sign in sign up
<<
Home , Double auction, Eric Maskin, Paul Milgrom, Reinhard Selten, Robert Aumann, Roger Myerson

dec04_Article 5 12/14/04 2:27 PM Page 1102

Journal of Economic Literature Vol. XVII (December 2004) pp. 1102–1115

The Unity of Theory: Milgrom’s Masterclass1

2

1. Introduction One explanation for this success is good timing. Many researchers started working y any standard measure, seriously on in the late 1970s and has been an enormous success. Even B 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- games of incomplete information (John erating rate;3 it has spawned much empirical Harsanyi 1967–68) and perfect equilibrium and experimental research;4 its tentacles ( 1975)—were becoming have spread into other disciplines; 5 and auc- widely known. Of course, numerous other 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 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 of game theory. Whereas enthusiasm for theo- 1 , Putting Auction Theory to Work. retical industrial organization has cooled Cambridge, Churchill Lectures in : Cambridge somewhat since the heady days of twenty University Press, 2004. NY, Melbourne, Pp. xxi, 368. ISBN years ago, research on auctions, as I have 0-521-55184-6. 2 Institute for Advanced Study and Princeton noted, continues apace. There are, I University. I thank the NSF (SES-0318103) for research believe, several reasons why auction theory support. has fared comparatively well. 3 To give just one indication: at the August 2004 joint meeting of the 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 not know the games that the players they sort of theory. 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 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), putational issues. and Robert Wilson (1969)—remained largely ignored.

1102 dec04_Article 5 12/14/04 2:27 PM Page 1103

Maskin: The Unity of Auction Theory 1103

study (e.g., firms or consumers) are actually most auction theory, numbers are small 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- into their own). bers as sealed bids; (ii) the winner is the bid- Finally, auction theory is a genuinely 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 Despite all these attractions, auctions helps put the auction theorist’s findings on a might have disappeared from the economic relatively strong footing; it also simplifies the theory scene had they not received an job of the experimentalist or empiricist. important rejuvenating boost from the Second, auction theory appeals to econo- worldwide impulse toward privatization that mists’ “social engineering” instincts. Many began in the early 1990s. This trend was 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 public ownership in the West. But state order to achieve a better —helps bureaucracy’s loss proved to be auction the- gratify that urge. No doubt, one reason 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 , provides implement the transfer of resources. More an attractive solution to an important social recently, enterprises such as problem: designing an efficient allocation eBay have provided further impetus for the mechanism. theory. Third, the basic transaction of an auc- tion—the transfer of a good from seller to 2. Milgrom’s Unified View buyer in exchange for a monetary pay- 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 Milgrom and Robert Weber 1982), but a foundation for understanding the workings 8 together with Robert Wilson, he had a major of competitive markets. Of course, there hand in designing the simultaneous ascend- are important differences: competitive theo- ing auction format used by the Federal ry usually supposes that there are large Communication Commission to sell off numbers of buyers and sellers, whereas in much of the radio spectrum in the . Thus, his book Putting Auction 7 Of course, auction theory is only a small part of a vast / literature. For Theory to Work has been eagerly awaited recent surveys of the literature from a general perspective see Thomas Palfrey (2002); Roberto Serrano (2004); and 9 , 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 (1977). things of real beauty. dec04_Article 5 12/14/04 2:27 PM Page 1104

1104 Journal of Economic Literature, Vol. XLII (December 2004)

since his 1995 Churchill lectures, on which it But, as they sometimes depend as much on is based. judgment (albeit very well-informed judg- The wait has clearly been worth it. The 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 the Vickrey auction (more precisely, its derives from Milgrom’s conception of auc- multigood generalization due to Theodore tion theory as a subspecies of demand theo- Groves 1973, and Edward Clarke 1971) is 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- on the design of real auctions. ory matters in practice. In chapter one, he Admittedly, the monograph is not the only shows that the 1990 New Zealand spectrum 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 design: separate simultaneous sealed-bid likely candidate for a graduate text on the auctions for each license. Specifically, he subject.11 Rather, its signal contribution is to points out why this auction form cannot lay out Milgrom’s unified view of the theory. properly accommodate substitutability or This vision is notably distinct from that of complementarity across licenses. And he other major auction scholars. For example, responds to those who argue that how gov- he must be nearly alone in deliberately ernment assets are sold off is irrelevant for avoiding the as an auc- efficiency (because, in their view, the “mar- tion-theoretic technique. But we quickly 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- more fun. ciency, once the assets are in private hands As for his ideas on how to apply (or not to (see Proposition 5 below). 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 in detail. 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 of the 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 compendium of theorems and proofs, together with less-formal observa- tions about their application to actual auctions. Except for the appendices of chapters 1 and 2, Klemperer’s is almost 3. Vickrey Auctions wholly nontechnical and consists largely of his views on the design of large-scale auctions in practice (although these In chapter two, Milgrom turns to the most views are informed by theory). famous example of modern auction design, 11 Indeed, Vijay Krishna, yet another prominent auc- the Vickrey (or “second-price”) auction (and tion theorist, has recently produced a beautifully lucid treatment (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 1105

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 vi for the good (the most he is willing to Q.E.D. pay for it). Thus if he pays p, his net payoff The key to the second-price auction’s is vi p. dominant- 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 make payments (which can in principle be to buyer i’s payment): negative) to the seller. An auction is efficient Proposition 2 (Jerry Green and Jean-Jacques if, in equilibrium (we need not worry about Laffont 1977; Bengt Holmström 1979; the precise concept of equilibrium at this Laffont and Maskin 1980; Milgrom’s

point), the winner is the buyer i with the Theorem 2.3): Suppose that, for all i, vi can highest valuation.12 assume any value in [0, 1]. Then an auction is Vickrey discovered that efficiency is efficient and truthfully is weakly dom- attained by a second-price auction: an auc- inant if and only if (a) the high bidder wins

tion in which buyers submit nonnegative and (b) for all i, buyer i’s payment pi satisfies numbers as bids, the winner is the high bid-  + maxbtbjii()− , if buyer i wins der (ties can be broken randomly), and the p =  ji≠ i  winner pays the second-highest bid (nobody tbii()− , if buyer i loses else pays anything). Formally, we have: for some function t , where b is the vector Proposition 1 (Vickrey 1961; Theorem 2.1 in i -i of bids other than b . Milgrom 2004): In a second-price auction, it i 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 i bidder must win (property (a)). As for (b), let buyers bid, it is optimal for buyer i to set a L t (b ,b ) be buyer i’s payment if he loses and bid of b v ). Furthermore, the auction is i i -i i i the bids are (b ,b ). If efficient. i -i L L Proof: Suppose that buyer i bids bi vi. The ti (bi,b-i) ti (bi,b-i) only circumstance in which the outcome for ′′′≤ for bids bbii, max bj , then buyer i is better i is changed by his bidding bi rather than vi is j ≠i when the highest bid b by other bidders sat- off bidding bwhen v b, contradicting the i i i isfies vi b bi. In that event, buyer i loses dominant-strategy property. Hence, we can by bidding b (for which his net payoff is 0) L i write ti (bi,b-i)as but wins by bidding vi (for which his net pay- t L(b ,b )t (b ). (1) off is vi b). Thus, he is worse off bidding i i -i i -i bi vi. By symmetric argument, he can only Similarly, we can write buyer i’s payment be worse off bidding bi vi. We conclude W ti (bi,b-i) if he wins as that bidding his valuation (truthful bidding) W ˆ is weakly dominant. Because it is optimal for ti (bi,b-i) ti(b-i). (2) = Now, if vi max≠ bj , buyer i’s winning or los- 12 Efficiency is often an important criterion in auction j i 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 ) puts “licenses into the hands of those who value indifferent between them. From (1) and (2), them the most” (see Milgrom 2004, p. 4). we have dec04_Article 5 12/14/04 2:27 PM Page 1106

1106 Journal of Economic Literature, Vol. XLII (December 2004)

−=−ˆ drawn independently from a distribution maxbtbjii (−− ) tb ii(). ji≠ with c.d.f. Fi and support [0, 1]. Then there Hence, exists an efficient and payment-balanced auc- tion in which bidding truthfully constitutes a ˆ =+ tbii()max− bjii tb ()− , ji≠ Bayesian equilibrium. Proof: For convenience, assume n2. In an i.e., (b) holds. Conversely, if (a) and (b) hold, auction where the high bidder wins and it is immediate that the auction is efficient buyer 1’s expected payment is P (b ) if he and, from Proposition 1, that truthful bidding 1 1 bids b , buyer 1 will choose b to maximize is dominant. Q.E.D. 1 1 b1 − Call the auctions of Proposition 2 “gener- ∫ vdFx12() P 11 ( b ), alized Vickrey” auctions. It is easy to see that 0 there is no generalized Vickrey auction in if buyer 2 bids truthfully. The first-order which payments “balance,” i.e., sum to zero. condition for this maximization is Proposition 3 (Green and Laffont 1977; bF′() b= P′ () b . Laffont and Maskin 1980; Milgrom’s Theorem 12 1 1 1 2.2): Under the hypotheses of Proposition 2 Thus if we set there exists no generalized Vickrey auction in = b1 ′ n Pb11()∫ xFxdx 2 () , ≡ 0 which the payments balance, i.e.,∑ pi 0 . i=1 buyer 1’s best reply to 2 is to bid truthfully (because the first-order condition holds at Proof: For convenience assume n2. b v , and so does the second-order condi- Consider a generalized Vickrey auction. 1 1 tion: F(v )0). Similarly, truthfulness is a best Choose v v . From Proposition 2 2 1 1 2 reply for buyer 2 if his payment function is b = 2 ′ p1 p2 v2 t1(v2) t2(v1). (3) Pb22()∫ xFxdx 1 () . 0 If the right-hand side of (3) equals zero for Now take as payment functions all v2, then pbb()()(), =− Pb Pb . 112 11 22 t1(v2) v2 k1, (4) pbP()()(), =− Pb Pb 212 22 11. where k1 is a constant. Similarly, for v2 v1, we obtain Then it is evident that the players’ payments sum to zero and that truthfulness remains an t (v )v k , (5) 2 1 1 2 equilibrium (the latter follows because sub-

for constant k2. From (4) and (5), we can tracting P2(b2) from buyer 1’s payment does rewrite (3) as not affect his incentives and similarly for buyer 2). Q.E.D. p p k k v , 1 2 1 2 1 Although balanced payments are consis-

which clearly cannot equal zero for all v1. tent with efficiency once we relax the solu- Hence, balanced payments are impossible. tion concept, we cannot also require Q.E.D. individual rationality if one of the players As Claude d’Aspremont and Louis-André already owns the good. More specifically, Gérard-Varet (1979) show, the failure of bal- suppose that player 1 owns the good and that ance in Proposition 3 can be overcome by n2. An efficient mechanism will transfer relaxing the from dominant- the good to player 2 if and only if v2 v1. strategy to Bayesian equilibrium: 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, vi 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 1107

1 − 1 We conclude that ∫∫pbv211()(), bv 22 () dFv 22() vdFv122 () 2 0 vv= 21 ˆ =−v1 + Pv11() k1 (8) ≥ 2 0 for all v1 (since p2 p1), (6) 2 ˆ =+v2 whereas individual rationality for player 2 Pv22() k2 , (9) (the “buyer”) is the condition 2 where k1 and k2 are constants of integration. v2 1 − () From (6) and (8) when v1 1, we obtain ∫∫= vdFv211() p211 b ()vbvdFv, 22 () 11 () v1 0 0 ≥ 1 ≥ k1 (10) 0 for all v2 , (7) 2 and from (7) and (9) when v2 0, we have where b1(v1) and b2(v2) are the (Bayesian) ≤ equilibrium bids by players 1 and 2 when k2 0 . (11) their valuations are v and v respectively. 1 2 By definition of Pˆ and Pˆ , Proposition 5 (Laffont and Maskin 1979; 1 2 1 ˆ = 1 ˆ and Mark Satterthwaite ∫∫Pvdv11() 1 Pvdv22() 2. 1983; Milgrom’s Theorem 3.6 ): Let n2. 0 0 Under the hypotheses of Proposition 4, Hence, from (8) and (9), we obtain there exists no efficient and payment-bal- 1 1 kk−= +, anced mechanism that is individually 126 6 rational for both players when Bayesian which contradicts (10) and (11). Q.E.D. equilibrium is the solution concept. Notice the striking contrast between 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- ()⋅⋅ bb12(), () is a Bayesian equilibrium. Let ference lies in the issue of ownership. In the ˆ = 1 former two propositions, no player yet owns Pv11()∫ p 2() bv12 ()12, bv () dv 2 0 the good. We can interpret the latter propo- and sition, however, as applying to the circum- ˆ = 1 stance in which there has already been an Pv22()∫ p 2() bv12 ()12, bv () dv 1, 0 auction that player 1 won—so that he now has the opportunity to resell. Together, these where pbv21()()12, bv 2 () is buyer 2’s equilibri- two sets of propositions validate Milgrom’s um payment when valuations are (v ,v ). 1 2 refutation of the claim that auctions are Hence, in equilibrium, player 1’s and player unnecessary for efficiency, that ex post free 2’s maximization problems are trade among the players will ensure the right  ˆ − 1  maxPv11 (ˆ ) ∫ vdv 1 2 allocation. According to this claim we might vˆ  vˆ  1 1 just as well assign assets like spectrum licens- and es randomly; firms can always exchange  vˆ2 − ˆ  them later to correct misallocations. But max∫ vdv21 P 22 ( vˆ ) . vˆ2  0  Proposition 5 demonstrates that once the licenses have been distributed, efficiency In Bayesian equilibrium, vˆ 1 v1 and vˆ 2 v2, and so we obtain first-order conditions may no longer be attainable. Q.E.D. ˆ ′ += Pv11() v 1 0 4. Auction Equivalences and A major achievement of auction theory is − ˆ ′ = vPv222() 0 . to have established equivalences between dec04_Article 5 12/14/04 2:27 PM Page 1108

1108 Journal of Economic Literature, Vol. XLII (December 2004)

very different auction forms. Milgrom pres- Clearly, Gi(vi) 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): revenue-equivalence theorem): In any Bayesian equilibrium a buyer’s proba- Proposition 6 (Vickrey 1961; Myerson 1981; bility of winning is a nondecreasing function John Riley and William Samuelson 1981; of his valuation. Milgrom’s Theorem 3.3): Under the 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 vi but he bids as though it werev ˆ i is (a) for all i and vi, the probability of winning G (ˆvi)(vi vˆ i). (15) for a buyer i with valuation vi is the same in both auctions, and (b) for all i, the amount But if G (vi) 0 for some vi then from that buyer i with valuation 0 pays is the same (15) the second-order condition for a in both auctions, then, for all i and vi, the maximum (G (vi) 0) is violated at vˆ i vi , a equilibrium expected payoff for buyer i with contradiction. Q.E.D. valuation vi is the same in both auctions. Notice that Proof: Choose one of the two auctions and =× let (b1(v1),…,bn(vn)) be Bayesian equilibri- Gvii() Fvji () (16) ji≠ um bids by the buyers when valuations are in equilibrium of the second-price auction (v1,…,vn). Because buyer i has the option of behaving as though his valuation isv ˆ because, from Proposition 1, buyers bid i truthfully, and so a buyer’s probability of when in fact it is vi, he, in effect, faces the maximization problem winning is simply the probability that the other buyers all have lower valuations (the []− max Gvviii()ˆˆ Pv ii (), (12) right-hand side of (16)). Furthermore, vˆi where Gi(ˆvi) buyer i’s probability of win- P ()00= i (17) ning and Pi(ˆvi) buyer i’s expected payment if he bids bi(ˆvi) 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 , the mech-

function bj(·). By definition of equilibrium, anism in which buyers call out bids openly, the maximizing choice of vˆ i in (12) is vˆ i vi, each successive bid must be higher than the and so we obtain first-order condition previous one, and the winner is the last buyer to bid (and pays his bid). To see this, notice Pv′()= Gvv′ () for all i. (13) ii iii that a buyer will continue to bid higher in the Integrating (13), we have English auction until the current price reach- =−vi + es his valuation, and so the high-valuation Pvii() vGv iii ()∫ Gxdxk i () i, (14) 0 buyer will win, i.e., (16) holds. Thus we have:

where ki is a constant of integration. Notice Proposition 8 (Vickrey 1961): The second- from (14) that buyer i’s expected payment if price and English auctions are payoff- vi 0 is ki. By hypothesis (b), this is true of equivalent. the other auction as well. Furthermore, by Remarkably, in the case of ex ante buyer hypothesis (a), i’s probability of winning in symmetry, i.e., where F1 … Fn, all the the other auction is Gi(vi) for all vi. Hence, “standard” auctions are equivalent. from (14), buyer i’s expected payment is Proposition 9 (Vickrey 1961; Riley and

Pi(vi) and his equilibrium expected payoff is Samuelson 1981; Myerson 1981; Milgrom’s Gi(vi)vi Pi(vi) in both auctions. Q.E.D. Theorems 4.6 and 4.9): When each vi is dec04_Article 5 12/14/04 2:27 PM Page 1109

Maskin: The Unity of Auction Theory 1109

drawn independently from a distribution We next examine how the auctions of 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 the seller’s revenue: all-pay auctions are payoff-equivalent. Proposition 10 (Riley and Samuelson 1981; Proof: We have already described the rules Myerson 1981; Milgrom’s Theorem 3.9): of all but the Dutch and all-pay auctions. In Assume the hypotheses of Proposition 9 and the , the auctioneer continu- suppose that ously lowers the price, starting from some J(v)0 for all v, (19) high level, until some buyer (the winner) agrees to buy at the current price. Notice 1 − Fv() where Jv()=− v . Then any of the that this is formally the same as the high-bid Fv′() 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 )0. submit sealed bids and the winner is the Proof: Given buyers’ ex ante symmetry, (14) high bidder, but all buyers pay their bids. implies that, for any symmetric auction (we Consider a 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 1  v  equilibrium exists because of the ex ante nvGvGxdxkdFv∫  ()−+∫ () (), symmetry of the buyers). From Proposition 0  0  7, b(·) must be nondecreasing. Suppose it is which can be rewritten as ∗ not strictly increasing, i.e., suppose b(v)b 1 ∗ ∗∗ ∗ ∗ for all v[v ,v ]. We have v b 0 nJvGvdFvnk∫ () () ()+ . (20) ∗ ∗ 0 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 ∗ ∗ G(v)0 and v b 0, the buyer would have a negative payoff). Hence we obtain 0G(v)1. (21) ∗∗ ∗ v b 0. (18) As Steven Matthews (1984) shows, it must also satisfy ∗∗ But if a buyer with reservation price v bids 1  n−1  ∗ nFxGxdFx∫ ()()− () ()≥ 0for all v. (22) b , he ties for the high bid with positive prob- v   ability. Thus if he slightly increases his bid, Consider the problem of maximizing (20) 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 k0, 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 choice of k is 0, i.e., that the high-bid auction is equivalent to the second-price auction. This same argument P(0)payment by a 0-valuatioin buyer0. applies also to the all-pay auction. Q.E.D. (23) From (19)–(22), the optimal choice of G(v) is  n−1 ∗ 13 This equivalence relies on the assumption that buy- () > =  Fv() , ifv v ers obey the usual axioms of expected utility; see Daisuke Gv() ∗ (24) Nakajima (2004).  0 ,,if v≤ v dec04_Article 5 12/14/04 2:27 PM Page 1110

1110 Journal of Economic Literature, Vol. XLII (December 2004)

∗ where J(v )0. 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 ui(vi–pi), ∗ v , satisfy (23) and (24), and so they are solu- where ui is strictly concave. Suppose that 14 tions to the seller’s problem. Q.E.D. buyers are ex ante symmetric, i.e., the vi’s are The fact that the Dutch and high-bid auc- drawn (independently) from the same distri- tions are equivalent is obvious from the bution with c.d.f. F and support [0, 1] and identical strategic structure of the two u1 … un u. Then the high-bid auction forms. Nor is the equivalence (Proposition generates higher expected revenue than the 8) between the English auction and the sec- second-price auction. 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). v solves n−1 max()Fv (ˆˆ ) uv()− bv(). 5. Departures from the Benchmark Model vˆ The first-order condition is therefore This deeper equivalence, however, relies − nn−−12′′+− ′ = on some restrictive hypotheses, viz., (i) buy- Fubn()10 FFu , ers’ risk neutrality, (ii) private values (to be and so defined below), (iii) independent valuations, ()()nFvuvbv− 1 ′ ()− () (iv) ex ante symmetry, and (v) financially bv′()= . (25) unconstrained buyers.16 We will now relax Fvu()′() v− bv () 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 become Maskin and Riley 2000; for relaxation of (v), ()()nFvvbv− 1 ′ ()− () see Milgrom’s Theorem 4.17, and Che and ′ = RN bvRN() . Gale 1998). Fv() Note first that in Propositions 6–10, we But from Proposition 9, the high-bid and 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 when buyers are risk neutral and ex ante we replace risk neutrality with risk aversion, symmetric. Hence, bRN(v) is also the expect- then in particular Proposition 9 no longer ed payment by a winning v-valuation buyer holds. in the second-price auction (whether he is Proposition 11 (Holt 1980; Maskin and Riley risk averse or not). Because u0, 1984; Matthews 1983; Milgrom’s Theorem uv()− bv() >−vbv() ′()− , 14 We have ignored the constraint G0 because it is sat- uv bv() isfied by the solution to the program in which it is omitted. and so 15 The reason a reserve price helps the seller is that it puts a lower bound on what buyers pay. Of course, by set- bv′()> bv′ ()whenever b () v= b () v . ting a positive reserve, the seller runs the risk of not selling RN RN at all, but this effect is outweighed by the lower-bound Because b(0)b (0)0, we conclude that consideration. To see this, imagine that there is just one RN buyer. Then in a high-bid auction, that buyer would bid bv()> b () v for all v>0 zero; the only way to get him to pay anything is to make RN the reserve positive. which implies that, for every v0, 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). price auction. Q.E.D. dec04_Article 5 12/14/04 2:27 PM Page 1111

Maskin: The Unity of Auction Theory 1111

Another important hypothesis in sections and ∂v ∂ϕ 2 and 3 is that buyers actually know their ∑ j i ∂ ≠ s ∂s valuations (more to the point, that their val- ji i1 ≠ i1 . (27) uations do not depend on the private infor- ∂v ∂ϕ ∑ j i mation of other buyers). This is called the ∂ ∂ ji≠ s si2 private values assumption. Let us now relax i2 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 s ues). Specifically, suppose that each buyer i i and s such that (s) (s). From (26), receives a private signal s and that his valu- i i i i i i buyer i’s preferences are identical for s s ation is a function of all the signals: i.e., his i i and s s and so, in equilibrium, he must be valuation is v (s , s ), where s is the vector of i i i i -i -i 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 s or s sholds will in general lead to dif- cient (Maskin 1992); the problem is that i i i i ferent efficient allocations—e.g., perhaps because buyers no longer know their valua- buyer i wins when s s and loses when tions, their bids (reflecting their expected i i s s. Thus buyer i cannot be indifferent valuations) do not guarantee that the high i i between the outcomes. Q.E.D. 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 sional (i.e., scalars), then, as Jacques case of single-good auctions. Specifically, Crémer and Richard McLean (1985) and introduce a one-dimensional “reduced” signal Maskin (1992) show, there exist mecha- r for buyer i and, for all j≠i, define nisms that ensure efficiency (provided that i vrsˆ ,,−−== E() vssϕ s r “single crossing” holds, i.e., buyer i’s signal ji() i si ji ()() i ii i, has a greater marginal effect on his own val- i.e.,v ˆ j(ri,s–i) is vj(si,s–i) expected over all those uation than on other buyers’ valuations: values si such that i(si) ri. Because we are ∂ ∂v vi > j back to one-dimensional signals, the extend- ∂ ∂ for j i). More concretely, Partha si si ed second-price auction mentioned above Dasgupta and Maskin (2000) and Motty will be efficient with respect to the Perry and Philip Reny (2002) show that, “reduced” valuations vi(ri,s–i) and {ˆvj} with single crossing, there is a way of (assuming that the single-crossing property extending the second-price auction to 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 si for cases, Maskin 1992 and Krishna 2003 show which i(si) ri, i.e., it is incentive efficient that the English auction is efficient with (see Dasgupta and Maskin 2000). Jehiel and one-dimensional signals). However, when Moldovanu (2001) show, however, that this signals are multidimensional, then efficien- “reduction” technique does not generalize to cy is no longer possible. more than one good. Proposition 12 (Maskin 1992; Philippe Jehiel Finally, let us explore what happens when and 2001; Milgrom’s we drop the assumption of independence in Theorem 3.8): Suppose that, for some buyer Proposition 9: 17 i, si (si1,si2), where Proposition 13 (Milgrom and Weber

vss()()(), −−=+ϕψ s s 1982; Milgrom’s Theorem 4.21): Suppose ii i ii i i (26) 18 that n 2, and that v1 and v2 are jointly 17 For convenience, suppose that all the other signals are one-dimensional. 18 Milgrom and Weber (1982) generalize this result to n3. dec04_Article 5 12/14/04 2:27 PM Page 1112

1112 Journal of Economic Literature, Vol. XLII (December 2004)

v symmetrically distributed with support [0, 1] 1 2 > ∫ vdF2 () v21| v and affiliated (positively correlated) in the 0 v sense that 1 2 ∫ vdFv22111()()|| v F v v ∂2 log Fv ( v ) > 0 . (34) 21> 0 , (28) Fv()11| v ∂∂vv 1 2 But (34) follows from affiliation, i.e., from where F(v2|v1) is the c.d.f. for v2 conditional (28). Q.E.D. on v1. Then, revenue from the second-price Notice that Propositions 11 and 13 pull in auction exceeds that from the high-bid opposite directions: the former favors the auction. high-bid auction, the latter the second-price Proof: Let P1(v1) be buyer 1’s equilibrium auction. This tension illustrates one of expected payment when his valuation is v1. Milgrom’s introductory points: that the kind We have of auction a seller will want to use depends v S = 1 heavily on the circumstances. Pv11()∫ vdFvv 2 ( 21| ) (29) 0 and 6. Theory versus Practice H = Pv11() Fvvbv ( 11| )() 1 , (30) I have already suggested that some of 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, S H that it has accomplished, auction theory still P1 (0) P1 (0). We wish to show that S H has not developed far enough to be directly P (v )P (v ) for all v 0. It suffices to show 1 1 1 1 1 applicable to situations as complex as, say, dPSH() v dP() v 11 11 S H the spectrum auctions. that> whenever P1(v1) P1 (v1). dv1 dv1 To begin with, those auctions involve mul- tiple goods. Observe, however, that all the Differentiating (29) and (30) we obtain results presented above are for single-good S v dP1 = 1 1 2 auctions. This is no coincidence; the litera- vF1 ()+() v11|| v∫ vdF 2 v21 v (31) 0 dv1 ture on auction theory has overwhelmingly and focused on the single-good case. Apart from H the efficiency of the multigood second-price dP1 = ′ 1 auction (the Vickrey-Clarke-Groves mecha- Fv()()11|| v b v 1 +()() bv 1 F v 11 v dv1 nism; see section 7) with private values, 2 there are very few general results for more +()bv111 F ( v| v ), than one good. The auction designer can where superscripts of F denote partial deri- attempt to extrapolate from well-analyzed vations. The second equation can be rewrit- environments (one good) to new circum- ten, using buyer 1’s first-order condition, as stances (multiple goods). But doing so is dP H hazardous (see, for example, Perry and 1 = vF12() v|| v+() bv F ( v v ) 1111 11(32) Reny 1999). dv1 H S Another difficulty for theory is that real When P (v )P (v ), 1 1 1 1 auctions impose constraints that are difficult v1 ∫ vdFv221()| v to formalize. Milgrom notes that prospective = 0 bv()1 . (33) bidders and sellers are typically nervous Fv()| v 11 about participating in auction mechanisms From (31)–(33), it remains to show that that seem unfamiliar or complicated. But dec04_Article 5 12/14/04 2:27 PM Page 1113

Maskin: The Unity of Auction Theory 1113

giving a precise meaning to “unfamiliar” or Instead, Milgrom worries about the fol- “complicated” is forbiddingly difficult. lowing sort of problem. Assume that there The upshot is that giving advice on real auc- are two goods A and B, and two potential tion design is, at this stage, far less a science buyers 1 and 2. Suppose that each buyer 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 in writing. that buyer 1 has a valuation of $100 for A and B together, but that buyer 2 has a pack- 7. The Vickrey-Clarke-Groves Mechanism age valuation of $200. If the buyers bid truthfully in the VCG mechanism, then I have noted that Milgrom voices serious buyer 2 will win both A and B (and pay $100, criticisms of the Vickrey-Clarke-Groves the winning bid were 2 not present). Buyer (VCG) mechanism, the generalization of the 1 will come away empty-handed. second-price auction to multiple goods. Suppose, however, that buyer 1 enters Indeed, his unhappiness with it has led him bids through two different proxy buyers, 1x to collaborate with Lawrence Ausubel and 1y. As buyer 1x he enters a bid of $201 (Ausubel and Milgrom 2002) on an interest- for good A (and zero for both B and the ing and ingenious alternative mechanism, package AB). As 1y, he enters a bid of $201 reported on in chapter eight. for good B (and zero for both A and AB). The VCG mechanisms works as follows: 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 an amount equal to the difference 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) while for 1. Unhappily, it generates no 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. the VCG auction. 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 of the winning valuations). quite sure that buyer 2 does not value A and One frequent objection to the VCG mech- B as single goods. As soon as there is a seri- anism is that it makes heavy demands on 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 (computing the winning allocation is a 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 Milgrom dwells on; in fact, the Ausubel- be awarded A and B with their $201 bids but Milgrom paper is subject to the same sort of buyer 1 would now pay $51$51$102 for criticism. a package worth only $100 to him. dec04_Article 5 12/14/04 2:27 PM Page 1114

1114 Journal of Economic Literature, Vol. XLII (December 2004)

Indeed, with sufficient uncertainty about Restricted Domains,” 47, pp. 1137–44. Holt, Charles. 1980. “Competitive Bidding for how other buyers will bid, it is not hard to see Contracts under Alternative Auction Procedures,” J. that nothing other than truthful bidding Polit. Econ. 88:3, pp. 433–45. makes sense for a buyer in a VCG auction. Jehiel, Philippe and Benny Moldovanu. 2001. “Efficient Design with Interdependent Valuations,” And since I would venture to say that consid- Econometrica 69:5, pp. 1237–59. erable uncertainty is quite common in real Klemperer, Paul. 2004. Auctions: Theory and Practice. auction settings, I believe that Milgrom is too Princeton: Princeton U. Press. Krishna, Vijay. 2002. Auction Theory. San Diego: harsh when he deems VCG “unsuitable” for Academic Press. most applications. ———. 2003. “Asymmetric English Auctions,” J. Econ. Theory 112:2, pp. 261–68. Laffont, Jean-Jacques and Eric Maskin. 1979. “A 8. Concluding Remarks Differential Approach to Expected Utility Maximizing Mechanisms,” in Aggregation and This is a minor reservation about a volume Revelation of Preferences. Jean-Jacques Laffont, ed. Amsterdam: North-Holland. that covers a cornucopia of material in mag- ———. 1980. “A Differentiable Approach to isterial fashion and gives us deep insight into Dominant Strategy Mechanisms,” Econometrica 48, the thinking of an outstanding theorist. The pp. 1507–20. Maskin, Eric. 1992. “Auctions and Privatization,” in book is not for everybody; one needs at least Privatization: Symposium in Honor of Herbert enough technique to be able to follow the Giersch. H. Siebert, ed. Tubingen: Mohr (Siebek), proofs of the propositions above. But, with pp. 115–36. ———. 2000. “Auctions, Development, and that qualification, I warmly commend it to Privitization: Efficient Auctions with Liquidity- all wishing to experience the beauty and Constrained Buyers,” European Econ. Rev. 44, pp. power of this remarkable theory. 667–81. Maskin, Eric and John Riley. 1984. “Optimal Auctions with Risk Averse Buyers,” Econometrica 52:6, pp. REFERENCES 1473–518. Ausubel, Lawrence and Paul Milgrom. 2002. Maskin, Eric and Tomas Sjöström. 2002. “Ascending Auctions with Package Bidding,” “Implementation Theory,” in Handbook of Social Frontiers Theoret. Econ. 1:1. Choice and Welfare, vol. I. , Amartya Che, Yeon-Koo and Ian Gale. 1998. “Standard Auctions Sen and Kataro Suzumara, eds. NY: with Financially Constrained Bidders,” Rev. Econ. Science, pp. 237–88. Stud. 65, pp. 1–21. Matthews, Steven. 1983. “Selling to Risk Averse Buyers Clarke, Edward. 1971. “Multipart Pricing of Public with Unobservable Tastes,” J. Econ. Theory 30, pp. Goods,” Public Choice 11, pp. 17–33. 370–400. Crémer, Jacques and Richard McLean. 1985. “Optimal ———. 1984. “On the Implementability of Reduced Selling Strategies under Uncertainly for a Form Auctions,” Econometrica 52, pp. 1519–22. Discriminating Monopolist When Demands Are Milgrom, Paul. 1981. “Rational Expectations, Interdependent,” Econometrica 53:2, pp. 345–61. Information Acquisition, and Competitive d’Aspremont, Claude and Louis-André Gérard-Varet. Equilibrium,” Econometrica 49, pp. 921–43. 1979. “Incentives and Incomplete Information,” J. ———. 2004. Putting Auction Theory to Work. Public Econ. 11, pp. 24–45. Cambridge: Cambridge U. Press. Dasgupta, Partha and Eric Maskin. 2000. “Efficient Milgrom, Paul and Robert Weber. 1982. “A Theory of Auctions,” Quart. J. Econ. 115:2, pp. 341–88. Auctions and Competitive Bidding,” Econometrica Green, Jerry and Jean-Jacques Laffont. 1977. 50, pp. 443–59. “Characterization of Satisfactory Mechanisms for the Myerson, Roger. 1981. “Optimal Auction Design,” Revelation of Preferences for Public Goods,” Math. Op. Res. 6, pp. 58–73. Econometrica 45, pp. 427–38. Myerson, Roger and Mark Satterthwaite. 1983. Griesmer, James; Richard Levitan and Martin Shubik. “Efficient Mechanisms for Bilateral Trade,” J. Econ. 1967. “Towards a Study of Bidding Process Part IV: Theory 29:2, pp. 265–81. Games with Unknown Costs,” Naval Res. Logistics Nakajima, Daisuke. 2004. “The First-Price and Dutch Quart. 14, pp. 415–33. Auctions with Allais-Paradox Bidders,” Princeton U. Groves, Theodore. 1973. “Incentives in Teams,” mimeo. Econometrica 61, pp. 617–31. Ortega Reichert, Armando. 1968. “A Harsanyi, John. 1967–68. “Games with Incomplete with Information Flow,” Tech. Report 8, dept. opera- Information Played by ‘Bayesian’ Players,” pts. 1–3, tions res. Stanford U. Manage. Sci. 14, pp. 159–82, 370–74, 486–502. Palfrey, Thomas. 2002. “Implementation Theory,” in Holmstrom, Bengt. 1979. “Groves Schemes on Handbook of Game Theory with Economic dec04_Article 5 12/14/04 2:27 PM Page 1115

Maskin: The Unity of Auction Theory 1115

Applications, vol. III. and Sergiu Double Auction as the Market Becomes Large,” Rev. Hart, eds. NY: Elsevier Science, pp. 2271–326. Econ. Stud. 56, pp. 477–98. Perry, Motty and Philip Reny. 1999. “On the Failure of Selten, Reinhard. 1975. “Reexamination of the the ,” Econometrica 67:4, pp. Perfectness for Equilibrium Points in Extensive 895–900. Games,” Int. J. Game Theory 4, pp. 25–55. ———. 2002. “An Efficient Auction,” Econometrica Serrano, Roberto. 2004. “The Theory of 70:3, pp. 1199–212. Implementation of Social Choice Rules,” SIAM Pesendorfer, Wolfgang and Jeroen Swinkels. 1997. Rev. 46, 377–414. “The Loser’s Curse and Information Aggregation in Vickrey, William. 1961. “Counterspeculation, Common Value Auctions,” Econometrica 65:6, pp. Auctions, and Competitive Sealed Tenders,” J. 1247-82. Finance 16, pp. 8–37. Riley, John and William Samuelson. 1981. “Optimal Wilson, Robert. 1969. “Competitive Bidding with Auctions,” Amer. Econ. Rev. 71, pp. 381–92. Disparate Information,” Manage. Sci. 15, pp. 446–48. Satterthwaite, Mark and Steven Williams. 1989. “The ———. 1977. “A Bidding Model of Perfect Rate of Convergence to Efficiency in the Buyer’s Bid ,” Rev. Econ. Stud. 44, pp. 511–18.