A set of dice is intransitive (or nontransitive) if it contains three dice, A, B, and C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but it is not true that A rolls higher than C more than half the time. In other words, a set of dice is intransitive if the binary relation – rolls a higher number than more than half the time – on its elements is not transitive. More simply, A normally beats B, B normally beats C, but A does not normally beat C. It is possible to find sets of dice with the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time. This is different in that instead of only "A does not normally beat C" it is now "C normally beats A" Using such a set of dice, one can invent games which are biased in ways that people unused to intransitive dice might not expect (see Example). Example Consider the following set of dice. Die A has sides 2, 2, 4, 4, 9, 9. Die B has sides 1, 1, 6, 6, 8, 8. Die C has sides 3, 3, 5, 5, 7, 7. The probability that A rolls a higher number than B, the probability that B rolls higher than C, and the probability that C rolls higher than A are all , so this set of dice is intransitive. In fact, it has the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time. Now, consider the following game, which is played with a set of dice. The first player chooses a die from the set. The second player chooses one die from the remaining dice. Both players roll their die; the player who rolls the higher number wins. If this game is played with a transitive set of dice, it is either fair or biased in favor of the first player, because the first player can always find a die that will not be beaten by any other dice more than half the time. If it is played with the set of dice described above, however, the game is