**Definition A.1** *A matrix game with matrix A*_{n×m} = (*a _{ij}*)

*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 a _{i*,j*}* =

**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 ...*

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