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frank@math.unimaas.nl March 5, 2007
Al Quds University, Jerusalem
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Game Theory:Sharing, Stability and Strategic Behaviour
Frank Thuijsman
Maastricht University
The Netherlands
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Al Quds University, Jerusalem
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John von Neumann Oskar Morgenstern
Theory of Games and Economic Behavior, Princeton, 1944
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Al Quds University, Jerusalem
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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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Al Quds University, Jerusalem
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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???
frank@math.unimaas.nl March 5, 2007
Al Quds University, Jerusalem
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Barry O’Neill
A problem of rights arbitration from the Talmud, Mathematical Social Sciences 2, 1982
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Al Quds University, Jerusalem
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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
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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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Al Quds University, Jerusalem
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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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Al Quds University, Jerusalem
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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
S Ø A B C AB AC BC ABC
v(S)
The nucleolus of the game
0 0 0 0 0 100 2000
The value of coalition S is the amount that remains,
if the others get their claims first.
100 200 300
100
200
300
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Al Quds University, Jerusalem
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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
The value of coalition S is the amount that remains,
if the others get their claims first.
S Ø A B C AB AC BC ABC
v(S)
The nucleolus of the game
0 0 0 0 0 0 0 100
100 200 300
A 100
B 200
C 300
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S Ø A B C AB AC BC ABC
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
S Ø A B C AB AC BC ABC
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
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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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S Ø A B C AB AC BC ABC
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)
S Ø A B C AB AC BC ABC
v(S) 0 6-2 7-2 7-2 9-2 11-2 11-2 14
(4,5,5) the nucleolus
S Ø A B C AB AC BC ABC
v(S) 0 6 7 7 9 11 11 14
S Ø A B C AB AC BC ABC
v(S) 0 6-x 7-x 7-x 9-x 11-x 11-x 14
S Ø A B C AB AC BC ABC
v(S) 0 4 5 5 7 9 9 14
Leeg
Φ = (4, 4.5, 5.5)
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Al Quds University, Jerusalem
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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
S Ø A B C AB AC BC ABC
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
S Ø A B C AB AC BC ABC
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
S Ø A B C AB AC BC ABC
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
S Ø A B C AB AC BC ABC
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
S Ø A B C AB AC BC ABC
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?
What if a fourth widow claims 400?
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Al Quds University, Jerusalem
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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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Al Quds University, Jerusalem
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Consistency100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
One claims 100, the other all,so 25 for the other;both claim the remains (100), so each gets half
jointly 125
100 50
300 25+50
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Al Quds University, Jerusalem
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Consistency100 200 300
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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Al Quds University, Jerusalem
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100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
Consistency100 200 300
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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Consistency100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
jointly 150
100
200
One claims 100, the other all,so 50 for the other;both claim the remains (100), so each gets half
jointly 150
100
200 50
jointly 150
100 50
200 50+50
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Consistency100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
jointly 200
100
300
One claims 100, the other all,so 100 for the other;both claim the remains (100), so each gets half
jointly 200
100
300 100
jointly 200
100 50
200 100+50
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Al Quds University, Jerusalem
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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?
What if a fourth widow claims 400?
Do we now really know how to do it?
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Al Quds University, Jerusalem
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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
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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 LLR LRL LRR RLL RLR RRL RRR
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
LLL LLR LRL LRR RLL RLR RRL RRR
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
LLL LLR LRL LRR RLL RLR RRL RRR
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”
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Equilibrium:
If players play
best responses to eachother,
then a stable situation arises
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45/571994: Nobel prize for Economics
John F. Nash John C. HarsanyiReinhard Selten
“A Beautiful Mind”
Non-cooperative games, Annals of Mathematics 54, 1951
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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
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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
“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 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
“Marriage Problems”
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Al Quds University, Jerusalem
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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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?
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GAME VER