Faster and Simpler Algorithm for Optimal Strategies of Blotto Game

In the Colonel Blotto game, which was initially introduced by Borel in 1921, two colonels simultaneously distribute their troops across different battlefields. The winner of each battlefield is determined independently by a winner-take-all rule. The ultimate payoff of each colonel is the number of battlefields he wins. The Colonel Blotto game is commonly used for analyzing a wide range of applications from the U.S presidential election, to innovative technology competitions, to advertisement, to sports, and to politics. There has been persistent efforts for finding the optimal strategies for the Colonel Blotto game. After almost a century Ahmadinejad, Dehghani, Hajiaghayi, Lucier, Mahini, and Seddighin [2] provided an algorithm for finding the optimal strategies in polynomial time. Ahmadinejad et al. [2] first model the problem by a Linear Program (LP) with both an exponential number of variables and an exponential number of constraints which makes the problem intractable. Then they project their solution to another space to obtain another exponential-size LP, for which they can use Ellipsoid method. However, despite the theoretical importance of their algorithm, it is highly impractical. In general, even Simplex method (despite its exponential running time in practice) performs better than Ellipsoid method in practice. In this paper, we provide the first polynomial-size LP formulation of the optimal strategies for the Colonel Blotto game. We use linear extension techniques. Roughly speaking, we project the strategy space polytope to a higher dimensional space, which results in lower number of facets for the polytope. In other words, we add a few variables to the LP, such that surprisingly the number of constraints drops down to a polynomial. We use this polynomial-size LP to provide a novel simpler and significantly faster algorithm for finding optimal strategies for the Colonel Blotto game. We further show this representation is asymptotically tight, which means there exists no other linear representation of the problem with less number of constraints. We also extend our approach to multi-dimensional Colonel Blotto games, where each player may have different sorts of budgets, such as money, time, human resources, etc. By implementing this algorithm we were able to run tests which were previously impossible to solve in a reasonable time. These informations, allow us to observe some interesting properties of Colonel Blotto; for example we find out the behaviour of players in the discrete model is very similar to the continuous model Roberson [34] solved. ∗Supported in part by NSF CAREER award CCF-1053605, NSF BIGDATA grant IIS-1546108, NSF AF:Medium grant CCF-1161365, DARPA GRAPHS/AFOSR grant FA9550-12-1-0423, and another DARPA SIMPLEX grant.