Baek Seung Ki, Son Seung-Woo, Jeong Hyeong-Chai
Department of Physics, Pukyong National University, Busan 608-737, Korea.
Department of Applied Physics, Hanyang University, Ansan 426-791, Korea.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Apr;91(4):042144. doi: 10.1103/PhysRevE.91.042144. Epub 2015 Apr 30.
We consider a tournament among four equally strong semifinalists. The players have to decide how much stamina to use in the semifinals, provided that the rest is available in the final and the third-place playoff. We investigate optimal strategies for allocating stamina to the successive matches when players' prizes (payoffs) are given according to the tournament results. From the basic assumption that the probability to win a match follows a nondecreasing function of stamina difference, we present symmetric Nash equilibria for general payoff structures. We find three different phases of the Nash equilibria in the payoff space. First, when the champion wins a much bigger payoff than the others, any pure strategy can constitute a Nash equilibrium as long as all four players adopt it in common. Second, when the first two places are much more valuable than the other two, the only Nash equilibrium is such that everyone uses a pure strategy investing all stamina in the semifinal. Third, when the payoff for last place is much smaller than the others, a Nash equilibrium is formed when every player adopts a mixed strategy of using all or none of its stamina in the semifinals. In a limiting case that only last place pays the penalty, this mixed-strategy profile can be proved to be a unique symmetric Nash equilibrium, at least when the winning probability follows a Heaviside step function. Moreover, by using this Heaviside step function, we study the tournament by using evolutionary replicator dynamics to obtain analytic solutions, which reproduces the corresponding Nash equilibria on the population level and gives information on dynamic aspects.
我们考虑一场有四名实力相当的半决赛选手参加的锦标赛。选手们必须决定在半决赛中使用多少耐力，前提是剩余的耐力可用于决赛和季军赛。当根据比赛结果给出选手的奖金（收益）时，我们研究了为连续比赛分配耐力的最优策略。基于获胜概率随耐力差异呈非递减函数这一基本假设，我们给出了一般收益结构下的对称纳什均衡。我们在收益空间中发现了纳什均衡的三个不同阶段。首先，当冠军获得的奖金比其他选手多得多时，只要所有四名选手共同采用，任何纯策略都可以构成纳什均衡。其次，当前两名的名次比后两名有价值得多时，唯一的纳什均衡是每个人都采用在半决赛中投入全部耐力的纯策略。第三，当最后一名的奖金比其他名次少得多时，当每个选手在半决赛中采用全部使用或不使用耐力的混合策略时，就会形成纳什均衡。在一种极限情况下，即只有最后一名会受到惩罚，这种混合策略配置至少在获胜概率遵循海维赛德阶跃函数时，可以被证明是唯一的对称纳什均衡。此外，通过使用这个海维赛德阶跃函数，我们利用进化复制动态来研究锦标赛以获得解析解，这在群体层面上再现了相应的纳什均衡，并给出了动态方面的信息。
