Signaling Econ 171 Breakfast Beer or quiche A Fable The original Fabulists are game theorists David Kreps and In Koo Cho Breakfast and the bully A new kid moves to town Other kids don t know if he is tough or weak Class bully likes to beat up weak kids but doesn t like to fight tough kids Bully gets to see what new kid eats for breakfast New kid can choose either beer or quiche Preferences Tough kids get utility of 1 from beer and 0 from quiche Weak kids get utility of 1 from quiche and 0 from beer Bully gets payoff of 1 from fighting a weak kid 1 from fighting a tough kid and 0 from not fighting New kid s total utility is his utility from breakfast minus w if the bully fights him and he is weak and utility from breakfast plus s if he is strong Nature Tough Weak New Kid New Kid Beer Quiche Fight 1 s 1 Beer Quiche Bully Don t 1 0 Fight Don t Bully B Fight s 1 w 0 0 Fight Don t 1 0 0 1 w 1 Don t 1 0 How many possible strategies are there for the bully A 2 B 4 C 6 D 8 What are the possible strategies for bully Fight if quiche Fight if beer Fight if quiche Don t if beer Fight if beer Don t if quiche Don t if beer Don t if quiche What are possible strategies for New Kid Beer if tough Beer if weak Beer if tough Quiche if weak Quiche if tough Beer if weak Quiche if tough Quiche if weak Separating equilibrium Suppose that Bully uses the strategy Fight if the New Kid has Quiche and Don t if the new kid has Beer And the new kid has Quiche if he is weak and Beer if he is strong When is this an equilibrium Best responses If bully will fight quiche eaters and not beer drinkers weak kid will get payoff of 0 if he has beer and 1w if he has quiche So weak kid will have quiche if w 1 Tough kid will get payoff of 1 if he has beer and s if he has quiche So tough kid will have quiche if s 1 Suppose w 1 and s 1 We see that if Bully fights quiche eaters and not beer drinkers the best responses are for the new kid to have quiche if he is weak and beer if he is strong If this is the new kid s strategy it is a best response for Bully to fight quiche eaters and not beer drinkers So the outcome where Bully uses strategy Fight if quiche Don t if beer and where New Kid uses strategy Quiche if weak Beer if tough is a Nash equilibrium If w 1 Then if Bully uses strategy Fight if quiche Don t if beer what will New Kid have for breakfast if he is weak Pooling equilibrium If w 1 is there an equilibrium in which the New Kid chooses to have beer for breakfast whether or not he is weak If everybody has beer for breakfast what will the Bully do Expected payoff from Fight if beer Don t if quiche depends on his belief about the probability that New Kid is tough or weak Payoff to Bully Let p be probability that new kid is tough If new kid always drinks beer and bully doesn t fight beer drinking new kids his expected payoff is 0 If Bully chooses strategy Fight if Beer Don t if Quiche his expected payoff will be 1 2p 1 2 1 p 1 2p Now 1 2p 0 if p 1 2 So if p 1 2 Don t fight if Beer fight if Quiche is a best response for Bully Pooling equilibrium If p 1 2 there is a pooling equilibrium in which the New Kid has beer even if he is weak and prefers quiche because that way he can conceal the fact that he is weak from the Bully If p 1 2 a best response for Bully is to fight the New Kid if he has quiche and not fight him if he has beer What if p 1 2 and w 1 There won t be a pure strategy equilibrium There will be a mixed strategy equilibrium in which a weak New Kid plays a mixed strategy that makes the Bully willing to use a mixed strategy when encountering a beer drinker What if s 1 Then tough New Kid would rather fight get in a fight with the Bully than have his favorite breakfast It would no longer be Nash equilibrium for Bully to fight quiche eaters and not beer drinkers because best response for tough New Kid would be to eat quiche Alice and Bob Revisited She loves me she loves me not Alice knowsNature where Bob goes She scorns him She loves him Bob Go to A Bob Go to B Go to A Alice Alice Alice Go to A 2 3 Go to B 0 0 Go to A 1 1 Go to B Go to B 3 2 Go to A 2 1 Alice Go to B Go to A 0 2 1 3 Go to B 3 0 What are strategies For Bob Go to A Go to B Either type of Alice could select any of these A if A and A if B A if A and B if B B if A and A if B B if B and A if B Alice could select any of these 4 strategies if she loves him and any of these 4 strategies if she scorns him This gives 16 possibilities Does she or doesn t she Simultaneous play Nature She scorns him She loves him Bob Go to A Bob Go to B Go to A Alice Alice Alice Go to A 2 3 Go to B 0 0 Go to A 1 1 Go to B Go to B 3 2 Go to A 2 1 Alice Go to B Go to A 0 2 1 3 Go to B 3 0 What are pure strategies Bob Go to A Go to B Alice doesn t know what Bob did so she can t make her action depend on his choice She can Go to A Go to B She can do either of these if she loves him and either of them if she scorns him Thus there are 4 possibilities A if love A if scorn A if love B if scorn B if love A if scorn B if love B if scorn Wyatt Earp and the Gun Slinger A Bayesian gunslinger game The gunfight game when the stranger is a a gunslinger or b a cowpoke What are the strategies Earp Draw Wait Stranger Draw if Gunslinger Draw if Cowpoke Draw if Gunslinger Wait if Cowpoke Wait if Gunslinger Draw if Cowpoke Wait if Gunslinger Wait if Cowpoke One Bayes Nash equilibrium Suppose that Earp waits and the other guy draws if …