Absent-minded driver
Dynamic games: Examples
1
(Hotel)
4
(Home)
Continue
1
Continue
Exit
1
Lectures in Game Theory
Spring 2011, Collection of examples 2
Exit
0
08.02.2011
G.B. Asheim, ECON4240-ex2
1
Battle of the Sexes with outside option
1
2
Opera Movie
Opera
2, 1
Movie 0, 0
1
2
Opera
Opera
1, 2
2, 1
0, 0
NC & M 0, 0
1, 2
C&O
3
2
,
3
2
3
2
,
3
2
C&M
3
2
,
3
2
3
2
,
3
2
Not call
1
1
Call
3
2
,
2
3
2
3
0, 3
Leave
LT
3, 0
0, 2
Take
TL
1, 0
1, 0
TT
1, 0
1, 0
3, 0
Leave
Take
1
0, 2
Take
1, 0
08.02.2011
G.B. Asheim, ECON4240-ex2
G.B. Asheim, ECON4240-ex2
0, 27.5
30, 0
20, 20
- 10, - 15
27.5, 0
- 15, - 10
- 85, - 85
4
Take-it-or-leave-it
Player 1 can take 10 kr or leave them.
If left, 10 kr is added & 2 can take 20 kr or leave them.
 If left, 10 kr is added & 1 can take 30 kr or leave them.
 If left, 10 kr is added & 2 can take 40 kr or leave them.
 If left, no more money is added & the left 40 kr goes to 1.
You will be player 1 or 22. Write a strategy for your player
on a piece of paper together with your name.
Alternatives … if player 1: 1T, 1LT, 1LL.
… if player 2: 2T, 2LT, 2LL.
Two pieces of paper will be drawn, one to pick player 1’s
strategy and one to pick player 2’s strategy. The game will
be played acc. to these strategies, and money will be paid.

2
Leave Take
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L
0, 0
0, 30

1
0, 2
0, 2
N
S
1, 2
Leave
2
0, 3
N
S
L
Movie
0, 2
LL
2
L
2
Leave Take
3, 0
S
Movie
2
Take
0, 0
0, 0
G.B. Asheim, ECON4240-ex2
0, 3
N
S
N
1
Take-it-or-leave-it
Leave
2
L
Movie
1
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1
Entry game
w/investment
2, 1
2
G.B. Asheim, ECON4240-ex2
2
0, 0
Opera Movie
NC & O
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(Slum)
5
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G.B. Asheim, ECON4240-ex2
6
1
Division of 50 kr
Division of 50 kr (results)
Half of you (proposers):
Other half (responders):
 Select an integer between 0  Select an integer between 0
and 50: your offer to the other and 50: smallest accept. sum
 Write this number, with
 Write this number, with
your name,
name on a slip of paper
paper. your name,
name on a slip of paper
paper.
Make your selections simultaneously and independently.
Two pieces of paper will be drawn, one from each pile.
If offered amount is at least as large as least accept. sum,
then the chosen responder receives the offered amount,
and the proposer the rest. If not, nothing is received.
08.02.2011
Ultimatum game
1
Demand
Responders:
(9.8)
(16.0)
(22.0)
(20.5)
(19 1)
(19.1)
(21.7)
(20.8)
(17.3)
(14.0)
(10.0)
(0,0)
0:
2
(23.5)
1:
1
(23.5)
10:
3
(23.4)
15:
1
(22.2)
20:
4
(22.2)
25:
2
(18.4)
35:
1
(7.4)
40:
1
(5.3)
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G.B. Asheim, ECON4240-ex2
Share
Proposers:
1:
2
10: 2
20: 1
22: 1
24: 1
25: 3
26: 2
30: 2
35: 1
40: 1
50: 1
Do people follow their selfish material interest?
– Game of trust – Game of punishment
2 Y 4, 4

Based on various papers by Ernst Fehr, U of Zurich.
N
2 Y
0, 0
7, 1
– P decides whether to impose p  4 on A if unfulfilled demand.
N
0, 0
– A receives
c v s 3x andd cchooses
s s to send
s d back
b c 0  y  3x to P.
– P gives 0  x  10 to A (& announces “demand” 0  d  3x).
– P’s payoff: 10  x  y A’s payoff: 10  3x  y ( punishm.)
YY YN NY NN
Share 4, 4 4, 4 0, 0 0, 0
What predictions if P and A follow their selfish material interests?
Experim. Larger x from P to A, leads to larger y from A to P.
results: If P threatens with an avail. punishm., then y is reduced.
Demand 7, 1 0, 0 7, 1 0, 0
If punishm. is avail., but is not used as a threat, then y is increased.
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G.B. Asheim, ECON4240-ex2
Finding the solution of Rubinstein’s bargaining model with an infinite time horizon
Equal bargaining weight for 1 i periods 1 and 3:
m1  m3
In period 1, 1 makes the best proposal that 2 accepts:
1  m1   (1  m2 )
In period 2, 2 makes the best proposal that 1 accepts:
m2   2 m3
Three equations in three unknown. Solution:
m1  m3 
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1 
G.B. Asheim, ECON4240-ex2
m2 

1 
11
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G.B. Asheim, ECON4240-ex2
2
Repeated Bertrand comp.
Coop.: Both set monopoly price:
p m  30  p1m  p2m
 1m   2m  450
1
If deviation, coop. breaks down:
b
b
b
b
b
b
p1  p2  0
1   2  0
q1  q2  30
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

 900  450 
( 450  0)  450   900   
2
1
P (Q )  60  Q
q1m  q2m  15
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G.B. Asheim, ECON4240-ex2
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2
Repeated Cournot comp.
2
Harsher punishments?
Coop.: In total, monopoly quantity.
q1m  q2m  15
 1m   2m  450
1
If deviation, coop. breaks down:
 1c   2c  400
q1c  q2c  20
p c  20
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


9
506.25  450 
( 450  400)  56.25   106.25   
1 
17
P (Q )  60  Q
p m  30
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G.B. Asheim, ECON4240-ex2
“Getting Even”
2
G.B. Asheim, ECON4240-ex2
An agreement between k participating countries based on:
Two observations:
– Not a subgame perfect NE if k is too low.
– Not “renegotiation-proof ” if k is too high.
08.02.2011
G.B. Asheim, ECON4240-ex2
G.B. Asheim, ECON4240-ex2
14

Based on Barrett, Journal of Theoretical Politics 11 (1999), 519–541.
Cooperate: Reduce the emission of greenhouse gases.
Defect: Do not reduce the
emission of greenh. gases.
In total N countries,, where
n countries cooperates.
Payoff function:
 C ( n)  c  dn
 D ( n)  bn
Conditions: c  d  b  0
b( N  1)  c  dN  0
Cooperation for climate control (cont.)
“Getting Even”: A participating country plays Cooperate
by reducing emission, except if another participating
country has been the sole deviator from “Getting Even”
in the previous period (in which case, the play of Defect in
the sense of no reduction of emission is specified).
Non-participating countries play Defect after any history.

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15
Cooperation for climate control (cont.)

Coop.: In total, monopoly quantity.
q1m  q2m  15
 1m   2m  450
1
If deviation, price = 0 in 1 period:
p0
1   2  0
q1  q2  30
Short-run gain  PV of loss in next period
If cooperation has not broken down:
1
 506.25  450   (450  0)  56.25   450   
8
If cooperation has broken down:
1
 225  0   ( 450  0)  225   450   
2
P (Q )  60  Q
p m  30
Cooperation for climate control
Coop.: In total, monopoly quantity.
P (Q )  60  Q
q1m  q2m  15
m
p  30
 1m   2m  450
1
If 1 deviates, 2 takes over for 1 period:
p  30
q1  0 & q2  30
 1  0 &  2  900
Short-run gain
 PV of loss in next period
If cooperation has not broken down:
1
 506.25  450   (450  0)  56.25   450   
8
If cooperation has broken down:
1
 225  0   ( 450  0)  225   450   
2
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2
17
Subgame perfect Nash equilibrium (NE in all subgames)?
Short-run gain  PV of loss in next period
If cooperation has not broken down:
 b( k  1)  dk  c    ( dk  d )  k  1  c  d
d  b  d
If cooperation has broken down:
cd

0  d  c    ( dk  d )

k  1
d
Are the punishing countries better off during punishment (so
they do not want to return to the original equilibrium at once)?
 b  dk  c  k  b  c b  d &  close to 1 :
d
c
c
 k  1
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G.B. Asheim, ECON4240-ex2
d
d
3