Computing Equilibria in Games with Stochastic Action Sets

2026-02-18

AI Summary

This work addresses the computational challenge of finding Nash equilibria in two-player zero-sum games with stochastic action sets (2p0s-GSAS), where each player’s available actions are constrained by external randomness, leading to exponential representation complexity in traditional approaches. Under the assumption that action availabilities are mutually independent, the paper proposes an efficient equilibrium learning algorithm based on sleep internal regret minimization. The method introduces a compact representation of the Nash equilibrium, reducing the equilibrium vector size from exponential to linear in the cardinality of the original action sets, thereby avoiding exponential blowup in game-dependent constants. The algorithm provably converges to an approximate Nash equilibrium at a rate of $O(\sqrt{\log|A_i|/T})$, significantly enhancing computational efficiency and scalability.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Computational social science

Abstract

The study of learning in games typically assumes that each player always has access to all of their actions. However, in many practical scenarios, arbitrary restrictions induced by exogenous stochasticity might be placed on a player's action set. To model this setting, for a game $\mathcal{G}_{\mathrm{orig}}$ with action set $A_i$ for each player $i$, we introduce the corresponding Game with Stochastic Action Sets (GSAS) which is parametrized by a probability distribution over the players' set of possible action subsets $\mathcal{S}_i \subseteq 2^{\vert A_i\vert}\backslash\{\varnothing\}$. In a GSAS, players' strategies and Nash equilibria (NE) admit prohibitively large representations, thus existing algorithms for NE computation scale poorly. Under the assumption that action availabilities are independent between players, we show that NE in two-player zero-sum (2p0s) GSAS can be compactly represented by a vector of size $\vert A_i\vert$, overcoming naive exponential sized representation of equilibria. Computationally, we introduce an efficient approach based on sleeping internal regret minimization and show that it converges to approximate NE in 2p0s-GSAS at a rate $O(\sqrt{\log\vert A_i\vert/T})$ with appropriate choice of stepsizes, avoiding the exponential blow-up of game-dependent constants.

Problem

• Stochastic Action Sets
• Nash Equilibrium
• Game Theory
• Zero-sum Games
• Equilibrium Computation

Innovation

• Stochastic Action Sets
• Nash Equilibrium
• Sleeping Regret Minimization
• Two-Player Zero-Sum Games
• Compact Representation