Strategies for Rolling the Efron Dice
Strategies for Rolling the Efron Dice
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掷埃夫隆骰子的策略
DOI：
10.1080/0025570x.2001.11953065
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发表时间：
2001
影响因子：
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通讯作者：
Christopher M. Rump
中科院分区：
文献类型：
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作者：
Christopher M. Rump
这些骰子的特殊之处在于它们在概率上是不可传递的[1，3]。这是因为骰子A击败骰子B的可能性是骰子A的两倍，骰子B击败骰子C的可能性是骰子C击败骰子D的可能性是骰子D击败骰子A的可能性的两倍！因此，在罗斯·洪斯伯格（Ross Honsberger）所说的“绅士游戏”中，我们会优雅地让对手选择一个骰子，这样我们就可以在三个骰子中选择一个能赢它两次的骰子。显然，“绅士游戏”对先选择的人没有吸引力。现在假设，每个玩家都有一套四个埃夫隆骰子，同时选择一个骰子，直到骰子被掷出，他们的选择才显示出来。在重复玩这个游戏时，玩家对掷骰子的选择并不那么清楚。很明显，一个确定性的策略，也就是说，一个涉及一个完全可预测的序列，如总是选择一个特定的骰子，可以被彻底击败。因此，玩家必须通过选择一个混合策略，在四个骰子中随机选择，让对手猜测。有人可能会怀疑，最佳策略是在四个概率相等（均匀）的骰子中随机选择。毕竟，这种类型的策略对于经典的非传递游戏石头剪刀布（石头击败剪刀，剪刀击败布，布击败石头）是最佳的。然而，这种策略对于Efron骰子游戏不是最佳的。
What is peculiar about these dice is that they are probabilistically non-transitive [1, 3]. This is due to the fact that die A is twice as likely to beat die B, die B is twice as likely to beat die C, die C is twice as likely to beat die D, and, paradoxically, die D is twice as likely to beat die A! Thus, in a "gentleman's game," as Ross Honsberger calls it [2], we graciously let our opponent choose a die to roll, so that we then can pick a die that beats it two times out of three. Clearly, a "gentleman's game" is not attractive for the person who chooses first. Suppose now, instead, that the players, each equipped with a personal set of four Efron dice, simultaneously choose a die to roll without revealing their selections until the dice are rolled. In repeated plays of this game, a player's choice of a die to roll is not so clear. What is clear is that a deterministic strategy, that is, one that involves a completely predictable sequence such as always choosing a particular die, can be soundly beaten. Thus, players must keep their opponents guessing by choosing a mixed strategy that randomly picks among the four dice. One might suspect that an optimal strategy would be to randomly choose among the four dice with equal (uniform) probability. After all, this type of strategy is optimal for the classic non-transitive game of rock-scissors-paper (rock beats scissors, scissors beats paper, paper beats rock) [6]. However, this strategy is not optimal for the Efron dice game.