A short history of equilibrium

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A short history of equilibrium

• John Nash and Game Theory

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Oskar Morgenstern
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Oskar Morgenstern

• Institut für Konjunkturforschung

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Oskar Morgenstern

• Institut für Konjunkturforschung
• Sherlock Holmes vs. Moriarty

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Oskar Morgenstern

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

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Holmes survival probability

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Holmes survival probability

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Holmes survival probability

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Holmes survival probability

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Holmes survival probability

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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
• Johnny gives up Oskar wins
• 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
• Johnny gives up Oskar wins
• Johnny raises stakes another dollar
• Oskar gives up Johnny wins
• Oskar raises Johnny shows card

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Poker for Beginners

• Johnny can
• bluff (raise even with king)

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Poker for Beginners

• Johnny can
• bluff (raise even with king)
• not bluff (raise only with ace)

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Poker for Beginners

• Johnny can
• bluff (raise even with king)
• not bluff (raise only with ace)
• Oskar can
• raise if Johnny raises

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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

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Johnnys expected gain

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Johnnys expected gain

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Johnnys expected gain

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Johnnys expected gain

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Johnnys expected gain

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Johnnys expected gain

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Johnnys expected gain

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Poker for Beginners

• Johnny maximize minimal payoff
• Johnny bluffs with probability 1/3

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Poker for Beginners

• Johnny maximize minimal payoff
• Johnny bluffs with probability 1/3
• Oskar maximize minimal payoff
• ( minimize Johnnys maximal payof)
• Oskar raises with probability 2/3

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Poker for Beginners

• Maximize minimal payoff
• Johnny bluffs with probability 1/3
• Oskar raises with probability 2/3
• none can improve

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Poker for Beginners
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Poker for Beginners

• Maximize minimal payoff
• Johnny bluffs with probability 1/3
• Oskar raises with probability 1/3
• none can improve
• Morgensterns example has a solution!

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But

• Why be a pessimist?

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But

• Why be a pessimist?
• Why only zero sum games?

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Chicken for Beginners

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Johnnys Payoff for Chicken

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Johnnys Payoff for Chicken

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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

• Arbitrarily many players
• each has arbitrarily many strategies

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Nash-Equilibrium

• Arbitrarily many players
• each has arbitrarily many strategies
• there always exists an equilibrium solution

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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

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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

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Nash-Equilibrium

• Presumes rational players

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Nash-Equilibrium

• Presumes rational players
• is unstable
• if others deviate, it may be better to also
  deviate

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Two-Person Games

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Mixed strategies

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Best reply

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Best reply

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Nash Equilibria

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Nash Equilibria

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Zero-sum Games

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Zero-sum Games

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Zero-sum Games

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Zero-sum Games

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Zero-sum Games

• Nash equilibria are maximin pairs!

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Zero-sum Games

• Nash equilibria are maximin pairs!
• (and vice versa)

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Prisoners Dilemma

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Repeated Prisoners Dilemma

• Throw dice, stop if 6, new round if not 6
• on average 6 rounds
• allow only two strategies
• Tit For Tat
• always defect

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Repeated Prisoners Dilemma

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Risky Coordination

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Public Goods Experiments

• Six players
• 20 Euros each
• invest into common pot
• this sum is tripled
• distributed equally among all six players

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Public Goods Experiments

• 50 cents return per invested euro
• Nash invest nothing!
• no public goods

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Evolutionary Game Theory

• adaptation
• best reply
• imitate successful players
• etc
• if convergence, then to Nash
• not necessarily convergence! (Hofbauer)
• local interaction (Nowak)
• transmission mechanisms and population structure

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Evolutionary Games

• Population dynamical viewpoint
• John Maynard Smith
• Peter Hammerstein
• Reinhard Selten
• 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

• Population dynamical viewpoint
• John Maynard Smith
• Peter Hammerstein
• Reinhard Selten
• Josef Hofbauer
• anticipated by John Nash mass action approach

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