Games with saddle points

The theorems in [7] dealing with classes 1A, 2A and 2B do not depend on the strategy sets being disjoint, and include all Silverman games where at least one player has an optimal pure strategy, except the symmetric 1 by 1 case: THEOREM 2.1. In the symmetric Silverman game (S,T,ν), suppose that there is an element c in S such that c < Tc_i for all c_i in S, and that S n (c,Tc) = Φ. Then pure strategy c is optimal.PROOF. Let A(x,y) be the payoff function. By symmetry the game value is 0. Since A(c,y) = 1,0 or ν according as y < c, y = c or y ≥ Tc, we have A(c, y) ≥ 0 for every y in S.