Games with ±1 as central diagonal element

Abstract

When the central diagonal element is ±1, the facts are considerably simpler. It again appears to be the case that unless both +1 and -1 occur on the diagonal, the game is irreducible. We shall show that when both do occur, the game always reduces to the 2 by 2 game $\left[ {\begin{array}{*{20}{c}} { - 1}&v \\ 1&{ - 1} \end{array}} \right]{\text{ }}or{\text{ }}\left[ {\begin{array}{*{20}{c}} 1&{ - 1} \\ { - v}&1 \end{array}} \right]$ according as the central diagonal element is +1 or -1. Let us denote the diagonal elements (x₁, x₂,..., x_2n+1).