纸牌游戏的最优策略:超越凹凸性

Soheil Behnezhad, Avrim Blum, M. Derakhshan, M. Hajiaghayi, C. Papadimitriou, Saeed Seddighin
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引用次数: 17

摘要

由Borel于1921年首次提出的“Blotto上校”游戏是博弈论的经典之作。两个上校各有一群军队,他们同时在一系列战场上分配。每个战场的赢家是派遣更多军队的上校,每个上校的整体效用是他/他赢得的战场的权重之和。在过去的一个世纪里,从广告到政治再到体育,上校布托游戏在许多不同形式的竞争中得到了应用。文献中提出了这个博弈的两个主要目标:(i)最大化保证的预期收益,(ii)最大化获得最小收益u的概率。前者对应于传统的效用最大化,后者涉及选举等场景,候选人的目标是最大化获得至少一半选票的概率(而不是预期的选票数量)。在本文中,我们考虑了这两个目标,并展示了如何获得(几乎)最优的解决方案,在他们的支持很少的策略。在Colonel Blotto游戏中获得有界支持策略的主要技术挑战之一是解空间变得非凸。这使我们无法使用凸规划技术来寻找最优策略,而这些策略本质上是文献中使用的主要工具。然而,我们通过一组结构结果表明,解空间可以,有趣的是,被划分成多项式许多可以独立考虑的不相交凸多面体。与许多其他组合观测相结合,这导致了上述两个目标的多项式时间近似方案。我们还提供了寻找Blotto类游戏的最大值的第一个复杂性结果:我们表明,计算Blotto上校游戏(我们称之为General Colonel Blotto)泛化的最大值是指数时间完备的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Optimal Strategies of Blotto Games: Beyond Convexity
The Colonel Blotto game, first introduced by Borel in 1921, is a well-studied game theory classic. Two colonels each have a pool of troops that they divide simultaneously among a set of battlefields. The winner of each battlefield is the colonel who puts more troops in it and the overall utility of each colonel is the sum of weights of the battlefields that s/he wins. Over the past century, the Colonel Blotto game has found applications in many different forms of competition from advertisements to politics to sports. Two main objectives have been proposed for this game in the literature: (i) maximizing the guaranteed expected payoff, and (ii) maximizing the probability of obtaining a minimum payoff u. The former corresponds to the conventional utility maximization and the latter concerns scenarios such as elections where the candidates' goal is to maximize the probability of getting at least half of the votes (rather than the expected number of votes). In this paper, we consider both of these objectives and show how it is possible to obtain (almost) optimal solutions that have few strategies in their support. One of the main technical challenges in obtaining bounded support strategies for the Colonel Blotto game is that the solution space becomes non-convex. This prevents us from using convex programming techniques in finding optimal strategies which are essentially the main tools that are used in the literature. However, we show through a set of structural results that the solution space can, interestingly, be partitioned into polynomially many disjoint convex polytopes that can be considered independently. Coupled with a number of other combinatorial observations, this leads to polynomial time approximation schemes for both of the aforementioned objectives. We also provide the first complexity result for finding the maximin of Blotto-like games: we show that computing the maximin of a generalization of the Colonel Blotto game that we call General Colonel Blotto is exponential time-complete.
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