MGF 1107/ Classroom examples/ Chapter 15

1. In early 1943, the Japanese controlled the northern half of the island of New Guinea, while the allies controlled the southern half. Intelligence reports indicated that the Japanese were assembling a convoy to reinforce their troops on the island. The convoy would take one of two routes:

(1) north of New Britain, where rain and bad visibility were predicted, or

(2) south, where the weather was expected to be fair.

It was estimated that the trip would take 3 days on either route. Upon receiving these intelligence estimates, the Supreme Allied Commander, General Douglas McArthur, ordered General George C. Kenney, commander of the Allied Air Forces in the Southwest Pacific Area, to inflict maximum possible damage on the Japanese convoy. Kenney had the choice of sending the bulk of his reconnaissance aircraft on either the southern or northern route. His objective was to maximize the expected number of days the convoy could be bombed, so he wanted to have his aircraft find the convoy as quickly as possible. Consequently, Kenney had two choices:

(1) use most of his aircraft to search the northern route, or

(2) focus his search in the south.

The payoff would then be measured by the expected number of days Kenney would have at his disposal to bomb the convoy. The overall situation facing the two commanders in what came to be known as the Battle of Bismarck Sea can be represented as follows, where the entries indicate the expected number of days of bombing by Kenney following detection of the Japanese convoy.

		Japanese
		Sail north	Sail South
Allies	Search North	2 days	2 days
	Search South	1 day	3 days

What is the best strategy for each commander?

2. Consider the following two-person game between players ROW and COLUMN. ROW has two possible strategies R₁ and R₂. COLUMN has the strategies C₁ and C₂.

a) What is the minimax of this game?

b) What is the maximin of this game?

c) Is the minimax equal to the maximin?

d) Is there an element of this matrix that is both a row minimum and a column maximum?

e) Does this game have a saddlepoint?

f) What strategies should ROW and COLUMN use?

g) What is the value of this game?

h) Is this a fair game?

3. Every weekend, Colonel Blotto’s forces attempt to defend Rome and Carthage, but his troops are strong enough to protect only one city. On the other hand, although the Huns constitute a horde 10,000 strong, they are capable of attacking only one city at a time. Colonel Blotto has estimated that if he defends the city where the Huns attack, he can save six million people. On the other hand, if he defends Carthage, but the Huns attack Rome, he can save five million people. Finally, if he defends Rome, but the Huns attack Carthage, then he can save only one million people. Find the best strategy for each player.

4. If both players use the optimal strategies in the previous example, what is the expected payoff for each play of the game?

5. Suppose a baseball pitcher throws only two pitches: a fastball and a curve. Many hitters try to guess what pitch will be thrown so the batter’s two strategies are also Fastball and Curve. The hitter’s batting averages are given in the matrix.

		Pitcher	throws
		Fastball	Curve
Batter	Fastball	.300	.200
guesses	Curve	.150	.250

Solve the game.

6. Find the value of the previous game.

7. There is a dispute between Florida and Mississippi over offshore drilling rights in the Gulf of Mexico. It is an election year and both parties have called simultaneous news conferences to announce their positions on the dispute. The leaders of both parties calculate what they think will happen under certain pairs of strategies and arrive at the following matrix, whose payoffs are the percentage of the vote that will go to the Republicans.

	Democrats take Florida’s side	Democrats take Mississippi’s side	Democrats remain neutral
Republicans take Florida’s side	45%	50%	40%
Republicans take Mississippi’s side	60%	55%	50%
Republican’s remain neutral	45%	55%	40%

What is each party’s best strategy?

8. What is the best strategy for each player in the following game?

	C1	C2	C3
R1	3	4	0
R2	5	6	2
R3	4	1	3

9. Solve the following game and then find its value.

	C1	C2	C3	C4	C5
R1	2	0	4	4	1
R2	4	5	5	3	4

10. Find the optimal strategy for each player in the following game:

	C1	C2
R1	-2	2
R2	-1	1
R3	2	0
R4	3	-1
R5	4	-2

11. Two persons accused of being partners in a crime are arrested and placed in separate cells so they cannot communicate with each other. Without a confession from one of the suspects, the district attorney has insufficient evidence to convict them of the crime. Both prisoners have the option of remaining silent or squealing on their partner. To extract a confession, the district attorney tells each suspect the following consequences of his and his partner’s actions.

a) If one suspect confesses and his partner does not, the one who confesses can go free and the other gets a stiff 10-year sentence.

b) If both suspects confess, they each get a reduced sentence of 5 years.

c) If both suspects remain silent, they each go to jail for one year on a lesser charge.

The payoff matrix is:

	Don’t confess	Confess
Don’t confess	(-1, -1)	(-10, 0)
Confess	(0, -10)	(-5, -5)

What is the best strategy for each prisoner?

12. Two countries, India and Pakistan, are involved in an arms race. They each have two strategies, continue (C) the arms race or desist (D). Looking at it from India’s point of view, there are four possible outcomes, which we rank from most desirable (4) to least desirable (1).

4. India chooses C and Pakistan chooses D giving India a big advantage in arms.

3. Both nations desist, so neither gains an arms advantage, and resources that would have been spent on armaments can now be diverted to socially useful projects.

2. Both nations continue, so neither gains an arms advantage, but budgets and diplomatic relations are strained.

1. India desists and Pakistan continues to arm, putting India at a military disadvantage.

Pakistan has the symmetrical ranking yielding the following payoff matrix.

	Pakistan D	Pakistan C
India D	(3, 3)	(1, 4)
India C	(4, 1)	(2, 2)

What is the best strategy for each country?

13. In the game called Chicken, two drivers drive towards each other on a collision course. Both players have the option of being a “chicken” by swerving to avoid the collision, or of continuing on the deadly course. There are four possible outcomes, which we rank from most desirable (4) to least desirable (1).

4. You don’t swerve and your opponent swerves. (You were brave, your opponent was chicken, and no one was hurt.)

3. You both swerve. (Both players lose face, but nobody is hurt.)

2. You swerve, but your opponent doesn’t. (Only you lose face, but nobody is hurt.)

1. Neither swerves. (Nobody loses face, but both may die.)

In the 1955 cult film Rebel Without a Cause, Jimbo, the character played by James Dean, and his high school nemesis Buzz played a version of Chicken, the only difference being that they both drove towards a cliff, the winner being the one to dive out of his car last before it went over the cliff. We use the rankings above to obtain the following payoff matrix:

	Buzz swerves	Buzz doesn’t swerve
Jimbo swerves	(3, 3)	(2, 4)
Jimbo doesn’t swerve	(4, 2)	(1, 1)

The first number in each ordered pair is the payoff to Jimbo, while the second number is the payoff to Buzz. What is the best strategy for each player?

14. In October 1962, the Soviet Union tried to place nuclear-armed missiles in Cuba. After considering several options, President Kennedy narrowed his strategies to two: a naval blockade or an air strike. Soviet Premier Khruschev also had two choices: withdraw the missiles or leave them in place. The four possible outcomes can be summarized as:

	USSR withdraws missiles	USSR maintains missiles
U.S. naval blockade	Compromise	USSR victory
U.S. air strike	U.S. victory	Nuclear war

Rank the four possible outcomes from best (4) to worst (1) for each country to obtain a payoff matrix.

15. Two drivers are waiting to enter an intersection at a 4-way stop. Each driver has a choice of driving (D) into the intersection or waiting (W) for the other to go first.

a) Rank the four possible outcomes from best (4) to worst (1) for each driver to obtain a payoff matrix.

b) Does either player have a dominant strategy?

c) What is the minimax strategy for each player?

d) Find all Nash equilibria.

e) Can the players benefit by cooperating?

16. A married couple has to choose between two options for their evening entertainment. The husband prefers going to a baseball game whereas his wife prefers going shopping at the mall. The problem is that they would both prefer going out together than alone.

a) Rank the four possible outcomes from best (4) to worst (1) for each spouse to complete the following payoff matrix.

	Wife chooses baseball	Wife chooses mall
Husband chooses baseball
Husband chooses mall

b) Does either player have a dominant strategy?

c) What is the minimax strategy for each player?

d) Find all Nash equilibria.

e) Can the players benefit by cooperating?

17. What is the best strategy for each player in the following game?

	C1	C2
R1	(4, 4)	(2, 3)
R2	(3, 2)	(1, 1)

18. An election has 3 voters X, Y, and Z and three alternatives x, y and z. The method of voting used is plurality and the chair X has the tie-breaking vote. The preference schedules for each voter are X prefers x to y to z, indicated by xyz, Y’s preference is yzx and Z’s preference is zxy. What is the outcome of this election under sophisticated voting?

19. In the previous example, which of the following outcomes are Nash equilibria? Justify your answer.

a) X votes for x, Y votes for z, and Z votes for z.

b) X votes for z, Y votes for z, and Z votes for x.

20. In the previous two examples, can tacit or revealed deception help the chair?