Robust PAC Learning of Concurrent Stochastic Games

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.

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.