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Game Theory: Sharing, Stability and Strategic Behaviour

Game Theory: Sharing, Stability and Strategic Behaviour. Frank Thuijsman Maastricht University The Netherlands. John von Neumann. Oskar Morgenstern. Theory of Games and Economic Behavior , Princeton, 1944. Programme. Three widows Cooperative games Strategic games Marriage problems.

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Game Theory: Sharing, Stability and Strategic Behaviour

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  1. Game Theory:Sharing, Stability and Strategic Behaviour Frank Thuijsman Maastricht University The Netherlands Al Quds University, Jerusalem

  2. John von Neumann Oskar Morgenstern Theory of Games and Economic Behavior, Princeton, 1944 Al Quds University, Jerusalem

  3. Programme • Three widows • Cooperative games • Strategic games • Marriage problems Al Quds University, Jerusalem

  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. Al Quds University, Jerusalem

  5. Estate 50 50 Widow 75 100 75 150 Equal ??? Proportional “Similarlyif three persons contributed to a joint fund and they had made a loss or a profitthen they share in the same manner.” How to share 400? What if a fourth widow claims 400? Al Quds University, Jerusalem

  6. Barry O’Neill A problem of rights arbitration from the Talmud, Mathematical Social Sciences 2, 1982 Al Quds University, Jerusalem

  7. Robert J. Aumann Thomas Schelling Michael Maschler Nobel prize for Economics, 12-10-2005 Game theoretic analysis of a bankruptcy problem from the Talmud, Journal of Economic Theory 36, 1985 Al Quds University, Jerusalem

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

  9. Cooperative games The value of coalition S is the amount that remains, if the others get their claims first. The nucleolus of the game 0 0 0 100 200 300 0 0 Al Quds University, Jerusalem

  10. Cooperative games The value of coalition S is the amount that remains, if the others get their claims first. The nucleolus of the game 0 0 100 200 0 0 0 0 Al Quds University, Jerusalem

  11. Cooperative games The value of coalition S is the amount that remains, if the others get their claims first. The nucleolus of the game 0 0 0 0 0 0 0 100 Al Quds University, Jerusalem

  12. Cooperative games Sharing costs or gains based upon the values of the coalitions Al Quds University, Jerusalem

  13. The core (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) Al Quds University, Jerusalem

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

  15. 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” Al Quds University, Jerusalem

  16. The Shapley-value 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 Al Quds University, Jerusalem

  17. David Schmeidler The nucleolus of a characteristic function game, SIAM Journal of Applied Mathematics 17, 1969 Al Quds University, Jerusalem

  18. The nucleolus (0,0,14) Φ = (4, 4.5, 5.5) (4,5,5) the nucleolus Leeg (14,0,0) (0,14,0) Al Quds University, Jerusalem

  19. The Talmud games (0,0,100) the nucleolus (100,0,0) (0,100,0) Al Quds University, Jerusalem

  20. The Talmud games (0,0,200) (200,0,0) (0,200,0) Al Quds University, Jerusalem

  21. The Talmud games (0,0,200) the nucleolus (200,0,0) (0,200,0) Al Quds University, Jerusalem

  22. The Talmud games (0,0,300) (300,0,0) (0,300,0) Al Quds University, Jerusalem

  23. The Talmud games (0,0,300) the nucleolus (300,0,0) (0,300,0) Al Quds University, Jerusalem

  24. Estate 50 50 Widow 75 100 75 150 “Similarlyif three persons contributed to a joint fund and they had made a loss or a profitthen they share in the same manner.” How to share 400? What if a fourth widow claims 400? Al Quds University, Jerusalem

  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 Al Quds University, Jerusalem

  26. Consistency One claims 100, the other all, so 25 for the other; both claim the remains (100), so each gets half Al Quds University, Jerusalem

  27. Consistency One claims 100, the other all, so 25 for the other; both claim the remains (100), so each gets half Al Quds University, Jerusalem

  28. Consistency Each claims all, so each gets half Al Quds University, Jerusalem

  29. Consistency Each claims all, so each gets half Al Quds University, Jerusalem

  30. Consistency One claims 100, the other all, so 50 for the other; both claim the remains (100), so each gets half Al Quds University, Jerusalem

  31. Consistency One claims 100, the other all, so 100 for the other; both claim the remains (100), so each gets half Al Quds University, Jerusalem

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

  33. Marek M. Kaminski ‘Hydraulic’ rationing, Mathematical Social Sciences 40, 2000 Al Quds University, Jerusalem

  34. Communicating Vessels 50 100 150 50 100 150 Al Quds University, Jerusalem

  35. Pouring in 100 33.33 33.33 33.33 Al Quds University, Jerusalem

  36. Pouring in 200 75 75 50 Al Quds University, Jerusalem

  37. Pouring in 300 150 100 50 Al Quds University, Jerusalem

  38. Pouring in 400 125 225 50 Al Quds University, Jerusalem

  39. 4 widows with 400 125 125 100 50 Al Quds University, Jerusalem

  40. Strategic games “game in extensive form” Strategy player 1: LLR Strategy player 2: RRR Al Quds University, Jerusalem

  41. Strategic games “Game in extensive form” Threat Strategy player 1: RLL Strategy player 2: RLL Al Quds University, Jerusalem

  42. “Game in strategic form” Al Quds University, Jerusalem

  43. “Game in strategic form” Al Quds University, Jerusalem

  44. Equilibrium: If players play best responses to eachother, then a stable situation arises Al Quds University, Jerusalem

  45. “A Beautiful Mind” Reinhard Selten John F. Nash John C. Harsanyi 1994: Nobel prize for Economics Non-cooperative games, Annals of Mathematics 54, 1951 Al Quds University, Jerusalem

  46. The iterated Prisoner’s Dilemma The Prisoner’s Dilemma be silent confess (-10,-1) (-2,-2) (-8,-8) (-1,-10) Al Quds University, Jerusalem

  47. Hawk-Dove (0,3) (2,2) (1,1) (3,0) Al Quds University, Jerusalem

  48. Hawk-Dove and Tit-for-Tat Tit-for-Tat: begin D and play the previous opponent’s action at every other stage Al Quds University, Jerusalem

  49. John Maynard Smith Robert Axelrod Anatol Rapoport Al Quds University, Jerusalem

  50. “Marriage Problems” Al Quds University, Jerusalem

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