Generalized Intransitive Dice: Mimicking an Arbitrary Tournament

2019/01/28 by Akin, Ethan
#05C20 #05C25 #05C62 #05C70 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.09477

Abstract

A generalized N-sided die is a random variable D on a sample space of N equally likely outcomes taking values in the set of positive integers. We say of independent N sided dice D_i, D_j that D_i beats D_j, written D_i → D_j, if Prob(D_i > D_j) > (1)/(2) . Examples are known of intransitive 6-sided dice, i.e. D₁ → D₂ → D₃ but D₃ → D₁. A tournament of size n is a choice of direction i → j for each edge of the complete graph on n vertices. We show that if R is tournament on the set \ 1, …, n \, then for sufficiently large N there exist sets of independent N-sided dice \ D₁, …, D_n \ such that D_i → D_j if and only if i → j in R.

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