How many strategies are possible for Hermoine? A)2 B)4 C)6 D) 8
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.
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
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.
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
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.
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 2323 0000 1111 3232 She loves him Nature She scorns him Go to A Go to B 2121 0202 1313 3030 Bob Alice Bob Alice
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.
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 doesnt she? Simultaneous Play Go to A Go to B Go to A Alice Go to B Go to A Go to B 2323 0000 1111 3232 She loves him Nature She scorns him Go to A Go to B 2121 0202 1313 3030 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?
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.
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?