The Paradox of Nontransitive Dice
The Paradox of Nontransitive Dice
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非传递骰子的悖论
DOI：
10.1080/00029890.1994.11996968
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发表时间：
1994
影响因子：
0.5
通讯作者：
Richard P. Savage
中科院分区：
文献类型：
--
作者：
Richard P. Savage
假设一个人拿了三个立方体，在每个立方体的每个面上写一个从1到18的数字，每个数字只写一次。这样，我们就可以构造出一些不寻常的骰子，我们用A、B和C来表示。有没有可能把骰子上的数字排列成，如果掷骰子，A赢B的概率大于1/2，B赢C的概率大于1/2，而C赢A的概率也大于1/2？答案是肯定的。例如，将18、9、8、7、6和5放在A上; 17、16、15、4、3和2放在B上; 14、13、12、11、10和1放在C上。很容易检验出A击败B的概率是21/36，B击败C的概率是21/36，C击败A的概率是25/36。这种“非传递骰子”的现象在《科学美国人》的马丁·加德纳专栏中得到了推广[7]。他举了一个例子，由于布拉德利埃夫隆与四个骰子，其中A击败B，B击败C，C击败D，D击败A每个概率2/3。非传递性骰子属于非传递性悖论的一般范畴，对此已有相当多的研究。最著名的非传递性悖论是投票悖论，孔多塞开始对它进行研究。在有三个候选人的情况下，观察到大多数选民可能更喜欢候选人A而不是候选人B，更喜欢B而不是C，更喜欢C而不是A。关于投票悖论的历史，参见布莱克[1]。非传递性的其他例子在[2]、[5]和[8]中讨论。让我们考虑下面的游戏。一家赌场制作了三个骰子，上面排列着数字。一个赞助人选择他想要的任何骰子，然后房子挑选剩下的骰子之一（击败赞助人选择的骰子），他们掷骰子，对结果下注。假设玩家将发挥自己的优势，赌场在安排数字时最初的目标必须是使三个概率中最小的概率尽可能大。在下面的讨论中，标准骰子有6个面的事实对分析并不重要，因此我们考虑n面骰子的问题。设X为随机变量，以骰子A上的数字为值，每个数字的概率为1/n。类似地为骰子B和C定义随机变量Y和Z。显然，X、Y和Z是独立的。施泰因豪斯和特里布拉[9][112]指出，如果X、Y和Z是随机变量，则P（X> Y）、P（Y> Z）和P（Z> X）都可能超过1/2，这一结果被称为施泰因豪斯-特里布拉悖论。此外，他们宣布，如果X，Y和Z是独立的，那么至少有一个概率不超过（1）/2。(The黄金分割!）因此，我们有以下结果。
Suppose one takes three cubes and writes a number from 1 to 18 on each face of each cube using each number once and only once. In this way one constructs some unusual dice we denote by A, B, and C. Is it possible to arrange the numbers on the dice such that if the dice are rolled, the probability that A beats B is greater than 1/2, the probability that B beats C is greater than 1/2, yet the probability that C beats A is also greater than 1/2? The answer is yes. For example, put 18, 9, 8, 7, 6, and 5 on A; 17, 16, 15, 4, 3, and 2 on B; and 14, 13, 12, 11, 10, and 1 on C. It is easily checked that the probability that A beats B is 21/36, that B beats C is 21/36, and that C beats A is 25/36. This phenomenon of "nontransitive dice" was popularized in Martin Gardner's column in Scientific American [7]. He gives an example due to Bradley Efron with four dice in which A beats B, B beats C, C beats D, and D beats A each with probability 2/3. Nontransitive dice fall into the general category of nontransitivity paradoxes about which there has been considerable study. The best known nontransitivity paradox is the voting paradox the study of which was begun by Condorcet [4]. In the case of three candidates this is the observation that a majority of voters may prefer candidate A to candidate B, B to C, and C to A. For a history of the voting paradox see Black [1]. Other examples of nontransitivity are discussed in [2], [5], and [8]. Let us consider the following game. A casino has constructed three dice with the numbers arranged on them. A patron chooses whichever die he wishes and the house then picks one of the remaining dice (the one which beats the patron's choice) and they roll the dice with a wager on the outcome. Assuming that the players will play to their own advantage, the goal that the casino has in arranging the numbers originally must be to make the smallest of the three probabilities as large as possible. In the following discussion the fact that a standard die has 6 faces is not important to the analysis, thus we consider the problem for n-faced dice. Let X be a random variable taking as values the numbers on die A each with probability 1/n. Define random variables Y and Z for dice B and C similarly. Clearly X, Y, and Z are independent. It was noted by Steinhaus and Trybula [9], [112 that if X, Y, and Z are random variables it is possible that P(X > Y), P(Y > Z), and P(Z > X) can all exceed 1/2, a result which is known as the Steinhaus-Trybula paradox. Further, they announced that if X, Y, and Z are independent then at least one of the probabilities is no more than ( 1)/2. (The golden ratio again!) Consequently we have the following result.