978-0393123982 Chapter 29 Solution Manual Part 2

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358 GAME THEORY (Ch. 29)

sonable than others? Why or why not? Although (Stay,

Stay) is a Nash equilibrium, it seems

(d) Let us change the game a little bit. Evangeline and Gabriel are still

desperate to find 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 payoff 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 find

each other at the huge party, the expected payoff to each of them is only

500. If they go to different parties, the payoff to both of them is zero.

The payoff matrix for this game is:

More Close Encounters

Gabriel

Little Party Big Party

(e) Does this game have a dominant strategy equilibrium? No. What

are the two Nash equilibria in pure strategies? (1) Both go to

(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? No. Can you think of any reason why the

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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?

If both know the game matrix and each

29.6 (1) The introduction to this chapter of Workouts, recounted the

sad tale of roommates Victoria and Albert and their dirty room. The

payoff matrix for their relationship was given as follows.

Domestic Life with Victoria and Albert

Victoria

Clean Don’t Clean

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 payoff 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 payoff 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 payoff for the whole day, Quarrel or Not Quarrel?

360 GAME THEORY (Ch. 29)

(b) Suppose that Victoria and Albert each believe that the other will try

to take the actions that will maximize his or her total payoff for the day.

Does either believe the other will start a quarrel? No Assuming

that each is trying to maximize his or her own payoff, given the actions

of the other, what would you expect each of them to do in the first stage

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 payoff of

each is diminished by 2. Given that there is certain to be a quarrel if

either does not clean, the payoff matrix for the game between Victoria

and Albert becomes:

Vengeful Victoria and Angry Albert

Victoria

Clean Don’t Clean

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

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

Clean Explain If one player does not clean,

(e) Does this game have a dominant strategy? No. Explain The

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

(g) Explain how it could happen that Albert and Victoria would both be

better off 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

off. If they both are easy to anger, there

are two equilibria, one that is better

(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? Albert does not clean and Victoria

29.7 (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 May-

nard’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 find the equilib-

rium attendance at Maynard’s? Y=10+.8Xand X=Y.

(b) What is the equilibrium nightly attendance? 50.

362 GAME THEORY (Ch. 29)

two equations you mentioned in Part (a). Label the equilibrium atten-

dance level.

0204060

X=Y

Y=10+.8X

Y=11+.8X

(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 different color ink to draw a new line representing the equa-

tion that changed. How many additional customers did the new steady

customer attract (besides himself)? 4.

(f) Suppose that everyone bases expectations about tonight’s attendance

on last night’s attendance and that last night’s attendance is public knowl-

edge. Then Xt=Yt−1,whereXtis expected attendance on day tand

Yt−1is actual attendance on day t−1. At any time t, Yt=10+.8Xt.

Suppose that on the first night that Maynard’s is open, attendance is 20.

29.8 (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 −2Xand 0. Thus Y=max{120 −2X, 0}.

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(a) Solve for the equilibrium attendance at Yogi’s. Draw a diagram de-

picting this equilibrium on the axes below.

0204060

X=Y

Y=120-2X

(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 first day of business. How many

will show up on the second day? 20. The third day? 80. The

fourth day? 0. The fifth day? 120. The sixth day?

0. The ninety-ninth day? 120. The hundredth day? 0.

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

They’d notice that last night’s attendance

is not a good predictor of tonight’s. If