Computing Equilibria in Games with Stochastic Action Sets
It addresses the problem of computing equilibria in games where players face random action restrictions, a common scenario in practice, but the contribution is incremental as it extends existing no-regret learning to a new setting.
This paper introduces Games with Stochastic Action Sets (GSAS) where players' available actions are randomly restricted. For two-player zero-sum GSAS, they show Nash equilibria can be compactly represented and propose an algorithm (SI-MWU) that converges to equilibrium with high probability at a rate of O(√(log|A_i|/T)).
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 naïve 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.