Computing Equilibria in Games with Stochastic Action Sets

TL;DR

This paper introduces a new framework for games with stochastic action sets, providing efficient algorithms for computing Nash equilibria in two-player zero-sum scenarios with compact representations.

Contribution

It models games with stochastic action restrictions, shows NE can be compactly represented, and develops an efficient algorithm with convergence guarantees.

Findings

NE in 2p0s-GSAS can be represented with size proportional to action set size

SI-MWU algorithm converges to NE with high probability

Procedure based on stochastic approximation recovers compact NE representations

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, players' available actions might be restricted due to exogenous stochasticity. 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, and 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 the na\"ive exponential-sized representation. Computationally, we introduce an efficient algorithm called SI-MWU that minimizes sleeping internal regret, converging to NE with high probability in 2p0s-GSAS with rate $O(\sqrt{\log\vert A_i\vert/T})$. Finally, using the SI-MWU iterates, we develop a procedure based on stochastic approximation to recover compactly represented NE.