An Algorithmic Solution to the Blotto Game Using Multimarginal Couplings

Abstract

We describe an efficient algorithm to compute solutions for the general two-player Blotto game on n battlefields with heterogeneous values. Whereas 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: a value-asymmetric game with an asymmetric budget with sufficient symmetry and homogeneity. The proposed algorithm rests on recent theoretical advances regarding Sinkhorn iterations for matrix and tensor scaling. An important case that had been out of reach of previous attempts is that of heterogeneous but symmetric battlefield values with asymmetric budgets. In this case, the Blotto game is constant-sum, so optimal solutions exist, and our algorithm samples from an $\epsilon$-optimal solution in time $\tilde{\mathcal{O}}(n^2+{\epsilon }^{-4})$, independent of budgets and battlefield values, up to some natural normalization. In the case of asymmetric values where optimal solutions need not exist but Nash equilibria do, our algorithm samples from an $\epsilon$-Nash equilibrium with similar complexity but where implicit constants depend on various parameters of the game such as battlefield values.

Funding: V. Perchet acknowledges support from the French National Research Agency (ANR) [Grant ANR-19-CE23-0026] as well as the support grant, and Investissements d’Avenir [Grant LabEx Ecodec/ANR-11-LABX-0047]. P. Rigollet is supported by the NSF [Grants IIS-1838071, DMS-2022448, and CCF-2106377].

Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2023.0049.