Applications of dynamic games:
Bargaining and reputation
Lectures in Game Theory
F ll 2011,
Fall
2011 Part
P 5
24.07.2011
G.B. Asheim, ECON3/4200-5
1
Modeling of

negotiations between parties with opposing
i t
interests:
t
Sequential bargaining games

non-cooperative cooperation between parties
that in the short run want to deviate:
R
Repeated
d games
24.07.2011
G.B. Asheim, ECON3/4200-5
2
1
Bargaining

u2
Agreement
creates value, but
how to divide?
A
d: default outcome,
disagreement point,
B
A, B: outcomes
d
A: efficient outcome if
transfers are possible


Efficient bargaining:
Agree on efficient outcome and divide surplus.
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Bilateral monopoly
Labor market negotiations
International negotiations
3
G.B. Asheim, ECON3/4200-5
The standard
bargaining solution

First: Normalize.

Then: Divide surplus
according to
bargaining weights, 1
and 2.

u1
Applications:
u2
1
2
1
1
u1
What do the bargaining weights depend on?
24.07.2011
G.B. Asheim, ECON3/4200-5
4
2
u2
1
1
m1
2
A m1 , 1  m1
R
m1 is 1’s
share.

0, 0
1
Ultimatum
bargaining
Bargaining weight 
for the proposer
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u1
5
G.B. Asheim, ECON3/4200-5
u2
1
m1
2
A m1 , 1  m1
R
2
mt is 1’s
share in
period t.t
 is the discount factor
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m2
1
R
0, 0
A
m2 ,  (1  m2 )

1
u1
Two-period
alternating offer
G.B. Asheim, ECON3/4200-5
6
3
u2
1
m1
2
A m1 , 1  m1
R
2
m2
mt is 1’s
share in
period t.t
1
 is the dis-
R
1
A
m2 ,  (1  m2 )
1
7
G.B. Asheim, ECON3/4200-5
u2
1

m1
2
u1
Three-period
alternating offer
count factor
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2 
Equal bargaining
weight for pl. 1
in periods 1 & 3.
A m1 , 1  m1
R
2
mt is 1’s
share in
period t.t
 is the discount factor
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m2
1
R
2
 2 m3  
A
m2 , (1  m2 )
 2 m3 ,  2 (1  m3 )
1
u1
Rubinstein’s
bargaining model
G.B. Asheim, ECON3/4200-5
8
4
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.
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G.B. Asheim, ECON3/4200-5
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.
24.07.2011
G.B. Asheim, ECON3/4200-5
2
u1
3
Yes, if gain
now  PV of
future loss
10
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
ti has
h nott broken
b k down:
d
Short-run gain  PV of long-run loss
1

 3 2 
 
( 2  1)  1   2
2
1 
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G.B. Asheim, ECON3/4200-5
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
24.07.2011
G.B. Asheim, ECON3/4200-5
u1
3
12
6