Chapter 9, Problem 17RE

(a)

To determine

The reason that the intersection of the lines $y=a_{11}r+a_{21}\left(1-r\right)$ and $y=a_{12}r+a_{22}\left(1-r\right)$ gives the value of $r$ which is best for $R$ if it uses the strategy $\left[\begin{array}{cc}r& 1-r\end{array}\right]$. The payoff matrix is provided by $\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$. In case $C$ chooses the column $1$ from the matrix then $R$ expects a return of $a_{11}r+a_{21}\left(1-r\right)$ and if $C$ chooses the column $2$ then $R$ expects a return of $a_{12}r+a_{22}\left(1-r\right)$.

(b)

To determine

The procedure to determine the best strategy for $C$ with the help of comparable argument. The two lines are $y=a_{11}r+a_{21}\left(1-r\right)$ and $y=a_{12}r+a_{22}\left(1-r\right)$ which gives the value of $r$ that is best for $R$. $R$ uses the strategy $\left[\begin{array}{cc}r& 1-r\end{array}\right]$. The payoff matrix is provided by $\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$. In case $C$ chooses the column $1$ from the matrix then $R$ expects a return of $a_{11}r+a_{21}\left(1-r\right)$ and if $C$ chooses the column $2$ then $R$ expects a return of $a_{12}r+a_{22}\left(1-r\right)$.