Upsets in Round Robin Tournaments

Abstract

Consider a round robin tournament in which each of n players is required to play precisely one game with each other player, and assume that each game ends in a win or a loss. The results of such a tournament can be conveniently recorded in a square (0, 1)-matrix T = (tij) of order n by setting tij = 1 if player i defeats player j , tij = 0 if player i loses to player j , and tii = 0. Thus T has 0's along the main diagonal, and in the off-diagonal positions T satisfies the "skew-symmetry" condition that tij = 1 if and only if tji = 0. We call such a (0, 1)-matrix T a tournament matrix.