Making the Best of Limited Memory in Multi-Player Discounted Sum Games

TL;DR

This paper proves the existence of optimal strategies with limited memory in multi-player discounted sum games, introduces a non-deterministic method to compute them, and explores how memory size affects optimality.

Contribution

It establishes the existence of bounded memory optimal strategies, introduces a novel non-deterministic computation approach, and analyzes the impact of memory size on strategy optimality.

Findings

Optimal bounded memory strategies exist in multi-player discounted sum games.

The decision problem for computing these strategies is NP-complete.

More memory can lead to better strategy profiles, and memory requirements cannot be inferred from game structure.

Abstract

In this paper, we establish the existence of optimal bounded memory strategy profiles in multi-player discounted sum games. We introduce a non-deterministic approach to compute optimal strategy profiles with bounded memory. Our approach can be used to obtain optimal rewards in a setting where a powerful player selects the strategies of all players for Nash and leader equilibria, where in leader equilibria the Nash condition is waived for the strategy of this powerful player. The resulting strategy profiles are optimal for this player among all strategy profiles that respect the given memory bound, and the related decision problem is NP-complete. We also provide simple examples, which show that having more memory will improve the optimal strategy profile, and that sufficient memory to obtain optimal strategy profiles cannot be inferred from the structure of the game.