Auction Theory. Jonathan Levin. October 2004

Transcription

1 Auction Theory Jonathan Levin October 2004 Our next topic i auction. Our objective will be to cover a few of the main idea and highlight. Auction theory can be approached from different angle from the perpective of game theory (auction are bayeian game of incomplete information), contract or mechanim deign theory (auction are allocation mechanim), market microtructure (auction are model of price formation), a well a in the context of different application (procurement, patent licening, public finance, etc.). We re going to take a relatively game-theoretic approach, but ome of thi richne hould be evident. 1 The Independent Private Value (IPV) Model 1.1 A Model The baic auction environment conit of: Bidder i =1,...,n Oneobjecttobeold Bidder i oberve a ignal S i F ( ), with typical realization i [, ], and aume F i continuou. Bidder ignal S 1,...,S n are independent. Bidder i value v i ( i )= i. Given thi baic et-up, pecifying a et of auction rule will give rie to a game between the bidder. Before going on, oberve two feature of the model that turn out to be important. Firt, bidder i information (her ignal) i independent of bidder j information. Second, bidder i value i independent of bidder j information o bidder j information i private in the ene that it doen t affect anyone ele valuation. 1

2 1.2 Vickrey (Second-Price) Auction In a Vickrey, or econd price, auction, bidder are aked to ubmit ealed bid b 1,...,b n. The bidder who ubmit the highet bid i awarded the object, and pay the amount of the econd highet bid. Propoition 1 In a econd price auction, it i a weakly dominant trategy to bid one value, b i ( i )= i. Proof. Suppoe i value i i, and he conider bidding b i > i. Let ˆb denote the highet bid of the other bidder j 6= i (from i perpective thi i a random variable). There are three poible outcome from i perpective: (i) ˆb >b i, i ;(ii) b i > ˆb > i ; or (iii) b i, i > ˆb. In the event of the firt or third outcome, i would have done equally well to bid i rather than b i > i. In (i) he won t win regardle, and in (ii) he will win, and will pay ˆb regardle. However, in cae (ii), i will win and pay more than her value if he bid ˆb, omething that won t happen if he bid i.thu,idoe better to bid i than b i > i. A imilar argument how that i alo doe better to bid i than to bid b i < i. Q.E.D. Since each bidder will bid their value, the eller revenue (the amount paid in equilibrium) will be equal to the econd highet value. Let S i:n denote the ith highet of n draw from ditribution F (o S i:n i a random variable with typical realization i:n ). Then the eller expected revenue i E S 2:n. The truthful equilibrium decribed in Propoition 1 i the unique ymmetric Bayeian Nah equilibrium of the econd price auction. There are alo aymmetric equilibria that involve player uing weakly dominated trategie. One uch equilibrium i for ome player i to bid b i ( i )=v and all the other player to bid b j ( j )=0. While Vickrey auction are not ued very often in practice, open acending (or Englih) auction are ued frequently. One way to model uch auction i to aume that the price rie continuouly from zero and player each can puh a button to drop out of the bidding. In an independent private value etting, the Nah equilibria of the Englih auction are the ame a the Nah equilibria of the Vickrey auction. In particular, the unique ymmetric equilibrium (or unique equential equilibrium) of the Englih auction ha each bidder drop out when the price reache hi value. In equilibrium, the auction end when the bidder with the econd-highet value drop out, o the winner pay an amount equal to the econd highet value. 2

3 1.3 Sealed Bid (Firt-Price) Auction In a ealed bid, or firt price, auction, bidder ubmit ealed bid b 1,...,b n. The bidder who ubmit the highet bid i awarded the object, and pay hi bid. Under thee rule, it hould be clear that bidder will not want to bid their true value. By doing o, they would enure a zero profit. By bidding omewhat below their value, they can potentially make a profit ome of the time. We now conider two approache to olving for ymmetric equlibrium bidding trategie. A. The Firt Order Condition Approach We will look for an equilibrium where each bidder ue a bid trategy that i a trictly increaing, continuou, and differentiable function of hi value. 1 To do thi, uppoe that bidder j 6= i ue identical bidding trategie b j = b( j ) with thee propertie and conider the problem facing bidder i. Bidder i expected payoff, a a function of hi bid b i and ignal i i: U(b i, i )=( i b i ) Pr [b j = b(s j ) b i, j 6= i] Thu, bidder i chooe b to olve: max ( i b i ) F n 1 b 1 (b i ). b i The firt order condition i: ( i b i )(n 1)F n 2 b 1 (b i ) f b 1 (b i ) 1 b 0 (b 1 (b i )) F n 1 b 1 (b i ) =0 At a ymmetric equilibrium, b i = b( i ), o the firt order condition reduce to a differential equation (here I ll drop the i ubcript): b 0 () =( b()) (n 1) f() F (). Thi can be olved, uing the boundary condition that b() =, to obtain: b() = R i F n 1 ( )d F n 1. () 1 In fact, it i poible to prove that in any ymmetric equilibrium each bidder mut ue a continuou and trictly increaing trategy. To prove thi, one how that in equilbrium there cannot be a gap in the range of bid offered in equilibrium (becaue then it would be ub-optimal to offer the bid jut above the gap) and there cannot be an atom in the equilibrium ditribution of bid (becaue then no bidder would make an offer jut below the atom, leading to a gap). I ll kip the detail though. 3

4 It i eay to check that b() i increaing and differentiable. So any ymmetric equilibrium with thee propertie mut involve bidder uing the trategy b(). B. The Envelope Theorem Approach A cloely related, and often convenient, approach to identify neceary condition for a ymmetric equilibrium i to exploit the envelope theorem. To thi end, uppoe b() i a ymmetric equilibrium in increaing differentiable trategie. Then i equilibrium payoff given ignal i i U ( i )=( i b( i )) F n 1 ( i ). (1) Alternatively, becaue i i playing a bet-repone in equilibrium: U( i )=max b i ( i b i ) F n 1 (b 1 (b i )). Applying the envelope theorem (Milgrom and Segal, 2002), we have: d d U () = F n 1 (b 1 (b( i )) = F n 1 ( i ) =i and alo, Z i U ( i )=U()+ F n 1 ( )d. (2) A b() i increaing, a bidder with ignal will never win the auction therefore, U() =0. Combining (1) and (2), we olve for the equilibrium trategy (again dropping the i ubcript): b() = R i F n 1 ( )d F n 1. () Again, we have howed neceary condition for an equilibrium (i.e. any increaing differentiable ymmetric equilibrium mut involve the trategy b()). To check ufficiency (that b() actually i an equilibrium), we can exploit the fact that b() i increaing and atifie the envelope formula to how that it mut be a election from i bet repone given the other bidder ue the trategy b(). (For detail, ee Milgrom 2004, Theorem 4.2 amd 4.6). 4

5 Remark 1 In mot auction model, both the firt order condition and the envelope approach can be ued to characterize an equilibrium. The trick i to figure out which i more convenient. What i the revenue from the firt price auction? It i the expected winning bid, or the expected bid of the bidder with the highet ignal, E b(s 1:n ).Toharpenthi,define G() =F n 1 (). Then G i the probability that if you take n 1draw from F, all will be below (i.e. it i the cdf of S 1:n 1 ). Then, R b() = F n 1 ( )d Z 1 F n 1 = () F n 1 df n 1 ( ) =E S 1:n 1 S 1:n 1. () That i, if a bidder ha ignal, he et hi bid equal to the expectation of thehighetoftheothern 1 value, conditional on all thoe value being le than hi own. Uing thi fact, the expected revenue i: E b S 1:n = E S 1:n 1 S 1:n 1 S 1:n = E S 2:n, equal to the expectation of the econd highet value. We have hown: Propoition 2 The firt and econd price auction yield the ame revenue in expectation. 1.4 Revenue Equivalence The reult above i a pecial cae of the celebrated revenue equivalence theorem due to Vickrey (1961), Myeron (1981), Riley and Samuelon (1981) and Harri and Raviv (1981). Theorem 1 (Revenue Equivalence) Suppoe n bidder have value 1,..., n identically and independently ditributed with cdf F ( ). Then all auction mechanim that (i) alway award the object to the bidder with highet value in equilibrium, and (ii) give a bidder with valuation zero profit, generate the ame revenue in expectation. Proof. We conider the general cla of auction where bidder ubmit bid b 1,...,b n. An auction rule pecifie for all i, x i : B 1... B n [0, 1] t i : B 1... B n R, 5

6 where x i ( ) give the probability i will get the object and t i ( ) givei required payment a a function of the bid (b 1,...,b n ). 2 Given the auction rule, bidder i expected payoff a a function of hi ignal and bid i: U i ( i,b i )= i E b i [x i (b i,b i )] E b i [t i (b i,b i )]. Let b i ( ),b i ( ) denote an equilibrium of the auction game. Bidder i equilibrium payoff i: U i ( i )=U i ( i,b( i )) = i F n 1 ( i ) E i [t i (b i ( i ),b i ( i )], whereweue(i) to write E i [x i (b( i ),b( i ))] = F n 1 ( i ). Uing the fact that b( i ) mut maximize i payoff given i and opponent trategie b i ( ), the envelope theorem implie that: d d U i () = E b i [x i (b i ( i ),b i ( i ))] = F n 1 ( i ), =i and alo Z i Z i U i ( i )=U i ()+ F n 1 ( )d = F n 1 ( )d, whereweue(ii) to write U i () =0. Combining our expreion for U i ( i ), we get bidder i expected payment given hi ignal: E i [t i (b i,b i )] = i F n 1 ( i ) Z i F n 1 ( )d = Z i df n 1 ( ), where the lat equality i from integration by part. Since x i ( ) doenot enter into thi expreion, bidder i expected equilibrium payment given hi ignal i the ame under all auction rule that atify (i) and (ii). Indeed, i expected payment given i i equal to: So the eller revenue i: E S 1:n 1 S 1:n 1 < i = E S 2:n S 1:n = i. E [Revenue] = ne i [i expected payment i ]=E S 2:n, 2 So in a firt price auction, x 1 (b 1,..., b n)equal1ifb 1 i the highet bid, and otherwie zero. Meanwhile t 1 (b 1,..., b n ) equal zero unle b 1 i highet, in which cae t 1 = b 1.Ina econd price auction, x 1 ( ) i the ame, and t 1 ( ) i zero unle b 1 i highet, in which cae t 1 equal the highet of (b 2,..., b n). 6

7 a contant. Q.E.D. The revenue equivalence theorem ha many application. One ueful trick i that it allow u to olve for the equilibrium of different auction, o long a we know that the auction will atifiy (i) and (ii). Here an example. Application: The All-Pay Auction. Conider the ame et-up bidder 1,..,n,withvalue 1,..., n, identically and independently ditributed with cdf F and conider the following rule. Bidder ubmit bid b 1,...,b n and the bidder who ubmit the highet bid get the object. However, bidder mut pay their bid regardle of whether they win the auction. (Thee rule might eem a little trange the all-pay auction i commonly ued a a model of lobbying or political influence). Suppoe thi auction ha a ymmetric equilibrium with an increaing trategy b A () ued by all player. Then, bidder i expected payoff given value i will be (if everyone play the equilibrium trategie): U( i )= i F n 1 ( i ) b A ( i )= Z i F n 1 ( )d So b A () =F n 1 () Z F n 1 ( )d In addition to the all-pay auction, many other auction rule alo atify the revenue equivalence aumption when bidder value are independently and identically ditributed. Two example are: 1. The Englih (oral acending) auction. All bidder tart in the auction with a price of zero. The price rie continuouly, and bidder may drop out at any point in time. Once they drop out, they cannot reenter. The auction end when only one bidder i left, and thi bidder pay the price at which the econd-to-lat bidder dropped out. 2. The Dutch (decending price) auction. The price tart at a very high level and drop continuouly. At any point in time, a bidder can top the auction, and pay the current price. Then the auction end. 7

8 2 Common Value Auction We would now like to generalize the model to allow for the poibility that (i) learning bidder j information could caue bidder i to re-ae hi etimate of how much he value the object, and (ii) the information of i and j i not independent (when j etimate i high, i i alo likely to be high). Thee feature are natural to incorporate in many ituation. For intance, conider an auction for a natural reource like a tract of timber. In uch a etting, bidder are likely to have different cot of harveting or proceing the timber. Thee cot may be independent acro bidder and private, much like in the above model. But at the ame time, bidder are likely to be unure exactly how much merchanteable timber i on the tract, and ue ome ort of tatitical ampling to etimate the quantity. Becaue thee etimate will be baed on limited ampling, they will be imperfect o if i learned that j had ampled a different area and got a low etimate, he would likely revie her opinion of the tract value. In addition, if the area ampled overlap, the etimate are unlikely to be independent. 2.1 A General Model Bidder i =1,...,n Signal S 1,...,S n with joint denity f( ) Signal are (i) exchangeable, and (ii) affiliated. Signal are exchangeable if 0 i a permutation of f() =f( 0 ). Signal are affiliated if f( 0 )f( 0 ) f( 0 )f() (i.e. j i ha monotone likelihood ratio property). Value to bidder i i v( i, i ). Example 1 The independent private value model above i a pecial cae: jut let v( i, i )= i, and uppoe that S 1,...,S n are independent. Example 2 Another common pecial cae i the pure common value model with conditionally independent ignal. In thi model, all bidder have the ame value, given by ome random variable V. The ignal S 1,..,S n are each correlated with V, but independent conditional on it (o for intance, S i = V + ε i,whereε 1,..,ε n are independent. Then v( i, i )=E[V 1,..., n ]. 8

9 Example 3 A commonly ued, but omewhat hard to motivate, variant of the general model i obtained by letting v i ( i, i )= i + β P j6=i j,with β 1, and auming that S 1,..., S n are independent. Thi model ha the feature that bidder have independent information (o a verion of the RET applie), but interdependent valuation (o there are winner cure effect). A new feature of the more general auction environment i that each bidder i will want to account for the fact that her opponent bid reveal omething about their ignal, information that i relevant for i own valuation. In particular, if i win, then the mere fact of winning reveal that her opponent value were not that high hence winning i bad new about i valuation. Thi feature i called the winner cure. 2.2 Second Price Auction Let look for a ymmetric increaing equilibrium bid trategy b() inthi more general environment. We tart by conidering the bidding problem facing i: Let i denote the highet ignal of bidder j 6= i. Bidder i will win if he bid b i b( i ), in which cae he pay b( i ) Bidder i problem i then: Z max ES i v(i,s i ) i,s i = i b( i ) 1 {b( b i ) b i }f( i i )d i i or Z b 1 (b i ) max ES i v(i,s i ) i,s i = i b( i ) df ( i i ) b i The firt order condition for thi problem i: 1 0= b 0 (b 1 ES i v(i,s i ) i,s i = b 1 (b i ) b(b 1 (b i )) f(b 1 (b i ) i ) (b i )) or implifying: b i = b( i )=E S i v( i,s i ) i, max j = i j6=i That i, in equilibrium, bidder i will bid her expected value conditional on her own ignal and conditional on all other bidder having a ignal le than her (with the bet of the ret equal to her). 9

10 Remark 2 While we will not pure it here, in thi more general environment the revenue equivalence theorem fail. Milgrom and Weber (1982) prove a very general reult called the linkage principle which baically tate that in thi general ymmetric etting, the more information on which the winner payment i baed, the higher will be the expected revenue. Thu, the firt price auction will have lower expected revenue than the econd price auction becaue the winner payment in the firt price auction i baed only on her own ignal, while in the econd price auction it i baed on her own ignal and the econd-highet ignal. 3 Large Auction & Information Aggregation An intereting quetion that ha been tudied in the auction literature arie if we think about auction model a a tory about how price are determined in Walraian market. In thi context, we might then ak to what extent price will aggregate the information of market participant. We now conider a erie of reult along thee line. For each reult, the baic et-up i the ame. There are a lot of bidder, and the auction ha pure common value with conditionally independent ignal. We conider econd price auction (or more generally, with k object, k +1 price auction, where all winning bidder pay the k + 1t highet bid). The baic quetion i: a the number of bidder get large, will the auction price converge to the true value of the object() for ale? 3.1 Wilon (1977, RES) Wilon conider information aggregation in a etting with a pecial information tructure. In hi etting, bidder learn a lower bound on the object value. The model ha: Bidder 1,...,n A ingle object with common value V U[0, 1] Bidder ignal S 1,...,S n are iid with S i U[0,v] (o ignal i V i ). Wilon how that a n, the expected price converge to v. Thii eay to ee: with lot of bidder, omeone will have a ignal cloe to v, and you have to bid very cloe to your ignal to have any chance of winning. 10

11 3.2 Milgrom (1979, EMA) Milgrom goe on to identify a neceary and ufficient condition for information aggregation with many bidder and a fixed number of object. Milgrom baic requirement i, in the limit a n, that for all v 0 Ω and all v<v 0,andM>0, there exit ome 0 S uch that P ( 0 v 0 ) P ( 0 v) >M. That i, for any poible value v 0, there mut be arbitrarily trong ignal that effectively rule out any value v<v 0. Thi condition build on Wilon, and the intuition i again quite eay. A n, there i a very trong winner cure. In a econd price auction, the high bid i: E V V 1:n, max S j = 1:n. j6=i While it i clear that 1:n i likely to be very high a n,theonlyway anyone would ever bid v would be to have a ignal o trong that conditioning on million of other ignal being lower than your own would not puh down your etimate too much. 3.3 Peendorfer and Swinkel (1997, EMA) Peendorfer and Swinkel conider a model that i quite imilar to Milgrom. n bidder, k object Each bidder ha value V F ( ) on[0, 1] Signal S i G( v) on[0, 1], where g( ) hathemlrp. Signal S 1,...,S n are independent conditional on V = v. Remark 3 Clearly with a large number of independent ignal, there i enough information to conitently etimate v. The quetion then i whether the bid will accurately aggregate thi information i.e. will the price be a conitent etimator? Peendorfer and Swinkel big idea i the following. While the winner cure will make a bidder with =1hadeherbidan with a fixed number of object k, a k,there i alo a loer cure: if lowering your bid ε matter there mut be k people with higher valuation. Thi operate againt the winner cure, counterbalancing it. 11

12 Propoition 3 The unique equilibrium of k+1 t price auction i ymmetric: b( i )=E V [V i,kth highet of other ignal i alo i ]. Definition 1 Aequenceofauction(n m,k m ) ha information aggregation if the price converge in probability to v a m. PS firt conider neceary condition for information aggregation. Clearly, a requirement for information aggregation i that: lim b m(0) = 0 and lim b m(1) = 1 m m Without thi, there will be no convergence when v =0orwhenv =1. Now, b m (1) = E V [V i =1,kth highet of other ignal i 1] If k m 9, then thi expectation will hrink below 1. So we mut have k m. And imilarly, looking at b m (0), we mut have n m k m. PS go on to how that thee condition are not jut neceary, but ufficient for information aggregation. Propoition 4 A equence of auction ha information aggregation if and only if n m k m and k m. The baic idea i that conditioning on k m 1 bidder having higher ignal than i give a very trong ignal about true value. So the bidder who actually ha the kth highet ighnal will tend to be right on when he bid E V [V i,kth highet of other ignal i i ]. Thu, we will end up with information aggregation. Reference [1] Harri, Milton and Arthur Raviv (1981) Allocation Mechanim and the Deign of Auction, Econometrica, 49, [2] Kremer, Ilan (2002) Information Aggregation in Common Value Auction, Econometrica. [3] Milgrom, Paul (1979) A Convergence Theorem for Bidding with Differential Information, Econometrica, 47. [4] Milgrom, Paul (2003) Putting Auction Theory to Work, Cambridge Univerity Pre. 12

13 [5] Milgrom, Paul and Robert Weber (1982) A Theory of Auction and Competitive Bidding, Econometrica, 50,. [6] Milgrom, Paul and Ilya Segal (2002) Envelope Theorem for Arbitrary Choice Set, Econometrica, 70, [7] Myeron, Roger (1981) Optimal Auction Deign, Math. Op. Re, 6, [8] Peendorfer, Wolfgang and Jeroen Swinkel (1997) The Loer Cure and Information Aggregation in Auction, Econometrica, 65,. [9] Riley, John and William Samuelon (1981) Optimal Auction, American Economic Review, 71, [10] Vickrey, William (1961) Counterpeculation, Auction and Competitive Sealed Tender, Journal of Finance, 16, [11] Wilon, Robert (1977) A Bidding Model of Perfect Competition, Review of Economic Studie. [12] Wilon, Robert (1992) Strategic Analyi of Auction, Handbook of Game Theory. 13