Commitment to Sparse Strategies in Two-Player Games

TL;DR

This paper explores the concept of $k$-sparse commitments in two-player games, providing structural insights, developing MILP-based algorithms, and demonstrating their effectiveness in security scenarios with small support sizes.

Contribution

It introduces a MILP-based framework for computing $k$-sparse strategies, analyzes their structural properties, and offers practical algorithms for large action spaces.

Findings

Achieves over 90% of Nash value with small support sizes

Structural properties show optimal support varies with $k$

Algorithms are effective in security-related applications

Abstract

While Nash equilibria are guaranteed to exist, they may exhibit dense support, making them difficult to understand and execute in some applications. In this paper, we study $k$-sparse commitments in games where one player is restricted to mixed strategies with support size at most $k$. Finding $k$-sparse commitments is known to be computationally hard. We start by showing several structural properties of $k$-sparse solutions, including that the optimal support may vary dramatically as $k$ increases. These results suggest that naive greedy or double-oracle-based approaches are unlikely to yield practical algorithms. We then develop a simple approach based on mixed integer linear programs (MILPs) for zero-sum games, general-sum Stackelberg games, and various forms of structured sparsity. We also propose practical algorithms for cases where one or both players have large (i.e., practically innumerable) action sets, utilizing a combination of MILPs and incremental strategy generation. We evaluate our methods on synthetic and real-world scenarios based on security applications. In both settings, we observe that even for small support sizes, we can obtain more than $90\%$ of the true Nash value while maintaining a reasonable runtime, demonstrating the significance of our formulation and algorithms.