Balanced Nontransitive Dice: Existence and Probability

초록

A triple (A, B, C) of dice is called nontransitive if each of P(A < B), P(B < C), and P(C < A) is greater than 21 and called balanced if P(A < B) = P(B < C) = P(C < A). From the result of Trybu la, it is known that P(A < B) is less than (-1+root 5)/(2) , the golden ratio, for every balanced nontransitive triple (A, B, C) of dice. Schaefer asked whether this upper bound is tight, and Hur and Kim conjectured that the upper bound can be reduced to (1)/(2) + (1)/(9). In this paper, we characterize all possible probabilities P(A < B) for balanced nontransitive triples (A, B, C) of dice. Precisely, we prove that, for every rational( 1)/(2) < q < (-1+root 5)/(2) , there exists a balanced nontransitive triple (A, B, C) of dice with P(A < B) = q, which disproves Hur and Kim's conjecture and answers Schaefer's question. We also characterize all triples (m, n, ) pound of positive integers such that there exists a balanced nontransitive triple (A, B, C) of dice, where A, B, and C are m-sided, n- sided, and -sided pound dice, respectively. This generalizes Schaefer and Schweig's result showing the existence of a balanced nontransitive triple of n-sided dice for every n >= 3.