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 NE Example1 Prisoner ’ s Dilemma Unique NE: (D,D)

5. Games of Unique NE Example2 Unique NE: (U,L)

6. Games of Unique NE Example2 Uniqueness: 1) Check each other strategy profile; 2) Proposition: If is a pure strategy NE of G then

7. Games of Unique NE Example3 Cournot game with linear demand and constant marginal cost Unique NE: intersection of the two BR functions

8. Games of Unique NE Example3 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 NE Example4 Bertrand Competition: 1) Positive price: 2) Constant marginal cost: 3) Demand curve: 4) Assume Unique NE:

10. Games of Unique NE Example4 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?  Players seem to believe that player 1 has an advantage by moving first, and they are more likely to ’ cave in ’.  17/25 choose the less desirable action(O).

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 ?  35/36 chose 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.Shapley "Non-Cooperative Games", 1951, Annals of Mathematics. "Two-Person Cooperative Games", 1953, Econometrica.