Transcript The Coase Theorem & Game Theory

The Coase Theorem &
Game Theory
Presented by
Dr. Elizabeth Hoffman
Professor of Economics and
CU President Emerita
IRLE
May 22, 2006
Coase Experiment
Coase Experiments Payoff Sheet
Number
A’s Payoff
B’s Payoff
1
$0.00
$12.00
2
4.00
10.00
3
7.50
4.00
4
7.50
4.00
5
9.00
2.50
6
10.50
1.00
7
12.00
0.00
Coase Theorem

Ronald Coase
–
–
–
–
British Economist
Born December 29,
1910
Won Nobel
Memorial Prize in
Economics in 1991
The Problem of
Social Cost, 1960
Coase Theorem

Owner of property right will manage
production or negotiate a price such that
those not owning property right will pay

Assign property rights:
– To pollute
– To breath clean air
– To a fishery
– To an oil pool
Coase Theorem

Coase argued outcome would be efficient, only the
distribution of resources would be affected.

Examples:
–
Farmer and rancher decide how much land to fence for
farming and how much to allow as open range for cattle
–
Railroad and farmer will decide how close to the tracks to
allow crops to grow and how much burning from sparks to
allow
Coase Theorem

Experimental Tests of the Coase
Theorem
–
–
–
Property rights by the flip of a coin
Large bargaining groups
Property rights by earned entitlement and
moral authority
Coase Theorem
Experimental Results
Number of
Decisions
(N)
Number
of Joint
Maxima
(N1)
Equal
Split
(N2)
1. No moral authority
22
20
10
2. Moral authority
20
19
9
1. No moral authority
22
18
9
2. Moral authority
22
21
4
86
78
32
Experiment
A. Coin flip:
B. Game trigger:
Total
Coase Theorem

Problems with the Coase Theorem
–
–
–
–
–
–
Assignment of property rights can be disputed in the legal
system (tort law)
Transaction costs may lead to different outcomes
depending on which side has property right
Imperfect or asymmetric information about valuations
Free rider problem comes into effect when more than one
person shares property right
Pooling can result in last owner holding out to get all the
profits
Others, not included in bargaining, may be affected by
outcome
Nash Equilibrium

John Nash
–
–
–
American
Mathematician
Born June 13, 1928
Won Nobel
Memorial Prize in
Economics in 1994
for Game Theory
Cournot-Nash Equilibrium



Cournot – 19th century economist who first
came up with an idea to formalize Adam
Smith’s theory of perfect competition
Games and Economic Behavior by von
Neumann and Morgenstern (1944), first real
book in game theory
Nash developed a simple way to illustrate
points made by both Cournot and von
Neumann and Morgenstern
Cournot-Nash Equilibrium

Definition:
–
Assuming players are maximizing in their own
interest, each player plays a strategy which
optimizes for that player, given what strategies the
other players are playing. A Nash equilibrium is a
set of strategies, such that when every one is
maximizing, given all other players’ strategies, no
player has an incentive to change his or her
strategy.
Simplest Example:
A Monopolist with zero marginal cost and many customers
P 60
30=Pm
MR
90=Qm
180
Q
Simplest Example:
A Monopolist with zero marginal cost and many customers
The demand curve intersects the quantity axis at 180
units. Therefore, the marginal revenue curve
intersects the quantity axis at ½ of 180, or 90. The
monopolist maximizes profits by equating a marginal
cost of 0 and a marginal revenue of 0, producing 90
units and charging a price that sells 90 units. Since
there are many consumers, no one consumer can
affect the market by unilaterally changing his or her
behavior. Thus, the monopolist’s decision is a Nash
equilibrium.
Enter another seller.
We call this a duopoly.
P 60
The duopolist (entrant) observes
that there are 90 units going unsold
in this market. What is his or her
best strategy for entering this
market?
30=Pm
MR
90=Qm
180
Q
Duopoly
P
60
The entrant observes that there is now residual
demand along the lower half of the demand
curve. To maximize against the monopolist’s
decision, the entrant would produce 45 units at a
price of 15, effectively giving the monopolist ½ of
the market and the entrant ¼ of the market.
Residual marginal revenue
30
15
90
45
180
Q
Is this a Nash Equilibrium?
- No, given the definition of a Nash
equilibrium, the monopolist’s strategy is no
longer optimal because the monopolist now
maximizes against the 45 as a given by the
entrant.
-However, the entrant will then change his or
her strategy in response to the change in
strategy by the original firm.
-One could imagine each firm changing its
strategy in response to the change in
strategy by the other firm until a Nash
equilibrium is reached.
P
60
112.5
45
Firm 2
¼
67.5
180
Firm 1
3/8
Q
¼ of market is not optimal
against 3/8th, for example.
What is the Nash Equilibrium?
If each firm produces 1/3 and
they share 2/3 of the market,
neither firm has an incentive
to change its output when
maximizing against the other
firm’s behavior.
P
60
20
120
Firm 1
60
Firm 2
60
180
Q
So ---what happens when more firms enter?
P
There is still a residual demand of 60 units.
The next firm could maximize on the residual
60 units, producing 30 units.
Is this Nash equilibrium?
180
Firm 1
Firm 2
Firm 3?
Q
Nash Equilibrium
Previous slide NOT Nash Equilibrium
because other firms would once again
respond.
What would be Nash equilibrium? Divide
in ¼’ths?
Nash Equilibrium
Suppose a thousand firms enter?
Suppose there are an infinite number of firms?
–
–
Quantity produced goes to 180, price goes to zero.
Obviously, if there is a positive marginal cost,
quantity will be driven to the quantity that equates
marginal cost and marginal revenue.
Therefore, the simplest Nash equilibrium with many firms
and many consumers is a competitive equilibrium!
So, what is a game in economics?

A set of players (firms, consumers,
governments), a set of alternative strategies
available to each player, and a set of payoffs
obtainable as a function of the strategies
simultaneously played by all the players.
Cooperative and non-cooperative
games:
A non-zero sum game is a game in which
there exists a joint-profit maximum if the
players can agree to play the game as a
cooperative game. Thus, the monopoly
game described above could be played as a
cooperative game if the players could agree
to split the monopoly profits and restrict
output to the monopoly output.
Cooperative and non-cooperative
games:
When a group of firms succeeds in colluding
we call it a cartel. OPEC is such a cartel.
Cartels were rendered illegal in the U.S. in
1890 under the Sherman Anti-trust Act after
the railroad consolidations and John D.
Rockefeller’s Standard Oil Trust.
Cooperative and non-cooperative
games:
A zero-sum game is a game in which one
player’s gain is always another player’s loss.
Zero-sum games can only be played as noncooperative games, since there exists no
joint maximizing solution.
Core:

The core of a cooperative game is a solution
that maximizes the joint profits and
guarantees each player at least as much as
he or she could earn alone or by cooperating
with any smaller group among the other
firms. We call any smaller group a coalition.
Back to the Coase Theorem
Experiments
Coase Experiments Payoff Sheet
Number
A’s Payoff
B’s Payoff
1
$0.00
$12.00
2
4.00
10.00
3
7.50
4.00
4
7.50
4.00
5
9.00
2.50
6
10.50
1.00
7
12.00
0.00
What is the Core and why?
–
–
Number 2 maximizes the joint profits but the
controller has to earn at least $12 to be as well off
as if he or she played the game as a noncooperative game.
If the players were to divide the $2 of surplus,
equally, we would call it a Nash bargaining
solution.
Thus, the Nash equilibrium of the Coase
game is to take the $12 and run, the core is
to agree to outcome 2, and give the
controller at least $12, and the Nash
bargaining solution is to agree to outcome 2
and split the $2.
The Prisoner’s Dilemma:
Another example of a non-zero sum game that can be played
cooperatively or non-cooperatively
In the prisoner’s dilemma, there are two suspects
accused of a crime. They are separated in two
rooms and grilled. If they stick together and don’t
confess, each might get off or get off lightly. Each is
offered a chance to rat on the other in return for no
conviction and a big conviction for the other person.
If both rat, they both get sentences, perhaps not
quite as bad as if only one rats.
Prisoner’s Dilemma
B’s Strategy
Don’t confess
Don’t
confess
Confess
A Serves 1 year
A Serves 15 years
B Serves 1 year
B Gets off (0 years)
A Gets off (0 years)
A Serves 10 years
B Serves 15 years
B Serves 10 years
A’s Strategy
Confess
Prisoner’s Dilemma
What is the Nash equilibrium strategy? For
prisoner A, if B doesn’t confess, he should
confess, because he gets off. Similarly, for
prisoner B, if A doesn’t confess, he should
confess, because he gets off. Thus, both confess
and get 10 years. We call this a dominant
strategy Nash equilibrium or a strong Nash
equilibrium because no other strategy dominates
unless they succeed in sticking together and
actively colluding.
We model cartels as:
Prisoner’s Dilemma
Firm B’s Strategies
Don’t cheat
Cheat
A gets $50
A gets $45
B gets $50
B gets $54
Firm A’s Strategies
A gets $54
A gets $48
Cheat
B gets $45
B gets $48
Don’t
Cheat
We model cartels as:
Prisoner’s Dilemma
The joint profit maximum is not to cheat and
split $100. But each has a dominant strategy
to cheat for the extra $4. The end result is
that they split $96, leaving $4 on the table.
Robert Axelrod

Robert Axelrod
–
–
–
Math and Political
Science
MacArthur
Fellowship
The Evolution of
Cooperation, 1984
Robert Axelrod
Robert Axelrod, The Evolution of Cooperation,
(1984) studied the Prisoner’s Dilemma when played
by the same two players repeatedly for many
periods. He formalized the type of strategic
interaction that would occur when the Prisoner’s
Dilemma is played repeatedly. In tit-for-tat, player 1
starts by cooperating and then each player simply
repeats what the other player does.
Robert Axelrod
Axelrod came up with this idea after running an
iterated prisoner’s game contest, in which game
theorists were invited to submit computer programs
for how to play an iterated prisoner’s dilemma game
against all other strategies. Anatol Rapaport
submitted the winning strategy, a four-line program,
which became known as tit-for-tat. Sometimes it is
played with “forgiveness”.
Robert Axelrod


Later, a team from Southampton University
(Nicholas Jennings, Rejdeep Dash, Sarvapali
Rachurn, Alex Rogers, and Perukrishnen
Vytelingum) introduced a more complicated,
more forgiving tit-for-tat strategy that beat
Rapaport’s. But, if it encounters a constant
defector, it always defects.
What this leads to is an understanding that
social norms of behavior have powerful effect
on how people actually play economic
games in the real world.
Robert Axelrod


Social norms favoring cooperation have
powerful evolutionary bases because our
species would not have survived the trials of
living in the African grasslands if we had not
learned to cooperate effectively and to
punish cheaters swiftly and effectively.
This brings us to work by Hoffman, McCabe,
and Smith on the ultimatum game.
Ultimatum Game
Experiment
Ultimatum Game Experiment


In the ultimatum game, there are two players who
must split a sum of money. One makes a proposal.
The other must agree or disagree. If the second
agrees, the division takes place. If the second
disagrees, they get nothing. The smallest offer is $1.
What is the Nash equilibrium and why?
In the dictator game, the first mover proposes a
division. The second mover has no recourse. What
is the Nash equilibrium and why?
FHSS Results
Figure 1
Ultimatum; FHSS Results,
Divide $10, N=24
Dictator; FHSS Results,
Divide $10, N=24
% Frequency
% Frequency
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
Offer $
% Offer
% Rejection
2
3
4
Offer $
5
6
7
8
9
Payoff Chart
Figure 2
Seller Chooses
PRICE (in $)
BUY
Buyer
Chooses
to
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
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7
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10
Seller Profit
10
9
8
7
6
5
4
3
2
1
0
Buyer Profit
0
0
0
0
0
0
0
0
0
0
0
Seller Profit
0
0
0
0
0
0
0
0
0
0
0
Buyer Profit
NOT
BUY
Figure 3
% Offer
% Rejection
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Ultimatum; Contest Entitlement,
FHSS Instructions, Divide $10, N=24
% Frequency
% Frequency
Ultimatum; Random Entitlement,
FHSS Instructions, Divide $10, N=24
0
1
2
3
4
5
6
7
8
9
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
Offer $
% Frequency
% Frequency
0
1
2
3
Offer $
4
5
6
7
8
9
2
3
4
5
6
7
8
9
8
9
Offer $
Ultimatum; Contest Entitlement,
Exchange, Divide $10, N=24
Ultimatum; Random Entitlement,
Exchange, Divide $10, N=24
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
Offer $
4
5
6
7
Figure 4
% Offer
% Rejection
Ultimatum; Contest Entitlement,
Exchange, $10, N=24
0.5
0.4
0.3
0.2
0.1
0
% Frequency
% Frequency
Ultimatum; Random Entitlement,
FHSS Instructions, Divide $10, N=24
0
1
2
3
4
5
6
7
8
0.5
0.4
0.3
0.2
0.1
0
9
0
1
Offer $
2
3
4
5
6
7
8
9
Offer $
Ultimatum; Random Entitlement,
FHSS Instructions, Divide $100, N=27
Ultimatum; Contest Entitlement,
Exchange, $100, N=23
0.4
% Frequency
% Frequency
0.5
0.3
0.2
0.1
0
0
10
20
Offer $
30
40
50
60
70
80
90
0.5
0.4
0.3
0.2
0.1
0
0
10 20 30 40 50 60 70 80 90
Offer $
Figure 5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Dictator; Contest Entitlement,
Exchange, N=24
% Frequency
% Frequency
Dictator; Random Entitlement,
Divide $10, N=24
0
1
2
3
4
5
6
7
8
9
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
Offer $
2
3
Offer $
5
6
7
8
9
Dictator; Random Entitlement,
Divide $10, Double Blind 2, N=41
% Frequency
% Frequency
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
4
Offer $
Dictator; Random Entitlement,
Divide $10, Double Blind 1, N=36
0
3
4
5
6
7
8
9
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
Offer $
3
4
5
6
7
8
9
Figure 6
100
Percent
80
DB1
DB2
SB1
SB2
FHSSV
FHSS R
60
40
20
0
0
1
2
3
Offer $
4
5