0.19 Game theory: homework  (Page 2/3)

Find the expected payoff for the given game matrix $G$ if the row player plays strategy $R$ , and column player plays strategy $C$ .

1. $G=\left[\begin{array}{cc}-3& 2\\ 1& -4\end{array}\right]R=\left[\begin{array}{cc}2/3& 1/3\end{array}\right]C=\left[\begin{array}{c}1/4\\ 3/4\end{array}\right]$

2. $G=\left[\begin{array}{cc}-1& 1\\ 1& -1\end{array}\right]R=\left[\begin{array}{cc}1/3& 2/3\end{array}\right]C=\left[\begin{array}{c}2/3\\ 1/3\end{array}\right]$

Two players play a game which involves holding out one or two fingers simultaneously. If the sum of the fingers is even, Player II pays Player I the sum of the fingers. If the sum of the fingers is odd, Player I pays Player II the sum of the fingers.

1. Write a payoff matrix for Player I.

2. Find the optimal strategies for both the row player and the column player, and the value of the game.

In December 1995, President Clinton ordered the first of 20,000 U. S. troops to be sent into Bosnia-Herzegovina as a peace keeping force. Unfortunately, the heavy fog made visibility very poor at the Tuzla airfield, and at the same time increased the threat of sniper attacks from the Serbian forces. U. S. Air Force Col. Neal Patton, and Lt. Col. Sid Kooyman, the advance specialists, had two choices: either to send in the troops by air with the difficulties already described or by road thus exposing the troops to ambush by the Serbian forces. The Serbian army, with its limited resources, had a choice of deploying its forces near the airport or along the road route.

If the U. S. lands its troops on the airfield in the fog while the Serbs are concentrating on the road route, the payoff for U. S. is 20 points. But if the U. S. lands its troops on the airfield, and Serbians are there hiding in the fog, U. S. wins only 5 points. On the other hand, if U. S. transports its troops by road and avoids Serbs its payoff is 35 points, but if U. S. meets Serb resistance on the road route, it loses 50 points.

1. Write a payoff matrix for the game.

2. If you were Air Force Col. Neal Patton's advisor, what advice would you give him?

Determine whether the games are strictly determined. If the games are strictly determined, find the optimal strategies for each player and the value of the game.

1. $\left[\begin{array}{cc}2& 3\\ 3& 4\end{array}\right]$

2. $\left[\begin{array}{ccc}0& 3& -1\\ 1& 3& -2\\ -1& 2& -5\end{array}\right]$

3. $\left[\begin{array}{ccc}3& 2& -1\\ 5& 3& 4\end{array}\right]$

4. $\left[\begin{array}{cc}4& 2\\ -1& 3\\ 4& 3\\ 1& -3\end{array}\right]$

Two players play a game which involves holding out a nickel or a dime simultaneously. If the sum of the coins is more than 10 cents, Player I gets both the coins; otherwise, Player II gets both the coins.

1. Write a payoff matrix for Player I.

2. Find the optimal strategies for each player and the value of the game.

Lacy's department store is thinking of having a major sale in the month of February, but does not know if its competitor store Hordstrom's is also planning one. If Lacy's has a sale and Hordstrom's does not, Lacy's sales go up by 30%, but if both stores have a sale simultaneously, Lacy's sales go up by only 5%. On the other hand, if Lacy's does not have a sale and Hordstrom's does, Lacy's loses 5% of its sales to Hordstrom's, and if neither of the stores has a sale, Lacy's experiences no gain in sales.

1. Write a payoff matrix for Lacy's.

2. Find the optimal strategies for both stores.