Circular chains of chinese dice

In this paper we study Chinese dice, mathematical objects similar to ordinary dice but allowing repetition among their face values. We say that a die A is preferred over a die B (written A > B) if A wins more frequently than B does. We study first the existence of circular chains of three dice A, B, C (those that A > B > C > A) using a mixed integer programming algorithm. Then we generalize the problem to n-dimensional dice—that is, dice of n faces, with n ≥ 4—and we search circular chains of length m (with m ≥ 3) using a simulated annealing algorithm. We compare some different objective functions and obtain good solutions to the problem with very efficient algorithms. Finally we obtain a theoretical result concerning the existence of circular chains in the general case.