An algorithmic solution to the Blotto game using multi-marginal couplings

TL;DR

This paper introduces an efficient algorithm for solving the general two-player Blotto game with heterogeneous values and budgets, leveraging Sinkhorn iterations for matrix and tensor scaling, and applies to complex asymmetric scenarios.

Contribution

The paper presents the first algorithmic solution for the most general Blotto game with value asymmetry and budget asymmetry, extending previous limited cases.

Findings

Algorithm samples from an ε-optimal solution in near-quadratic time.

Applicable to symmetric value cases with asymmetric budgets, a previously intractable scenario.

Samples from an ε-Nash equilibrium in asymmetric value cases with similar efficiency.

Abstract

We describe an efficient algorithm to compute solutions for the general two-player Blotto game on n battlefields with heterogeneous values. While explicit constructions for such solutions have been limited to specific, largely symmetric or homogeneous, setups, this algorithmic resolution covers the most general situation to date: value-asymmetric game with asymmetric budget. The proposed algorithm rests on recent theoretical advances regarding Sinkhorn iterations for matrix and tensor scaling. An important case which had been out of reach of previous attempts is that of heterogeneous but symmetric battlefield values with asymmetric budget. In this case, the Blotto game is constant-sum so optimal solutions exist, and our algorithm samples from an $\varepsilon$-optimal solution in time $\tilde{\mathcal{O}}(n^2 + \varepsilon^{-4})$, independently of budgets and battlefield values. In the case of asymmetric values where optimal solutions need not exist but Nash equilibria do, our algorithm samples from an $\varepsilon$-Nash equilibrium with similar complexity but where implicit constants depend on various parameters of the game such as battlefield values.