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.
Online decision making often requires navigating a landscape shaped by both dynamic contexts and strategic interactions. In competitive pricing, for example, hotels must account for both dynamic contextual factors and rivals' strategic responses. Existing approaches address only part of this challenge: contextual bandits optimize single-agent decisions using observable features but ignore multi-player interactions, while online matrix games capture strategic behavior through Nash equilibrium but assume fixed payoffs, ignoring contextual information. How should agents act then when strategic payoffs evolve with contextual signals? We introduce \emph{online contextual matrix games} to integrate contextual information into multi-player online games. We further propose \emph{OnGameLearn}, an online learning algorithm that efficiently balances exploration and exploitation across both player actions and contexts. This approach comes with statistical guarantees: tail bounds for the estimated payoff matrix, the convergence of the estimated Nash equilibrium, the asymptotic normality of the parameter estimators, and the sublinear regret bound. We also develop the notion of \emph{policy value} in matrix games and develop a doubly robust, $\sqrt{T}$-consistent estimator for it. Across simulated studies and a real-world hotel pricing application, we find that OnGameLearn effectively navigates the intertwined challenges of strategic and contextual decision-making.
Multi-agent reinforcement learning (MARL) is a powerful framework for solving complex collaborative tasks, but it relies heavily on well-defined global reward functions. Designing such rewards is challenging, especially in systems with heterogeneous agents, where a single scalar objective may fail to capture diverse behaviors. In this paper, we introduce Multi-AGent Preference-Integrated lEarning (MAGPIE), which addresses these challenges through agent-specific preference modeling. Each agent is evaluated by a dedicated expert through preference signals, eliminating the need for global evaluation. We theoretically prove that optimizing these decentralized preferences converges to a Nash equilibrium policy. To integrate local preferences into a coherent global objective, we construct agent-specific reward models from preference data and combine them via a monotonic aggregation mechanism. We further prove that optimizing this aggregate reward model is equivalent to training the Nash equilibrium policy. Extensive experiments on benchmark multi-agent tasks and a sequential production line task show that MAGPIE achieves performance comparable to reward-engineered baselines, demonstrating its potential to facilitate policy learning in scenarios where precise reward engineering is impractical.
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.
In two-player zero-sum games whose Nash equilibria form a convex set, regularized solvers such as Regularized Nash Dynamics (R-NaD) empirically select the maximum-entropy member: the information projection (I-projection) of a uniform reference onto the Nash set. On a panel of small games this match is exact, with one apparent exception: in Kuhn poker R-NaD lands at bluff coordinate 0.180 while the maximum-entropy member sits at 0.201, a coordinate gap of about 0.021, even though R-NaD attains 99.7 percent of the maximum entropy. We ask whether this gap is a genuine selection bias or an artifact, and answer it quantitatively. We show that for selection on a one-dimensional Nash manifold the coordinate gap factorizes as $\mathrm{gap} \approx \sqrt{2δ/κ}$, where $δ$ is the entropy shortfall of the solver and $κ$ is the curvature of the entropy landscape at its peak. Across five games this relation holds to within $2 \times 10^{-4}$ (under 1 percent relative error). The four matrix games have $δ\approx 0$ (R-NaD reaches the maximum-entropy member exactly) and therefore no gap regardless of curvature; only the sequential game (Kuhn) has $δ> 0$. A causal sweep of the magnet strength drives $δ\to 0$ and the gap toward zero along the predicted curve (fitted scaling exponent 0.50, $R^2 > 0.999999$, against the exact prediction of 1/2), until the dynamics destabilize at a stability floor: behavior consistent with a removable shortfall and inconsistent with a fixed bias. We quantify the curvature half of the law from measured curvatures and flag a moving-target pitfall in the natural Tsallis-entropy experiment. The Kuhn gap is thus the curvature shadow of a small, removable entropy shortfall on an unusually flat peak; the I-projection account is upheld up to a flatness-limited residual.
Many important games have more than two players and imperfect information. Existing approaches for computing Nash equilibrium, the central game-theoretic solution concept, in such games either lack scalability or obtain poor performance. In this paper we introduce a new algorithm called projected exploitability descent (PED) for approximating Nash equilibria in multiplayer games of imperfect information. The algorithm works by running projected subgradient descent minimizing a proxy for the multiplayer generalized exploitability function. The objective is nonconvex and nonsmooth, but can be represented as the sum of the maxima of linear functions, for which a subgradient can easily be computed and projected to the polytope of feasible sequence-form strategies. We explore performance of PED on a generalized version of the well-studied benchmark game three-player Kuhn poker. No prior exact algorithms scale to the version of the game with deck size larger than 4, and we compare performance to the popular algorithms of fictitious play (FP) and counterfactual regret minimization (CFR). We find that PED obtains a consistent near-monotonic improvement throughout all runs, though both FP and CFR perform significantly better in the initial iterations. This inspires a hybrid algorithm FP-PED that runs FP for an initial burn-in period before switching to PED for stable long-run refinement. We can alternatively view this as a multi-step algorithm that runs FP as a pre-processing step to obtain a strong initialization for PED.
Many two-player zero-sum games admit not a unique Nash equilibrium but a convex set of them: a polytope of profiles that all share the minimax value V* yet prescribe different behaviour. Standard solvers each converge to some equilibrium and are treated as interchangeable. We ask whether they instead select different members of the Nash set, systematically as a function of the algorithm rather than the seed. Using a tabular, exactly solvable testbed of six games with analytically known Nash sets -- including a two-dimensional Nash polytope and Kuhn poker -- we find that (i) selection is determined by the algorithm, not the seed, but families differ only on asymmetric Nash sets; (ii) regularized last-iterate methods (R-NaD, magnetic mirror descent) select the maximum-entropy member, the information projection of their uniform reference onto the Nash set -- exactly on the 2-D polytope and at 99.7% of maximum entropy in Kuhn -- while regret-averaging methods (CFR, CFR+, fictitious play) drift to a lower-entropy face; we confirm this on a randomized 180-game ensemble, where R-NaD attains the maximum-entropy member in 100% of converged games while CFR+ sits strictly below it in 94% (paired Wilcoxon p < 10^-27); (iii) the selected member has downstream consequences against sub-optimal opponents that scale with sequential/hidden-information structure but stay bounded -- in Kuhn the max-entropy member is a strictly better hedge, whereas on the matrix games the members differ without either dominating. We also report two negative results correcting common intuitions: removing CFR's positive-orthant (max(R,0)) projection does not eliminate boundary drift; and R-NaD's selection is anchor-following, not initialization-independent. We state the maximum-entropy / I-projection characterization as a strongly data-supported conjecture, checked throughout against analytic ground truth.