Understanding Optimal Portfolios of Strategies for Solving Two-player Zero-sum Games

TL;DR

This paper formalizes the problem of constructing optimal strategy portfolios in two-player zero-sum games, proves its NP-hardness, and evaluates heuristics, highlighting their game-dependent effectiveness.

Contribution

It establishes a theoretical foundation for portfolio optimization, proves NP-hardness, and provides an evaluation framework for heuristics in large-scale zero-sum games.

Findings

NP-hardness of portfolio optimization problem

Heuristics can be highly suboptimal

Heuristic success varies across different games

Abstract

In large-scale games, approximating the opponent's strategy space with a small portfolio of representative strategies is a common and powerful technique. However, the construction of these portfolios often relies on domain-specific knowledge or heuristics with no theoretical guarantees. This paper establishes a formal foundation for portfolio-based strategy approximation. We define the problem of finding an optimal portfolio in two-player zero-sum games and prove that this optimization problem is NP-hard. We demonstrate that several intuitive heuristics-such as using the support of a Nash Equilibrium or building portfolios incrementally - can lead to highly suboptimal solutions. These negative results underscore the problem's difficulty and motivate the need for robust, empirically-validated heuristics. To this end, we introduce an analytical framework to bound portfolio quality and propose a methodology for evaluating heuristic approaches. Our evaluation of several heuristics shows that their success heavily depends on the specific game being solved. Our code is publicly available.