Topics in Game Theory

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Topics in Game TheoryTopics in Game Theory

SS 2008SS 2008

Avner ShakedAvner Shaked

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http://www.wiwi.uni-bonn.de/shaked/topics/http://www.wiwi.uni-bonn.de/shaked/topics/

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K. BinmoreK. Binmore Fun & GamesFun & Games A Text on Game TheoryA Text on Game Theory D.C. Heath & Co., 1992 D.C. Heath & Co., 1992

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M. Osborne & A. RubinsteinM. Osborne & A. Rubinstein Bargaining and MarketsBargaining and Markets Academic Press, 1990Academic Press, 1990

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K. BinmoreK. Binmore Fun & GamesFun & Games A Text on Game TheoryA Text on Game Theory D.C. Heath & Co., 1992 D.C. Heath & Co., 1992

M. Osborne & A. RubinsteinM. Osborne & A. Rubinstein Bargaining and MarketsBargaining and Markets Academic Press, 1990Academic Press, 1990

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A Bargaining Problem

• S - a feasible set• d - a disagreement point

Nash Bargaining TheoryNash Bargaining TheoryNash VerhandlungstheorieNash Verhandlungstheorie

John Nash

d S s S s d , ,

2 is compact & convexS

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Nash Bargaining TheoryNash Bargaining Theory2 is compact & convexS

u2

u1

S

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Nash Bargaining TheoryNash Bargaining Theory

u2

u1

bounded

closedS


limn nn

x S x S

9

αA+ 1 - α B

0 α 1


u2

u1

A

BS


S

A,B S

10


d S s S s d , ,

u2

u1

d

S


11


d S s S s d , ,

is a bargaining problem< S,d >

is a bargaining problem{ }= < S,d > < S,d >B

u2

u1

d

S


12


d

A Nash Bargaining Solutionis a function

2:

( , )S d S

f

f

Bu2

u1

S

is a bargaining problem{ }= < S,d > < S,d >B

13


A Nash Bargaining Solutionis a function

2: f B

u2

u1

S

( ) ( )

( )

f S,d f S x | x d ,d

f S x | x d ,d S x | x d

( , )S d dfd

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Axioms A1-A4

A1 (Pareto)

if then x > f(S,d) x S

A2 (Symmetry)

d

S

&i i i ii x y i x y x > y

1 2 2 1( , ) ( , )x x x xα

f(S,d)

( , ) α α αf S d f S,d

S

α S

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Axioms A1-A4

A3 (Invariance to affine transformation)

A4 (Independence of Irrelevant Alternatives IIAIIA)

1 2 1 2( , ) ( , ) , 0x x x x α


d S T

f T,d S f S,d f T,d

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Axioms A1-A4


d S T

f T,d S f S,d f T,d

u2

u1

d

f T,dT S = f S,d

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Axioms A1-A4


d S T

f T,d S f S,d f T,d

Gives f(T,d) a flavour of maximum

PastaFishMeat

IIA IIA is violated whenis violated when

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A1 ParetoA2 SymmetryA3 InvarianceA4 IIA

A1 ParetoA2 SymmetryA3 InvarianceA4 IIA

First, we show that there exists a function satisfying the axioms.

There exists a unique

satisfying A1- A4

2: f BTheorem:

Proof:

19d

,S,dFor any given bargaining problem

define g S,d

S

=

= Does such a point always exist ??Is it unique ??

Yes !!!

Proof:

20

?

d

g S,d

S

=

=

Proof:

does satisfy A1-A4 ?? g S,d

Pareto

Symmetry

IIA

Invariance

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(divide the $)

0

Proof:

Consider the bargaining problem ,Δ,0 f S,d g S,d

Uniqueness: If satisfies the axioms then: f S,d

(1,0)

(0,1)

1 12 2f ,0 ,By Pareto + Symmetry:

A2 (Symmetry)

1 2 2 1( , ) ( , )x x x xα


( , ) α α αf 0 f ,0 f ,0

By definition: 1 12 2g ,0 ,

1 12 2f ,0 g ,0 ,

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Proof:

For a given bargaining problem ,S,d

d

S

g S,d

(a,d2)

(d1,b)

(d1, d2)

=

=

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Proof:

For a given bargaining problem ,S,d

d

S

g S,d f S,d

If is a degenerate ProblemS,d

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Proof:

For a given (nondegenerate) bargaining problem ,S,d

0 (1,0)

(0,1)

d

S

g S,d

(a,d2)

(d1,b)

(d1, d2)

Consider the bargaining problem ,Δ,0Find an affine transformation α

α

α

α

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Proof:

0 (1,0)

(0,1)

d

S

g S,d

(a,d2)

(d1,b)

(d1, d2)

Find an affine transformation α

α

α

α

α x, y = μx + ν, μy + ν

1 2α 0,0 = ν,ν = d ,d 2α 1,0 = μ + ν ν = a,d, 1α 0,1 = ν, μ ν = d+ ,b

,1 2

1 2

ν,ν

μ, μ

= d ,d

a - d b - d

,1 2a - d b - d > 0,0

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Proof:

0 (1,0)

(0,1)

d

S

g S,d

(a,d2)(d1, d2)

α

α

α Δ = S

S

αf Δ,0 = f α ,α 0

f S,d 1 12 2α ,

1 12 2,

==

(d1,b)α

g S,d??

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Proof:

0 (1,0)

(0,1)

d

S

g S,d

(a,d2)(d1, d2)

α

α

S

αf Δ,0 = f α ,α 0

f S,d 1 12 2α ,

1 12 2,

==

(d1,b)α

g S,d??

1 12 2α , 1 1

2 2α 1,0 + 0,1

1 1 1 12 12 2 2 2α 1,0 + α 0,1 a,d d ,b

= g S,d

g S,d f S,d S

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Proof:

d

S

g S,d f S,dS

g S,d f S,d S

By IIA f S,d f S,d

g S,d f S,d

f S,d

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Proof:

d

S

g S,d f S,d

g S,d f S,d

f S,d

satisfying

g f,

! f A1 - A4

end of proof

300

A Generalization

Changing A2 (Symmetry)

A2 (nonsymmetric)

for some f(Δ,0) = α,1 - α 0 < α < 1.

Δ

(1,0)

(0,1)

A

B

α measures the strength of Player 1

α,1 - α

A

B

1 - α=

α

31d

,S,dFor any given bargaining problem

define g S,d

SA

B

With the new A2, define a different g S,d

A

B

1 - α=

α

32d

g S,dS

Does such a point always exist ??Is it unique ??

Yes !!!

A

B

Following the steps of the previous theorem,

g S,d is the unique function satisfying the 4 axioms.

Yes !!!Yes !!!

A

B

1 - α=

α

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A brief mathematical InterludeA brief mathematical Interlude

Consider the (implicit) functionα 1-αx y = K

Find a tangent at a point (x0,y0) on the curve

α-1 1-α α -ααx y + 1 - α x y y = 0

0 0αy + 1 - α x y = 0

0

0

αyy = -

1 - α x

differentiating

y

x

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Find a tangent at a point (x0,y0) on the curve

0

0

αy-

1 - α x0

0

y - y=

x - x

x

y

The tangent’s equation:

The intersections with the axis (x=0, y=0)

0 0x y,0 , 0,

α 1 - α

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A

B


x

y

0 00 0

x yx , y = α ,0 + β 0,

α 1 - α

(x0/α,0)

(0, y0/(1-α))

(x0 , y0)

A

B

1 - α=

α

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A

B


x

y

α 1-αx y = KAny tangent of the function

is split by the tangency point in the ratio 1 - α

α

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x

y

α 1-α

(x,y) Smax x y

For any convex set S, by maximizing

S

end of mathematical Interludeend of mathematical Interlude

We find the unique point in S in which the tangent is split

in the ratio 1 - α

α

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α 1-α

1 2(x,y) Smax x - d y - d

To find the Nash Bargaining Solution of a bargaining problem ,S,d

S

d

Nash Bargaining Solution

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All axioms were used in the proof

But are they necessary?



A1. Without Pareto, h(S,d) = dsatisfies the other axioms.

A2. Without Symmetry,

satisfies the other axioms.

α 1-α

1 2(x,y) Smax x - d y - d

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A3. Without Invariance,

satisfies the other axioms.

(x, y) S

h(S,d) = max x + y

S

(0,1)

(0,0) (2,0)d

(x, y) Smax xy

h(S,d)

(1, 0.5)

Nash Bargaining Solution

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S

d

A4. Without IIA, the following function satisfies the other axioms.

The Kalai Smorodinsky solution

(S,d)K

1 2max max, (S,d)K

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S

d

A4. Without IIA, the following function satisfies the other axioms.

The Kalai Smorodinsky solution

(S,d)K

1 2max max,(S,d)K

is on the Pareto front of S

1

2

max

max

(S,d) = (x, y)

x=

y

K is on the Pareto front of S

1

2

max

max

(S,d) = (x, y)

x=

y

K

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Topics in Game Theory