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

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A short history of equilibrium John Nash and Game Theory

Oskar Morgenstern

n Institut für Konjunkturforschung

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

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

Holmes‘ survival probability

John von Neumann Zur Theorie der Gesellschaftsspiele (1928)

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

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

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

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

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

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

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

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

Johnnys expected gain

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

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

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

Poker for Beginners

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!

But: n Why be a pessimist?

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

Chicken for Beginners

Johnnys Payoff for Chicken

Payoff for Chicken

Chicken for Beginners Maximin: yield

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

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

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

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

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

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

Nash-Equilibrium n Presumes rational players

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

Two-Person Games n

Mixed strategies n

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Nash Equilibria n

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Zero-sum Games n

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n Nash equilibria are maximin pairs!

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

Prisoner‘s Dilemma

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

Repeated Prisoner‘s Dilemma

Risky Coordination

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

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

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

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

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

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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