Colonel Blotto Games with Favoritism: Competitions with Pre-allocations and Asymmetric Effectiveness

TL;DR

This paper extends the Colonel Blotto game to include favoritism, pre-allocations, and asymmetrical effectiveness, providing equilibrium characterizations and an efficient approximation algorithm.

Contribution

It introduces a novel favoritism extension to the Colonel Blotto game, characterizes equilibrium in related all-pay auctions, and develops an algorithm for computing approximate equilibria.

Findings

Explicit equilibrium constructions in special cases

Existence of optimal univariate distributions for equilibrium

Efficient algorithm for approximate equilibrium computation

Abstract

We introduce the Colonel Blotto game with favoritism, an extension of the famous Colonel Blotto game where the winner-determination rule is generalized to include pre-allocations and asymmetry of the players' resources effectiveness on each battlefield. Such favoritism is found in many classical applications of the Colonel Blotto game. We focus on the Nash equilibrium. First, we consider the closely related model of all-pay auctions with favoritism and completely characterize its equilibrium. Based on this result, we prove the existence of a set of optimal univariate distributions -- which serve as candidate marginals for an equilibrium -- of the Colonel Blotto game with favoritism and show an explicit construction thereof. In several particular cases, this directly leads to an equilibrium of the Colonel Blotto game with favoritism. In other cases, we use these optimal univariate distributions to derive an approximate equilibrium with well-controlled approximation error. Finally, we propose an algorithm -- based on the notion of winding number in parametric curves -- to efficiently compute an approximation of the proposed optimal univariate distributions with arbitrarily small error.