Bounded-Memory Strategies in Partial-Information Games

TL;DR

This paper investigates the computational complexity of finding bounded-memory strategies in partial-information stochastic games, providing complexity bounds and algorithms for approximating equilibria and game values.

Contribution

It establishes NP-hardness results and offers polynomial space algorithms for deciding the existence of bounded-memory equilibria in multi-player games.

Findings

NP-hardness for approximating zero-sum values with memoryless strategies

Polynomial space decision procedure for bounded-memory $oldsymbol{ ext{$orall$}k}$-player games

Algorithms for finding $oldsymbol{ ext{$orall$}k}$-player and 2-player bounded-memory $oldsymbol{ ext{$orall$}b}$-strategy equilibria

Abstract

We study the computational complexity of solving stochastic games with mean-payoff objectives. Instead of identifying special classes in which simple strategies are sufficient to play $\epsilon$-optimally, or form $\epsilon$-Nash equilibria, we consider general partial-information multiplayer games and ask what can be achieved with (and against) finite-memory strategies up to a {given} bound on the memory. We show $NP$-hardness for approximating zero-sum values, already with respect to memoryless strategies and for 1-player reachability games. On the other hand, we provide upper bounds for solving games of any fixed number of players $k$. We show that one can decide in polynomial space if, for a given $k$-player game, $\epsilon\ge 0$ and bound $b$, there exists an $\epsilon$-Nash equilibrium in which all strategies use at most $b$ memory modes. For given $\epsilon>0$, finding an $\epsilon$-Nash equilibrium with respect to $b$-bounded strategies can be done in $FN[NP]$. Similarly for 2-player zero-sum games, finding a $b$-bounded strategy that, against all $b$-bounded opponent strategies, guarantees an outcome within $\epsilon$ of a given value, can be done in $FNP[NP]$. Our constructions apply to parity objectives with minimal simplifications. Our results improve the status quo in several well-known special cases of games. In particular, for $2$-player zero-sum concurrent mean-payoff games, one can approximate ordinary zero-sum values (without restricting admissible strategies) in $FNP[NP]$.