Exam 9: Optimization Online Only

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, $\frac { 3 } { 4 }$ of the supplies will get through, but 9 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. What is the optimal strategy for the guerrillas if they want to inflict maximum losses on the army?

Find all saddle points for the matrix game. - $\left[ \begin{array} { r r } - 7 & 4 \\3 & 6\end{array} \right]$

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

Find the value of the strategy. -Let $M$ be the matrix game having payoff matrix $\left[ \begin{array} { r r r } 0 & - 4 & 2 \\ 1 & 5 & - 4 \\ - 2 & 1 & 0 \end{array} \right]$ . Find $v ( \mathbf { y } )$ when $\mathbf { y } = \left[ \begin{array} { c } \frac { 1 } { 3 } \\ \frac { 1 } { 2 } \\ \frac { 1 } { 6 } \end{array} \right]$ .

Find the value of the strategy. -Let $M$ be the matrix game having payoff matrix $\left[ \begin{array} { r r r } 0 & - 2 & 3 \\ - 2 & 1 & 5 \\ - 2 & 1 & 0 \end{array} \right]$ . Find $v ( x )$ when $x = \left[ \begin{array} { c } \frac { 1 } { 3 } \\ \frac { 1 } { 2 } \\ \frac { 1 } { 6 } \end{array} \right]$ .

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

Write the payoff matrix for the game. -In the traditional Japanese childrenʹs game janken (or ʺstone, scissors, paperʺ), at a given signal, each of two players shows either no fingers (stone), two fingers (scissors), or all five fingers (paper). Stone beats scissors, scissors beats paper, and paper beats stone. In the case of a tie, there Is no payoff. In the case of a win, the winner collects 50 yen.

Write the payoff matrix for the game. -Player R has two cards: a red 2 and a black 7. Player C has three cards: a red 5, a black 6, 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.

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 row strategy.

Find the value of the strategy. -Let $M$ be the matrix game having payoff matrix $\left[ \begin{array} { r r r } 0 & - 4 & 3 \\ - 4 & 1 & 3 \\ 2 & - 2 & 0 \end{array} \right]$ . Find $v ( x )$ when $x = \left[ \begin{array} { c } \frac { 1 } { 4 } \\ \frac { 1 } { 4 } \\ \frac { 1 } { 2 } \end{array} \right]$ .

Find all saddle points for the matrix game. - $\left[ \begin{array} { r r r r } 0 & 3 & - 1 & 4 \\0 & - 4 & - 1 & 5 \\2 & 8 & 2 & 5\end{array} \right]$

Solve the problem. -A matrix game has payoff matrix $\left[ \begin{array} { l l l } a _ { 11 } & a _ { 12 } & a _ { 13 } \\a _ { 21 } & a _ { 22 } & a _ { 23 }\end{array} \right]$ in which entry a 21 is a saddle point. What are the optimal strategies for player $\mathrm { R }$ and player C?

Find the value of the matrix game. -Let $\mathrm { M }$ be the matrix game having payoff matrix $\left[ \begin{array} { r r r r } 2 & 2 & - 2 & 5 \\ - 4 & 2 & 6 & 2 \end{array} \right]$ .

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 4 army soldiers will be lost. If the convoy goes overland and encounters the guerrillas, $\frac { 1 } { 3 }$ of the supplies will get through, but 5 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. What is the optimal strategy for the army if it wants to maximize the amount of supplies it gets to its troops?

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 4 army soldiers will be lost. If the convoy goes overland and encounters the guerrillas, $\frac { 1 } { 4 }$ of the supplies will get through, but 6 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. What is the optimal strategy for the army if it wants to minimize its casualties?

Find the optimal row or column strategy of the matrix game. -Let $\mathrm { M }$ be the matrix game having payoff matrix $\left[ \begin{array} { r r } 1 & 0 \\ - 3 & 6 \end{array} \right]$ . Find the optimal row strategy.

Solve the problem. -In a matrix game with payoff matrix A, how can you find the value ν(y)of a strategy y to player C?

Write the payoff matrix for the game. -Each player has a supply of nickels, dimes, and quarters. At a given signal, both players display one coin. If the displayed coins are not the same, then the player showing the higher valued coin Gets to keep both. If they are both nickels or dimes, then player R keeps both; but if they are both Quarters, then player C keeps both.

Solve the problem. -Each player has a supply of nickels, dimes, and quarters. At a given signal, both players display one coin. If the displayed coins are not the same, then the player showing the higher valued coin Gets to keep both. If they are both nickels or dimes, then player R keeps both; but if they are both Quarters, then player C keeps both. Find the optimal strategy for player C.

Find the value of the matrix game. -Let $M$ be the matrix game having payoff matrix $\left[ \begin{array} { c r c } 4 & - 2 & 4 \\ - 1 & 5 & 5 \\ - 7 & 0 & 4 \end{array} \right]$ .