Intransitive Dice

摘要

We consider n-sided dice whose face values lie between 1 and n and whose faces sum to n（n+l）/2. For two dice A and B, define A > B if it is more likely for A to show a higher face than B. Suppose k such dice A_1, ..., A_k are randomly selected. We conjecture that the probability of ties goes to 0 as n grows. We conjecture and provide some supporting evidence that—contrary to intuition—each of the 2(k2) assignments of > or < to each pair is equally likely asymptotically. For a specific example, suppose we randomly select k dice A_1, ..., A_k and observe that A_1 > A_2 > ...> A_k. Then our conjecture asserts that the outcomes A_k > A_1 and A_1 > A_k both have probability approaching 1/2 as n → ∞.