Equilibrium Computation and Robust Optimization in Zero Sum Games with Submodular Structure

TL;DR

This paper introduces a pseudopolynomial-time algorithm for computing approximate equilibria in zero-sum games with submodular structure, enabling scalable solutions for complex security and optimization problems.

Contribution

It presents the first scalable algorithm with guaranteed approximation for equilibrium computation in large zero-sum submodular games.

Findings

Algorithm achieves a (1 - 1/e)^2 approximation ratio.

Experimental results show scalability to larger instances.

Method outperforms previous exponential-time algorithms.

Abstract

We define a class of zero-sum games with combinatorial structure, where the best response problem of one player is to maximize a submodular function. For example, this class includes security games played on networks, as well as the problem of robustly optimizing a submodular function over the worst case from a set of scenarios. The challenge in computing equilibria is that both players' strategy spaces can be exponentially large. Accordingly, previous algorithms have worst-case exponential runtime and indeed fail to scale up on practical instances. We provide a pseudopolynomial-time algorithm which obtains a guaranteed $(1 - 1/e)^2$-approximate mixed strategy for the maximizing player. Our algorithm only requires access to a weakened version of a best response oracle for the minimizing player which runs in polynomial time. Experimental results for network security games and a robust budget allocation problem confirm that our algorithm delivers near-optimal solutions and scales to much larger instances than was previously possible.