淘汰赛常规赛各队夺冠的最佳策略
Best Strategy for Each Team in The Regular Season to Win Champion in The Knockout Tournament
摘要：
在J. Schwenk。（2018）“什么是种子淘汰赛的正确方法？”从美国数学月刊检索]，Schwenk发现了一个令人惊讶的弱点，在标准的方法播种一个单一的消除（或淘汰赛）比赛。特别是，他表明，对于游戏结果的特定概率模型，可能是顶级种子球队赢得比赛的可能性低于第二种子球队。这就提出了一种可能性，即在某些情况下，球队故意输掉一场比赛，试图在锦标赛中获得更优（尽管可能更低）的种子，这可能是有利的。我们在一个四队联赛的背景下研究这个问题，该联赛包括一个循环赛的“常规赛”，然后是一个单淘汰赛，淘汰赛由常规赛的结果决定[4]。使用与Schwenk相同的概率模型，我们证明了在某些情况下，球队故意输球确实是最优的。此外，我们展示了一个团队如何决定是否应该故意输掉比赛。我们做了两个详细的分析。一个是其他球队总是试图赢得每场比赛的情况。另一种是其他球队足够聪明的情况，即如果必要的话，他们也可以故意输掉一些比赛。分析涉及概率和（多玩家）对策论的计算。
英文摘要：
In J. Schwenk.(2018) ['What is the Correct Way to Seed a Knockout Tournament?' Retrieved from The American Mathematical Monthly], Schwenk identified a surprising weakness in the standard method of seeding a single elimination (or knockout) tournament. In particular, he showed that for a certain probability model for the outcomes of games it can be the case that the top seeded team would be less likely to win the tournament than the second seeded team. This raises the possibility that in certain situations it might be advantageous for a team to intentionally lose a game in an attempt to get a more optimal (though possibly lower) seed in the tournament. We examine this question in the context of a four team league which consists of a round robin "regular season" followed by a single elimination tournament with seedings determined by the results from the regular season [4]. Using the same probability model as Schwenk we show that there are situations where it is indeed optimal for a team to intentionally lose. Moreover, we show how a team can make the decision as to whether or not it should intentionally lose. We did two detailed analysis. One is for the situation where other teams always try to win every game. The other is for the situation where other teams are smart enough, namely they can also lose some games intentionally if necessary. The analysis involves computations in both probability and (multi-player) game theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
