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  1. In the classical discrete Colonel Blotto game—introducedby Borel in 1921—two colonels simultaneously distributetheir troops across multiple battlefields. The winner of eachbattlefield is determined by a winner-take-all rule, independentlyof other battlefields. In the original formulation, eachcolonel’s goal is to win as many battlefields as possible. TheBlotto game and its extensions have been used in a widerange of applications from political campaign—exemplifiedby the U.S presidential election—to marketing campaign,from (innovative) technology competition to sports competition.Despite persistent efforts, efficient methods for findingthe optimal strategies in Blotto games have been elusivefor almost a century—due to exponential explosion inthe organic solution space—until Ahmadinejad, Dehghani,Hajiaghayi, Lucier, Mahini, and Seddighin developed thefirst polynomial-time algorithm for this fundamental gametheoreticalproblem in 2016. However, that breakthroughpolynomial-time solution has some structural limitation. Itapplies only to the case where troops are homogeneous withrespect to battlegruounds, as in Borel’s original formulation:For each battleground, the only factor that matters to the winner’spayoff is how many troops as opposed to which sets oftroops are opposing one another in that battleground.In this paper, we consider a more general setting of thetwo-player-multi-battleground game, in which multifacetedresources (troops) may have different contributions to differentbattlegrounds. In the case of U.S presidential campaign,for example, one may interpret this as different typesof resources—human, financial, political—that teams can investin each state. We provide a complexity-theoretical evidencethat, in contrast to Borel’s homogeneous setting, findingoptimal strategies in multifaceted Colonel Blotto gamesis intractable. We complement this complexity result witha polynomial-time algorithm that finds approximately optimalstrategies with provable guarantees. We also study a furthergeneralization when two competitors do not have zerosum/constant-sum payoffs. We show that optimal strategiesin these two-player-multi-battleground games are as hard tocompute and approximate as Nash equilibria in general noncooperative games and economic equilibria in exchange markets. 
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