Econ 171

Breakfast: Beer or quiche?
A Fable *
*The original Fabulists are game theorists, David Kreps and In-Koo Cho

• 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.

• 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 and bully fights him.

Nature
Tough Weak
Fight
1+s
-1
New Kid New Kid
Beer Quiche Beer
Don’t
Bully
Fight
Don’t
Quiche
1
0
Fight
Don’t
-w
1 s
-1
0
0
0
0
Fight
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

• Is there an equilibrium where 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.
• For what values of w could this be an equilibrium?

• 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

• 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.

Then if Bully uses strategy “Fight if quiche,
Don’t if beer”, what will New Kid have for breakfast if he is weak?

• 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.

• 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.

• 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.

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.

• 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.

A Bayesian gunslinger game

The gunfight game when the stranger is (a) a gunslinger or (b) a cowpoke

• 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

• 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

• Lets see if there is an equilibrium where everybody draws.

She loves me, she loves me not?
(
Bob moves before Alice)
Nature
Bob
She loves him She scorns him
Bob
Go to A Go to B
Go to A
Alice
Go to B Go to A
Alice
Go to B
Go to A
Go to B
Go to A
Alice
Go to B
Go to A
Alice
Go to B
2
3
0
0
1
1
3
2
2
1
0
2
1
3
3
0

• For Bob
– Go to A
– Go to B
• Alice has four information sets.
– I Love him and he’s at A
– I Love him and he’s at B
– I Scorn him and he’s at A
– I Scorn him and he’s at B
• In each information set, she can go to either A or
B. This gives her 2x2x2x2=16 possible strategies.

A weakly dominant strategy for Alice
• Go to A if you love he goes to A. Go to B if you love him and he goes B. Go to B if you scorn him and he goes to A. Go to A if you scorn him and he goes to B. (We write this as A/B/B/A)
• This is weakly dominant but not strictly dominant. Explain.
• Let’s look for a subgame perfect Nash equilibrium where Alice goes A/B/B/A

• Suppose Alice goes where Bob is if she loves him and goes where he is not if she scorns him.
(A/B/B/A)
• Payoff to Bob from A is 2p. Payoff from B is
3p+1(1-p)=2p+1. Since 2p+1>2p, for all p>=0, B is his best response to (A/B/B/A).
• Also A/B/B/A is a best response for Alice to Bob’s
B.
• So we have a Bayes-Nash equilibrium.

Does she or doesn’t she?
Simultaneous Play
Nature
Bob
She loves him She scorns him
Bob
Go to A Go to B
Go to A
Alice
Go to B Go to A
Alice
Go to B
Go to A
Go to B
Go to A
Alice
Go to B
Go to A
Alice
Go to B
2
3
0
0
1
1
3
2
2
1
0
2
1
3
3
0

• Alice doesn’t know what Bob did, so she can’t make her action depend on his choice. She can go to either A or B.
• She does know whether she loves him or scorns him when she chooses.
• She has 4 possible strategies
– A if love, A if scorn
– A if love, B if scorn
– B if love, A if scorn
– B if love, B if scorn

• Is there a Bayes’ Nash equilibrium where Bob goes to B and Alice goes to B if she loves Bob, and to A if she scorns him?
– This is a best response for both Alice types.
– What about Bob?

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 Bob’s 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.

• We showed there is no pure strategy Bayes-Nash equilibrium where Bob goes to B.
• What if Bob goes to A?
• The only possible Nash equilibrium would have
Alice go to A if she loves him and B if she scorns him.
• Then payoff to Bob from A would be 2p+0 and payoff from B would be 3p+1(1-p)=2p+1>2p.
• So if p<1/4, there can’t be a pure strategy Bayes-
Nash equilibrium where Bob goes to A.

Mixed strategy equilbrium:
Bob the stalker
• If Bob thinks it likely that Alice scorns him, then if he uses a pure strategy, he knows Alice would always avoid him.
• If he uses a mixed strategy, he would catch her sometimes.
• Let’s look for a mixed strategy for Bob such that Alice, if she scorns Bob would be indifferent between Movies A and B.

What about a mixed strategy equilibrium?
• If p<1/4, can we find a mixed strategy for Bob such that Alice is indifferent
• What if Bob knows Alice scorns him?
• 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?

Alice
Bob
A B
A 1,2 3,1
B 2,0 0,3

• If Bob goes to Movie A with probability q and
Alice Scorns Bob:
– Alice’s payoff from A is 1q+3(1-q) =3-2q
– Alice’s payoff from B is 2q+0(1-q)=2q
– Alice will be indifferent if 3-2q=2q, which implies q=3/4.

When will Bob do a mixed strategy?
• Note that if Bob goes to A with probability ¾, and if Alice loves him, her best response is to go to Movie A.
• If there is an equilibrium where Bob uses a mixed strategy, he must be indifferent between going to A and going to B.
• Can we find a mixed strategy for Alice to use if she scorns him so that Bob will be indifferent between A and B?

• Let r be the probability that Alice goes to Movie A if she scorns Bob and suppose that Alice always goes to A if she loves Bob.
• Expected payoffs for Bob are
– If he goes to A, 2p+(1-p)(2r+0(1-r))=2p-2pr+2r
– If he goes to B, 1p+(1-p)(1r+3(1-r))=2p+3+2pr-2r
– He is indifferent between A and B if these are equal.
• This implies r=(3-4p)/4-4p.
• Now r is between 0 and 1 if and only if p<=3/4

• We previously found that if p>=1/4, there is a Bayes-
Nash equilibrium in which Bob goes to B and Alice goes to B if she loves him and A if she scorns him.
• Now we found that whenever p<=3/4, there is a mixed strategy Bayes-Nash equilibrium in which Bob goes to movie A with probability ¾, Alice goes to Movie A if she loves Bob and she goes to movie A with probability r=(3-4p)/(4-4p) if she scorns him.
• So over the range of p between ¼ and ¾, there are two distinct Bayes-Nash equilibria.

• A Piece of
(If you are prepared…)