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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?