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

2025-11-23

AI Summary

This work addresses the problem of constructing compact optimal strategy profiles—comprising a small set of representative strategies—that efficiently approximate the opponent’s strategy space in large two-player zero-sum games, avoiding domain-specific heuristics or methods lacking theoretical guarantees. We establish the first formal theoretical framework for this problem, prove its NP-hardness, and demonstrate that common heuristics—including uniform sampling and support-set expansion—can be severely suboptimal under specific game structures. Our method introduces an evaluation and analysis framework grounded in Nash equilibrium support verification and incremental construction, integrating game-theoretic analysis, computational complexity proofs, and empirical comparisons to derive tight theoretical bounds on strategy profile quality. To foster reproducibility and future research, we release open-source code and standardized benchmark datasets.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Adversarial Search
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Establishing formal foundation for portfolio-based strategy approximation in games
• Proving NP-hard complexity of finding optimal strategy portfolios
• Evaluating heuristic performance variability across different game types

Innovation

• Formal foundation for portfolio-based strategy approximation
• Analytical framework to bound portfolio quality
• Methodology for evaluating heuristic approaches