Generalized Colonel Blotto Game

Abstract

Competitive resource allocation between adversarial decision makers arises in a wide spectrum of realworld applications such as in communication systems, cyber-physical systems security, as well as financial and political competition. Hence, developing analytical tools to model and analyze competitive resource allocation is crucial for devising optimal allocation strategies and anticipating the potential outcomes of the competition. To this end, the Colonel Blotto game is one of the most popular game-theoretic frameworks for modeling and analyzing such competitive resource allocation problems. However, in many practical competitive situations, the Colonel Blotto game does not admit solutions in deterministic strategies and, hence, one must rely on analytically complex mixed-strategies with their associated tractability, applicability, and practicality challenges. In this regard, in this paper, a generalization of the Colonel Blotto game which enables the derivation of deterministic, practical, and implementable equilibrium strategies is proposed while accounting for scenarios with heterogeneous battlefields. In addition, the proposed generalized game factors in the consumed/destroyed resources in each battlefield, a feature that is not considered in the classical Blotto game. For this generalized game, the existence of a Nash equilibrium in pure strategies is shown. Then, closed-form analytical expressions of the equilibrium strategies are derived and the outcome of the game is characterized, based on the number of each player's resources and each battlefield's valuation. The generated results provide invaluable insights on the outcome of the competition. For example, the results show that, when both players are fully rational, the more resourceful player can achieve a better total payoff at the Nash equilibrium, a result that is not mimicked in the classical Blotto game.