Download presentation

Presentation is loading. Please wait.

Published byDaphne Week Modified over 2 years ago

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

Similar presentations

OK

Nash’s Theorem Theorem (Nash, 1951): Every finite game (finite number of players, finite number of pure strategies) has at least one mixed-strategy Nash.

Nash’s Theorem Theorem (Nash, 1951): Every finite game (finite number of players, finite number of pure strategies) has at least one mixed-strategy Nash.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google