Games with convex payoff function in the first variable (Q1113812)

The game \(\Gamma\) over the unit square whose payoff function M(x,y) is bounded and convex in x for each y is studied. Another game \(\Gamma\) ' is constructed which has the same value and whose optimal strategies correspond to optimal strategies in \(\Gamma\). The payoff function in the game \(\Gamma\) ' is concave in the first variable and continuous. Therefore the results dual to those published by \textit{F. Bohnenblust, S. Karlin} and \textit{L. S. Shapley} [in: Contributions to the Theory of Games, Ann. Math. Studies 24, 181-192 (1950)] for games convex in y can be applied to the game \(\Gamma\) '. Thus characteristics of optimal strategies in the game \(\Gamma\) and a constructive method to find these strategies can be obtained.