# Bayes-Nash equilibrium with Incomplete Information

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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?
1 2 4 6

The Yule Ball

How many strategies are possible for Hermoine?
2 4 6 D) 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 Ron Set 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 Ron So she has 8 possible strategies.

Dating Dilemma Victor Asks Hermione Y,Y,Y Y,Y,N Y,N,Y Y,N,N N,Y,Y
8,3,6 1,8*,8* 3,2,4 Don’t 7*,6*,5* 2,5,3 2,5*,3 Ron Victor Doesn’t Ask Hermione 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* Don’t 5,4,1 Ron

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 which Hermione always says “yes” to Ron if she hasn’t accepted a date from Victor.

Victor Asks Hermione’s strategy Yes to Victor No to Victor Ask 8,3,6 1,8*,8* Don’t Ask 7*,6*,5* 2,5,3 Ron’s Strategy Victor Doesn’t Ask Hermione’s strategy Yes to Victor No to Victor Ask 4,7*,7* 4*,7*,7* Don’t Ask 5,4,1* Ron’s Strategy

What are the strategies used in subgame perfect equilibria?

She loves me, she loves me not?
Nature She loves him She scorns him Bob Bob Go to A Go to B Go to A Go to B Alice Alice Alice Alice Go to B Go to B Go to A Go to A Go to B Go to A Go to A Go to B 1 3 2 2 3 3 2 1 2 1 3

Whats New here? Incomplete information: Bob doesn’t know Alice’s payoffs In previous examples we had “Imperfect Information”. Players Knew 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
Nature She loves him She scorns him Bob Bob Go to A Go to B Go to A Go to B Alice Alice Alice Alice Go to B Go to B Go to A Go to A Go to B Go to A Go to A Go to B 1 3 2 2 3 3 2 1 2 1 3

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 is 3p+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?
Yes, Alice goes to B if she loves Bob and A if she scorns him and Bob goes to B. Yes, Alice goes to A if she loves Bob and B if she scorns him and Bob goes to B. Yes there is one, where Alice always goes to A. 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?