发表机构
The Hong Kong Polytechnic University（香港理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究策略证明锦标赛规则抵抗配对操纵的能力，提出BlockBonusedWinStrengths规则，满足孔多塞一致性、单调性及2-NM₂，大幅改进上界至下界两倍以内。
AI 中文摘要
锦标赛规则将$n$支队伍之间所有配对比赛的结果映射为一个可能随机的获胜者。理想的规则应满足孔多塞一致性和单调性，同时还要抵抗联盟的操纵。先前的工作大多通过$k$-强不可操纵性（在$\alpha$意义下，记为$k$-SNM-$\alpha$）来加性地衡量这种操纵，即不存在规模为$k$的联盟能够通过操纵彼此之间的比赛，使其总获胜概率增加$\alpha$。最近，引入了两种新的不可操纵性概念。乘法不可操纵性（$k$-MNM-$\delta$）类似地定义，但使用乘法因子。$\lambda$-不可操纵性（$k$-NM$_\lambda$）刻画了队伍的自私性，它限制联盟的收益小于其成员所牺牲获胜概率的$\lambda$倍。在这项工作中，我们首先建立了这三种概念之间的严格层级关系：NM$_\lambda$强于MNM，而MNM又强于SNM。这促使我们考虑这两种更强但研究较少的概念：配对乘法不可操纵性和$\lambda$的$2$-不可操纵性。我们证明了随机死亡竞赛是$2$-MNM-$3/2$的，并且最优地匹配了下界。然后，我们引入了BlockBonusedWinStrengths规则，该规则满足孔多塞一致性、单调性和$2$-NM$_2$。这一规则大幅改进了先前$\lambda=11$的上界，并达到了下界$\lambda=1$的两倍以内。
英文摘要
A tournament rule maps the outcomes of all pairwise matches among $n$ teams to a possibly randomized winner. Desirable rules should be Condorcet consistent and monotone, yet also resistant to manipulation among coalition. Prior work mostly measures such manipulation additively through $k$-strongly non-manipulable at $α$ ($k$-SNM-$α$), meaning that no coalition of size $k$ can fix the matches among themselves to increase their total winning probability by $α$. Very recently, two new notions of non-manipulability were introduced. Multiplicative non-manipulability ($k$-MNM-$δ$) is defined analogously, using the multiplicative factor instead. Non-manipulability for $λ$ ($k$-NM$_λ$) characterizes the selfishness of a team, which restricts a coalition's gain to be less than $λ$ times the winning probability sacrificed by its members. In this work, we begin with a strict hierarchy among these three notions: NM$_λ$ is stronger than MNM, which is then stronger than SNM. This motivates us to consider those two notions that are stronger but less studied: pairwise multiplicative non-manipulability and $2$-non-manipulability for $λ$. We show that Randomized Death Match is $2$-MNM-$3/2$ and optimally matches the lower bound. Then, we introduce the BlockBonusedWinStrengths rule, which is Condorcet consistent, monotone, and $2$-NM$_2$. This rule substantially improves the previous upper bound of $λ=11$ and comes within a factor of two of the lower bound $λ=1$.
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