10How many strategies are possible for Hermoine? 246D) 8
11What 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.
13Simplifying the GameIf 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.
14Victor AsksHermione’s strategyYes to VictorNo to VictorAsk8,3,61,8*,8*Don’t Ask7*,6*,5*2,5,3Ron’s StrategyVictor Doesn’t AskHermione’s strategyYes to VictorNo to VictorAsk4,7*,7*4*,7*,7*Don’t Ask5,4,1*Ron’s Strategy
15What are the strategies used in subgame perfect equilibria? Equilibrium 1)Victor asksRon doesn’t askHermoine 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 askRon AsksHermoine 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.
16She loves me, she loves me not? NatureShe loves himShe scorns himBobBobGo to AGo to BGo to AGo to BAliceAliceAliceAliceGo to BGo to BGo to AGo to AGo to BGo to AGo to AGo to B13223321213
17Whats New here?Incomplete information: Bob doesn’t know Alice’s payoffsIn previous examples we had “Imperfect Information”. PlayersKnew each others payoffs, but didn’t know the other’s move.
18Bayes-Nash Equilibrium Alice could be one of two types. “loves Bob”“scorns BobWhichever 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.
19Expected 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 is2p+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 then3p+1(1-p)=2p+1.For any p, his best choice is movie B.
20Does she or doesn’t she? Simultaneous Play NatureShe loves himShe scorns himBobBobGo to AGo to BGo to AGo to BAliceAliceAliceAliceGo to BGo to BGo to AGo to AGo to BGo to AGo to AGo to B13223321213
21Bayes’ 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?
22Bob’s CalculationsIf 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 is2(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.
23Is 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.
24What 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?