March 5, 2007Al Quds University, Jerusalem 1/57 Game Theory: Sharing, Stability and Strategic Behaviour Frank Thuijsman Maastricht University The Netherlands

March 5, 2007Al Quds University, Jerusalem 2/57 John von NeumannOskar Morgenstern Theory of Games and Economic Behavior, Princeton, 1944

March 5, 2007Al Quds University, Jerusalem 3/57 1.Three widows 2.Cooperative games 3.Strategic games 4.Marriage problems Programme

March 5, 2007Al Quds University, Jerusalem 4/57 “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.” Kethuboth, Fol. 93a, Babylonian Talmud, Epstein, ed, 1935 So: 100 is shared equally, each gets So: 200 is shared as So: 300 is shared proportionally as

March 5, 2007Al Quds University, Jerusalem 5/ Estate Widow “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? EqualProportional???

March 5, 2007Al Quds University, Jerusalem 6/57 Barry O’Neill A problem of rights arbitration from the Talmud, Mathematical Social Sciences 2, 1982

March 5, 2007Al Quds University, Jerusalem 7/57 Robert J. Aumann Michael Maschler Game theoretic analysis of a bankruptcy problem from the Talmud, Journal of Economic Theory 36, 1985 Nobel prize for Economics, Thomas Schelling

March 5, 2007Al Quds University, Jerusalem 8/57 Robert J. Aumann Michael Maschler Game theoretic analysis of a bankruptcy problem from the Talmud, Journal of Economic Theory 36, 1985

March 5, 2007Al Quds University, Jerusalem 9/ S Ø ABCABACBCABC v(S) The nucleolus of the game The value of coalition S is the amount that remains, if the others get their claims first Cooperative games

March 5, 2007Al Quds University, Jerusalem 10/ Cooperative games S Ø ABCABACBCABC v(S) The nucleolus of the game The value of coalition S is the amount that remains, if the others get their claims first

March 5, 2007Al Quds University, Jerusalem 11/ A B C Cooperative games A B C The value of coalition S is the amount that remains, if the others get their claims first. S Ø ABCABACBCABC v(S) The nucleolus of the game A 100 B 200 C 300

March 5, 2007Al Quds University, Jerusalem 12/57 S Ø ABCABACBCABC v(S) Cooperative games Sharing costs or gains based upon the values of the coalitions

March 5, 2007Al Quds University, Jerusalem 13/57 The core S Ø ABCABACBCABC v(S) (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

March 5, 2007Al Quds University, Jerusalem 14/57 Lloyd S. Shapley A value for n-person games, In: Contribution to the Theory of Games, Kuhn and Tucker (eds), Princeton, 1953

March 5, 2007Al Quds University, Jerusalem 15/57 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”

March 5, 2007Al Quds University, Jerusalem 16/57 S Ø ABCABACBCABC v(S) The Shapley-value ABC A-B-C A-C-B B-A-C B-C-A C-A-B C-B-A Sum: Φ:Φ: Marginal contributions

March 5, 2007Al Quds University, Jerusalem 17/57 David Schmeidler The nucleolus of a characteristic function game, SIAM Journal of Applied Mathematics 17, 1969

March 5, 2007Al Quds University, Jerusalem 18/57 The nucleolus (14,0,0)(0,14,0) (0,0,14) S Ø ABCABACBCABC v(S) (4,5,5) the nucleolus S Ø ABCABACBCABC v(S) S Ø ABCABACBCABC v(S)06-x7-x 9-x11-x 14 S Ø ABCABACBCABC v(S) Leeg Φ = (4, 4.5, 5.5)

March 5, 2007Al Quds University, Jerusalem 19/ A B C The Talmud games S Ø ABCABACBCABC v(S) (100,0,0) (0,100,0) (0,0,100) the nucleolus

March 5, 2007Al Quds University, Jerusalem 20/ The Talmud games S Ø ABCABACBCABC v(S) (200,0,0) (0,200,0) (0,0,200)

March 5, 2007Al Quds University, Jerusalem 21/ The Talmud games S Ø ABCABACBCABC v(S) (200,0,0) (0,200,0) (0,0,200) the nucleolus

March 5, 2007Al Quds University, Jerusalem 22/ The Talmud games S Ø ABCABACBCABC v(S) (300,0,0) (0,300,0) (0,0,300)

March 5, 2007Al Quds University, Jerusalem 23/ The Talmud games S Ø ABCABACBCABC v(S) (300,0,0) (0,300,0) (0,0,300) the nucleolus

March 5, 2007Al Quds University, Jerusalem 24/ Estate Widow “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?

March 5, 2007Al Quds University, Jerusalem 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

March 5, 2007Al Quds University, Jerusalem 26/57 Consistency jointly One claims 100, the other all, so 25 for the other; both claim the remains (100), so each gets half jointly jointly

March 5, 2007Al Quds University, Jerusalem 27/57 Consistency One claims 100, the other all, so 25 for the other; both claim the remains (100), so each gets half jointly

March 5, 2007Al Quds University, Jerusalem 28/57 Consistency jointly Each claims all, so each gets half

March 5, 2007Al Quds University, Jerusalem 29/ Consistency jointly Each claims all, so each gets half jointly

March 5, 2007Al Quds University, Jerusalem 30/57 Consistency jointly One claims 100, the other all, so 50 for the other; both claim the remains (100), so each gets half jointly jointly

March 5, 2007Al Quds University, Jerusalem 31/57 Consistency jointly One claims 100, the other all, so 100 for the other; both claim the remains (100), so each gets half jointly jointly

March 5, 2007Al Quds University, Jerusalem 32/ How to share 400? What if a fourth widow claims 400? Do we now really know how to do it?

March 5, 2007Al Quds University, Jerusalem 33/57 Marek M. Kaminski ‘Hydraulic’ rationing, Mathematical Social Sciences 40, 2000

March 5, 2007Al Quds University, Jerusalem 34/ Communicating Vessels

March 5, 2007Al Quds University, Jerusalem 35/57 Pouring in

March 5, 2007Al Quds University, Jerusalem 36/57 Pouring in

March 5, 2007Al Quds University, Jerusalem 37/57 Pouring in

March 5, 2007Al Quds University, Jerusalem 38/57 Pouring in

March 5, 2007Al Quds University, Jerusalem 39/57 4 widows with

March 5, 2007Al Quds University, Jerusalem 40/57 Strategic games Strategy player 1: LLR Strategy player 2: RRR “game in extensive form”

March 5, 2007Al Quds University, Jerusalem 41/57 Strategy player 1: RLL Strategy player 2: RLL Threat Strategic games “Game in extensive form”

March 5, 2007Al Quds University, Jerusalem 42/57 LLLLLRLRLLRRRLLRLRRRLRRR LLL LLR 2,2 LRL LRR RLL 3,4 RLR RRL RRR “Game in strategic form”

March 5, 2007Al Quds University, Jerusalem 43/57 LLLLLRLRLLRRRLLRLRRRLRRR LLL 6,1 2,2 LLR 6,1 2,2 LRL 4,3 2,2 LRR 4,3 2,2 RLL 3,4 1,3 3,4 1,3 RLR 2,14,21,3 2,14,21,3 RRL 3,4 1,3 3,4 1,3 RRR 2,14,21,3 2,14,21,3 LLLLLRLRLLRRRLLRLRRRLRRR LLL 6,1 2,2 LLR 6,1 2,2 LRL 4,3 2,2 LRR 4,3 2,2 RLL 3,4 1,3 3,4 1,3 RLR 2,14,21,3 2,14,21,3 RRL 3,4 1,3 3,4 1,3 RRR 2,14,21,3 2,14,21,3 LLLLLRLRLLRRRLLRLRRRLRRR LLL 6,1 2,2 LLR 6,1 2,2 LRL 4,3 2,2 LRR 4,3 2,2 RLL 3,4 1,3 3,4 1,3 RLR 2,14,21,3 2,14,21,3 RRL 3,4 1,3 3,4 1,3 RRR 2,14,21,3 2,14,21,3 “Game in strategic form”

March 5, 2007Al Quds University, Jerusalem 44/57 Equilibrium: If players play best responses to eachother, then a stable situation arises

March 5, 2007Al Quds University, Jerusalem 45/ : Nobel prize for Economics John F. NashJohn C. HarsanyiReinhard Selten “A Beautiful Mind” Non-cooperative games, Annals of Mathematics 54, 1951

March 5, 2007Al Quds University, Jerusalem 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

March 5, 2007Al Quds University, Jerusalem 47/57 D H DHDH 2,20,3 3,01,1 Hawk-Dove (2,2) (3,0) (0,3) (1,1)

March 5, 2007Al Quds University, Jerusalem 48/57 D H DHDH 2,20,3 3,01,1 Hawk-Dove and Tit-for-Tat Tit-for-Tat: begin D and play the previous opponent’s action at every other stage DHT D20 H31 T DHT D202 H311 T212

March 5, 2007Al Quds University, Jerusalem 49/57 Robert AxelrodAnatol RapoportJohn Maynard Smith

March 5, 2007Al Quds University, Jerusalem 50/ AnnyFreddyHarryKennyGerryLenny BettyGerryKennyFreddyHarryLenny ConnyLennyHarryGerryFreddyKenny DollyHarryLennyFreddyGerryKenny EmmyHarryKennyGerryLennyFreddy ConnyBettyAnnyEmmyDolly GerryDollyAnnyBettyEmmyConny HarryEmmyAnnyDollyBettyConny KennyEmmyConnyAnnyDollyBetty LennyEmmyAnnyBettyConnyDolly “Marriage Problems”

March 5, 2007Al Quds University, Jerusalem 51/ AnnyFreddyKennyGerryLenny BettyGerryKennyFreddyLenny ConnyLennyGerryFreddyKenny DollyLennyFreddyGerryKenny GerryLennyFreddy ConnyBettyAnnyDolly GerryDollyAnnyBettyConny AnnyDollyBettyConny KennyConnyAnnyDollyBetty LennyAnnyBettyConnyDolly “Marriage Problems”

March 5, 2007Al Quds University, Jerusalem 52/ AnnyFreddyKennyGerryLenny BettyGerryKennyFreddyLenny ConnyLennyGerryFreddyKenny DollyLennyGerryKenny GerryLennyFreddy ConnyBettyAnny GerryDollyAnnyBettyConny AnnyDollyBettyConny KennyConnyAnnyDollyBetty LennyAnnyBettyConnyDolly “Marriage Problems”

March 5, 2007Al Quds University, Jerusalem 53/57 Lloyd S. Shapley College admissions and the stability of marriage, American Mathematical Monthly 69, 1962 David Gale

March 5, 2007Al Quds University, Jerusalem 54/ AnnyFreddyHarryKennyGerryLenny BettyGerryKennyFreddyHarryLenny ConnyLennyHarryGerryFreddyKenny DollyHarryLennyFreddyGerryKenny EmmyHarryKennyGerryLennyFreddy ConnyBettyAnnyEmmyDolly GerryDollyAnnyBettyEmmyConny HarryEmmyAnnyDollyBettyConny KennyEmmyConnyAnnyDollyBetty LennyEmmyAnnyBettyConnyDolly Gale-Shapley Algorithm

March 5, 2007Al Quds University, Jerusalem 55/57 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”

March 5, 2007Al Quds University, Jerusalem 56/57 ?

March 5, 2007Al Quds University, Jerusalem 57/57 GAME VER