# Private Information and Auctions

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Private Information and Auctions

Auction Situations Private Value Common Value
Everybody knows their own value for the object Nobody knows other people’s values. Common Value The object has some ``true value’’ that it would be worth to anybody Nobody is quite sure what it is worth. Different bidders get independent hints.

First Price Sealed Bid Auction
Suppose that everyone knows their own value V for an object, but all you know is that each other bidder has a value that is equally likely to be any number between 1 and 100. A strategy is an instruction for what you will do with each possible value. Let’s look for a symmetric Nash equilibrium.

Case of two bidders. Let’s see if there is an equilibrium where everyone bids some fraction a of their values. Let’s see what that fraction would be. Suppose that you believe that if the other guy’s value is X, he will bid aX. If you bid B, the probability that you will be the high bidder is the probability that B>aX. The probability that B>aX is the probability that X<B/a.

Two bidder case We have assumed that the probability distribution of the other guy’s value is uniform on the interval [0,100]. For number X between 0 and 100, the probability that his value is less than X is just X/100. The probability that X<B/a is therefore equal to B/(100 a). This is the probability that you win the object if you bid B.

So what’s the best bid? If you bid B, you win with probability B/(100a). Your profit is V-B if you win and 0 if you lose. So your expected profit if you bid B is (V-B) times B/(100a)=(1/100a)(VB-B2). To maximize expected profit, set derivative equal to zero. We have V-2B=0 or B=V/2. This means that if the other guy bids proportionally to his value, you will too, and your proportion will be a=1/2.

In a first-price sealed bid private value auction, where each believes that the other’s value is uniformly distributed on the interval [0,100], there is a Bayes-Nash equilibrium in which each bidder bids half of her value.

Back to the oil fields

Single bidder version A new oil field has come up for lease.
The current owner has done no exploration of the oilfield. All that the current owner knows is that the field has two halves such that each half is worth either \$0 or \$3 million dollars with independent probabilities of ½ for each value for each half. It is common knowledge that there is only one possible buyer and that buyer has learned the value of one half of the field but not the other.

Expected Values What is the expected value of the whole oilfield to the bidder if he knows that his own half is worth \$3 million? \$1.5 million \$3 million \$4 million \$4.5 million \$6 million

Expected Values What is the expected value of the whole oilfield to the bidder if he knows that his own half is worth \$0 ? \$0 \$1 million \$1.5 million \$2 million \$3 million

Seller posts a price The seller posts a price for the oilfield.
The buyer either buys the field at the posted price or refuses to buy. If the buyer pays the posted price, her profits are the actual value of the oilfield minus the price she paid. The seller’s profits are the price received. If buyer doesn’t buy the field, buyer and seller both get 0 payoff.

Bayes-Nash equilibrium
Strategy for Seller--Posts a price. Strategy for Buyer—Rule that specifies whether she will buy or not at each possible price if the side she knows is worth \$0 and if it is worth \$3 million. Buyer tries to maximize her expected profit. Buyer will buy if and only if the price is less than or equal to her expected value, given her information. When seller posts his price, he doesn’t know whether the side the buyer explored is worth 0 or \$3 million.

Clicker question If the seller posts a price of \$4.5 million, what is her expected profit? \$1 million \$1.5 million \$2 million \$2.25 million \$3 million

Clicker question If the seller posts a price of \$1.5 million, what is her expected profit? \$.5 million \$.75 million \$1 million \$1.25 million \$1.5 million

In a Bayes-Nash Equilibrium
A) Seller will post a price of \$4.5 million. Buyer will buy if buyer’s side is worth \$3 million and will reject if her side is worth \$0. B) Seller will post a price of \$1.5 million. Buyer will buy whether her side is worth \$3 million or 0. C) Seller will post a price of \$1.5 million. Buyer will buy only if her side is worth \$3 million.

In Bayes-Nash equilibrium for this game, the expected revenue of the seller is
\$1 million \$2 million \$2.25 million \$3 million \$4.5 million

What if there are two bidders?
Each has explored a different half of the oil field and knows the value of the half she explored. The value of each side is either \$3 million or 0, which depended on the flip of a fair coin. Total value of field is the sum of the two sides Each bidder knows what her side is worth, but not the other bidder’s side.

Posted Price The seller posts a price.
If one bidder accepts and the other declines, oilfield is sold to that bidder at posted price. If both bidders accept, the seller tosses a fair coin to decide which bidder to sell to. If neither bidder accepts, oilfield remains undeveloped and seller and both buyers receive 0 payoff.

Symmetric Bayes-Nash equilibrium?
Is there a symmetric equilibrium where the seller asks \$4.5 million and both buyers play the strategy: Don’t buy if you see 0 and buy if you see \$3 million? Yes No

Buyer’s calculation If I see \$3 million on my side, then the whole field is worth \$3 million with probability ½ and \$6 million with probability ½. But if it is worth \$6 million, then the other buyer’s side is worth \$3 million. Suppose that her strategy is to offer to buy if her side is worth \$3 million and not offer if her side is worth 0. Then if I accept the offer when my side is worth \$3 million, I will be sure to get the field if the field is worth only \$3 million and I will get it with probability ½ if it is worth \$6 million.

Buyer calculates If my side is worth and I pay \$4.5 million
Then with probability ½ oilfield is worth \$3 million and I pay \$4.5 million. With probabilty ¼, oilfield is worth \$6 million and I get it for \$4.5 million. With probability ¼ oilfield is worth \$6 million and the other buyer gets it. So my expected payoff would be ½(3-4.5)+¼ (6-4.5)+¼0=-3/4+3/8=-3/8.

Winner’s curse in common value auctions
If you win something in an auction or other bidding contest, makes it more likely that the object is of low value to other bidders. If low value to other bidders implies lower value to you, then it is a mistake to bid as much as your expected value given only your own information.

What is it worth if you win it?
In this example, if you get the oilfield for \$4.5 million, the probability is 2/3= .5/(.5+.25) that it is worth \$3 million and 1/3 = .25/(.5+.25) that it is worth \$6 million. So in symmetric solution, the expected value conditional on your getting it is 2/3x3+1/3x6=\$4 million.

There would be a Bayes-Nash equilibrium in which seller sets price at \$4 million and buyers will offer to buy if their side is worth \$3 million and will not offer to buy if their side is worth 0. Expected revenue of seller would then be (1/4)x0+(3/4)x4 million= \$3 million

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

If stranger is a gunslinger, he will always draw. If stranger is a cowboy, he will prefer to draw if Earp draws and wait if Earp waits.

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. If Earp always draws, both cowpoke and gunslinger are better off drawing. Let p be probability stranger is gunslinger. If both types always draw, payoff to Earp from draw is 2p+5(1-p)=5-3p and payoff to Earp from wait is p+6(1-p)=6-5p Now 5-3p>6-5p if p>1/2.

If Earp always draws, best response for stranger of either type is to draw.
If stranger always draws, best response for Earp is to always , whenever he thinks stranger is a gunslinger with p>1/2. Note that this is so, even though if he knew stranger was a cowpoke, it would be dominant strategy to wait.

Problem 10.2 Curly FD FW SD SW 20,20 30,-40 -40,30 50,50 40,-30 Bat
Table shows payoffs to Bat and Curly conditional on their actions and type. Note: This is not a strategic form representation. A strategy takes the form x/y where x is what you do if you are fast and y is what you do if you are slow. Is D/D for both players a SPNE? Need to show that if Curly always draws, Bat will prefer to draw if he is fast and also if he is slow. Also need to show that if Bat always draws, Curly will prefer to draw if he is fast also if he is slow.

Suppose Curly always draws
If Bat draws when he is fast his expected payoff is .6 x x 30=24. If Bat waits when he is fast, his expected payoff is .5x(-40)+.4x20=-12 So if Bat is fast, Bat’s best response is draw. If Bat draws when he is slow, his expected payoff is .6x(-40)+.4x20=-16. If Bat waits when he is slow, his expected payoff is .6 x(-40)+.4x(-40)=-40. So if Bat is slow, his best response is draw. Thus if Curly always draws, Bat’s best response is to always draw.

Suppose Bat always draws
Need to show that Curly is better off drawing than waiting whether he is fast or slow. Similar calculation to previous one except that probability that Bat is fast is .65

Problem 10.5 Find Bayes-Nash equilibrium
Note that if Player 1 is a Low type, he will prefer y to x, whether Player 2 plays a or b. So in any SPNE, if Player 1 is a low type, he will choose y.

Problem 10.5 Strategic form (leaving out dominated strategies for Player 1) a b x/y 3, 1 1,3p y/y 2, 1 5, 2p Note that ( x/y, a) is a Bayes- Nash equilibrium if and only if 3p≤ 1, equivalently, p≤1/3 Also, (y/y, b) is a Nash equilibrium if and only if 2p≥1, equivalently, p≥1/2. If p≤1/3, the only Bayes-Nash equilibrium has Player 1 goes x if he’s high, and y if he’s low, while Player 2 goes a. If p≥1/2, the only Bayes-Nash has Player 1 always goes y and Player 2 goes b.

Second Midterm, next Tuesday