Applications of dynamic games:
Bargaining and reputation
Lectures in Game Theory
Spring 2011, Part 5
10.12.2010
G.B. Asheim, ECON4240-5
1
Modeling of

negotiations between parties with opposing
interests:
Sequential bargaining games

non-cooperative cooperation between parties
that in the short run want to deviate:
Repeated games
10.12.2010
Bargaining

u2
Agreement
creates value, but
how to divide?
A
d: default outcome,
disagreement point,
B
A B:
A,
B outcomes
d
A: efficient outcome if
transfers are possible

Efficient bargaining:
Agree on efficient outcome and divide surplus.
10.12.2010
2
G.B. Asheim, ECON4240-5

Applications:
u1
Bilateral monopoly
Labor market negotiations
International negotiations
G.B. Asheim, ECON4240-5
3
1
The standard
bargaining solution

First: Normalize.

Then: Divide surplus
d to
according
bargaining weights, 1
and 2.

u2
1
2
1
1
u1
What do the bargaining weights depend on?
10.12.2010
4
G.B. Asheim, ECON4240-5
u2
1
1
m1
2
A m1 , 1  m1
R
m1 is
i 1’s
1’
share.

0, 0
1
Ultimatum
bargaining
Bargaining weight 
for the proposer
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u1
5
G.B. Asheim, ECON4240-5
u2
1
m1
2
A m1 , 1  m1
R
2
mt is
i 1’s
1’
share in
period t.
 is the discount factor
10.12.2010
m2
1
R
A
m2 ,  (1  m2 )
0, 0
G.B. Asheim, ECON4240-5

1
u1
Two-period
alternating offer
6
2
u2
1
m1
2
A m1 , 1  m1
R
2
m2
mt is
i 1’s
1’
share in
period t.
1
A
u1
Three-period
alternating offer
count factor
10.12.2010
1
m2 ,  (1  m2 )
R
1
 is the dis-
2
7
G.B. Asheim, ECON4240-5
u2
1

m1
2
Equal bargaining
weight for pl. 1
in periods 1 & 3.
A m1 , 1  m1
R
2
mt is
i 1’s
1’
share in
period t.
 is the discount factor
10.12.2010
m2
1
R
 2 m3  2 
A
m2 , (1  m2 )
 m3 ,  2 (1  m3 )
2
1
u1
Rubinstein’s
bargaining model
G.B. Asheim, ECON4240-5
8
Solution of Rubinstein’s bargaining model
with an infinite time horizon
Unique subgame perfect Nash equilibrium:
1
1 

1 accepts if and only if mt 
1



1
2 offers mt 
 1
1 
1 

1
2 accepts if and only if (1 - mt ) 
 1
1 
1 
1 demands mt 
1’s proposal at time t  1 is accepted.
1st mover advantage: 1’s share is larger than 2’s.
If disc. factors are different: It pays to be patient.
10.12.2010
G.B. Asheim, ECON4240-5
9
3
Repeated games with an infinite time horizon
Prisoners’
dilemma
D
D 1, 1
C
3, 0
C 0, 3 2, 2
u2
3
2
1
Can repetition discipline
1
the players to cooperate?
Deviating yields a short run gain.
Deviating yields a loss of reputation
that undermines future cooperation.
10.12.2010
u1
2
3
Yes, if gain
now  PV of
future loss
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G.B. Asheim, ECON4240-5
Cooperation in infinitely rep. 3
" Trigger strategy" : Prisoners’ Dil. 2
Play D if D has been used earlier;
u2
1
otherwise play C . (Start with C .)
1
2
u1
3
If cooperation breaks down, it will never be restarted.
Subgame perfect Nash equilibrium (NE in all subgames)?
If cooperation has broken down: NE in the subgames.
If cooperation has not broken down:
Short-run gain  PV of long-run loss
1

 3 2 
 
(2  1)  1   2
2
1
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G.B. Asheim, ECON4240-5
u2
“Getting Even”
3
Play C , unless " permitted" to play D.
2
" Permitted" to play D if the opponent
1
played D w/o " perm" in previous period.
1
2
If 1 deviates, then 1 is punished in the next period.
Short-run gain  PV of loss in next period
If cooperation has not broken down:
1
 3  2   ( 2  0)  1   2   
2
If cooperation has broken down:
1
 1  0   ( 2  0)  1   2   
2
10.12.2010
G.B. Asheim, ECON4240-5
u1
3
12
4