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A short history of equilibrium John Nash and Game Theory

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Oskar Morgenstern

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n Institut für Konjunkturforschung

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

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

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Holmes‘ survival probability

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John von Neumann Zur Theorie der Gesellschaftsspiele (1928)

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

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

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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

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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

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

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

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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

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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

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Johnnys expected gain

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

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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

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

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Poker for Beginners

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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!

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But: n Why be a pessimist?

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

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Chicken for Beginners

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Johnnys Payoff for Chicken

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Payoff for Chicken

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Chicken for Beginners Maximin: yield

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

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

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

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

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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

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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

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Nash-Equilibrium n Presumes rational players

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

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Two-Person Games n

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Mixed strategies n

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Best reply n

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n

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Nash Equilibria n

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Zero-sum Games n

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n

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n Nash equilibria are maximin pairs!

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

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Prisoner‘s Dilemma

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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

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Repeated Prisoner‘s Dilemma

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Risky Coordination

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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

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

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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

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

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

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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

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