Bidding on an Antique

An antique auction An antique is sold in a sealed-bid auction. It will go to the high bidder at the price the high bidder bids. You do not know if it is a fake or not. You know that 20% of all antiques that look like this one are fakes. If it is a fake, it will be worthless to you. If it is real, you would be willing to pay $1,000 for it.

What is your expected value of this antique? B) $100 C) $800 D) $1000 E) $1200

Another bidder You know that an expert dealer has also looked at it. This dealer can always tell a fake and so would never bid on a fake. You know that if the table is genuine, the dealer will either bid $800 for it or $500 for it, depending on whether he has similar items in stock. You believe that if the table is genuine, the dealer is equally likely to bid $500 or to bid $800.

Diagram of Events Event A : Antique is a Fake and Dealer doesn’t bid Event C P(C0=.4 Event B P(B)=.4 P(A)=.2 Event A : Antique is a Fake and Dealer doesn’t bid Event B: Dealer bids $500 Event C: Dealier bids $800

If you bid B where 0<B<$500 and you win the auction, what is the probability the antique is a fake? 1 ½ 1/10 2/10

If you bid $100 for the antique and you get it, what is the expected value of the antique? $100 $200 $400 $800

If you bid $501 for the antique, what is the probability you get it? ½ .6 .8 1

When do you get it? You get the antique with a bid of $501 if it is a fake, and also if it is real and the dealer bids only $500. Let A be the event that the antique is a fake. Then P(A)=.2 Let B be the event Dealer bids $500. Then P(B)=.4 You will get it with a bid of $501. either if Event A occurs or Event B occurs This happens with probability P(A or B)=P(A)+P(B)= .2+.4=.6.

If you bid $501 for the antique and you get it, what is the probability that it is a fake? 1/10 ¼ 1/3 ½ 1

Diagram of Events Event A Event C P=.4 Event B P=.4 P=.2 A bid of $501 gets the object if Event A or Event B occurs, but not if Event C occurs. Given that Event A or B occurs, the conditional Probability that A occurs is P(A)/((P(A)+P(B))=.2/(.2+.4)=1/3

The conditional probability that you get a fake if you bid $501 If you bid $501 the probability that you get it is P(A or B)=.6. The probability that it is a fake, given that you get it is P(A|AorB)= P(A)/ P(AorB)=.2/.6=1/3

Your expected profit if you bid $501 If you bid $501, and get the antique, the probability that it is good is 2/3 Your expected net gain if you get the antique is (1000)2/3-501 =165.66. A bid of $501 will get the object with probability .6. So your expected profit from a bid of $501 is .6 times $165.66=$99.396.

What is the probability that the antique is fake if you bid $801 and you get it. 1/5 2/5 ½ 1

What is your expected profit if you bid $801? -$100 -$1 $100 $199

Bidding strategy If you bid

Where does Bayes’ Nash come in? Where does Bayes’ Nash come in? You don’t know the dealer’s type. That is, you don’t know whether he knows the antique is a fake or what he will bid if it is real. You have some a prior probability distribution of the dealer’s type and you assume that the dealer does what is in his best interest, given his type. Notice that your expected value of the antique conditional on winning the auction is not the same as your expected value of the antique.

Some lessons If you are bidding against experts for an object of unknown quality, a low bid is a poor idea, since you are likely to get stuck with the item if it is bad. If you are bidding against experts, it doesn’t pay to bid at all, unless you believe the object is worth “enough” more to you than to the experts.

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 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? 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 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? 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 separating equilibrium A pooling equilibrium 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<1/2 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.