Combinatorial Theory

An \(n\)-sided die is an \(n\)-tuple of positive integers. We say that a die \((a_1,\dots,a_n)\) beats a die \((b_1,\dots,b_n)\) if the number of pairs \((i,j)\) such that \(a_i›b_j\) is greater than the number of pairs \((i,j)\) such that \(a_i‹b_j\). We show that for a natural model of random \(n\)-sided dice, if \(A, B\) and \(C\) are three random dice then the probability that \(A\) beats \(C\) given that \(A\) beats \(B\) and \(B\) beats \(C\) is approximately 1/2. In other words, the information that \(A\) beats \(B\) and \(B\) beats \(C\) has almost no effect on the probability that \(A\) beats \(C\). This proves a statement that was conjectured by Conrey, Gabbard, Grant, Liu and Morrison for a different model.

Mathematics Subject Classifications: 60C05

Keywords: Intransitive dice, central limit theorems