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Introduction to Nash Equilibrium

Introduction to Nash Equilibrium. Presenter: Guanrao Chen Nov. 20, 2002. Outline. Definition of Nash Equilibrium (NE) Games of Unique NE Games of Multiple NE Interpretations of NE Reference. Definition of Nash Equilibrium. Pure strategy NE A pure strategy NE is strict if

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Introduction to Nash Equilibrium

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  1. Introduction to Nash Equilibrium Presenter: Guanrao Chen Nov. 20, 2002

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

  3. Definition of Nash Equilibrium • Pure strategy NE • A pure strategy NE is strict if • ->Neither player can increase his expected payoff by unilaterally changing his strategy

  4. Games of Unique NEExample1 • Prisoner’s Dilemma • Unique NE: (D,D)

  5. Games of Unique NEExample2 • Unique NE: (U,L)

  6. Games of Unique NEExample2 • Uniqueness: • 1) Check each other strategy profile; • 2) • Proposition: If is a pure strategy NE of G then

  7. Games of Unique NEExample3 • Cournot game with linear demand and constant marginal cost • Unique NE: intersection of the two BR functions

  8. Games of Unique NEExample3 • Proof: is a NE iff. for all i. • ->Any NE has to lie on the best response function of both players. • Best response functions: • =>

  9. Games of Unique NEExample4 • Bertrand Competition: • 1) Positive price: • 2) Constant marginal cost: • 3) Demand curve: • 4) Assume • Unique NE:

  10. Games of Unique NEExample4 • Proof: 1) is a NE. • 2) Uniqueness: • Case 1: • Case 2: • Case 3: If deviate: Profit before: Profit after: Gain:

  11. Multiple Equilibria I - Simple Coordination Games • The problem: How to select from different equilibria • New-York Game • Two NEs: (E,E) and (C,C)

  12. 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:

  13. 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

  14. Multiple Equilibria II - Battle of the Sexes

  15. 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…

  16. 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’.

  17. 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 ? • 35/36chose the less desirable action. • Announcement strengthens beliefs that the other player will choose O.

  18. 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? • <12 choose the less desirable action. • Silence = weakness??

  19. Multiple Equilibria III - Coordination &Risk Dominance • Given the following game: • What action, A or B, will you choose?

  20. 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

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

  22. Interpretations of NE • 1) Play Prescription: • 2) Preplay communication: • 3) Rational Introspection: • 4) Focal Point: • 5) Learning: • 6) Evolution: • Remarks:

  23. 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.

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