Properties of Tournaments among Well-Matched Players

1. TOURNAMENTS. In an n-player round robin tournament, each player plays one match against each of the other n 1 players. The win-loss outcomes of these matches can be conveniently recorded in a tournament matrix A = [aij] as follows: First label the players in any order as 1, 2,.. ., n. For each pair i and j, set ai1 = 1 if player i defeats player j, and set aij = 0 otherwise. If i j , then exactly one of aij and aji is nonzero; when i = j, aii = 0. What properties of the matrix A are related to the strengths of the players? The simplest measure of strength is the number of matches that the player wins, and the row sums of A count the number of matches won by each player. We are interested in understanding tournaments among players who are well matched in the sense that each player wins about half of the matches played. If the number of players is odd, many properties of A are very well understood. If the number of players is even, however, the properties of A are far less well understood. Indeed, there are many easily stated questions that lead to hard, open problems. Some of these problems are the focus of this paper. We let I denote the identity matrix, we let J denote the square matrix all of whose entries are ones, and we let e denote the column vector whose entries are all ones. A matrix A whose entries are zeros and ones is a tournament matrix exactly when