The Asymmetric Colonel Blotto Game

TL;DR

This paper investigates the Nash equilibria of a modified Colonel Blotto game where force distributions are nondecreasing across battlefields, providing explicit solutions for two and three battlefield scenarios.

Contribution

It introduces the asymmetric variant of the Colonel Blotto game and derives explicit Nash equilibria and payoff characterizations for specific cases.

Findings

Explicit Nash equilibria for three battlefields with equal force levels.

Unique equilibrium payoff for two battlefields across all force levels.

Partial results for asymmetric force levels in three battlefields.

Abstract

This paper explores the Nash equilibria of a variant of the Colonel Blotto game, which we call the Asymmetric Colonel Blotto game. In the Colonel Blotto game, two players simultaneously distribute forces across $n$ battlefields. Within each battlefield, the player that allocates the higher level of force wins. The payoff of the game is the proportion of wins on the individual battlefields. In the asymmetric version, the levels of force distributed to the battlefields must be nondecreasing. In this paper, we find a family of Nash equilibria for the case with three battlefields and equal levels of force and prove the uniqueness of the marginal distributions. We also find the unique equilibrium payoff for all possible levels of force in the case with two battlefields, and obtain partial results for the unique equilibrium payoff for asymmetric levels of force in the case with three battlefields.