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358 GAME THEORY (Ch. 29)
(c) Do any of the pure Nash equilibria that you found seem more rea-
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
(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 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)
(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.
0204060
80
20
40
60
x
y
80
X=Y
Y=10+.8X
e
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
80
20
40
60
x
y
80
e
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.
(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?
They’d notice that last night’s attendance
is not a good predictor of tonight’s. If
