Non-Transitive Dominance

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.