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dec04_Article 5 12/14/04 2:27 PM Page 1102 Journal of Economic Literature Vol. XVII (December 2004) pp. 1102–1115 seriously on auctions in the late 1970s and early 1980s,6 just when the right game-the- after 25 years of intensive work, the literature oretic methods for studying this subject— continues to grow at a prodigious, even accel- erating rate;3 it has spawned much empirical Harsanyi 1967–68) and perfect equilibrium and experimental research;4 its tentacles have spread into other disciplines; 5 and auc- tion theorists have been influential in the fields, e.g., industrial organization (I.O.), design of mechanisms for the privatization of benefited from the same symbiosis of tech- public assets (such as spectrum bands) and nique and application; collectively, they for the allocation of electricity and other resulted in the game theory revolution. But goods (they have also often served as con- the study of auctions has had more staying sultants to the bidders in those mechanisms).
power than many other applications ofgame theory. Whereas enthusiasm for theo- 1 Paul Milgrom, Putting Auction Theory to Work.
retical industrial organization has cooled Cambridge, Churchill Lectures in Economics: Cambridge University Press, 2004. NY, Melbourne, Pp. xxi, 368. ISBN years ago, research on auctions, as I have 2 Institute for Advanced Study and Princeton University. I thank the NSF (SES-0318103) for research believe, several reasons why auction theory 3 To give just one indication: at the August 2004 joint meeting of the Econometric Society and European First, theorists of I.O. and other applied Economic Association, there were seven separate sessions fields labor under the constraint that they do on auction theory, a figure well beyond that for any other 4 Again, to cite only one, conference-related datum, the organizers of the 2005 Econometric Society World 6 Auction theory actually began well before then.
Congress, who attempt to invite special talks on the most Indeed, the seminal contribution was William Vickrey lively and interesting developments in recent economics, (1961). But until game theory came into its own fifteen are planning a set of talks on empirical auctions work.
years later, Vickrey’s work—as well as that of other early 5 So, for example, there is now a sizeable computer-sci- pioneers such as James Griesmer, Richard Levitan, and ence literature on auction theory, often focusing on com- Martin Shubik (1967), Armando Ortega Reichert (1968), and Robert Wilson (1969)—remained largely ignored.
dec04_Article 5 12/14/04 2:27 PM Page 1103 Maskin: The Unity of Auction Theory study (e.g., firms or consumers) are actually playing; models are at best approximations (indeed, one implication of the papers of of reality. By contrast, auction theorists typi- footnote 8 is that, as numbers grow, most cally know the rules that their players follow reasonable sorts of auctions converge in per- precisely. If, for example, a high-bid auction formance; only in the small-numbers case do is the object of study, the theorist knows that the differences between auction forms come (i) the bidders submit nonnegative real num- bers as sealed bids; (ii) the winner is the bid- der submitting the highest bid; and (iii) the beautiful edifice:9 many of its major proposi- winner pays his bid (of course, there may tions deliver remarkably powerful conclu- still be uncertainty about how the buyers sions from apparently modest hypotheses.
behave under these rules). This precision helps put the auction theorist’s findings on a relatively strong footing; it also simplifies the job of the experimentalist or empiricist.
worldwide impulse toward privatization that mists’ “social engineering” instincts. Many people go into economics at least in part brought on by the fall of communism in the because they want to improve the world. The East—and the consequent need to sell off mechanism-design 7 aspect of auction theo- state assets—and the disenchantment with ry—tinkering with the rules of the game in order to achieve a better outcome—helps bureaucracy’s loss proved to be auction the- ory’s gain, as auction mechanisms, to great Vickrey’s work is so celebrated is that his public acclaim, were increasingly invoked to famous creation, the Vickrey auction, provides implement the transfer of resources. More an attractive solution to an important social recently, online auction enterprises such as problem: designing an efficient allocation eBay have provided further impetus for the tion—the transfer of a good from seller to Paul Milgrom has played a starring role in ment—is fundamental to all of economics, auction theory’s success story. Not only has and so auction theory has been nourished by he been a seminal contributor to the theo- its connection with other theoretical areas.
retical literature (e.g., Milgrom 1981; and For example, it has sometimes been used as a foundation for understanding the workings together with Robert Wilson, he had a major of competitive markets.8 Of course, there hand in designing the simultaneous ascend- are important differences: competitive theo- numbers of buyers and sellers, whereas in much of the radio spectrum in the UnitedStates. Thus, his book Putting Auction 7 Of course, auction theory is only a small part of a vast mechanism design/implementation theory literature. For Theory to Work has been eagerly awaited recent surveys of the literature from a general perspectivesee Thomas Palfrey (2002); Roberto Serrano (2004); and 9 Economists, being a hard-boiled lot, sometimes deny Eric Maskin and Tomas Sjöström (2002).
that esthetics have anything to do with what they are up to.
8 See for example Milgrom (1981), Wolfgang But this sentiment belies the fact that the most important Pesendorfer and Jeroen Swinkels (1997); Mark economic ideas, e.g., the first welfare theorem of compet- Satterthwaite and Steven Williams (1989); and Wilson itive theory or the principle of comparative advantage, are dec04_Article 5 12/14/04 2:27 PM Page 1104 Journal of Economic Literature, Vol. XLII (December 2004) since his 1995 Churchill lectures, on which it judgment (albeit very well-informed judg- ment) as logic, they occasionally contrast jar- book covers a great deal of theoretical mate- ringly with the authority and precision of the rial and does so with extraordinary economy theory. For example, in Milgrom’s opinion, (without sacrificing rigor). This economy derives from Milgrom’s conception of auc- tion theory as a subspecies of demand theo- ry, in which a few key tools—the envelope “unsuitable for most applications”—a con- theorem in particular—do most of the work.
clusion that is far from being a theorem and Indeed, once these tools are in place, he that I will come back to in section 7.
establishes most theorems with just a few But putting such quibbles aside, I should lines of proof. As the title suggests, he also emphasize that Milgrom is completely per- discusses the extent to which the results bear suasive on the general point that auction the- ory matters in practice. In chapter one, he Admittedly, the monograph is not the only current volume of reflections by a leading auction’s failure to raise the revenue antici- auction expert on theory and practice.10 Nor, pated can be traced to its seriously flawed despite its unfailing clarity, is it the most likely candidate for a graduate text on the auctions for each license. Specifically, he subject.11 Rather, its signal contribution is to lay out Milgrom’s unified view of the theory.
This vision is notably distinct from that of other major auction scholars. For example, responds to those who argue that how gov- ernment assets are sold off is irrelevant for avoiding the revelation principle as an auc- efficiency (because, in their view, the “mar- ket” will correct any misallocation after- learn to enjoy seeing things his way. In this wards) with the theoretical riposte that, sense, the book is more a “master class” (to under incomplete information, there exists quote Al Roth’s blurb on the back cover) no nonconfiscatory mechanism (market- than a text. And, of course, a master class is based or otherwise) capable of attaining effi- ciency, once the assets are in private hands As for his ideas on how to apply (or not to apply) the theory to actual auctions, these are The heart of chapters 2–8 consists of a suc- certainly most welcome and enlightening.
cession of formal results, almost all proved indetail. I will try to reinforce the book’s 10 Coincidentally, Paul Klemperer—like Milgrom, a important lesson that auction theorems are theorist of the first rank and also a principal architect ofthe United Kingdom’s 3G mobile-phone auction—has easy to prove by stating and proving some of almost simultaneously produced his own take on the sub- them below (although I will not attempt to ject (Klemperer 2004). The two books differ markedly in replicate Milgrom’s rigor or generality).
style and substance. Milgrom’s is primarily a compendiumof theorems and proofs, together with less-formal observa-tions about their application to actual auctions. Except forthe appendices of chapters 1 and 2, Klemperer’s is almost wholly nontechnical and consists largely of his views on thedesign of large-scale auctions in practice (although these In chapter two, Milgrom turns to the most 11 Indeed, Vijay Krishna, yet another prominent auc- the Vickrey (or “second-price”) auction (and tion theorist, has recently produced a beautifully lucidtreatment (Krishna 2002), that in its organization and cov- its Groves–Clarke extension). Suppose that erage may be more suitable as an introductory textbook.
there is one unit of an indivisible good for dec04_Article 5 12/14/04 2:27 PM Page 1105 Maskin: The Unity of Auction Theory sale. There are n potential buyers, indexed buyers to bid truthfully and the high bidder by i = 1,…, n, and each buyer i has a valua- wins, the second-price auction is efficient. tion v for the good (the most he is willing to pay for it). Thus if he pays p, his net payoff dominant-strategy property is the fact that a An auction is a game in which (i) buyers winning buyer’s payment does not depend make “bids” for the good (for now we will be on his bid. Next, we show that, under mild permissive about what a bid can be), on the hypotheses, the second-price auction is the basis of which (ii) the good is allocated to (at only efficient auction with this property most) one of the buyers, and (iii) buyers (modulo adding a term not depending on bi negative) to the seller. An auction is efficient Proposition 2 (Jerry Green and Jean-Jacques if, in equilibrium (we need not worry about the precise concept of equilibrium at this point), the winner is the buyer i with the Theorem 2.3): Suppose that, for all i, v can assume any value in [0, 1]. Then an auction is efficient and bidding truthfully is weakly dom- attained by a second-price auction: an auc- inant if and only if (a) the high bidder wins and (b) for all i, buyer i’s payment p satisfies numbers as bids, the winner is the high bid- der (ties can be broken randomly), and the winner pays the second-highest bid (nobody for some function t , where b is the vector Proposition 1 (Vickrey 1961; Theorem 2.1 in Milgrom 2004): In a second-price auction, it Proof: Consider an efficient auction in which is (weakly) dominant for each buyer i to bid truthful bidding is dominant. Then, the high his valuation v (i.e., regardless of how other bidder must win (property (a)). As for (b), let buyers bid, it is optimal for buyer i to set a tL(b ,b ) be buyer i’s payment if he loses and bid of b ϭv ). Furthermore, the auction is Proof: Suppose that buyer i bids b Ͻv . The t (bЈ,b )Ͼt (bЉ,b ) only circumstance in which the outcome fori is changed by his bidding b rather than v is for bids b′,b′ ≤ max b when the highest bid b by other bidders sat- off bidding bЉwhen v ϭbЈ, contradicting the isfies v Ͼ bϾ b . In that event, buyer i loses dominant-strategy property. Hence, we can by bidding b (for which his net payoff is 0) but wins by bidding v (for which his net pay- off is v Ϫb). Thus, he is worse off bidding t (b ,b- )ϭt (b- ).
b Ͻv . By symmetric argument, he can only Similarly, we can write buyer i’s payment be worse off bidding b Ͼv . We conclude tW(b ,b ) if he wins as that bidding his valuation (truthful bidding) is weakly dominant. Because it is optimal for tW(b ,b )ϭˆt (b ).
j , buyer i’s winning or los- Efficiency is often an important criterion in auction design, particularly in the case of privatization. Indeed, the ing are both efficient, and so for truthful U.S. Congress directed the FCC to choose an auction bidding to be dominant, buyer i must be design for allocating spectrum licenses that (to quote AlGore) puts “licenses into the hands of those who value indifferent between them. From (1) and (2), them the most” (see Milgrom 2004, p. 4).
dec04_Article 5 12/14/04 2:27 PM Page 1106 Journal of Economic Literature, Vol. XLII (December 2004) max b − tˆ (b ) = −t (b ) with c.d.f. F and support [0, 1]. Then there exists an efficient and payment-balanced auc-tion in which bidding truthfully constitutes a ˆt (b ) = max b + t (b ) Proof: For convenience, assume nϭ2. In an i.e., (b) holds. Conversely, if (a) and (b) hold, it is immediate that the auction is efficient and, from Proposition 1, that truthful bidding 1, buyer 1 will choose b1 to maximize Call the auctions of Proposition 2 “gener- 1 v dF (x) − P (b ) alized Vickrey” auctions. It is easy to see that there is no generalized Vickrey auction in if buyer 2 bids truthfully. The first-order which payments “balance,” i.e., sum to zero.
Proposition 3 (Green and Laffont 1977; b F′(b ) = P′(b ) Laffont and Maskin 1980; Milgrom’s Theorem2.2): Under the hypotheses of Proposition 2 there exists no generalized Vickrey auction in buyer 1’s best reply to 2 is to bid truthfully(because the first-order condition holds at Proof: For convenience assume nϭ2.
Consider a generalized Vickrey auction.
2 (v1)Ն0). Similarly, truthfulness is a best reply for buyer 2 if his payment function is p2 v2 t1(v2)ϩt2(v1).
If the right-hand side of (3) equals zero for p (b ,b ) = P (b ) − P (b ) p (b ,P ) = P (b ) − P (b ) 1 is a constant. Similarly, for v2 Then it is evident that the players’ paymentssum to zero and that truthfulness remains an equilibrium (the latter follows because sub- for constant k2. From (4) and (5), we can tracting P2(b2) from buyer 1’s payment does not affect his incentives and similarly forbuyer 2). which clearly cannot equal zero for all v1.
tent with efficiency once we relax the solu- Hence, balanced payments are impossible. individual rationality if one of the players already owns the good. More specifically, Gérard-Varet (1979) show, the failure of bal- suppose that player 1 owns the good and that nϭ2. An efficient mechanism will transfer relaxing the solution concept from dominant- the good to player 2 if and only if v Ͼ Thus, in a payment-balanced and efficient Proposition 4 (d’Aspremont and Gérard- mechanism, individual rationality for player Varet 1979): Suppose that, for all i, v is 1 (the “seller”) is the requirement that dec04_Article 5 12/14/04 2:27 PM Page 1107 Maskin: The Unity of Auction Theory ∫ (b (v ),b (v ))dF (v )− whereas individual rationality for player 2 where k1 and k2 are constants of integration.
v dF (v ) − p (b (v ),b (v ))dF (v ) 1(v1) and b2(v2) are the (Bayesian) their valuations are v1 and v2 respectively.
Proposition 5 (Laffont and Maskin 1979; 1983; Milgrom’s Theorem 3.6 ): Let nϭ2.
there exists no efficient and payment-bal- equilibrium is the solution concept.
Proof: The proof is considerably simplified Propositions 1 and 4 on the one hand (which by supposing that F1 and F2 are uniform dis- exhibit efficient auctions) and Proposition 5 tributions on [0, 1]. Consider a balanced- on the other (which denies the existence of payment and efficient mechanism for which such a mechanism). The reason for the dif- ference lies in the issue of ownership. In the former two propositions, no player yet owns ∫ (b (v ),b (v ))dv the good. We can interpret the latter propo- sition, however, as applying to the circum- stance in which there has already been an ∫ (b (v ),b (v ))dv auction that player 1 won—so that he now where p (b (v ),b (v )) has the opportunity to resell. Together, these two sets of propositions validate Milgrom’s um payment when valuations are (v1,v2).
refutation of the claim that auctions are Hence, in equilibrium, player 1’s and player unnecessary for efficiency, that ex post free trade among the players will ensure the right allocation. According to this claim we might just as well assign assets like spectrum licens- them later to correct misallocations. But licenses have been distributed, efficiency v − Pˆ′(v ) = 0 dec04_Article 5 12/14/04 2:27 PM Page 1108 Journal of Economic Literature, Vol. XLII (December 2004) very different auction forms. Milgrom pres- Clearly, G (v ) must be lie between 0 and ents his view of this material in chapters 3 1, but there are other restrictions on it as and 4. The central result is what he calls the well. In particular, it must be nondecreasing.
payoff-equivalence theorem (which implies Proposition 7 (Myerson 1981, Riley and the considerably weaker but more familiar Samuelson 1981, Milgrom’s Theorem 4.1): In any Bayesian equilibrium a buyer’s proba- Proposition 6 (Vickrey 1961; Myerson 1981; bility of winning is a nondecreasing function Proof: From (12) and (13), the derivative of hypotheses of Proposition 4, if there are two buyer i’s equilibrium expected payoff if his val- auctions such that, in Bayesian equilibrium, uation is v but he bids as though it were ˆv is (a) for all i and v , the probability of winning for a buyer i with valuation v is the same in GЈ(ˆv )(v Ϫ ˆv ).
both auctions, and (b) for all i, the amount But if GЈ(v )Ͻ0 for some v then from that buyer i with valuation 0 pays is the same in both auctions, then, for all i and v , the maximum (GЈ(v )Ն0) is violated at ˆv ϭv , a equilibrium expected payoff for buyer i with valuation v is the same in both auctions.
Proof: Choose one of the two auctions andlet (b 1(v1),…,b (v )) be Bayesian equilibri- um bids by the buyers when valuations are(v in equilibrium of the second-price auction 1,…,v ). Because buyer i has the option of behaving as though his valuation is ˆv when in fact it is v , he, in effect, faces the truthfully, and so a buyer’s probability of winning is simply the probability that theother buyers all have lower valuations (the where G ( ˆv )ϭbuyer i’s probability of win- ning and P ( ˆv )ϭbuyer i’s expected payment if he bids b ( ˆv ) and each of the other buyers in that auction. But (16) and (17) also hold for j bids according to the equilibrium bid equilibrium of the English auction, the mech- function b (·). By definition of equilibrium, anism in which buyers call out bids openly, the maximizing choice of ˆv in (12) is ˆv ϭv , each successive bid must be higher than the previous one, and the winner is the last buyer P′(v ) = G′(v )v to bid (and pays his bid). To see this, notice that a buyer will continue to bid higher in the English auction until the current price reach- P (v ) = v G (v ) − es his valuation, and so the high-valuation buyer will win, i.e., (16) holds. Thus we have: where k is a constant of integration. Notice Proposition 8 (Vickrey 1961): The second- from (14) that buyer i’s expected payment if v ϭ0 is k . By hypothesis (b), this is true of the other auction as well. Furthermore, by hypothesis (a), i’s probability of winning in the other auction is G (v ) for all v . Hence, “standard” auctions are equivalent.
from (14), buyer i’s expected payment is Proposition 9 (Vickrey 1961; Riley and P (v ) and his equilibrium expected payoff is Samuelson 1981; Myerson 1981; Milgrom’s G (v )v ϪP (v ) in both auctions.
Theorems 4.6 and 4.9): When each v is dec04_Article 5 12/14/04 2:27 PM Page 1109 Maskin: The Unity of Auction Theory with c.d.f. F and support [0, 1], then the Proposition 9 can be modified to maximize high-bid, second-price, English, Dutch, and all-pay auctions are payoff-equivalent.
Proposition 10 (Riley and Samuelson 1981; Proof: We have already described the rules of all but the Dutch and all-pay auctions. In Assume the hypotheses of Proposition 9 and the Dutch auction, the auctioneer continu- ously lowers the price, starting from some JЈ(v)Ͼ0 for all v, high level, until some buyer (the winner)agrees to buy at the current price. Notice that this is formally the same as the high-bid where J(v) = v − − auction, since the price at which a buyer auctions of Proposition 9 maximizes the sell- agrees to buy in the Dutch auction is the er’s expected revenue provided that the sell- same as the bid he would make in the high- er sets a reserve price v∗ (i.e., he refuses to bid auction.13 In the all-pay auction, buyers sell for less than v∗), where J(v∗)ϭ0.
Proof: Given buyers’ ex ante symmetry, (14) high bidder, but all buyers pay their bids.
implies that, for any symmetric auction (we Consider a symmetric equilibrium b(·) of restrict to symmetric auctions without loss of the high-bid auction; i.e., b(v) is the bid any generality), the seller’s expected revenue is buyer with valuation v makes (a symmetric equilibrium exists because of the ex ante vG(v) − G(x)dx + k dF(v), symmetry of the buyers). From Proposition 7, b(·) must be nondecreasing. Suppose it is not strictly increasing, i.e., suppose b(v)ϭb∗ for all vʦ[v∗,v∗∗]. We have v∗Ϫb∗Ն0 ) (v)dF(v)+n .
because, thanks to the atom at b∗, a bid of b∗ We have already noted that G(v) must satisfy wins with positive probability (and thus if v∗Ϫb∗Ͻ0, the buyer would have a negativepayoff). Hence we obtain As Steven Matthews (1984) shows, it mustalso satisfy But if a buyer with reservation price v∗∗ bids b∗, he ties for the high bid with positive prob- − G(x) dF(x) 0 for all v. (22) ability. Thus if he slightly increases his bid, he discontinuously raises his chances of win- subject to (21) and (22). Note from (12) and ning (since a tie then has zero probability), (14) that kՅ0, otherwise a buyer with zero which is worthwhile in view of (18). We con- valuation has a negative expected payoff (an clude that b(·) must be strictly increasing, impossibility, since he always has the option which means that the high-valuation buyer of not participating). Hence the maximizing always wins. Thus, Proposition 6 implies that the high-bid auction is equivalent to thesecond-price auction. This same argument P(0)ϭpayment by a 0-valuatioin buyerϭ0.
applies also to the all-pay auction.
From (19)–(22), the optimal choice of G(v) is This equivalence relies on the assumption that buy-  F(v) , if v v∗ ers obey the usual axioms of expected utility; see Daisuke dec04_Article 5 12/14/04 2:27 PM Page 1110 Journal of Economic Literature, Vol. XLII (December 2004) where J(v∗)ϭ0. But all of the auctions of 4.12): Assume that buyers are risk averse; Proposition 9, modified by a reserve price of i.e., buyer i’s utility from winning is u (v –p ), v∗, satisfy (23) and (24), and so they are solu- where u is strictly concave. Suppose that buyers are ex ante symmetric, i.e., the v ’s are The fact that the Dutch and high-bid auc- drawn (independently) from the same distri- bution with c.d.f. F and support [0, 1] and forms. Nor is the equivalence (Proposition generates higher expected revenue than the 8) between the English auction and the sec- ond-price auction very deep. But the sense Proof: First, observe that risk aversion does in which all four auctions are equivalent not affect behavior in the second-price auc- (Proposition 9) is more interesting, as is the tion; it is still optimal to bid truthfully. If b(·) idea that any of them—modified by setting a is a symmetric-equilibrium bid function in reserve price15—can be used to maximize the high-bid auction, a buyer with valuation the seller’s revenue (Proposition 10).
5. Departures from the Benchmark Model on some restrictive hypotheses, viz., (i) buy- −F u′b′ + (n − 1 F ers’ risk neutrality, (ii) private values (to be defined below), (iii) independent valuations, ) ′(v)u(v − b(v )) (iv) ex ante symmetry, and (v) financially unconstrained buyers.16 We will now relax F(v)u′(v − b(v )) each of (i)–(iii) in turn (for relaxation of (iv), Now, if buyers were risk neutral, (25) would see Milgrom’s Theorems 4.24–4.27, and Maskin and Riley 2000; for relaxation of (v), ) ′(v)(v − b (v) see Milgrom’s Theorem 4.17, and Che and Note first that in Propositions 6–10, we suppose that buyer i’s objective function is second-price auctions are payoff-equivalent given by (12), i.e., that he is risk neutral. If we replace risk neutrality with risk aversion, symmetric. Hence, b (v) is also the expect- then in particular Proposition 9 no longer ed payment by a winning v-valuation buyer in the second-price auction (whether he is Proposition 11 (Holt 1980; Maskin and Riley risk averse or not). Because uЉϽ0, u(v − b(v )) v b(v), We have ignored the constraint GЈՆ0 because it is sat- isfied by the solution to the program in which it is omitted.
15 The reason a reserve price helps the seller is that it puts a lower bound on what buyers pay. Of course, by set- b′(v) > b′ (v) whenever b(v) = b (v) ting a positive reserve, the seller runs the risk of not sellingat all, but this effect is outweighed by the lower-bound Because b(0)ϭb (0)ϭ0, we conclude that consideration. To see this, imagine that there is just one buyer. Then in a high-bid auction, that buyer would bid b(v) > b (v) zero; the only way to get him to pay anything is to makethe reserve positive.
which implies that, for every vϾ0, a buyer 16 The efficiency of the second-price auction (Proposition 1) invokes (ii), as we will see in Proposition 12, and (v) (see pays more in the high-bid than in the second- Maskin 2000), although it does not demand (i), (iii), or (iv).
dec04_Article 5 12/14/04 2:27 PM Page 1111 Maskin: The Unity of Auction Theory 2 and 3 is that buyers actually know their valuations (more to the point, that their val- j≠i ∂si1 ≠ ∂si1 uations do not depend on the private infor- mation of other buyers). This is called the private values assumption. Let us now relax Then, if Bayesian equilibrium is the solution it to accommodate interdependent values concept, there exists no efficient auction.
(sometimes called generalized common val- Proof (sketch): Choose parameter values sЈ ues). Specifically, suppose that each buyer i and sЉ such that ␸ (sЈ)ϭ␸ (sЉ). From (26), receives a private signal s and that his valu- buyer i’s preferences are identical for s ϭsЈ ation is a function of all the signals: i.e., his and s ϭsЉ and so, in equilibrium, he must be valuation is v (s , s ), where s is the vector of indifferent between the outcomes that result other buyers’ signals. In such a setting, the from the two cases. But, from (27), which of second-price auction will no longer be effi- s ϭsЈ or s ϭsЉholds will in general lead to dif- ferent efficient allocations—e.g., perhaps because buyers no longer know their valua- buyer i wins when s ϭsЈ and loses when tions, their bids (reflecting their expected s ϭsЉ. Thus buyer i cannot be indifferent valuations) do not guarantee that the high bidder actually has the highest valuation.
Despite this negative result, matters are Nevertheless, if the signals are one-dimen- not as bleak as they may seem, at least in the case of single-good auctions. Specifically, introduce a one-dimensional “reduced” signal r for buyer i and, for all j≠i, define nisms that ensure efficiency (provided that “single crossing” holds, i.e., buyer i’s signal ( ,s ) = E v (s ,s ) ϕ (s ) = r has a greater marginal effect on his own val- i.e., ˆv (r ,s ) is v (s ,s ) expected over all those uation than on other buyers’ valuations: values s such that ␸ (s )ϭr . Because we are j for j i). More concretely, Partha back to one-dimensional signals, the extend- “reduced” valuations v (r ,s ) and { ˆv } (assuming that the single-crossing property above holds). That is, the auction is efficient accommodate contingent bids so that effi- subject to the constraint that buyer i behaves ciency is restored (in a more limited class of the same way for any signal values s for which ␸ (s )ϭr , i.e., it is incentive efficient that the English auction is efficient with (see Dasgupta and Maskin 2000). Jehiel and Moldovanu (2001) show, however, that this signals are multidimensional, then efficien- “reduction” technique does not generalize to Proposition 12 (Maskin 1992; Philippe Jehiel Finally, let us explore what happens when we drop the assumption of independence in Theorem 3.8): Suppose that, for some buyer v (s ,s ) = ϕ (s ) + ψ (s ) that n ϭ 2,18 and that v1 and v2 are jointly 17 For convenience, suppose that all the other signals 18 Milgrom and Weber (1982) generalize this result to nՆ3.
dec04_Article 5 12/14/04 2:27 PM Page 1112 Journal of Economic Literature, Vol. XLII (December 2004) symmetrically distributed with support [0, 1] and affiliated (positively correlated) in the v1 v dF(v | v )F2(v | v ) But (34) follows from affiliation, i.e., from where F(v2|v1) is the c.d.f. for v2 conditional on v1. Then, revenue from the second-price Notice that Propositions 11 and 13 pull in opposite directions: the former favors the high-bid auction, the latter the second-price Proof: Let P1(v1) be buyer 1’s equilibrium expected payment when his valuation is v1.
Milgrom’s introductory points: that the kind of auction a seller will want to use depends PH(v ) = F(v | v ) b(v ), Milgrom’s observations about auctions in where the superscripts S and H denote sec- practice are a good deal less compelling than ond-price and high-bid auctions, respective- the book’s theoretical results. But this con- ly, and b(·) is the symmetric equilibrium bid trast is not primarily his fault. In spite of all function in the high-bid auction. Clearly, that it has accomplished, auction theory still 1 (0) ϭP1 (0). We wish to show that has not developed far enough to be directly 1 (v1)ϾP1 (v1) for all v1 applicable to situations as complex as, say, dPS (v ) dPH (v ) 1 (v1)ϭP1 (v1).
To begin with, those auctions involve mul- tiple goods. Observe, however, that all the results presented above are for single-good auctions. This is no coincidence; the litera- ture on auction theory has overwhelmingly focused on the single-good case. Apart fromthe efficiency of the multigood second-price ′(v ) + b(v )F1 auction (the Vickrey-Clarke-Groves mecha- nism; see section 7) with private values, + b(v )F2(v | v ), there are very few general results for more where superscripts of F denote partial deri- attempt to extrapolate from well-analyzed vations. The second equation can be rewrit- ten, using buyer 1’s first-order condition, as stances (multiple goods). But doing so is Another difficulty for theory is that real auctions impose constraints that are difficult to formalize. Milgrom notes that prospective bidders and sellers are typically nervous about participating in auction mechanisms From (31)–(33), it remains to show that dec04_Article 5 12/14/04 2:27 PM Page 1113 Maskin: The Unity of Auction Theory giving a precise meaning to “unfamiliar” or “complicated” is forbiddingly difficult.
lowing sort of problem. Assume that there The upshot is that giving advice on real auc- tion design is, at this stage, far less a science than an art. And the essence of an art is far wants these goods only as a package, i.e., his harder than a science to convey convincingly valuation for A or B alone is zero. Suppose that buyer 1 has a valuation of $100 for Aand B together, but that buyer 2 has a pack- 7. The Vickrey-Clarke-Groves Mechanism age valuation of $200. If the buyers bidtruthfully in the VCG mechanism, then buyer 2 will win both A and B (and pay $100, the winning bid were 2 not present). Buyer (VCG) mechanism, the generalization of the second-price auction to multiple goods.
Indeed, his unhappiness with it has led him bids through two different proxy buyers, 1x and 1y. As buyer 1x he enters a bid of $201 (Ausubel and Milgrom 2002) on an interest- package AB). As 1y, he enters a bid of $201 for good B (and zero for both A and AB).
Then 1x and 1y will be the winners of A and (i) each buyer makes a bid not just for each B respectively, and so 1 will obtain both good but for each combination (or “pack- goods. Furthermore, notice that had 1x not age”) of goods; (ii) goods are allocated to bid at all, 1y would still be the winner of buyers in the way that maximizes the sum of good B (good A would just be thrown away), the winning bids (a bid for a package is “win- and so the sum of the other buyers’ winning ning” if the buyer making that bid is allocat- bids is the same (namely, $201) whether 1x ed the package); (iii) each winning buyer i participates or not. Thus, by VCG rules, 1x pays nothing at all (and, similarly, neither between (a) the sum of the bids that would does 1y), which means that the ploy of pass- win if i were not a participant in the auction ing himself off as multiple buyers is worth- and (b) the sum of the other buyers’ (actual) winning bids. Following the line of argu- revenue for the seller and leads to an ineffi- ment in the proof of Proposition 1, one can cient allocation (1 wins the goods, rather show that truthful bidding (reporting one’s than 2), which is why Milgrom is led to reject true valuation for each package) is dominant.
Thus the auction results in an efficient allo- But notice that having 1x and 1y enter cation (an allocation that maximizes the sum these bids makes sense for 1 only if he is quite sure that buyer 2 does not value A and B as single goods. As soon as there is a seri- ous risk that 2 will make single-good bids both buyers (placing bids on each package that add up to $101 or more, buyer 1 will can be an onerous task) and the auctioneer come out behind with this strategy (relative to truthful bidding). If, for example, buyer 2 potentially difficult maximization problem).
bid $51 for each of A and B alone (as well as This, however, is not the shortcoming that $200 for the package), 1x and 1y would still be awarded A and B with their $201 bids but Milgrom paper is subject to the same sort of dec04_Article 5 12/14/04 2:27 PM Page 1114 Journal of Economic Literature, Vol. XLII (December 2004) Indeed, with sufficient uncertainty about Restricted Domains,” Econometrica 47, pp. 1137–44.
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