# Chapter 9: Chapter 9

Problem 5

Determine which of the matrices are stochastic. $\left[\begin{array}{rrr}.3 & .2 & .4 \\ .4 & .7 & .3 \\ .3 & .1 & .2\end{array}\right]$

Problem 5

Determine the maximin and minimax strategies for each two-person, zero-sum matrix game. $$ \left[\begin{array}{rrr} 3 & 2 & 1 \\ 1 & -2 & 3 \\ 6 & 4 & 1 \end{array}\right] $$

Problem 6

Determine which of the matrices are stochastic. $\left[\begin{array}{rrr}\frac{1}{3} & \frac{1}{4} & \frac{1}{2} \\\ \frac{1}{3} & 0 & -\frac{1}{2} \\ \frac{1}{4} & \frac{3}{4} & \frac{1}{2}\end{array}\right]$

Problem 6

Determine whether the matrix is an absorbing stochastic matrix. $\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & \frac{5}{8} & 0 & \frac{1}{6} \\\ 0 & \frac{1}{8} & 1 & 0 \\ 0 & \frac{1}{4} & 0 & \frac{5}{6}\end{array}\right]$

Problem 6

Find the expected payoff \(E\) of each game whose payoff matrix and strategies \(P\) and \(Q\) (for the row and column players, respectively) are given. $\left[\begin{array}{rrr}1 & -4 & 2 \\ 2 & 1 & -1 \\ 2 & -2 & 0\end{array}\right], P=\left[\begin{array}{lll}.2 & .3 & .5\end{array}\right], Q=\left[\begin{array}{r}.6 \\ .2 \\ .2\end{array}\right]$

Problem 6

Determine which of the matrices are regular. $$ \left[\begin{array}{lll} 1 & .3 & .1 \\ 0 & .4 & .8 \\ 0 & .3 & .1 \end{array}\right] $$

Problem 6

Determine the maximin and minimax strategies for each two-person, zero-sum matrix game. $$ \left[\begin{array}{rr} 1 & 4 \\ 2 & -2 \\ 3 & 0 \end{array}\right] $$

Problem 7

Determine whether the matrix is an absorbing stochastic matrix. $\left[\begin{array}{llll}1 & 0 & .3 & 0 \\ 0 & 1 & .2 & 0 \\ 0 & 0 & .1 & .5 \\ 0 & 0 & .4 & .5\end{array}\right]$

Problem 7

Determine which of the matrices are regular. $$ \left[\begin{array}{rrr} .7 & .2 & .3 \\ .3 & .8 & .3 \\ 0 & 0 & .4 \end{array}\right] $$

Problem 7

The payoff matrix for a game is given by $$ \left[\begin{array}{rr} 1 & -2 \\ -2 & 3 \end{array}\right] $$ Compute the expected payoffs of the game for the pairs of strategies in parts (a-d). Which of these strategies is most advantageous to \(R\) ? a. $P=\left[\begin{array}{ll}1 & 0\end{array}\right], Q=\left[\begin{array}{l}1 \\ 0\end{array}\right]$ b. $P=\left[\begin{array}{ll}0 & 1\end{array}\right], Q=\left[\begin{array}{l}1 \\ 0\end{array}\right]$ c. $P=\left[\begin{array}{ll}\frac{1}{2} & \frac{1}{2}\end{array}\right], Q=\left[\begin{array}{l}\frac{1}{2} \\ \frac{1}{2}\end{array}\right]$ d. $P=\left[\begin{array}{ll}.5 & .5\end{array}\right], Q=\left[\begin{array}{l}.8 \\ .2\end{array}\right]$