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? 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? 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?