Chapter 8: Strategy and Game Theory
55
CHAPTER 8
STRATEGY AND GAME THEORY
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
bytwo 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
Provides practice in finding pure and mixedstrategy Nash equilibria using a
simple payoff matrix.
The threebythree payoff matrix makes the problem
slightly harder than the simplest case of a twobytwo matrix.
Although this
problem points the student where to look for the mixedstrategy equilibrium, in
other cases there may be many possibilities that need to be checked for mixed
strategy equilibria.
In a game represented by a threebythree matrix, each
player has four combinations of two or more actions, and so there are 16
possible types of mixedstrategy 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 “gametheory calculator.” One useful
classroom exercise would be have students solve some of the problems on a
gametheory 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 mixedstrategy Nash equilibria.
8.3
Provides practice in converting the payoff matrix for a simultaneous game into
one for a sequential game.
Illustrates the application of subgameperfect
equilibrium in the simple case of the famous Chicken game.
8.4
The 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
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response functions are upwardsloping).
Players’ best responses are computed
using calculus, and the resulting equations are then solved simultaneously.
8.5
Asks students to solve for the mixedstrategy 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.
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 Spring '08
 Buddin
 Game Theory, The Land, Veer, firstorder condition

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