Chapter 11.2, Problem 50E

Viewer ratings A city has two competitive television stations, station $R$ and station $C$. Every month, each station makes exactly one choice for the Thursday $8-9\text{ P}\text{.M}$. time slot from the program categories shown in the following matrix. Each matrix entry is an average viewer index rating gain (or loss) established by a rating firm using data collected over the past $5$ years. (Any $R$ gain for station is a loss for station $C$, and vice versa.)

$\begin{array}{l}C\\ \begin{array}{cccc}\begin{array}{l}\text{Nature}\\ \text{films}\end{array}& \begin{array}{l}\text{Talks}\\ \text{Shows}\end{array}& \begin{array}{l}\text{Sports}\\ \text{Events}\end{array}& \begin{array}{l}\\ \text{Movies}\end{array}\end{array}\\ R\begin{array}{c}\text{Travel}\\ \text{News}\\ \text{Sitcoms}\\ \text{Soaps}\end{array}\left[\begin{array}{c}0\\ 1\\ 2\\ 1\end{array}\begin{array}{c}2\\ 0\\ 3\\ -2\end{array}\begin{array}{c}-1\\ -1\\ -1\\ 0\end{array}\begin{array}{c}0\\ -2\\ 1\\ 0\end{array}\right]\end{array}$

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

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

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

(D) What is the expected value of the game for $R$ if station $R$ always chooses sitcoms and station $C$ always chooses sports events?