A Generalized Vickrey Auction

Transcription

1 A Generalzed Vckrey Aucton Lawrence M. Ausubel* Unversty of Maryland September 1999 Abstract In aucton envronments where bdders have pure prvate values, the Vckrey aucton (Vckrey, 1961) provdes a smple mechansm for effcently allocatng homogeneous goods. However, n envronments where bdders have nterdependent values, the Vckrey aucton does not generally yeld effcency. Ths manuscrpt defnes a generalzed Vckrey aucton whch yelds effcency when bdders have nterdependent values. Each bdder reports her type to the auctoneer. Gven the reports, the auctoneer determnes the allocaton that maxmzes surplus. The payment rule s the followng extenson of Vckrey aucton prcng: a bdder s charged for a gven unt that she wns accordng to valuatons evaluated at the mnmum sgnal that she could have reported and stll won that unt. JEL No.: D44 (Auctons) Keywords: Auctons, Vckrey Auctons, Effcency, Interdependent Values, Mechansms Send comments to: Professor Lawrence M. Ausubel Department of Economcs Unversty of Maryland College Park, MD ausubel@econ.umd.edu (301) *The analyss presented here was orgnally contaned n Appendx B of Ausubel (1997). Several colleagues advsed me that t was bured and completely mssed amd the long paper, and they urged me to repackage t nto a more vsble form. Fnally they have shamed me nto removng the appendx from that paper, and wrtng a separate manuscrpt. I am grateful to Peter Cramton, Ted Groves, Phlppe Jehel, Erc Maskn, Benny Moldovanu, Motty Perry and Phl Reny for extremely helpful dscussons at varous stages of the development of ths manuscrpt. All remanng errors are my sole responsblty.

2 A Generalzed Vckrey Aucton Lawrence M. Ausubel INTRODUCTION. In aucton envronments where bdders have pure prvate values, the Vckrey aucton (Vckrey, 1961) provdes a smple mechansm for effcently allocatng M dentcal objects. Qute straghtforwardly, bdders smultaneously and ndependently submt up to M bds each; and the M hghest bds wn. More sophstcatedly, the payment rule s that, f bdder s to be assgned k objects, then she s charged the k th hghest rejected bd (submtted by another bdder) for her frst unt, the (k 1) st hghest rejected bd for her second unt,, and the hghest rejected bd for her k th unt. As s well known, t then becomes a (weakly) domnant strategy for each bdder to submt bds equalng her true margnal values, yeldng effcency, when bdders have dmnshng pure prvate values. However, n envronments where bdders have nterdependent values meanng that one bdder s value depends on another bdder s sgnal the Vckrey aucton as defned n the prevous paragraph does not generally yeld effcency. Whle effcency obtans n sngle-object envronments where bdders are completely symmetrc (Mlgrom and Weber, 1982) and n two-bdder auctons generally (Maskn, 1992), the Vckrey aucton does not generally yeld effcency n sngle-object envronments wth three or more asymmetrc bdders (Maskn, 1992), nor n multple-object envronments wth symmetrc bdders (Ausubel, 1997). For the case of a sngle object where bdders have nterdependent values, Maskn (1992) defned a modfed second-prce aucton whch extends the standard second-prce aucton to yeld effcency. In the same sprt, n ths manuscrpt, we shall generalze Maskn s approach by defnng a generalzed Vckrey aucton for multple dentcal objects whch yelds effcency when bdders have nterdependent values. The three papers most closely related to the current manuscrpt are Dasgupta and Maskn (forthcomng), Perry and Reny (1999), and Jehel and Moldovanu (1999). The frst two papers dffer from the current one n ther basc nformatonal perspectve: they assume that the mappng from sgnals to valuatons s commonly known by bdders, but not known by the auctoneer; here t s assumed that the mappng from sgnals to valuatons s commonly known by bdders and the auctoneer. Dasgupta and Maskn s work contemporaneous wth Ausubel (1997, Appendx B), the orgnal verson of the current manuscrpt provdes an extremely general, but rather complcated procedure: each bdder communcates her valuaton, as a functon of each possble realzaton of all other bdders valuatons, and the auctoneer computes fxed ponts. Perry and Reny s work subsequent to Ausubel (1997, Appendx B) restrcts attenton, as here, to the case of homogeneous objects and dmnshng returns, and then obtans a tworound bddng procedure whch s less computatonally ntensve for the auctoneer. In the frst round, bdders smultaneously submt bds, whch become publc nformaton and whch are fully revealng of the bdders sgnals. In the second round, each bdder submts bds { b jl } representng what bdder k 1

3 would submt n a two-bdder, second-prce, sngle-object aucton for the k th unt of bdder versus the l th unt of bdder j. Bdders may need to submt a farly volumnous collecton of bds, as j runs through all bdders (j ) whle k and l run through all k + l M + 1, where M s the number of objects. However, the auctoneer s merely requred to pck out the hgh bds. The current manuscrpt, by assumng that the auctoneer also knows the payoff structure, s nstead able to get by wth a very smple drect mechansm. Each bdder reports her type, and the auctoneer then determnes an effcent allocaton and payments remnscent of Vckrey s rule. Ths makes the mechansm qute ntutve and transparent, and the analyss qute short and smple. Of course, ther procedures and mne must be outcome-equvalent. The stronger nformatonal requrements placed on the auctoneer than n Dasgupta-Maskn and Perry-Reny can perhaps be defended by embeddng the drect mechansm as the second stage of a twostage procedure. In the frst stage, each bdder reports the mappng from sgnals to valuatons both for herself and all other bdders. If the bdders make consstent reports, then the auctoneer proceeds to carry out the generalzed Vckrey aucton; f the bdders make nconsstent reports, then the auctoneer sends everybody home empty-handed. Jehel and Moldovanu s work also subsequent to Ausubel (1997, Appendx B) shows that wth mult-dmensonal sgnals, an effcent drect mechansm s mpossble. Ths s consstent wth the current manuscrpt, as I assume a sngle-dmensonal sgnal space. For the case of a sngle-dmensonal sgnal space and under the hypothess that each bdder s utlty s a lnear functon of her own and other bdders sgnals Jehel and Moldovanu extend the drect mechansm of ths manuscrpt to yeld effcent outcomes n envronments wth allocatve externaltes (.e., unlke n the other cted papers, they allow each bdder s utlty to also depend on the assgnments to other bdders). The current manuscrpt as well as the three related papers set effcency as the sole objectve. In the real world, sellers often set reserve prces n auctons. It then becomes an nterestng queston whether t s possble to extend the sellng procedure heren so as to be constraned-effcent subject to the reserve prce (.e., to effcently assgn all objects that are sold, but to neffcently wthhold some of the objects from the market). At the same tme, ths manuscrpt (and the three related papers) assumes that payoffs are realzed wthout the possblty for further trade n the auctoned tem followng the concluson of the aucton. In the real world, agents often engage n post-aucton resale. It also becomes an nterestng queston whether t s possble to embed the effcent equlbrum of the aucton nto the larger game consstng of an aucton round followed by a resale round. Both of these problems are affrmatvely solved n Ausubel and Cramton (1999). THE MODEL AND THE RESULTS. A seller has a quantty M of a homogeneous good to sell to n bdders, N {1,,n}. The good may be assumed to be ether n dscrete unts or perfectly dvsble, wth lttle effect on the analyss. In the 2

4 dscrete case, each bdder can consume any quantty q {0, 1,, M}. In the perfectly-dvsble case, each bdder can consume any quantty q [0,M]. Let q (q 1,,q n ), and let Q {q q M} be the set of all feasble assgnments. Each bdder s value for the good may depend on the prvate nformaton of all the bdders. Let t T [0,t max ] be bdder s type ( s prvate nformaton), t (t 1,,t n ) T T 1 T n, and t t ~ t. (Type may be dscrete or contnuous.) A bdder s value V (t,q ) for the quantty q depends on her own type t and the other bdders types t. A bdder s utlty s her value less the amount, X, she pays: V (t,q ) X. Let v (t,q ) denote the margnal value for bdder, gven the vector t of types and quantty q. Ths s defned so that, n the dscrete case, q k = 1 V(, t q ) = v (, t k), and n the perfectly-dvsble case, V(, t q ) = v (, t y) dy. We make the followng two assumptons on v (t,q ): q 0 Value monotoncty. For all, t, t, and q, v (t,q ) 0, v (t,q ) s strctly ncreasng n t, v (t,q ) s weakly ncreasng n t j (j ), and v (t,q ) s weakly decreasng n q. Value regularty. For all, j, q, q j, t, and t > t, v (t,t,q ) > v j (t,t,q j ) v (t,t,q ) > v j (t,t,q j ) and v (t,t,q ) < v j (t,t,q j ) v (t,t,q ) < v j (t,t,q j ). Value monotoncty mples that types have a natural order, and that bdders have (weakly) dmnshng margnal valuatons. Observe that, wthout dmnshng margnal valuatons, the standard Vckrey aucton does not yeld effcency even wth pure prvate values. Value regularty s effectvely a sngle-crossng property: t mples that an effcent assgnment rule may be selected so that each bdder s quantty s weakly ncreasng n t. Value regularty holds f an ncrease n bdder s type rases s margnal value at least as much as that of any other bdder. Wthout value regularty, Perry and Reny (1999) show that there may not exst any effcent mechansm. Let q ( t) ( q ( t),!, q ( t)) denote an ex post effcent assgnment rule for the M objects,.e., * * * 1 n q * () t maxmzes Σ V(, t q()) t subject to Σ q() t M, for all type realzatons t (t 1,,t n ) T. (When the effcent assgnment s not unque due to flat regons n the aggregate demand curve, q * (t) s chosen so that each q * () t s weakly ncreasng n t.) Gven effcent assgnment rule q * () t, let us defne: * { } tˆ ( t, y) = nf t q ( t, t ) y. (1) Thus, tˆ ( t, y) s the mnmum report that bdder can make and stll receve at least y unts n the effcent allocaton, n the event that her opponents report t. Fnally, t s useful to defne v (t,q ) as the margnal value of the (q ) th unt to bdders (and gven that the unts are allocated effcently among bdders ). The generalzed Vckrey aucton s now defned to be the drect mechansm n whch objects are assgned accordng to q * () t and the payment rule s defned as the followng extenson of Vckrey aucton 3

5 prcng: bdder pays the ( q * () 1 t + k) th hghest rejected value (other than her own) for her k th object where, crucally, values are evaluated for ths calculaton usng tˆ ( t, k) as the sgnal for bdder and usng t (the vector of actual reports) as the sgnals for bdder s opponents. 1 Observe that ths statc mechansm has the same general flavor as the Vckrey aucton. Any bdder s submtted bd does not determne the prce she pays (condtonal on wnnng the object), snce: (1) à la Vckrey, her payment s determned only by the opportunty cost of provdng her wth the object; and (2) n computng the opportunty cost, the bdder s actual reported sgnal s not used, but rather the lowest sgnal whch would enable her to wn the object. More formally, we defne: DEFINITION 1. Gven any effcent assgnment rule q * (t) such that q * () t s nondecreasng n t for each, the generalzed Vckrey aucton s the drect mechansm n whch bdders smultaneously report ther types and each bdder s assgned q * () t unts and s charged a payment X * () t computed by: n the case of dscrete unts, and computed by: X () t = v ( t ( t, k ), t, M + 1 k ), (2) * q * () t ˆ k= 1 n the case of perfectly-dvsble unts. We easly have the followng theorem: q * () * t ˆ 0 X () t = v ( t ( t, y), t, M y) dy, (3) THEOREM 1. For any valuaton functons v (t,q ) satsfyng value monotoncty and value regularty, and for any effcent assgnment rule q * (t) such that q * () t s nondecreasng n t for each, the generalzed Vckrey aucton has sncere bddng as an ex post equlbrum. PROOF. Snce q * (t) has the property that q * () t s nondecreasng n t for each, tˆ ( t, y) defned by Eq. (1) s nondecreasng n y. For the case of perfectly-dvsble unts, substtutng Eq. (3) nto the expresson, V (t,q ) X, for bdder s utlty yelds the followng ntegral for bdder s utlty from reportng her type as t when her true type s t and the other bdders true and reported types are t : q * ( t, t ) U (, ) (,, ) ( ˆ t t t = v t t y v t( t, y), t, M y) dy. 0 (4) 1 In ths paragraph, for the case of perfectly-dvsble unts, bdder then pays the ( q * () t y) (other than her own) for her y th object. th hghest rejected value 4

6 Observe that the ntegrand of Eq. (4) s ndependent of t, bdder s reported type; t enters nto Eq. (4) only through the upper lmt on the ntegral. Moreover, by value monotoncty, the ntegrand of Eq. (4) s nonnegatve for all y q * ( t, t ) and s nonpostve for all y maxmzed when the upper lmt on the ntegral equals q * ( t, t ). Hence, U( t t, t ) s q * ( t, t ), whch s attaned by sncere bddng. For the case of dscrete unts, the argument s analogous.! In the case of a perfectly-dvsble good, the expresson taken by the payment rule of the Vckrey aucton becomes stll smpler f the type space s contnuous, the valuaton functons v (t,q ) are contnuous for each, and f the zero types have zero margnal valuaton for the good. In that event, q * (0, t ) may always be taken to be zero, and for y > 0, we have that tˆ ( t, y) exactly satsfes: v ( tˆ( t, y), t, M y) = v ( tˆ( t, y), t, y). (5) Consequently, we mmedately have: PROPOSITION 1. For the case of contnuous types and a perfectly-dvsble good, consder any valuaton functons v (t,q ) satsfyng value monotoncty, value regularty, contnuty, and v (0,t,q ) = 0, for all, t and q, and any effcent assgnment rule q * (t) such that q * (0, t ) = 0 and nondecreasng n t for each and t. Then the generalzed Vckrey aucton has the payment rule: q * () t s q * () * t ˆ 0 X () t = v ( t ( t, y), t, y) dy. (6) Fnally, observe gven Eq. (6) that, under the assumptons of Proposton 1, Eq. (4) reduces to: q * ( t, t ) U ( t t, t ) = (,, ) ( ˆ v t t y v t ( t, y), t, y) dy. (7) 0 Eq. (7) has an emnently smple nterpretaton, n close keepng wth the tradtonal mechansm-desgn lterature. Bdder s precsely permtted to retan her nformatonal rents : her value for the y th unt s v ( t, t, y) ; she s requred to pay only v ( ˆ t( t, y), t, y), whch would be exactly her value f she were just the mnmal type who s assgned a y th unt. It s nterestng to observe that the equlbrum of the generalzed Vckrey aucton s not only a Bayesan-Nash equlbrum, but also an ex post equlbrum: gven that bdder knows the announcement, t, that bdders wll make (and beleves the announcement), bdder stll fnds t a best response to announce her true type. Gven that bdders wll not possess domnant strateges n an envronment wth nterdependent values, ths s about the strongest result we can hope for. Moreover, snce ths s an ex post equlbrum, observe that the outcome s ndependent of the jont dstrbuton of types. 5

7 Fnally, let us relax the nformatonal requrements placed on the auctoneer by specfyng a twostage revelaton procedure n whch the auctoneer need not know the mappng from sgnals to valuatons. (The mappng stll needs to known by all the bdders.) In the frst stage, each bdder smultaneously reports the mappng from sgnals to valuatons both for herself and all other bdders. If all of the bdders reports agree, then the auctoneer proceeds to calculate and carry out the generalzed Vckrey aucton for the reported mappng from sgnals to valuatons; f the bdders reports do not agree, then the auctoneer sends everybody home wth zero goods assgned and zero payments. Snce every bdder s nterm payoff n the generalzed Vckrey aucton s nonnegatve (n fact, every bdder s ex post payoff s nonnegatve), we easly have: THEOREM 2. For any valuaton functons v (t,q ) satsfyng value monotoncty and value regularty, and for any effcent assgnment rule q * (t) such that q * () t s nondecreasng n t for each, the two-stage procedure has truthful reportng n the frst stage and sncere bddng n the second stage as an equlbrum. CONCLUSION: COMPARISON WITH ASCENDING-BID AUCTIONS. For some symmetrc models wth nterdependent values, there exst ascendng-bd aucton procedures whch also yeld effcent allocatons. For a sngle ndvsble object, ths s provded by the Englsh aucton (Mlgrom and Weber, 1982). For M objects and unt demands, ths s provded by an ascendng-clock aucton whch ends at the moment the (M+1) st bdder drops out. For M objects and flat demands, ths s provded by my effcent ascendng-bd aucton desgn (Ausubel, 1997). It s nterestng to now observe how the outcome of the generalzed Vckrey aucton compares wth the outcome of the effcent ascendng-bd aucton under these crcumstances. Wth a sngle ndvsble object, the wnner s payment n the generalzed Vckrey aucton equals v ( ˆ t ( t,1), t,1), concdng wth the payment n the Englsh aucton. Thus, the equlbra are outcomeequvalent. However, f there are M 2 objects and f effcency requres that a postve quantty of objects be awarded to two or more of the bdders, then the outcomes dffer n a subtle way. In the generalzed Vckrey aucton, all of the prvate sgnals have been revealed to the medator, and the payment s then allowed to depend on all (n 1) prvate sgnals of the other bdders. By way of contrast, n the effcent ascendng-bd aucton desgns, some or all of the objects are awarded at a tme when two or more bdders reman n the aucton. Consequently, the payment then depends on (n 2) or fewer of the prvate sgnals of the other bdders. By the same logc as n Mlgrom and Weber (1982), n envronments where the bdders sgnals are strctly afflated, the generalzed Vckrey aucton uses more prvate sgnals and hence yelds hgher expected revenues. Indeed, as shown by Perry and Reny (1999), t yelds an upper bound for the expected revenues of an effcent ex post equlbra, and thus, t s the approprate benchmark for comparng the revenues of the effcent ascendng-bd aucton desgns. 6

8 At the same tme, the generalzed Vckrey aucton has a serous dsadvantage relatve to the effcent ascendng-bd auctons. To paraphrase Maskn (1992, p. 127, footnote 3), the reader should notce that the rules of the generalzed Vckrey aucton are defned n terms of the mappngs from sgnals to valuatons. That s, the aucton desgner must know these mappngs (à la Theorem 1), a demandng requrement, or ask them and obtan consstent responses (à la Theorem 2), stll a reasonably mplausble task. By contrast, the desgner can reman gnorant of the mappngs f he uses the effcent ascendng-bd desgn, a dstnct advantage that may well more than offset the theoretcal dsadvantage n expected revenues. References AUSUBEL, LAWRENCE M, An Effcent Ascendng-Bd Aucton for Multple Objects, Workng Paper No , Unversty of Maryland, Department of Economcs, June AUSUBEL, LAWRENCE M. AND PETER C. CRAMTON, Vckrey Auctons wth Reserve Prcng, Mmeo, Unversty of Maryland, Department of Economcs, June DASGUPTA, PARTHA AND ERIC S. MASKIN, Effcent Auctons, forthcomng, Quarterly Journal of Economcs. JEHIEL, PHILIPPE AND BENNY MOLDOVANU, Effcent Auctons wth Interdependent Valuatons, Workng Paper No , Unversty of Mannhem, July MASKIN, ERIC S., Auctons and Prvatzaton, n Horst Sebert, ed., Prvatzaton: Symposum n Honor of Herbert Gersch, Tubngen: Mohr (Sebeck), 1992, pp MILGROM, PAUL R. AND ROBERT J. WEBER, A Theory of Auctons and Compettve Bddng, Econometrca, September 1982, 50(5), pp PERRY, MOTTY AND PHILIP J. RENY, An Ex-Post Effcent Aucton, Workng Paper, Hebrew Unversty and Unversty of Chcago, July VICKREY, WILLIAM, Counterspeculaton, Auctons, and Compettve Sealed Tenders, Journal of Fnance, March 1961, 16(1), pp