Epsilon-Nash Equilibria in History-Dependent SA-MDPs

Abstract

We study state-adversarial Markov decision processes (SA-MDPs) as games of observation-space attacks: at each step, an agent selects an action from a received observation while an adversary$\unicode{x2014}$who knows the true state the agent is in$\unicode{x2014}$chooses a perturbed observation within a state-dependent proximity set. While existing work focuses on Markovian policies, we develop a solution concept and computational approach for SA-MDPs under history dependence. History dependence can materially change equilibrium outcomes and can force both the agent and the adversary to adapt their strategies. First, we prove the non-existence of universal (agnostic of the initial state distribution) history-dependent equilibrium policies. Our main result presents the first algorithmic route to computing $\epsilon$-approximations of initial-state dependent equilibria. We do so by reducing SA-MDPs to a strategically equivalent constrained zero-sum one-sided partially observable stochastic game. We test our algorithm on small analytically verifiable games and show that it scales to larger, more realistic benchmarks, including Atari Freeway rollouts with a 12-period ahead horizon.