Page 625 - Microeconomics, Fourth Edition

c14gametheoryandstrategicbehavior.qxd 8/6/10 8:22 AM Page 599

PROBLEMS 599
you can see from the following table, there are some ben- b) Does either student have any dominated strategies?
efits if they end up doing the same thing. Ignoring mixed c) What is the Nash equilibrium in this game?
strategies, is there a Nash equilibrium in this game? If so, d) Suppose that Jack and Jill each could borrow money
what is it? from the other students in the class, so that each of them
Ricky had a total of $11 to bid. Would ($11, $11) be a Nash
Ballet Boxing Match equilibrium?
14.15. Consider the following game between Sony, a
Lucy Ballet 100, 30 90, 90 manufacturer of video cassette players, and Columbia
Boxing Match 90, 90 30, 100 Pictures, a movie studio. Each firm must decide whether to
use the VHS or Beta format—Sony to make video players,
14.12. Suppose market demand is P 130 Q. Columbia to release its movies for rental or purchase.
a) If two firms compete in this market with marginal cost Columbia Pictures
c 10, find the Cournot equilibrium output and profit
per firm. Beta VHS
b) Find the monopoly output and profit if there is only Sony Beta 20, 10 0, 0
one firm with marginal cost c 10. VHS 0, 0 10, 20
c) Using the information from parts (a) and (b), construct
a 2 2 payoff matrix where the strategies available to a) Restrict attention to pure strategies. Does either firm
each of two players are to produce the Cournot equilib- have a dominant strategy? What is (are) the Nash equi-
rium quantity or half the monopoly quantity. librium (equilibria) of this game?
d) What is the Nash equilibrium (or equilibria) of the b) Is there a mixed strategy Nash equilibrium in this
game you constructed in part (c)? game? If so, what is it?
c) Restrict attention again to pure strategies, but now
14.13. Consider the following game, where x 0:
focus on a sequential-move game in which Sony chooses
Firm 2 its strategy first. What is (are) the Nash equilibrium
(equilibria) of this game?
High Price Low Price
High Price 140, 140 20, 160 14.16. In a World Series game, Tim Lincecum is pitch-
Firm 1 ing and Joe Mauer is batting. The count on Mauer is 3 balls
Low Price 90 x , 90 x 50, 50
and 2 strikes. Lincecum has to decide whether to throw a
fastball or a curveball. Mauer has to decide whether to
a) For what values of x do both firms have a dominant swing or not swing. If Lincecum throws a fastball and
strategy? What is the Nash equilibrium (or equilibria) in Mauer doesn’t swing, the pitch will almost certainly be a
these cases? strike, and Mauer will be out. If Mauer does swing, how-
b) For what values of x does only one firm have a domi- ever, there is a strong likelihood that he will get a hit. If
nant strategy? What is the Nash equilibrium (or equilib- Lincecum throws a curve and Mauer swings, there is a
ria) in these cases? strong likelihood that Mauer will strike out. But if
Lincecum throws a curve and Mauer doesn’t swing, there is
c) Are there any values of x such that neither firm has a
dominant strategy? Ignoring mixed strategies, is there a a good chance that it will be ball four and Mauer will walk
Nash equilibrium in such cases? (assume that a walk is as good as a hit in this instance).
The following table shows the payoffs from each
14.14. Professor Nash announces that he will auction pair of choices that the two players can make:
off a $20 bill in a competition between Jack and Jill, two
students chosen randomly at the beginning of class. Each Joe Mauer
student is to privately submit a bid on a piece of paper; Swing Do Not Swing
whoever places the highest bid wins the $20 bill. (In the 100, 100 100, 100
event of a tie, each student gets $10.) The catch, however, Tim Lincecum Fastball
is that each student must pay whatever he or she bid, re- Curveball 100, 100 100, 100
gardless of who wins the auction. Suppose that each student
has only two $1 bills in his or her wallet that day, so the a) Is there a Nash equilibrium in pure strategies in this
available strategies to each student are to bid $0, $1, or $2. game?
a) Write down a 3 3 payoff matrix describing this b) Is there a mixed strategy Nash equilibrium in this
game. game? If so, what is it?