Some Lecture Notes on Auctions

Transcription

1 Some Lecture Notes on Auctions John Morgan Haas School of Business and Department of Economics Uniersity of California, Berkeley Preliminaries Perhaps the most fruitful area for the application of optimal screening contracts is in auction theory principle, many bidders. The goal might be to allocate e ciently (and maximize reenues while doing so) Or it might just be to maximize reenues. The main result we will establish is: Reenue Equialence Theorem (RET): Assume each of a gien number of risk-neutral potential buyers has a priately-known aluation independently drawn from a strictly-increasing atomless distribution, and that no buyer wants more than one of the k identical indiisible prizes. Then any mechanism in which (i) the prizes always go to the k buyers with the highest aluations and (ii) any bidder with the lowest feasible aluation expects zero surplus, yields the same expected reenue (and results in each bidder making the same expected payment as a function of her aluation).. Model Throughout, we study the archetypal auction model: n ex ante identical potential bidders independent priate alues, i ; drawn from some atomless distribution F on [; ] : Single object to be auctioned. Seller has commonly known aluation :

2 Game form Seller chooses the contract (i.e. auction form). Bidders bid. Payo s are realized. 2 Second Price Auction Consider an auction where the winning bidder pays the second highest bid (introduced by Vickrey) Proposition Suppose bidders hae priate alues, then bidding one s aluation is a weakly dominant strategy. Proof. General and informal. Suppose you bid b > ; then the outcome compared to the putatie eqm strategy is changed only when highest (other than s) is between and b: But in these circumstances you incur losses. Suppose you bid b < ; then the outcome compared to the putatie eqm strategy is changed only when highest bid (other than s) is between b and : In these circumstances you miss out on pro ts. Less general, more mathematical: Suppose eeryone else is bidding according to the increasing bidding strategy () :Let F (y) be the distribution function of the highest of n draws from F: That is y = max f 2 ; 3 ; :::; n g Bidder wins if b (y) ; so E ( ; b) = Di erentiating wrt b: (b) [ (y)] df (y) : [ b] f (b) d db (b) = Since f (b) 6= and d db (b) 6= ; it then follows that b = : It then follows that: Proposition 2 The Vickrey auction is e cient when bidders hae priate alues. Experimental ndings sometimes di er from this. 2

3 Reenues h The seller s expected reenue is simple E n draws from F: With uniform distributions, this becomes h i E Y (n) 2 Y (n) 2 = n n + i ; i.e., the second highest of so it s obious that reenues are increasing in n and conerge to where the seller obtains all the surplus. We can get at reenues a di erent way: What is the expected payment of a bidder with aluation x? Under the uniform distribution P II (x) = Pr fwing E fpaymentjwing = Pr fwing E fpayment I wing Pr fwing = ydf (y) yx = x yd y n = n n xn The ex ante expected payment is then Expected reenue P II = = P II (x) df (x) n n (n + ) More generally Integrate by parts P II (x) = ER = np II = n n + yx P II (x) = xf (x) = x x ydf (y) x F (y) dy dy 3

4 3 First Price Auctions Bidder, gien aluation ; chooses b to maximize Di erentiate E ( ; b) = (b) [ b] df (y) = [ b] F (b) [ b] df (b) d db (b) F (b) = Under a symmetric equilibrium, b = ( ) ; hence [ ()] f () () F () = Rewriting + A () = A () where A () = f () : F () Multiply by e R A : Hence e R A + Ae R A = Ae R A d e R A = Ae R A d be R A = Ae R A d + c b () = e R A Ae R A d + c Since b () = ; then Now b () = e R A ta (t) e R A dt e R R A F (x) n 2 f(x) = e = e R (n (F (x)) n dx )f(x) F (x) dx ln F (x) = e = F (x) n 4

5 Hence Integrate by parts b () = = F () n x f (x) F (x) F (x)n xd b () = Compute expected payment: dx dx P I (x) = b (x) (x) x = x Notice that it is exactly the same as the Second price auction. Thus, the expected reenue to the seller in either of these auctions is simply ER = E (n) 2 [] dy i.e. the expectation of the second highest of n draws. Thus, both types of auctions are equally good (Vickrey 962). Strategic equialence with other auction forms. Di erence between strat equialence in rst-price ersus second-price case. But experiments do not bear this out. 4 The Reenue Equialence Theorem Consider the following set of auction contracts. Announce a minimum opening bid b : High bidder wins Rules are anonymous Strictly increasing symmetric bidding strategy in auction. Non-negatie returns to bidding. 5

6 Use the reelation principle to restrict attention to direct mechanisms. Bidder s Problem Gien a aluation ; choose a message ^ to maximize (^; ) = (^) P A (^) where P A is the expected payment from pretending to be a ^ type in auction form A Optimize with respect to ^ : In equilibrium, = ^ (^; ) = 2 (^) f (^) P A (^) 2 () f () = P A () Boundary condition: Find a type soling P A ( ) = F ( ) n Now sole the di erential equation PA (x) dx = xd (x) Notice that the RHS is independent of the auction form! With algebra for all Expected reenue P A () = () ER = ne [P A ()] = n (x) dx [f () + F () ] () d So we hae proed the reenue equialence theorem! 5 Applications Knowing the RET can be helpful in directly computing bidding sstrategies. First-price auction () P I () = Pr fy g b () (x) dx = () b () 6

7 So b () = All-pay Auction so () P AP () = () () = () And so on. (x) dx: (x) dx 6 Empirical Tests of the RET Turkish treasury auctions Structural estimation literature EBay experiments 7 Reenue Maximization Using the RET, the principal s problem is to choose to maximize ER = F ( ) n + n [f () + F () ] () d Rewrite this ER = F ( ) n + n F () d () d f () call the term in the square brackets the marginal reenue to the seller. Optimizing df ( ) n F ( ) df ( ) n = f ( ) Which implies or MR=MC. Further = F ( ) f ( ) = + F ( ) f ( ) so reenue maximization and allocatie e ciency are in con ict. 7

8 For rst and second price auctions this means that the optimal auction is simply to choose an opening bid of : This opening bid is equal for these two auction forms An entry fee will do the trick as well (same entry fee for all auction forms). A small increase in the resere aboe its lowest leel always raises reenues. Optimal resere price is indenpendent of n: Negotiation To see the monopoly interpretation, consider the case where n = : In this case, the auctioneer is simply a monopolist facing the problem of choosing an o er to maximize E = p ( F (p)) + F (p) so F is the demand cure. Optimizing ( F ) pf + f = Diiding and rearranging or p F f = MR = MC which is of course the same as the optimal resere price in an n player auction. 8 Tort Reform US society is too litigious How to reduce incenties to sue? Toy model: Both parties priately obsere the alue of winning the case. Each decides how much to spend on lawyers. Higher spender wins. This is just the all-pay auction we analyzed earlier. European system: Loser pays winner s expenses. Notice that the expected payo from the lowest type is negatie So the RET does not hold So what happens? 8

9 Bidder s problem is to choose a bid b to maximize Optimizing In equilibrium, ^ = Rewriting where P () = Usual trick: E (^; ) = F (^) (^) ( F (^)) ^ (t) f (t) dt f (^) (^) ( F (^)) + (^) f (^) + (^) f (^) = f F f ( F ) + 2f = + P () = Q () and Q () = f () = e F : R P Qe R P dx and since P = ln ( F ) : Then () = F No upper bound on bidding. Expected payment EP EUR () = () ( F ()) + = xf (x) dx + xf (x) dx t (t) f (t) dt Recall that expected payment in a Vickrey auction is Thus, for all Hence EP II () = xf (x) dx EP EUR () > EP II () EP EUR > EP II f (t) xf (x) dx F dt so the European legal system is more expensie than the American system. Quayle plan: Losing party pays the winner an amount equal to his own expenses. Here the RET applies. So the Quayle plan costs exactly the same as the current plan. Same incenties to initiate litigation. 9