Find the steady-state vector for the transition matrix. $$ \left[\begin{array}{ll} \frac{4}{5} & \frac{3}{5} \\ \frac{1}{5} & \frac{2}{5} \end{array}\right] $$

Find the optimal strategies, \(P\) and \(Q\), for the row and column players, respectively. Also compute the expected payoff \(E\) of each matrix game and determine which player it favors, if any, if the row and column players use their optimal strategies. \(\left[\begin{array}{rr}2 & 5 \\ -2 & 4\end{array}\right]\)

Find the steady-state vector for the transition matrix. $$ \left[\begin{array}{lll} .1 & .2 & .3 \\ .1 & .2 & .3 \\ .8 & .6 & .4 \end{array}\right] $$

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

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?

From data collected over the past decade by the Association of Realtors of a certain city, the following transition matrix was obtained. The matrix describes the buying pattern of home buyers who buy single-family homes \((S)\) or condominiums \((C)\). Currently, \(80 \%\) of the homeowners live in single- family homes, whereas \(20 \%\) live in condominiums. If this trend continues, what percentage of homeowners in this city will own single-family homes and condominiums 2 decades from now? In the long run?