setting under smooth and convex loss functions and (ii) the optimal O(1{T}) last-iterate convergence rate to a Nash equilibrium in multi-player smooth monotone games. As a byproduct of the accelerated last-iterate convergence rate, we further show that each player suffers only an O(log T) individual worst-case dynamic regret, providing an exponential improvement over the previous state-of-the-art O(T) bound.

Implementing quantum game theory on real hardware is challenging due to noise, decoherence, and limited qubit connectivity, yet such demonstrations are essential to validate theoretical predictions. We present one of the first full experimental realizations of the Battle of the Sexes game under the Eisert-Wilkens-Lewenstein (EWL) framework on IBM Quantum's ibm sherbrooke superconducting processor. Four quantum strategies (I, H, R(pi/4), R(pi)) were evaluated across 31 entanglement values gamma in [0, pi] using 2048 shots per configuration, enabling a direct comparison between analytical predictions and hardware execution. To mitigate noise and variability, we introduce a Guided Circuit Mapping (GCM) method that dynamically selects qubit pairs and optimizes routing based on real-time topology and calibration data. The analytical model forecasts up to 108% payoff improvement over the classical equilibrium, and despite hardware-induced deviations, experimental results with GCM preserve the expected payoff trends within 3.5%-12% relative error. These findings show that quantum advantages in strategic coordination can persist under realistic NISQ conditions, providing a pathway toward practical applications of quantum game theory in multi-agent, economic, and distributed decision-making systems.

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.

Reinforcement Learning with Human Feedback (RLHF) has achieved great success in aligning large language models (LLMs) with human preferences. Prevalent RLHF approaches are reward-based, following the Bradley-Terry (BT) model assumption, which may not fully capture the complexity of human preferences. In this paper, we explore RLHF under a general preference framework and approach it from a game-theoretic perspective. Specifically, we formulate the problem as a two-player game and propose a novel algorithm, iterative Nash policy optimization (INPO). The key idea is to let the policy play against itself via no-regret learning, thereby approximating the Nash policy. Unlike previous methods, INPO bypasses the need for estimating the expected win rate for individual responses, which typically incurs high computational or annotation costs. Instead, we introduce a new loss objective that is directly minimized over a preference dataset. We provide theoretical analysis for our approach and demonstrate its effectiveness through experiments on various representative benchmarks. With an LLaMA-3-8B-based SFT model, INPO achieves a 41.5% length-controlled win rate on AlpacaEval 2.0 and a 38.3% win rate on Arena-Hard, showing substantial improvement over the state-of-the-art iterative algorithm [Dong et al., 2024] under the BT model assumption. Additionally, our ablation study highlights the benefits of incorporating KL regularization for response length control.

We investigate the mechanism design problem faced by a principal who hires multiple agents to gather and report costly information. Then, the principal exploits the information to make an informed decision. We model this problem as a game, where the principal announces a mechanism consisting in action recommendations and a payment function, a.k.a. scoring rule. Then, each agent chooses an effort level and receives partial information about an underlying state of nature based on the effort. Finally, the agents report the information (possibly non-truthfully), the principal takes a decision based on this information, and the agents are paid according to the scoring rule. While previous work focuses on single-agent problems, we consider multi-agents settings. This poses the challenge of coordinating the agents' efforts and aggregating correlated information. Indeed, we show that optimal mechanisms must correlate agents' efforts, which introduces externalities among the agents, and hence complex incentive compatibility constraints and equilibrium selection problems. First, we design a polynomial-time algorithm to find an optimal incentive compatible mechanism. Then, we study an online problem, where the principal repeatedly interacts with a group of unknown agents. We design a no-regret algorithm that provides mathcal{O}(T^{2/3}) regret with respect to an optimal mechanism, matching the state-of-the-art bound for single-agent settings.

Normal-form games (NFGs) are the fundamental model of strategic interaction. We study their representation using neural networks. We describe the inherent equivariance of NFGs -- any permutation of strategies describes an equivalent game -- as well as the challenges this poses for representation learning. We then propose the NfgTransformer architecture that leverages this equivariance, leading to state-of-the-art performance in a range of game-theoretic tasks including equilibrium-solving, deviation gain estimation and ranking, with a common approach to NFG representation. We show that the resulting model is interpretable and versatile, paving the way towards deep learning systems capable of game-theoretic reasoning when interacting with humans and with each other.

Decision-making, a complicated task requiring various types of abilities, presents an excellent framework for assessing Large Language Models (LLMs). Our research investigates LLMs' decision-making capabilities through the lens of a well-established field, Game Theory. We focus specifically on games that support the participation of more than two agents simultaneously. Subsequently, we introduce our framework, GAMA-Bench, including eight classical multi-agent games. We design a scoring scheme to assess a model's performance in these games quantitatively. Through GAMA-Bench, we investigate LLMs' robustness, generalizability, and enhancement strategies. Results reveal that while GPT-3.5 shows satisfying robustness, its generalizability is relatively limited. However, its performance can be improved through approaches such as Chain-of-Thought. Additionally, we conduct evaluations across various LLMs and find that GPT-4 outperforms other models on GAMA-Bench, achieving a score of 60.5. Moreover, Gemini-1.0-Pro and GPT-3.5 (0613, 1106, 0125) demonstrate similar intelligence on GAMA-Bench. The code and experimental results are made publicly available via https://github.com/CUHK-ARISE/GAMABench.

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.

We consider learning to play multiplayer imperfect-information games with simultaneous moves and large state-action spaces. Previous attempts to tackle such challenging games have largely focused on model-free learning methods, often requiring hundreds of years of experience to produce competitive agents. Our approach is based on model-based planning. We tackle the problem of partial observability by first building an (oracle) planner that has access to the full state of the environment and then distilling the knowledge of the oracle to a (follower) agent which is trained to play the imperfect-information game by imitating the oracle's choices. We experimentally show that planning with naive Monte Carlo tree search does not perform very well in large combinatorial action spaces. We therefore propose planning with a fixed-depth tree search and decoupled Thompson sampling for action selection. We show that the planner is able to discover efficient playing strategies in the games of Clash Royale and Pommerman and the follower policy successfully learns to implement them by training on a few hundred battles.

We propose a game-based formulation for learning dimensionality-reducing representations of feature vectors, when only a prior knowledge on future prediction tasks is available. In this game, the first player chooses a representation, and then the second player adversarially chooses a prediction task from a given class, representing the prior knowledge. The first player aims is to minimize, and the second player to maximize, the regret: The minimal prediction loss using the representation, compared to the same loss using the original features. For the canonical setting in which the representation, the response to predict and the predictors are all linear functions, and under the mean squared error loss function, we derive the theoretically optimal representation in pure strategies, which shows the effectiveness of the prior knowledge, and the optimal regret in mixed strategies, which shows the usefulness of randomizing the representation. For general representations and loss functions, we propose an efficient algorithm to optimize a randomized representation. The algorithm only requires the gradients of the loss function, and is based on incrementally adding a representation rule to a mixture of such rules.

This paper generalises the treatment of compositional game theory as introduced by the second and third authors with Ghani and Winschel, where games are modelled as morphisms of a symmetric monoidal category. From an economic modelling perspective, the existing notion of an open game is not expressive enough for many applications. This includes stochastic environments, stochastic choices by players, as well as incomplete information regarding the game being played. The current paper addresses these three issue all at once. To achieve this we make significant use of category theory, especially the 'coend optics' of Riley.

The process of revising (or constructing) a policy at execution time -- known as decision-time planning -- has been key to achieving superhuman performance in perfect-information games like chess and Go. A recent line of work has extended decision-time planning to imperfect-information games, leading to superhuman performance in poker. However, these methods involve solving subgames whose sizes grow quickly in the amount of non-public information, making them unhelpful when the amount of non-public information is large. Motivated by this issue, we introduce an alternative framework for decision-time planning that is not based on solving subgames, but rather on update equivalence. In this update-equivalence framework, decision-time planning algorithms replicate the updates of last-iterate algorithms, which need not rely on public information. This facilitates scalability to games with large amounts of non-public information. Using this framework, we derive a provably sound search algorithm for fully cooperative games based on mirror descent and a search algorithm for adversarial games based on magnetic mirror descent. We validate the performance of these algorithms in cooperative and adversarial domains, notably in Hanabi, the standard benchmark for search in fully cooperative imperfect-information games. Here, our mirror descent approach exceeds or matches the performance of public information-based search while using two orders of magnitude less search time. This is the first instance of a non-public-information-based algorithm outperforming public-information-based approaches in a domain they have historically dominated.

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.

The Shapley value is arguably the most popular approach for assigning a meaningful contribution value to players in a cooperative game, which has recently been used intensively in explainable artificial intelligence. The meaningfulness is due to axiomatic properties that only the Shapley value satisfies, which, however, comes at the expense of an exact computation growing exponentially with the number of agents. Accordingly, a number of works are devoted to the efficient approximation of the Shapley values, most of them revolve around the notion of an agent's marginal contribution. In this paper, we propose with SVARM and Stratified SVARM two parameter-free and domain-independent approximation algorithms based on a representation of the Shapley value detached from the notion of marginal contributions. We prove unmatched theoretical guarantees regarding their approximation quality and provide empirical results including synthetic games as well as common explainability use cases comparing ourselves with state-of-the-art methods.

Assessing and comparing player skill in online multiplayer gaming environments is essential for fair matchmaking and player engagement. Traditional ranking models like Elo and Glicko-2, designed for two-player games, are insufficient for the complexity of multi-player, asymmetric team-based matches. To address this gap, the OpenSkill library offers a suite of sophisticated, fast, and adaptable models tailored for such dynamics. Drawing from Bayesian inference methods, OpenSkill provides a more accurate representation of individual player contributions and speeds up the computation of ranks. This paper introduces the OpenSkill library, featuring a Python implementation of the Plackett-Luce model among others, highlighting its performance advantages and predictive accuracy against proprietary systems like TrueSkill. OpenSkill is a valuable tool for game developers and researchers, ensuring a responsive and fair gaming experience by efficiently adjusting player rankings based on game outcomes. The library's support for time decay and diligent documentation further aid in its practical application, making it a robust solution for the nuanced world of multiplayer ranking systems. This paper also acknowledges areas for future enhancement, such as partial play and contribution weighting, emphasizing the library's ongoing development to meet the evolving needs of online gaming communities.

Despite incredible advances, deep learning has been shown to be susceptible to adversarial attacks. Numerous approaches have been proposed to train robust networks both empirically and certifiably. However, most of them defend against only a single type of attack, while recent work takes steps forward in defending against multiple attacks. In this paper, to understand multi-target robustness, we view this problem as a bargaining game in which different players (adversaries) negotiate to reach an agreement on a joint direction of parameter updating. We identify a phenomenon named player domination in the bargaining game, namely that the existing max-based approaches, such as MAX and MSD, do not converge. Based on our theoretical analysis, we design a novel framework that adjusts the budgets of different adversaries to avoid any player dominance. Experiments on standard benchmarks show that employing the proposed framework to the existing approaches significantly advances multi-target robustness.

We introduce an Implicit Game-Theoretic MPC (IGT-MPC), a decentralized algorithm for two-agent motion planning that uses a learned value function that predicts the game-theoretic interaction outcomes as the terminal cost-to-go function in a model predictive control (MPC) framework, guiding agents to implicitly account for interactions with other agents and maximize their reward. This approach applies to competitive and cooperative multi-agent motion planning problems which we formulate as constrained dynamic games. Given a constrained dynamic game, we randomly sample initial conditions and solve for the generalized Nash equilibrium (GNE) to generate a dataset of GNE solutions, computing the reward outcome of each game-theoretic interaction from the GNE. The data is used to train a simple neural network to predict the reward outcome, which we use as the terminal cost-to-go function in an MPC scheme. We showcase emerging competitive and coordinated behaviors using IGT-MPC in scenarios such as two-vehicle head-to-head racing and un-signalized intersection navigation. IGT-MPC offers a novel method integrating machine learning and game-theoretic reasoning into model-based decentralized multi-agent motion planning.

We study the problem of online learning in a two-player decentralized cooperative Stackelberg game. In each round, the leader first takes an action, followed by the follower who takes their action after observing the leader's move. The goal of the leader is to learn to minimize the cumulative regret based on the history of interactions. Differing from the traditional formulation of repeated Stackelberg games, we assume the follower is omniscient, with full knowledge of the true reward, and that they always best-respond to the leader's actions. We analyze the sample complexity of regret minimization in this repeated Stackelberg game. We show that depending on the reward structure, the existence of the omniscient follower may change the sample complexity drastically, from constant to exponential, even for linear cooperative Stackelberg games. This poses unique challenges for the learning process of the leader and the subsequent regret analysis.

We present a new type of game, the Liquidity Game. We draw inspiration from the UK government bond market and apply game theoretic approaches to its analysis. In Liquidity Games, market participants (agents) use non-cooperative games where the players' utility is directly defined by the liquidity of the game itself, offering a paradigm shift in our understanding of market dynamics. Each player's utility is intricately linked to the liquidity generated within the game, making the utility endogenous and dynamic. Players are not just passive recipients of utility based on external factors but active participants whose strategies and actions collectively shape and are shaped by the liquidity of the market. This reflexivity introduces a level of complexity and realism previously unattainable in conventional models. We apply Liquidity Game theoretic approaches to a simple UK bond market interaction and present results for market design and strategic behavior of participants. We tackle one of the largest issues within this mechanism, namely what strategy should market makers utilize when uncertain about the type of market maker they are interacting with, and what structure might regulators wish to see.

Playing games has a long history of describing intricate interactions in simplified forms. In this paper we explore if large language models (LLMs) can play games, investigating their capabilities for randomisation and strategic adaptation through both simultaneous and sequential game interactions. We focus on GPT-4o-Mini-2024-08-17 and test two games between LLMs: Rock Paper Scissors (RPS) and games of strategy (Prisoners Dilemma PD). LLMs are often described as stochastic parrots, and while they may indeed be parrots, our results suggest that they are not very stochastic in the sense that their outputs - when prompted to be random - are often very biased. Our research reveals that LLMs appear to develop loss aversion strategies in repeated games, with RPS converging to stalemate conditions while PD shows systematic shifts between cooperative and competitive outcomes based on prompt design. We detail programmatic tools for independent agent interactions and the Agentic AI challenges faced in implementation. We show that LLMs can indeed play games, just not very well. These results have implications for the use of LLMs in multi-agent LLM systems and showcase limitations in current approaches to model output for strategic decision-making.

In this work we create agents that can perform well beyond a single, individual task, that exhibit much wider generalisation of behaviour to a massive, rich space of challenges. We define a universe of tasks within an environment domain and demonstrate the ability to train agents that are generally capable across this vast space and beyond. The environment is natively multi-agent, spanning the continuum of competitive, cooperative, and independent games, which are situated within procedurally generated physical 3D worlds. The resulting space is exceptionally diverse in terms of the challenges posed to agents, and as such, even measuring the learning progress of an agent is an open research problem. We propose an iterative notion of improvement between successive generations of agents, rather than seeking to maximise a singular objective, allowing us to quantify progress despite tasks being incomparable in terms of achievable rewards. We show that through constructing an open-ended learning process, which dynamically changes the training task distributions and training objectives such that the agent never stops learning, we achieve consistent learning of new behaviours. The resulting agent is able to score reward in every one of our humanly solvable evaluation levels, with behaviour generalising to many held-out points in the universe of tasks. Examples of this zero-shot generalisation include good performance on Hide and Seek, Capture the Flag, and Tag. Through analysis and hand-authored probe tasks we characterise the behaviour of our agent, and find interesting emergent heuristic behaviours such as trial-and-error experimentation, simple tool use, option switching, and cooperation. Finally, we demonstrate that the general capabilities of this agent could unlock larger scale transfer of behaviour through cheap finetuning.

We propose and evaluate a number of solutions to the problem of calculating the cost to serve each location in a single-vehicle transport setting. Such cost to serve analysis has application both strategically and operationally in transportation. The problem is formally given by the traveling salesperson game (TSG), a cooperative total utility game in which agents correspond to locations in a traveling salesperson problem (TSP). The cost to serve a location is an allocated portion of the cost of an optimal tour. The Shapley value is one of the most important normative division schemes in cooperative games, giving a principled and fair allocation both for the TSG and more generally. We consider a number of direct and sampling-based procedures for calculating the Shapley value, and present the first proof that approximating the Shapley value of the TSG within a constant factor is NP-hard. Treating the Shapley value as an ideal baseline allocation, we then develop six proxies for that value which are relatively easy to compute. We perform an experimental evaluation using Synthetic Euclidean games as well as games derived from real-world tours calculated for fast-moving consumer goods scenarios. Our experiments show that several computationally tractable allocation techniques correspond to good proxies for the Shapley value.

We study the problem of designing mechanisms for information acquisition scenarios. This setting models strategic interactions between an uniformed receiver and a set of informed senders. In our model the senders receive information about the underlying state of nature and communicate their observation (either truthfully or not) to the receiver, which, based on this information, selects an action. Our goal is to design mechanisms maximizing the receiver's utility while incentivizing the senders to report truthfully their information. First, we provide an algorithm that efficiently computes an optimal incentive compatible (IC) mechanism. Then, we focus on the online problem in which the receiver sequentially interacts in an unknown game, with the objective of minimizing the cumulative regret w.r.t. the optimal IC mechanism, and the cumulative violation of the incentive compatibility constraints. We investigate two different online scenarios, i.e., the full and bandit feedback settings. For the full feedback problem, we propose an algorithm that guarantees mathcal O(sqrt T) regret and violation, while for the bandit feedback setting we present an algorithm that attains mathcal O(T^{alpha}) regret and mathcal O(T^{1-alpha/2}) violation for any alphain[1/2, 1]. Finally, we complement our results providing a tight lower bound.

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.

Understanding and analyzing markets is crucial, yet analytical equilibrium solutions remain largely infeasible. Recent breakthroughs in equilibrium computation rely on zeroth-order policy gradient estimation. These approaches commonly suffer from high variance and are computationally expensive. The use of fully differentiable simulators would enable more efficient gradient estimation. However, the discrete allocation of goods in economic simulations is a non-differentiable operation. This renders the first-order Monte Carlo gradient estimator inapplicable and the learning feedback systematically misleading. We propose a novel smoothing technique that creates a surrogate market game, in which first-order methods can be applied. We provide theoretical bounds on the resulting bias which justifies solving the smoothed game instead. These bounds also allow choosing the smoothing strength a priori such that the resulting estimate has low variance. Furthermore, we validate our approach via numerous empirical experiments. Our method theoretically and empirically outperforms zeroth-order methods in approximation quality and computational efficiency.

Mean-field games have been used as a theoretical tool to obtain an approximate Nash equilibrium for symmetric and anonymous N-player games. However, limiting applicability, existing theoretical results assume variations of a "population generative model", which allows arbitrary modifications of the population distribution by the learning algorithm. Moreover, learning algorithms typically work on abstract simulators with population instead of the N-player game. Instead, we show that N agents running policy mirror ascent converge to the Nash equilibrium of the regularized game within mathcal{O}(varepsilon^{-2}) samples from a single sample trajectory without a population generative model, up to a standard O(1{N}) error due to the mean field. Taking a divergent approach from the literature, instead of working with the best-response map we first show that a policy mirror ascent map can be used to construct a contractive operator having the Nash equilibrium as its fixed point. We analyze single-path TD learning for N-agent games, proving sample complexity guarantees by only using a sample path from the N-agent simulator without a population generative model. Furthermore, we demonstrate that our methodology allows for independent learning by N agents with finite sample guarantees.

We consider the task of identifying the Copeland winner(s) in a dueling bandits problem with ternary feedback. This is an underexplored but practically relevant variant of the conventional dueling bandits problem, in which, in addition to strict preference between two arms, one may observe feedback in the form of an indifference. We provide a lower bound on the sample complexity for any learning algorithm finding the Copeland winner(s) with a fixed error probability. Moreover, we propose POCOWISTA, an algorithm with a sample complexity that almost matches this lower bound, and which shows excellent empirical performance, even for the conventional dueling bandits problem. For the case where the preference probabilities satisfy a specific type of stochastic transitivity, we provide a refined version with an improved worst case sample complexity.

Having explored an environment, intelligent agents should be able to transfer their knowledge to most downstream tasks within that environment. Referred to as "zero-shot learning," this ability remains elusive for general-purpose reinforcement learning algorithms. While recent works have attempted to produce zero-shot RL agents, they make assumptions about the nature of the tasks or the structure of the MDP. We present Proto Successor Measure: the basis set for all possible solutions of Reinforcement Learning in a dynamical system. We provably show that any possible policy can be represented using an affine combination of these policy independent basis functions. Given a reward function at test time, we simply need to find the right set of linear weights to combine these basis corresponding to the optimal policy. We derive a practical algorithm to learn these basis functions using only interaction data from the environment and show that our approach can produce the optimal policy at test time for any given reward function without additional environmental interactions. Project page: https://agarwalsiddhant10.github.io/projects/psm.html.

We introduce the use of generative adversarial learning to compute equilibria in general game-theoretic settings, specifically the generalized Nash equilibrium (GNE) in pseudo-games, and its specific instantiation as the competitive equilibrium (CE) in Arrow-Debreu competitive economies. Pseudo-games are a generalization of games in which players' actions affect not only the payoffs of other players but also their feasible action spaces. Although the computation of GNE and CE is intractable in the worst-case, i.e., PPAD-hard, in practice, many applications only require solutions with high accuracy in expectation over a distribution of problem instances. We introduce Generative Adversarial Equilibrium Solvers (GAES): a family of generative adversarial neural networks that can learn GNE and CE from only a sample of problem instances. We provide computational and sample complexity bounds, and apply the framework to finding Nash equilibria in normal-form games, CE in Arrow-Debreu competitive economies, and GNE in an environmental economic model of the Kyoto mechanism.

We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.

In order for agents in multi-agent systems (MAS) to be safe, they need to take into account the risks posed by the actions of other agents. However, the dominant paradigm in game theory (GT) assumes that agents are not affected by risk from other agents and only strive to maximise their expected utility. For example, in hybrid human-AI driving systems, it is necessary to limit large deviations in reward resulting from car crashes. Although there are equilibrium concepts in game theory that take into account risk aversion, they either assume that agents are risk-neutral with respect to the uncertainty caused by the actions of other agents, or they are not guaranteed to exist. We introduce a new GT-based Risk-Averse Equilibrium (RAE) that always produces a solution that minimises the potential variance in reward accounting for the strategy of other agents. Theoretically and empirically, we show RAE shares many properties with a Nash Equilibrium (NE), establishing convergence properties and generalising to risk-dominant NE in certain cases. To tackle large-scale problems, we extend RAE to the PSRO multi-agent reinforcement learning (MARL) framework. We empirically demonstrate the minimum reward variance benefits of RAE in matrix games with high-risk outcomes. Results on MARL experiments show RAE generalises to risk-dominant NE in a trust dilemma game and that it reduces instances of crashing by 7x in an autonomous driving setting versus the best performing baseline.

Recent advances in generative models have significantly impacted game generation. However, despite producing high-quality graphics and adequately receiving player input, existing models often fail to maintain fundamental game properties such as numerical and spatial consistency. Numerical consistency ensures gameplay mechanics correctly reflect score changes and other quantitative elements, while spatial consistency prevents jarring scene transitions, providing seamless player experiences. In this paper, we revisit the paradigm of generative games to explore what truly constitutes a Model as a Game (MaaG) with a well-developed mechanism. We begin with an empirical study on ``Traveler'', a 2D game created by an LLM featuring minimalist rules yet challenging generative models in maintaining consistency. Based on the DiT architecture, we design two specialized modules: (1) a numerical module that integrates a LogicNet to determine event triggers, with calculations processed externally as conditions for image generation; and (2) a spatial module that maintains a map of explored areas, retrieving location-specific information during generation and linking new observations to ensure continuity. Experiments across three games demonstrate that our integrated modules significantly enhance performance on consistency metrics compared to baselines, while incurring minimal time overhead during inference.

In reinforcement learning, specification gaming occurs when AI systems learn undesired behaviors that are highly rewarded due to misspecified training goals. Specification gaming can range from simple behaviors like sycophancy to sophisticated and pernicious behaviors like reward-tampering, where a model directly modifies its own reward mechanism. However, these more pernicious behaviors may be too complex to be discovered via exploration. In this paper, we study whether Large Language Model (LLM) assistants which find easily discovered forms of specification gaming will generalize to perform rarer and more blatant forms, up to and including reward-tampering. We construct a curriculum of increasingly sophisticated gameable environments and find that training on early-curriculum environments leads to more specification gaming on remaining environments. Strikingly, a small but non-negligible proportion of the time, LLM assistants trained on the full curriculum generalize zero-shot to directly rewriting their own reward function. Retraining an LLM not to game early-curriculum environments mitigates, but does not eliminate, reward-tampering in later environments. Moreover, adding harmlessness training to our gameable environments does not prevent reward-tampering. These results demonstrate that LLMs can generalize from common forms of specification gaming to more pernicious reward tampering and that such behavior may be nontrivial to remove.

For reinforcement learning systems to be widely adopted, their users must understand and trust them. We present a theoretical analysis of explaining reinforcement learning using Shapley values, following a principled approach from game theory for identifying the contribution of individual players to the outcome of a cooperative game. We call this general framework Shapley Values for Explaining Reinforcement Learning (SVERL). Our analysis exposes the limitations of earlier uses of Shapley values in reinforcement learning. We then develop an approach that uses Shapley values to explain agent performance. In a variety of domains, SVERL produces meaningful explanations that match and supplement human intuition.

The entropic fictitious play (EFP) is a recently proposed algorithm that minimizes the sum of a convex functional and entropy in the space of measures -- such an objective naturally arises in the optimization of a two-layer neural network in the mean-field regime. In this work, we provide a concise primal-dual analysis of EFP in the setting where the learning problem exhibits a finite-sum structure. We establish quantitative global convergence guarantees for both the continuous-time and discrete-time dynamics based on properties of a proximal Gibbs measure introduced in Nitanda et al. (2022). Furthermore, our primal-dual framework entails a memory-efficient particle-based implementation of the EFP update, and also suggests a connection to gradient boosting methods. We illustrate the efficiency of our novel implementation in experiments including neural network optimization and image synthesis.

Recently, remarkable progress has been made by approximating Nash equilibrium (NE), correlated equilibrium (CE), and coarse correlated equilibrium (CCE) through function approximation that trains a neural network to predict equilibria from game representations. Furthermore, equivariant architectures are widely adopted in designing such equilibrium approximators in normal-form games. In this paper, we theoretically characterize benefits and limitations of equivariant equilibrium approximators. For the benefits, we show that they enjoy better generalizability than general ones and can achieve better approximations when the payoff distribution is permutation-invariant. For the limitations, we discuss their drawbacks in terms of equilibrium selection and social welfare. Together, our results help to understand the role of equivariance in equilibrium approximators.

Large Language Models (LLMs) are transforming society and permeating into diverse applications. As a result, LLMs will frequently interact with us and other agents. It is, therefore, of great societal value to understand how LLMs behave in interactive social settings. Here, we propose to use behavioral game theory to study LLM's cooperation and coordination behavior. To do so, we let different LLMs (GPT-3, GPT-3.5, and GPT-4) play finitely repeated games with each other and with other, human-like strategies. Our results show that LLMs generally perform well in such tasks and also uncover persistent behavioral signatures. In a large set of two players-two strategies games, we find that LLMs are particularly good at games where valuing their own self-interest pays off, like the iterated Prisoner's Dilemma family. However, they behave sub-optimally in games that require coordination. We, therefore, further focus on two games from these distinct families. In the canonical iterated Prisoner's Dilemma, we find that GPT-4 acts particularly unforgivingly, always defecting after another agent has defected only once. In the Battle of the Sexes, we find that GPT-4 cannot match the behavior of the simple convention to alternate between options. We verify that these behavioral signatures are stable across robustness checks. Finally, we show how GPT-4's behavior can be modified by providing further information about the other player as well as by asking it to predict the other player's actions before making a choice. These results enrich our understanding of LLM's social behavior and pave the way for a behavioral game theory for machines.

Deep neural networks coupled with fast simulation and improved computation have led to recent successes in the field of reinforcement learning (RL). However, most current RL-based approaches fail to generalize since: (a) the gap between simulation and real world is so large that policy-learning approaches fail to transfer; (b) even if policy learning is done in real world, the data scarcity leads to failed generalization from training to test scenarios (e.g., due to different friction or object masses). Inspired from H-infinity control methods, we note that both modeling errors and differences in training and test scenarios can be viewed as extra forces/disturbances in the system. This paper proposes the idea of robust adversarial reinforcement learning (RARL), where we train an agent to operate in the presence of a destabilizing adversary that applies disturbance forces to the system. The jointly trained adversary is reinforced -- that is, it learns an optimal destabilization policy. We formulate the policy learning as a zero-sum, minimax objective function. Extensive experiments in multiple environments (InvertedPendulum, HalfCheetah, Swimmer, Hopper and Walker2d) conclusively demonstrate that our method (a) improves training stability; (b) is robust to differences in training/test conditions; and c) outperform the baseline even in the absence of the adversary.

Recent work has shown that, under certain assumptions, zero-shot reinforcement learning (RL) methods can generalise to any unseen task in an environment after reward-free pre-training. Access to Markov states is one such assumption, yet, in many real-world applications, the Markov state is only partially observable. Here, we explore how the performance of standard zero-shot RL methods degrades when subjected to partially observability, and show that, as in single-task RL, memory-based architectures are an effective remedy. We evaluate our memory-based zero-shot RL methods in domains where the states, rewards and a change in dynamics are partially observed, and show improved performance over memory-free baselines. Our code is open-sourced via: https://enjeeneer.io/projects/bfms-with-memory/.

Game theory, as an analytical tool, is frequently utilized to analyze human behavior in social science research. With the high alignment between the behavior of Large Language Models (LLMs) and humans, a promising research direction is to employ LLMs as substitutes for humans in game experiments, enabling social science research. However, despite numerous empirical researches on the combination of LLMs and game theory, the capability boundaries of LLMs in game theory remain unclear. In this research, we endeavor to systematically analyze LLMs in the context of game theory. Specifically, rationality, as the fundamental principle of game theory, serves as the metric for evaluating players' behavior -- building a clear desire, refining belief about uncertainty, and taking optimal actions. Accordingly, we select three classical games (dictator game, Rock-Paper-Scissors, and ring-network game) to analyze to what extent LLMs can achieve rationality in these three aspects. The experimental results indicate that even the current state-of-the-art LLM (GPT-4) exhibits substantial disparities compared to humans in game theory. For instance, LLMs struggle to build desires based on uncommon preferences, fail to refine belief from many simple patterns, and may overlook or modify refined belief when taking actions. Therefore, we consider that introducing LLMs into game experiments in the field of social science should be approached with greater caution.

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.

Since DeepMind's AlphaZero, Zero learning quickly became the state-of-the-art method for many board games. It can be improved using a fully convolutional structure (no fully connected layer). Using such an architecture plus global pooling, we can create bots independent of the board size. The training can be made more robust by keeping track of the best checkpoints during the training and by training against them. Using these features, we release Polygames, our framework for Zero learning, with its library of games and its checkpoints. We won against strong humans at the game of Hex in 19x19, which was often said to be untractable for zero learning; and in Havannah. We also won several first places at the TAAI competitions.

The generalization of decision-making agents encompasses two fundamental elements: learning from past experiences and reasoning in novel contexts. However, the predominant emphasis in most interactive environments is on learning, often at the expense of complexity in reasoning. In this paper, we introduce CivRealm, an environment inspired by the Civilization game. Civilization's profound alignment with human history and society necessitates sophisticated learning, while its ever-changing situations demand strong reasoning to generalize. Particularly, CivRealm sets up an imperfect-information general-sum game with a changing number of players; it presents a plethora of complex features, challenging the agent to deal with open-ended stochastic environments that require diplomacy and negotiation skills. Within CivRealm, we provide interfaces for two typical agent types: tensor-based agents that focus on learning, and language-based agents that emphasize reasoning. To catalyze further research, we present initial results for both paradigms. The canonical RL-based agents exhibit reasonable performance in mini-games, whereas both RL- and LLM-based agents struggle to make substantial progress in the full game. Overall, CivRealm stands as a unique learning and reasoning challenge for decision-making agents. The code is available at https://github.com/bigai-ai/civrealm.

Learning in strategy games (e.g. StarCraft, poker) requires the discovery of diverse policies. This is often achieved by iteratively training new policies against existing ones, growing a policy population that is robust to exploit. This iterative approach suffers from two issues in real-world games: a) under finite budget, approximate best-response operators at each iteration needs truncating, resulting in under-trained good-responses populating the population; b) repeated learning of basic skills at each iteration is wasteful and becomes intractable in the presence of increasingly strong opponents. In this work, we propose Neural Population Learning (NeuPL) as a solution to both issues. NeuPL offers convergence guarantees to a population of best-responses under mild assumptions. By representing a population of policies within a single conditional model, NeuPL enables transfer learning across policies. Empirically, we show the generality, improved performance and efficiency of NeuPL across several test domains. Most interestingly, we show that novel strategies become more accessible, not less, as the neural population expands.

Large Language Models (LLMs) show significant potential in economic and strategic interactions, where communication via natural language is often prevalent. This raises key questions: Do LLMs behave rationally? Can they mimic human behavior? Do they tend to reach an efficient and fair outcome? What is the role of natural language in the strategic interaction? How do characteristics of the economic environment influence these dynamics? These questions become crucial concerning the economic and societal implications of integrating LLM-based agents into real-world data-driven systems, such as online retail platforms and recommender systems. While the ML community has been exploring the potential of LLMs in such multi-agent setups, varying assumptions, design choices and evaluation criteria across studies make it difficult to draw robust and meaningful conclusions. To address this, we introduce a benchmark for standardizing research on two-player, sequential, language-based games. Inspired by the economic literature, we define three base families of games with consistent parameterization, degrees of freedom and economic measures to evaluate agents' performance (self-gain), as well as the game outcome (efficiency and fairness). We develop an open-source framework for interaction simulation and analysis, and utilize it to collect a dataset of LLM vs. LLM interactions across numerous game configurations and an additional dataset of human vs. LLM interactions. Through extensive experimentation, we demonstrate how our framework and dataset can be used to: (i) compare the behavior of LLM-based agents to human players in various economic contexts; (ii) evaluate agents in both individual and collective performance measures; and (iii) quantify the effect of the economic characteristics of the environments on the behavior of agents.

Reinforcement learning from human feedback (RLHF) has become essential for improving language model capabilities, but traditional approaches rely on the assumption that human preferences follow a transitive Bradley-Terry model. This assumption fails to capture the non-transitive nature of populational human preferences. Nash learning from human feedback (NLHF), targeting non-transitive preferences, is a problem of computing the Nash equilibrium (NE) of the two-player constant-sum game defined by the human preference. We introduce Extragradient preference optimization (EGPO), a novel algorithm for NLHF achieving last-iterate linear convergence to the NE of KL-regularized games and polynomial convergence to the NE of original games, while being robust to noise. Unlike previous approaches that rely on nested optimization, we derive an equivalent implementation using gradients of an online variant of the identity preference optimization (IPO) loss, enabling more faithful implementation for neural networks. Our empirical evaluations demonstrate EGPO's superior performance over baseline methods when training for the same number of epochs, as measured by pairwise win-rates using the ground truth preference. These results validate both the theoretical strengths and practical advantages of EGPO for language model alignment with non-transitive human preferences.

In recent years there have been many successes of using deep representations in reinforcement learning. Still, many of these applications use conventional architectures, such as convolutional networks, LSTMs, or auto-encoders. In this paper, we present a new neural network architecture for model-free reinforcement learning. Our dueling network represents two separate estimators: one for the state value function and one for the state-dependent action advantage function. The main benefit of this factoring is to generalize learning across actions without imposing any change to the underlying reinforcement learning algorithm. Our results show that this architecture leads to better policy evaluation in the presence of many similar-valued actions. Moreover, the dueling architecture enables our RL agent to outperform the state-of-the-art on the Atari 2600 domain.

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.

Equipping agents with the capacity to justify made decisions using supporting evidence represents a cornerstone of accountable decision-making. Furthermore, ensuring that justifications are in line with human expectations and societal norms is vital, especially in high-stakes situations such as healthcare. In this work, we propose the use of a debate-based reward model for reinforcement learning agents, where the outcome of a zero-sum debate game quantifies the justifiability of a decision in a particular state. This reward model is then used to train a justifiable policy, whose decisions can be more easily corroborated with supporting evidence. In the debate game, two argumentative agents take turns providing supporting evidence for two competing decisions. Given the proposed evidence, a proxy of a human judge evaluates which decision is better justified. We demonstrate the potential of our approach in learning policies for prescribing and justifying treatment decisions of septic patients. We show that augmenting the reward with the feedback signal generated by the debate-based reward model yields policies highly favored by the judge when compared to the policy obtained solely from the environment rewards, while hardly sacrificing any performance. Moreover, in terms of the overall performance and justifiability of trained policies, the debate-based feedback is comparable to the feedback obtained from an ideal judge proxy that evaluates decisions using the full information encoded in the state. This suggests that the debate game outputs key information contained in states that is most relevant for evaluating decisions, which in turn substantiates the practicality of combining our approach with human-in-the-loop evaluations. Lastly, we showcase that agents trained via multi-agent debate learn to propose evidence that is resilient to refutations and closely aligns with human preferences.

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

To optimally coordinate with others in cooperative games, it is often crucial to have information about one's collaborators: successful driving requires understanding which side of the road to drive on. However, not every feature of collaborators is strategically relevant: the fine-grained acceleration of drivers may be ignored while maintaining optimal coordination. We show that there is a well-defined dichotomy between strategically relevant and irrelevant information. Moreover, we show that, in dynamic games, this dichotomy has a compact representation that can be efficiently computed via a Bellman backup operator. We apply this algorithm to analyze the strategically relevant information for tasks in both a standard and a partially observable version of the Overcooked environment. Theoretical and empirical results show that our algorithms are significantly more efficient than baselines. Videos are available at https://minknowledge.github.io.

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

In this work, we consider the problem of collaborative multi-user reinforcement learning. In this setting there are multiple users with the same state-action space and transition probabilities but with different rewards. Under the assumption that the reward matrix of the N users has a low-rank structure -- a standard and practically successful assumption in the offline collaborative filtering setting -- the question is can we design algorithms with significantly lower sample complexity compared to the ones that learn the MDP individually for each user. Our main contribution is an algorithm which explores rewards collaboratively with N user-specific MDPs and can learn rewards efficiently in two key settings: tabular MDPs and linear MDPs. When N is large and the rank is constant, the sample complexity per MDP depends logarithmically over the size of the state-space, which represents an exponential reduction (in the state-space size) when compared to the standard ``non-collaborative'' algorithms.

In this paper, we evaluate whether LLMs learn to make human-like preference judgements in strategic scenarios as compared with known empirical results. Solar and Mistral are shown to exhibit stable value-based preference consistent with humans and exhibit human-like preference for cooperation in the prisoner's dilemma (including stake-size effect) and traveler's dilemma (including penalty-size effect). We establish a relationship between model size, value-based preference, and superficiality. Finally, results here show that models tending to be less brittle have relied on sliding window attention suggesting a potential link. Additionally, we contribute a novel method for constructing preference relations from arbitrary LLMs and support for a hypothesis regarding human behavior in the traveler's dilemma.

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.

Recent advances in LLMs have enabled their use as autonomous agents across a range of tasks, yet they continue to struggle with formulating and adhering to coherent long-term strategies. In this paper, we investigate whether LLM agents can self-improve when placed in environments that explicitly challenge their strategic planning abilities. Using the board game Settlers of Catan, accessed through the open-source Catanatron framework, we benchmark a progression of LLM-based agents, from a simple game-playing agent to systems capable of autonomously rewriting their own prompts and their player agent's code. We introduce a multi-agent architecture in which specialized roles (Analyzer, Researcher, Coder, and Player) collaborate to iteratively analyze gameplay, research new strategies, and modify the agent's logic or prompt. By comparing manually crafted agents to those evolved entirely by LLMs, we evaluate how effectively these systems can diagnose failure and adapt over time. Our results show that self-evolving agents, particularly when powered by models like Claude 3.7 and GPT-4o, outperform static baselines by autonomously adopting their strategies, passing along sample behavior to game-playing agents, and demonstrating adaptive reasoning over multiple iterations.

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.

In multi-agent reinforcement learning, the behaviors that agents learn in a single Markov Game (MG) are typically confined to the given agent number. Every single MG induced by varying the population may possess distinct optimal joint strategies and game-specific knowledge, which are modeled independently in modern multi-agent reinforcement learning algorithms. In this work, our focus is on creating agents that can generalize across population-varying MGs. Instead of learning a unimodal policy, each agent learns a policy set comprising effective strategies across a variety of games. To achieve this, we propose Meta Representations for Agents (MRA) that explicitly models the game-common and game-specific strategic knowledge. By representing the policy sets with multi-modal latent policies, the game-common strategic knowledge and diverse strategic modes are discovered through an iterative optimization procedure. We prove that by approximately maximizing the resulting constrained mutual information objective, the policies can reach Nash Equilibrium in every evaluation MG when the latent space is sufficiently large. When deploying MRA in practical settings with limited latent space sizes, fast adaptation can be achieved by leveraging the first-order gradient information. Extensive experiments demonstrate the effectiveness of MRA in improving training performance and generalization ability in challenging evaluation games.

Since the introduction of ChatGPT and GPT-4, these models have been tested across a large number of tasks. Their adeptness across domains is evident, but their aptitude in playing games, and specifically their aptitude in the realm of poker has remained unexplored. Poker is a game that requires decision making under uncertainty and incomplete information. In this paper, we put ChatGPT and GPT-4 through the poker test and evaluate their poker skills. Our findings reveal that while both models display an advanced understanding of poker, encompassing concepts like the valuation of starting hands, playing positions and other intricacies of game theory optimal (GTO) poker, both ChatGPT and GPT-4 are NOT game theory optimal poker players. Profitable strategies in poker are evaluated in expectations over large samples. Through a series of experiments, we first discover the characteristics of optimal prompts and model parameters for playing poker with these models. Our observations then unveil the distinct playing personas of the two models. We first conclude that GPT-4 is a more advanced poker player than ChatGPT. This exploration then sheds light on the divergent poker tactics of the two models: ChatGPT's conservativeness juxtaposed against GPT-4's aggression. In poker vernacular, when tasked to play GTO poker, ChatGPT plays like a nit, which means that it has a propensity to only engage with premium hands and folds a majority of hands. When subjected to the same directive, GPT-4 plays like a maniac, showcasing a loose and aggressive style of play. Both strategies, although relatively advanced, are not game theory optimal.

Graphon games have been introduced to study games with many players who interact through a weighted graph of interaction. By passing to the limit, a game with a continuum of players is obtained, in which the interactions are through a graphon. In this paper, we focus on a graphon game for optimal investment under relative performance criteria, and we propose a deep learning method. The method builds upon two key ingredients: first, a characterization of Nash equilibria by forward-backward stochastic differential equations and, second, recent advances of machine learning algorithms for stochastic differential games. We provide numerical experiments on two different financial models. In each model, we compare the effect of several graphons, which correspond to different structures of interactions.

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.

Coordinating multiple large language models (LLMs) to solve complex tasks collaboratively poses a fundamental trade-off between the computation costs and collective performance compared with individual model. We introduce a novel, game-theoretically grounded reinforcement learning (RL) framework, the Multi-Agent Cooperation Sequential Public Goods Game (MAC-SPGG), to systematically incentivize cooperation in multi-LLM ensembles. In MAC-SPGG, LLM agents move in sequence, observing predecessors' outputs and updating beliefs to condition their own contributions. By redesigning the public-goods reward, effortful contributions become the unique Subgame Perfect Nash Equilibrium (SPNE), which eliminates free-riding under traditional SPGG or PGG. Its sequential protocol replaces costly round-based information exchanges with a streamlined decision flow, cutting communication overhead while retaining strategic depth. We prove the existence and uniqueness of the SPNE under realistic parameters, and empirically show that MAC-SPGG-trained ensembles outperform single-agent baselines, chain-of-thought prompting, and other cooperative methods, even achieving comparable performance to large-scale models across reasoning, math, code generation, and NLP tasks. Our results highlight the power of structured, incentive-aligned MAC-SPGG cooperation for scalable and robust multi-agent language generation.

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.

We propose a new framework for estimating generative models via an adversarial process, in which we simultaneously train two models: a generative model G that captures the data distribution, and a discriminative model D that estimates the probability that a sample came from the training data rather than G. The training procedure for G is to maximize the probability of D making a mistake. This framework corresponds to a minimax two-player game. In the space of arbitrary functions G and D, a unique solution exists, with G recovering the training data distribution and D equal to 1/2 everywhere. In the case where G and D are defined by multilayer perceptrons, the entire system can be trained with backpropagation. There is no need for any Markov chains or unrolled approximate inference networks during either training or generation of samples. Experiments demonstrate the potential of the framework through qualitative and quantitative evaluation of the generated samples.

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.

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.

We consider the problem of fairly allocating a sequence of indivisible items that arrive online in an arbitrary order to a group of n agents with additive normalized valuation functions. We consider both the allocation of goods and chores and propose algorithms for approximating maximin share (MMS) allocations. When agents have identical valuation functions the problem coincides with the semi-online machine covering problem (when items are goods) and load balancing problem (when items are chores), for both of which optimal competitive ratios have been achieved. In this paper, we consider the case when agents have general additive valuation functions. For the allocation of goods, we show that no competitive algorithm exists even when there are only three agents and propose an optimal 0.5-competitive algorithm for the case of two agents. For the allocation of chores, we propose a (2-1/n)-competitive algorithm for n>=3 agents and a square root of 2 (approximately 1.414)-competitive algorithm for two agents. Additionally, we show that no algorithm can do better than 15/11 (approximately 1.364)-competitive for two agents.

Evaluation has traditionally focused on ranking candidates for a specific skill. Modern generalist models, such as Large Language Models (LLMs), decidedly outpace this paradigm. Open-ended evaluation systems, where candidate models are compared on user-submitted prompts, have emerged as a popular solution. Despite their many advantages, we show that the current Elo-based rating systems can be susceptible to and even reinforce biases in data, intentional or accidental, due to their sensitivity to redundancies. To address this issue, we propose evaluation as a 3-player game, and introduce novel game-theoretic solution concepts to ensure robustness to redundancy. We show that our method leads to intuitive ratings and provide insights into the competitive landscape of LLM development.

Progress in fields of machine learning and adversarial planning has benefited significantly from benchmark domains, from checkers and the classic UCI data sets to Go and Diplomacy. In sequential decision-making, agent evaluation has largely been restricted to few interactions against experts, with the aim to reach some desired level of performance (e.g. beating a human professional player). We propose a benchmark for multiagent learning based on repeated play of the simple game Rock, Paper, Scissors along with a population of forty-three tournament entries, some of which are intentionally sub-optimal. We describe metrics to measure the quality of agents based both on average returns and exploitability. We then show that several RL, online learning, and language model approaches can learn good counter-strategies and generalize well, but ultimately lose to the top-performing bots, creating an opportunity for research in multiagent learning.

We present a Spades bidding algorithm that is superior to recreational human players and to publicly available bots. Like in Bridge, the game of Spades is composed of two independent phases, bidding and playing. This paper focuses on the bidding algorithm, since this phase holds a precise challenge: based on the input, choose the bid that maximizes the agent's winning probability. Our Bidding-in-Spades (BIS) algorithm heuristically determines the bidding strategy by comparing the expected utility of each possible bid. A major challenge is how to estimate these expected utilities. To this end, we propose a set of domain-specific heuristics, and then correct them via machine learning using data from real-world players. The \BIS algorithm we present can be attached to any playing algorithm. It beats rule-based bidding bots when all use the same playing component. When combined with a rule-based playing algorithm, it is superior to the average recreational human.

We introduce a framework for translating game descriptions in natural language into extensive-form representations in game theory, leveraging Large Language Models (LLMs) and in-context learning. Given the varying levels of strategic complexity in games, such as perfect versus imperfect information, directly applying in-context learning would be insufficient. To address this, we introduce a two-stage framework with specialized modules to enhance in-context learning, enabling it to divide and conquer the problem effectively. In the first stage, we tackle the challenge of imperfect information by developing a module that identifies information sets along and the corresponding partial tree structure. With this information, the second stage leverages in-context learning alongside a self-debugging module to produce a complete extensive-form game tree represented using pygambit, the Python API of a recognized game-theoretic analysis tool called Gambit. Using this python representation enables the automation of tasks such as computing Nash equilibria directly from natural language descriptions. We evaluate the performance of the full framework, as well as its individual components, using various LLMs on games with different levels of strategic complexity. Our experimental results show that the framework significantly outperforms baseline models in generating accurate extensive-form games, with each module playing a critical role in its success.

Traditional Reinforcement Learning from Human Feedback (RLHF) often relies on reward models, frequently assuming preference structures like the Bradley-Terry model, which may not accurately capture the complexities of real human preferences (e.g., intransitivity). Nash Learning from Human Feedback (NLHF) offers a more direct alternative by framing the problem as finding a Nash equilibrium of a game defined by these preferences. In this work, we introduce Nash Mirror Prox (Nash-MP), an online NLHF algorithm that leverages the Mirror Prox optimization scheme to achieve fast and stable convergence to the Nash equilibrium. Our theoretical analysis establishes that Nash-MP exhibits last-iterate linear convergence towards the beta-regularized Nash equilibrium. Specifically, we prove that the KL-divergence to the optimal policy decreases at a rate of order (1+2beta)^{-N/2}, where N is a number of preference queries. We further demonstrate last-iterate linear convergence for the exploitability gap and uniformly for the span semi-norm of log-probabilities, with all these rates being independent of the size of the action space. Furthermore, we propose and analyze an approximate version of Nash-MP where proximal steps are estimated using stochastic policy gradients, making the algorithm closer to applications. Finally, we detail a practical implementation strategy for fine-tuning large language models and present experiments that demonstrate its competitive performance and compatibility with existing methods.

Large Language Models (LLMs) have achieved remarkable progress in reasoning, yet sometimes produce responses that are suboptimal for users in tasks such as writing, information seeking, or providing practical guidance. Conventional alignment practices typically assume that maximizing model reward also maximizes user welfare, but this assumption frequently fails in practice: models may over-clarify or generate overly verbose reasoning when users prefer concise answers. Such behaviors resemble the prisoner's dilemma, where individually rational choices lead to socially suboptimal outcomes. The fundamental challenge is the lack of a principled decision making mechanism that mutually benefits both the LLM and the user. We propose Game-Theoretic Alignment (GTAlign), an alignment framework that integrates game-theoretic decision making into both reasoning and training. During reasoning, the model explicitly treats user-LLM interaction as a strategic game: it constructs payoff matrices within its reasoning chain to estimate welfare for both itself and the user, and then selects actions that are mutually beneficial. During training, we introduce a mutual welfare reward that reinforces cooperative responses, aligning model behavior with socially efficient outcomes. In addition, we introduce an inference technique that leverages game-theoretic reasoning to dynamically adapt LLM's response when pricing policies of LLM service change. Extensive experiments demonstrate that GTAlign substantially improves reasoning efficiency, answer quality, and mutual welfare compared to baselines across diverse tasks. The code is available at https://github.com/ulab-uiuc/GTAlign .

In a recent work, Laforgue et al. introduce the model of last switch dependent (LSD) bandits, in an attempt to capture nonstationary phenomena induced by the interaction between the player and the environment. Examples include satiation, where consecutive plays of the same action lead to decreased performance, or deprivation, where the payoff of an action increases after an interval of inactivity. In this work, we take a step towards understanding the approximability of planning LSD bandits, namely, the (NP-hard) problem of computing an optimal arm-pulling strategy under complete knowledge of the model. In particular, we design the first efficient constant approximation algorithm for the problem and show that, under a natural monotonicity assumption on the payoffs, its approximation guarantee (almost) matches the state-of-the-art for the special and well-studied class of recharging bandits (also known as delay-dependent). In this attempt, we develop new tools and insights for this class of problems, including a novel higher-dimensional relaxation and the technique of mirroring the evolution of virtual states. We believe that these novel elements could potentially be used for approaching richer classes of action-induced nonstationary bandits (e.g., special instances of restless bandits). In the case where the model parameters are initially unknown, we develop an online learning adaptation of our algorithm for which we provide sublinear regret guarantees against its full-information counterpart.

In this paper, we provide a general framework for studying multi-agent online learning problems in the presence of delays and asynchronicities. Specifically, we propose and analyze a class of adaptive dual averaging schemes in which agents only need to accumulate gradient feedback received from the whole system, without requiring any between-agent coordination. In the single-agent case, the adaptivity of the proposed method allows us to extend a range of existing results to problems with potentially unbounded delays between playing an action and receiving the corresponding feedback. In the multi-agent case, the situation is significantly more complicated because agents may not have access to a global clock to use as a reference point; to overcome this, we focus on the information that is available for producing each prediction rather than the actual delay associated with each feedback. This allows us to derive adaptive learning strategies with optimal regret bounds, even in a fully decentralized, asynchronous environment. Finally, we also analyze an "optimistic" variant of the proposed algorithm which is capable of exploiting the predictability of problems with a slower variation and leads to improved regret bounds.

Unlike perfect information games, where all elements are known to every player, imperfect information games emulate the real-world complexities of decision-making under uncertain or incomplete information. GPT-4, the recent breakthrough in large language models (LLMs) trained on massive passive data, is notable for its knowledge retrieval and reasoning abilities. This paper delves into the applicability of GPT-4's learned knowledge for imperfect information games. To achieve this, we introduce Suspicion-Agent, an innovative agent that leverages GPT-4's capabilities for performing in imperfect information games. With proper prompt engineering to achieve different functions, Suspicion-Agent based on GPT-4 demonstrates remarkable adaptability across a range of imperfect information card games. Importantly, GPT-4 displays a strong high-order theory of mind (ToM) capacity, meaning it can understand others and intentionally impact others' behavior. Leveraging this, we design a planning strategy that enables GPT-4 to competently play against different opponents, adapting its gameplay style as needed, while requiring only the game rules and descriptions of observations as input. In the experiments, we qualitatively showcase the capabilities of Suspicion-Agent across three different imperfect information games and then quantitatively evaluate it in Leduc Hold'em. The results show that Suspicion-Agent can potentially outperform traditional algorithms designed for imperfect information games, without any specialized training or examples. In order to encourage and foster deeper insights within the community, we make our game-related data publicly available.