68 Chapter Highlights
Chapter 29
Game Theory
This is a fun chapter. Students like it a lot, and faculty usually enjoy teaching
it. Game theory is hot stuff in economics these days, and this chapter tries to
convey some of the reasons why.
The first two equilibrium concepts, that of a dominant strategy equilibrium
and that of a Nash equilibrium, are reasonably easy to convey. The idea of a
Nash equilibrium in mixed strategies is a little harder. Here’s an example that
will motivate the idea.
Consider the game of baseball. The pitcher has two strategies: pitch high or
pitch low. Likewise, the batter has two strategies, swing high or swing low. If
the batter connects, he gets a payoff of 1 and the pitcher gets zero. If the batter
misses, the pitcher gets a payoff of 1.
What are the Nash equilibria in this game? If the pitcher always pitches
high, the batter will always swing high, and if the pitcher always pitches low,
then the batter will always swing low. It is clear from this observation—and from
observing baseball games—that the equilibrium strategy must involve a mixed
strategy. The pitcher will flip a coin and decide whether to pitch high or low,
and the batter flips a coin to decide whether to swing high or low. The batter
will connect 50% of the time. Here students are very willing to accept that the
optimal strategy must involve randomization.
If you really want to get them buzzing, you can talk about the following
paradox. If the batter really believes that the pitcher will really randomize 50–
50, then he might as well swing high all the time. But of course, once the pitcher
detects this departure from randomizing, he will modify his own behavior to
exploit the batter’s sloppiness. This example drives home the important point
that what keeps the players at the Nash equilibrium is the desire to avoid being
psyched out by their opponents.
Most students have heard of the prisoners’ dilemma by now, but they haven’t
seen the analysis of the repeated game. The reason why the repeated game
is different from the one-shot game is that in the repeated game, the strategy
choice at time tcan depend on the entire history of the game up until t.Thus
choices at time t−1 may have some influence on choices at time t. This opens
the possibility of tit-for-tat and other strategies that can allow for cooperative
solutions.
Chapter 29 69
The analysis of the sequential games, and especially the game of entry
deterrence, is a very interesting topic. Students really get excited about this
kind of analysis since they think that it will help them be better managers.
(Well, who knows, maybe it will!)
Game Theory
A. Game theory studies strategic interaction, developed by von Neumann and
Morgenstern around 1950
B. How to depict payoffs of game from different strategies
1. two players
2. two strategies
3. example
Row
Column
Left Right
Top 1,2 0,1
Bottom 2,1 1,0
C. Nash equilibrium
1. what if there is no dominant strategy?
2. in this case, look for strategy that is best if the other player plays his best
strategy
3. note the “circularity” of definition
Row
Column
Left Right
Top 2,1 0,0
Bottom 0,0 1,2
7. Nash equilibrium in pure strategies may not exist.
Row
Column
Left Right
Top 0,0 0,−1
Bottom 1,0−1,3
8. but if allow mixed strategies (and people only care about expected payoff),
then Nash equilibrium will always exist
70 Chapter Highlights
D. Prisoner’s dilemma
1. 2 prisoners, each may confess (and implicate other) or deny
2. gives payoff matrix
Row
Column
Left Right
Top −3,−3 0,−6
Bottom −6,0−1,−1
3. note that (confess, confess) is unique dominant strategy equilibrium, but
E. Repeated games
1. if game is repeated with same players, then there may be ways to enforce
2. suppose PD is repeated 10 times and people know it
3. suppose that PD is repeated an indefinite number of times
4. Axelrod’s experiment: tit-for-tat
F. Example – enforcing cartel and price wars
G. Sequential game — time of choices matters
H. Example:
Row
Column
Left Right
Top 1,9 1,9
Bottom 0,0 2,1
1. (Top, Left) and (Bottom, Right) are both Nash equilibria
I. Example: entry deterrence
1. stay out and fight