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 ->Neither player can increase his expected payoff by unilaterally changing his strategy

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

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

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

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

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: =>

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

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:

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

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:

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

Multiple Equilibria II - Battle of the Sexes

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 …

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).

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.

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??

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

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

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

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

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.