Game-Theoretic RL

Online robust zero-sum Markov games pose a coupled operator-learning and game-solving problem: from nominal interaction data, the learner must recover a worst-case minimax Bellman operator while maintaining strategic coverage and controlling stage-game approximation error. Existing robust reinforcement-learning theory does not yet provide such an operator-learning perspective beyond specialized multi-agent settings. We develop an optimistic dual robust Bellman framework that combines dual robust Bellman fitting, optimistic minimax planning, backward-validated confidence sets, and exploiter-mix data collection. Our analysis is organized around the robust Bellman residuals induced by the candidate value class and yields a regret reduction that separates exploration complexity, robust operator-estimation error, stage-game solution error, and misspecification. Under explicit interpretable assumptions, we obtain conditional theorem interfaces for general and projected function approximation, together with closed results in several structured regimes, including tabular, epochwise cross-fit linear, self-normalized pure-online linear / finite-rank kernel, and projected RKHS truncation. Small-scale numerical experiments illustrate robustness calibration, strategic coverage, and the return-gap trade-off predicted by the framework.

The rapid proliferation of distributed energy resources (DERs) aggregated by virtual power plants (VPPs) elevates cybersecurity risk by expanding the attack surface. A coordinated cyberattack has the potential to disrupt distribution-system, bulk-system, and market operations. Existing cyberphysical defense strategies largely assume access to grid network models and operating conditions, which may not be available to third-party aggregators. This paper proposes a novel hybrid, perturbation-based cyber-physical defense for VPP-operated DERs that blends game-theoretic planning with model-free reinforcement learning (RL) under a Bayesian belief-update framework. Topology agnosticism is defined from the VPP information and implementation perspective: the VPP does not require access to the feeder topology, line parameters, switching configuration, or full system state. This work advances the state of the art through three contributions: (1) a multi-output Gaussian process with Hamiltonian Monte Carlo for discrete attacker-type inference, probability of detection, and confidence of detection for modeling the belief distribution; (2) moment-matched Gaussian propagation of posterior beliefs across episodes; and (3) an RL policy that adaptively integrates Stackelberg leader (VPP) actions with exploration. Furthermore, we establish equilibrium existence and derive an episode-level closed-loop utility bound connecting attacker-belief uncertainty with GT-RL action alignment. Evaluations on IEEE 13- and 123-Bus feeders with DER portfolios demonstrate effective detection under attacker-model uncertainty and evaluated feeder re-configuration conditions, while requiring no explicit feeder-topology knowledge. Overall, the results indicate that combining bi-level game-theoretic planning, nonparametric Bayesian inference, and adaptive RL is a promising direction for VPP cybersecurity.

Zero-sum Markov games arise in a wide variety of sequential decision-making problems such as adversarial learning and planning against modeled uncertainties. However, prior work on finite-sample guarantees on the learned state-action value function ($Q$-function) for zero-sum Markov games is largely restricted to finite-action settings, or to continuous-action games in which agents have linear dynamics and quadratic rewards (i.e. linear-quadratic, or LQ, games). A central challenge in extending such guarantees to more general continuous-action zero-sum Markov games is that the associated Bellman operator involves a minimax problem that need not admit a tractable saddle-point solution. To this end, we first introduce a class of neural network function approximators for the $Q$-function that is convex-concave in the players' actions, which guarantees that this minimax problem admits a pure-strategy saddle point. We then study an online variant of fitted $Q$-iteration employing this function class and establish, to the best of our knowledge, the first finite-sample guarantees for non-LQ zero-sum Markov games with continuous states and actions. Our code can be found at https://github.com/CLeARoboticsLab/QCanPlayThatGame.

In this paper, we propose GERIS, a game-theoretic framework for instance selection in the data augmentation phase of license plate recognition systems. During augmentation, synthetic license plate images are generated and transformed using stochastic noise to simulate real-world conditions. However, certain noise configurations lead to highly distorted, unreadable images that degrade model performance by introducing instance-dependent label noise. GERIS formulates a non-cooperative game in which each noise vector competes for inclusion in the training set based on its similarity to labeled data and its contribution to model reliability. By identifying and pruning low-quality instances, GERIS improves the overall quality of the augmented dataset. Unlike traditional black-box learning methods, GERIS offers a transparent, theoretically grounded mechanism for data filtering. Experimental results demonstrate that GERIS outperforms existing instance selection methods in terms of classification accuracy and robustness.

Reinforcement learning agents trained to maximize their own reward in repeated interactions can converge to supra-competitive outcomes resembling explicit collusion, without communication or shared design. Existing mitigation approaches are largely tied to specific economic settings, like two-sided platforms and auctions, leaving open how to design interventions for general repeated games. We address this gap by formalizing the connection between empirical observations from prior work on Q-learning collusion and classical theory of Simple Penal Codes (SPCs). We show any non-trivial SPC induces a quantifiable conditional dependence in agents' policies, detectable via the total variation distance between an agent's action distributions across cooperation and defection histories. Building on this connection, we propose CURB (Collusion Unwinding via Reward shaping and Belief injection), a reward-shaping framework that penalizes this Total Variation (TV) distance signal during Q-learning and is guaranteed to convert any SPC fixed point of the dynamics into a trivial one, thus precluding collusive equilibria sustained by punishment threats. Empirically, CURB substantially reduces collusion by Q-learning agents in both Bertrand and Cournot Competition Repeated Games. We further demonstrate that CURB extends to deep Q-network agents in Bertrand competition, suggesting the mechanism generalizes beyond tabular Q-learning.

Model-based reinforcement learning (MBRL) has achieved remarkable results in single-agent domains, yet its extension to competitive imperfect information games (IIGs) remains underexplored. In multi-agent settings, opponent-induced non-stationarity complicates the learning process, and decentralized model learning faces severe identifiability barriers, which we argue make centralized model learning a mathematical necessity. Building on this analysis, we propose NashDreamer, a principled MBRL framework for two-player zero-sum IIGs. It introduces a centralized Multi-Agent Recurrent State-Space Model (MARSSM) that decouples environment dynamics from the effect of players' strategies on their individual observations. NashDreamer is designed to use arbitrary policy gradient algorithms and inherits their convergence guarantees towards Nash equilibria under an idealized model. Empirical evaluations across four benchmark games demonstrate that NashDreamer substantially improves sample efficiency over model-free baselines early in the training. Finally, we theoretically analyze the architecture's optimization landscape, identifying the vulnerability of the Dreamer family of algorithms to posterior collapse in stochastic environments. We leave it as an open challenge.

In this work, we study radically uncoupled learning in discounted general-sum Markov games. Assuming ``$\mathsf{ETH}$ for $\mathsf{PPAD}$", we show that, for every fixed discount factor, there is no polynomial-time algorithm for computing inverse-polynomially accurate coarse correlated equilibria in discounted general-sum Markov games when players learn independently in decentralized settings. Complementing this hardness result, we provide what appears to be the first radically uncoupled algorithm with sub-exponential convergence guarantees to coarse correlated equilibria in discounted general-sum Markov games without imposing any structural restrictions on the game. Our algorithm is a layered variant of optimistic mirror descent with an increasing step-size schedule tailored to the multi-agent setting. Finally, we develop both full-feedback and partial feedback versions of the aforementioned algorithm and establish sub-exponential convergence guarantees for each case.

Reinforcement learning (RL) fine-tuning is widely used in language model training to improve performance on a target task while limiting drift from a reference policy. A standard way to balance this trade-off is via a KL-regularized RL objective, although this formulation does not by itself provide a principled way to set the regularization coefficient. In practice, the coefficient is typically chosen heuristically or via hyperparameter search, which can lead to unnecessary overhead in training cost or undesirable reward-retention trade-offs. We instead propose a game-theoretic framework that gives this trade-off an explicit statistical interpretation. Specifically, we study a sequential game in which an agent chooses a policy to maximize cumulative reward while a monitor observes policy outputs over time and tests for deviations from the reference policy. Although not originating from the same perspective, we show that the resulting equilibrium policy can nonetheless be expressed as the solution to a KL-regularized RL problem for an optimal regularization parameter that can be viewed as maximizing reward per unit of statistical distinguishability. Drawing on classical results from concave-convex fractional programming, we provide a principled method for learning this equilibrium coefficient via reduction to the KL-regularized RL objective, thus allowing for flexible integration into standard fine-tuning pipelines. In experiments with Qwen3-8B and Llama-3.2-1B, we show that our methods result in competitive reward-retention trade-offs in a continual learning setting, and illustrate how our framework may be used to audit API providers serving open-source models.

Reinforcement learning (RL) provides a framework for sequential decision making under explicit objectives. In its classical form, RL studies how an agent should act to maximise long-term reward in a dynamic environment. In richer settings, the problem extends beyond a single agent and fixed environment: intelligent behavior may require strategic interaction, adaptation to uncertainty, and reasoning over high-dimensional worlds. This thesis studies RL from two perspectives: algorithms in games and RL in the era of foundation models. The first part focuses on multi-agent RL in games. It examines how incentives, policies, and equilibrium concepts interact in competitive and general-sum environments, spanning two-player zero-sum games, large-scale video games, and multi-player settings with general structure. These works investigate learning in multi-agent systems and the behavior of RL methods in interactive environments. The second part studies RL with generative and foundation models, motivated by the idea that prior knowledge can enrich sequential decision making. Pretrained generative models and learned world models serve as representation tools and structured priors for planning, control, and policy optimization. The thesis develops diffusion-based world models, investigates RL for efficient video generation, explores generative models as policy classes, and studies interactive video world models in which actions shape future observations. It also addresses long-horizon modeling through architectures with memory. Together, these contributions present a unified view of RL as objective-driven adaptation in complex sequential domains. From strategic games to generative world models, the thesis highlights how RL connects decision making, environment modeling, and emerging foundation-model capabilities, offering a broader perspective on the principles underlying intelligent behavior.

We study inverse reinforcement learning for discrete-time, infinite-horizon mean-field games (MFGs) under an average-reward criterion. Expert demonstrations are assumed to arise from a stationary mean-field equilibrium under an unknown reward, and the goal is to recover a policy explaining the observed behaviour via the maximum causal entropy principle. We formulate the inverse problem by enforcing consistency with the expert mean-field term and long-run feature expectations, treating two reward classes within a unified occupation-measure framework. For finite-dimensional linear rewards, we give a convex dual reformulation with an explicit log-partition objective, and prove smoothness and curvature properties justifying constant-step-size gradient descent. For infinite-dimensional RKHS rewards, we develop a Lagrangian relaxation whose inner-maximising policy is characterised by a soft Bellman equation. The main obstacle is the absence of a discount-factor contraction. We resolve this by introducing a minorisation-based sub-stochastic kernel that yields a strict contraction of the soft Bellman operator. We establish Fréchet differentiability and Lipschitz smoothness of the log-likelihood score, leading to a gradient ascent algorithm with convergence guarantees. Two numerical examples, a malware-spread MFG and an RKHS-based consumer-choice model, show that the recovered policies closely match expert behaviour.

LLM-based multi-agent systems are increasingly used for strategic decision-making tasks. In such settings, performance depends not only on individual model capabilities, but also on the policies by which agents interact and adapt. Multi-agent reinforcement learning can optimise these interaction policies, but its reward design often remains task-specific and weakly grounded in interaction structure. To address this gap, we propose GARL, a GAme-theoretic Reinforcement Learning framework for multi-agent strategic prioritisation. GARL formalises strategic prioritisation as a two-stage game: competing agents first allocate strategic resources over a shared candidate set, and a higher-level arbiter then produces the final ranking. The resulting game-theoretic utilities are converted into role-specific reinforcement signals, allowing policy optimisation to be guided by structured interaction. We instantiate GARL on issues-in-dispute ranking, where the goal is to prioritise core issues in legal proceedings. Experiments show that GARL improves ranking performance, enables small open-source LLMs to become competitive with a strong closed-source LLM under the same candidate-ranking setting, and yields gains in legal-domain competence and broader strategic decision-making. Overall, GARL demonstrates how game-theoretic interaction structure can be turned into reinforcement-learning objectives, providing a principled approach to policy optimisation in multi-agent strategic prioritisation.

Language models fine-tuned with reinforcement learning typically optimize for task reward, ignoring multi-agent strategic structure. Because these agents condition on natural language game-state descriptions and emit actions through free-form generation, strategic failure modes -- exploiting weaker opponents, coordinating on harmful equilibria, and externalizing costs are inseparable from the language interface itself. We propose Safe Equilibrium Policy Optimization (\sepo{}), a training objective that augments expected payoff with explicit penalties for exploitability, collusion risk, and externality cost. We implement \sepo{} as a reward signal for Group Relative Policy Optimization (GRPO), applied to Gemma4 E4B-it and Qwen3.5-4B after supervised fine-tuning (SFT). Evaluated across five strategic domains: Iterated Prisoner's Dilemma, repeated auctions, two negotiation variants, and Kuhn Poker. \sepo{} achieves zero exploit-pool advantage in Kuhn Poker for both models, outperforms the base model on safety in four domains, and corrects the over-cooperative behavior introduced by SFT. In negotiation, \sepo{} achieves a positive-safety outcome and only the positive normalized relative advantage of any negotiation configuration. Ablation experiments confirm that per-rollout exploit computation is necessary: a shared constant penalty cancels in GRPO advantage normalization (constant control-variate property), producing zero gradient. To support further research in strategic safety for agents, we release our code and SFT datasets.

We study the problem of learning Nash equilibria in offline two-player zero-sum Markov games. While existing approaches often rely on explicit pessimism to address distribution shift, we show that KL regularization alone suffices to stabilize learning and guarantee convergence. We first introduce Regularized Offline Sequential Equilibrium (ROSE), a theoretical framework that achieves a fast $\widetilde{\mathcal{O}}(1/n)$ convergence rate under unilateral concentrability, improving over the standard $\widetilde{\mathcal{O}}(1/\sqrt{n})$ rates in unregularized settings. We then propose Sequential Offline Self-play Mirror Descent (SOS-MD), a practical model-free algorithm based on least-squares value estimation and iterative self-play updates. We prove that the last iterate of SOS-MD attains the same $\widetilde{\mathcal{O}}(1/n)$ statistical rate up to a vanishing optimization error of order $\widetilde{\mathcal{O}}(1/\sqrt{T})$ in the number of self-play iterations $T$.

In this paper we study team-symmetric games with $m\ge 2$ teams. Players within a team have symmetric identity and have a common payoff function. We show that team-symmetric games always have a team-symmetric Nash equilibrium. We develop and solve a linear complementarity problem of team-symmetric Nash equilibria. We propose an actor-critic based multi-agent reinforcement learning algorithm for team-symmetric games. Through simulations, we show that this multi-agent reinforcement learning algorithm performs much better than many existing algorithms.

We show that a threshold in decision capacity determines whether self-play reinforcement learning agents collapse under asymmetric rule perturbations. Across poker variants, matrix games, a dice game, and multiple learning algorithms, eliminating all positive-reach contingent decisions causes rapid convergence to a deterministic exploitation attractor, a fixed point at near-maximal loss. Preserving even a single positive-reach contingent decision point prevents this collapse. A frozen baseline and fixed-opponent control confirm that the mechanism is co-adaptation under constraint, not the perturbation itself. The phenomenon is timing-invariant, fully reversible upon action restoration, and intensifies under function approximation. These results establish a sharp threshold at zero reach-weighted contingent action capacity, with severity scaling continuously via reach-weighted capacity in the tested domains.

Cooperative equilibria are fragile. When agents learn alongside each other rather than in a fixed environment, the process of learning destabilizes the cooperation they are trying to sustain: every gradient step an agent takes shifts the distribution of actions its partner will play, turning a cooperative partner into a source of stochastic noise precisely where the cooperation decision is most sensitive. We study how this co-learning noise propagates through the structure of coordination games, and find that the cooperative equilibrium, even when strongly Pareto-dominant, is exponentially unstable under standard risk-neutral learning, collapsing irreversibly once partner noise crosses the game's critical cooperation threshold. The natural response to apply distributional robustness to hedge against partner uncertainty makes things strictly worse: risk-averse return objectives penalize the high-variance cooperative action relative to defection, widening the instability region rather than shrinking it, a paradox that reveals a fundamental mismatch between the domains where robustness is applied and instability originates. We resolve this by showing that robustness should target the policy gradient update variance induced by partner uncertainty, not the return distribution. This distinction yields an algorithm whose gradient updates are modulated by an online measure of partner unpredictability, provably expanding the cooperation basin in symmetric coordination games. To unify stability, sample complexity, and welfare consequences of this approach, we introduce the Price of Paranoia as the structural dual of the Price of Anarchy. Together with a novel Cooperation Window, it precisely characterizes how much welfare learning algorithms can recover under partner noise, pinning down the optimal degree of robustness as a closed-form balance between equilibrium stability and sample efficiency.

Reinforcement learning has been successful both empirically and theoretically in single-agent settings, but extending these results to multi-agent reinforcement learning in general-sum Markov games remains challenging. This paper studies the convergence of Stackelberg Q-value iteration in two-player general-sum Markov games from a control-theoretic perspective. We introduce a relaxed policy condition tailored to the Stackelberg setting and model the learning dynamics as a switching system. By constructing upper and lower comparison systems, we establish finite-time error bounds for the Q-functions and characterize their convergence properties. Our results provide a novel control-theoretic perspective on Stackelberg learning. Moreover, to the best of the authors' knowledge, this paper offers the first finite-time convergence guarantees for Q-value iteration in general-sum Markov games under Stackelberg interactions.

Embodied Visual Tracking (EVT), a core dynamic task in embodied intelligence, requires an agent to precisely follow a language-specified target. Yet most existing methods rely on single-agent imitation learning, suffering from costly expert data and limited generalization due to static training environments. Inspired by competition-driven capability evolution, we propose CoMaTrack, a competitive game-theoretic multi-agent reinforcement learning framework that trains agents in a dynamic adversarial setting with competitive subtasks, yielding stronger adaptive planning and interference-resilient strategies. We further introduce CoMaTrack-Bench, the first open-source Habitat-based benchmark protocol and episode set for language-conditioned competitive EVT featuring dynamic dueling, featuring game scenarios between a tracker and adaptive opponents across diverse environments and instructions, enabling standardized robustness evaluation under active adversarial interactions. Experiments show that CoMaTrack achieves state-of-the-art results on both standard benchmarks and CoMaTrack-Bench. Notably, a 3B VLM trained with our framework surpasses previous single-agent imitation learning methods based on 7B models on the challenging EVT-Bench, achieving 92.1% in STT, 74.2% in DT, and 57.5% in AT. The benchmark code will be available at https://github.com/wlqcode/CoMaTrack-Bench.

We study Bayesian learning in episodic, finite-horizon zero-sum Markov games with unknown transition and reward models. We investigate a posterior algorithm in which each player maintains a Bayesian posterior over the game model, independently samples a game model at the beginning of each episode, and computes an equilibrium policy for the sampled model. We analyze two settings: (i) Both players use the posterior sampling algorithm, and (ii) Only one player uses posterior sampling while the opponent follows an arbitrary learning algorithm. In each setting, we provide guarantees on the expected regret of the posterior sampling agent. Our notion of regret compares the expected total reward of the learning agent against the expected total reward under equilibrium policies of the true game. Our main theoretical result is an expected regret bound for the posterior sampling agent of order $O(HS\sqrt{ABHK\log(SABHK)})$ where $K$ is the number of episodes, $H$ is the episode length, $S$ is the number of states, and $A,B$ are the action space sizes of the two players. Experiments in a grid-world predator--prey domain illustrate the sublinear regret scaling and show that posterior sampling competes favorably with a fictitious-play baseline.

Game theory provides the gold standard for analyzing adversarial engagements, offering strong optimality guarantees. However, these guarantees often become brittle when assumptions such as perfect information are violated. Reinforcement learning (RL), by contrast, is adaptive but can be sample-inefficient in large, complex domains. This paper introduces a hybrid approach that leverages game-theoretic insights to improve RL training efficiency. We study a border defense game with limited perceptual range, where defender performance depends on both search and pursuit strategies, making classical differential game solutions inapplicable. Our method employs the Apollonius Circle (AC) to compute equilibrium in the post-detection phase, enabling early termination of RL episodes without learning pursuit dynamics. This allows RL to concentrate on learning search strategies while guaranteeing optimal continuation after detection. Across single- and multi-defender settings, this early termination method yields 10-20% higher rewards, faster convergence, and more efficient search trajectories. Extensive experiments validate these findings and demonstrate the overall effectiveness of our approach.

Provably efficient and robust equilibrium computation in general-sum Markov games remains a core challenge in multi-agent reinforcement learning. Nash equilibrium is computationally intractable in general and brittle due to equilibrium multiplicity and sensitivity to approximation error. We study Risk-Sensitive Quantal Response Equilibrium (RQRE), which yields a unique, smooth solution under bounded rationality and risk sensitivity. We propose RQRE-OVI, an optimistic value iteration algorithm for computing RQRE with linear function approximation in large or continuous state spaces. Through finite-sample regret analysis, we establish convergence and explicitly characterize how sample complexity scales with rationality and risk-sensitivity parameters. The regret bounds reveal a quantitative tradeoff: increasing rationality tightens regret, while risk sensitivity induces regularization that enhances stability and robustness. This exposes a Pareto frontier between expected performance and robustness, with Nash recovered in the limit of perfect rationality and risk neutrality. We further show that the RQRE policy map is Lipschitz continuous in estimated payoffs, unlike Nash, and RQRE admits a distributionally robust optimization interpretation. Empirically, we demonstrate that RQRE-OVI achieves competitive performance under self-play while producing substantially more robust behavior under cross-play compared to Nash-based approaches. These results suggest RQRE-OVI offers a principled, scalable, and tunable path for equilibrium learning with improved robustness and generalization.

Multi-agent reinforcement learning (MARL) is increasingly used to design learning-enabled agents that interact in shared environments. However, training MARL algorithms in general-sum games remains challenging: learning dynamics can become unstable, and convergence guarantees typically hold only in restricted settings such as two-player zero-sum or fully cooperative games. Moreover, when agents have heterogeneous and potentially conflicting preferences, it is unclear what system-level objective should guide learning. In this paper, we propose a new MARL pipeline called Near-Potential Policy Optimization (NePPO) for computing approximate Nash equilibria in mixed cooperative--competitive environments. The core idea is to learn a player-independent potential function such that the Nash equilibrium of a cooperative game with this potential as the common utility approximates a Nash equilibrium of the original game. To this end, we introduce a novel MARL objective such that minimizing this objective yields the best possible potential function candidate and consequently an approximate Nash equilibrium of the original game. We develop an algorithmic pipeline that minimizes this objective using zeroth-order gradient descent and returns an approximate Nash equilibrium policy. We empirically show the superior performance of this approach compared to popular baselines such as IPPO and MAPPO.

Offline learning of strategies takes data efficiency to its extreme by restricting algorithms to a fixed dataset of state-action trajectories. We consider the problem in a mixed-motive multiagent setting, where the goal is to solve a game under the offline learning constraint. We first frame this problem in terms of selecting among candidate equilibria. Since datasets may inform only a small fraction of game dynamics, it is generally infeasible in offline game-solving to even verify a proposed solution is a true equilibrium. Therefore, we consider the relative probability of low regret (i.e., closeness to equilibrium) across candidates based on the information available. Specifically, we extend Policy Space Response Oracles (PSRO), an online game-solving approach, by quantifying game dynamics uncertainty and modifying the RL objective to skew towards solutions more likely to have low regret in the true game. We further propose a novel meta-strategy solver, tailored for the offline setting, to guide strategy exploration in PSRO. Our incorporation of Conservatism principles from Offline reinforcement learning approaches for strategy Exploration gives our approach its name: COffeE-PSRO. Experiments demonstrate COffeE-PSRO's ability to extract lower-regret solutions than state-of-the-art offline approaches and reveal relationships between algorithmic components empirical game fidelity, and overall performance.

The long-tail distribution, where a few head labels dominate while rare tail labels abound, poses a persistent challenge for large-scale Multi-Label Classification (MLC) in real-world data mining applications. Existing resampling and reweighting strategies often disrupt inter-label dependencies or require brittle hyperparameter tuning, especially as the label space expands to tens of thousands of labels. To address this issue, we propose Curiosity-Driven Game-Theoretic Multi-Label Learning (CD-GTMLL), a scalable cooperative framework that recasts long-tail MLC as a multi-player game - each sub-predictor ("player") specializes in a partition of the label space, collaborating to maximize global accuracy while pursuing intrinsic curiosity rewards based on tail label rarity and inter-player disagreement. This mechanism adaptively injects learning signals into under-represented tail labels without manual balancing or tuning. We further provide a theoretical analysis showing that our CD-GTMLL converges to a tail-aware equilibrium and formally links the optimization dynamics to improvements in the Rare-F1 metric. Extensive experiments across 7 benchmarks, including extreme multi-label classification datasets with 30,000+ labels, demonstrate that CD-GTMLL consistently surpasses state-of-the-art methods, with gains up to +1.6% P@3 on Wiki10-31K. Ablation studies further confirm the contributions of both game-theoretic cooperation and curiosity-driven exploration to robust tail performance. By integrating game theory with curiosity mechanisms, CD-GTMLL not only enhances model efficiency in resource-constrained environments but also paves the way for more adaptive learning in imbalanced data scenarios across industries like e-commerce and healthcare.

Since Isaac Newton first studied the Kissing Number Problem in 1694, determining the maximal number of non-overlapping spheres around a central sphere has remained a defining challenge in discrete geometry. As the local analogue of Hilbert's 18th problem, it has profound implications across geometry, number theory and information theory. Although lattices and codes have achieved significant progress, the field is confined to isolated extremal configurations, leaving underlying geometric principles obscured. Here we shift the object to the broader extremal configuration space, thereby opening a new path for the Kissing Number Problem. Accordingly, we recast this problem as a cooperative matrix-completion game, and train a reinforcement learning system, PackingStar, to solve it. One player fills cosine entries while the other corrects suboptimal ones, making explosive geometric complexity tractable. Working within extremal configuration spaces, PackingStar discovers new interpretable geometric structures that improve 15 strong bounds held for decades in kissing numbers and their generalizations, several of them provably optimal under natural inner products. These findings reveal the first explicit spherical-code realization of the Fischer group Fi22, extend the classical Euclidean representation of subgroup structure, and directly inspire subsequent breakthroughs by mathematicians. Overall, the work provides an early example of AI-driven progress on a Hilbert-calibre problem, showing how reinforcement learning advances mathematical discovery by unlocking more expressive objects.

We propose and study distributionally robust Markov games (DR-MGs) with the average-reward criterion as a crucial framework for multi-agent decision-making under model mismatches and over extended horizons. Under a standard irreducible assumption, we first derive a correspondence between the optimal policies and the solutions of the robust Bellman equation, based on which we further show the existence of a stationary Nash Equilibrium (NE) of the game. We further study DR-MGs under a more general weakly communicating setting. We construct a set-valued map based on the constant-gain optimal robust Bellman operator and show that its value is a subset of the best-response policies. We further prove that this map admits a fixed point, which implies the existence of NE. We then design two algorithms, Robust Nash-Iteration and robust TD Descent, with provably convergent guarantees. Finally, we show that the NE under average-reward can be approximated by the ones for the discounted DR-MGs as the discount factor approaches one. Our studies provide a comprehensive theoretical and algorithmic foundation for decision-making in complex, uncertain, and long-running multi-player environments.

Multi-agent learning is intrinsically harder, more unstable and unpredictable than single agent optimization. For this reason, numerous specialized heuristics and techniques have been designed towards the goal of achieving convergence to equilibria in self-play. One such celebrated approach is the use of dynamically adaptive learning rates. Although such techniques are known to allow for improved convergence guarantees in small games, it has been much harder to analyze them in more relevant settings with large populations of agents. These settings are particularly hard as recent work has established that learning with fixed rates will become chaotic given large enough populations.In this work, we show that chaos persists in large population congestion games despite using adaptive learning rates even for the ubiquitous Multiplicative Weight Updates algorithm, even in the presence of only two strategies. At a technical level, due to the non-autonomous nature of the system, our approach goes beyond conventional period-three techniques Li-Yorke by studying fundamental properties of the dynamics including invariant sets, volume expansion and turbulent sets. We complement our theoretical insights with experiments showcasing that slight variations to system parameters lead to a wide variety of unpredictable behaviors.

Many real-world domains--including e-commerce platform design, security planning, and multi-agent coordination--feature leader-follower problems where one decision-maker commits to a policy and others react strategically. We develop a reinforcement learning framework for such interactions in sequential environments with partial observations and multiple followers. Followers may adapt through no-regret learning or reinforcement learning, potentially departing from equilibrium behavior. The framework embeds follower adaptation into the leader's environment to construct a single-agent partially observable Markov decision process--the Stackelberg POMDP. For policy-interactive response algorithms, which access the leader's policy through queries, we prove that an optimal policy based only on the leader's game history yields an optimal commitment under the specified response procedure. We use proximal policy optimization with a centralized critic and train contextual meta-followers to respond across leader policies. In indirect mechanism design, mechanisms using buyer messages achieve higher social welfare than optimal standard sequential price mechanisms across all tested type counts, with responses certified as approximate Bayesian coarse correlated equilibria. In platform design, learned display rules increase mean consumer surplus by 8.4% over an optimized fixed price cap while accommodating hidden seller costs. In Atari bilateral trade, meta-learned follower responses support joint learning of visual gameplay and economic decisions; assigning leadership to the seller or buyer shifts transaction prices and payoffs in that agent's favor. Controlled ablations examine how response credit, policy consistency, and reward timing affect learning.