Game Theory: Sharing, Stability and Strategic Behaviour

[email protected] March 5, 2007

Al Quds University, Jerusalem

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Game Theory:Sharing, Stability and Strategic Behaviour

Frank Thuijsman

Maastricht University

The Netherlands



2/57

John von Neumann Oskar Morgenstern

Theory of Games and Economic Behavior, Princeton, 1944



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1.Three widows

2.Cooperative games

3.Strategic games

4.Marriage problems

Programme



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“If a man who was married to three wives died and the kethubah ofone was 100 zuz, of the other 200 zuz, and of the third 300 zuz, andthe estate was worth only 100 zuz, then the sum is divided equally.

If the estate was worth 200 zuz then the claimant of the 100 zuzreceives 50 zuz and the claimants respectively of the 200 and the 300zuz receive each 75 zuz.

If the estate was worth 300 zuz then the claimant of the 100 zuzreceives 50 zuz and the claimant of the 200 zuz receives 100 zuz whilethe claimant of the 300 zuz receives 150 zuz.

Similarly if three persons contributed to a joint fund and theyhad made a loss or a profit then they share in the same manner.”

Kethuboth, Fol. 93a, Babylonian Talmud, Epstein, ed, 1935

So: 100 is shared equally, each gets 33.33.

So: 200 is shared as 50 - 75 - 75.

So: 300 is shared proportionally as 50 - 100 - 150.



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100 200 300

100 33.33

200 33.33

300 33.33

Estate

Widow

50

75

75

50

100

150

“Similarly if three persons contributed to a joint fund and theyhad made a loss or a profit then they share in the same manner.”

How to share 400?

What if a fourth widow claims 400?

Equal Proportional???



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Barry O’Neill

A problem of rights arbitration from the Talmud, Mathematical Social Sciences 2, 1982



7/57

Robert J. Aumann Michael Maschler

Game theoretic analysis of a bankruptcy problem from the Talmud, Journal of Economic Theory 36, 1985Nobel prize for Economics, 12-10-2005

Thomas Schelling



8/57

Robert J. Aumann Michael Maschler

Game theoretic analysis of a bankruptcy problem from the Talmud, Journal of Economic Theory 36, 1985



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100 200 300

100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

100 200 300

100 50

200 100

300 150

S Ø A B C AB AC BC ABC

v(S)

The nucleolus of the game

0 0 0 0 0 100 200 300

The value of coalition S is the amount that remains,

if the others get their claims first.

100 200 300

100

200

300

Cooperative games



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100 200 300

100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

Cooperative games100 200 300

100 50

200 75

300 75


v(S)


0 0 0 0 0 100 2000



100 200 300

100

200

300



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100 200 300

A 100 33.33 50 50

B 200 33.33 75 100

C 300 33.33 75 150

Cooperative games100 200 300

A 100 33.33

B 200 33.33

C 300 33.33




v(S)


0 0 0 0 0 0 0 100

100 200 300

A 100

B 200

C 300



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v(S) 0 6 7 7 9 11 11 14

Cooperative games

Sharing costs or gains based upon the valuesof the coalitions



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The core


v(S) 0 6 7 7 9 11 11 14

(14,0,0) (0,14,0)

(0,0,14)

(6,0,8)

(6,8,0)

(0,7,7)

(7,7,0)

(7,0,7)

Empty



14/57

Lloyd S. Shapley

A value for n-person games, In: Contribution to the Theory of Games, Kuhn and Tucker (eds), Princeton, 1953



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The Shapley-value

For cooperative games there is ONLY ONE

solution concept that satisfies the properties:

- Anonimity

- Efficiency

- Dummy

- Additivity

Φ : the average of the “marginal contributions”



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v(S) 0 6 7 7 9 11 11 14

The Shapley-value

A B C

A-B-C

A-C-B

B-A-C

B-C-A

C-A-B

C-B-A

Sum:

Φ:

6 3 5

6 3 5

2 7 5

3 7 4

4 3 7

3 4 7

24 27 33

4 4.5 5.5

Marginal contributions



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David Schmeidler

The nucleolus of a characteristic function game, SIAM Journal of Applied Mathematics 17, 1969



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The nucleolus

(14,0,0) (0,14,0)

(0,0,14)


v(S) 0 6-2 7-2 7-2 9-2 11-2 11-2 14

(4,5,5) the nucleolus


v(S) 0 6 7 7 9 11 11 14


v(S) 0 6-x 7-x 7-x 9-x 11-x 11-x 14


v(S) 0 4 5 5 7 9 9 14

Leeg

Φ = (4, 4.5, 5.5)



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100 200 300

A 100 33.33 50 50

B 200 33.33 75 100

C 300 33.33 75 150

The Talmud games


v(S) 0 0 0 0 0 0 0 100

(100,0,0) (0,100,0)

(0,0,100)

the nucleolus



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100 200 300

100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

The Talmud games


v(S) 0 0 0 0 0 0 100 200

(200,0,0) (0,200,0)

(0,0,200)



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100 200 300

100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

The Talmud games


v(S) 0 0 0 0 0 0 100 200

(200,0,0) (0,200,0)

(0,0,200)

the nucleolus



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100 200 300

100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

The Talmud games


v(S) 0 0 0 0 0 100 200 300

(300,0,0) (0,300,0)

(0,0,300)



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100 200 300

100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

The Talmud games


v(S) 0 0 0 0 0 100 200 300

(300,0,0) (0,300,0)

(0,0,300)

the nucleolus



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100 200 300

100 33.33

200 33.33

300 33.33

Estate

Widow

50

75

75

50

100

150

“Similarly if three persons contributed to a joint fund and theyhad made a loss or a profit then they share in the same manner.”

How to share 400?




25/57

The Answer

Another part of the Talmud:

“Two hold a garment; one claims it all, the other claims half. Then one gets 3/4 , while the other gets 1/4.”

Baba Metzia 2a, Fol. 1, Babylonian Talmud, Epstein, ed, 1935



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Consistency100 200 300

100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

jointly 125

100

200

One claims 100, the other all,so 25 for the other;both claim the remains (100), so each gets half

jointly 125

100

200 25

jointly 125

100 50

200 25+50



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100 33.33 50 50

200 33.33 75 100

300 33.33 75 150


jointly 125

100 50

300 25+50



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100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

jointly 150

200 75

300 75

Each claims all,so each gets half



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100 200 300

100 33.33 50 50

200 33.33 75 100

300 33.33 75 150


100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

jointly 66.66

100

200

Each claims all,so each gets half

jointly 66.66

100 33.33

200 33.33



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100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

jointly 150

100

200


jointly 150

100

200 50

jointly 150

100 50

200 50+50



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100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

jointly 200

100

300


jointly 200

100

300 100

jointly 200

100 50

200 100+50



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100 200 300

100 33.33 50 50

200 33.33 75 100

300 33.33 75 150

How to share 400?


Do we now really know how to do it?



33/57

Marek M. Kaminski

‘Hydraulic’ rationing, Mathematical Social Sciences 40, 2000



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50

50 100

100

150

150

Communicating Vessels



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Pouring in 100

33.3333.33 33.33



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Pouring in 200

7550

75



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Pouring in 300

150

100

50



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Pouring in 400

50

125 225



39/57

4 widows with 400

125100

50

125



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Strategic games

A1

B1

A22,2

6,1 4,3

B2

A3

3,4

1,3

B3

2,1 4,2

Strategy player 1: LLR

Strategy player 2: RRR

“game in extensive form”



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A1

B1

A22,2

6,1 4,3

B2

A3

3,4

1,3

B3

2,1 4,2

Strategy player 1: RLL

Strategy player 2: RLL

Threat

Strategic games

“Game in extensive form”



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LLL LLR LRL LRR RLL RLR RRL RRR

LLL

LLR 2,2LRL

LRR

RLL 3,4RLR

RRL

RRR

“Game in strategic form”



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LLL 6,1 6,1 6,1 6,1 2,2 2,2 2,2 2,2LLR 6,1 6,1 6,1 6,1 2,2 2,2 2,2 2,2LRL 4,3 4,3 4,3 4,3 2,2 2,2 2,2 2,2LRR 4,3 4,3 4,3 4,3 2,2 2,2 2,2 2,2RLL 3,4 3,4 1,3 1,3 3,4 3,4 1,3 1,3RLR 2,1 4,2 1,3 1,3 2,1 4,2 1,3 1,3RRL 3,4 3,4 1,3 1,3 3,4 3,4 1,3 1,3RRR 2,1 4,2 1,3 1,3 2,1 4,2 1,3 1,3





“Game in strategic form”



44/57

Equilibrium:

If players play

best responses to eachother,

then a stable situation arises



45/571994: Nobel prize for Economics

John F. Nash John C. HarsanyiReinhard Selten

“A Beautiful Mind”

Non-cooperative games, Annals of Mathematics 54, 1951



46/57

Player 2

Player 1-2,-2 -10,-1

-1,-10 -8,-8

The Prisoner’s Dilemma

(-2,-2)

(-1,-10)

(-10,-1)

(-8,-8)

The iterated Prisoner’s Dilemma

be silent

confess



47/57

D H

D

H

2,2 0,3

3,0 1,1

Hawk-Dove

(2,2)

(3,0)

(0,3)

(1,1)



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D H

D

H

2,2 0,3

3,0 1,1

Hawk-Dove and Tit-for-Tat

Tit-for-Tat: begin D and play the previous opponent’s action at every other stage

D H T

D 2 0

H 3 1

T

D H T

D 2 0 2

H 3 1 1

T 2 1 2



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Robert Axelrod Anatol RapoportJohn Maynard Smith



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1 2 3 4 5

Anny Freddy Harry Kenny Gerry Lenny

Betty Gerry Kenny Freddy Harry Lenny

Conny Lenny Harry Gerry Freddy Kenny

Dolly Harry Lenny Freddy Gerry Kenny

Emmy Harry Kenny Gerry Lenny Freddy

1 2 3 4 5

Freddy Conny Betty Anny Emmy Dolly

Gerry Dolly Anny Betty Emmy Conny

Harry Emmy Anny Dolly Betty Conny

Kenny Emmy Conny Anny Dolly Betty

Lenny Emmy Anny Betty Conny Dolly

“Marriage Problems”



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1 2 3 4 5

Anny Freddy Kenny Gerry Lenny

Betty Gerry Kenny Freddy Lenny

Conny Lenny Gerry Freddy Kenny

Dolly Lenny Freddy Gerry Kenny

Kenny Gerry Lenny Freddy

1 2 3 4 5

Freddy Conny Betty Anny Dolly

Gerry Dolly Anny Betty Conny

Anny Dolly Betty Conny

Kenny Conny Anny Dolly Betty

Lenny Anny Betty Conny Dolly




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1 2 3 4 5

Anny Freddy Kenny Gerry Lenny

Betty Gerry Kenny Freddy Lenny

Conny Lenny Gerry Freddy Kenny

Dolly Lenny Gerry Kenny

Kenny Gerry Lenny Freddy

1 2 3 4 5

Freddy Conny Betty Anny

Gerry Dolly Anny Betty Conny

Anny Dolly Betty Conny

Kenny Conny Anny Dolly Betty

Lenny Anny Betty Conny Dolly




53/57

Lloyd S. Shapley

College admissions and the stability of marriage, American Mathematical Monthly 69, 1962

David Gale



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1 2 3 4 5

Anny Freddy Harry Kenny Gerry Lenny

Betty Gerry Kenny Freddy Harry Lenny

Conny Lenny Harry Gerry Freddy Kenny

Dolly Harry Lenny Freddy Gerry Kenny

Emmy Harry Kenny Gerry Lenny Freddy

1 2 3 4 5

Freddy Conny Betty Anny Emmy Dolly

Gerry Dolly Anny Betty Emmy Conny

Harry Emmy Anny Dolly Betty Conny

Kenny Emmy Conny Anny Dolly Betty

Lenny Emmy Anny Betty Conny Dolly

Gale-Shapley Algorithm

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9



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Gale-Shapley Algorithm

- Also applicable if the groups are not equally big

- Also applicable if not everyone wants to be matched to anybody

- Also applicable for “college admissions”

- Gives the best stable matching for the “proposers”



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[email protected]

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GAME VER

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Game Theory: Sharing, Stability and Strategic Behaviour

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