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##### A short history of equilibrium

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**A short history of equilibrium**John Nash and Game Theory**Oskar Morgenstern**• Institut für Konjunkturforschung**Oskar Morgenstern**• Institut für Konjunkturforschung • Sherlock Holmes vs. Moriarty**Oskar Morgenstern**• Institut für Konjunkturforschung • Sherlock Holmes vs. Moriarty • London -- Canterbury -- Dover**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 • Johnny gives up: Oskar wins • 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 • Johnny gives up: Oskar wins • Johnny raises stakes: another dollar • Oskar gives up: Johnny wins • Oskar raises: Johnny shows card**Poker for Beginners**• Johnny can • bluff (raise even with king)**Poker for Beginners**• Johnny can • bluff (raise even with king) • not bluff (raise only with ace)**Poker for Beginners**• Johnny can • bluff (raise even with king) • not bluff (raise only with ace) • Oskar can • raise if Johnny raises**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**Poker for Beginners**• Johnny: maximize minimal payoff • Johnny bluffs with probability 1/3**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**Poker for Beginners**• Maximize minimal payoff • Johnny bluffs with probability 1/3 • Oskar raises with probability 2/3 • none can improve**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!**But:**• Why be a pessimist?**But:**• Why be a pessimist? • Why only zero sum games?**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**• Arbitrarily many players • each has arbitrarily many strategies**Nash-Equilibrium**• Arbitrarily many players • each has arbitrarily many strategies • there always exists an equilibrium solution**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**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**Nash-Equilibrium**• Presumes rational players**Nash-Equilibrium**• Presumes rational players • is unstable: • if others deviate, it may be better to also deviate