Definition A.1 A matrix game with matrix An×m = (aij) has the lower value
and the upper value
The lower value υ− is the smallest amount that player I is guaranteed to receive (υ− is player I’s gain floor), and the upper value υ+ is the guaranteed greatest amount that player II can lose (υ+ is player II’s loss ceiling). The game has a value if υ− = υ+, and we write it as υ = υ(A) = υ+ = υ−. This means that the smallest max and the largest min must be equal and the row and column i*, j* giving the payoffs ai*,j* = υ+ = υ− are optimal, or a saddle point in pure strategies.
Definition A.2 We call a particular row i* and column j* a saddle point in pure strategies of the game if
Theorem A.3 Let f : C × D → be a continuous function. Let C ⊂ and D ⊂ be convex, closed, and bounded. Suppose that x f (x, y) is concave ...