# Chapter 9: Chapter 9

Problem 7

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

Problem 8

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

Problem 8

Determine whether the matrix is an absorbing stochastic matrix. $\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & .2 & .6 \\\ 0 & 0 & .8 & .4\end{array}\right]$

Problem 8

The payoff matrix for a game is $$ \left[\begin{array}{rrr} 3 & 1 & 1 \\ 0 & 2 & 0 \\ -1 & 0 & 2 \end{array}\right] $$ Compute the expected payoffs of the game for the pairs of strategies in parts \((\mathrm{a}-\mathrm{d})\). Which of these strategies is most advantageous to \(R ?\) a. $P=\left[\begin{array}{lll}\frac{1}{3} & \frac{1}{3} & \frac{1}{3}\end{array}\right], Q=\left[\begin{array}{l}\frac{1}{3} \\\ \frac{1}{3} \\ \frac{1}{3}\end{array}\right]$ b. $P=\left[\begin{array}{lll}\frac{1}{4} & \frac{1}{2} & \frac{1}{4}\end{array}\right], Q=\left[\begin{array}{l}\frac{1}{8} \\\ \frac{3}{8} \\ \frac{1}{2}\end{array}\right]$ c. $P=\left[\begin{array}{lll}.4 & .3 & .3\end{array}\right], Q=\left[\begin{array}{l}.6 \\ .2 \\ .2\end{array}\right]$ d. $P=\left[\begin{array}{lll}.1 & .5 & .4\end{array}\right], Q=\left[\begin{array}{c}.3 \\ .3 \\ .4\end{array}\right]$

Problem 8

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

Problem 9

Determine whether the two-person, zero-sum matrix game is strictly determined. If a game is strictly determined, a. Find the saddle point(s) of the game. b. Find the optimal strategy for each player. c. Find the value of the game. d. Determine whether the game favors one player over the other. $$ \left[\begin{array}{rr} 2 & 3 \\ 1 & -4 \end{array}\right] $$

Problem 9

Find the steady-state vector for the transition matrix. $$ \left[\begin{array}{ll} \frac{1}{3} & \frac{1}{4} \\ \frac{2}{3} & \frac{3}{4} \end{array}\right] $$

Problem 9

Rewrite each absorbing stochastic matrix so that the absorbing states appear first, partition the resulting matrix, and identify the submatrices \(R\) and \(S\). \(\left[\begin{array}{ll}.6 & 0 \\ .4 & 1\end{array}\right]\)

Problem 9

The payoff matrix for a game is $$ \left[\begin{array}{rrr} -3 & 3 & 2 \\ -3 & 1 & 1 \\ 1 & -2 & 1 \end{array}\right] $$ a. Find the expected payoff to the row player if the row player \(R\) uses the maximin pure strategy and the column player \(C\) uses the minimax pure strategy. b. Find the expected payoff to the row player if \(R\) uses the maximin strategy \(50 \%\) of the time and chooses each of the other two rows \(25 \%\) of the time, while \(C\) uses the minimax strategy \(60 \%\) of the time and chooses each of the other columns \(20 \%\) of the time. c. Which of these pairs of strategies is most advantageous to the row player?

Problem 9

Determine which of the matrices are stochastic. \(\left[\begin{array}{rr}.2 & .3 \\ .3 & .1 \\ .5 & .6\end{array}\right]\)