1. A short history of equilibrium John Nash and Game Theory

3. Oskar Morgenstern

4. n Institut für Konjunkturforschung

5. Oskar Morgenstern n Institut für Konjunkturforschung n Sherlock Holmes vs. Moriarty

6. Oskar Morgenstern n Institut für Konjunkturforschung n Sherlock Holmes vs. Moriarty n London -- Canterbury -- Dover

7. Holmes‘ survival probability

13. John von Neumann Zur Theorie der Gesellschaftsspiele (1928)

14. Poker for Beginners Two players, Johnny and Oskar Two cards, King and Ace

15. Poker for Beginners Two players, Johnny and Oskar Two cards, King and Ace Stakes one dollar each Johnny draws a card

16. Poker for Beginners Two players, Johnny and Oskar Two cards, King and Ace Stakes one dollar each Johnny draws a card l Johnny gives up: Oskar wins l Johnny raises stakes: another dollar

17. Poker for Beginners Two players, Johnny and Oskar Two cards, King and Ace Stakes one dollar each Johnny draws a card l Johnny gives up: Oskar wins l Johnny raises stakes: another dollar l Oskar gives up: Johnny wins l Oskar raises: Johnny shows card

18. Poker for Beginners Johnny can l bluff (raise even with king)

19. Poker for Beginners Johnny can l bluff (raise even with king) l not bluff (raise only with ace)

20. Poker for Beginners Johnny can l bluff (raise even with king) l not bluff (raise only with ace) Oskar can l raise if Johnny raises

21. Poker for Beginners Johnny can l bluff (raise even with king) l not bluff (raise only with ace) Oskar can l raise if Johnny raises l give up if Johnny raises

22. Johnnys expected gain

29. Poker for Beginners Johnny: maximize minimal payoff l Johnny bluffs with probability 1/3

30. Poker for Beginners Johnny: maximize minimal payoff l Johnny bluffs with probability 1/3 Oskar: maximize minimal payoff (= minimize Johnny‘s maximal payof) l Oskar raises with probability 2/3

31. Poker for Beginners Maximize minimal payoff l Johnny bluffs with probability 1/3 l Oskar raises with probability 2/3 none can improve

32. Poker for Beginners

33. Maximize minimal payoff l Johnny bluffs with probability 1/3 l Oskar raises with probability 1/3 none can improve Morgenstern‘s example has a solution!

34. But: n Why be a pessimist?

35. But: n Why be a pessimist? n Why only zero sum games?

36. Chicken for Beginners

37. Johnnys Payoff for Chicken

40. Payoff for Chicken

41. Chicken for Beginners Maximin: yield

42. Chicken for Beginners Maximin: yield not consistent! If the co-player yields, escalate!

43. Chicken for Beginners Maximin: yield not consistent! If the co-player yields, escalate! If both yield with probability 9/10, none can improve

44. Nash-Equilibrium n Arbitrarily many players n each has arbitrarily many strategies

45. Nash-Equilibrium n Arbitrarily many players n each has arbitrarily many strategies n there always exists an equilibrium solution

46. Nash-Equilibrium n Arbitrarily many players n each has arbitrarily many strategies n there always exists an equilibrium solution n no player can improve payoff by deviating n each strategy best reply to the others

47. Nash-Equilibrium n Arbitrarily many players n each has arbitrarily many strategies n there always exists an equilibrium solution n no player can improve payoff by deviating n each strategy best reply to the others n if zero-sum game: maximin solution

48. Nash-Equilibrium n Presumes rational players

49. Nash-Equilibrium n Presumes rational players n is unstable: if others deviate, it may be better to also deviate

50. Two-Person Games n

51. Mixed strategies n

52. Best reply n

53. n

54. Nash Equilibria n

55. n

56. Zero-sum Games n

57. n

58. n

59. n

60. n

61. n Nash equilibria are maximin pairs!

62. Zero-sum Games n Nash equilibria are maximin pairs! n (and vice versa)

63. Prisoner‘s Dilemma

64. Repeated Prisoner‘s Dilemma n Throw dice, stop if 6, new round if not 6 n on average 6 rounds n allow only two strategies: n Tit For Tat n always defect

65. Repeated Prisoner‘s Dilemma

66. Risky Coordination

83. Public Goods Experiments n Six players n 20 Euros each n invest into common pot n this sum is tripled n distributed equally among all six players

84. Public Goods Experiments n 50 cents return per invested euro n Nash: invest nothing! n no ‚public goods‘

85. Evolutionary Game Theory n adaptation n best reply n imitate successful players n etc n if convergence, then to Nash n not necessarily convergence! (Hofbauer) n local interaction (Nowak) n transmission mechanisms and population structure

86. Evolutionary Games n Population dynamical viewpoint n John Maynard Smith n Peter Hammerstein n Reinhard Selten n Josef Hofbauer

87. Evolutionary Games The greatest conceptual revolution in biology...the replacement of typological thinking by population thinking. Ernst Mayr

88. Evolutionary Games n Population dynamical viewpoint n John Maynard Smith n Peter Hammerstein n Reinhard Selten n Josef Hofbauer n anticipated by John Nash: mass action approach