Chapter 10.2, Problem 39E

You $\left(R\right)$ and a friend $\left(C\right)$ are playing the following matrix game, where the entries indicate your winnings from $C$ in dollars .In order to encourage your friend to play, since you cannot lose as the matrix is written, you pay her $\text{\$}3$ before each game.

$\begin{array}{l}C\\ R\left[\begin{array}{cccc}0& 2& 1& 0\\ 4& 3& 5& 4\\ 0& 2& 6& 1\\ 0& 1& 0& 3\end{array}\right]\end{array}$

(A) If selection of a particular row or a particular column is made at random, what is your expected value?

(B) If you make whatever row choice you wish and your opponent continues to select a column at random, what is your expected value, assuming that you optimize your choice?

(C) If you both make your own choices, what is your expected value, assuming that you both optimize your choices?