 Award ID(s):
 1331175
 NSFPAR ID:
 10057357
 Date Published:
 Journal Name:
 Proceedings of the 29th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2018)
 ISSN:
 21601445
 Page Range / eLocation ID:
 22912310
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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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 winnertakesall rule. The ultimate payoff for each colonel is the number of battlefields won. 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 advertising, sports, and politics. There are persistent efforts to find the optimal strategies for the Colonel Blotto game. However, the first polynomialtime algorithm for that has very recently been provided by Ahmadinejad, Dehghani, Hajiaghayi, Lucier, Mahini, and Seddighin. Their algorithm consists of an exponential size linear program (LP), which they solve using the ellipsoid method. Because of the use of the ellipsoid method, despite its significant theoretical importance, this algorithm is highly impractical. In general, even the simplex method (despite its exponential running time in practice) performs better than the ellipsoid method in practice. In this paper, we provide the first polynomialsize LP formulation of the optimal strategies for the Colonel Blotto game using linear extension techniques. Roughly speaking, we consider the natural representation of the strategy space polytope and transform it to a higher dimensional strategy space, which interestingly has exponentially fewer facets. 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 polynomialsize LP to provide a simpler and significantly faster algorithm for finding optimal strategies of the Colonel Blotto game. We further show this representation is asymptotically tight, which means there exists no other linear representation of the strategy space with fewer constraints. We also extend our approach to multidimensional Colonel Blotto games, in which players may have different sorts of budgets, such as money, time, human resources, etc. By implementing this algorithm, we are able to run tests that were previously impossible to solve in a reasonable time. This information allows us to observe some interesting properties of Colonel Blotto; for example, we find out the behavior of players in the discrete model is very similar to the continuous model Roberson solved.more » « less

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 winnertakeall 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 polynomialtime algorithm for this fundamental gametheoreticalproblem in 2016. However, that breakthroughpolynomialtime 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 thetwoplayermultibattleground 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 complexitytheoretical evidencethat, in contrast to Borel’s homogeneous setting, findingoptimal strategies in multifaceted Colonel Blotto gamesis intractable. We complement this complexity result witha polynomialtime algorithm that finds approximately optimalstrategies with provable guarantees. We also study a furthergeneralization when two competitors do not have zerosum/constantsum payoffs. We show that optimal strategiesin these twoplayermultibattleground games are as hard tocompute and approximate as Nash equilibria in general noncooperative games and economic equilibria in exchange markets.more » « less

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Abstract A set of players delegate playing a game to a set of representatives, one for each player. We imagine that each player trusts their respective representative’s strategic abilities. Thus, we might imagine that per default, the original players would simply instruct the representatives to play the original game as best as they can. In this paper, we ask: are there safe Pareto improvements on this default way of giving instructions? That is, we imagine that the original players can coordinate to tell their representatives to only consider some subset of the available strategies and to assign utilities to outcomes differently than the original players. Then can the original players do this in such a way that the payoff is guaranteed to be weakly higher than under the default instructions for all the original players? In particular, can they Paretoimprove without probabilistic assumptions about how the representatives play games? In this paper, we give some examples of safe Pareto improvements. We prove that the notion of safe Pareto improvements is closely related to a notion of outcome correspondence between games. We also show that under some specific assumptions about how the representatives play games, finding safe Pareto improvements is NPcomplete.