Chapter 9.1, Problem 25E

Consider the matrix game $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ where A has no saddle point.

1. a. Find a formula for the optimal strategies $\hat{\text{x}}$ for R and $\hat{\text{y}}$ for C. What is the value of the same?
2. b. Let $J=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ and let α and β be real numbers with α ≠ 0. Use your answer in part (a) to show that the optimal strategies for the matrix game B = αA + βJ are the same as for A. In particular, note that the optimal strategies for A and A + βJ are the same.