Commitment to Sparse Strategies in Two-Player Games

TL;DR

This work introduces and validates the notion of $k$-sparse commitments in two-player games, where one player is restricted to strategies with support at most $k$. It systematically analyzes the limitations of naive sparsification and demonstrates that optimal sparse supports can be disjoint, with non-submodular value structures, motivating a robust MILP-based framework. The authors develop scalable approaches for zero-sum, general-sum Stackelberg, and structured-sparsity scenarios, including single-oracle and MILP-representable-space methods, and extend to large action spaces via combined strategies. Empirically, the method achieves around 90% of the unrestricted Nash value with small $k$ across random and security-inspired domains, often outperforming $k$-uniform strategies and remaining tractable in sizable problems. This yields practical, interpretable near-optimal strategies for security applications such as patrolling and air-defense placement, with clear extensions to broader game classes and online learning.

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.

Figures (19)

• Figure 1: Average normalized runtime and relative utility for solving random zero-sum (left) and general-sum (right) games.
• Figure 2: Average (red) normalized runtime and relative utility over 30 instances (gray) of a PE on a university campus. We impose vanilla sparsity on path distribution (left) and structured sparsity on the starting point distribution of Player 1 (right).
• Figure 3: Average normalized relative utility for solving randomly generated zero-sum and general-sum games using optimal $k$-sparse (red) and $k$-uniform (blue) commitments. Top: zero-sum games. Bottom: general-sum games.
• Figure 4: Average normalized runtime (left) and relative utility (right) for solving a pursuit-evasion game on a university campus, imposing structured sparsity on the distribution over starting points ($k_1$) and paths ($k_2$) of Player 1. The smallest values of $k_1$ and $k_2$ at which the Nash value is attained is marked by "$\times$".
• Figure 5: Average normalized relative utility curves for optimal $k$-sparse (red) and $k$-uniform (blue) commitments. Results for individual instances using optimal $k$-sparse commitments are in gray. Top row: sparse player has a large strategy space. Middle row: non-sparse player has a large strategy space. Bottom row: both players have large strategy spaces.

Theorems & Definitions (10)

• Remark 1
• Proposition 1
• Proposition 2: non-diminishing marginal returns
• Proposition 3: non-submodularity
• Proposition 4
• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof