70
These problems cover a variety of different concepts introduced in the chapter. They
range in difficulty from the simplest exercise of finding the Nash equilibrium in a two-
by-two matrix to characterizing equilibrium when players have continuous actions and
payoffs with general functional forms. Practice with problems may be the primary way
for students to master the material on game theory.
Comments on Problems
8.1 This problem provides practice in finding pure- and mixed-strategy Nash
equilibria using a simple payoff matrix. The three-by-three payoff matrix makes
the problem slightly harder than the simplest case of a two-by-two matrix.
Although this problem points the student where to look for the mixed-strategy
equilibrium, in other cases there may be many possibilities that need to be
checked for mixed-strategy equilibria. In a game represented by a three-by-three
matrix, each player has four combinations of two or more actions, and so there
are 16 possible types of mixed-strategy equilibria to check. Software, called
Gambit, has been developed that can solve for all the Nash equilibria of games
the user specifies in extensive or normal form. Gambit is freely available on the
Internet. It is easy to use, almost functioning as a “game–theory calculator.” One
useful classroom exercise would be to have students solve some of the problems
on a game-theory problem set using Gambit, either alone or in teams.
McKelvey, R. D., A. M. McLennan, and T. L. Turocy (2007) Gambit: Software
Tools for Game Theory, Version 0.2007.01.30.
http://econweb.tamu.edu/gambit
8.2 A slight generalization of payoffs in the Battle of the Sexes provides students
with further practice in computing mixed-strategy Nash equilibria.
8.3 This problem provides practice in converting the payoff matrix for a
simultaneous game into one for a sequential game. Illustrates the application of
subgame-perfect equilibrium in the simple case of the famous Chicken game.
8.4 This problem provides practice in computing the Nash equilibrium in a game
with continuous actions (similar to the Tragedy of the Commons in this chapter
and in Chapter 15 with the Cournot game, except in this problem the best-
CHAPTER 8:
Strategy and Game Theory
71 Chapter 8: Strategy and Game Theory
response functions are upward-sloping). Players’ best responses are computed
using calculus, and the resulting equations are then solved simultaneously.
8.5 This problem asks students to solve for the mixed-strategy Nash equilibrium
with a general number of players
n
. The “punchline” to the problem that the
blond is less likely to be approached as the number of males increases is a
paradoxical result characteristic of such games. The problem is based on a scene
in the Academy Award winning movie, A Beautiful Mind, about the life of John
Nash, in which the Nash character discovers his equilibrium concept (the one
scene in the movie that involves any game theory). If the classroom facilities
allow, it is worthwhile to show students this scene (Scene 5: “Governing
Dynamics”) when covering this problem.
8.6 This problem gives the student practice with the repeated version of the
Prisoners’ Dilemma, adjusting the payoffs in the version given in the text.
8.7 A simultaneous game of incomplete information providing practice in finding
the Bayesian–Nash equilibrium. Similar to the Tragedy of the Commons in
Example 8.6.
8.8 This problem asks students to solve for a hybrid perfect Bayesian equilibrium.
Students may find the application interesting given the growth in popularity of
poker on television, in particular Texas Hold ‘Em (to which the name “Blind
Texan” in the problem is meant to be a tongue–in-cheek reference). In typical
intermediate microeconomics courses, instructors will have only a short time to
cover signaling games, and in such courses it would be perfectly reasonable to
omit this problem, focusing exclusively on the simpler computations associated
with separating and pooling equilibria. Two reasons to delve into hybrid
equilibria if there is sufficient time—say in an advanced course with an
extensive game theory component—are, first, that games like Blind Texan do
not have separating and pooling equilibria, only a hybrid one and, second, that
the full power of Bayes Rule in a signaling game is only apparent with a hybrid
equilibrium since the application of the rule with separating and pooling
equilibria is fairly trivial.
Analytical Problems
8.9 Alternatives to Grim Strategy. This problem provides further practice with the
discounting calculations associated with infinitely repeated games. Demonstrates
the value of harsh punishments in sustaining cooperation by examining the
difficulty in sustaining cooperation with less than grim-strategy punishments.
8.10 Refinements of perfect Bayesian equilibrium. Part (a) is standard. Given it is
the simplest problem on signaling games, all instructors who cover the topic
should consider including it in the problem set. Part (b) goes beyond the material
72 Chapter 8: Strategy and Game Theory
in the chapter in exploring the intuitive criterion, a refinement of perfect Bayesian
equilibrium, which restricts posterior beliefs to be “reasonable.”
Behavioral Problems
8.11 Fairness in the Ultimatum Game. This problem was added for professors
interested in including some behavioral economics in their course. The problem
covers the canonical Fehr–Schmidt fairness model, one of the simplest models in
which players care not just about their own payoffs but about the payoffs of
others. It applies the analysis to help understand experimental results from the
Ultimatum and Dictator games. The absolute value sign in the payoff function
gives some students trouble, so the answers dwell on how to deal with that in the
calculations.
8.12 Rotten Kid Theorem. This problem analyzes altruism, included in the
behavioral problems because it departs from the standard, selfish preferences.
Perhaps the most challenging problem in the chapter since it works with general
functional forms, so requires the application of the implicit function theorem
rather than the computation of explicit derivatives. Shows how subgame-perfect
equilibrium concept can be used to derive one of Nobel-prize winner Gary
Becker’s famous results. The parent-child application may hold interest for
students.
Solutions
b. Let
and
−1
be the probabilities that player 1 plays
A
and
,B
c. Players each earn 4 in the pure-strategy equilibrium. Player 2 earns
73 Chapter 8: Strategy and Game Theory
d.
8.2 Let
and
−1
be the wife’s probabilities, respectively, of playing ballet and
boxing. The husband’s expected payoff from ballet then is
74 Chapter 8: Strategy and Game Theory
8.3 a.
c. Let
and
−1
be teen 1’s probabilities, respectively, of veering and not.
2
Veer
2, 2
75 Chapter 8: Strategy and Game Theory
d. Teen 2 has four contingent strategies: always veer, take the same action as
teen 1, do the opposite of teen 1, never veer. The normal and extensive
forms for the game are as follows.
Teen 2’s
probability
of veering
1
BR2
E1
76 Chapter 8: Strategy and Game Theory
d. There are three Nash equilibria: 1 veers and 2 never veers, 1 doesn’t veer
e. The game has three subgames: the game itself and the subgames starting
8.4 a. Homeowner 1’s objective function is
2
Veer
2, 2
.b
77 Chapter 8: Strategy and Game Theory
b.
c. The change is indicated by the shift (following the arrow) in Homeowner
8.5 a. If all play blond, then one would prefer to deviate to brunette to obtain a
b. Playing brunette provides the male with a certain payoff of
.b
Playing
c. The probability the blond is approached by at least one male equals 1
minus the probability no males approach her:
78 Chapter 8: Strategy and Game Theory
8.6 a. Using the underlining algorithm or other method, one can verify that
b. Cooperation on silent is best sustained using grim strategies as described
in the text. In this cooperative equilibrium, each player earns present
discounted value of 1 each period:
The player earns 3 in the deviation period from his/her surprise fink, but
then players revert to the static Nash equilibrium of (fink, fink) from then
on. Cooperation is sustainable if
8.7 a. The best-response function is
2
3.5 4
LC
ll=+
for the low-cost type of
b. Player 2 best responds to the average best response across the two types of
player 1, given by the dashed line between the two best responses, and
79 Chapter 8: Strategy and Game Theory
l2
BRHC(l2)
BRLC(l2)
80 Chapter 8: Strategy and Game Theory
8.8 a.
b. In a hybrid equilibrium, at least some type of some player plays a mixed
strategy. If player 1 sees the low card, she prefers the pure strategy of
staying. So it must be that player 1 randomizes after seeing a high card.
In order for player 2 to be willing to randomize, he must be
indifferent between staying and folding. His expected payoff from staying
is
Stay
2
n1
Stay
Fold
50, –50
-100, 100
81 Chapter 8: Strategy and Game Theory
67
Equating this last expression with
Pr( |stay) 1 4H=
and solving yields
c. The low type’s expected payoff is
Analytical Problems
8.9 Alternatives to Grim Strategy
a. Cooperating gives a stream of per-period payoffs of 2, for a present
82 Chapter 8: Strategy and Game Theory
If players use two periods of punishment, the present discounted
value from deviating is
b. The required condition is that the present discounted value of the payoffs
from cooperating,
)1/(2
−
, exceed that from deviating,
0.02
0.03
83 Chapter 8: Strategy and Game Theory
8.10 Refinements of perfect Bayesian equilibrium
a. The key condition is for the firm to be willing to offer a job to an
uneducated worker. (Regarding the other player, the worker, all worker
types obtain the highest payoffs possible, since they are hired and don’t
have to expend the cost of education.) The firm’s expected payoff from J
is
b. For the firm to prefer not to offer a job to an uneducated worker,
calculations similar to those in Part (a) (but with the inequalities reversed)
Behavioral Problems
8.11 Fairness in the Ultimatum Game
a. Solve using backward induction, starting with the responder. The
responder certainly accepts any offer of
0.r
0.r=
The remaining question is
84 Chapter 8: Strategy and Game Theory
c. (1) The answer here is a bit technical because of the absolute value
sign in the utility function, requiring the analysis of two cases. We
will avoid this technicality by noting that the proposer would never
(2) The proposer obtains payoff
85 Chapter 8: Strategy and Game Theory
(3) In the Dictator Game, for
1 2,a
1 2,a
the proposer makes the lowest
8.12 Rotten Kid Theorem
In the second stage, the parent chooses
L
to maximize
In the first state, the child maximizes
*
11
( ( ) ( )),U Y r L r−
yielding first-order
condition