Paradoxes with Dice and Elections

There are two players playing the game, player U and player I. Each player chooses a (distinct) die and rolls – the player with the higher value is the winner. I kindly gives player U the first choice of any one of the three dice. Let us say that U chooses die X. After some thought, I chooses die Z, and then both players roll their dice. What are the possibilities? Because both dice have 6 sides there are 6 × 6 = 36 possibilities, however because of repetitions in the labeling of the dice, many of these outcomes are indistinguishable. In Table 2, the columns are labeled by the six faces values of die X and the rows by the face values of die Z. In a row with Z face value z and X face value x, we have put a Z if z > x and an X if x > z , in other words, the X’s and Z’s in the content of the table correspond to possible wins by the respective dice.