Zero--Sum Differential Game

In multi-agent environments, coordinating agents to prevent interference and ensure robust individual performance is a critical challenge. Previous research on social laws for multi-agent systems has primarily focused on deterministic, goal-based settings. This paper extends the concept of social laws to stochastic, reward-based environments, proposing a formalism for defining and verifying their robustness under various conditions. We introduce the notion of $α$-robustness, a measure of the guaranteed utility each agent retains while pursuing its optimal single agent policy, assuming all agents obey the social law. We then present an approach for robustness verification of social laws in stochastic settings, based on a reduction to solving a series of Markov decision processes. Empirical evaluations on toy environments illustrate the potential of our framework.

We study information design in games where players choose from a continuum of actions and have continuously differentiable payoffs. We show that an information structure is optimal when the equilibrium it induces can also be implemented in a principal-agent contracting problem. Building on this result, we characterize optimal information structures in symmetric linear-quadratic games. With common values, targeted disclosure is robustly optimal across all priors. With interdependent and normally distributed values, linear disclosure is uniquely optimal. We illustrate our findings with applications in venture capital, Bayesian polarization, and price competition.

We study decentralized learning of Nash equilibria (NE) in infinite-horizon discounted Markov games under bandit feedback, focusing on Markov $α$-potential games. We develop KL-projected natural policy gradient (NPG) algorithms in two settings: an episodic setting with frozen policies during sampling and a fully online setting in which players receive a single realized cost sample per time step and update their policies asynchronously along a continuing trajectory. We establish finite-time high-probability NE regret bounds of order $\widetilde O(T^{-1/4})$ and $\widetilde O(T^{-2/15})$ for the episodic and fully online settings, respectively, up to fixed approximation terms. Crucially, our bounds eliminate the distribution-mismatch coefficient, which can scale prohibitively with the size of the state space, while accommodating potential approximation, estimation-oracle bias, and transition sensitivity. We further identify a state-wise potential structure that yields sharper guarantees with additive dependence on the potential approximation error $α$. We specialize the framework to independent-resource Markov congestion games (IMCGs), establish their approximate-potential and transition-sensitivity properties, and construct decentralized estimation oracles from realized costs. As an application, we introduce strategic online job scheduling on stochastic machines and obtain a scalable decentralized algorithm for learning stable dispatching policies. Overall, our results provide the first finite-time high-probability NE regret guarantees for fully online asynchronous decentralized learning in Markov $α$-potential games, remove distribution-mismatch coefficients from the regret bounds, accommodate fixed estimation-oracle bias, and provide scalable decentralized learning with finite-time guarantees for IMCGs.

We introduce the first Probably Approximately Correct (PAC) learning framework for general-sum concurrent stochastic games (CSGs) with transition uncertainty, while addressing the challenge of Nash equilibrium (NE) existence. Our algorithm maintains data-driven $L^1$ confidence sets over transition kernels and solves a robust CSG to compute a social-welfare optimal $\varepsilon$-NE, using a robust MDP-based exploration mechanism to drive joint state-action coverage. Crucially, we introduce a Nash margin characterisation that enables principled reasoning about equilibrium existence: the framework either returns an $\varepsilon$-approximate NE whose social-welfare value is $\varepsilon$-close to optimal, or provides a sound certificate that no exact NE exists. Under a minimum reachability condition $p_{\mathrm{reach}}>0$ over relevant state-action pairs, the algorithm terminates after a polynomial number of trajectory samples, with sample complexity $\widetilde{O}\left( {R_{\max}^2 H^4 |S|^2 |A| / (p_{\mathrm{reach}} \varepsilon^2)} \right)$. Empirical results on benchmark CSGs demonstrate near-optimal performance, correct handling of equilibrium (non-)existence, and sample complexity consistent with theory.

We introduce a novel Divide and Conquer control design methodology leveraging differential games in single-agent, multi-objective dynamical systems. The proposed framework associates each control objective with a virtual input and establishes a non-cooperative, finite or infinite horizon differential game among representative players. Each player optimizes a distinct virtual cost function tailored to its specific goal, the full system state, and the other virtual inputs, while accounting for the remaining players' optimal policies. By establishing a Nash Equilibrium for this game, we synthesize a composite controller that achieves a stable balance across competing objectives, providing control engineers with an intuitive and modular framework for parameter re-tuning throughout the design cycle. We provide formal mathematical derivations for both continuous-time and discrete-time dynamical systems, targeting large-scale single-agent applications where complex, dynamically conflicting control objectives make global weighting intractable. To demonstrate the methodology, we developed an open-source Python package implementing a novel numerical algorithm for solving Coupled Algebraic Riccati Equations arising in infinite-horizon differential games. We evaluate the approach on two benchmark case studies: an inverted pendulum on a cart and a non-linear hierarchically controlled quadrotor. The resulting closed-loop performance is compared against the classical Linear Quadratic Regulator (LQR) across various transient and steady-state control metrics, demonstrating superior trajectory tracking and robust multi-objective regulation.

We present a robust safety-filtering framework for input-constrained underactuated linear systems subject to unknown disturbances. A baseline H-$\infty$ input is derived from a zero-sum differential game, while a disturbance observer supplies an estimate and a transient error bound. The baseline input is adjusted using the disturbance estimate, while the estimate and its error bound are used to define robust high-order control barrier function constraints; forward invariance holds as long as the admissible-input set remains nonempty. For scalar-input systems, pointwise feasibility is determined from an exact input interval, and the interval width defines the feasibility margin. A finite-horizon H-$\infty$ performance balance accounts for the accumulated deviation of the applied input from the baseline H-$\infty$ policy. Simulations on a linearized two-wheeled balancing robot show how position and body-pitch constraints compete for the same bounded wheel-torque input.

As firms increasingly deploy machine learning for strategic decision-making, understanding algorithmic interactions has become central to operations research and economics. This paper studies learning in infinite-horizon, nonzero-sum linear-quadratic stochastic games under a radically uncoupled information structure, where players are either unaware of opponents or strategically oblivious, observing only a common state and their own action history. Under this minimal information, we analyze an asynchronous decentralized learning process in which each player independently runs a single-agent $ε$-greedy iterated least-squares algorithm. We prove that, despite being unable to identify the system parameters, players' learning dynamics converge almost surely to the complete-information Nash equilibrium and characterize the convergence rate. We then apply the framework to a dynamic Cournot competition with sticky prices. Numerical experiments validate the theoretical results and show that learning under limited information reduces firm profits under both low and high price stickiness, while total surplus declines and market concentration increases when price stickiness is high. Publicly revealing aggregate market output substantially accelerates convergence and mitigates these welfare losses.

This paper studies feedback Nash equilibrium (FBNE) seeking for multi-agent trajectory planning in nonlinear dynamical systems with unknown agents' objectives and state-dependent inter-agent coupling. While dynamic game theory provides a principled framework for such problems, existing approaches typically assume fully rational agents with known objectives or rely on fixed regularization, limiting their ability to capture bounded rationality and spatially varying interaction intensity in safety-critical settings. To this end, we propose a KL-regularized dynamic game with a state-dependent weight that adaptively balances optimality and behavioral priors. To infer unknown cost parameters from demonstrated behaviors, we develop a context-aware inverse game module based on maximum-entropy inverse reinforcement learning with physics-informed regularization, ensuring structural consistency with the forward game. We establish per-iteration well-posedness of the regularized local game and show that the adaptive weighting function remains Lipschitz continuous under bounded nominal-trajectory updates. Numerical simulations and multi-robot experiments on cooperative navigation and merging scenarios validate the effectiveness of the proposed framework.

Concurrent games are a standard model for multi-agent systems, with Nash equilibrium as their central solution concept. The associated \emph{constrained existence problem}---does a game admit a Nash equilibrium whose expected payoff lies within a prescribed interval for every player?---is undecidable, and remains so even for 10-player \emph{stopping} games, in which a terminal state is reached almost surely under every strategy profile. We give two routes to tractability. We first relax exactness and consider the problem of approximate constrained existence problem, parametrised by $\varepsilon$-NE, which decides whether an $\varepsilon$-Nash equilibrium with the prescribed payoffs exists. The algorithm runs in exponential time, and only polynomially in the bit-size of $\varepsilon$. We complement it with a \PSPACE-hardness lower bound that holds already for turn-based games, and for pure equilibria as well. We then relax the solution concept, turning to \emph{extreme risk-sensitive equilibria} (XRSE), recently introduced for turn-based stochastic games. Here the players are partitioned into optimists and pessimists, who evaluate a strategy profile by the best, respectively the worst, payoff attainable with positive probability, instead of the expected payoff. We prove that the constrained existence problem for XRSE is \NP-complete on concurrent games, as for turn-based games.

Reachability is the most fundamental logical objective, yet it is notoriously difficult to learn in reinforcement learning settings: even for Markov decision processes, PAC learning of reachability is impossible without additional assumptions. This difficulty also holds in turn-based stochastic games (TBSGs), where two adversarial players interact on a finite state space. In this work, we consider turn-based stochastic games with reachability objectives. For such settings, adversarial learning, in which players are adversarial even in the learning phase, is impossible. Therefore, the goal is to consider learning, in which both players learn the unknown model together. In this spirit, previous literature on PAC learning in TBSGs considers (a)~public information shared by both players; and (b)~centralized learning, which means that players share the same learning algorithm. In this work, our contribution is two-fold. First, we relax these strong assumptions and ensure learning: (i)~with private information not shared with the other player; and (ii)~decentralized learning where the players do not share the same learning algorithm. To the best of our knowledge, this work is the first positive result for decentralized and private information learning of TBSGs with reachability objectives. Second, we introduce a game-theoretic generalization of the Expected Conditional Distance (ECD) parameter, which measures the expected length of reaching the target set. We establish a polynomial-sample complexity bound with respect to the number of states, actions, ECD parameter, and inverses of error tolerance and failure probability.

Exploration is essential for reliable autonomy in multi-agent systems, yet it remains unclear whether large language model (LLM) agents can explore effectively when interacting with one another. We show that modern LLM agents fail to do so, often exhibiting myopic and polarized interaction patterns that lead to suboptimal coordination and increased regret. We formalize this challenge as the Multi-Agent Exploration problem, modeling it as a partially observable stochastic game (POSG) problem in which agents must probe peers to infer their capabilities and identify effective interaction strategies. To address this, we introduce Multi- Agent Contextual Exploration (MACE), a lightweight framework that explicitly promotes exploration through structured peer selection. Across both contextual and parametric diversity settings, MACE substantially improves exploration behavior and downstream task performance. We further show theoretically that the value of exploration increases with agent diversity. Overall, our results highlight a fundamental limitation of current LLM agents and underscore the importance of explicitly guided exploration for reliable multi-agent autonomy. Code will be released in https://github.com/deeplearning-wisc/mace

Stackelberg differential games (SDGs) provide a powerful framework for hierarchical decision-making in stochastic and continuous-time environments, yet their solution remains computationally challenging due to the complexity of traditional dynamic programming and Hamilton-Jacobi-Bellman-Isaacs (HJBI) methods, especially in high-dimensional systems. This paper proposes an entropy-regularized reinforcement learning (ERRL) approach for linear-quadratic SDGs (LQ-SDGs) within a continuous-time diffusion framework governed by Markovian regime switching. The key innovation lies in deriving exploratory weakly-coupled HJBI equations with entropy regularization, which promotes stochastic policies that actively avoid suboptimal equilibria -- a limitation of classical SDG methods. Neural networks are integrated to approximate regime-dependent value functions and solve high-dimensional partial differential equations (PDEs) efficiently, while a novel sampling technique enhances computational tractability. Numerical results demonstrate the effectiveness of the framework compared to conventional approaches, particularly in escaping suboptimal traps through exploratory policies. The study highlights the critical role of entropy regularization and neural network approximations in achieving robust solutions for hierarchical decision-making problems under abrupt environmental shifts.

To address parameter misspecification and sudden structural environmental changes in conventional stochastic differential game (SDG) frameworks, this paper introduces a distributional control approach that characterizes optimal strategies as probability distributions over actions, conditioned on the continuous state, the discrete regime state, and parameters. This forms a reinforcement learning framework for entropy-regularized zero-sum stochastic differential games (ERRL-ZSSDGs) in a regime-switching jump-diffusion process. Using the dynamic programming principle (DPP), we derive the associated coupled systems of Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, from which equilibrium strategies are expressed via gradients of the value function. For linear-quadratic problems, semi-analytical solutions for both value function and equilibrium strategies are obtained by solving a system of coupled ordinary differential equations (ODEs). In more general settings, an Actor-Critic policy improvement algorithm is developed to approximate the value functions and equilibrium policies across different regimes. The method is applied to an investment game, and numerical examples illustrate the effect of the temperature parameter and regime transitions on optimal policies and values.

Solving multi-agent optimal control problems in aerospace such as pursuit-evasion and contested space operations can be modeled as non-convex differential games for which, there are limited algorithms. In this work, a relaxation of generalized Nash Equilibrium problems (GNEPs) to exclude inter-agent control coupling in dynamics, which is representative of many multi-agent systems is introduced. The main contribution is an algorithm for solving a broad class of differential games named FALCON: Fast Augmented Lagrangian Convexification for Open-loop Nash equilibria is presented. Methodologically, sequential convex programming (SCP) is utilized to create tractable convex sub-games which can then be solved via standard convex programming methods involving a potential game reformulation. FALCON is demonstrated to have global convergence guarantees to an open-loop Nash equilibrium for non-convex differential games under mild assumptions. This is numerically shown through both cooperative and competitive differential games.

Mean field games efficiently approximate a very large population of strategic agents. While these games can aid the understanding of complex systems, their deployment in real-world settings is challenged by the specification of their parameters: mean field games (MFGs) often involve hidden preferences, constraints, and interactions that can rarely be theoretically derived or directly observed. To address this gap, we present a neural network-based framework for learning parametric, finite-state MFGs from observed population dynamics. To do so, we formulate the parameter calibration as an inverse problem and use implicit differentiation to backpropagate through the games' equilibrium. The resulting approach is fully differentiable and enables us to estimate flexible trajectory-wise parameter paths, including state- and time-dependent specifications without requiring observations of the individual agents' actions or rewards. We provide a proof for the exactness of the gradient computation in a discrete-time formulation. We validate our framework through numerical experiments across four systems of increasing complexity, ranging from synthetic linear-quadratic benchmarks to real-world urban mobility datasets.

Deploying multi-agent reinforcement learning (MARL) in the real world is often limited by model mismatches between the training simulators and the true environment, which could be further amplified through strategic interactions and result in severe performance degradation upon deployment. Distributional robustness offers a principled response by optimizing policies against worst-case transition models drawn from an uncertainty set, but standard robust MARL frameworks become increasingly intractable as the number of agents grows. This paper develops an infinite-horizon, stationary mean-field game framework that incorporates distributional model uncertainty directly into the population-coupled dynamics. We establish a robust dynamic programming principle with a contractive Bellman operator and prove the existence of a stationary robust mean-field equilibrium via a fixed-point argument. We further develop the first concrete algorithm with convergence guarantees. We then connect the mean-field solution to a finite-population robust game whose ambiguity sets depend on the empirical distribution, showing that the mean-field equilibrium policy induces approximate equilibrium behavior as the population size increases. Under a contractive robust-dynamics regime, we further obtain explicit non-asymptotic error bounds. Numerical experiments further illustrate the qualitative and quantitative impact of robustness under multiple uncertainty models, validating our theoretical findings.

This paper investigates mobile target search under imperfect perceptions caused by sensor limitations, malicious jamming, or communication noise. Searchers and targets operate in a grid-shaped area with bounded mobility, leading to a dynamic interplay between search and evasion. To capture this adversarial interaction under imperfect perceptions, we adopt the partially observable stochastic game (POSG) approach, which generalizes partially observable Markov decision processes (POMDPs) by incorporating target intelligence. To handle false alarms and missed detections caused by perceptual uncertainties, we propose a novel detectability concept to determine whether a search strategy guarantees eventual detection, and provide sufficient detectability criteria based on stochastic recurrence analysis. We further develop a server-assisted distributed algorithm that utilizes the aggregative potential game structure for searchers and a KL-divergence-based reduction for target prediction. Numerical simulations validate the effectiveness of the proposed algorithm and support the detectability analysis.

Uncrewed Aerial Systems (UASs) have become a growing threat to the security of critical infrastructure, exploiting spatiotemporal gaps in sensor perimeters to infiltrate restricted airspace undetected. We formulate this interaction as a two-player zero-sum differential game between an adversarial UAS and a heterogeneous sensor network of directional and omnidirectional sensors. Unlike earlier game-theoretic approaches that restrict the defender to discrete placement graphs or fixed configurations, we introduce a continuous sensor redeployment technique in which each sensor slides freely along the convex building boundaries. This is enforced via a log-sum-exp smooth approximation that preserves differentiability at polygon vertices, enabling optimization with gradient-based methods. The attacker's best response is computed via a two-step approach combining STP-RRT* for feasible trajectory initialization and nonlinear programming for detection-minimization refinement. The joint optimization converges to a Local Nash Equilibrium (LNE) via alternating bilevel optimization, with analytical first-order stationarity conditions derived for both players, thereby establishing a deployable baseline for heterogeneous sensor placements in CUAS missions.

This article presents a Hamilton--Jacobi (HJ) reachability framework for a two--satellite collision avoidance problem operating in the same circular orbit, where relative motion is modeled in the radial--tangential--normal (RTN) frame using planar Hill--Clohessy--Wiltshire (HCW) dynamics. We define the target state space as unsafe relative configurations in the orbit plane corresponding to minimum separation requirements consistent with Federal Communications Commission (FCC) orbital standards. The interaction between spacecraft is formulated as a zero--sum differential game, where Player 1 is the controlled satellite and Player 2 is modeled as a bounded adversarial disturbance with unknown intent. We present the HJ formulation and compute backward reachable sets that characterize relative states from which collision cannot be avoided under worst-case disturbances, while states outside this set admit provably collision-free trajectories. These reachable sets are integrated with supervisory hybrid control logic to determine when evasive maneuvers must be initiated, enabling mathematically grounded safety guarantees for scalability.

Responsibility allocation -- determining the extent to which agents are accountable for outcomes -- is a fundamental challenge in the design and analysis of multi-agent systems. In this work, we model such systems as concurrent stochastic multi-player games and introduce a notion of retrospective (backward) counterfactual responsibility, which quantifies an agent's accountability for outcomes resulting from a given strategy profile. To allocate responsibility among agents, we utilise the Shapley value and formally show that this method satisfies key desirable properties, including fairness and consistency. Building on this foundation, we propose a formal framework that supports both verification and strategic reasoning in responsibility-aware multi-agent systems. Furthermore, by adopting Nash equilibrium as the solution concept, we demonstrate how to compute stable strategy profiles in which agents trade off responsibility against expected reward.

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.

Reachability games are two-player games played on a graph, where the objective of $\texttt{REACH}$ player is to reach the target set whereas the objective of $\texttt{SAFE}$ player is to stay away from the target set. Reachability games have important applications in artificial intelligence and reactive synthesis, and many of these applications give rise to infinite-state reachability games. In this paper, we study turn-based reachability games on infinite-state graphs defined over valuations of a finite set of real variables. We consider the problem of determining the existence of and computing a winning strategy for $\texttt{REACH}$ player. Our contributions are twofold. First, we propose ranking certificates for reachability games, a sound and complete proof rule for proving that $\texttt{REACH}$ player has a winning strategy from the specified initial state. Second, we consider polynomial reachability games, where transitions and objectives are described by polynomial constraints over real variables, and propose a fully automated algorithm for computing a winning strategy for $\texttt{REACH}$ player together with a formal correctness witness in the form of a ranking certificate. The algorithm is sound, semi-complete, and runs in sub-exponential time. Our experiments demonstrate the ability of our method to solve challenging examples from the literature that were out of the reach of existing methods. Specifically, for the classical Cinderella-Stepmother game, we are able to compute an optimal winning strategy for an arbitrary precision parameter for the first time.

Current research applying N-level Stackelberg Game to multi-agent systems often uses the default decision order of agents provided by the environment. However, this raises the question: does the order of agents necessarily affect the final equilibrium point of the game? To address this, we formally analyze the N-level Stackelberg Game, where changing the order in which agents make decisions typically leads to an overdetermined system. As a result, the equilibrium point shifts unless special structural conditions are satisfied. Based on this analysis, we propose the Hierarchical Priority Adjustment (HPA) method, which adjusts and selects the agents' decision order. At the upper level, an upper policy dynamically selects the optimal decision order of agents based on the current game state. At the lower level, agents execute strategies in the Spatio-Temporal Sequential Markov Game (STMG) according to the selected order. To coordinate learning across time scales, we employ a slow-fast update scheme with shared intrinsic rewards derived from the advantage function of the upper policy. Experimental results on high-precision control tasks, including multi-agent MuJoCo, show that HPA outperforms benchmark algorithms and robustly adapts to changing environments. These results highlight the crucial role of optimizing the agents' decision order in N-level Stackelberg Game.

We cast the design of parameterized quantum circuits as a four-player potential game whose state is a circuit directed acyclic graph (DAG) and whose players encode trainability, non-stabilizerness, task performance, and hardware cost. Per-player restricted action sets factorize the move space into append, remove, retype, and rewire operations; a block-coordinate $\varepsilon$-Nash residual $δ_\text{Nash}$ certifies that no single player can improve unilaterally. A single weight sweep on MaxCut $K_4$ traces a Pareto frontier from a Clifford endpoint $(M_2/n,\langle H\rangle)=(0,4.00)$ to a non-Clifford endpoint $(0.48,3.30)$. On three four-qubit hardware topologies (heavy-hex, $2\times 2$ grid, Rydberg all-to-all), Nash search achieves the highest mean potential; on the $2\times 2$ grid Nash reaches the theoretical ceiling $Φ_\text{max}=4.10$ on two of five seeds while the simulated-annealing baseline does so on one; paired Wilcoxon tests over five seeds cannot reject the null on any single topology ($p\ge 0.22$). On LiH/STO-3G, seeding Nash from a 58-gate Givens-doubles ansatz produces a 48-operation, depth-25 circuit retaining $97.7\%$ of the correlation energy while simultaneously reducing gate count, increasing non-stabilizerness, and controlling trainability. The framework is complementary to energy-only searches such as ADAPT-VQE and k-UpCCGSD, which reach chemical accuracy with fewer operations but do not optimize the other three axes.

Multi-agent games in dynamic nonlinear settings are challenging due to the time-varying interactions among the agents and the non-stationarity of the (potential) Nash equilibria. In this paper we consider model-free games, where agent transitions and costs are observed without knowledge of the transition and cost functions that generate them. We propose a novel distributed policy structure that follows the communication constraints in multi-team games, with multiple agents per team, and learned through policy gradients. Our formulation is inspired by the structure of distributed policies in linear quadratic games, which take the form of time-varying linear feedback gains. In the nonlinear case, we model the policies as nonlinear feedback gains, parameterized by self-attention layers to account for the time-varying multi-agent communication topology. We demonstrate that our approach achieves strong performance in several settings, including distributed linear and nonlinear regulation, and simulated and real multi-robot pursuit-and-evasion games.

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.

This paper addresses conservatism, limited numerical accuracy, and high computational effort in existing methods ensuring safety by design in online model-based motion generation. The presented method employs a novel captivity-escape zero-sum differential game to adapt the planning model's performance so that resulting reference trajectories are trackable within a prescribed safety margin by a jointly synthesized safety controller. A numerical example demonstrates orders-of-magnitude faster computation and improved numerical accuracy compared to the state of the art.

We present a finite-sample analysis of decentralized learning in two-player zero-sum matrix games and stochastic games, with a focus on best-response-based learning algorithms. In matrix games, the learning algorithm is payoff-based and symmetric: each player updates its policy using only its own payoff observations, incrementally moving toward an estimated smoothed best response to the opponent's latest policy. For stochastic games, we build on this matrix-game primitive to develop a learning algorithm called value iteration with smoothed best response (VI-SBR), which combines smoothed-best-response learning in induced matrix games with a decentralized, model-free approximation of minimax value iteration. We establish finite-sample guarantees in both settings. For matrix games, our results imply a sample complexity of $\mathcal{O}(ε^{-1})$ for finding an $ε$-Nash distribution and, with explicit exploration, $\tilde{\mathcal{O}}(ε^{-8})$ for finding an $ε$-Nash equilibrium. For stochastic games, we prove that the exploration-enhanced VI-SBR algorithm achieves a sample complexity of $\tilde{\mathcal{O}}(ε^{-8})$ for finding an $ε$-Nash equilibrium. Technically, our analysis develops a coupled Lyapunov-drift framework. This framework simultaneously handles stochastic iterative algorithms with multiple interacting stochastic iterates, the non-zero-sum auxiliary games generated by independently updated value functions, and the time-inhomogeneous Markovian noise induced by time-varying policies. The resulting tools may be useful more broadly for analyzing learning algorithms with coupled stochastic iterates and nonstationary sampling processes.