Introduction to Nash Equilibrium

Presenter: Guanrao ChenNov. 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

),(),( ***iiiiii ssussu

),(),( ***iiiiii ssussu

Games of Unique NEExample1 Prisoner’s Dilemma

Unique NE: (D,D)

Games of Unique NEExample2

Unique NE: (U,L)

Games of Unique NEExample2 Uniqueness: 1) Check each other strategy

profile; 2) Proposition: If is a pure

strategy NE of G then

}{&}{ 21 LSUS

Ss**s

Games of Unique NEExample3 Cournot game with linear demand an

d constant marginal cost Unique NE: intersection of the two BR

functions

Games of Unique NEExample3 Proof: is a NE iff. for all i. ->Any NE has to lie on the best respon

se function of both players. Best response functions:

=>

),( *2

*1 qq )(*

iii qBRq

Games of Unique NEExample4 Bertrand Competition: 1) Positive price: 2) Constant marginal cost: 3) Demand curve:

4) Assume Unique NE:

Games of Unique NEExample4 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 e

quilibria 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) seem

s better than (B,B). 2) BUT (B,B) is more frequently select

ed. 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.

"Non-Cooperative Games", 1951, Annals of Mathematics.

"Two-Person Cooperative Games", 1953, Econometrica.