Game Theory: Sharing, Stability and Strategic Behaviour

1
Game TheorySharing, Stability and Strategic
Behaviour

• Frank Thuijsman
• Maastricht University
• The Netherlands

2
John von Neumann
Oskar Morgenstern
Theory of Games and Economic Behavior, Princeton,
1944
3
Programme

• Three widows
• Cooperative games
• Strategic games
• Marriage problems

4
Kethuboth, Fol. 93a, Babylonian Talmud, Epstein,
ed, 1935
If a man who was married to three wives died and
the kethubah of one was 100 zuz, of the other 200
zuz, and of the third 300 zuz, and the 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 zuz receives 50 zuz and
the claimants respectively of the 200 and the
300 zuz receive each 75 zuz. If the estate was
worth 300 zuz then the claimant of the 100
zuz receives 50 zuz and the claimant of the 200
zuz receives 100 zuz while the claimant of the
300 zuz receives 150 zuz. Similarly if three
persons contributed to a joint fund and they had
made a loss or a profit then they share in the
same manner.
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.
5
Estate
100 200 300
100 33.33
200 33.33
300 33.33
50
50
Widow
75
100
75
150
Equal
Proportional
???
Similarly if three persons contributed to a
joint fund and they had made a loss or a profit
then they share in the same manner.
How to share 400?
What if a fourth widow claims 400?
6
Barry ONeill
A problem of rights arbitration from the Talmud,
Mathematical Social Sciences 2, 1982
7
Robert J. Aumann
Michael Maschler
Thomas Schelling
Game theoretic analysis of a bankruptcy problem
from the Talmud, Journal of Economic Theory 36,
1985
Nobel prize for Economics, 12-10-2005
8
Robert J. Aumann
Michael Maschler
Game theoretic analysis of a bankruptcy problem
from the Talmud, Journal of Economic Theory 36,
1985
9
Cooperative games
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
100 200 300
100
200
300
The value of coalition S is the amount that
remains, if the others get their claims first.
The nucleolus of the game
S Ø A B C AB AC BC ABC
v(S)
0
0
0
100
200
300
0
0
10
Cooperative games
100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
100 200 300
100 50
200 75
300 75
100 200 300
100
200
300
The value of coalition S is the amount that
remains, if the others get their claims first.
The nucleolus of the game
S Ø A B C AB AC BC ABC
v(S)
0
100
200
0
0
0
0
0
11
Cooperative games
100 200 300
A 100 33.33 50 50
B 200 33.33 75 100
C 300 33.33 75 150
100 200 300
A 100 33.33
B 200 33.33
C 300 33.33
100 200 300
A 100
B 200
C 300
The value of coalition S is the amount that
remains, if the others get their claims first.
The nucleolus of the game
S Ø A B C AB AC BC ABC
v(S)
0
0
0
0
0
0
0
100
12
Cooperative games
S Ø A B C AB AC BC ABC
v(S) 0 6 7 7 9 11 11 14
Sharing costs or gains based upon the values of
the coalitions
13
The core
S Ø A B C AB AC BC ABC
v(S) 0 6 7 7 9 11 11 14
(0,0,14)
(6,0,8)
(7,0,7)
(0,7,7)
Empty
(6,8,0)
(7,7,0)
(14,0,0)
(0,14,0)
14
Lloyd S. Shapley
A value for n-person games, In Contribution to
the Theory of Games, Kuhn and Tucker (eds),
Princeton, 1953
15
The Shapley-value
For cooperative games there is ONLY ONE
solution concept that satisfies the
properties- Anonimity - Efficiency - Dummy -
Additivity
F the average of the marginal contributions
16
The Shapley-value
S Ø A B C AB AC BC ABC
v(S) 0 6 7 7 9 11 11 14
A B C
A-B-C
A-C-B
B-A-C
B-C-A
C-A-B
C-B-A
Sum
F
Marginal contributions
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
17
David Schmeidler
The nucleolus of a characteristic function game,
SIAM Journal of Applied Mathematics 17, 1969
18
The nucleolus
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
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
(0,0,14)
F (4, 4.5, 5.5)
(4,5,5) the nucleolus
Leeg
(14,0,0)
(0,14,0)
19
The Talmud games
(0,0,100)
100 200 300
A 100 33.33 50 50
B 200 33.33 75 100
C 300 33.33 75 150
the nucleolus
(100,0,0)
(0,100,0)
S Ø A B C AB AC BC ABC
v(S) 0 0 0 0 0 0 0 100
20
The Talmud games
(0,0,200)
100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
(200,0,0)
(0,200,0)
S Ø A B C AB AC BC ABC
v(S) 0 0 0 0 0 0 100 200
21
The Talmud games
(0,0,200)
100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
the nucleolus
(200,0,0)
(0,200,0)
S Ø A B C AB AC BC ABC
v(S) 0 0 0 0 0 0 100 200
22
The Talmud games
(0,0,300)
100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
(300,0,0)
(0,300,0)
S Ø A B C AB AC BC ABC
v(S) 0 0 0 0 0 100 200 300
23
The Talmud games
(0,0,300)
100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
the nucleolus
(300,0,0)
(0,300,0)
S Ø A B C AB AC BC ABC
v(S) 0 0 0 0 0 100 200 300
24
Estate
100 200 300
100 33.33
200 33.33
300 33.33
50
50
Widow
75
100
75
150
Similarly if three persons contributed to a
joint fund and they had made a loss or a profit
then they share in the same manner.
How to share 400?
What if a fourth widow claims 400?
25
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
26
Consistency
100 200 300
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
200
jointly 125
100
200 25
jointly 125
100 50
200 2550
27
Consistency
100 200 300
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 2550
28
Consistency
100 200 300
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
29
Consistency
100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
100 200 300
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
30
Consistency
100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
One claims 100, the other all, so 50 for the
other both claim the remains (100), so each
gets half
jointly 150
100
200
jointly 150
100
200 50
jointly 150
100 50
200 5050
31
Consistency
100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
One claims 100, the other all, so 100 for the
other both claim the remains (100), so each
gets half
jointly 200
100
300
jointly 200
100
300 100
jointly 200
100 50
200 10050
32
100 200 300
100 33.33 50 50
200 33.33 75 100
300 33.33 75 150
Do we now really know how to do it?
How to share 400?
What if a fourth widow claims 400?
33
Marek M. Kaminski
Hydraulic rationing, Mathematical Social
Sciences 40, 2000
34
Communicating Vessels
50
100
150
50
100
150
35
Pouring in 100
33.33
33.33
33.33
36
Pouring in 200
75
75
50
37
Pouring in 300
150
100
50
38
Pouring in 400
125
225
50
39
4 widows with 400
125
125
100
50
40
Strategic games
game in extensive form
Strategy player 1 LLR
Strategy player 2 RRR
41
Strategic games
Game in extensive form
Threat
Strategy player 1 RLL
Strategy player 2 RLL
42
Game in strategic form
LLL LLR LRL LRR RLL RLR RRL RRR
LLL
LLR 2,2
LRL
LRR
RLL 3,4
RLR
RRL
RRR
43
Game in strategic form
LLL LLR LRL LRR RLL RLR RRL RRR
LLL 6,1 6,1 6,1 6,1 2,2 2,2 2,2 2,2
LLR 6,1 6,1 6,1 6,1 2,2 2,2 2,2 2,2
LRL 4,3 4,3 4,3 4,3 2,2 2,2 2,2 2,2
LRR 4,3 4,3 4,3 4,3 2,2 2,2 2,2 2,2
RLL 3,4 3,4 1,3 1,3 3,4 3,4 1,3 1,3
RLR 2,1 4,2 1,3 1,3 2,1 4,2 1,3 1,3
RRL 3,4 3,4 1,3 1,3 3,4 3,4 1,3 1,3
RRR 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 2,2
LLR 6,1 6,1 6,1 6,1 2,2 2,2 2,2 2,2
LRL 4,3 4,3 4,3 4,3 2,2 2,2 2,2 2,2
LRR 4,3 4,3 4,3 4,3 2,2 2,2 2,2 2,2
RLL 3,4 3,4 1,3 1,3 3,4 3,4 1,3 1,3
RLR 2,1 4,2 1,3 1,3 2,1 4,2 1,3 1,3
RRL 3,4 3,4 1,3 1,3 3,4 3,4 1,3 1,3
RRR 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 2,2
LLR 6,1 6,1 6,1 6,1 2,2 2,2 2,2 2,2
LRL 4,3 4,3 4,3 4,3 2,2 2,2 2,2 2,2
LRR 4,3 4,3 4,3 4,3 2,2 2,2 2,2 2,2
RLL 3,4 3,4 1,3 1,3 3,4 3,4 1,3 1,3
RLR 2,1 4,2 1,3 1,3 2,1 4,2 1,3 1,3
RRL 3,4 3,4 1,3 1,3 3,4 3,4 1,3 1,3
RRR 2,1 4,2 1,3 1,3 2,1 4,2 1,3 1,3
44
Equilibrium
If players play best responses to eachother,
then a stable situation arises
45
A Beautiful Mind
John F. Nash
John C. Harsanyi
Reinhard Selten
1994 Nobel prize for Economics
Non-cooperative games, Annals of Mathematics 54,
1951
46
The Prisoners Dilemma
The iterated Prisoners Dilemma
Player 2 Player 2
Player 1 -2,-2 -10,-1
Player 1 -1,-10 -8,-8
be silent
confess
(-10,-1)
(-2,-2)
(-8,-8)
(-1,-10)
47
Hawk-Dove
D H D H
D H 2,2 0,3
D H 3,0 1,1
(0,3)
(2,2)
(1,1)
(3,0)
48
Hawk-Dove and Tit-for-Tat
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
D H D H
D H 2,2 0,3
D H 3,0 1,1
Tit-for-Tat begin D and play the previous
opponents action at every other stage
49
Robert Axelrod
Anatol Rapoport
John Maynard Smith
50
Marriage Problems
  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
51
Marriage Problems
  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
52
Marriage Problems
  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
Lloyd S. Shapley
David Gale
College admissions and the stability of marriage,
American Mathematical Monthly 69, 1962
54
Gale-Shapley Algorithm
  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
9
3
4
6
7
8
5
  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
1
7
2
8
4
5
9
3
6
55
Gale-Shapley Algorithm
- Gives the best stable matching for the
proposers
- 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
56
?
frank_at_math.unimaas.nl
57
GAME VER