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Appendix A: Matrix Two-Person Games

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 Dbe convex, closed, and bounded. Suppose that x f (x, y) is concave ...

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