Prisoners’ Dilemma
Static games: Examples
1
Lectures in Game Theory
Fall 2011, Collection of examples 1
09.11.2011
G.B. Asheim, ECON3/4200-ex1
1
Prisoners’ Dilemma as a collective good game
2
C
D
C
2, 2
0, 3
D
3, 0
1, 1
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2
G.B. Asheim, ECON3/4200-ex1
Pigs
C : Contributi ng to a collective good
Gain for each : 2 Cost for contributo r : 3
1
2
C
D
C
2, 2
0, 3
D
3, 0
1, 1
D
S
P
D
P
4, 2
2, 3
D
6, - 1
0, 0
Payoff for each if both contribute s : 1  2  2 - 3  2
Payoff if only self contribute s : 1  2 - 3  0
Payoff if only other contribute s : 1  2  3
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G.B. Asheim, ECON3/4200-ex1
What should the players do?
1
2
L
C
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What should the players do?
1
R
2
Frequency for 2 :
39
14
14
L
C
R Exp utility for 1 :
U 20, 40 10, 30 80, 20
Fre -
11 U 20, 40 10, 30 80, 20
26.7
M 50, 30 10, 40 30, 30
quency 24 M 50, 30 10, 40 30, 30
for 1 :
4 D 30, 0 20, 10 40, 0
22.5
D 30, 0 20, 10 40, 0
Exp utility for 2 : 29.7
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G.B. Asheim, ECON3/4200-ex1
G.B. Asheim, ECON3/4200-ex1
5
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34.1
26.3
24.1
G.B. Asheim, ECON3/4200-ex1
6
1
Is strategy D strictly dominated?
Is strategy D strictly dominated?
Payoff for 1 if
2 chooses R
Payoff for 1 if
2 chooses R
2
1
L
R
U
6, 3
0, 1
5
M
2, 1
5, 0
4
3, 2
3, 1
M
5
4
3
Mixtures off
Mi
M and U
D
D
M
Mixtures
of M and
U
2
1
U
1
2
3
4
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5
Payoff for 1 if
2 chooses L
6
7
L
R
6, 3
0, 1
M
2, 1
5, 0
D
4, 2
3, 1
Indifference curve if 1 U
puts prob 5/9 on L
and 4/9 on R
M
5
2
4
D
3
Mixtures
of M and
U
2
1
2
3
4
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5
6, 3
0, 1
M
2, 1
5, 0
D
4, 2
3, 1
2
3
4
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5
6 U 2 chooses L
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G.B. Asheim, ECON3/4200-ex1
 Selects an integer between 1 and 100.
 Write this number, with your name, on a slip of paper.
Make your
o r selections simultaneously
sim ltaneo sl and independently.
independentl
The average of the your numbers is then computed.
The one whose number is closest to 70 percent of this
average wins 50 kroner. (If two or more tie, then the
prize will be shared in equal proportions.)
6 U 2 chooses L
G.B. Asheim, ECON3/4200-ex1
U
Each of you:
Payoff for 1 if
1
R
70 percent game
Is strategy D a best response?
1
L
Payoff for 1 if
1
G.B. Asheim, ECON3/4200-ex1
Payoff for 1 if
2 chooses R
2
D
3
2
1
1
9
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G.B. Asheim, ECON3/4200-ex1
10
Hawk-Dove/Chicken as an R&D game
Hawk-Dove/Chicken
H : Develop new technology . Cost for developer : 4
Gain if the other has not developed : 5 for self, - 1 for other
1
2
Gain if the other has developed : 3 for self, - 3 for other
2
1
H
D
H
D
H
0, 0
3, 1
H
0, 0
3, 1
D
1, 3
2, 2
D
1, 3
2, 2
Payoff for each if both develop : 2  5 - 4 - 3  2 - 1  3 - 4  0
Payoff for a sole developer : 2  5 - 4  3
Payoff if only the other develops : 2 - 1  1
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G.B. Asheim, ECON3/4200-ex1
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2
Battle of the Sexes
1
2
Opera
M
Opera Movie
0, 0
Movie 0, 0
1, 2
O
O
Pareto Coordination
Player 1’s best
response fn
2
3
2, 1
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Player 2’s best
response fn
1
1
3
G.B. Asheim, ECON3/4200-ex1
M
13
Pareto Coordination (a variant)
1
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2
A
2
A
B
A
2, 2
0, 0
B
0, 0
1, 1
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Stag Hunt
1
B
H
2
Stag
Hare
A
2, 2
0,
3
2
Stag
5, 5
0, 4
B
3
,
2
1, 1
Hare
4, 0
4, 4
0
G.B. Asheim, ECON3/4200-ex1
15
A game of trust
14
G.B. Asheim, ECON3/4200-ex1
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Player 2’s best
response fn
Player 1’s best
response fn
1
5
S
S
G.B. Asheim, ECON3/4200-ex1
H
1
5
16
A game of trust
Each of you is to choose A or B indep. of each other.
2
1
A player who chooses A, receives 100 kr from UofO.
Choose A Choose B
100, 110
Choose A 100, 100 100,-500
A player who chooses B, receives 200 kr from UofO
if all others also choose B, but must pay 500 kr if at
least one other player chooses A.
Will a
guarantee
help?
Choose B -110,
500,100
100 200, 200
Write A or B on a piece of paper!
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G.B. Asheim, ECON3/4200-ex1
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G.B. Asheim, ECON3/4200-ex1
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3
Illustrating the best response property and
Illustrating the best response property and
best response completeness
best response completeness
1
2
L
C
R
U
0, 4
4, 0
5, 1
M
4, 0
0, 4
D
1, 5
1, 6
1
L
C
R
U
0, 4
4, 0
5, 1
6, 1
M
4, 0
0, 4
6, 1
6, 6
D
1, 5
1, 6
6, 6
{D} {R} has the best resp. property, but is not best resp. complete.
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G.B. Asheim, ECON3/4200-ex1
Illustrating the best response property and
2
L
C
R
U
0, 4
4, 0
5, 1
M
4, 0
0, 4
D
1, 5
1, 6
{U, M} {L, C} has the best resp. property and is best resp. complete.
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1
C
R
U
0, 4
4, 0
5, 3
6, 1
M
4, 0
0, 4
5, 3
6, 6
D
3, 5
3, 5
6, 6
21
G.B. Asheim, ECON3/4200-ex1
Matching Pennies
T
2
H
T
1, - 1
- 1, 1
T - 1, 1
1, - 1
H
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G.B. Asheim, ECON3/4200-ex1
H
H
{U, M} {L, C} has the best resp. property.
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Player 1’s best
response fn
Police
1
2
T
Drivers
Legal Illegal
Monitoring - 1, 0 - 1, - 4
Not monitoring 0, 0
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G.B. Asheim, ECON3/4200-ex1
Monitoring (of traffic)
Player 2’s best
response fn
1
2
1
2
L
{U} {L} neither has the best resp. property nor is best resp. complete.
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20
G.B. Asheim, ECON3/4200-ex1
Even in games with only one Nash equilibrium,
rational players may not coordinate
best response completeness
1
2
I
Drivers’ best
response fn
Police’s best
response fn
1
3
L
N
1
5
M
- 3, 1
G.B. Asheim, ECON3/4200-ex1
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4
LUPI (lowest unique positive integer) game
A homogeneous
good market
price
Each of you:
 Selects a positive integer (“whole number”).
 2 (q1, q2 )  p (Q )q2
 Write this number on a slip of paper.
 a  b( q1  q2 ) q2
Inverse
demand fn.:
p
y
Make yyour selections simultaneouslyy and independently.
The one choosing the lowest integer that nobody else
chooses wins 50 kr.
p (Q )  a  bQ
"Testing Game Theory in the Field: Swedish LUPI Lottery Games"
Robert Östling, Joseph Tao-yi Wang, Eileen Chou & Colin Camerer,
American Economic Journal: Microeconomics 3 (2011), 1–33.
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G.B. Asheim, ECON3/4200-ex1
Simultaneous quantity setting: Cournot comp.
q1
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Nash equilibrium: Are there quantities for the two
firms so that no firm will regret its own quantity
when told of the quantity of the other firm?
1' s best respo
p nse fn : q1  (a  bqq2 ) / 2b
2' s best respo nse fn : q2  ( a  bq1 ) / 2b
q1c  q2c  a / 3b
Q c  q1c  q2c  2a / 3b p (Q c )  a / 3
 1 (q1c , q2c )   2 (q1c , q2c )  a 2 / 9b
Firm 1 chooses quantity first (1 is the leader)
Stackelberg comp. (cont.)
27
The follower’s
best response fn
p
The leader' s problem :
max a  b( q1 
q2
q1
a bq1
2b
) q1
 max a  bq1 q1
q1
Is it best to be a leader or a follower?
Th follower'
The
f ll
r' s problem
pr bl m :
max a  b(q1  q2 ) q2
q2
FOC : a  bq1  2bq2  0
1
2
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q2
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q2
a  60 b  1
q1c  q2c  20
c
Q  40 p (Q c )  20
c c
 1 (q1 , q2 )   2 (q1c , q2c )  400
q1m  q2m  30
Q  30 p (Q m )  30
 1 (q1m , q2m ) 
p (Q s )  a / 4
m
q1
 2 ( q1s , q2s )  p (Q s ) q2s  a 2 / 16b
29
28
G.B. Asheim, ECON3/4200-ex1
q1s  30 q2s  15
Q  45 p(Q
(Q s )  15
s
s
s
 1 ( q1 , q2 )  450  2 (q1 , q2s )  225
q2s  a  bq2s  / 2b  a / 4b
G.B. Asheim, ECON3/4200-ex1
q2  f 2 ( q1 )  ( a  bq1 ) / 2b
s
Q  1 ( q1s , q2s )  p (Q s ) q1s  a 2 / 8b
q1
q2
A numerical example
i e y1s  a / 2b
i.e.,
Q s  q1s  q2s  3a / 4b
Firm 2' s best respo nse fn :
Q
Iso profit curve FOC : a / 2  bq  0
1
for the leader
q1
Firm 2 chooses quantity after having
observed firm 1’s quantity (2 is the follower)
p
q1
G.B. Asheim, ECON3/4200-ex1
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Sequential quantity setting: Stackelberg comp.
q1
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quantity
G.B. Asheim, ECON3/4200-ex1
Firms 1 and 2 choose quantities simultaneously & independently.
q2
Q
q2
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G.B. Asheim, ECON3/4200-ex1
 2 (q1m , q2m )  900
30
5