game theory lecture 3
DESCRIPTION
Game Theory Lecture 3. Game Theory Lecture 3. 0. 6. 1. 1/6. 4. 5. 3. 2. 1. 1. 1. 1. 1. 1. N.S. S. D,W. D,W. D,W. D,W. D,W. D,W. L,W. 2. 2. 2. 2. 2. W,D. W,D. W,D. W,D. W,D. W,L. 1. 1. 1. 1. D,W. D,W. D,W. D,W. L,W. 2. 2. 2. W,D. W,D. W,D. W,L. - PowerPoint PPT PresentationTRANSCRIPT
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Game TheoryLecture 3
Game TheoryLecture 3
Game TheoryLecture 3
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Game Theory WS 2003
Problem Set 3 From Binmore's Fun and Games p. 266 Exercises
6, 8 p. 120 Exercises
7, 9, 10, 16, 17, (15,18, 19)
S
N.S.
0
1
2
1 1 1 1 1
1
2
2
1
1/6
2
1
2
1
2
1
2
2
1
2
D,W D,WD,WD,WD,WD,W
W,DW,DW,DW,DW,D
D,WD,WD,WD,W
W,DW,DW,D
D,WD,W
W,DW,L
W,L
L,W
W,L
L,W
L,W
14 2356
S
N.S.
N.S.
S
2
2
1
2
1
2
W,DW,DW,D
D,WD,W
W,D
W,L
W,L
L,W
W,D
W,L
N.S.
S.
1/2 1/2
L W
D
A Lottery
N.S.
S.
1/2 1/2
L W
D
A Lottery?
1 -
L W
α α 0 α 1
consider the lottery
for
1 - α α L α = 0
L W W α = 1
1 - α α
L W αassume that
L D W
S
N.S.
2
2
1
2
1
2
W,DW,DW,D
D,WD,W
W,D
W,L
W,L
L,W N.S.
S.
1/2 1/2
L W
D
S
N.S.
2
2
1
2
1
2
W,DW,DW,D
D,WD,W
W,D
W,L
W,L
L,W
N.S. D
S.
W,L
L,W
W,D
1/3 2
1/2 1/
3
2
D W
/
L
1/3 1/3 1/3
L D W
N.S. D
S.
1/3 2
1/2 1/
3
2
D W
/
L
1/3 1/3 1/3
L D W
1/3 1/3 1/3
L D W
?
von Neumann - Morgenstern utility functions
von Neumann - Morgenstern utility functions
A consumer has preferences over a set of prizes and preferences over the set of all lotteries over the prizes
:prizes 1 2 n-1 n w w .....w w
:lotteries
1 2 n
1 2 n
p p . . p
w w . . w
:and compound lotteries
1 2
1 2 n 1 2 n
1 2 n 1 2 n
p p . . .
q q . . q q' q' . . q'. . .
w w . . w w w . . w
:
The preferences over the lotteries
satisfy the following assumptions
w
1
w
for each prize wj there exists a unique j s.t.
j jj
1 n
1 - α α
w ww
1 n
1 - α α
w w
1.
2.
3.
if
1 2
1 2 n 1 2 n
1 2 n 1 2 n
1 2
1 2 n
1 2 n
p p . . .
q q . . q q' q' . . q'. . .
w w . . w w w . . w
p p . . .
q' q' . . q'w . . .
w w . . w
1 2 n
1 2 n
q q . . qw
w w . . w
then:
1 2 n
1 2 n
q q . . q
w w . . w
w
4.
... ...
1 2
1 2 n 1 2 n
1 2 n 1 2 n
1 1 2 1 1 2 2 2
1 2
p p . . .
q q . . q q' q' . . q'. . .
w w . . w w w . . w
p q + p q' p q + p q' . . .
w w . .
5.
:define j j u w = α
j jj
1 n
1 - α α
w ww
represents the preference
on the prizes
u
.
we now look for a utlity function representing the preferences over the lotteries
1 2 n
1 2 n
p p . . p
w w . . w
take a lottery:
j jj j
1 n 1 n
j1 - u w u w1 - α α
w w w ww
Replace each prize with an equivalent lottery
1 1 2 2 n n
1 n 1 n 1 n
1 2 n
1 2 n
1 2 n
1 - u w u w 1 - u w
p p
u w 1 - u w u w
w w w w
. . p
w w . . w
p p . . p
w w. .
1 2 n
1 2 n
p p . . p
w w . . w
1 2 n
1 1 2 2 n n
1 n 1 n 1 n
1 - u w u w 1 - u w u w 1 - u w u w
w w w w
p p . . p
w w. .
n n
j j j jj=1 j=1
n1
p 1- u w p u w
w w
n n
j j j jj=1 j=1
1 -
n1
p u w p u w
w w
n n
j j j jj=1 j=1
1 -
n1
p u w p u w
w w
define:
1 2 n
1 2 n
n
j jj=1
p p . . p
w w . . wp u wu
1 2 n
1 2 n
p p . . p
w w . . w
n n
j j j jj=1 j=1
1 -
n1
p u w p u w
w w
the expected utility of the lottery
clearly U represents the preferences on the lotteries
if
1 2 n 1 2 n
1 2 n 1 2 n
q q . . q p p . . p
w w . . w w w . . w
n n n n
j j j j j j j jj=1 j=1 j=1 j=1
1 - 1 -
n n1 1
q u w q u w p u w p u w
w w w w
n
j jj=1
q u w n
j jj=1
p u w
<
1 2 n
1 2 n
p p . . p=
w w . . wu
1 2 n
1 2 n
q q . . q=
w w . . wu
A utility function on prizes u wis called a von Neumann - Morgenstern utility function if
the expected utility function :
1 2 n
1 2 n
n
j jj=1
p p . . p
w w . . wp u wu
represents the preferences over the lotteries.
i.e. if U is a utility function for lotteries.
If u is a vN-M utility function then
for some α β v = u + α > 0
vis a vN-M utility function iff
If u is a vN-M utility function then
for some α β v = u + α > 0
vis a vN-M utility function iff
2. Let v() be a vN-M utility function. Choose a>0 ,b s.t.
1 n
f
f w = 0, f
= v + b
w
a
= 1
1. It is easy to show that if u( ) is a vN-M utility function then so is au( )+b a>0
since f( ) is a vN-M utility function, and since for all j
j jj
1 n
1 - α α
w ww
j 1 j n j1 - α f w + α f w = f w
It follows that:
But by the definition of f( )
j 1 j n j j1 - α f w + α f w = α = f w
hence:
f u av + b
j jw = αu
John von Neumann1903-1957
John von Neumann1903-1957
Oskar Morgenstern1902-1976
Oskar Morgenstern1902-1976
Neumann Janos
Kurt Gödel
Information Sets and
Simultaneous Moves
1
22
Some (classical) examples of simultaneous games
Cnot confess
D confess
C
not confess -3 , -3 -6 , 0
D confess 0 , -6 -5 , -5
Prisoners’ Dilemma
+6
3 , 3 0 , 6
6 , 0 1 , 1
Free Rider
(Trittbrettfahrer) CCooperate
D defect
C cooperate
3 , 3 0 , 6
D defect 6 , 0 1 , 1
Some (classical) examples of simultaneous games
CCooperate
D defect
C cooperate
3 , 3 0 , 6
D defect 6 , 0 1 , 1
Prisoners’ Dilemma
The ‘D strategy dominates the C strategy
Strategy s1 strictly dominates strategy s2 if for all strategies t of the other player
G1(s1,t) > G1(s2,t)
1 , 5 2 , 3 7 , 4
3 , 3 4 , 7 5 , 2
X
X
Nash Equilibrium(saddle point)
Successive deletion of dominated strategies