Bayes-Nash equilibrium with Incomplete Information

Econ 171

First some problems

• The Goblins.• Working backwards.

• What if there are 100 Goblins

Todd and Steven Problem

Problem 1 p 281

How many proper subgames are there?

A) 0B) 1C) 2D) 4E) 6

The Yule Ball

How many strategies are possible for Hermoine?

A) 2B) 4C) 6D) 8

What are the strategies?

Victor and Ron each have only one information set and only two possible actions, ask or don’t ask. Hermione has 3 information sets at which she must choose a move. A strategy specifies whether she will say yes or no in each of them. Set 1: Victor has asked: Say yes or no to RonSet 2: Victor has asked, Hermione said no, Ron asked: Say yes or no to Ron Set 3: Victor didn’t ask and Ron asked: Say yes or No to RonSo she has 8 possible strategies.

Dating Dilemma

Ron

Hermione

Victor Asks

Y,Y,Y Y,Y,N Y,N,Y Y,N,N N,Y,Y N,Y,N N,N,Y N,N,N

Ask 8,3,6 8,3,6 8,3,6 8,3,6 1,8*,8* 1,8*,8* 3,2,4 3,2,4

Don’t 7*,6*,5* 7*,6*,5* 7*,6*,5* 7*,6*,5* 2,5,3 2,5,3 2,5*,3 2,5*,3

Hermione

Victor Doesn’t Ask

Y,Y,Y Y,Y,N Y,N,Y Y,N,N N,Y,Y N,Y,N N,N,Y N,N,N

Ask 4,7*,7* 6,1,2 4,7*,7* 6,1,2 *4,7*,7* 6,1,2 *4,7*,7* 6,1,2

Don’t 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1 5,4,1Ron

Simplifying the Game

If Hermione ever reaches either of the two nodes where Ron gets to ask her, she would say Yes. So a subgame perfect equilibrium must be a Nash equilbrium for the simpler game in whichHermione always says “yes” to Ron if she hasn’t accepted a date from Victor.

Yes to Victor No to Victor

Ask 8,3,6 1,8*,8*

Don’t Ask 7*,6*,5* 2,5,3

Victor Asks

Hermione’s strategy

Ron’s Strategy

Yes to Victor No to Victor

Ask 4,7*,7* 4*,7*,7*

Don’t Ask 5,4,1* 5,4,1*

Hermione’s strategy

Victor Doesn’t Ask

Ron’s Strategy

What are the strategies used in subgame perfect equilibria?

Equilibrium 1)– Victor asks– Ron doesn’t ask– Hermoine says yes to V if V asks, Yes to Ron if she says No to V

and Ron asks, Yes to Ron if Ron asks and Victor doesn’t ask.Equilibrium 2)– Victor doesn’t ask– Ron Asks– Hermoine would say No to V if Victor asked, Yes to Ron

and Victor asked and she said no to V, Yes to Ron if Ron asked and Victor didn’t.

She loves me, she loves me not?

Go to A Go to B

Go to A

AliceAlice

Go to B Go to A

Go to B

23

00

11

32

She loves him

Nature

She scorns him

Go to A

Go to A Go to AGo to BGo to B

Go to B

21

02

13

30

Bob

Alice

Bob

Alice

Whats New here?

Incomplete information: Bob doesn’t know Alice’s payoffs

In previous examples we had “Imperfect Information”. PlayersKnew each others payoffs, but didn’t know the other’s move.

Bayes-Nash Equilibrium

• Alice could be one of two types. “loves Bob”“scorns Bob• Whichever type she is, she will choose a best

response. • Bob thinks the probability that she is a loves

Bob type is p. • He maximized his expected payoff, assuming

that Alice will do a best response to his action.

Expected payoffs to Bob

• If he goes to movie A, he knows that Alice will go to A if she loves him, B if she scorns him.

His expected payoff from A is 2p+0(1-p)=2p.• If he goes to movie B, he knows that Alice will

go to B if she loves him, A if she scorns him. His expected from B is then

3p+1(1-p)=2p+1.• For any p, his best choice is movie B.

Does she or doesn’t she?Simultaneous Play

Go to A Go to B

Go to A

AliceAlice

Go to B Go to A

Go to B

23

00

11

32

She loves him

Nature

She scorns him

Go to A

Go to A Go to AGo to BGo to B

Go to B

21

02

13

30

Bob

Alice

Bob

Alice

Bayes’ Nash equilibrium

• Is there a Bayes’ Nash equilibrium where Bob goes to B and Alice goes where Alice goes to B if she loves him, and to A if she scorns him?– This is a best response for both Alice types.– What about Bob?

Bob’s Calculations

If Bob thinks the probability that Alice loves him is p and Alice will go to B if she loves him and A if she scorns him:– His expected payoff from going to B is3p+1(1-p)=1+2p.– His expected payoff from going to A is 2(1-p)+0p=2-2p.Going to B is Bob’s best response to the strategies of the Alice types if 1+2p>=2-2p. Equivalently p>=1/4.

Is there a Bayes-Nash equilibrium in pure strategies if p<1/4?

A) Yes, Alice goes to B if she loves Bob and A if she scorns him and Bob goes to B.

B) Yes, Alice goes to A if she loves Bob and B if she scorns him and Bob goes to B.

C) Yes there is one, where Alice always goes to A.

D) No there is no Bayes-Nash equilibrium in pure strategies.

What about a mixed strategy equilibrium?

• Can we find a mixed strategy for Bob that makes one or both types of Alice willing to do a mixed strategy?

• Consider the Alice type who scorns Bob. If Bob goes to movie A with probability q, When will Alice be indifferent between going to the two movies?