Non-Transitive Dominance
Non-Transitive Dominance
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非传递性显性
DOI：
10.1080/0025570x.1976.11976556
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发表时间：
1976
期刊：
影响因子：
--
通讯作者：
C. C. Foster
中科院分区：
文献类型：
--
作者：
R. L. Tenney;C. C. Foster
如果艾尔比比尔高，比尔比查理高，我们就可以得出艾尔比查理高的结论。这一事实在数学上被抽象为关系“比”高“是一个传递关系。许多其他关系也是可传递的：例如，“大于”、“小于”、“同构于”和“等于”。当然，如果所有关系都是传递性的，那么传递性将不是一个值得研究的有趣性质。“不分”关系(书面A‘)不是传递性的，因为从34’5和54‘12的事实来看，它并不能得出34’12的结论。直觉上，人们认为与支配地位有关的关系应该是传递性的，比如“比”或“赢于”或“比比更聪明”。但并不是所有人都是，这令人惊讶。国际象棋大师的故事比比皆是，他们可以击败任何人，除了某个死对头。这个死对头可能是一个相当二流的棋手，经常被国际象棋大师击败的许多棋手击败。因此，我们有一个例子，A(主人)打败B，B打败C(死对头)，但C打败A。这个例子在一定程度上是不令人满意的，因为它的原因不清楚，或者至少这个问题可能不是数学问题。也许错误在于定义中的某些不准确。因此，让我们考虑一种更好地定义的情况，一种涉及可以详细描述的对象和事件，一种涉及一些有趣的数学。我们将考虑两个玩家之间的一场游戏，其中包括三个骰子，红色、白色和蓝色，以便于识别。每位玩家选择一枚骰子并掷出，掷出的数字较大者获胜。骰子是专门为这个游戏制作的，每个面上都有一个介于1到9之间的整数；每个骰子的对面是相同的；并且骰子是公平的，因为每一方的可能性都是相等的。令人惊讶的是，这场听起来非常公平的游戏，可以被操纵成这样一种方式，选择第一个死亡的玩家将平均输掉九场比赛中的五场。
If Al is taller than Bill and Bill is taller than Charlie, we may conclude that Al is taller than Charlie. This fact is abstracted mathematically by the statement that the relation "is taller than" is a transitive relation. Many other relations are also transitive: e.g., "greater than", "less than", "is isomorphic to", and "equals". Certainly, if all relations were transitive, transitivity would not be an interesting property to study. The relation "does not divide" (written A') is not transitive, for from the facts 34' 5 and 54' 12, it does not follow that 34' 12. Intuitively one feels that relations having to do with dominance like "is better than" or "wins at chess from" or "is wiser than" should be transitive. But not all of them are, and this is surprising. Stories abound of chess masters who can beat everybody but a certain nemesis. This nemesis may be a rather second-rate player and be beaten regularly by many of the players that the chess master beats. Thus we have a case where A (the master) beats B and B beats C (the nemesis), but C beats A. This example is to a certain extent unsatisfying because the reasons for it are unclear, or at least the problem may not be mathematical. Perhaps the fault lies in some imprecision in the definitions. So let us consider a better defined situation, one involving objects and events that can be described in detail, and one which involves some interesting mathematics. We will consider a game between two players involving three dice, colored red, white, and blue for purposes of identification. Each player chooses a die and rolls it, and the one who rolls the higher number wins. The dice have been specially made for the game, and each face has an integer between one and nine on it; opposite faces of each die are identical; and the dice are fair in the sense that each side is equally likely. Surprisingly, this game, which sounds perfectly fair, can be rigged in such a way that the player who chooses the first die will lose an average of five out of nine games.