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

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The Yule Ball

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How many strategies are possible for Hermoine? A)2 B)4 C)6 D) 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 dont 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 didnt ask and Ron asked: Say yes or No to Ron So she has 8 possible strategies.

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Dating Dilemma Ron Hermione Victor Asks Y,Y,YY,Y,NY,N,YY,N,NN,Y,YN,Y,NN,N,YN,N,N Ask8,3,6 1,8*,8* 3,2,4 Dont 7*,6*,5* 2,5,3 2,5*,3 Hermione Victor Doesnt Ask Y,Y,YY,Y,NY,N,YY,N,NN,Y,YN,Y,NN,N,YN,N,N Ask 4,7*,7*6,1,24,7*,7*6,1,2*4,7*,7*6,1,2*4,7*,7* 6,1,2 Dont5,4,1 Ron

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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 which Hermione always says yes to Ron if she hasnt accepted a date from Victor.

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Yes to VictorNo to Victor Ask8,3,61,8*,8* Dont Ask7*,6*,5*2,5,3 Victor Asks Hermiones strategy Rons Strategy Yes to VictorNo to Victor Ask4,7*,7*4*,7*,7* Dont Ask5,4,1* Hermiones strategy Victor Doesnt Ask Rons Strategy

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What are the strategies used in subgame perfect equilibria? Equilibrium 1) – Victor asks – Ron doesnt 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 doesnt ask. Equilibrium 2) – Victor doesnt 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 didnt.

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

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Whats New here? Incomplete information: Bob doesnt know Alices payoffs In previous examples we had Imperfect Information. Players Knew each others payoffs, but didnt know the others 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 doesnt she? Simultaneous Play Go to A Go to B Go to A Alice Go to B Go to A Go to B She loves him Nature She scorns him Go to A Go to B 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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Bobs 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 Bobs 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?

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