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

Appendix B: Solution Methods for Matrix Games

Theorem B.1    In the 2 × 2 game with matrix A, assume that there are no pure optimal strategies. If we set

then X* = (x*, 1 −x*), Y* = (y*, 1 − y*) are optimal mixed strategies for players I and II, respectively. The value of the game is

Theorem B.2    Assume that

1. A_n×n has an inverse A⁻¹;

2.

3. v(A) ≠ 0.

Set X = (x₁, . . ., x_n), Y = (y₁, . . ., y_m), and

If x_i ≥ 0, i = 1, . . . ,n and y_j ≥ 0, j = 1, . . ., n, we have that v = v(A) is the value of the game with matrix A and (X,Y) is a saddle point in mixed strategies.

Definition B.3    A game is completely mixed if every saddle point consisting of strategies X = (x₁, . . ., x_n) S_n, Y = (y₁, . . ., y_m) S_m satisfies the property x_i > 0, i = 1, 2, . . ., n and y_j > 0, j = 1, 2, . . ., m. Every row and every column is used with positive probability.

B.1   Linear Programming Methods

B.1.1   METHOD ...