Varian Microeconomics Workout

(d) In general, what relationship between B and N must hold for \Keep"

to be a dominant strategy?

.

(e) Sometimes the action that maximizes a player's absolute payo®, does not maximize his relative payo®. Consider the example of a voluntary public goods game as described above, where B = 6 and N = 5. Suppose that four of the ¯ve players in the group contribute their $10, while the ¯fth player keeps his $10. What is the payo® of each of the four contrib-

utors?			What is the payo® of the player who keeps his
$10?				Who has the highest payo® in the group?
					What would be the payo® to the ¯fth

player if instead of keeping his $10, he contributes, so that all ¯ve players

contribute. If the other four players contribute, what

should the ¯fth player to maximize his absolute payo®?

What should he do to maximize his payo® relative to that of the other

players? .

(f) If B = 6 and N = 5, what is the dominant strategy equilibrium for this

game? Explain your answer.

.

28.4 (1) The Stag Hunt game is based on a story told by Jean Jacques Rousseau in his book Discourses on the Origin and Foundation of Inequality Among Men (1754). The story goes something like this: \Two hunters set out to kill a stag. One has agreed to drive the stag through the forest, and the other to post at a place where the stag must pass. If both faithfully perform their assigned stag-hunting tasks, they will surely kill the stag and each will get an equal share of this large animal. During the course of the hunt, each hunter has an opportunity to abandon the stag hunt and to pursue a hare. If a hunter pursues the hare instead of the stag he is certain to catch the hare and the stag is certain to escape. Each hunter would rather share half of a stag than have a hare to himself."

The matrix below shows payo®s in a stag hunt game. If both hunters hunt stag, each gets a payo® of 4. If both hunt hare, each gets 3. If one hunts stag and the other hunts hare, the stag hunter gets 0 and the hare hunter gets 3.

The Stag Hunt Game
		Hunter B
		Hunt Stag	Hunt Hare
	Hunt Stag
Hunter A		4; 4	0; 3
	Hunt Hare	3; 0	3; 3

(a) If you are sure that the other hunter will hunt stag, what is the best

thing for you to do?

.

(b) If you are sure that the other hunter will hunt hare, what is the best

thing for you to do?					.
(c) Does either hunter have a dominant strategy in this game?				If
so, what is it? If not explain why not.
					.

(d) This game has two pure strategy Nash equilibria. What are they?

.

(e) Is one Nash equilibrium better for both hunters than the other?

If so, which is the better equilibrium?

.

(f) If a hunter believes that with probability 1/2 the other hunter will hunt stag and with probability 1/2 he will hunt hare, what should this

hunter do to maximize his expected payo®?

.

28.5 (1) Evangeline and Gabriel met at a freshman mixer. They want desperately to meet each other again, but they forgot to exchange names or phone numbers when they met the ¯rst time. There are two possible strategies available for each of them. These are Go to the Big Party or Stay Home and Study. They will surely meet if they both go to the party,

and they will surely not otherwise. The payo® to meeting is 1,000 for each of them. The payo® to not meeting is zero for both of them. The payo®s are described by the matrix below.

	Close Encounters of the Second Kind
			Gabriel
			Go to Party	Stay Home
		Go to Party
	Evangeline		1000; 1000	0; 0
		Stay Home	0; 0	0; 0

(a) A strategy is said to be a weakly dominant strategy for a player if the payo® from using this strategy is at least as high as the payo® from using any other strategy. Is there any outcome in this game where both

players are using weakly dominant strategies?

.

(b) Find all of the pure-strategy Nash equilibria for this game.

.

(c) Do any of the pure Nash equilibria that you found seem more reason-

able than others? Why or why not?

.

(d) Let us change the game a little bit. Evangeline and Gabriel are still desperate to ¯nd each other. But now there are two parties that they can go to. There is a little party at which they would be sure to meet if they both went there and a huge party at which they might never see each other. The expected payo® to each of them is 1,000 if they both go to the little party. Since there is only a 50-50 chance that they would ¯nd each other at the huge party, the expected payo® to each of them is only 500. If they go to di®erent parties, the payo® to both of them is zero. The payo® matrix for this game is:

More Close Encounters
		Gabriel
		Little Party		Big Party
Evangeline	Little Party	1000; 1000		0; 0
	Big Party	0; 0		500; 500

(e) Does this game have a dominant strategy equilibrium?

What

are the two Nash equilibria in pure strategies?

.

(f) One of the Nash equilibria is Pareto superior to the other. Suppose that each person thought that there was some slight chance that the other would go to the little party. Would that be enough to convince

them both to attend the little party? Can you think of any reason why the Pareto superior equilibrium might emerge if both players understand the game matrix, if both know that the other understands it, and each knows that the other knows that he or she understands the

game matrix?

.

28.6 (1) The introduction to this chapter of Workouts, recounted the sad tale of roommates Victoria and Albert and their dirty room. The payo® matrix for their relationship was given as follows.

Domestic Life with Victoria and Albert

			Victoria
		Clean		Don't Clean
Albert	Clean	5; 5		2; 6
	Don't Clean	6; 2		3; 3

Suppose that we add a second stage to this game in which Victoria and Albert each have a chance to punish the other. Imagine that at the

end of the day, Victoria and Albert are each able to see whether the other has done any housecleaning. After seeing what the other has done, each has the option of starting a quarrel. A quarrel hurts both of them, regardless of who started it. Thus we will assume that if either or both of them starts a quarrel, the day's payo® for each of them is reduced by 2. (For example if Victoria cleans and Albert doesn't clean and if Victoria, on seeing this result, starts a quarrel, Albert's payo® will be 6 ¡ 2 = 4 and Victoria's will be 2 ¡ 2 = 0.)

(a) Suppose that it is evening and Victoria sees that Albert has chosen not to clean and she thinks that he will not start a quarrel. Which strategy will give her a higher payo® for the whole day, Quarrel or Not Quarrel?

.

(b) Suppose that Victoria and Albert each believe that the other will try to take the actions that will maximize his or her total payo® for the day.

Does either believe the other will start a quarrel? Assuming that each is trying to maximize his or her own payo®, given the actions of the other, what would you expect each of them to do in the ¯rst stage

of the game, clean or not Clean? .

(c) Suppose that Victoria and Albert are governed by emotions that they cannot control. Neither can avoid getting angry if the other does not clean. And if either one is angry, they will quarrel so that the payo® of each is diminished by 2. Given that there is certain to be a quarrel if either does not clean, the payo® matrix for the game between Victoria and Albert becomes:

Vengeful Victoria and Angry Albert

			Victoria
		Clean		Don't Clean
Albert	Clean	5; 5		0; 4
	Don't Clean	4; 0		1; 1

(d) If the other player cleans, is it better to clean or not clean?

If the other player does not clean, is it better to clean or not clean.

Explain

.

(e) Does this game have a dominant strategy?

Explain

.

(f) This game has two Nash equilibria. What are they?

.

(g) Explain how it could happen that Albert and Victoria would both be better o® if both are easy to anger than if they are rational about when to get angry, but it might also happen that they would both

be worse o®.

.

(h) Suppose that Albert and Victoria are both aware that Albert will get angry and start a quarrel if Victoria does not clean, but that Victoria is level-headed and will not start a quarrel. What would be the

equilibrium outcome?

.

28.7 (1) Maynard's Cross is a trendy bistro that specializes in carpaccio and other uncooked substances. Most people who come to Maynard's come to see and be seen by other people of the kind who come to Maynard's. There is, however, a hard core of 10 customers per evening who come for the carpaccio and don't care how many other people come. The number of additional customers who appear at Maynard's depends on how many people they expect to see. In particular, if people expect that the number of customers at Maynard's in an evening will be X, then the number of people who actually come to Maynard's is Y = 10 + :8X: In equilibrium, it must be true that the number of people who actually attend the restaurant is equal to the number who are expected to attend.

(a) What two simultaneous equations must you solve to ¯nd the equilib-

rium attendance at Maynard's?

.

(b) What is the equilibrium nightly attendance?		.

(c) On the following axes, draw the lines that represent each of the two equations you mentioned in Part (a). Label the equilibrium atten-

dance level.

y

80

60

40

20

0	20	40	60	80
				x

(d) Suppose that one additional carpaccio enthusiast moves to the area. Like the other 10, he eats at Maynard's every night no matter how many others eat there. Write down the new equations determining attendance at Maynard's and solve for the new equilibrium number of customers.

.

(e) Use a di®erent color ink to draw a new line representing the equation that changed. How many additional customers did the new steady

customer attract (besides himself)?

.

(f) Suppose that everyone bases expectations about tonight's attendance on last night's attendance and that last night's attendance is public knowledge. Then Xt = Yt¡1, where Xt is expected attendance on day t and Yt¡1 is actual attendance on day t ¡ 1. At any time t, Yt = 10 + :8Xt. Suppose that on the ¯rst night that Maynard's is open, attendance is 20.

What will be attendance on the second night?		.

(g) What will be the attendance on the third night?		.

(h) Attendance will tend toward some limiting value. What is it?		.

28.8 (0) Yogi's Bar and Grill is frequented by unsociable types who hate crowds. If Yogi's regular customers expect that the crowd at Yogi's will be X, then the number of people who show up at Yogi's, Y , will be the larger of the two numbers, 120 ¡2X and 0. Thus Y = maxf120 ¡2X; 0g:

(a) Solve for the equilibrium attendance at Yogi's. Draw a diagram depicting this equilibrium on the axes below.

y

80

60

40

20

0	20	40	60	80
				x

(b) Suppose that people expect the number of customers on any given night to be the same as the number on the previous night. Suppose that 50 customers show up at Yogi's on the ¯rst day of business. How many

will show up on the second day?

The third day?

The

fourth day?

The ¯fth day?

The sixth day?

The ninety-ninth day?

The hundredth day?

.

(c) What would you say is wrong with this model if at least some of Yogi's

customers have memory spans of more than a day or two?

.

As we have seen, some games do not have a \Nash equilibrium in pure strategies." But if we allow for the possibility of \Nash equilibrium in mixed strategies," virtually every game of the sort we are interested in will have a Nash equilibrium.

The key to solving for such equilibria is to observe that if a player is indi®erent between two strategies, then he is also willing to choose randomly between them. This observation will generally give us an equation that determines the equilibrium.

In the game of baseball, a pitcher throws a ball towards a batter who tries to hit it. In our simpli¯ed version of the game, the pitcher can pitch high or pitch low, and the batter can swing high or swing low. The ball moves so fast that the batter has to commit to swinging high or swinging low before the ball is released.

Let us suppose that if the pitcher throws high and batter swings low, or the pitcher throws low and the batter swings high, the batter misses the ball, so the pitcher wins.

If the pitcher throws high and the batter swings high, the batter always connects. If the pitcher throws low and the batter swings low, the batter will connect only half the time.

This story leads us to the following payo® matrix, where if the batter hits the ball he gets a payo® of 1 and the pitcher gets 0, and if the batter misses, the pitcher gets a payo® of 1 and the batter gets 0.

	Simpli¯ed Baseball
		Batter
		Swing Low	Swing High
	Pitch High
Pitcher		1; 0	0; 1
	Pitch Low	:5; :5	1; 0

This game has no Nash equilibrium in pure strategies. There is no combination of actions taken with certainty such that each is making the best response to the other's action. The batter always wants to swing the same place the pitcher throws, and the pitcher always wants to throw to the opposite place. What we can ¯nd is a pair of equilibrium mixed strategies.

In a mixed strategy equilibrium each player's strategy is chosen at random. The batter will be willing to choose a random strategy only if

the expected payo® to swinging high is the same as the expected payo® to swinging low.

The payo®s from swinging high or swinging low depend on what the pitcher does. Let ¼P be the probability that the pitcher throws high and 1 ¡ ¼P be the probability that he throws low. The batter realizes that if he swings high, he will get a payo® of 0 if the pitcher throws low and 1 if the pitcher throws high. The expected payo® to the batter is therefore

¼P .

If the pitcher throws low, then the only way the batter can score is if pitcher pitches low, which happens with probability 1 ¡ ¼P . Even then the batter only connects half the time. So the expected payo® to the batter from swinging low is :5(1 ¡ ¼P ).

These two expected payo®s are equalized when ¼P = :5(1 ¡ ¼P ). If we solve this equation, we ¯nd ¼P = 1=3. This has to be the probability that the pitcher throws high in a mixed strategy equilibrium.

Now let us ¯nd the probability that the batter swings low in a mixed strategy equilibrium. In equilibrium, the batter's probability ¼B from swinging low must be such that the pitcher gets the same expected payo® from throwing high as from throwing low The expected payo® to the pitcher is the probability that the batter does not score.

If the pitcher throws high, then the batter will not connect if he swings low, but will connect if he swings high, so the expected payo® to the pitcher from pitching high is ¼B.

If the pitcher throws low, then with probability (1 ¡ ¼B), the batter will swing swing high, in which case the pitcher gets a payo® of 1. But when the pitcher throws low, the batter will swing low with probability ¼B and connect half the time, giving a payo® to the pitcher of :5¼B. Therefore the expected payo® to the pitcher from throwing low is (1 ¡¼B) + :5¼B = 1 ¡ :5¼B. Equalizing the payo® to the pitcher from throwing high and throwing low requires ¼B = 1 ¡ :5¼B. Solving this equation we ¯nd that in the equilibrium mixed strategy, ¼B = 2=3.

Summing up, the pitcher should throw low two-thirds of the time, and the batter should swing low two-thirds of the time.

29.1 (2) Two software companies sell competing products. These products are substitutes, so that the number of units that either company sells is a decreasing function of its own price and an increasing function of the other product's price. Let p1 be the price and x1 the quantity sold of product 1 and let p2 and x2 be the price and quantity sold of product

2. Then x1 = 1000			90 ¡ 21 p1 + 41 p2	and x2 = 1000	90 ¡ 21 p2 + 41 p1 .
Each company has	incurred a ¯xed cost for designing their software and
		¡	¢	¡	¢

writing the programs, but the cost of selling to an extra user is zero. Therefore each company will maximize its pro¯ts by choosing the price that maximizes its total revenue.

(a) Write an expression for the total revenue of company 1, as a function

of the its price p1 and the other company's price p2.

.