Reduction of balanced games to odd order

Abstract

Recall that for balanced Silverman games the payoff matrix is completely determined by the diagonal, and that every diagonal element is 1, 0 or -1. The evidence strongly suggests that unless both 1 and -1 occur (and therefore all three of 1, 0, -1), the game is irreducible. If both 1 and -1 occur, with one of them in the middle position, then the game reduces to 2 by 2, as we show in Section 10. In this section and the next three, we examine the reduction for all other diagonals; i.e. those where each of 1, 0 and -1 occur on the diagonal and the middle element is 0. Those which reduce to an odd order game are treated in the present section and those reducing to even order in Section 9.