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 Beer Bully Quiche Fight Don’t Don’t Fight B Bully 1 1+s -1 Fight -w 1 Don’t Don’t Fight 1 s -1 1-w 1

How many possible strategies are there for the bully? 2 4 6 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 1-w 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 knows where Bob goes 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

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 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

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 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 he is a gunslinger, waits if he is a cowpoke. Stranger in either case is doing a best response. If stranger follows this rule, is waiting best for Earp? Earp’s Payoff from waiting is 3/4x1+1/4x8=2.75 Earp’s Payoff from drawing, given these strategies for the other guys is (¾)2+(1/4) 4=2.5 So this is a Bayes Nash equilibrium

There is another equilibrium Lets see if there is an equilibrium where everybody draws.

What will exam be like? A Piece of If you are prepared.