Daily Papers

Large language model pretraining is compute-intensive, yet many tokens contribute marginally to learning, resulting in inefficiency. We introduce Efficient Selective Language Modeling (ESLM), a risk-aware algorithm that improves training efficiency and distributional robustness by performing online token-level batch selection. ESLM leverages per-token statistics (e.g., entropy or loss) and applies value-at-risk thresholding to retain only the most informative tokens per batch. This data-centric mechanism reshapes the training loss, prioritizing high-risk tokens and eliminating redundant gradient computation. We frame ESLM as a bilevel game: the model competes with a masking adversary that selects worst-case token subsets under a constrained thresholding rule. In the loss-based setting, ESLM recovers conditional value-at-risk loss minimization, providing a principled connection to distributionally robust optimization. We extend our approach to Ada-ESLM, which adaptively tunes the selection confidence during training. Experiments on GPT-2 pretraining show that ESLM significantly reduces training FLOPs while maintaining or improving both perplexity and downstream performance compared to baselines. Our approach also scales across model sizes, pretraining corpora, and integrates naturally with knowledge distillation.

In this paper, we develop an approximation scheme for solving bilevel programs with equilibrium constraints, which are generally difficult to solve. Among other things, calculating the first-order derivative in such a problem requires differentiation across the hierarchy, which is computationally intensive, if not prohibitive. To bypass the hierarchy, we propose to bound such bilevel programs, equivalent to multiple-followers Stackelberg games, with two new hierarchy-free problems: a T-step Cournot game and a T-step monopoly model. Since they are standard equilibrium or optimization problems, both can be efficiently solved via first-order methods. Importantly, we show that the bounds provided by these problems -- the upper bound by the T-step Cournot game and the lower bound by the T-step monopoly model -- can be made arbitrarily tight by increasing the step parameter T for a wide range of problems. We prove that a small T usually suffices under appropriate conditions to reach an approximation acceptable for most practical purposes. Eventually, the analytical insights are highlighted through numerical examples.

The optimal policy in various real-world strategic decision-making problems depends both on the environmental configuration and exogenous events. For these settings, we introduce Contextual Bilevel Reinforcement Learning (CB-RL), a stochastic bilevel decision-making model, where the lower level consists of solving a contextual Markov Decision Process (CMDP). CB-RL can be viewed as a Stackelberg Game where the leader and a random context beyond the leader's control together decide the setup of many MDPs that potentially multiple followers best respond to. This framework extends beyond traditional bilevel optimization and finds relevance in diverse fields such as RLHF, tax design, reward shaping, contract theory and mechanism design. We propose a stochastic Hyper Policy Gradient Descent (HPGD) algorithm to solve CB-RL, and demonstrate its convergence. Notably, HPGD uses stochastic hypergradient estimates, based on observations of the followers' trajectories. Therefore, it allows followers to use any training procedure and the leader to be agnostic of the specific algorithm, which aligns with various real-world scenarios. We further consider the setting when the leader can influence the training of followers and propose an accelerated algorithm. We empirically demonstrate the performance of our algorithm for reward shaping and tax design.

Bilevel optimization enjoys a wide range of applications in hyper-parameter optimization, meta-learning and reinforcement learning. However, bilevel optimization problems are difficult to solve. Recent progress on scalable bilevel algorithms mainly focuses on bilevel optimization problems where the lower-level objective is either strongly convex or unconstrained. In this work, we tackle the bilevel problem through the lens of the penalty method. We show that under certain conditions, the penalty reformulation recovers the solutions of the original bilevel problem. Further, we propose the penalty-based bilevel gradient descent (PBGD) algorithm and establish its finite-time convergence for the constrained bilevel problem without lower-level strong convexity. Experiments showcase the efficiency of the proposed PBGD algorithm.

Generative models form the backbone of modern machine learning, underpinning state-of-the-art systems in text, vision, and multimodal applications. While Maximum Likelihood Estimation has traditionally served as the dominant training paradigm, recent work have highlighted its limitations, particularly in generalization and susceptibility to catastrophic forgetting compared to Reinforcement Learning techniques, such as Policy Gradient methods. However, these approaches depend on explicit reward signals, which are often unavailable in practice, leaving open the fundamental problem of how to align generative models when only high-quality datasets are accessible. In this work, we address this challenge via a Bilevel Optimization framework, where the reward function is treated as the optimization variable of an outer-level problem, while a policy gradient objective defines the inner-level. We then conduct a theoretical analysis of this optimization problem in a tractable setting and extract insights that, as we demonstrate, generalize to applications such as tabular classification and model-based reinforcement learning. We release the code at https://github.com/abenechehab/nll_to_po .

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.

Bilevel optimization is an important formulation for many machine learning problems. Current bilevel optimization algorithms assume that the gradient of the upper-level function is Lipschitz. However, recent studies reveal that certain neural networks such as recurrent neural networks (RNNs) and long-short-term memory networks (LSTMs) exhibit potential unbounded smoothness, rendering conventional bilevel optimization algorithms unsuitable. In this paper, we design a new bilevel optimization algorithm, namely BO-REP, to address this challenge. This algorithm updates the upper-level variable using normalized momentum and incorporates two novel techniques for updating the lower-level variable: initialization refinement and periodic updates. Specifically, once the upper-level variable is initialized, a subroutine is invoked to obtain a refined estimate of the corresponding optimal lower-level variable, and the lower-level variable is updated only after every specific period instead of each iteration. When the upper-level problem is nonconvex and unbounded smooth, and the lower-level problem is strongly convex, we prove that our algorithm requires mathcal{O}(1/epsilon^4) iterations to find an epsilon-stationary point in the stochastic setting, where each iteration involves calling a stochastic gradient or Hessian-vector product oracle. Notably, this result matches the state-of-the-art complexity results under the bounded smoothness setting and without mean-squared smoothness of the stochastic gradient, up to logarithmic factors. Our proof relies on novel technical lemmas for the periodically updated lower-level variable, which are of independent interest. Our experiments on hyper-representation learning, hyperparameter optimization, and data hyper-cleaning for text classification tasks demonstrate the effectiveness of our proposed algorithm.

Many problems in machine learning involve bilevel optimization (BLO), including hyperparameter optimization, meta-learning, and dataset distillation. Bilevel problems consist of two nested sub-problems, called the outer and inner problems, respectively. In practice, often at least one of these sub-problems is overparameterized. In this case, there are many ways to choose among optima that achieve equivalent objective values. Inspired by recent studies of the implicit bias induced by optimization algorithms in single-level optimization, we investigate the implicit bias of gradient-based algorithms for bilevel optimization. We delineate two standard BLO methods -- cold-start and warm-start -- and show that the converged solution or long-run behavior depends to a large degree on these and other algorithmic choices, such as the hypergradient approximation. We also show that the inner solutions obtained by warm-start BLO can encode a surprising amount of information about the outer objective, even when the outer parameters are low-dimensional. We believe that implicit bias deserves as central a role in the study of bilevel optimization as it has attained in the study of single-level neural net optimization.

Bilevel optimization recently has received tremendous attention due to its great success in solving important machine learning problems like meta learning, reinforcement learning, and hyperparameter optimization. Extending single-agent training on bilevel problems to the decentralized setting is a natural generalization, and there has been a flurry of work studying decentralized bilevel optimization algorithms. However, it remains unknown how to design the distributed algorithm with sample complexity and convergence rate comparable to SGD for stochastic optimization, and at the same time without directly computing the exact Hessian or Jacobian matrices. In this paper we propose such an algorithm. More specifically, we propose a novel decentralized stochastic bilevel optimization (DSBO) algorithm that only requires first order stochastic oracle, Hessian-vector product and Jacobian-vector product oracle. The sample complexity of our algorithm matches the currently best known results for DSBO, and the advantage of our algorithm is that it does not require estimating the full Hessian and Jacobian matrices, thereby having improved per-iteration complexity.

As machine learning models grow in complexity and increasingly rely on publicly sourced data, such as the human-annotated labels used in training large language models, they become more vulnerable to label poisoning attacks. These attacks, in which adversaries subtly alter the labels within a training dataset, can severely degrade model performance, posing significant risks in critical applications. In this paper, we propose FLORAL, a novel adversarial training defense strategy based on support vector machines (SVMs) to counter these threats. Utilizing a bilevel optimization framework, we cast the training process as a non-zero-sum Stackelberg game between an attacker, who strategically poisons critical training labels, and the model, which seeks to recover from such attacks. Our approach accommodates various model architectures and employs a projected gradient descent algorithm with kernel SVMs for adversarial training. We provide a theoretical analysis of our algorithm's convergence properties and empirically evaluate FLORAL's effectiveness across diverse classification tasks. Compared to robust baselines and foundation models such as RoBERTa, FLORAL consistently achieves higher robust accuracy under increasing attacker budgets. These results underscore the potential of FLORAL to enhance the resilience of machine learning models against label poisoning threats, thereby ensuring robust classification in adversarial settings.

We introduce a framework based on bilevel programming that unifies gradient-based hyperparameter optimization and meta-learning. We show that an approximate version of the bilevel problem can be solved by taking into explicit account the optimization dynamics for the inner objective. Depending on the specific setting, the outer variables take either the meaning of hyperparameters in a supervised learning problem or parameters of a meta-learner. We provide sufficient conditions under which solutions of the approximate problem converge to those of the exact problem. We instantiate our approach for meta-learning in the case of deep learning where representation layers are treated as hyperparameters shared across a set of training episodes. In experiments, we confirm our theoretical findings, present encouraging results for few-shot learning and contrast the bilevel approach against classical approaches for learning-to-learn.

Federated bilevel optimization has received increasing attention in various emerging machine learning and communication applications. Recently, several Hessian-vector-based algorithms have been proposed to solve the federated bilevel optimization problem. However, several important properties in federated learning such as the partial client participation and the linear speedup for convergence (i.e., the convergence rate and complexity are improved linearly with respect to the number of sampled clients) in the presence of non-i.i.d.~datasets, still remain open. In this paper, we fill these gaps by proposing a new federated bilevel algorithm named FedMBO with a novel client sampling scheme in the federated hypergradient estimation. We show that FedMBO achieves a convergence rate of Obig(1{nK}+1{K}+sqrt{n}{K^{3/2}}big) on non-i.i.d.~datasets, where n is the number of participating clients in each round, and K is the total number of iteration. This is the first theoretical linear speedup result for non-i.i.d.~federated bilevel optimization. Extensive experiments validate our theoretical results and demonstrate the effectiveness of our proposed method.

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.

We present a novel unified bilevel optimization-based framework, PARL, formulated to address the recently highlighted critical issue of policy alignment in reinforcement learning using utility or preference-based feedback. We identify a major gap within current algorithmic designs for solving policy alignment due to a lack of precise characterization of the dependence of the alignment objective on the data generated by policy trajectories. This shortfall contributes to the sub-optimal performance observed in contemporary algorithms. Our framework addressed these concerns by explicitly parameterizing the distribution of the upper alignment objective (reward design) by the lower optimal variable (optimal policy for the designed reward). Interestingly, from an optimization perspective, our formulation leads to a new class of stochastic bilevel problems where the stochasticity at the upper objective depends upon the lower-level variable. To demonstrate the efficacy of our formulation in resolving alignment issues in RL, we devised an algorithm named A-PARL to solve PARL problem, establishing sample complexity bounds of order O(1/T). Our empirical results substantiate that the proposed PARL can address the alignment concerns in RL by showing significant improvements (up to 63\% in terms of required samples) for policy alignment in large-scale environments of the Deepmind control suite and Meta world tasks.

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.

Bilevel optimization has been recently revisited for designing and analyzing algorithms in hyperparameter tuning and meta learning tasks. However, due to its nested structure, evaluating exact gradients for high-dimensional problems is computationally challenging. One heuristic to circumvent this difficulty is to use the approximate gradient given by performing truncated back-propagation through the iterative optimization procedure that solves the lower-level problem. Although promising empirical performance has been reported, its theoretical properties are still unclear. In this paper, we analyze the properties of this family of approximate gradients and establish sufficient conditions for convergence. We validate this on several hyperparameter tuning and meta learning tasks. We find that optimization with the approximate gradient computed using few-step back-propagation often performs comparably to optimization with the exact gradient, while requiring far less memory and half the computation time.

Stackelberg games and their resulting equilibria have received increasing attention in the multi-agent reinforcement learning literature. Each stage of a traditional Stackelberg game involves a leader(s) acting first, followed by the followers. In situations where the roles of leader(s) and followers can be interchanged, the designated role can have considerable advantages, for example, in first-mover advantage settings. Then the question arises: Who should be the leader and when? A bias in the leader selection process can lead to unfair outcomes. This problem is aggravated if the agents are self-interested and care only about their goals and rewards. We formally define this leader selection problem and show its relation to fairness in agents' returns. Furthermore, we propose a multi-agent reinforcement learning framework that maximizes fairness by integrating mediators. Mediators have previously been used in the simultaneous action setting with varying levels of control, such as directly performing agents' actions or just recommending them. Our framework integrates mediators in the Stackelberg setting with minimal control (leader selection). We show that the presence of mediators leads to self-interested agents taking fair actions, resulting in higher overall fairness in agents' returns.

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter sigma > 0. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be O(sigma)-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as O(sigma)-approximation of the original BO, we propose first-order algorithms that find an epsilon-stationary solution by optimizing the penalty formulation with sigma = O(epsilon). When the perturbed lower-level problem uniformly satisfies the small-error proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an epsilon-stationary point of the penalty function, using in total O(epsilon^{-3}) and O(epsilon^{-7}) accesses to first-order (stochastic) gradient oracles when the oracle is deterministic and oracles are noisy, respectively. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, i.e., with O(1) samples per iteration, and achieves the improved oracle-complexity of O(epsilon^{-3}) and O(epsilon^{-5}), respectively.

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.

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.

Current reinforcement learning from human feedback (RLHF) pipelines for large language model (LLM) alignment typically assign scalar rewards to sequences, using the final token as a surrogate indicator for the quality of the entire sequence. However, this leads to sparse feedback and suboptimal token-level credit assignment. In this work, we frame reward shaping as an optimization problem focused on token-level credit assignment. We propose a reward-shaping function leveraging explainability methods such as SHAP and LIME to estimate per-token rewards from the reward model. To learn parameters of this shaping function, we employ a bilevel optimization framework that integrates Bayesian Optimization and policy training to handle noise from the token reward estimates. Our experiments show that achieving a better balance of token-level reward attribution leads to performance improvements over baselines on downstream tasks and finds an optimal policy faster during training. Furthermore, we show theoretically that explainability methods that are feature additive attribution functions maintain the optimal policy as the original reward.

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.

Effective macroeconomic policies play a crucial role in promoting economic growth and social stability. This paper models the optimal macroeconomic policy problem based on the Stackelberg Mean Field Game (SMFG), where the government acts as the leader in policy-making, and large-scale households dynamically respond as followers. This modeling method captures the asymmetric dynamic game between the government and large-scale households, and interpretably evaluates the effects of macroeconomic policies based on microfoundations, which is difficult for existing methods to achieve. We also propose a solution for SMFGs, incorporating pre-training on real data and a model-free Stackelberg mean-field reinforcement learning (SMFRL) algorithm, which operates independently of prior environmental knowledge and transitions. Our experimental results showcase the superiority of the SMFG method over other economic policies in terms of performance, efficiency-equity tradeoff, and SMFG assumption analysis. This paper significantly contributes to the domain of AI for economics by providing a powerful tool for modeling and solving optimal macroeconomic policies.

Bilevel optimization has shown its utility across various machine learning settings, yet most algorithms in practice require second-order information, making it challenging to scale them up. Only recently, a paradigm of first-order algorithms has emerged in the theoretical literature, capable of effectively addressing bilevel optimization problems. Nevertheless, the practical efficiency of this paradigm remains unverified, particularly in the context of large language models (LLMs). This paper introduces the first scalable instantiation of this paradigm called ScaleBiO, focusing on bilevel optimization for large-scale LLM data reweighting. By combining with a recently proposed memory-efficient training technique called LISA, our novel algorithm allows the paradigm to scale to sim30B-sized LLMs on 8timesH100 GPUs, marking the first successful application of bilevel optimization under practical scenarios for large-sized LLMs. Empirically, extensive experiments on data reweighting verify the effectiveness of ScaleBiO for different-scaled models, including Llama-3-8B, Gemma-2-9B, Qwen-2-7B, and Qwen-2.5-32B, where bilevel optimization succeeds in instruction-following and math reasoning tasks, outperforming several popular baselines, including uniform sampling, influence-aware data filtering, and reference-model-based sampling methods. Theoretically, ScaleBiO ensures the optimality of the learned data weights, along with a convergence guarantee matching the conventional first-order bilevel optimization paradigm on smooth and strongly convex objectives.

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.

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

The 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.

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.

The design of effective reward functions presents a central and often arduous challenge in reinforcement learning (RL), particularly when developing autonomous agents for complex reasoning tasks. While automated reward optimization approaches exist, they typically rely on derivative-free evolutionary heuristics that treat the reward function as a black box, failing to capture the causal relationship between reward structure and task performance. To bridge this gap, we propose Differentiable Evolutionary Reinforcement Learning (DERL), a bilevel framework that enables the autonomous discovery of optimal reward signals. In DERL, a Meta-Optimizer evolves a reward function (i.e., Meta-Reward) by composing structured atomic primitives, guiding the training of an inner-loop policy. Crucially, unlike previous evolution, DERL is differentiable in its metaoptimization: it treats the inner-loop validation performance as a signal to update the Meta-Optimizer via reinforcement learning. This allows DERL to approximate the "meta-gradient" of task success, progressively learning to generate denser and more actionable feedback. We validate DERL across three distinct domains: robotic agent (ALFWorld), scientific simulation (ScienceWorld), and mathematical reasoning (GSM8k, MATH). Experimental results show that DERL achieves state-of-the-art performance on ALFWorld and ScienceWorld, significantly outperforming methods relying on heuristic rewards, especially in out-of-distribution scenarios. Analysis of the evolutionary trajectory demonstrates that DERL successfully captures the intrinsic structure of tasks, enabling selfimproving agent alignment without human intervention.

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

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.

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 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.

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

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.

Can classical game-theoretic frameworks be extended to capture the bounded rationality and causal reasoning of AI agents? We investigate this question by extending Causal Normal Form Games (CNFGs) to sequential settings, introducing Sequential Causal Multi-Agent Systems (S-CMAS) that incorporate Pearl's Causal Hierarchy across leader-follower interactions. While theoretically elegant -- we prove PSPACE-completeness, develop equilibrium refinements, and establish connections to signaling theory -- our comprehensive empirical investigation reveals a critical limitation: S-CNE provides zero welfare improvement over classical Stackelberg equilibrium across all tested scenarios. Through 50+ Monte Carlo simulations and hand-crafted synthetic examples, we demonstrate that backward induction with rational best-response eliminates any strategic advantage from causal layer distinctions. We construct a theoretical example illustrating conditions where benefits could emerge (ε-rational satisficing followers), though implementation confirms that even relaxed rationality assumptions prove insufficient when good instincts align with optimal play. This negative result provides valuable insight: classical game-theoretic extensions grounded in rational choice are fundamentally incompatible with causal reasoning advantages, motivating new theoretical frameworks beyond standard Nash equilibrium for agentic AI.

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 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.

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.

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.

We study the incentivized information acquisition problem, where a principal hires an agent to gather information on her behalf. Such a problem is modeled as a Stackelberg game between the principal and the agent, where the principal announces a scoring rule that specifies the payment, and then the agent then chooses an effort level that maximizes her own profit and reports the information. We study the online setting of such a problem from the principal's perspective, i.e., designing the optimal scoring rule by repeatedly interacting with the strategic agent. We design a provably sample efficient algorithm that tailors the UCB algorithm (Auer et al., 2002) to our model, which achieves a sublinear T^{2/3}-regret after T iterations. Our algorithm features a delicate estimation procedure for the optimal profit of the principal, and a conservative correction scheme that ensures the desired agent's actions are incentivized. Furthermore, a key feature of our regret bound is that it is independent of the number of states of the environment.

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.

We consider stochastic unconstrained bilevel optimization problems when only the first-order gradient oracles are available. While numerous optimization methods have been proposed for tackling bilevel problems, existing methods either tend to require possibly expensive calculations regarding Hessians of lower-level objectives, or lack rigorous finite-time performance guarantees. In this work, we propose a Fully First-order Stochastic Approximation (F2SA) method, and study its non-asymptotic convergence properties. Specifically, we show that F2SA converges to an epsilon-stationary solution of the bilevel problem after epsilon^{-7/2}, epsilon^{-5/2}, and epsilon^{-3/2} iterations (each iteration using O(1) samples) when stochastic noises are in both level objectives, only in the upper-level objective, and not present (deterministic settings), respectively. We further show that if we employ momentum-assisted gradient estimators, the iteration complexities can be improved to epsilon^{-5/2}, epsilon^{-4/2}, and epsilon^{-3/2}, respectively. We demonstrate even superior practical performance of the proposed method over existing second-order based approaches on MNIST data-hypercleaning experiments.

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.

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.

Beyond minimizing a single training loss, many deep learning estimation pipelines rely on an auxiliary objective to quantify and encourage desirable properties of the model (e.g. performance on another dataset, robustness, agreement with a prior). Although the simplest approach to incorporating an auxiliary loss is to sum it with the training loss as a regularizer, recent works have shown that one can improve performance by blending the gradients beyond a simple sum; this is known as gradient surgery. We cast the problem as a constrained minimization problem where the auxiliary objective is minimized among the set of minimizers of the training loss. To solve this bilevel problem, we follow a parameter update direction that combines the training loss gradient and the orthogonal projection of the auxiliary gradient to the training gradient. In a setting where gradients come from mini-batches, we explain how, using a moving average of the training loss gradients, we can carefully maintain this critical orthogonality property. We demonstrate that our method, Bloop, can lead to much better performances on NLP and vision experiments than other gradient surgery methods without EMA.

We propose a novel independent and payoff-based learning framework for stochastic games that is model-free, game-agnostic, and gradient-free. The learning dynamics follow a best-response-type actor-critic architecture, where agents update their strategies (actors) using feedback from two distinct critics: a fast critic that intuitively responds to observed payoffs under limited information, and a slow critic that deliberatively approximates the solution to the underlying dynamic programming problem. Crucially, the learning process relies on non-equilibrium adaptation through smoothed best responses to observed payoffs. We establish convergence to (approximate) equilibria in two-agent zero-sum and multi-agent identical-interest stochastic games over an infinite horizon. This provides one of the first payoff-based and fully decentralized learning algorithms with theoretical guarantees in both settings. Empirical results further validate the robustness and effectiveness of the proposed approach across both classes of games.

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.

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.

Computing stable partitions in hedonic games is a challenging task because there exist games in which stable outcomes do not exist. Even more, these No-instances can often be leveraged to prove computational hardness results. We make this impression rigorous in a dynamic model of cardinal hedonic games by providing meta theorems. These imply hardness of deciding about the possible or necessary convergence of deviation dynamics based on the mere existence of No-instances. Our results hold for additively separable, fractional, and modified fractional hedonic games (ASHGs, FHGs, and MFHGs). Moreover, they encompass essentially all reasonable stability notions based on single-agent deviations. In addition, we propose dynamics as a method to find individually rational and contractually individual stable (CIS) partitions in ASHGs. In particular, we find that CIS dynamics from the singleton partition possibly converge after a linear number of deviations but may require an exponential number of deviations in the worst case.

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.

Gradient-based optimization has been critical to the success of machine learning, updating a single set of parameters to minimize a single loss. A growing number of applications rely on a generalization of this, where we have a bilevel or nested optimization of which subsets of parameters update on different objectives nested inside each other. We focus on motivating examples of hyperparameter optimization and generative adversarial networks. However, naively applying classical methods often fails when we look at solving these nested problems on a large scale. In this thesis, we build tools for nested optimization that scale to deep learning setups.

Various tasks in data science are modeled utilizing the variational regularization approach, where manually selecting regularization parameters presents a challenge. The difficulty gets exacerbated when employing regularizers involving a large number of hyperparameters. To overcome this challenge, bilevel learning can be employed to learn such parameters from data. However, neither exact function values nor exact gradients with respect to the hyperparameters are attainable, necessitating methods that only rely on inexact evaluation of such quantities. State-of-the-art inexact gradient-based methods a priori select a sequence of the required accuracies and cannot identify an appropriate step size since the Lipschitz constant of the hypergradient is unknown. In this work, we propose an algorithm with backtracking line search that only relies on inexact function evaluations and hypergradients and show convergence to a stationary point. Furthermore, the proposed algorithm determines the required accuracy dynamically rather than manually selected before running it. Our numerical experiments demonstrate the efficiency and feasibility of our approach for hyperparameter estimation on a range of relevant problems in imaging and data science such as total variation and field of experts denoising and multinomial logistic regression. Particularly, the results show that the algorithm is robust to its own hyperparameters such as the initial accuracies and step size.

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.

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.

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.

We study the problem of learning a Nash equilibrium (NE) in an imperfect information game (IIG) through self-play. Precisely, we focus on two-player, zero-sum, episodic, tabular IIG under the perfect-recall assumption where the only feedback is realizations of the game (bandit feedback). In particular, the dynamic of the IIG is not known -- we can only access it by sampling or interacting with a game simulator. For this learning setting, we provide the Implicit Exploration Online Mirror Descent (IXOMD) algorithm. It is a model-free algorithm with a high-probability bound on the convergence rate to the NE of order 1/T where T is the number of played games. Moreover, IXOMD is computationally efficient as it needs to perform the updates only along the sampled trajectory.

This paper investigates the evaluation of learned multiagent strategies in the incomplete information setting, which plays a critical role in ranking and training of agents. Traditionally, researchers have relied on Elo ratings for this purpose, with recent works also using methods based on Nash equilibria. Unfortunately, Elo is unable to handle intransitive agent interactions, and other techniques are restricted to zero-sum, two-player settings or are limited by the fact that the Nash equilibrium is intractable to compute. Recently, a ranking method called α-Rank, relying on a new graph-based game-theoretic solution concept, was shown to tractably apply to general games. However, evaluations based on Elo or α-Rank typically assume noise-free game outcomes, despite the data often being collected from noisy simulations, making this assumption unrealistic in practice. This paper investigates multiagent evaluation in the incomplete information regime, involving general-sum many-player games with noisy outcomes. We derive sample complexity guarantees required to confidently rank agents in this setting. We propose adaptive algorithms for accurate ranking, provide correctness and sample complexity guarantees, then introduce a means of connecting uncertainties in noisy match outcomes to uncertainties in rankings. We evaluate the performance of these approaches in several domains, including Bernoulli games, a soccer meta-game, and Kuhn poker.

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.

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.

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.

Self-play, characterized by agents' interactions with copies or past versions of itself, has recently gained prominence in reinforcement learning. This paper first clarifies the preliminaries of self-play, including the multi-agent reinforcement learning framework and basic game theory concepts. Then it provides a unified framework and classifies existing self-play algorithms within this framework. Moreover, the paper bridges the gap between the algorithms and their practical implications by illustrating the role of self-play in different scenarios. Finally, the survey highlights open challenges and future research directions in self-play. This paper is an essential guide map for understanding the multifaceted landscape of self-play in RL.

Two-player, constant-sum games are well studied in the literature, but there has been limited progress outside of this setting. We propose Joint Policy-Space Response Oracles (JPSRO), an algorithm for training agents in n-player, general-sum extensive form games, which provably converges to an equilibrium. We further suggest correlated equilibria (CE) as promising meta-solvers, and propose a novel solution concept Maximum Gini Correlated Equilibrium (MGCE), a principled and computationally efficient family of solutions for solving the correlated equilibrium selection problem. We conduct several experiments using CE meta-solvers for JPSRO and demonstrate convergence on n-player, general-sum games.

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.

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.

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.

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.

Achieving robust coordination and cooperation is a central challenge in multi-agent reinforcement learning (MARL). Uncovering the mechanisms underlying such emergent behaviors calls for a dynamical understanding of learn processes. In this work, we investigate the dynamics of actor-critic agents in stochastic games, focusing on the impact of entropy regularization. By leveraging time-scale separation, we derive the system's evolution equations, which are then formally analyzed using dynamical systems theory. We find that in the constant-sum game of Matching Pennies, the system exhibits chaotic behavior. Entropy regularization mitigates this chaos and drives the dynamics toward convergence to fair cooperation. In contrast, in the general-sum game of the Prisoner's Dilemma, the system displays multistability. Interestingly, the three stable equilibria of the system correspond to the well-known ALLC (Always Cooperate), ALLD (Always Defect), and GRIM (Grim Trigger) strategies from evolutionary game theory (EGT). Entropy regularization strengthens system resilience by enlarging the basin of attraction of the cooperative equilibrium. Our findings reveal a close link between the mechanism of direct reciprocity in EGT and how cooperation emerges in MARL, offering insights for designing more robust and collaborative multi-agent systems.

Recent AI algorithms are blackbox models whose decisions are difficult to interpret. eXplainable AI (XAI) seeks to address lack of AI interpretability and trust by explaining to customers their AI decision, e.g., decision to reject a loan application. The common wisdom is that regulating AI by mandating fully transparent XAI leads to greater social welfare. This paper challenges this notion through a game theoretic model for a policy-maker who maximizes social welfare, firms in a duopoly competition that maximize profits, and heterogenous consumers. The results show that XAI regulation may be redundant. In fact, mandating fully transparent XAI may make firms and customers worse off. This reveals a trade-off between maximizing welfare and receiving explainable AI outputs. We also discuss managerial implications for policy-maker and firms.

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.

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.

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.

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.

Fair machine learning seeks to mitigate model prediction bias against certain demographic subgroups such as elder and female. Recently, fair representation learning (FRL) trained by deep neural networks has demonstrated superior performance, whereby representations containing no demographic information are inferred from the data and then used as the input to classification or other downstream tasks. Despite the development of FRL methods, their vulnerability under data poisoning attack, a popular protocol to benchmark model robustness under adversarial scenarios, is under-explored. Data poisoning attacks have been developed for classical fair machine learning methods which incorporate fairness constraints into shallow-model classifiers. Nonetheless, these attacks fall short in FRL due to notably different fairness goals and model architectures. This work proposes the first data poisoning framework attacking FRL. We induce the model to output unfair representations that contain as much demographic information as possible by injecting carefully crafted poisoning samples into the training data. This attack entails a prohibitive bilevel optimization, wherefore an effective approximated solution is proposed. A theoretical analysis on the needed number of poisoning samples is derived and sheds light on defending against the attack. Experiments on benchmark fairness datasets and state-of-the-art fair representation learning models demonstrate the superiority of our attack.

We investigate how reinforcement learning can be used to train level-designing agents. This represents a new approach to procedural content generation in games, where level design is framed as a game, and the content generator itself is learned. By seeing the design problem as a sequential task, we can use reinforcement learning to learn how to take the next action so that the expected final level quality is maximized. This approach can be used when few or no examples exist to train from, and the trained generator is very fast. We investigate three different ways of transforming two-dimensional level design problems into Markov decision processes and apply these to three game environments.

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.

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.

Optimization problems over dynamic networks have been extensively studied and widely used in the past decades to formulate numerous real-world problems. However, (1) traditional optimization-based approaches do not scale to large networks, and (2) the design of good heuristics or approximation algorithms often requires significant manual trial-and-error. In this work, we argue that data-driven strategies can automate this process and learn efficient algorithms without compromising optimality. To do so, we present network control problems through the lens of reinforcement learning and propose a graph network-based framework to handle a broad class of problems. Instead of naively computing actions over high-dimensional graph elements, e.g., edges, we propose a bi-level formulation where we (1) specify a desired next state via RL, and (2) solve a convex program to best achieve it, leading to drastically improved scalability and performance. We further highlight a collection of desirable features to system designers, investigate design decisions, and present experiments on real-world control problems showing the utility, scalability, and flexibility of our framework.

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.

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

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.

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.

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.

Modeling the interaction between traffic agents is a key issue in designing safe and non-conservative maneuvers in autonomous driving. This problem can be challenging when multi-modality and behavioral uncertainties are engaged. Existing methods either fail to plan interactively or consider unimodal behaviors that could lead to catastrophic results. In this paper, we introduce an integrated decision-making and trajectory planning framework based on Bayesian game (i.e., game of incomplete information). Human decisions inherently exhibit discrete characteristics and therefore are modeled as types of players in the game. A general solver based on no-regret learning is introduced to obtain a corresponding Bayesian Coarse Correlated Equilibrium, which captures the interaction between traffic agents in the multimodal context. With the attained equilibrium, decision-making and trajectory planning are performed simultaneously, and the resulting interactive strategy is shown to be optimal over the expectation of rivals' driving intentions. Closed-loop simulations on different traffic scenarios are performed to illustrate the generalizability and the effectiveness of the proposed framework.

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.

We present a new algorithm to optimize distributions defined implicitly by parameterized stochastic diffusions. Doing so allows us to modify the outcome distribution of sampling processes by optimizing over their parameters. We introduce a general framework for first-order optimization of these processes, that performs jointly, in a single loop, optimization and sampling steps. This approach is inspired by recent advances in bilevel optimization and automatic implicit differentiation, leveraging the point of view of sampling as optimization over the space of probability distributions. We provide theoretical guarantees on the performance of our method, as well as experimental results demonstrating its effectiveness in real-world settings.

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.

Fair division considers the allocation of scarce resources among agents in such a way that every agent gets a fair share. It is a fundamental problem in society and has received significant attention and rapid developments from the game theory and artificial intelligence communities in recent years. The majority of the fair division literature can be divided along at least two orthogonal directions: goods versus chores, and divisible versus indivisible resources. In this survey, besides describing the state of the art, we outline a number of interesting open questions and future directions in three mixed fair division settings: (i) indivisible goods and chores, (ii) divisible and indivisible goods (mixed goods), and (iii) indivisible goods with subsidy which can be viewed like a divisible good.

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.

Hierarchical reinforcement learning (RL) can accelerate long-horizon decision-making by temporally abstracting a policy into multiple levels. Promising results in sparse reward environments have been seen with skills, i.e. sequences of primitive actions. Typically, a skill latent space and policy are discovered from offline data, but the resulting low-level policy can be unreliable due to low-coverage demonstrations or distribution shifts. As a solution, we propose fine-tuning the low-level policy in conjunction with high-level skill selection. Our Skill-Critic algorithm optimizes both the low and high-level policies; these policies are also initialized and regularized by the latent space learned from offline demonstrations to guide the joint policy optimization. We validate our approach in multiple sparse RL environments, including a new sparse reward autonomous racing task in Gran Turismo Sport. The experiments show that Skill-Critic's low-level policy fine-tuning and demonstration-guided regularization are essential for optimal performance. Images and videos are available at https://sites.google.com/view/skill-critic. We plan to open source the code with the final version.

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.

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 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.

The game show Lucky 13 differs from other television game shows in that contestants are required to place a bet on their own knowledge of trivia by selecting a range that contains the number of questions that they answered correctly. We present a model for this game show using binomial random variables and generate tables outlining the optimal range the player should select based on maximization of two different utility functions. After analyzing the decisions made by some actual contestants on this show, we present a numerical simulation for how many questions an average player is expected to answer correctly based on question categories observed for two sample contestants.

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 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.

Designing reward functions for efficiently guiding reinforcement learning (RL) agents toward specific behaviors is a complex task. This is challenging since it requires the identification of reward structures that are not sparse and that avoid inadvertently inducing undesirable behaviors. Naively modifying the reward structure to offer denser and more frequent feedback can lead to unintended outcomes and promote behaviors that are not aligned with the designer's intended goal. Although potential-based reward shaping is often suggested as a remedy, we systematically investigate settings where deploying it often significantly impairs performance. To address these issues, we introduce a new framework that uses a bi-level objective to learn behavior alignment reward functions. These functions integrate auxiliary rewards reflecting a designer's heuristics and domain knowledge with the environment's primary rewards. Our approach automatically determines the most effective way to blend these types of feedback, thereby enhancing robustness against heuristic reward misspecification. Remarkably, it can also adapt an agent's policy optimization process to mitigate suboptimalities resulting from limitations and biases inherent in the underlying RL algorithms. We evaluate our method's efficacy on a diverse set of tasks, from small-scale experiments to high-dimensional control challenges. We investigate heuristic auxiliary rewards of varying quality -- some of which are beneficial and others detrimental to the learning process. Our results show that our framework offers a robust and principled way to integrate designer-specified heuristics. It not only addresses key shortcomings of existing approaches but also consistently leads to high-performing solutions, even when given misaligned or poorly-specified auxiliary reward functions.

In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve mgg 1 lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling I blocks and sampling B samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of O(mepsilon^{-3I(I<m)}{II} + mepsilon^{-3}{IB}) for finding an epsilon-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms.

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.

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.

This study evaluates Artificial Intelligence (AI) agents for Dhumbal, a culturally significant multiplayer card game with imperfect information, through a systematic comparison of rule-based, search-based, and learning-based strategies. We formalize Dhumbal's mechanics and implement diverse agents, including heuristic approaches (Aggressive, Conservative, Balanced, Opportunistic), search-based methods such as Monte Carlo Tree Search (MCTS) and Information Set Monte Carlo Tree Search (ISMCTS), and reinforcement learning approaches including Deep Q-Network (DQN) and Proximal Policy Optimization (PPO), and a random baseline. Evaluation involves within-category tournaments followed by a cross-category championship. Performance is measured via win rate, economic outcome, Jhyap success, cards discarded per round, risk assessment, and decision efficiency. Statistical significance is assessed using Welch's t-test with Bonferroni correction, effect sizes via Cohen's d, and 95% confidence intervals (CI). Across 1024 simulated rounds, the rule-based Aggressive agent achieves the highest win rate (88.3%, 95% CI: [86.3, 90.3]), outperforming ISMCTS (9.0%) and PPO (1.5%) through effective exploitation of Jhyap declarations. The study contributes a reproducible AI framework, insights into heuristic efficacy under partial information, and open-source code, thereby advancing AI research and supporting digital preservation of cultural games.

We focus on a simple, one-dimensional collective decision problem (often referred to as the facility location problem) and explore issues of strategyproofness and proportionality-based fairness. We introduce and analyze a hierarchy of proportionality-based fairness axioms of varying strength: Individual Fair Share (IFS), Unanimous Fair Share (UFS), Proportionality (as in Freeman et al, 2021), and Proportional Fairness (PF). For each axiom, we characterize the family of mechanisms that satisfy the axiom and strategyproofness. We show that imposing strategyproofness renders many of the axioms to be equivalent: the family of mechanisms that satisfy proportionality, unanimity, and strategyproofness is equivalent to the family of mechanisms that satisfy UFS and strategyproofness, which, in turn, is equivalent to the family of mechanisms that satisfy PF and strategyproofness. Furthermore, there is a unique such mechanism: the Uniform Phantom mechanism, which is studied in Freeman et al. (2021). We also characterize the outcomes of the Uniform Phantom mechanism as the unique (pure) equilibrium outcome for any mechanism that satisfies continuity, strict monotonicity, and UFS. Finally, we analyze the approximation guarantees, in terms of optimal social welfare and minimum total cost, obtained by mechanisms that are strategyproof and satisfy each proportionality-based fairness axiom. We show that the Uniform Phantom mechanism provides the best approximation of the optimal social welfare (and also minimum total cost) among all mechanisms that satisfy UFS.

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.

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 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.

Learning to play optimally against any mixture over a diverse set of strategies is of important practical interests in competitive games. In this paper, we propose simplex-NeuPL that satisfies two desiderata simultaneously: i) learning a population of strategically diverse basis policies, represented by a single conditional network; ii) using the same network, learn best-responses to any mixture over the simplex of basis policies. We show that the resulting conditional policies incorporate prior information about their opponents effectively, enabling near optimal returns against arbitrary mixture policies in a game with tractable best-responses. We verify that such policies behave Bayes-optimally under uncertainty and offer insights in using this flexibility at test time. Finally, we offer evidence that learning best-responses to any mixture policies is an effective auxiliary task for strategic exploration, which, by itself, can lead to more performant populations.

The core of Transferable Utility (T.U.) games is a well-known solution concept from cooperative game theory yielding a cost allocation among n agents (called players) forming a coalition that is stable (i.e. no subset of players has an interest to deviate). In this paper, inspired by a practical application in the context of a decision support system for collaborative transportation in a Short Food Supply Chain (SFSC), we mainly focus on Traveling Salesman Games (TSGs), where the objective is to allocate the cost of a Traveling Salesman Problem (TSP) with n locations and 1 depot to n players, each linked to exactly one of the locations. Given the computational complexity of computing an element of the core and the cost of a TSP, we study semicore allocations: a relaxation of the core that only requires that the subsets of size n -1 and of size 1 do not wish to deviate from the coalition. In the literature, instances of TSGs with empty cores and semicores are found. Hence, this paper first surveys the methods to approximate stability whenever the core is empty, such as the cost of stability (computing the minimum amount of money to subsidize the coalition with to attain stability) and the ε-core (which is a set of allocations that allow subsets of players to exceed their actual cost, but at most of a value of ε). We prove that these two solution

Classical convergence guarantees for gradient-based learning in games require the pseudo-gradient to be (strongly) monotone in Euclidean geometry as shown by rosen(1965), a condition that often fails even in simple games with strong cross-player couplings. We introduce Small-Gain Nash (SGN), a block small-gain condition in a custom block-weighted geometry. SGN converts local curvature and cross-player Lipschitz coupling bounds into a tractable certificate of contraction. It constructs a weighted block metric in which the pseudo-gradient becomes strongly monotone on any region where these bounds hold, even when it is non-monotone in the Euclidean sense. The continuous flow is exponentially contracting in this designed geometry, and projected Euler and RK4 discretizations converge under explicit step-size bounds derived from the SGN margin and a local Lipschitz constant. Our analysis reveals a certified ``timescale band'', a non-asymptotic, metric-based certificate that plays a TTUR-like role: rather than forcing asymptotic timescale separation via vanishing, unequal step sizes, SGN identifies a finite band of relative metric weights for which a single-step-size dynamics is provably contractive. We validate the framework on quadratic games where Euclidean monotonicity analysis fails to predict convergence, but SGN successfully certifies it, and extend the construction to mirror/Fisher geometries for entropy-regularized policy gradient in Markov games. The result is an offline certification pipeline that estimates curvature, coupling, and Lipschitz parameters on compact regions, optimizes block weights to enlarge the SGN margin, and returns a structural, computable convergence certificate consisting of a metric, contraction rate, and safe step-sizes for non-monotone games.

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 consider strategy proof mechanisms for facility location which maximize equitability between agents. As is common in the literature, we measure equitability with the Gini index. We first prove a simple but fundamental impossibility result that no strategy proof mechanism can bound the approximation ratio of the optimal Gini index of utilities for one or more facilities. We propose instead computing approximation ratios of the complemented Gini index of utilities, and consider how well both deterministic and randomized mechanisms approximate this. In addition, as Nash welfare is often put forwards as an equitable compromise between egalitarian and utilitarian outcomes, we consider how well mechanisms approximate the Nash welfare.

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

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.

We study how to learn ε-optimal strategies in zero-sum imperfect information games (IIG) with trajectory feedback. In this setting, players update their policies sequentially based on their observations over a fixed number of episodes, denoted by T. Existing procedures suffer from high variance due to the use of importance sampling over sequences of actions (Steinberger et al., 2020; McAleer et al., 2022). To reduce this variance, we consider a fixed sampling approach, where players still update their policies over time, but with observations obtained through a given fixed sampling policy. Our approach is based on an adaptive Online Mirror Descent (OMD) algorithm that applies OMD locally to each information set, using individually decreasing learning rates and a regularized loss. We show that this approach guarantees a convergence rate of mathcal{O}(T^{-1/2}) with high probability and has a near-optimal dependence on the game parameters when applied with the best theoretical choices of learning rates and sampling policies. To achieve these results, we generalize the notion of OMD stabilization, allowing for time-varying regularization with convex increments.

Learning the behavior of large agent populations is an important task for numerous research areas. Although the field of multi-agent reinforcement learning (MARL) has made significant progress towards solving these systems, solutions for many agents often remain computationally infeasible and lack theoretical guarantees. Mean Field Games (MFGs) address both of these issues and can be extended to Graphon MFGs (GMFGs) to include network structures between agents. Despite their merits, the real world applicability of GMFGs is limited by the fact that graphons only capture dense graphs. Since most empirically observed networks show some degree of sparsity, such as power law graphs, the GMFG framework is insufficient for capturing these network topologies. Thus, we introduce the novel concept of Graphex MFGs (GXMFGs) which builds on the graph theoretical concept of graphexes. Graphexes are the limiting objects to sparse graph sequences that also have other desirable features such as the small world property. Learning equilibria in these games is challenging due to the rich and sparse structure of the underlying graphs. To tackle these challenges, we design a new learning algorithm tailored to the GXMFG setup. This hybrid graphex learning approach leverages that the system mainly consists of a highly connected core and a sparse periphery. After defining the system and providing a theoretical analysis, we state our learning approach and demonstrate its learning capabilities on both synthetic graphs and real-world networks. This comparison shows that our GXMFG learning algorithm successfully extends MFGs to a highly relevant class of hard, realistic learning problems that are not accurately addressed by current MARL and MFG methods.

How have individuals of social animals in nature evolved to learn from each other, and what would be the optimal strategy for such learning in a specific environment? Here, we address both problems by employing a deep reinforcement learning model to optimize the social learning strategies (SLSs) of agents in a cooperative game in a multi-dimensional landscape. Throughout the training for maximizing the overall payoff, we find that the agent spontaneously learns various concepts of social learning, such as copying, focusing on frequent and well-performing neighbors, self-comparison, and the importance of balancing between individual and social learning, without any explicit guidance or prior knowledge about the system. The SLS from a fully trained agent outperforms all of the traditional, baseline SLSs in terms of mean payoff. We demonstrate the superior performance of the reinforcement learning agent in various environments, including temporally changing environments and real social networks, which also verifies the adaptability of our framework to different social settings.

In this paper, we tackle the problem of learning to play 3v3 multi-drone volleyball, a new embodied competitive task that requires both high-level strategic coordination and low-level agile control. The task is turn-based, multi-agent, and physically grounded, posing significant challenges due to its long-horizon dependencies, tight inter-agent coupling, and the underactuated dynamics of quadrotors. To address this, we propose Hierarchical Co-Self-Play (HCSP), a hierarchical reinforcement learning framework that separates centralized high-level strategic decision-making from decentralized low-level motion control. We design a three-stage population-based training pipeline to enable both strategy and skill to emerge from scratch without expert demonstrations: (I) training diverse low-level skills, (II) learning high-level strategy via self-play with fixed low-level skills, and (III) joint fine-tuning through co-self-play. Experiments show that HCSP achieves superior performance, outperforming non-hierarchical self-play and rule-based hierarchical baselines with an average 82.9% win rate and a 71.5% win rate against the two-stage variant. Moreover, co-self-play leads to emergent team behaviors such as role switching and coordinated formations, demonstrating the effectiveness of our hierarchical design and training scheme. The project page is at https://sites.google.com/view/hi-co-self-play.

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.

While Explainable Artificial Intelligence (XAI) is increasingly expanding more areas of application, little has been applied to make deep Reinforcement Learning (RL) more comprehensible. As RL becomes ubiquitous and used in critical and general public applications, it is essential to develop methods that make it better understood and more interpretable. This study proposes a novel approach to explain cooperative strategies in multiagent RL using Shapley values, a game theory concept used in XAI that successfully explains the rationale behind decisions taken by Machine Learning algorithms. Through testing common assumptions of this technique in two cooperation-centered socially challenging multi-agent environments environments, this article argues that Shapley values are a pertinent way to evaluate the contribution of players in a cooperative multi-agent RL context. To palliate the high overhead of this method, Shapley values are approximated using Monte Carlo sampling. Experimental results on Multiagent Particle and Sequential Social Dilemmas show that Shapley values succeed at estimating the contribution of each agent. These results could have implications that go beyond games in economics, (e.g., for non-discriminatory decision making, ethical and responsible AI-derived decisions or policy making under fairness constraints). They also expose how Shapley values only give general explanations about a model and cannot explain a single run, episode nor justify precise actions taken by agents. Future work should focus on addressing these critical aspects.

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.

A successful tactic that is followed by the scientific community for advancing AI is to treat games as problems, which has been proven to lead to various breakthroughs. We adapt this strategy in order to study Rocket League, a widely popular but rather under-explored 3D multiplayer video game with a distinct physics engine and complex dynamics that pose a significant challenge in developing efficient and high-performance game-playing agents. In this paper, we present Lucy-SKG, a Reinforcement Learning-based model that learned how to play Rocket League in a sample-efficient manner, outperforming by a notable margin the two highest-ranking bots in this game, namely Necto (2022 bot champion) and its successor Nexto, thus becoming a state-of-the-art agent. Our contributions include: a) the development of a reward analysis and visualization library, b) novel parameterizable reward shape functions that capture the utility of complex reward types via our proposed Kinesthetic Reward Combination (KRC) technique, and c) design of auxiliary neural architectures for training on reward prediction and state representation tasks in an on-policy fashion for enhanced efficiency in learning speed and performance. By performing thorough ablation studies for each component of Lucy-SKG, we showed their independent effectiveness in overall performance. In doing so, we demonstrate the prospects and challenges of using sample-efficient Reinforcement Learning techniques for controlling complex dynamical systems under competitive team-based multiplayer conditions.