Nontransitive Dice With Equal Means∗
نویسندگان
چکیده
We analyze a game in which two players choose in turn and roll an n-sided die, each player having his choice of numbering of the faces, subject to certain constraints. The player who rolls the larger number wins. We say die B “dominates” die A if P [B > A] > P [A > B]. The nontransitivity of this dominance relation for dice has been studied by several authors. We analyze the case of n-sided dice for which the faces are equally likely to be uppermost when rolled, the values of the numbers on the faces sum to n(n+1)/2, and are chosen with replacement from the integers 1 through n. Our analysis shows that in contests which are scored numerically, the relation “A dominates B” is nontransitive even though all participants have the same expected score. We illustrate this with an example of m money managers (m ≥ 3) who all have the same mean performance but for each i = 0, 1, . . . ,m− 1, manager i is expected to outperform manager i+ 1 (mod m).
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