Oil Lease Auctions and the Winners’ Curse

Geologists’ estimates of value differ widely Company that makes highest estimate bids the highest. Often loses money. Oil companies became aware of this in the 1960’s and now take it into account.

From industry experts An article in the Journal of Petroleum Technology, 1971 by three employees of Atlantic Richfield Oil Company reports that many oil companies have found, in retrospect that they bid too much for offshore oil leases, particularly those in the Gulf of Mexico. `` a lease winner tends to be the bidder who most overestimates reserves potential… ``successful” bidders may not be so successful after all.” “Competitive Bidding in High-Risk Situations”, By Capen, Clapp, and Campbell 1971

Our simplified set-up You own an oil company. A new field has come up for lease. There are two bidders. You and another firm. Each of you has explored half of the oil field and knows the value of the half they explored. The value of each side is either $3 million or 0, which nature determined by the flip of a fair coin. Total value of field is the sum of the two sides You know what your side is worth, but not the other company’s side.

The Auction The lease for the entire field is up for auction. A bid must be an integer number (possibly 0) of million $. There are two bidders, you and the company that explored the other side. You know what your side is worth. Entire field will be leased to the higher bidder in a sealed bid auction. If there are tie bids, winner is chosen by coin flip. If you win the auction, your profit or loss is the value of the total field minus your bid.

A strategy A strategy states the amount you will bid if your side is worth $0 and the amount you will bid if your side is worth $3 million.

Finding a symmetric Bayes-Nash equilibrium A symmetric Bayes-Nash equilibria is one in which all players use the same strategy. Note that a strategy tells what you do for each type you could be. This means that two players using the same strategy might take different actions (because they turn out to be of different types. In this game, two players using the same strategy might bid differently because one saw $3 million on his half and the other saw $0.

Lets find your strategy: What would you bid if your side is worth $0? A) $0 B) $1 million C) $2 million D) $3 million E) $4 million

What would you bid if your side is worth $3 million? A) $1 million B) $2 million C) $3 million D) $4 million E) $5 million

Some things to think about What would be your expected profit if the company you bid against uses the same strategy that you do? If your side is worth $0 and you win the auction, what do you expect the total oilfield to be worth? What does this tell us about symmetric equilibrium strategies?

Is (0,4) a symmetric Bayes-Nash equilibrium? Suppose other guy bids 0 when he sees 0 and $4 million when he sees $3 million on his own side. If (0,4) is a symmetric Bayes-Nash equilibrium, your best response has to be (0,4)? Lets see if it is – What if you see 0? – What if you see $3 million?

What is your best response if you see $3 million? If the other guy bids 0 when he sees 0 and $4m when he sees $3m on his own side. If I bid $4m when I see $3m – If other guy sees 0, he bids 0. The object is worth $3 and I pay $4. So I would lose $1m. – If other guy sees $3m, he bids $4. Bids are a tie. If I win coin toss, I make a profit of $6m-4m=$2m. With probability ½ I lose $1m. With probability ¼ I make $2m. My expected winnings if I see 3 are - 1/2x1+1/4x2=0.

Suppose I bid only $1m when I see $3m If the other guy bids 0 when he sees 0 and $4m when he sees $3m on his own side. If I bid $1m when I see $3m – If other guy sees 0, he bids 0. The object is worth $3m. I pay $1m. I get a profit of $2m. – If other guy sees $3m, he bids $4. I don’t get field. My profit is 0. My expected winnings are ½ 2 +1/2 0=1. This is a better payoff than I get if I bid $4m when I see $3m.

What if I bid just $1m when I see $3m? If the other guy bids 0 when he sees 0 and $4 million when he see $3m on his own side. If I see $3 million, then I am sure to get the object if he sees 0. If instead I played (0,1), I would get the object only when I see $3 million and the other guy sees 0. This happens with probability ¼ and then my profit would 2. So my expected profit would be 1/2. Therefore (0,4) is not a best response to (0,4). So (0,4) is not a symmetric Bayes-Nash equilibrium.

Is (0,3) a symmetric Bayes-N.E? If other guy is playing (0,3) and I play (0,3), I will get the object if I saw 3 and he saw 0 OR if we both saw 3 and the coin flip came out my way. I make 0 profit if I saw 3 and he saw 0. I make a profit of 3 if we both saw 3 and I won the coin flip, which happens with probability 1/8. So my expected payoff would be 3/8. If I play (0,1), I get the object if I saw 3 and he saw 0. This happens with probability ¼ and in this case my profit is 3-1=2. So my expected profit if I play (0,1) is 2x(1/4)=1/2. So (0,3) is not a best response to (0,3). Hence (0,3) is not a symmetric NE strategy.

What about (0,2). Suppose other guy is playing (0,2). If I play (0,2) my expected payoff is ½x0+½(½x1+½x½x4)=¾. If I play (0,1) my expected payoff is ½ If I play (0,3) my expected payoff is ¾. You can check out that (0,x) is worse for any other x and so (0,2) for both players is a symmetric Bayes-Nash equilibrium.

Signaling Games Econ 171

General form Two players– a sender and receiver. Sender knows his type. Receiver does not. It is not necessarily in the sender’s interest to tell the truth about his type. Sender chooses an action that receiver observes Receiver observes senders action, which may influence his belief about receiver’s type. Receiver takes action

Perfect Bayes Nash equilibrium for signaling game Sender’s strategy specifies an action for each type that she could be. Her action maximizes her expected payoff for that type, given the way the receiver will respond. For each action of the sender, receiver’s strategy specifies an action that maximizes his expected payoff. Receiver’s beliefs about sender’s type, conditional on actions observed are consistent.

Types of equilibria. Separating equilibria. Different types of senders take different actions. Pooling equilibria Different types of senders take same actions.

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 ToughWeak Beer Quiche Fight Don’t B Bully 1+s 1010 s w w

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? 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 beer if s<1 – Tough kid would have quiche if s>1. (explain)

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.

Clicker question The equilibrium in which the new kid has quiche if weak and beer if tough and where the bully fights quiche-eaters but doesn’t fight beer-drinkers is A)A separating equilibrium B)A pooling equilibrium C)Neither of these

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 has 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 quiche, Don’t if beer 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 chooses Don’t Fight if Beer, Fight if Quiche, Bullie’s payoff is 0. If Bully chooses a strategy that Fight if Beer, (anything) if Quiche, Bullie’s expected payoff is -1xp+1x(1-p)=1-2p. If p>1/2, Fight if Beer, Don’t 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? 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.

An Education 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 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 equal to the productivity of an average worker: $1250 per month.

Enter Professor Drywall

Drywall claims My 10-lecture course raises worker productivity by 20%!

One employer believes that Drywall’s lectures are useful and requires its workers attend 10 monthly lectures by Professor Drywall and pays wages 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.