Is There a Strongest Die in a Set of Dice with the Same Mean Pips?

Abstract

Jan-ken, a.k.a. rock-paper-scissors, is a cerebrated example of a non-transitive game with three (pure) strategies, rock, paper and scissors. Interestingly, any Jan-ken generalized to four strategies contains at least one useless strategy unless it allows a tie between distinct pure strategies. Non-transitive dice could be a stochastic analogue of Jan-ken: the stochastic transitivity does not hold on some sets of dice, e.g., Efron's dice. Including the non-transitive dice, this paper is interested in dice sets which do not contain a useless die. In particular, we are concerned with the existence of a strongest (or weakest, symmetrically) die in a dice set under the two conditions that (1) any number appears on at most one die and at most one side, i.e., no tie break between two distinct dice, and (2) the mean pips of dice are the same. We firstly prove that a strongest die never exist if a set of n dice of m-sided is given as a partition of the set of numbers {1,…,mn}. Next, we show some sufficient conditions that a strongest die exists in a dice set which is not a partition of a set of numbers. We also give some algorithms to find a strongest die in a dice set which includes given dice.