Solutions Manual to Accompany Game Theory: An Introduction, 2nd Edition

The Main Definitions and Theorems

Appendix A: Matrix Two-Person Games

Definition A.1    A matrix game with matrix A_n×m = (a_ij) 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 a_i*,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 ...