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

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.