1

Game TheoryLecture 3

Game TheoryLecture 3

Game TheoryLecture 3

22

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