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

2 is compact & convexS

limn nn

x S x S

9

αA+ 1 - α B

0 α 1

Nash Bargaining TheoryNash Bargaining Theory

u2

u1

A

BS

2 is compact & convexS

S

A,B S

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Nash Bargaining TheoryNash Bargaining Theory

d S s S s d , ,

u2

u1

d

S

2 is compact & convexS

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Nash Bargaining TheoryNash Bargaining Theory

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

2 is compact & convexS

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Nash Bargaining TheoryNash Bargaining Theory

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

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Nash Bargaining TheoryNash Bargaining Theory

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

( , ) α α αf S d f S,d

d S T

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

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Axioms A1-A4

A4 (Independence of Irrelevant Alternatives IIAIIA)

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

A4 (Independence of Irrelevant Alternatives IIAIIA)

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 S d f S,d

( , ) 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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A brief mathematical InterludeA brief mathematical Interlude

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

A brief mathematical InterludeA brief mathematical Interlude

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

A brief mathematical InterludeA brief mathematical Interlude

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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A brief mathematical InterludeA brief mathematical Interlude

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?

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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All axioms were used in the proof

But are they necessary?

All axioms were used in the proof

But are they necessary?

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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All axioms were used in the proof

But are they necessary?

All axioms were used in the proof

But are they necessary?

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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All axioms were used in the proof

But are they necessary?

All axioms were used in the proof

But are they necessary?

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