1

Spieltheorie IISpieltheorie II

SS 2005SS 2005

Avner ShakedAvner Shaked

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Game Theory IIGame Theory II

SS 2005SS 2005

Avner ShakedAvner Shaked

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http://www.wiwi.uni-bonn.de/shaked/ST-II/http://www.wiwi.uni-bonn.de/shaked/ST-II/

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Game Theory IIGame Theory II

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

Game Theory IIGame Theory II

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

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

Game Theory IIGame Theory II

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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

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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