Nontransitive dice sets realizing Paley tournaments for solving Schütte’s tournament problem

Abstract

The problem of a multiple player dice tournament is discussed and solved in the paper. A die has a finite number of faces with real numbers written on each. Finite dice sets are proposed which have the following property, defined by Schutte for tournaments: for an arbitrary subset of k dice there is at least one die that beats each of the k with a probability greater than 1/2. It is shown that the proposed dice set realizes the Paley tournament, that is known to have the Schutte property (for a given k) if the number of vertices is large enough. The proof is based on Dirichlet's theorem, stating that the sum of quadratic nonresidues is strictly larger than the sum of quadratic residues.