Gusher or Dry Hole?

Classroom Exercise

• Two companies bid for oil field.• Each bidder explores half the oilfield and

determines what his half is worth. (Either $3 million or 0)

• Neither will know what other half is worth.• Total value is sum of the values of the two halves.• Oil field is sold in sealed bid second price auction.

(If both bid the same amount, a coin is flipped to decide who gets it.)

What is the field worth?

• If you see 0 on your half, what is the expected value of the whole oil field, given your information?

• If you see 3 on your half, what is the expected value of the whole oil field, given your information.

What if you were the only bidder?

• What would be the expected value of the oil field if you saw 0 on your half?

• What if you saw 1 on your half.

If you just wanted to maximize expected wealth, what would you do?

How should you bid if there are two bidders?

• What if you both bid expected value, given what you have seen?

• Then you would bid 1.5 million if you see 0 and 4.5 million if you see 3.

• What would be your expected winnings?

Expected Payoff if both bid expected values and you see 0

• If you saw 0, and bid 1.5, then – with probability ½ other guy saw 3 and bids 4.5.– With probability ½ other guy saw 0 and bid 1.5.

• In this case, if the other guy sees 3, you don’t get the oilfield and your profit is 0.

• If the other guy sees 0, then with probability ½ you get the oilfield for 1.5 million, but it is worthless.

• Expected loss if you is $750,000.

What have we learned so far.

• Suppose that both players bid more if their own side is worth 3 million than if it is worth 0.

• Then if you see 0, you will win the auction only if the other guys saw 0 too.

• But this means that the oilfield will be worthless if you win it.

• So only possibile equilibrium is bid 0 when you see 0.

A puzzle for the curious

• Suppose bidders bid 0 if they see 0.• Is there a symmetric mixed strategy equilibrium

in which bidders who see 3 bid the same amount x?

• If bids must be integer number of millions, yesWhat if both bid 0 if they see 0 and 4 million if they see 3 million? • If you can bid any number, the only equilibrium is

in mixed strategies.

Results Table Value Seen Bid Profit

Bidder A 0 2 0Bidder B 0 3 -2Total 0 x -2

Results Table Value Seen Bid Profit

Bidder A 3 6 0Bidder B 0 3 0Total 3 x 0

Results Table Value Seen Bid Profit

Bidder A 3 3.01 3.01Bidder B 3 2.99 0Total 6 x 0

Results Table Value Seen Bid Profit

Bidder A 3 3 0Bidder B 0 3 0Total 3 x 0

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 and bully fights him.

Nature

New Kid New Kid

Tough Weak

Beer BeerQuiche

QuicheFight

Fight

Fight

Fight

Don’t

Don’t

Don’t

Don’t

B

Bully

Bully1+s-1

10

s-1 0

0

-w1

00

1-w1

10

How many possible strategies are there for the bully?A) 2B) 4C) 6D) 8

What are the possible strategies for bully?

Fight if quiche, Fight if beerFight if quiche, Don’t if beerFight if beer, Don’t if quicheDon’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?

• 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?

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.

Signaling Game Problems

Problem 1, p 348Quality Probability Value to Seller Value to Buyer

Good Car q 10,000 12,000

Lemon 1-q 6,000 7,000

their Expected Value of a random car is12000q+7000(1-q)=7,000+5,000q

If Buyers believe that the fraction of good cars on market is q,

• In this case, we can expect all used cars to sell for about PU=7,000+5,000q. • If q>3/5, then PU=7000+5000q> 10,000 and so owners of lemons and of good cars and of will be willing to sell at price PU. • Thus the belief that the fraction q of all used cars are goodIs confirmed. We have a pooling equilibrium.

There is also a separating equilibriumQuality Probability Value to Seller Value to Buyer

Good Car q 10,000 12,000

Lemon 1-q 6,000 7,000

Suppose that buyers all believe that the only used cars on the market. Then they all believe that a used car is only worth $7000. The price will not be higher than $7000.

At this price, nobody would sell his good car, since good used cars are worth $10,000 to their current owners. Buyer’s beliefs are confirmed by experience. This is a separating equilibrium. Good used car owners act differently from lemon owners.

Problem 3, page 348• Suppose that buyers believe that product with no warranty is

low quality and that with warranty is high quality.• High quality items work with probability H and low quality

items work with probability L. Consumer values a working item at V.

• Buyers are willing to pay up to LV an item that works with probability L.

• Buyers are willing to pay up to V for any item with a money back guarantee. (If it works, their net gain is V-P

and if it fails they get their money back so their net gain is 0. Therefore they will buy if P<V.)

Equilibrium• If the item with warranty sells for just under V and that with

no warranty sells for just under LV, buyers will take either one.

• Given these consumer beliefs, V is the highest price that sellers can get for high quality with warranty and LV is the highest price for the low quality without warranty.

• Seller’s profits from high quality sales with guarantee are hV-c and profits from low quality without guaranty are LV-c.

• If seller put a guarantee on low quality items and sold them for V, his profit would be LV-c, which is no better than he does without a guarantee on these.

Equilibrium

• If buyers believe that only the good items have guarantees, the Nash equilibrium outcome confirms this belief.

• If fraction of items sold that are of high quality is r, then retailer’s average profit per unit sold

Is rHV+(1-r)LV.• Retailer can not do better with a pooling

equilbrium in which he guaranteed nothing, or in one in which he guaranteed everything.

Can you show this?

Problem 5, p 350• Students are of 3 types, High, medium, and low. Cost of

getting a college degree to a student is 2 if high, 4 if medium, and 6 if low.

• 1/6 of students are of high type, ½ of medium type, 1/3 are of low type.

• Salaries for managers are 15, and 10 for clerks.• An employer has one clerk’s job to fill and one manager’s

job to fill. Employer’s profits (net of wages) are 7 from hiring anyone as a clerk,

4 from hiring a low type as a manager, 6 from hiring a medium type as manager, 14 from hiring a high type as manager.

Equilibrium where high and medium types go to college, low does not.

• If high and medium types go to college, what is the expected profit from hiring a college grad as a manager?

• Find probability p that someone is of high type given college:

• P(H|C)=P(H and C)/P( C)=(1/6) / (1/6+1/2)=1/4• Expected profit is 1/4x14+3/4x6=8.• If you hire a college grad as clerk, expected profit

is 7. So better off to hire her as manager.

Equilibrium for workers.

• High types get paid 15 as manager have college costs of 2. So net wage is 13. That’s better than the 10 that nondegree people get as clerks.

• Medium type get paid 15 as manager have college costs 4, net wage of 11, so they prefer college and managing to no college and clerk.

• Low types would get 15 as manager with college costs of 6. Net pay of 15-6=9 is less than they would get with no college and being a clerk.

Professor Drywall’s Lectures

Another fable

• Imagine that the labor force consists of two types of workers: Able and Middling with equal proportions of each.

• Employers are not able to tell which type they are when they hire them.

• A worker is worth $1500 a month to his boss if he is Able and $1000 a month if he is Middling.

• Average worker is worth • $ ½ 1500 + ½ 1000=$1250 per month.

Competitive labor market

• The labor market is competitive and since employers can’t tell the Able from the Middling, all laborers are paid a wage of $1250 per month.

• One employer believes that Drywall’s lectures are useful and requires its workers attend 10 monthly lectures by Professor Drywall and payswages of $100 per month above the average wage.– Middling workers find Drywall’s lectures excruciatingly

dull. Each lecture is as bad as losing $20.– Able workers find them only a little dull. To them, each

lecture is as bad as losing $5.• Which laborers stay with the firm?• What happens to the average productivity of

laborers?

Other firms see what happened

• Professor Drywall shows the results of his lectures for productivity at the first firm.

• Firms decide to pay wages of about $1500 for people who have taken Drywall’s course.

• Now who will take Drywall’s course? • What will be the average productivity of

workers who take his course? Do we have an equilibrium now?

Professor Drywall responds

• Professor Drywall is not discouraged.• He claims that the problem is that people have

not heard enough lectures to learn his material. • Firms believe him and Drywall now makes his

course last for 30 hours a month. • Firms pay almost $1500 wages for those who

take his course and $1000 for those who do not.

Separating Equilibrium

• Able workers will prefer attending lectures and getting a wage of $1500, since to them the cost of attending the lectures is $5x30=$150 per month.

• Middling workers will prefer not attending lectures since they can get $1000 if they don’t attend. Their cost of attending the lectures would be $20x30=$600, leaving them with a net of $900.

So there we are.

Problem 5, page 350

George Bush and Saddam Hussein

The story

• Bush believes that probability Hussein has WMDs is w<3/5.

• When is there a perfect Bayes-Nash equilibrium with strategies?

• Hussein: If WMD, Don’t allow, if no WMD allow with probability h.

• Bush: If allow and WMD, Invade. If allow and no WMD, Don’t invade, If don’t allow, invade with probability b.

Payoffs for Hussein if he has no WMDs

Payoff from not allow is 2b+8(1-b)=8-6bPayoff from allow is 4, since if he allows Bush will not invade.Hussein is indifferent if 4=8-6b or equivalentlyb=2/3.So he would be willing to use a mixed strategy if he thought that Bush would invade with probability 2/3 if Hussein doesn’t allow inspections.

Probability that Hussein has WMD’s if he uses mixed strategy

• If Hussein does not allow inspections, what is probability that he has WMDs?

• Apply Bayes’ law. P(WMD|no inspect)=P(WMD and no inspect)/P(no inspect)=w/(w+(1-w)(1-h))

Bush’s payoffs if Hussein refuses inspections

• If Bush does not invade: 1 w/(w+(1-w)(1-h)) +9(1-(w/(w+(1-w)(1-h))) • If Bush invades:3 w/(w+(1-w)(1-h)) +6(1-w/(w+(1-w)(1-h)) Bush will use a mixed strategy only if these two payoffs are equal.We need to solve the equation 1 w/(w+(1-w)(1-h)) +9(1-(w/(w+(1-w)(1-h))) =3 w/(w+(1-w)(1-h)) +6(1-w/(w+(1-w)(1-h)) for h.

Solution

• Solving equation on previous slide, we see that if Saddam refuses inspections, Bush is indifferent between invading and not if h=3-5w/3(1-w). (Remember we assumed w<3/5) so 0<h<1)

• If Saddam has no WMD’s, he is indifferent between allowing and not allowing inspections Bush would invade with probability 4/5 if there are no inspections.

Describing equilibrium strategies

Saddam: Do not allow inspections if he has WMD. Allow inspections with probability h=3-5w/3(1-w) if he has no WMD. (e.g. if w=1/2, h=1/3. If w=1/3, h=2/3.)Bush: Invade if Saddam has WMD and allows inspections, Don’t invade if Saddam has no WMD and allows inspections. Invade with probability 4/5 if Saddam does not allow inspections.