Balanced 5 by 5 games

Abstract

Subject to our restriction that the first nonzero diagonal element is -, there are exactly 50 balanced 5 by 5 games. We may list them in lexicographic order of diagonals from 0 0 0 0 0 to - - - - - (with the ordering 0 < - < +). Of these fifty, the five with diagonals of the form — 0 + x y reduce to 2 by 2 games of type A, as may be seen from Theorem 10.1 below. They are numbers 34–38 in our ordering. The four with diagonals x y — 0 + similarly reduce to 2 by 2 games of type A’, as implied by Theorem 10.2. They are numbers 7, 19, 31 and 48. The four having diagonals — x 0 y +, numbers 24, 28, 41 and 45, reduce to 3 by 3, as implied by Theorem 8.1.