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
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
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
CHAPTER 8:
Strategy and Game Theory
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
8.9 Alternatives to Grim Strategy. This problem provides further practice with the
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
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
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.
b. Let
and
1
be the probabilities that player 1 plays
A
and
,B
respectively.
playing
E
is
8 6(1 ).
For player 2 to be indifferent between
D
and
E
and
c. Players each earn 4 in the pure-strategy equilibrium. Player 2 earns
**
6 8(1 ) 6(1 2) 8(1 2) 7
in the mixed-strategy equilibrium. Similar
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
)1)(0())(1(
and from boxing is
8.3 a.
2
1
A
B
C
D
E
F
7, 6
5, 8
0, 0
2
D
E
F
5, 8
7, 6
1, 1
2
D
E
F
0, 0
1, 1
4, 4
b. (Don’t veer, veer) and (veer, don’t veer).
c. Let
and
1
be teen 1’s probabilities, respectively, of veering and not. Teen
2’s expected payoff from veering then is
1)1)(1())(2(
and from not
2
1
Veer
Veer
veer
veer
2, 2
1, 3
2
Veer
veer
3, 1
0, 0
0
probability
of veering
probability
of veering
1/2 1
1/2
1
BR2
BR1
E1
E2
Mixed-strategy
equilibrium
must be to do the opposite of 1 in a subgame-perfect equilibrium. Teen 1 thus
8.4 a. Homeowner 1’s objective function is
1 1 2 1
(10 2) 4 .l l l l
21
12
b.
c. The change is indicated by the shift (following the arrow) in Homeowner 1’s best–
8.5 a. If all play blond, then one would prefer to deviate to brunette to obtain a
.b
Playing blond
provides a payoff of
a
with probability
1
(1 ) .
n
p
This is the
1
1
*1.
n
b
pa