In a set nontransitive dice, each die is superior to another die, but is inferior to a third. It is similar to the game of rock-paper-scissors. Here is one example:
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
Die A has a 5/9ths chance of rolling a higher number than B, which itself has a 5/9ths chance of rolling a higher number than C, which itself has a 5/9ths chance of rolling a higher number than A. It is a circle with no overall winner.
Suppose there is a simple dice game where one person picks a die from the set of three nontransitive dice. A second person then picks another die from the set. The players then roll their dice, and the person who rolls the higher number wins. If there is a tie, the players simply roll again until there is a winner.
If this game is played with the above set of dice, then the second player will always be able to pick a superior die, and will win the game 5/9ths of the time.
What is one possible set of nontransitive dice that maximizes the unfairness of this game? One additional requirement is that the second player's odds of winning must not be affected by the first player's choice of die.