Bayes-Nash equilibrium with Incomplete Information Econ 171.
Transcript of Bayes-Nash equilibrium with Incomplete Information Econ 171.
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Bayes-Nash equilibrium with Incomplete Information
Econ 171
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First some problems
• The Goblins.• Working backwards.
• What if there are 100 Goblins
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Todd and Steven Problem
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Problem 1 p 281
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How many proper subgames are there?
A) 0B) 1C) 2D) 4E) 6
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The Yule Ball
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How many strategies are possible for Hermoine?
A) 2B) 4C) 6D) 8
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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.
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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
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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.
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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
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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.
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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
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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.
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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.
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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.
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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
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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?
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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.
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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.
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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?