We consider the manipulability of tournament rules which map the results of $\binom{n}{2}$ pairwise matches and select a winner. Prior work designs simple tournament rules such that no pair of teams can manipulate the outcome of their match to improve their probability of winning by more than $1/3$, and this is the best possible among any Condorcet-consistent tournament rule (which selects an undefeated team whenever one exists) [Schneider et al., 2017, Schvartzman et al., 2020]. These lower bounds require the manipulators to know precisely the outcome of all future matches. We take a beyond worst-case view and instead consider tournaments which are "close to uniform": the outcome of all matches are independent, and no team is believed to win any match with probability exceeding $1/2+\varepsilon$. We show that Randomized Single Elimination Bracket [Schneider et al., 2017] and a new tournament rule we term Randomized Death Match have the property that no pair of teams can manipulate the outcome of their match to improve their probability of winning by more than $\varepsilon/3 + 2\varepsilon^2/3$, for all $\varepsilon$, and this is the best possible among any Condorcet-consistent tournament rule. Our main technical contribution is a recursive framework to analyze the manipulability of certain forms of tournament rules. In addition to our main results, this view helps streamline previous analysis of Randomized Single Elimination Bracket, and may be of independent interest.
翻译:我们认为,映射$\binom{n ⁇ 2}美元双向匹配结果并选择一个赢家的锦标赛规则的可操纵性。 先前的工作设计了简单的锦标赛规则,让任何一对球队都无法操纵比赛结果来提高以1/3美元以上赢得比赛的概率。 这是任何康多采特一致的锦标规则(只要有球队就选择一个不败球队)[Schneider等人,2017年,Schvartzman等人,2020年]中最下限的规则。 这些下限规则要求操控者确切了解未来比赛的结果。 我们超越了最坏的球队观点,而是考虑“接近统一”的锦标赛结果:所有比赛的结果都是独立的,而且相信没有一个球队能够赢得任何超过1/2 ⁇ vareprepslon的比赛规则。 我们的简化单一消除赛程[Schneider等人和Al.,201717年]和一个新的锦标赛规则可能具有以下的特性, 任何一组球队不能操纵其匹配结果来提高他们的比值的比值, 美元比正3的比赛的比值的比值的比值, 的比值比值比值比值比值比值比值比值比值比值比值比值比值比值比值比值比值比值比值比值比值比值比值比值更多。