The Vckrey-Target Strategy and the Core n Ascendng Combnatoral Auctons Ryuj Sano ISER, Osaka Unversty Prelmnary Verson December 26, 2011 Abstract Ths paper consders a general class of combnatoral auctons wth ascendng prces, whch ncludes the Vckrey-Clarke-Groves mechansm and core-selectng auctons. We analyze ncentves n ascendng combnatoral auctons under complete and perfect nformaton. We show that n every ascendng aucton, the Vckrey-target strategy consttutes a subgame perfect equlbrum wth a restrcted strategy space. The equlbrum outcome s n the bdder-optmal core and unque under some crtera. Ths mples equlbrum selecton s done by an ascendng prce scheme from many equlbra of sealed-bd auctons. The equlbrum outcome s unfar n the sense that wnners wth low valuatons tend to earn hgh profts. The payoff non-monotoncty may lead to neffcency n the equlbrum under unrestrcted strategy space. JEL classfcaton: D44, C7 Keywords: combnatoral aucton, ascendng prce, the Vckrey aucton, coreselectng aucton, core Insttute of Socal and Economc Research, Osaka Unversty, 6-1 Mhogaoka, Ibarak, Osaka 567-0047, Japan. E-mal: r-sano@ser.osaka-u.ac.jp 1
1 Introducton Ths paper formulates a broad class of aucton mechansms wth ascendng prces. We ntroduce the ascendng Vckrey-reserve auctons and analyze the subgame perfect equlbrum under complete nformaton. The Vckrey-reserve auctons are a class of combnatoral (or package) auctons, whch ncludes the Vckrey-Clarke-Groves mechansm and core-selectng auctons. Snce the U.S. Federal Communcatons Commsson conducted a spectrum lcense aucton n 1994, the theory of mult-object auctons has been attracted a great deal of attenton. In recent decades, a consderable number of studes have been conducted on the desgns and analyses of mult-object auctons, especally combnatoral auctons. Combnatoral auctons are those n whch bdders can make bds for bundles or packages of goods, not just each ndvdual good. Although such auctons are generally complcated n practcal uses, they are actually beng mplemented n spectrum lcense auctons n several countres and proposed for auctons of arport landng slots n these days. 1 In the lterature of combnatoral aucton, ncentves and equlbra n several aucton mechansms have been examned under sealed-bd formats, or drect revelaton mechansms. The Vckrey-Clarke-Groves mechansm (the Vckrey aucton) s an mportant benchmark. The Vckrey aucton s known as an effcent mechansm such that s ncentve compatble n domnant strategy (Green and Laffont, 1977; Holmstrom, 1979). However, some studes pont that the Vckrey aucton has several dsadvantages such as low revenue and vulnerablty to collusve bddngs (Ausubel and Mlgrom, 2006). In practcal combnatoral aucton desgns, the Vckrey aucton has hardly been used ever. Core-selectng auctons are recent attractve alternatves to the Vckrey aucton. A core-selectng aucton selects an outcome n the core wth respect to the reported valuatons. The core-selectng property avods some dsadvantages of the Vckrey auctons. Although core-selectng auctons are not ncentve compatble, Day and Mlgrom (2008) show that they acheve an outcome n the core n a Nash equlbrum under complete nformaton. Ths fact forms a theoretcal foundaton for applyng core-selectng auctons. Recently, they have been used for spectrum lcenses auctons n the U.K. and several European countres. 1 See Cramton (2009) for the applcatons of combnatoral auctons to spectrum lcense auctons. 2
On the other hand, from the vewpont of practcal aucton desgn, ascendngprce auctons are frequently preferred to sealed-bd auctons. Dynamc, open-bd format s transparent and economzes revealed nformaton about valuatons durng the aucton. For example, the U.S. Federal Communcaton Commsson frst adopted the smultaneous ascendng aucton (SAA) for spectrum auctons, n whch tems are put on sale smultaneously usng an ascendng-prce rule. Bdders can submt new bds for any tem f a new bd s submtted for some tem. An ascendng-prce format s standard for spectrum lcense auctons n many countres. The ascendng-prce, open-bd format s more mportant when package bds are allowed, snce a sealedbd format often requres bdders to submt an exponental number of package bds. Many studes nvestgate and propose varous combnatoral aucton desgns wth ascendng-prce formats. 2 Most studes try to formulate ascendng-prce Vckrey auctons for multple objects;.e., ascendng auctons whch termnate wth the Vckrey-Clarke-Groves outcome. Such auctons correspond to a standard Englsh aucton of sgle object, and bdders reveal true nformaton on valuatons n the equlbrum. Parkes and Ungar (2000), Ausubel and Mlgrom (2002), and de Vres et al. (2007) formulate ascendng combnatoral auctons wth non-lnear and non-anonymous prces. Ther auctons are ascendng Vckrey auctons for substtute goods, whereas they are not for general valuatons. However, they are ascendng core-selectng auctons for general valuatons. These auctons wll be desrable f we prefer ascendng-prce formats and coreselectng auctons. However, a natural queston s what the subgame perfect equlbrum of an ascendng aucton s, f t s not an ascendng Vckrey aucton. Unfortunately, the equlbrum of those ascendng combnatoral auctons has never been examned. In ther semnal paper, Ausubel and Mlgrom (2002) formulate an ascendng combnatoral aucton, however, they consder proxy bddng n the equlbrum analyss and do not examne the ncentves durng the ascendng-prce procedure. Is there a subgame perfect equlbrum whch acheves an outcome n the core n ascendng core-selectng auctons? Ths paper answers ths queston and shows that every ascendng core-selectng aucton has a subgame perfect equlbrum n the bdder-optmal core wth a lmted 2 See Parkes (2006) for a revew of several desgns and the advantages of ascendng auctons over sealed-bd auctons. 3
strategy space. Moreover, we show that the dentcal equlbrum exsts n a broader class of ascendng combnatoral auctons. We consder a general form of ascendng prce combnatoral aucton wth a sngle prce path of non-lnear and non-anonymous prce vector. We allow arbtrary ascendng prce scheme and possble fnal dscounts;.e., payments may be dfferent from the termnal prces. We ntroduce ascendng Vckrey-reserve auctons, n whch bdders pay at least ther Vckrey payments wth respect to the revealed nformaton on valuatons. Our model ncludes most of auctons n the lterature, such as Parkes and Ungar (2000), Ausubel and Mlgrom (2002), de Vres et al. (2007), and Mshra and Parkes (2007). 3 We focus on a class of dynamc strategy, sem-truthful strategy, whch corresponds to the truncaton strategy by Day and Mlgrom (2008) n sealed-bd auctons. In a sem-truthful strategy, a bdder ether reports hs true demand or stops bddng at each perod. The stoppng tmngs of bdders generally depend on the others behavor. In most of the paper, we consder the subgame perfect equlbrum (SPE) wth the strategy space restrcted to sem-truthful strateges. We have three man results. Frst, we show that a partcular strategy, whch we call the Vckrey-target strategy, consttutes an subgame perfect equlbrum. In ths strategy, a bdder ams to bd up to hs constraned Vckrey prce. Ths strategy s free from the specfcatons of the aucton rules. The equlbrum outcome s n the bdder-optmal core 4 wth respect to the true valuatons. Ths result s smlar to that of Day and Mlgrom (2008), who show a partcular strategy profle as a Nash equlbrum of every core-selectng aucton. Second, we show that the specfed equlbrum outcome s a unque equlbrum outcome under certan condtons n every strct Vckrey-reserve aucton, n whch wnners pay amounts strctly more than the Vckrey prce. Ths result contrasts wth the fact that there are possbly many Nash equlbra n sealed-bd Vckreyreserve auctons. Equlbrum selecton s done to some extent by ntroducng an ascendng-prce format and subgame perfecton. Thrd, although the outcome by the Vckrey-target strateges s n the bdderoptmal core, t s unfar n the sense that the lower the valuaton of a wnner, the hgher are the profts he tends to earn. The payoff non-monotoncty leads to a possblty that the Vckrey-target strategy may not consttute an SPE wth unre- 3 Ausubel s (2006) aucton uses multple prce paths, and t s an excepton. 4 A core outcome s bdder-optmal f t s Pareto-optmal among bdders. 4
strcted strategy space. Moreover, we show that an SPE wth unrestrcted strategy space may be neffcent. The ntuton of these results s smple. In an ascendng aucton, the prces of goods ncrease gradually from the ntal low values. Bdders decde whether to contnue bddng or not at each perod. Note that there exsts a best core outcome for each bdder, n whch he obtans the Vckrey payoff. If a bdder stops bddng at hs Vckrey prce, and f the aucton fnally selects the effcent allocaton, he wll be able to wn the goods wth the Vckrey prce. Hence, by stoppng at the Vckrey prce, he wll defntely earn the Vckrey payoff, whch s the best payoff n the core. The Vckrey payments of the wnners wth lower valuatons are generally lower as compared to those of the hgh-value wnners. Hence, the prces frst reach the Vckrey prces of low-value wnners, and hence, low-value wnners acheve ther most preferred outcomes. When a wnner stops bddng, the remanng bdders need to rase ther bds even further to wn. Hgh-value bdders tend to pay dearly and earn lttle net proft. It s qute restrctve to focus only on sem-truthful strateges. However, the analyss of ths paper s appled under the unrestrcted strategy space f bdders are sngle-mnded,.e., they are nterested only n a partcular bundle and place only bds for that bundle. The contrbuton of the paper s as follows. Frst, we consder a general class of combnatoral auctons wth ascendng prce and show an equvalence n an equlbrum strategy under complete nformaton. Wth a restrcted strategy space, every ascendng combnatoral aucton has a subgame perfect equlbrum n bdder-optmal core wth respect to true valuatons. Ths corresponds to the precedng results on sealed-bd combnatoral auctons by Bernhem and Whnston (1986), Ausubel and Mlgrom (2002), Day and Mlgrom (2008). Second, we show that the equlbrum outcome s unque wth some crtera. Ths contrasts wth the multple equlbra of sealed-bd combnatoral auctons. As Mlgrom (2007) dscusses, the precedng analyses are not satsfyng even f we accept the strong assumpton of complete nformaton, snce there are many plausble equlbra. Our result can be nterpreted as an equlbrum selecton and ndcate whch outcome n the core s the most plausble, f we stll assume complete nformaton. Fnally, we show some negatve propertes such as non-monotoncty of the equlbrum payoff and possble neffcency n an equlbrum wth unrestrcted strategy space. In practcal uses, we need to consder 5
that the ascendng prce formats do not always perform well. 1.1 Related Lterature As we have mentoned, varous ascendng-prce auctons are proposed by Parkes and Ungar (2000), Ausubel and Mlgrom (2002), Ausubel (2006), de Vres et al. (2007), and Mshra and Parkes (2007). All these auctons termnate wth the Vckrey outcome and ncentve compatble for substtutes goods. Even for general valuatons, they are core-selectng auctons except Ausubel (2006) and Mshra and Parkes (2007). de Vres et al. (2007) show that t s mpossble to desgn an ascendng aucton whch converges to the Vckrey outcome under general valuatons. Mshra and Parkes (2007) ntroduce fnal dscounts after ascendng prce procedure and provde a class of the ascendng Vckrey auctons for general valuatons. The condtons for the Vckrey outcome beng n the core are studed by Bkhchancan and Ostroy (2002) and Ausubel and Mlgrom (2002). On nformaton requrement, Mshra and Parkes (2007) show the necessary and suffcent condton for computng the Vckrey outcome from the aucton outcome. Matsushma (2011) provdes another necessary and suffcent condton for mplementng the Vckrey outcome usng a general prce-based scheme. He also shows the necessary and suffcent condton for mplementng a strategy-proof and nterm ndvdually ratonal mechansm usng a prce-based scheme. Blumrosen and Nsan (2010) show that we need non-lnear and non-anonymous prces to acheve effcency by ascendng auctons n general valuatons. Equlbrum analyses of combnatoral auctons are conducted manly under complete nformaton and sealed-bd formats. Bernhem and Whnston (1986) consder the frst-prce combnatoral aucton called menu aucton. They show that there exst possbly many full-nformaton Nash equlbra n the core. Ausubel and Mlgrom (2002) consder the ascendng proxy aucton, where bdders report ther valuatons n advance to ther proxy agents. They show that Bernhem and Whnston s (1986) Nash equlbrum s also a Nash equlbrum n the ascendng proxy aucton. Day and Mlgrom (2008) generalze these results to every sealed-bd core-selectng aucton. Sano (2011b) further generalze Day and Mlgrom (2008). Sano shows that Bernhem and Whnston s (1986) Nash equlbrum exsts f and only f bdders pay at least ther Vckrey payments. 6
There are several studes on the analyss of ascendng-prce non-package auctons. Ausubel and Schwartz (1999) and Grmm et al. (2003) study the subgame perfect equlbrum n a mult-unt ascendng aucton wth complete nformaton. They consder a mult-unt, ascendng unform-prce aucton wth two bdders. They show that there s a unque low-prce subgame perfect equlbrum. Ther low-prce equlbrum stems from the demand reducton or mplct collusons by bdders n mult-unt unform-prce auctons (Engelbrecht-Wggans and Kahn, 1998; Ausubel and Cramton, 2002). The remander of ths paper proceeds as follows. In secton 2, we provde a smple example and explan the ntuton of the results. In secton 3, we formulate the model and the aucton. We defne a Vckrey-reserve aucton and ntroduce an ascendng-prce format. In secton 4, we show that the Vckrey-target strateges consttute an equlbrum and lead to an outcome n the bdder-optmal core. Then, we examne the unqueness of the equlbrum and the equlbrum selecton. In secton 5, we dscuss the results. If an ascendng aucton s core-selectng, t s robust to collusve overbddngs. In addton, we show the non-monotoncty of the equlbrum payoff. The payoff non-monotoncty leads to neffcency n the SPE wth unrestrcted strategy space. 2 An Illustraton We frst look at a smple example of two goods and three bdders. Example 1. There are two goods {A, B}, and three bdders {1, 2, 3}. Suppose that bdder 1 wants only good A, whereas bdder 2 wants only B. Bdder 3 wants both A and B. Bdder 1 s wllngness to pay for A s 7, and 2 s wllngness to pay for B s 8. Bdder 3 s wllngness to pay s 10 for AB, and 0 for each good. In the effcent allocaton, bdders 1 and 2 get A and B, respectvely. The core of the aucton game s descrbed as follows: p 1 (A) 7, p 2 (B) 8, and p 1 (A) + p 2 (B) 10, where p (k) denotes the payment of for good(s) k. In the bdder-optmal core, p 1 (A) + p 2 (B) = 10. Frst, consder a sealed-bd core-selectng aucton. Assumng bdder 3 truthfully places a bd 10 for the package AB, every bd profle (b 1, b 2 ) such that b 1 + b 2 = 10, b 1 7, and b 2 8, s a Nash equlbrum. In the equlbrum, each wnnng bdder pays b ( = 1, 2) by the core-selectng prcng rule. Thus, any bdder-optmal core 7
Fgure 1: The subgame perfect equlbrum path outcome s acheved n a Nash equlbrum (Day and Mlgrom, 2008). 5 Notably, these strategy profles are also Nash equlbra of the Vckrey aucton (Sano, 2011b). Next, consder an ascendng aucton by Parkes and Ungar (2000) and Ausubel and Mlgrom (2002). The aucton starts from zero prces, and bdders gradually rase the bds. 6 Suppose that bdders 1 and 2 submt bds for each sngle good, p t 1 (A) and p t 2 (B), and that bdder 3 can submt package bds for AB, pt 3 (AB). At perod 1, each bdder places bds of 1 for hs nterest: p 1 1 (A) = p1 2 (B) = p1 3 (AB) = 1. Then, bdders 1 and 2 are tentatve wnners, so that bdder 3 rases the bd at perod 2: p 2 3 (AB) = 2. If bdder 3 becomes the tentatve wnner at perod 2, bdders 1 and 2 rase the bds at perod 3, and so on. Suppose that all bdders behave truthfully and rase the bds untl ther true values. Then, the aucton termnates when p 1 (A) = p 2 (B) = 5 and p 3 (AB) = 10. Bdders 1 and 2 wn goods A and B, respectvely wth the prce of 5. Note that the outcome s n the core. 7 Let us consder the subgame perfect equlbrum of the aucton. It s natural to assume that once a bdder stops bddng at t, he can no longer rase the bds. We can easly obtan the equlbrum by the standard manner of the backward nducton. Suppose that bdder 3 behaves truthfully and rases the bds untl p 3 (AB) = 10. 5 The specfcaton of a prcng rule does not matter. The core wth respect to the reported bds s unquely determned n the equlbrum. 6 Bdders are not allowed to jump bds. 7 The fnal prces may dffer by the bd ncrement. However, we gnore t. 8
Consder a subgame n whch bdder 2 frst stops bddng at 3 p 2 (B) < 5. Snce 10 p 2 (B) 7, bdder 1 successfully wns the good by bddng untl 10 p 2 (B). Note that the prce 10 p 2 (B) s 1 s Vckrey prce, gven that 2 s value s p 2 (B). Hence, for bdder 1, the Vckrey-target strategy s to bd up untl 10 p 2 (B), and t s optmal when p 2 (B) 3. When bdder 2 stops bddng at p 2 (B) < 3, bdder 1 has to pay 10 p 2 (B) > 7 n the case of wnnng. Hence, t s optmal for 1 to stop at p 1 (A) = 7 and lose. Smlarly, for bdder 2, the Vckrey-target strategy, bddng untl 10 p 1 (A), s optmal when p 1 (A) 2. Now, go back to a subgame where nobody has stopped yet. Applyng the consderaton above, bdder 1 wll wn as long as he bds untl p 1 (A) 10 8 = 2. On the other hand, bdder 2 wll wn when he frst stops at p 2 (B) 10 7 = 3. To mnmze the payment, bdder 1 s best strategy s to bd untl p 1 (A) = 2, and 2 s untl p 2 (B) = 3. These prces are ther Vckrey prces gven true values. Thus, t s a perfect equlbrum to stop at the Vckrey prces that are computed from bdders true values and stoppng prces. In the equlbrum outcome, bdder 1 stops frst at p 1 (A) = 2 and bdder 2 rases bds untl p 2 (B) = 8. Ths outcome s n the bdder-optmal core. In addton, by nspecton, ths s a unque subgame perfect equlbrum outcome as long as bdder 3 behaves truthfully. Fgure 1 llustrates the equlbrum path of ths example. 3 The Model A seller wants to allocate multple ndvsble objects, and K denotes the set of the goods. Let N {0, 1, 2,..., n} be the set of all players. I = {1,..., n} s the set of all bdders and 0 denotes the seller. Let X 2 K be the set of admssble bundles for bdder. For each I, a null bundle s denoted by x, and x X. X X 1 X n denotes the set of feasble allocatons. All bdders have quaslnear utltes. Suppose that valuatons for bundles of goods are nteger-valued. Let u : X Z + be a bdder s valuaton functon. Suppose each u s monotone and u (x ) = 0 for all I. Bdder earns a payoff π = u (x ) p where x X denotes goods allocated to and p s the monetary transfer to the seller. The seller s payoff s the revenue from the aucton: π 0 = I p. Let X (u) X be the set of effcent allocatons wth respect to the profle of 9
valuaton functons u = (u ) I : X (u) arg max x X u (x ). (1) I The coaltonal value functon V s defned by max x X J V (J, u) = u (x ) f 0 J 0 f 0 J, (2) where J N and u 0 ( ) 0. V (J, u) denotes the total surplus that a coalton J can produce. We sometmes use the notaton of V ( ) nstead of V (, u). Gven a valuaton profle u, a payoff profle π R n+1 s feasble f N π V (N). A payoff profle π s ndvdually ratonal f π 0. The core of the aucton game s Core(N, V ) = { π 0 π = V (N) and ( J N) π V (J) }. (3) N J A payoff profle π Core(N, V ) s bdder-optmal f there s no π Core(N, V )\{π} such that π π for all I. Let BOC(N, V ) Core(N, V ) be the set of bdderoptmal core payoff profles. 3.1 Vckrey-Reserve Auctons Before we defne a class of ascendng-prce auctons, we ntroduce sealed-bd auctons or drect revelaton mechansms. In a sealed-bd aucton (ḡ, p), each bdder reports a valuaton functon û. For a profle of valuaton functons û = (û ) I, the outcome of the aucton s (ḡ(û), ( p (û)) I ) (X, R n +), whch specfes the choce of an allocaton x = ḡ(û) and payments p (û) R +. A sealed-bd aucton (ḡ, p) s effcent f for all û, ḡ(û) X (û). In addton, (ḡ, p) s ndvdually ratonal f for all û and x = ḡ(û), p (û) û (x ) for all I. 8 Bdder s sad to be a wnner f ḡ (û) x. Conversely, s sad to be a loser f ḡ (û) = x. Let ˆV ( ) V (, û), whch s the coaltonal value functon wth respect to û. Gven an aucton mechansm (ḡ, p) and a report profle û, let ˆπ û (ḡ (û)) p (û) for each bdder and ˆπ 0 π 0 = p (û) for the seller. The aucton mechansms n the exstng lterature are defned as follows. 8 Let û (x ) 0 for all I. Snce p R +, ndvdual ratonalty mples that every bdder assgned the null bundle pays 0. 10
Defnton 1 A sealed-bd aucton (ḡ, p V ) s the Vckrey aucton f t s effcent and for each I, p V (û) = ˆV (N ) û j (ḡ j (û)). (4) j In addton, π denotes bdder s Vckrey payoff : π u (ḡ (u)) p V (u) = V (N, u) V (N, u). Defnton 2 A sealed-bd aucton (ḡ, p) s core-selectng f t satsfes û, ˆπ Core(N, ˆV ). Defnton 3 A sealed-bd aucton (ḡ, p) s of Vckrey-reserve f t s effcent, ndvdually ratonal, and û, p(û) p V (û). In addton, t s a strct Vckrey-reserve aucton f t satsfes p (û) > p V (û) as long as pv (û) < û (ḡ ). Note that each bdder s payoff n the truth-tellng equlbrum n the Vckrey aucton π s the upper bound of the payoffs acheved n the core. 9 Ths mples that every core-selectng aucton s a Vckrey-reserve aucton. Bkhchandan and Ostroy (2002) show that f goods are substtutes, core-selectng auctons and Vckrey-reserve auctons are equvalent. 10 When goods may be complements, the equvalence does not hold. 3.2 Ascendng Auctons An ascendng-prce format s ntroduced to Vckrey-reserve auctons. Followng Parkes and Ungar (2000), de Vres et al. (2003), and Mshra and Parkes (2007), we consder complex prces, whch are non-lnear and non-anonymous. Ths means that a prce of a bundle x for, whch s denoted by p (x ), does not have to be the sum of prces of each ndvdual object. Moreover, the prce for a bundle can be dfferent between bdders. A non-lnear and non-anonymous prce vector p s n P X R+. We suppose p (x ) 0 for all. Blumrosen and Nsan (2010) show that we need a complex prce vector to conduct an ascendng aucton whch fnds an effcent allocaton. Gven a prce vector p, let D (p) be s (true) demand set: D (p) {x X u (x ) p (x ) u (y ) p (y ) y X }. (5) 9 See Ausubel and Mlgrom (2002) and Bkhchandan and Ostroy (2002). 10 To be precse, the equvalence holds f and only f bdders are substtutes. 11
In an ascendng aucton, the auctoneer proposes a prce vector p t at each perod t. Each bdder responds wth hs demand set ˆD (p t ). The auctoneer then adjusts the prce vector and repeats the process. Bdder s sad to be actve at t f for all τ t, x ˆD (p τ ). Let I t I be the set of all actve bdders at t. Actve bdders are defned above because f x D, he has a non-null bundle x whch earns a postve payoff under the current prce: u (x ) p (x ) > 0. Thus, he can afford to pay more for that bundle. In ths paper, we defne ascendng combnatoral auctons n a general form n the sense that we do not specfy the detal of the rule n the followng three ways. Frst, although we fx the prce ncrement by unty, the selectons of bdders facng prce ncreases at each perod are arbtrary. Second, we do not specfy when the aucton termnates. We allow varous condtons for stoppng prce ncreases to consder both the Vckrey and core-selectng prcng. Thrd, bdders payments can be dfferent from the prces n the termnal perod. Bdders payments may be dscounted from the termnal prces, and the dscountng rule s arbtrary wth mld condtons. Our defnton of the aucton extends Mshra and Parkes (2007) n the thrd way. Now we defne ascendng combnatoral auctons n a general form. Our defnton follows that of Mshra and Parkes (2007). 1. The auctoneer ntalzes the prce vector as p 1 = (0,..., 0). 2. At each perod t = 1, 2,..., each bdder reports hs demand set ˆD (p t ). The auctoneer chooses a set of actve bdders J t I t. If J t and f x ˆD (p t ), then p t+1 (x ) = p t (x ) + 1. Else, let p t+1 (x ) = p t (x ). 3. Repeat the process. It termnates at T T, when I T =. The auctoneer selects an allocaton x X and determnes bdders payments p R n +. Let (g, (p ) I ) be the mechansm of the ascendng aucton, whch decdes the fnal allocaton g(h) X and the payments (p (h)) I R n +, where h H denotes a hstory throughout the ascendng aucton. 11 Note that bdders payments do not have to be the posted prces at the termnal perod. We focus on auctons whch lead to an effcent allocaton wth respect to reported nformaton. Although we do not specfy the condton for the termnaton 11 Mshra and Parkes (2007) and de Vres et al. (2007) defne an ascendng aucton as (g, p) s determned from only (p T, (D (p T )) I ). Our defnton allows the auctoneer to determne an outcome usng all the nformaton durng the aucton. 12
of the prce adjustment, we need at the termnal perod T, the auctoneer specfes a compettve equlbrum from the hstory of prces and demand sets (Mshra and Parkes, 2007). Ths means that the auctoneer can select an allocaton x X such that for all I, and x ˆD (p T ) x arg max p T (x ). X However, t s not suffcent to mplement the Vckrey outcome, and the auctoneer needs to contnue prce ncrease even f p t s a compettve equlbrum. 12 dfferent termnal condtons wll be adopted by dfferent goals. Thus, Defne û : X R + for each I as follows. If x ˆD (p T ), let û ( ) p T ( ). If x ˆD (p T ), then let p T û (x ) (x ) + 1 f x ˆD (p T ). (6) 0 otherwse Obvously, each û s consstent wth the all demand sets reported by, and û can be nterpreted as the representatve valuaton functon (Matsushma, 2011). The effcency and the ndvdual ratonalty n the ascendng aucton are defned wth respect to û smlarly to sealed-bd auctons. In addton, ˆV and ˆπ are also smlarly defned. Defnton 4 An ascendng aucton s an ascendng Vckrey aucton f t s effcent and p(h) = p V (û) for all h H. An ascendng aucton s core-selectng f h, ˆπ Core(N, ˆV ). An ascendng aucton s of Vckrey-reserve f t s effcent, ndvdually ratonal, and p(h) p V (û) for all h. The defnton of û follows from the termnaton of the aucton. Hence, for each aucton, a proper termnal condton s appled. In order to avod confuson regardng the termnal condtons and the correspondng defntons of û, t s convenent to modfy the defnton of the termnaton of the ascendng prces as follows: Modfed Termnal Condton. The aucton termnates at T f I T =. 12 To mplement the Vckrey outcome, the auctoneer needs to fnd a unversal compettve equlbrum (Mshra and Parkes, 2007). 13
We allow the fnal outcome to be determned from all the nformaton durng the aucton. Hence, the formerly defned aucton mechansms are compatble to the modfed defnton by nterpretng the perods after the true termnaton, T + 1, T +2,..., T, as a fcttous game whch s rrelevant to the fnal outcome. Under the modfed termnaton, we suppose that all bdders have to fully reveal ther valuaton functons. Our defnton of ascendng auctons are motvated not by proposng a specfc ascendng aucton desgn, but rather by provdng a general model for analyzng the proposed desgns. Selectons of J t specfes the ascendng prce procedure n detal. A specfcaton of J t s selectng tentatve losng bdders. The auctoneer selects a revenuemaxmzng allocaton x(t) X ( ( ˆD 1 {x 1 }) ( ˆD n {x n }) ) at each perod. Then J t s defned as J t = {j I t x j (t) = x j }. Ths specfcaton s ntutve and proposed by Parkes and Ungar (2000) and Ausubel and Mlgrom (2002). Other studes specfy J t as the mnmally undersuppled bdders (de Vres et al. 2007). Durng the aucton, bdders are restrcted by the followng actvty rule n order that there exsts a valuaton functon consstent wth a bdder s behavor. We follow the actvty rule consdered by Mshra and Parkes (2007). Assumpton 1 (Actvty Rule) Each bdder must satsfy the followngs: 1. If p s = pt, ˆD (p s ) = ˆD (p t ). 2. For all t, ˆD (p t ) ˆD (p t+1 ). 3. If x x and x ˆD (p t ), then x ˆD (p t ). The frst rule requres that f the prces reman the same for a bdder, he must report the same demand set. Equvalently, only bdders who face prce ncreases make new decsons at each perod. The second one should be satsfed when there s a valuaton functon û consstent wth the collecton of demand sets. Every bundle demanded at t has to be demanded at t + 1 because the prce of the bundle s ncreased by only the mnmum ncrement. The thrd one requres that reports have to be consstent wth monotoncty of valuaton functons. 13 To smplfy the analyss, we assume that bdders make choces sequentally. Ths assumpton s crucal for the unqueness result of the equlbrum. However, as we wll dscuss later, t wll not be essental to the results. 13 See Mshra and Parkes (2007) for the suffcency of ths actvty rule. 14
Assumpton 2 (Sequental Decsons) Bdders make choces sequentally from 1 to n. Each bdder observes all actons made before hs decson at each perod. 3.3 Strategy and Equlbrum In sealed-bd auctons, precedng studes (Bernhem and Whnston, 1986; Ausubel and Mlgrom, 2002; Day and Mlgrom, 2008) focus on a class of strateges: truncaton strateges. 14 A strategy û s sad to be α truncaton of u f α 0, x X, û (x ) = max{u (x ) α, 0}. That s, a bdder understates a value for each bundle of the goods by a fxed amount α. For every Vckrey-reserve aucton, a profle of truncaton strategy consttutes a Nash equlbrum. Proposton 1 (Sano, 2011b) For every u and every π BOC(N, V ), the profle of π truncatons of u s a Nash equlbrum of every sealed-bd Vckrey-reserve aucton. The assocated equlbrum payoff profle s π. In an ascendng aucton, bdder s (pure) strategy σ s a mappng from hs nformaton sets or s decson nodes E to 2 X. A feasble strategy s one satsfyng the Actvty Rule. Σ denotes the set of feasble strateges for. Let Σ Σ 1 Σ n be the set of profles of feasble strateges. We focus on the followng sem-truthful strateges, whch corresponds to the concept of truncaton strategy. 15 Defnton 5 A strategy σ Σ s sem-truthful f t satsfes t, ˆD (p t ) {D (p t ), X }. (7) Let Σ Σ be the set of sem-truthful strateges, and let Σ Σ 1 Σ n. The termnology truthful s adopted snce bdder reports hs true demand set as long as he s actve. Once reports ˆD = X ( x ), he cannot renew hs demand set any longer. Hence, bdder s sad to stop at t f ˆD (p t 1 ) X and f ˆD (p t ) = X. A sem-truthful strategy does not necessarly report the true valuatons, snce bdders 14 The termnology of truncaton strategy s adopted n Day and Mlgrom (2008). The truncaton strategy s also called truthful strategy (Bernhem and Whnston, 1986), sem-sncere strategy, and proft-target strategy (Ausubel and Mlgrom, 2002). 15 Ausubel and Mlgrom (2002) use the termnology of lmted straghtforward bddng for our sem-truthful strategy. 15
may stop before prces reach ther true valuatons. Sem-truthful strateges n an ascendng aucton correspond to truncaton strateges n a sealed-bd aucton. Lemma 1 A bdder follows σ Σ x X, Proof. See Appendx B. f and only f there exsts α 0 and for û (x ) = max{u (x ) α, 0}. (8) In every sem-truthful strategy, bdders report ther true valuatons or understate and never bd over ther true values. We allow bdders playng overbddng strateges consstent wth sem-truthful strateges as follows: ˆD (p t ) {X \ {x }, X } f x D (p t ). Let Σ + Σ be the set of sem-truthful strateges and overbddng strateges consstent wth sem-truthful strateges, and let Σ + Σ + 1 Σ + n. In the most of the paper, we restrct each bdder s strategy space to Σ or Σ +. We consder the truthful perfect equlbrum as an equlbrum concept. A truthful perfect equlbrum s an SPE wth respect to Σ +. Defnton 6 A strategy profle σ Σ + s a truthful perfect equlbrum (TPE) f t s a subgame perfect equlbrum under the condton whch each bdder s strategy space s restrcted to Σ +. 4 Man Results Snce each bdder makes choces sequentally, the aucton s a perfect nformaton game. To make t clear, we relabel the tme by each bdder s decson node. We refer to bdder s descon node at perod s as perod t (= n(s 1) + ). 4.1 Further Notatons and Assumptons Let u t : X R + be the provsonal valuaton functon at t, whch s the possble valuaton functon gven bddng behavor up to t: for each x X u t max{u (x ), p t (x ) (x ) + 1 {x ˆD (p t )} } f s actve at t. (9) p t (x ) otherwse When the strategy space s restrcted to Σ, the prce vector never exceeds the true valuaton functon. Then, the provsonal valuaton functon s equvalent to s true 16
valuaton functon f he s actve at t, and otherwse t concdes wth the reported valuaton û. Gven u t = (u t ) I, let V t ( ) V (, u t ) for smplcty. In addton, let π t be bdder s Vckrey payoff wth respect to ut : π t = V t (N) V t (N ). Let X t X (u t ) be the set of effcent allocatons wth respect to u t, and let X t {x X x X t }. We mpose two addtonal assumptons. One s regardng the aucton rule. To smplfy the analyss and sharpen the results, we consder the followng te-breakng rule. Assumpton 3 For each x X (û), defne t(x) mn{t ( s t) x X s }. Then, g(h) arg mn x X (û) t(x). In aucton models wth complete nformaton, tes are lkely to occur, and an equlbrum may fal to exst wth random te-breakng when strategy space s contnuous. Hence, tes are tradtonally broken n a way that depends on bdders values and not only on ther bds. For example, n a frst-prce aucton of a sngle object, the hghest two bdders submt the same bd n a Nash equlbrum, whch s the value of the second hghest bdder. In the analyss, we assume that the bdder wth the hgher value s chosen n the result of te-breakng. Ths practce s accepted because the selected outcome s the lmt of an equlbrum of an aucton n whch bddng s dscrete wth an ncrement ɛ > 0. 16 Snce the strategy space n our model s dscrete, we do not have to care about tes actually. However, from a vewpont that our model can be converted nto a contnuous case by takng a lmt of small prce ncrement, we follows ths practce. Indeed, wth Assumpton 3, we sharpen the results and have a strkng property wth respect to bdder-optmalty. In Appendx A, we construct a TPE wthout Assumpton 3. Another assumpton s regardng bdders behavor. Assumpton 4 Let (x, p ) ndcate obtanng x wth a payment p. Suppose that for any non-null bundle x, there s a set of alternatves C {(x, 0), (x, u (x ))}. Then, every bdder chooses (x, 0) wth probablty 0. Assumpton 4 mples that f a bdder expects that he can wn a bundle x by placng the bd of u (x ), he actually does. We do not need Assumpton 4 for the exstence 16 For the ways of te-breakng and the related topcs, see Reny (1999), Smon and Zame (1990), Ausubel and Mlgrom (2002), and Day and Mlgrom (2008). 17
of an equlbrum (Theorem 1). However, t s crtcal for the unqueness of the equlbrum outcome. It can be justfed when we addtonally requre tremblng-hand perfecton for the equlbrum concept. A bdder may wn some goods wth a lower prce wth a small probablty when other bdders stop earler than the predcton. 4.2 The Vckrey-Target Strategy The followng proposton states that any effcent allocaton accordng to u t, x X t, remans effcent later on n any TPE. It smplfes the backward nducton. Bdders never choose an acton whch changes X t f no bdder s restrcted to bd over the true values. Proposton 2 Suppose Assumptons 1, 2, 3, and 4. Suppose that each bdder s strategy space s restrcted to Σ. Then, any TPE satsfes Xt 1 X t for all t, both on and off equlbrum paths. Proof. See Appendx B. Proposton 2 mples X (u) = X 0 X T = X (û). Hence, any TPE s effcent as long as no one overbds. Suppose that n an effcent allocaton, bdder obtans a non-null bundle x. By Proposton 2, gven u t, t s optmal for to stop bddng at the least prce pt such that x X (p t, ut ), snce s payment never exceed pt (x ) by the ndvdual ratonalty. Such a prce vector satsfes j u t j(x j) + p t (x ) = max X u t j(x j ), (10) j hence, p t (x ) = V t (N ) j u t j(x j), (11) whch s the Vckrey payment. Thus, t s a TPE for each bdder to stop at the Vckrey payment wth respect to u t. Formally, we defne the Vckrey-target strategy as follows. Defnton 7 A sem-truthful strategy σ Σ s sad to be the Vckrey-target 18
strategy f t 1 and p t, D (p t ) ˆD (p t ) = otherwse X f p t (x ) < u (x ) π t 1 or f X t 1 = {x } for all x ( x ) X t 1,. (12) Theorems 1 and 2 are our man theorems of the current paper. The Vckrey-target strategy consttutes a TPE of every ascendng Vckrey-reserve aucton. Moreover, the equlbrum outcome s n the bdder-optmal core wth respect to the true valuatons. Let π be the correspondng payoff allocaton assocated wth σ. Theorem 1 Suppose Assumptons 2 and 3. 17 The profle of the Vckrey-target strateges σ Σ + s a TPE of every ascendng Vckrey-reserve aucton. Proof. See Appendx B. Theorem 2 Suppose that p V (u) > 0 for all wnners. Then, the outcome assocated wth σ, π, s n the bdder-optmal core wth respect to the true values. Proof. See Appendx B. At the ntal perod, every bdder s actve and π t = π. Hence, bdders frst seek to stop bddng at ther Vckrey payments. Note that once a bdder stops, he cannot renew the bds any longer. Hence, the stoppng bdder s reported utlty functon s revealed. Each bdder recomputes hs Vckrey payoff, regardng the prce vector for the stoppng bdder as hs true valuaton functon. Ths recomputaton weakly decreases the Vckrey payoffs of bdders. Remanng bdders contnue bddng and am for the revsed Vckrey prces. When the Vckrey outcoome s n the core, t s a unque bdder-optmal outcome. Hence, the TPE outcome concdes wth the Vckrey outcome. Note that σ s a TPE regardless of any specfcaton of J t, the termnal condton, and fnal dscounts. Theorem 1 shows an equvalence n equlbrum strategy of ascendng Vckrey-reserve auctons. It s smlar to the results of Day and Mlgrom (2008) and Sano (2011b), whch show a partcular strategy profle s a Nash equlbrum of every core-selectng or Vckrey-reserve auctons. However, the equlbrum outcome π can dffer between the rules. 17 Assumpton 1 s automatcally satsfed as long as we focus on sem-truthful strateges. 19
We consder a general valuatons structure and a restrcted strategy space. Another way of the analyss s to formulate a restrcted valuatons doman wth unrestrcted strateges. If bdders are sngle-mnded;.e., they are nterested only n a specfc bundle of goods, then Theorems 1 and 2 hold wth unrestrcted strateges. A bdder s sngle-mnded f there s a non-null bundle y X and f v f y x u (x ) =. 0 otherwse If a bdder s sngle-mnded, he can make proft by bddng for y (or larger bundles). It s meanngless to bd for bundles that do not contan y. Hence, t s obvous that the bdder s stratgy must be sem-truthful. Corollary 1 If each bdder s sngle-mnded, Theorems 1 and 2 hold wth unrestrcted strategy space. Remark 1 Assumpton 2 s not crucal for Theorem 1. We have Theorem 1 wthout Assumpton 2 by slghtly modfyng the Vckrey-target strategy. If two or more bdders (say, {, j,... } M) smultaneously reach ther stoppng prces at t wth π t 1, π j t 1,..., then we take a maxmal set M M that satsfes the followngs: (a) Each M stops at t, (b) the others reman actve at t, and (c) X t 1 X t. 4.3 Equlbrum Selecton Under certan crtera, π s a unque TPE outcome. When goods complementartes exst, there are many outcome n the bdder-optmal core n general. As Day and Mlgrom (2008) and Proposton 1 show, any payoff profle n the bdder-optmal core s acheved n a Nash equlbrum. Subgame perfecton (restrcted to Σ ) selects one from the set of those Nash equlbra. We focus on the equlbrum outcome n whch losers behave truthfully. There are n fact many equlbrum outcomes snce t s optmal for losers to stop at any perod n the aucton as long as they lose. The restrcton s natural and some precedng studes also focus on such an equlbrum n sealed-bd formats (Bernhem and Whnston, 1986; Ausubel and Mlgrom, 2002). Also, ths restrcton can be justfed when we addtonally requre tremblng-hand perfecton. 18 We assume losers 18 If a loser stops under the true values, he loses a chance to wn wth a small probablty. Conversely, f he bds over the true values, he may suffer a loss. 20
follow the Vckrey-target strategy σ. Theorem 3 Suppose Assumpton 1, 2, 3, and 4. Further suppose that p V (u) > 0 for all wnners. If each bdder s strategy space s restcted to Σ and f all losng bdders follow σ, then π s a unque TPE outcome n every ascendng strct Vckrey-reserve aucton. Proof. Under the assumptons, any TPE s effcent by Proposton 2. Snce all losers reveal true utlty functons, for any wnnng bdder, π t s nonncreasng n t n any equlbrum by the argument n the proof of Theorem 1. By Proposton 2, every wnner must stop when p t (x ) u (x ) π t 1. If û (x ) > u (x ) π t 1, s payment s p > u (x ) π t 1 by the strct Vckrey-reserve prcng and monotoncty of π t. On the other hand, f bdder follows σ, hs payment p = u (x ) π t 1. Hence, σ s a unque optmal strategy for each wnnng bdder. The ntuton of Theorem 3 s straghtforward. By Proposton 2, each bdder always chooses X t -preservng actons. Hence, each wnner can mnmze the payment by stoppng at the earlest perod such that X t unchanges even f stops. Such a strategy s the Vckrey-target strategy. Assumpton 2 s crtcal to Theorem 3. As we dscussed n Remark 1, we need to coordnate the behavor f two or more bdders smultaneously reach ther target prces at t. There wll be several possble selectons of M, and each of them wll lead to dfferent equlbrum outcome. Although we focus on only strct Vckrey-reserve auctons, Theorem 3 s appled to ascendng auctons wthout fnal dscounts, such as Parkes and Ungar (2000), Ausubel and Mlgrom (2002), de Vres et al. (2007). If the payments are equal to the fnal prces of the bundles, t s clearly suboptmal to bd over the true values and wn. Hence, Theorem 3 holds wthout restrctng to Σ. Corollary 2 Suppose Assumptons 1, 2, 3, and 4, and suppose that p V (u) > 0 for all wnners. If all losng bdders follow σ, then π s a unque TPE outcome n every ascendng auctons wth no fnal dscount. 21
5 Dscussons 5.1 Resstance to Collusve Overbddng Theorem 3 holds for every ascendng strct Vckrey-reserve aucton when bdders are not allowed to overbd. As Day and Mlgrom (2008) pont, there may exst an neffcent equlbrum n whch some bdders collusvely overstate ther values and outbd the effcent allocaton. For example, consder the same stuaton as Example 1. Suppose that bdder 1, who wants good A, values 4 and that bdder 2 values 4 for good B. Suppose that bdder 3 wants the package of A and B and values 10. In ths case, n the subgame perfect equlbrum σ, bdder 3 wns both goods wth the payment 8. Ths outcome concdes wth the Vckrey outcome. However, there s an neffcent Nash equlbrum n the sealed-bd Vckrey aucton. Suppose that all the bdders submt 10. Then, bdders 1 and 2 wns each good wth zero payment, and t s an equlbrum. Such an equlbrum exsts n some strct Vckrey-reserve auctons. In addton, smlar TPE exsts n some ascendng strct Vckrey-reserve auctons as well. Such a collusve overbddng equlbrum s excluded by mposng core-selectng prcng (Day and Mlgrom, 2008). Smlarly, ascendng core-selectng auctons prevent bdders from bddng over the true valuatons collusvely. Let G be the set of wnnng bdders assocated wth σ. let σ J = (σ j ) j J. Theorem 4 In any ascendng core-selectng aucton, there s no group of bdders G G such that σ G Σ + G, πσ σ = (σ G, σi\g ). Proof. See Appendx B. > π, and û (ˆx ) > 0 for all G under 5.2 Ascendng Vckrey Aucton Ascendng Vckrey-reserve auctons nclude ascendng Vckrey auctons. truth-tellng s also an SPE n ther auctons. Clearly, Proposton 3 Suppose that the Vckrey outcome s not n the core. Then, there are at least two TPE outcomes n every ascendng Vckrey aucton: the Vckrey outcome π and the core-mplementng outcome π. 22
Aucton desgners expect that n an ascendng Vckrey aucton, bdders behave truthfully, and thus that the Vckrey outcome s actually mplemented. However, when the Vckrey outcome s not n the core, ascendng Vckrey auctons have another equlbrum that leads to an outcome n the core. Moreover, the core-mplementng equlbrum seems more robust n the followng senses. Frst, the Vckrey-target strategy σ s obvously a best response among Σ to both σ and truth-tellng strateges. Conversely, the truth-tellng strategy s not the best response f the other players follow σ. Second, σ s an equlbrum even f the aucton s slghtly dfferent from the Vckrey auctons. Truth-tellng, however, s not an equlbrum of such an almost-vckrey aucton. 5.3 Payoff Non-Monotoncty and Free-Rder Problem It seems to be a postve result that the ascendng auctons have a unque TPE lyng n the core. However, the equlbrum outcome may not be necessarly desrable. In the TPE σ, when a wnner has a low valuaton, he tends to obtan a large proft. Ths s because the Vckrey payments for the low-value bdders are low and ther prces reach the Vckrey prces earler. The hgher the value a wnner has, the lower are the profts he tends to get n the equlbrum. Example 1 (contnued). Remember Parkes and Ungar s (2000) aucton wth 2 goods and 3 bdders. When bdders actual values are (7, 8, 10), the equlbrum payoff allocaton s (π 0, π 1, π 2, π 3 ) = (10, 5, 0, 0). Note that bdder 1, who has a lower value than bdder 2, earns all the gans, whereas bdder 2 earns zero net payoff. Suppose that bdder 1 s value for A s 9, wth everythng else remanng unchanged. Then, n the equlbrum, bdder 2 stops at the prce of 1 and bdder 1 behaves truthfully. The equlbrum payoff allocaton s now (10, 0, 7, 0). The equlbrum payoff of bdder 1 decreases as hs valuaton ncreases (Fgure 2). As ths example shows, the TPE outcome s on the corner of the bdder-optmal core. Moreover, the wnner wth a low value earns the Vckrey payoff, whle the hgh-value wnner earns 0. 19 Ths stuaton s qute smlar to a standard free-rder problem. Suppose a prvate 19 The equlbrum payoff allocaton depends on the aucton rule, and payoff non-monotoncty does not necessarly arse. 23
Fgure 2: Non-Monotonc Equlbrum Payoffs provson game of a publc good, n whch margnal values for a publc good are heterogeneous among agents. Then n a unque equlbrum, only one agent wth the hghest value provdes the good and the others do not provde at all. 20 In our aucton stuaton, bdders wth low values free-rde on other bdders wth hgher values. Indeed, the ncentve problem n core-selectng auctons s referred to as a knd of free-rder problem (Mlgrom, 2000), whle t s called the threshold problem. In a sealed-bd format, the threshold problem s often nterpreted as a knd of coordnaton falure by bdders (Bykowsky et al., 2000). However, n an ascendng prce open-bd format, t seems more approprate to nterpret the ncentve problem as a free-rder problem. Ths free-rder problem appears n a strkng form when valuatons are prvate nformaton of each bdder. Suppose the same stuaton as Example 1 and asymmetrc nformaton. Even when bdder 1 has a low value, he wll have a certan amount of expected payoff because he can free-rde on bdder 2. Conversely, when bdder 1 has a hgh value, hs expected payoff may be low because bdder 2 may have a low value and free-rde on hm. Hence, t may be good for bdder 1 to behave as a low value bdder even when he has a hgh value. Thus, both bdders 1 and 2 behave as a low value bdder. Ths wll lead to neffcency and low revenue. 21 20 See Mas-Collel et al. (1995) for the free-rder problem n the publc goods game. 21 For the ncomplete nformaton case, see Sano (2011b). 24
5.4 Ineffcency under the Unrestrcted Strateges Even wth complete nformaton, the payoff non-monotoncty provdes neffcency n the case where strategy space s unrestrcted. In a sealed-bd format, a truncaton strategy s a best response among all strateges (Day and Mlgrom, 2008; Sano 2011b). However, n the ascendng auctons, the Vckrey-target strategy s not necessarly a best response among Σ. The TPE σ s not an SPE n general. Moreover, an SPE may be neffcent. The followng example shows an SPE s not effcent. Example 2. Suppose that there are 3 goods {A, B, C} and 7 bdders. All bdders except bdder 7 are nterested only n a unque bundle of the goods. Each of the values for these sx bdders s u 1 (ABC) = 12, u 2 (A) = 7, u 3 (B) = u 4 (B) = 1, and u 5 (C) = u 6 (C) = 1 respectvely. Bdder 7 s nterested n goods B and C. Hs valuaton functon s such that u 7 (B) = u 7 (BC) = 8 and u 7 (C) = 6. In the effcent allocaton, bdders 2, 5 (or 6), and 7 wn the sngle tem A, C, B, respectvely. The aucton s Parkes and Ungar s (2000) ascendng aucton. Snce bdders 3, 4, 5, and 6 are completely compettve, they bd untl ther true values n any equlbrum. The Vckrey payments for bdders 2 and 7 are 3 and 4, respectvely. Hence, n the TPE σ, bdder 2 stops earler than bdder 7. In the equlbrum, p 2 (A) = 3, p 5 (C) = 1, and p 7 (B) = 8. Bdder 7 s TPE payoff s 0. Now, consder that bdder 7 follows the Vckrey-target strategy wth respect to the followng valuaton functon: ũ 7 (B) < ũ 7 (C) = ũ 7 (BC) = 6. Under (u 7, ũ 7 ), bdder 7 obtans good C n the effcent allocaton. And, bdder 2 s Vckrey payment changes to 5, whereas that of bdder 7 for C remans the same. In the TPE outcome, bdder 7 wns tem C wth p 7 (C) = 4, whereas bdder 2 pays 7 for tem A. By nspecton, ths s an SPE and the equlbrum outcome s neffcent. Ths neffcency stems from the payoff non-monotoncty. In an effcent allocaton, a bdder obtans some goods whose value s suffcently large. However, the true value s so large that other bdders stop earler and he may have to pay too much. On the other hand, f he focuses on another good whose value s not so hgh, he may wn t wth a lower prce, and t may be more proftable. Proposton 4 The TPE σ s not an SPE wth the unrestrcted strategy doman n general. Moreover, An SPE s not effcent n general. Remark 2 Ineffcent subgame perfect equlbrum can exst because of another 25
logc. Suppose that there are 2 goods, A and B, and 2 bdders. Both bdders have dentcal valuatons of u (A) = u (B) = 6 and u (AB) = 12. In any TPE outcome, both bdders have to earn zero payoff. However, f both bdders report ˆD = {A, B, AB} at the ntal perod, then both can get one good wth zero payment. And, ths consttutes an equlbrum. Ths s an mplct colluson, n whch bdders splt up goods between each other and end up the aucton wth low prces. Such an equlbrum s consdered also by Ausubel and Schwartz (1999) and Grmm et al. (2003) n an ascendng aucton wthout package bddng. 6 Concluson We formulate a general class of ascendng-prce auctons. The Vckrey-target strategy consttutes a perfect equlbrum of every ascendng Vckrey-reserve aucton wth restrcted strategy space. The equlbrum outcome s n the bdder-optmal core and unque n ascendng strct Vckrey-reserve auctons f losng bdders follow the Vckrey-target strategy. These results are postve fndngs, as sealed-bd Vckrey-reserve auctons may have multple Nash equlbra. Although ascendng Vckrey auctons have both truth-tellng and core-mplementng equlbra, the coremplementng equlbrum seems more robust under complete nformaton and restrcted strategy space. The equlbrum outcome, however, can be unfar n the sense that bdders wth lower values tend to obtan hgher payoffs. Ths stuaton s smlar to a standard free-rder problem, and t wll lead to neffcency under ncomplete nformaton. The payoff non-monotonc property also provdes neffcency n the case of unrestrcted strategy space. An nterestng future research s to consttute an SPE wth unrestrcted strategy space. It s also an open queston what class of valuaton functons assures SPE n the core. A Theorem 1 wthout Assumpton 3 In ths paper, we suppose that bdders strateges are dscrete, however, tes are broken n a way that partcular bdders are favored. Ths assumpton s made for analytcal purpose. We also have a correspondng result when the specal te-breakng rule s not assumed. We redefne the Vckrey-target strategy as one decreasng the 26