The heterogeneous Colonel Blotto Game

The Colonel Blotto game, proposed by Borel in 1921, is a fundamental model of strategic resource allocation. Two players allocate an exogenously given amount of resources to a fixed number of battlefields with given values. Each battlefield

is then won by the player who allocated more resources to it, and each player maximizes the aggregate value of battlefields he wins. This game allows modeling many practical problems of resource allocation in various strategic settings ranging

from international war to competition for attention in social networks; it is particularly useful in security to model the allocation of defense resources on different potential targets. The scope of applications, however, has been limited by the lack of solutions of the game in realistic scenarios. Indeed, despite its apparent simplicity, the Colonel Blotto game is very intricate and it remains unsolved in the case with asymmetric players (i.e., with different resources) and with an arbitrary number of battlefields that can have different values. In this paper, we propose a solution of the heterogeneous Colonel Blotto game with asymmetric players and heterogeneous battlefield values, under the assumption that there is a sufficient number of battlefields of each possible value relative to the players' resources asymmetry. In particular, our assumption implies that there must be at least three battlefields of each

possible value. Then, we characterize the unique equilibrium payoffs and univariate marginal distributions, along with the proof that there exist n-variate joint distributions with such marginals. Our results expand the scope of potential applications of the Colonel Blotto game, and mark a new step towards a complete solution of the game.