The Lovely but Lonely Vickrey Auction


 Barbara Dorsey
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1 The Lovey but Loney Vickrey Auction Lawrence M. Ausube and Pau Migro 1. Introduction Wiia Vickrey s (1961) inquiry into auctions and counterspecuation arked the first serious attept by an econoist to anayze the detais of arket rues and to design new rues to achieve superior perforance. He deonstrated that a particuar pricing rue akes it a doinant strategy for bidders to report their vaues truthfuy, even when they know that their reported vaues wi be used to aocate goods efficienty. Vickrey s discovery was argey ignored for a decade, but the foodgates have since opened. Dozens of studies have extended his design to new environents, deveoped his associated theory of bidding in auctions, and tested its ipications using aboratory experients and fied data. Despite the enthusias that the Vickrey echanis and its extensions generate aong econoists, practica appications of Vickrey s design are rare at best. The cassic Engish auction of Sotheby s and Christie s, in which bidders iterativey subit successivey higher bids and the fina bidder wins the ite in return for a payent equaing his fina bid, is cosey reated to Vickrey s secondprice seaedbid auction, but ong predates it. Onine auctions such as ebay, in which bidders coony utiize proxy bids authorizing the auctioneer to bid up to specified prices on their behaf, ore neary resebe the Vickrey design for a singe ite; however, these reain true dynaic auctions, as onine bidders who subit proxy bids generay retain the abiity to raise their proxy bids ater. The ost genera and nove version of Vickrey s design, which 1
2 appies to saes in which different bidders ay want utipe units of hoogeneous goods or packages of heterogeneous goods reains argey unused. Why is the Vickrey auction design, which is so ovey in theory, so oney in practice? The answer, we beieve, is a cautionary tae that ephasizes the iportance of anayzing practica designs fro any perspectives. Vickrey s design has soe ipressive theoretica virtues, but it aso suffers fro weaknesses that are frequenty decisive. This chapter reviews the theoretica puses and inuses of the Vickrey design, highighting issues that cannot be ignored in deveoping practica auction designs. 2. Description of the genera Vickrey (VCG) design Vickrey s origina inquiry treated both auctions of a singe ite and auctions of utipe identica ites, providing a echanis in which it is a doinant strategy for bidders to report their vaues truthfuy and in which outcoes are efficient. For a singe ite, the echanis is often referred to as the secondprice seaedbid auction, or sipy the Vickrey auction. Bidders siutaneousy subit seaed bids for the ite. The highest bidder wins the ite, but (unike standard seaedbid tenders) the winner pays the aount of the secondhighest bid. For exape, if the winning bidder bids 10 and the highest osing bid is 8, the winner pays 8. With these rues, a winning bidder can never affect the price it pays, so there is no incentive for any bidder to isrepresent his vaue. Fro bidder n s perspective, the aount he bids deterines ony whether he wins, and ony by bidding his true vaue can he be sure to win exacty when he is wiing to pay the price. In Vickrey s origina treatent of utipe units of a hoogeneous good, which ay be avaiabe in either continuous or discrete quantities, each bidder is assued to have onotonicay nonincreasing argina vaues for the good. The bidders siutaneousy 2
3 subit seaed bids coprising deand curves. The seer cobines the individua deand curves in the usua way to deterine an aggregate deand curve and a cearing price for S units. Each bidder wins the quantity he deanded at the cearing price. However, rather than paying the prices he bid or the cearing price for his units, a winning bidder pays the opportunity cost for the units won. In the case of discrete units, an equivaent way to describe the utiunit Vickrey auction is that each bidder subits a nuber of separate bids, each representing an offer to buy one unit. These individua bids describe the bidder s deand curve. The auctioneer accepts the S highest bids. If bidder n wins K units, then he pays the su of the K highest rejected bids by other bidders. For exape, if a bidder wins 2 units and the highest rejected bids by his copetitors are 12 and 11, then the bidder pays 23 for his two units. Another way to describe the rue is that the price a bidder pays for his r th unit is the cearing price that woud have resuted if bidder n had restricted his deand to r units (a other bidders behaviors hed fixed). This equivaent description akes cear the opportunitycost interpretation of the winners payents. The tota payent for bidder n is coputed by suing this payent over a ites won, in the case of discrete units, or by integrating this payent fro 0 to the quantity won, in the case of continuous units. The echanis can be used as either a echanis to se (a standard auction) or as a echanis to buy (a reverse auction). Described as a standard auction, the buyers generay pay a discount as copared to the cearing price. Described as a reverse auction, the seers generay receive a preiu as copared to the cearing price. Indeed, the ain point of Vickrey s seina artice was that the governent cannot 3
4 estabish a arketing agency to ipeent a doinantstrategy echanis in twosided arkets without providing a subsidy: The basic drawback to this schee is, of course, that the arketing agency wi be required to ake payents to suppiers in an aount that exceeds, in the aggregate, the receipts fro purchasers (Vickrey 1961, p. 13). Since Vickrey s origina contribution, his auction design has been eded with the CarkeGroves design for pubic goods probes. 1 The resuting auction design works for heterogeneous goods as we as hoogeneous goods and does not require that bidders have nonincreasing argina vaues. As with Vickrey s origina design, this echanis sti assigns goods efficienty and sti charges bidders the opportunity cost of the ites they win. The ain difference is that the aounts paid cannot generay be expressed as the sus of bids for individua ites. The extended Vickrey echanis goes by various naes. Here, we ca it the VickreyCarkeGroves or VCG echanis. Foray, the VCG echanis is described as foows. Let x be a vector of goods that a seer has on offer and et v ( x ) denote bidder n s vaue for any nonnegative n n vector x n. Each bidder n = 1,..., N reports a vaue function v ˆn to the auctioneer. The auctioneer then coputes a vaueaxiizing aocation: * x ˆ x1,..., x v N n n xn arg ax ( ) subject to n x x. The price paid by a bidder n is then * p ˆ ( ) n n = α n v n x, where { vˆ x x x}. Notice that n α n = ax ( ) n n of the other bidders and not on what bidder n reports. α depends ony on the vaue reports 1 The CarkeGroves design was introduced in Carke (1971) and Groves (1973). 4
5 To iustrate the VCG echanis, suppose that there are two ites for sae (A and B) and two bidders. Each bidder n = 1,2 subits bids: v ˆ n(a) for ite A; v ˆ n(b) for ite B; and v ˆ n(ab) for the two ites together. Assue without oss of generaity that vˆ ˆ 1(AB) v2(ab) and vˆ1(a) + vˆ ˆ ˆ 2(B) v1(b) + v2(a). If vˆ1(ab) > vˆ ˆ 1(A) + v2(b), then the outcoe is that bidder 1 wins both ites. Appying the forua, his payent is v ˆ 2(AB). However, if vˆ1(ab) < vˆ ˆ 1(A) + v2(b), then the outcoe is that bidder 1 wins ite A (with an associated payent of vˆ ˆ 2(AB) v2(b) ) and bidder 2 wins ites B (with an associated payent of vˆ ˆ 1(AB) v1(a) ). In each case, the winner pays the opportunity cost of the ites won, and his payent depends ony on his opponent s reports. The first theore confirs that the genera VCG echanis sti has the properties that it is a doinant strategy for each bidder to report its vaues truthfuy and that the outcoe in that event is an efficient one. Theore 1. Truthfu reporting is a doinant strategy for each bidder in the VCG echanis. Moreover, when each bidder reports truthfuy, the outcoe of the echanis is one that axiizes tota vaue. Proof. Consider any fixed profie of reports, { ˆ } v n, for the bidders besides n. Suppose that when bidder n reports truthfuy, the resuting aocation and payent vectors are denoted by * x and ˆx and ˆp. When bidder n reports v ˆn, its payoff is: * p, but when bidder n reports v ˆn, the resuting vectors are 5
6 n α { v x vˆ x x n x} * ˆ * α v ( xˆ ) pˆ = v ( xˆ ) + vˆ ( xˆ ) n n n n n n ax ( ) + ( ) α n n n = v ( x ) + v ( x ) n n n n * * n( n) n. = v x p (1) The ast ine is bidder n s payoff fro truthfu reporting, so truthfu reporting is aways optia. We oit the tedious but routine check that no other report is aways optia. The ast stateent foows by construction of the echanis. 3. Virtues of the VCG Mechanis The VCG echanis has severa iportant virtues. The first is the doinantstrategy property, which reduces the costs of the auction by aking it easier for bidders to deterine their optia bidding strategies and by eiinating bidders incentives to spend resources earning about copetitors vaues or strategies. Such spending is pure waste fro a socia perspective, since it is not needed to identify the efficient aocation, yet it can be encouraged by auction forats in which each bidder s best strategy depends on its opponents ikey actions. The doinant strategy property aso has the apparent advantage of adding reiabiity to the efficiency prediction, because it eans that the concusion is not sensitive to assuptions about what bidders ay know about each others vaues and strategies. This is a distinctive virtue of the VCG echanis. Theores by Green and Laffont (1979) and by Hostro (1979) show that, under weak assuptions, the VCG echanis is the unique direct reporting echanis with doinant strategies, efficient outcoes, and zero payents by osing bidders. Here, we report a version of Hostro s theore. To prove it, we need one extra assuption that has not been needed so far, naey, that the 6
7 set V of possibe vaue functions is soothy path connected. This eans that given any two functions in V, there is a soothy paraeterized faiy of functions { vxt (, )} that ies whoy in V, and connects the two functions. More precisey, for any two eeents v(,0) i and v(,1) i in V, there exists a path { v(,) i t t [0,1]} such that v is differentiabe in its second arguent and such that the derivative satisfies 1 x v x t dt<, where sup (, ) 2 0 v v t here denotes the partia derivative of v with respect to the second arguent. 2 / Theore 2. If the set of possibe vaue functions V is soothy path connected and contains the zero function, then the unique direct reveation echanis for which truthfu reporting is a doinant strategy, the outcoes are aways efficient, and there are no payents by or to osing bidders is the VCG echanis. Proof. Fix any vaues for the bidders besides bidder n and consider any echanis satisfying the assuptions of the theore. If n reports the zero function, then his VCG aocation is zero and his payoff is aso zero. Suppose that n reports soe vaue function v(,1) i and et v(,0) i be the zero function. By construction, a bidder with vaues of zero for every package is a osing bidder at any efficient aocation. Let { v(,) i t t [0,1]} be a sooth path of vaue functions, as defined above. Denote the totavaueaxiizing aocation when n reports v(, i t) by Vt () denote n s corresponding payoff in the VCG echanis: x * () t and et Vt = vx s t p s. By the enveope theore in integra for (Migro and * () ax s ( n(),) n() * Sega (2002)), V(1) V(0) = v ( x ( t), t) dt
8 Let pˆ( t ) be the payents ade under any other direct reveation echanis for which truthfu reporting is a doinant strategy, the outcoes are aways efficient, and there are no payents by or to osing bidders. Let π () t denote n s payoff in the aternate * echanis when his vaue function is v(, i t) : π () t = ax v( x (),) s t pˆ () s. By the s n n * enveope theore in integra for, π(1) π(0) = v2( x ( t), t) dt = V(1) V(0). 1 0 Since there are no payents by or to osing bidders, π (0) = V (0) = 0, so vx pˆ = π = V = vx p. Hence, p (1) = pˆ (1) ; the payent * * ( (1),1) n(1) (1) (1) ( (1),1) n(1) n n rue ust be the sae as for the VCG echanis. Another virtue of the VCG echanis is its scope of appication. Theores 1 and 2 above do not ipose any restrictions on the bidders possibe rankings of different outcoes. The basic rues of the Vickrey auction can be further adapted if the auctioneer wishes to ipose soe extra constraints. For exape, the governent seer in a spectru auction ay wish to iit the concentration of spectru ownership according to soe easure. Or, the buyer in a procureent auction ight want to iit its tota purchases fro firsttie bidders or ight want to ensure security by requiring that the tota reevant capacity of its suppiers is at east 200% of the aount ordered. One can repace the constraint that x by any constraint of the for x X without x affecting the preceding theory or arguents in any essentia way. A fina virtue of the Vickrey auction is that its average revenues are not ess than that fro any other efficient echanis, even when the notion of ipeentation is 8
9 expanded to incude Bayesian equiibriu. A fora stateent of this faous revenue equivaence theore is given beow. Theore 3. Consider a Bayesian ode in which the support of the set of possibe vaue functions, V, is soothy path connected and contains the zero function. Suppose the bidder vaue functions are independenty drawn fro V. If, for soe echanis, the BayesianNash equiibriu outcoes are aways efficient and there are no payents by or to osing bidders, then the expected payent of each bidder n, conditiona on his vaue function v n V, is the sae as for the VCG echanis. In particuar, the seer s revenue is the sae as for the VCG echanis Weaknesses of the VCG Mechanis Despite the attractiveness of the doinantstrategy property, the VCG echanis aso has severa possibe weaknesses: ow (or zero) seer revenues; nononotonicity of the seer s revenues in the set of bidders and the aounts bid; vunerabiity to cousion by a coaition of osing bidders; and vunerabiity to the use of utipe bidding identities by a singe bidder. 2 The proof of Theore 3 is siiar to that for Theore 2, reying on the enveope theore to deterine the expected payoff of each type of each bidder. Wiias (1999) proves ore generay that optiization over the cass of doinantstrategy echaniss yieds the sae objective as optiization over the (arger) cass of Bayesian echaniss. Krishna and Perry (1997) show that the VCG echanis axiizes the seer s revenues over the cass of efficient echaniss satisfying incentive copatibiity and individua rationaity. 9
10 It wi be seen ater in this chapter that a sipe and intuitive condition on individua bidders characterizes whether these deficiencies are present. In econoic environents where every bidder has substitutes preferences, the aboveisted weaknesses wi never occur. However, if there is but a singe bidder whose preferences vioate the substitutes condition, then with an appropriate choice of vaues for the reaining bidders (even if the atter vaues are restricted to be additive), a of the aboveisted weaknesses wi be present. In what foows, we wi iit attention to auctions of utipe ites, in which different bidders ay want different nubers or different packages of ites. One obvious reason for the disuse of the VCG echanis for argescae appications with diverse ites is the sae as for other cobinatoria or package auctions: copexity in a aspects of its ipeentation. The chapters of this book on the winner deterination probe give soe insights into the probe facing the auctioneer. There are aso iportant difficuties facing the bidder in such auctions. Copexity, however, cannot be the whoe expanation of the rarity of the VCG echanis. Cobinatoria auctions of other kinds have been adopted for a variety of procureent appications, fro schoo ik progras in Chie to bus route services in London, Engand. Sascae cobinatoria auctions are technicay feasibe, so we are forced to concude that VCG rues have not been epoyed even when they are feasibe. Anayses of the VCG echanis often excude any discussion of the auction revenues. This is an iportant oission. For private seers, revenues are the priary concern. Even in the governentrun spectru auctions, in which priorities ike aocationa efficiency, prooting ownership by sa businesses or woen or inority 10
11 owned businesses, rewarding innovation, and avoiding concentration of ownership are weighty considerations, one of the perforance easures ost ephasized by the pubic and poiticians aike is the fina auction revenue. 3 Against this background, it is particuary troubing that the Vickrey auction revenues can be very ow or zero, even when the ites being sod are quite vauabe and copetition is ape. For exape, 4 consider a hypothetica auction of two spectru icenses to three bidders. Suppose that bidder 1 wants ony the package of two icenses and is wiing to pay $2 biion, whie bidders 2 and 3 are both wiing to pay $2 biion for a singe icense. The VCG echanis assigns the icenses efficienty to bidders 2 and 3. The price paid by bidder 2 is the difference in the vaue of one icense or two icenses to the reaining bidders. Since that difference is zero, the price is zero! The sae concusion appies syetricay to bidder 3, so the tota auction revenues are zero. Notice that, in this exape, if the governent had sod the two icenses as an indivisibe whoe, then it woud have had three bidders each wiing to pay $2 biion for the cobined icense. That is ape copetition: an ascending auction woud have been expected to ead to a price of $2 biion. This revenue deficiency of the VCG echanis is decisive to reject it for ost practica appications. Cosey reated to the revenue deficiency of the Vickrey auction is the nononotonicity of seer revenues both in the set of bidders and in the aounts bid. In the preceding exape, if the third bidder were absent or if its vaue for a icense were reduced fro $2 biion to zero, then the seer s revenues in the VCG auction woud 3 However, Ausube and Craton (1999) argue that in the extree case of an auction foowed by a resae arket in which a avaiabe gains fro trade are reaized, a focus soey on efficiency is appropriate. 4 This and a subsequent exapes are drawn fro Ausube and Migro (2002). 11
12 increase fro $0 to $2 biion. This nononotonicity is not ony a potentia pubic reations probe but aso creates oophoes and vunerabiities that bidders can soeties expoit. One is the Vickrey design s vunerabiity to cousion, even by osing bidders. This can be iustrated by a variant of the preceding exape, in which bidder 1 s vaues are unchanged but bidders 2 and 3 have their vaues reduced so that each is wiing to pay ony $0.5 biion to acquire a singe icense. The change in vaues aters the efficient aocation, aking bidders 2 and 3 osers in the auction. However, notice that if the bidders each bid $2 biion for their individua icenses, then the outcoe wi be the sae as above: each wi win one icense at a price of zero! This shows that the VCG echanis has the unusua vunerabiity that even osing bidders ay possess profitabe joint deviations, faciitating cousion in the auction. A cosey reated stratage is a version of shi bidding a bidder s use of utipe identities in the auction as discussed in chapter 7 of this book. 5 To construct this exape, et us repace bidders 2 and 3 of the preceding exape by a singe cobined bidder whose vaues are $0.5 biion for a singe icense and $1.0 biion for the pair of icenses. Again, the efficient aocation assigns the icenses to bidder 1, so the cobined bidder is destined to ose. However, if the auctioneer cannot keep track of the bidders rea identities, then the cobined bidder coud participate in the auction under two naes bidder 2 and bidder 3 each of who bids $2 biion for a singe icense. As we have aready seen, the resut is that bidders 2 and 3 win at a price of zero, so the cobined bidder wins both icenses at a tota price of zero. 5 See Sakurai, Yokoo and Matsubara (1999) and Yokoo, Sakurai and Matsubara (2000, 2004) for the origina treatents of what they have caed fase nae bidding. 12
13 A reated difficuty with the VCG echanis arises even if a the bidders pay their doinant strategies, with no cousion or utipe identities. Suppose again that bidders 2 and 3 each vaue a singe icense at $2 biion, but that by cobining operations they coud increase the vaue of the package fro $4 biion to $4+X biion. If a bidders bid their vaues as in the origina Vickrey anaysis, then the unerged bidders 2 and 3 woud pay zero and earn a tota profit of $4 biion. A erger by bidders 2 and 3 before the auction woud raise the price fro zero to $2 biion, so the firs woud find it profitabe to erge ony if X>$2 biion. Thus, whie the doinant strategy soution of the VCG echanis provides the proper incentives for bidding with a fixed set of bidders, it distorts preauction decisions reated to erging bidders. Notice that, in each of these exapes, bidder 1 has vaue ony for the entire package. If we odified the exapes to ake the icenses substitutes for bidder 1, for exape, by specifying that bidder 1 was wiing to pay $1 biion for each icense rather than just $2 biion for the pair, then a of our concusions woud change. The seer s tota revenue in the first exape woud be $2 biion, which is a copetitive payent in the sense that the payoff outcoe ies in the core. 6 This odification reverses a of the reated exapes, eiinating nononotonicities in seer revenues, profitabe joint deviations by osing bidders, the profitabe use of utipe identities in bidding, and the bias against vauecreating ergers. The substitutes condition turns out to pay a decisive roe in characterizing the perforance of the Vickrey auction. The fora anaysis beow deveops that concusion. 6 In the unodified exape, the Vickrey outcoe was not in the core. The coaition consisting of the seer and bidder 1 coud bock the Vickrey outcoe since, by theseves, they earn a coaition payoff of $2 biion, but the Vickrey outcoe gave the ony a payoff of zero. 13
14 We show that the kinds of probes described above cannot arise if a bidders are iited to having substitutes preferences. Moreover, if the possibe bidder vaues incude the additive vaues and if they are not restricted ony to substitutes preferences, then there exist vaues profies that exhibit a of the kinds of probes described above. Before turning to that fora anaysis, we iustrate soe other, siper drawbacks of the VCG auction design. The Vickrey theory incorporates the very restrictive assuption that bidders payoffs are quasiinear. This requires that payoffs can be expressed as the vaue of the ites received inus the payent ade. In particuar, it requires that there is no effective budget iit to constrain the bidders and that the buyer, in a procureent auction, does not have any overa iit on its cost of procureent. Athough we have no data on how frequenty these assuptions are satisfied, it appears that faiures ay be coon in practice. It is easy to see that the doinant strategy property breaks down when bidders have iited budgets. 7 For exape, consider an auction in which each bidder has a budget of $1.2 biion to spend. Vaues are deterined separatey fro budgets: they refect the increent to the net present vaue of the bidder s profits if it acquires the specified spectru icenses. Bidder A has vaues of $1 biion for a singe icense or $2 biion for the pair. Bidders B and C each want ony one icense. For bidder B, the vaue of a icense is $800 iion. Bidder C s vaue is unknown to the other bidders, because it depends on whether C is abe to acquire a particuar substitute outside of the auction. Depending on 7 Che and Gae (1998) anayze revenue differences aong firstprice and secondprice auctions in the presence of budget constraints. 14
15 the circustances, bidder C ay be wiing to pay either $1.1 biion or zero for a icense. In a Vickrey auction, bidder A shoud win either two icenses or one, depending on the ast bidder s decision. In either case, its tota payent wi be $800 iion, so its budget is aways adequate to ake its Vickrey payent. Yet, if A s budget iit constrains it to bid no ore than $1.2 biion for any package, then it has no doinant strategy. For suppose that bidders B and C adopt their doinant strategies, subitting bids equa to their vaues. Then, if bidder C bids zero, then A ust bid ess than $400 iion for a singe icense (and, say, $1.2 biion for the package) to win both icenses and axiize its payoff. If instead bidder C bids $1.1 biion, then A ust bid ore than $800 iion for a singe icense to win a singe icense and axiize its payoff. Since A s best bid depends on C s bid, A has no doinant strategy. Vickrey auctions are soeties viewed as unfair because two bidders ay pay different prices for identica aocations. To iustrate, suppose there are two bidders and two identica ites. Bidder 1 bids 12 for a singe ite and 13 for the package, whie bidder 2 bids 12 for a singe ite and 20 for the package. The resut is that each bidder wins one ite, but the prices they pay are 8 and 1, respectivey, even though the ites are identica and each has ade the sae bid for the ite it wins. Rothkopf, Teisberg and Kahn (1990) have argued that the VCG auction design presents a privacy preservation probe. Bidders ay rationay be reuctant to report their true vaues, fearing that the inforation they revea wi ater be used against the. For exape, the pubic has soeties been outraged when bidders for governent assets are peritted to pay significanty ess than their announced axiu prices in a Vickrey auction (McMian (1994)), and such reactions can ead to canceations, 15
16 renegotiations, or at east negative pubicity. Modern systes of encryption ake it technicay feasibe to verify the auction outcoe without reveaing the actua bids, so the privacy concern ay see ess fundaenta than the others weaknesses reported above. Moreover, the widespread use of proxy bidders, in which bidders are asked to report their vaues or axiu bids, in eectronic auctions conducted by Googe (for ad paceents), ebay, Aazon and others, estabish that privacy concerns are not aways decisive. Our entire discussion above has assued the socaed private vaues ode, in which each bidder s payoff depends soey on its own estiate of vaue and not on its opponents estiates of vaue. This is appropriate, inasuch as the cassic theory of the Vickrey auction and the VCG echanis is deveoped for a private vaues ode. Without private vaues, they iediatey ose their doinantstrategy property. 8 Moreover, serious probes ay arise when Vickrey auctions are appied in other frequenty studied odes. In the coon vaue ode of Migro (1981) and the aost coon vaue ode of Keperer (1998), revenues can be neary zero for a different reason than above, reated to the extree sensitivity of equiibria in the secondprice auction to the winner s curse. In the ode of Buow, Huang and Keperer (1999), the aocationa efficiency of the second price auction design discourages entry by bidders who are unikey to be part of the efficient aocation. The paucity of entry can be another reason for very ow prices. 8 A generaization of the utiunit Vickrey auction to environents where bidders receive onediensiona signas, and each bidder s vaues ay depend on its opponents signas, is described in Ausube (1999). With bidders vaues depending on opponents signas, the efficient echanis cannot possess the doinantstrategy property, but it retains the soewhat weaker property that truthfu reporting is an ex post Nash equiibriu. 16
17 5. VCG Outcoes and the Core Our discussion of the weaknesses of the VCG echanis began with a threebidder exape in which the auction revenues were zero. Severa of the subsequent weaknesses were buit by varying this basic exape. Since the zero revenues exape was very specia, a basic question wi be: how ow ust revenue be before it is unacceptaby ow? We wi adopt a copetitive standard: the payoff outcoe ust ie in the core. To justify the core as a copetitive standard, suppose that two or ore riskneutra brokers copete to purchase the services of the auction participants. The broker s profits are defined to be the vaue of the hired coaition inus the wages paid to those hired. It then foows that the copetitive equiibriu price vectors of such an econoy are the sae as the payoff vectors in the core of the gae. The reason is sipe. First, the requireent of equiibriu that the brokers do not ose oney is identica with the core requireent that the payoff aocation (or iputation ) is feasibe. Second, the requireent of equiibriu that no broker can earn positive profits by any feasibe strategy coincides exacty with the core requireent that no coaition can bock the payoff aocation. The conditions iposed by equiibriu on the price vector are thus the sae as those iposed by the core on payoff profies. To investigate the reationship between VCG outcoes and the core, we introduce soe notation. We first define the coaitiona gae (L,w) that is associated with the trading ode. The set of payers is L = {0,..., L 1}, with payer 0 being the seer. The set of feasibe aocations is X, for exape, ( ) The coaitiona vaue function is defined for coaitions S { : 0 and } X = x x x x. L 0 L 0 L as foows: 17
18 ax v( x), if 0 S, S ( ) x X ws = 0, if 0 S. (2) The vaue of a coaition is the axiu tota vaue the payers can create by trading aong theseves. If the seer is not incuded in the coaition, that vaue is zero. The core of a gae with payer set L and coaitiona vaue function w() i is defined as foows: { π π π }. Core ( L, w ) = : w ( L ) =, w ( S ) for a S L L S Thus, the core is the set of profit aocations that are feasibe for the coaition of the whoe and unbocked by any coaition. Let π denote the Vickrey payoff vector: π = wl ( ) wl ( ) for bidders L 0 and π = wl ( ) π for the seer. The next severa theores and their proofs are taken 0 L 0 fro Ausube and Migro (2002). Theore 4. A bidder s Vickrey payoff π is s highest payoff over a points in the core. That is, for a L 0 : π ax { π π (, )} = Core L w. Proof. The payoff vector defined by π 0 = wl ( ), π = wl ( ) wl ( ), and π = 0, for j 0,, satisfies Core( L, w) π. Hence, π ax { π π (, )} Core L w. j Now suppose that π is a feasibe payoff aocation with π > π for soe 0. Then k π k = wl ( ) π < wl ( ), so coaition L bocks the aocation. Hence, π Core( L, w). 18
19 A payoff vector in the core is bidder optia if there is no other core aocation that a bidders prefer. It is bidder doinant if it is the bidders unaniousy ost preferred point in the core. More precisey, et π Core( L, w). We say that π is bidder optia if there is no π Core( L, w) with π π and π π for every bidder. We say that π is bidder doinant if every π Core( L, w) satisfies π π for every bidder. Theore 5. If the Vickrey payoff vector π is in the core, then it is the bidderdoinant point in the core. If the Vickrey payoff vector π is not in the core, then there is no bidderdoinant point in the core and the seer s Vickrey payoff is stricty ess than the saest of the seer s core payoffs. Proof. By Theore 4, π π for a π Core( L, w) and L 0. Hence, if the Vickrey payoff π is in the core, then it is bidder doinant. Conversey, suppose that π Core( L, w) and consider any ˆ π Core( L, w) and any j for who ˆ π j π j. By Theore 4, ˆ π j < π j, so ˆ π is not bidder doinant, and ˆ π π for a L 0. So, π = wl ( ) π 0 L 0 < wl ( ) π L = π 0 0 ˆ ˆ. An equivaent way of expressing Theore 5 is that, if the Vickrey payoff is in the core, then it is the unique bidderoptia point in the core. Otherwise, there is a utipicity of bidderoptia points in the core none of which is bidder doinant. We thus find that the seer s revenues fro the VCG echanis are ower than the copetitive benchark uness the core contains a bidderdoinant point. Ony then is the Vickrey payoff in the core, and ony then does the seer get copetitive payoff for its goods. The next tasks are to investigate when the Vickrey payoff is in the core and to 19
20 show that, as in the exapes, the VCG echanis s other weaknesses hinge on these sae conditions. The stye of our anaysis is to treat conditions that are robust to certain variations in the underying ode. One ight otivate this by iagining that the auction designer does not know who wi bid in the auction or precisey what vaues the bidders wi have. The next two subsections dea with the uncertainties in sequence. 6. Conditions on the Coaitiona Vaue Function For this section, it is hepfu to iagine that the seer knows the set of potentia bidders L but does not know which of the wi actuay participate in the auction. The question we ask concerns conditions on the coaitiona vaue function sufficient to ensure that the Vickrey outcoe eets the copetitive benchark, that is, ies in the core, regardess of which bidders decide to participate. This sort of robustness property is both practicay vauabe and anayticay usefu, as we wi see beow. To expore the question described above, we introduce the concept of a restricted Vickrey payoff vector π ( S), which appies to the cooperative gae in which participation is restricted to the ebers of coaition S. Thus, π ( S) w( S) w( S ) for S 0 and π ( S ) w ( S ) π ( ) 0 S 0 S. We aso introduce our ain condition for this section: Definition. The coaitiona vaue function w is biddersuboduar if for a L 0 and a coaitions S and S satisfying 0 S S, ws ( { }) ws ( ) ws ( { }) ws ( ). 20
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