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Chapter 30 71
Chapter 30
Game Applications
This is a chapter for those who want more game theory. Basically it is a set
of examples of game theory in action. Most of the examples are self-contained
so you can pick and choose what you want.
There are 4 major themes: cooperation (games of coordination), competition
(zero-sum games), coexistence (evolutionary games), and commitment (subgame
perfection).
I start with a description of best response functions, since they are a useful tool
for calculating Nash equilibria. This part just formalizes some of the material
already introduced in the last chapter.
I then describe the basic coordination games: battle of the sexes, prisoner’s
dilemma, assurance games, and chicken. The idea of a focal point and of the
importance of observability and committment are introduced here.
I then move on to games of competition: zero-sum games, using a soccer free
kick as an example. This section illustrates the importance of mixed strategies.
The next section is on coexistence, which is a very brief introduction to
evolutionary game theory. I fudge the topic of ESS since I didn’t want to
introduce all the details.
Finally I look at a number of sequential games where committment is
important.
Note that Schelling’s game of the “kindly kidnapper” has a plot line similar
to that of the movie Nasty People. A kidnapper grabs a victim and then gets
cold feet. The problem is that if he releases his victim, the rational strategy for
the victim is to go to the police and identify the kidnapper. The problem with
this sequential game is that the victim has no way to commit to staying away
from the police.
Schelling’s solution is characteristically inventive: he suggests that the victim
allow the kidnapper to photograph him in some unspeakably disgusting act.
This gives the kidnapper a threat—if the victim ever exposes the kidnapper, the
kidnapper can release the photo. The students think that this game is great
fun. You can ask them to suggest various “unspeakably disgusting acts” that
the victim might suggest.
72 Chapter Highlights
Game Applications
A. Last chapter presented the basics, here we apply them
B. Best response curves
1. two-person game (for simplicity)
2. best response curve: choice that maximizes one player’s payoff, given the
other player’s choice
3.
Column
Left Right
4. note that (Top, Left) is mutually consistent
C. Definition of Nash equilibrium
D. Examples: Cournot equilibrium is Nash in output; Bertrand equilibrium is
Nash in prices
E. Mixed strategies
F. Games of coordination
1. Battle of the sexes
d) three equilibria: both Art, both Action, or each chooses their preferred
2. Prisoner’s dilemma
3. Assurance game
a) U.S.-U.S.S.R missile deployment
Chapter 30 73
4. Chicken
a) two cars race towards each other
5. Commitment
a) one player moves first
G. Competition
1. zero-sum games: what one gets the other loses
2. penalty kick in soccer
a) kicker: kick left, kick right
b) goalie: defend left, defend right
H. Coexistence
1. evolutionary forces adjust population size
2. mixed equilibrium comes from different behaviors in population
3. example: hawk-dove game
a) wild dogs can either fight or share
b) fighting works well when other dog is dovish
I. Commitment
1. frog and scorpion
2. kindly kidnapper
3. pigs playing games
4. savings and Social Security
74 Chapter Highlights
5. Hold up
a) contractor: charge high price, charge cost
b) client: pay the money, look for another painter
J. Bargaining
1. Rubinstein bargaining model
a) one dollar to divide
2. ultimatum game
a) looks at last move of Rubinstein model
b) how do people actually behave?
