Exam 9: Optimization Online Only

Solve the problem. -The optimal strategy for a $2 \times 5$ matrix game can be found as follows: obtain 5 linear functions by finding the inner product of $x ( t ) = \left[ \begin{array} { c } 1 - t \\ t \end{array} \right]$ with each of the columns of the payoff matrix A. Graph the 5 linear functions on a t-z coordinate system. Then $v ( x ( t ) )$ is the minimum value of the 5 linear functions which will be seen on the graph as a polygonal path. The z-coordinate of any point on this path is the minimum of the corresponding $z$ coordinates of points on the 5 lines. The highest point on the path $v ( x ( t ) )$ is M. Suppose that only the lines corresponding to columns 1 and 4 of matrix $A$ pass through the point $\mathrm { M }$ . What can be said about the optimal column strategy $\hat { y }$ ?

Find the value of the strategy. -Let $M$ be the matrix game having payoff matrix $\left[ \begin{array} { r r r r } 0 & 3 & - 1 & - 3 \\ 3 & 0 & 3 & 0 \\ - 1 & 2 & 0 & - 1 \end{array} \right]$ . Find $v ( x )$ when $x = \left[ \begin{array} { c } 0 \\ \frac { 2 } { 3 } \\ \frac { 1 } { 3 } \end{array} \right]$

Solve the problem. -In certain situations, a matrix game can be reduced to a smaller game by deleting certain rows and/or columns from the payoff matrix. The optimal strategy for the reduced game will then Determine the optimal strategy for the original game. In what circumstances may a row or column Be deleted from the payoff matrix?

Find the optimal row or column strategy of the matrix game. -Let $\mathrm { M }$ be the matrix game having payoff matrix $\left[ \begin{array} { r r r } 5 & - 4 & 5 \\ - 3 & 4 & 0 \\ - 9 & - 3 & - 2 \end{array} \right]$ . Find the optimal row strategy.

Determine whether the statement is true or false. -In a matrix game, if row s is dominant to some other row in payoff matrix A, then row s will not be used in some optimal strategy for row player R.

Find all saddle points for the matrix game. - $\left[ \begin{array} { r r r } 9 & 2 & - 9 \\5 & 2 & - 4 \\- 1 & 4 & - 6\end{array} \right]$

Solve the problem. -If the payoff matrix of a matrix game contains a saddle point, what is the optimal strategy for the 45) row player?

Solve the problem. -Player R has two cards: a red 2 and a black 8. Player C has three cards: a red 3, a black 5, and a black 10. They each show one of their cards. If the cards are the same color, C receives the larger of The two numbers. If the cards are of different colors, R receives the sum of the two numbers. The Payoff matrix is : $\text { Find the optimal strategy for player } R \text {. }$

Find the expected payoff. -Let $M$ be the matrix game having payoff matrix $\left[ \begin{array} { r r r } - 1 & 5 & 3 \\ 0 & - 4 & 2 \\ - 2 & 1 & 0 \end{array} \right]$ . Find $\mathrm { E } ( \mathbf { x } , \mathbf { y } )$ when $\mathbf { x } = \left[ \begin{array} { c } \frac { 1 } { 6 } \\ \frac { 1 } { 2 } \\ \frac { 1 } { 3 } \end{array} \right]$ and $\mathbf { y } = \left[ \begin{array} { c } \frac { 1 } { 4 } \\ \frac { 1 } { 4 } \\ \frac { 1 } { 2 } \end{array} \right]$

Solve the problem. -A certain army is engaged in guerrilla warfare. It has two ways of getting supplies to its troops: it can send a convoy up the river road or it can send a convoy overland through the jungle. On a given day, the guerrillas can watch only one of the two roads. If the convoy goes along the river and the guerrillas are there, the convoy will have to turn back and 6 army soldiers will be lost. If the convoy goes overland and encounters the guerrillas, $\frac { 1 } { 4 }$ of the supplies will get through, but 8 army soldiers will be lost. Each day a supply convoy travels one of the roads, and if the guerrillas are watching the other road, the convoy gets through with no losses. If the army chooses the optimal strategy to maximize the amount of supplies it gets to its troops and the guerrillas choose the optimal strategy to prevent the most supplies from getting through, then what portion of the supplies will get through?

Find all saddle points for the matrix game. - $\left[ \begin{array} { r r r } 3 & - 2 & - 9 \\5 & 5 & 9 \\- 2 & - 1 & 2\end{array} \right]$

Find the optimal row or column strategy of the matrix game. -Let $\mathrm { M }$ be the matrix game having payoff matrix $\left[ \begin{array} { r r r r } 1 & 1 & - 1 & 5 \\ 0 & 3 & 3 & 4 \end{array} \right]$ . Find the optimal row strategy.

Find the expected payoff. -Let $M$ be the matrix game having payoff matrix $\left[ \begin{array} { r r r } - 2 & 0 & 2 \\ 2 & - 1 & 3 \\ - 2 & 1 & 0 \end{array} \right]$ . Find $E ( x , y )$ when $x = \left[ \begin{array} { c } \frac { 1 } { 4 } \\ \frac { 1 } { 2 } \\ \frac { 1 } { 4 } \end{array} \right]$ and $\mathbf { y } = \left[ \begin{array} { c } \frac { 1 } { 6 } \\ \frac { 1 } { 2 } \\ \frac { 1 } { 3 } \end{array} \right]$

Find the optimal row or column strategy of the matrix game. -Let $M$ be the matrix game having payoff matrix $\left[ \begin{array} { r r } 4 & - 1 \\ - 4 & 5 \end{array} \right]$ . Find the optimal column strategy.

Find the optimal row or column strategy of the matrix game. -Let $M$ be the matrix game having payoff matrix $\left[ \begin{array} { r r r r r } 1 & 1 & 3 & - 1 & 4 \\ 1 & - 2 & - 1 & 1 & - 2 \\ 2 & - 1 & 0 & 3 & - 2 \end{array} \right]$ . Find the optimal column strategy.