A Colonel Blotto Gladiator Game

TL;DR

This paper analyzes a stochastic zero-sum game called the gladiator game, where two teams allocate strengths among gladiators to compete in sequential fights, deriving Nash equilibria and game value based on team strengths and sizes.

Contribution

It introduces a stochastic variant of the Blotto game, providing new Nash equilibrium solutions and analyzing their dependence on team parameters using novel probability inequalities.

Findings

Nash equilibria characterized for the gladiator game.

Game value depends on team strengths and number of gladiators.

Majorization inequalities for Gamma distributions developed.

Abstract

We consider a stochastic version of the well-known Blotto game, called the gladiator game. In this zero-sum allocation game two teams of gladiators engage in a sequence of one-to-one fights in which the probability of winning is a function of the gladiators' strengths. Each team's strategy consists of the allocation of its total strength among its gladiators. We find the Nash equilibria and the value of this class of games and show how they depend on the total strength of teams and the number of gladiators in each team. To do this, we study interesting majorization-type probability inequalities concerning linear combinations of Gamma random variables. Similar inequalities have been used in models of telecommunications and research and development.