Computational Analyses of the Electoral College: Campaigning Is Hard But Approximately Manageable

In the classical discrete Colonel Blotto game—introduced by Borel in 1921—two colonels simultaneously distribute their troops across multiple battlefields. The winner of each battlefield is determined by a winner-take-all rule, independently of other battlefields. In the original formulation, each colonel’s goal is to win as many battlefields as possible. The Blotto game and its extensions have been used in a wide range of applications from political campaign—exemplified by the U.S presidential election—to marketing campaign, from (innovative) technology competition to sports competition. Despite persistent efforts, efficient methods for finding the optimal strategies in Blotto games have been elusive for almost a century—due to exponential explosion in the organic solution space—until Ahmadinejad, Dehghani, Hajiaghayi, Lucier, Mahini, and Seddighin developed the first polynomial-time algorithm for this fundamental gametheoretical problem in 2016. However, that breakthrough polynomial-time solution has some structural limitation. It applies only to the case where troops are homogeneous with respect to battlegruounds, as in Borel’s original formulation: For each battleground, the only factor that matters to the winner’s payoff is how many troops as opposed to which sets of troops are opposing one another in that battleground.

In this paper, we consider a more general setting of the two-player-multi-battleground game, in which multifaceted resources (troops) may have different contributions to different battlegrounds. In the case of U.S presidential campaign, for example, one may interpret this as different types of resources—human, financial, political—that teams can invest in each state. We provide a complexity-theoretical evidence that, in contrast to Borel’s homogeneous setting, finding optimal strategies in multifaceted Colonel Blotto games is intractable. We complement this complexity result with a polynomial-time algorithm that finds approximately optimal strategies with provable guarantees. We also study a further generalization when two competitors do not have zerosum/ constant-sum payoffs. We show that optimal strategies in these two-player-multi-battleground games are as hard to compute and approximate as Nash equilibria in general noncooperative games and economic equilibria in exchange markets.