Equilibria of the Colonel Blotto Games with Costs

TL;DR

This paper introduces a generalized Colonel Blotto game incorporating costs, demonstrating its strategic equivalence to a zero-sum game with an extra battlefield, and providing a polynomial-time method to compute Nash equilibria.

Contribution

It extends the classic Colonel Blotto game by including costs and shows how to efficiently compute equilibria through strategic equivalence.

Findings

Game with costs is equivalent to a zero-sum game with one additional battlefield.

Nash equilibria can be computed in polynomial time.

The model captures strategic importance of resource and assignment costs.

Abstract

This paper studies a generalized variant of the Colonel Blotto game, referred to as the Colonel Blotto game with costs. Unlike the classic Colonel Blotto game, which imposes the use-it-or-lose-it budget assumption, the Colonel Blotto game with costs captures the strategic importance of costs related both to obtaining resources and assigning them across battlefields. We show that every instance of the Colonel Blotto game with costs is strategically equivalent to an instance of the zero-sum Colonel Blotto game with one additional battlefield. This enables the computation of Nash equilibria of the Colonel Blotto game with costs in polynomial time with respect to the game parameters: the number of battlefields and the number of resources available to the players.