A short history of equilibrium

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

3. Oskar Morgenstern

4. Oskar Morgenstern • Institut für Konjunkturforschung

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

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

7. Holmes‘ survival probability

8. Holmes‘ survival probability

9. Holmes‘ survival probability

10. Holmes‘ survival probability

11. Holmes‘ survival probability

12. 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 • Johnny gives up: Oskar wins • 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 • Johnny gives up: Oskar wins • Johnny raises stakes: another dollar • Oskar gives up: Johnny wins • Oskar raises: Johnny shows card

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

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

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

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

22. Johnnys expected gain

23. Johnnys expected gain

24. Johnnys expected gain

25. Johnnys expected gain

26. Johnnys expected gain

27. Johnnys expected gain

28. Johnnys expected gain

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

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

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

32. Poker for Beginners

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

34. But: • Why be a pessimist?

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

36. Chicken for Beginners

37. Johnnys Payoff for Chicken

38. Johnnys Payoff for Chicken

39. 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 • Arbitrarily many players • each has arbitrarily many strategies

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

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

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

48. Nash-Equilibrium • Presumes rational players

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

50. Two-Person Games