Introduction to Nash Equilibrium

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Introduction to Nash Equilibrium

• Presenter Guanrao Chen
• Nov. 20, 2002

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Outline

• Definition of Nash Equilibrium (NE)
• Games of Unique NE
• Games of Multiple NE
• Interpretations of NE
• Reference

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Definition of Nash Equilibrium

• Pure strategy NE
• A pure strategy NE is strict if
• -gtNeither player can increase his expected payoff
  by unilaterally changing his strategy

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Games of Unique NEExample1

• Prisoners Dilemma
• Unique NE (D,D)

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Games of Unique NEExample2

• Unique NE (U,L)

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Games of Unique NEExample2

• Uniqueness
• 1) Check each other strategy profile
• 2)
• Proposition If is a pure strategy NE
  of G then

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Games of Unique NEExample3

• Cournot game with linear demand and constant
  marginal cost
• Unique NE intersection of the two BR functions

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Games of Unique NEExample3

• Proof is a NE iff. for all i.
• -gtAny NE has to lie on the best response function
  of both players.
• Best response functions
• gt

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Games of Unique NEExample4

• Bertrand Competition
• 1) Positive price
• 2) Constant marginal cost
• 3) Demand curve
• 4) Assume
• Unique NE

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Games of Unique NEExample4

• Proof 1) is a NE.
• 2) Uniqueness
• Case 1
• Case 2
• Case 3
• If deviate
• Profit before
• Profit after
• Gain

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Multiple Equilibria I - Simple Coordination Games

• The problem How to select from different
  equilibria
• New-York Game
• Two NEs (E,E) and (C,C)

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Multiple Equilibria I - Simple Coordination Games

• Voting Game 3 players, 3 alternatives, if 1-1-1,
  alternative A is retained
• Preferences
• Has several NEs (A,A,A),(B,B,B),(C,C,C),(A,B,A),(
  A,C,C)..
• Informal proof

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Multiple Equilibria Focal Point

• A focal point is a NE which stands out from the
  set of NEs.
• Knowledge information which is not part of the
  formal description of game.
• Example Drive on the right

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Multiple Equilibria II - Battle of the Sexes

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Multiple Equilibria II - Battle of the Sexes

• Class Experiment
• You are playing the battle of the sexes. You are
  player2. Player 1 will make his choice first but
  you will not know what that move was until you
  make your own. What will you play?

• 18/25 men vs. 6 out of 11 women

• Men are more aggressive creatures

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Multiple Equilibria II - Battle of the Sexes

• Class Experiment
• You are player 1. Player 2 makes the first move
  and chooses an action. You cannot observe her
  action until you have chosen your own action.
• Which action will you choose?

• 17/25 choose the less desirable action(O).

• Players seem to believe that player 1 has an
  advantage by moving first, and they are more
  likely to cave in.

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Multiple Equilibria II - Battle of the Sexes

• Class Experiment
• You are player 1. Before the game, your opponent
  (player 2) made an announcement. Her announcement
  was I will play O. You could not make a
  counter-announcement.
• What will you play ?

• 35/36 chose the less desirable action.

• Announcement strengthens beliefs that the other
  player will choose O.

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Multiple Equilibria II - Battle of the Sexes

• Class Experiment
• You are player 1. Before the game, player 2 (the
  wife) had an opportunity to make a short
  announcement. Player 2 choose to remain silent.
• What will you play?

• lt12 choose the less desirable action.

• Silence weakness??

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Multiple Equilibria III - Coordination Risk
Dominance

• Given the following game
• What action, A or B, will you choose?

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Multiple Equilibria III - Coordination Risk
Dominance

• Observation
• 1) Two NEs (A,A) and (B,B). (A,A) seems better
  than (B,B).
• 2) BUT (B,B) is more frequently selected.
• Risk-dominance u(A)-3 while u(B)7.5

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Interpretations of NE

• In NE, players have precise beliefs about the
  play of other players.
• Where do these beliefs come from?

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Interpretations of NE

• 1) Play Prescription
• 2) Preplay communication
• 3) Rational Introspection
• 4) Focal Point
• 5) Learning
• 6) Evolution
• Remarks

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References

• "Equilibrium points in N-Person Games", 1950,
  Proceedings of NAS.
• "The Bargaining Problem", 1950, Econometrica.
• "A Simple Three-Person Poker Game", with L.S.
  Shapley, 1950, Annals of Mathematical Statistics.
  
• "Non-Cooperative Games", 1951, Annals of
  Mathematics.
• "Two-Person Cooperative Games", 1953,
  Econometrica.