Abstract: We will discuss the simplest possible kind of dice game: two players roll one die apiece and if the results differ, the winner is the player whose die has rolled the higher value. People have played such games (and much more complicated ones too) since antiquity, but the idea of playing a simple dice game with different dice seems to have been introduced in living memory -- Gardner's Scientific American articles attribute the idea to Efron [unpublished]. Gardner's discussion centers on the two following observations:

1. In such games the dice need not tend to have equal numbers of wins and losses in the long run, even if their rolls have the same expected value. For instance, the 3-sided die (3,3,4) is stronger than the 3-sided die (2,2,6). Moreover, the relative strengths of dice need not obey transitivity. For instance, the die (1,4,5) will lose to (2,2,6) more often than not, though it is stronger than (3,3,4).
2. The second observation naturally suggests the celebrated theorem of Arrow. Indeed, dice games provide a natural way to judge multicandidate elections; if several candidates are assessed by n voters then each candidate has an associated n-sided die, which simply lists the scores the various voters have given that candidate. The relative strengths of two candidates may be assessed by playing a dice game with their associated dice.