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

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

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 2/57 John von NeumannOskar Morgenstern Theory of Games and Economic Behavior, Princeton, 1944

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

frank@math.unimaas.nl 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 33.33. So: 200 is shared as 50 - 75 - 75. So: 300 is shared proportionally as 50 - 100 - 150.

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 5/57 100200300 10033.33 20033.33 30033.33 Estate Widow 50 75 50 100 150 “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? EqualProportional???

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

frank@math.unimaas.nl 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, 12-10-2005 Thomas Schelling

frank@math.unimaas.nl 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

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 9/57 100200300 10033.3350 20033.3375100 30033.3375150 100200300 10050 200100 300150 S Ø ABCABACBCABC v(S) The nucleolus of the game 0 00 00100200300 The value of coalition S is the amount that remains, if the others get their claims first. 100200300 100 200 300 Cooperative games

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 10/57 100200300 10033.3350 20033.3375100 30033.3375150 Cooperative games 100200300 10050 20075 30075 S Ø ABCABACBCABC v(S) The nucleolus of the game 0000 01002000 The value of coalition S is the amount that remains, if the others get their claims first. 100200300 100 200 300

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 11/57 100200300 A 10033.3350 B 20033.3375100 C 30033.3375150 Cooperative games 100200300 A 10033.33 B 20033.33 C 30033.33 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 0000000100 200300 A 100 B 200 C 300

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 12/57 S Ø ABCABACBCABC v(S)0677911 14 Cooperative games Sharing costs or gains based upon the values of the coalitions

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 13/57 The core S Ø ABCABACBCABC v(S)0677911 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

frank@math.unimaas.nl 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

frank@math.unimaas.nl 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”

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 16/57 S Ø ABCABACBCABC v(S)0677911 14 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: 635 635 275 374 437 347 242733 44.55.5 Marginal contributions

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

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 18/57 The nucleolus (14,0,0)(0,14,0) (0,0,14) S Ø ABCABACBCABC v(S)06-27-2 9-211-2 14 (4,5,5) the nucleolus S Ø ABCABACBCABC v(S)0677911 14 S Ø ABCABACBCABC v(S)06-x7-x 9-x11-x 14 S Ø ABCABACBCABC v(S)045579914 Leeg Φ = (4, 4.5, 5.5)

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 19/57 100200300 A 10033.3350 B 20033.3375100 C 30033.3375150 The Talmud games S Ø ABCABACBCABC v(S)0000000100 (100,0,0) (0,100,0) (0,0,100) the nucleolus

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 20/57 100200300 10033.3350 20033.3375100 30033.3375150 The Talmud games S Ø ABCABACBCABC v(S)000000100200 (200,0,0) (0,200,0) (0,0,200)

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 21/57 100200300 10033.3350 20033.3375100 30033.3375150 The Talmud games S Ø ABCABACBCABC v(S)000000100200 (200,0,0) (0,200,0) (0,0,200) the nucleolus

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 22/57 100200300 10033.3350 20033.3375100 30033.3375150 The Talmud games S Ø ABCABACBCABC v(S)00000100200300 (300,0,0) (0,300,0) (0,0,300)

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 23/57 100200300 10033.3350 20033.3375100 30033.3375150 The Talmud games S Ø ABCABACBCABC v(S)00000100200300 (300,0,0) (0,300,0) (0,0,300) the nucleolus

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 24/57 100200300 10033.33 20033.33 30033.33 Estate Widow 50 75 50 100 150 “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?

frank@math.unimaas.nl 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

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 26/57 Consistency 100200300 10033.3350 20033.3375100 30033.3375150 jointly125 100 200 One claims 100, the other all, so 25 for the other; both claim the remains (100), so each gets half jointly125 100 20025 jointly125 10050 20025+50

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 27/57 Consistency 100200300 10033.3350 20033.3375100 30033.3375150 One claims 100, the other all, so 25 for the other; both claim the remains (100), so each gets half jointly125 10050 30025+50

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 28/57 Consistency 100200300 10033.3350 20033.3375100 30033.3375150 jointly150 20075 30075 Each claims all, so each gets half

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 29/57 100200300 10033.3350 20033.3375100 30033.3375150 Consistency 100200300 10033.3350 20033.3375100 30033.3375150 jointly66.66 100 200 Each claims all, so each gets half jointly66.66 10033.33 20033.33

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 30/57 Consistency 100200300 10033.3350 20033.3375100 30033.3375150 jointly150 100 200 One claims 100, the other all, so 50 for the other; both claim the remains (100), so each gets half jointly150 100 20050 jointly150 10050 20050+50

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 31/57 Consistency 100200300 10033.3350 20033.3375100 30033.3375150 jointly200 100 300 One claims 100, the other all, so 100 for the other; both claim the remains (100), so each gets half jointly200 100 300100 jointly200 10050 200100+50

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 32/57 100200300 10033.3350 20033.3375100 30033.3375150 How to share 400? What if a fourth widow claims 400? Do we now really know how to do it?

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

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 34/57 50 100 150 Communicating Vessels

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 35/57 Pouring in 100 33.33

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 36/57 Pouring in 200 75 50 75

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 37/57 Pouring in 300 150 100 50

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 38/57 Pouring in 400 50 125225

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 39/57 4 widows with 400 125 100 50 125

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

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

frank@math.unimaas.nl 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”

frank@math.unimaas.nl 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”

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

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 45/57 1994: Nobel prize for Economics John F. NashJohn C. HarsanyiReinhard Selten “A Beautiful Mind” Non-cooperative games, Annals of Mathematics 54, 1951

frank@math.unimaas.nl 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

frank@math.unimaas.nl 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)

frank@math.unimaas.nl 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

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 49/57 Robert AxelrodAnatol RapoportJohn Maynard Smith

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 50/57 12345 AnnyFreddyHarryKennyGerryLenny BettyGerryKennyFreddyHarryLenny ConnyLennyHarryGerryFreddyKenny DollyHarryLennyFreddyGerryKenny EmmyHarryKennyGerryLennyFreddy 12345 ConnyBettyAnnyEmmyDolly GerryDollyAnnyBettyEmmyConny HarryEmmyAnnyDollyBettyConny KennyEmmyConnyAnnyDollyBetty LennyEmmyAnnyBettyConnyDolly “Marriage Problems”

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 51/57 12345 AnnyFreddyKennyGerryLenny BettyGerryKennyFreddyLenny ConnyLennyGerryFreddyKenny DollyLennyFreddyGerryKenny GerryLennyFreddy 12345 ConnyBettyAnnyDolly GerryDollyAnnyBettyConny AnnyDollyBettyConny KennyConnyAnnyDollyBetty LennyAnnyBettyConnyDolly “Marriage Problems”

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 52/57 12345 AnnyFreddyKennyGerryLenny BettyGerryKennyFreddyLenny ConnyLennyGerryFreddyKenny DollyLennyGerryKenny GerryLennyFreddy 12345 ConnyBettyAnny GerryDollyAnnyBettyConny AnnyDollyBettyConny KennyConnyAnnyDollyBetty LennyAnnyBettyConnyDolly “Marriage Problems”

frank@math.unimaas.nl 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

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 54/57 12345 AnnyFreddyHarryKennyGerryLenny BettyGerryKennyFreddyHarryLenny ConnyLennyHarryGerryFreddyKenny DollyHarryLennyFreddyGerryKenny EmmyHarryKennyGerryLennyFreddy 12345 ConnyBettyAnnyEmmyDolly GerryDollyAnnyBettyEmmyConny HarryEmmyAnnyDollyBettyConny KennyEmmyConnyAnnyDollyBetty LennyEmmyAnnyBettyConnyDolly Gale-Shapley Algorithm 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

frank@math.unimaas.nl 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”

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 56/57 frank@math.unimaas.nl ?

frank@math.unimaas.nl March 5, 2007Al Quds University, Jerusalem 57/57 GAME VER

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