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. An×n has an inverse A−1;
2.
3. v(A) ≠ 0.
Set X = (x1, . . ., xn), Y = (y1, . . ., ym), and
If xi ≥ 0, i = 1, . . . ,n and yj ≥ 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, . . ., xn) Sn, Y = (y1, . . ., ym) Sm satisfies the property xi > 0, i = 1, 2, . . ., n and yj > 0, j = 1, 2, . . ., m. Every row and every column is used with positive probability.
B.1 Linear Programming Methods
B.1.1 METHOD ...
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