Private Information and Auctions Auction Situations Private Value

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Private Information and Auctions

Private Information and Auctions

Auction Situations • Private Value – Everybody knows their own value for the object

Auction Situations • Private 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

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

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 a. X. • If you bid B, the probability that you will be the high bidder is the probability that B>a. X. • The probability that B>a. X is the probability that X<B/a.

Two bidder case • We have assumed that the probability distribution of the other

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

So what’s the best bid? • If you bid B, you win with probability B/(100 a). • 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/(100 a)=(1/100 a)(VB-B 2). To maximize expected profit, set derivative equal to zero. We have V-2 B=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.

What if there are n bidders? • Suppose that the other bidders each bid

What if there are n bidders? • Suppose that the other bidders each bid the same fraction a of their values. • If you bid B, you will be high bidder if each of them bids less than B. • If others bid a. X when there values are X, the probability that you outbid any selected bidder is the probability that a. X<B, which is B/(100 a).

Winning the object • You get the object only if you outbid all other

Winning the object • You get the object only if you outbid all other bidders. The probability that with bid B you outbid all n-1 other guys is (B/100 a)n-1. • If you bid B and get the object, you win V-B. • So your expected winnings if you bid B are (V-B) (B/100 a)n-1=(1/100 a)n-1(V Bn-1 -Bn) • To maximize expected winnings set derivative with respect to B equal to 0.

Equilbrium bid-shading • Derivative of (1/100 a)n-1(V Bn-1 -Bn) is equal to zero if

Equilbrium bid-shading • Derivative of (1/100 a)n-1(V Bn-1 -Bn) is equal to zero if • (n-1)VBn-2 -n. Bn-1=0 • This implies that (n-1)V=n. B and hence B= V(n-1)/n Therefore if everybody bids a fraction a of their true value, it will be in the interest of everybody to bid the fraction (n-1)/n of their true value.

Wyatt Earp and the Gun Slinger

Wyatt Earp and the Gun Slinger

A Bayesian gunslinger game

A Bayesian gunslinger game

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

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

What are the strategies? • Earp – Draw – Wait • Stranger – –

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

One Bayes Nash equilibrium • Suppose that Earp waits and the other guy draws

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 3/4 x 1+1/4 x 8=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

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 2 p+5(1 -p)=5 -3 p and payoff to Earp from wait is p+6(1 -p)=6 -5 p • Now 5 -3 p>6 -5 p if p>1/2.

 • If Earp always draws, best response for stranger of either type is

• 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.