Chapter 10.2, Problem 47E

In Problems 47 and 48, derive the formulas of Theorem 4 for the solution of any $2\times 2$ non strictly determined matrix game by rewriting and analyzing

$E\left(P,Q\right)=a{p}_1{q}_2+b{p}_1{q}_2+c{p}_2{q}_1+d{p}_2{q}_2$ (4)

(See the solution of the two-finger Morra game on pages G9-G11.)

(A) Let $p_2=1-p_1$ and $q_2=1-q_1$ and simplify (4) to show that

$E\left(P,Q\right)=\left[D{p}_1-\left(d-c\right)\right]q_1+\left(b-d\right)p_1+d$ where $D=\left(a+d\right)-\left(b+c\right)$.

(B) Show that if $p_1$ is chosen so that $D{p}_1-\left(d-c\right)=0$, then $v=\frac{ad-bc}{D}$, regardless of the value of $p_1$.