Chapter 11.2, Problem 49E

Bank promotion A town has only two banks, bank $R$ band bank $C$, and both compete equally for the town’s business. Every week, each bank decides on the use of one, and only one, of the following means of promotion: TV, radio, newspaper. and mail. A market research firm provided the following payoff matrix, which indicates the percentage of market gain or loss for each choice of action by $R$ and by $C$ (we assume that any gain by $R$ is a loss by $C$, and vice versa):

$\begin{array}{l}C\\ \begin{array}{cccc}\text{TV}& \text{Radio}& \text{Paper}& \text{Mail}\end{array}\\ R\begin{array}{c}\text{TV}\\ \text{Radio}\\ \text{Paper}\\ \text{Mail}\end{array}\text{}\left[\begin{array}{c}0\\ 1\\ 0\\ -1\end{array}\begin{array}{c}-1\\ 2\\ -1\\ -1\end{array}\begin{array}{c}-1\\ -1\\ 0\\ -1\end{array}\begin{array}{c}0\\ -1\\ 1\\ 0\end{array}\right]\end{array}$

(A) Find optimal strategies for bank $R$ and bank $C$. What is the value of the game?

(B) What is the expected value of the game for $R$ if bank $R$ always chooses TV and bank $C$ uses its optimal strategy?

(C) What is the expected value of the game for $R$ if bank $C$ always chooses radio and bank $R$ uses its optimal strategy?

(D) What is the expected value of the game for $R$ if both banks always use the newspaper?