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

  2. Oskar Morgenstern

  3. Oskar Morgenstern • Institut für Konjunkturforschung

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

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

  6. Holmes‘ survival probability

  7. Holmes‘ survival probability

  8. Holmes‘ survival probability

  9. Holmes‘ survival probability

  10. Holmes‘ survival probability

  11. Holmes‘ survival probability

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

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

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

  15. 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

  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 • Oskar gives up: Johnny wins • Oskar raises: Johnny shows card

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

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

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

  20. 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

  21. Johnnys expected gain

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

  29. 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

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

  31. Poker for Beginners

  32. 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!

  33. But: • Why be a pessimist?

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

  35. Chicken for Beginners

  36. Johnnys Payoff for Chicken

  37. Johnnys Payoff for Chicken

  38. Johnnys Payoff for Chicken

  39. Payoff for Chicken

  40. Chicken for Beginners • Maximin: yield

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

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

  43. Nash-Equilibrium • Arbitrarily many players • each has arbitrarily many strategies

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

  45. 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

  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 • if zero-sum game: maximin solution

  47. Nash-Equilibrium • Presumes rational players

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

  49. Two-Person Games