An Ascending Vickrey Auction for Selling Bases of a Matroid

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1 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod Sushl Bkhchandan Sven de Vres James Schummer Rakesh V. Vohra July 27, 2010 Abstract Consder sellng bundles of ndvsble goods to buyers wth concave utltes that are addtvely separable n money and goods. We propose an ascendng aucton for the case when the seller s constraned to sell bundles whose elements form a bass of a matrod. It extends easly to polymatrods. Applcatons nclude schedulng (Demange, Gale, and Sotomayor, 1986), allocaton of homogeneous goods (Ausubel, 2004), and spatally dstrbuted markets (Babaoff, Nsan, and Pavlov, 2004). Our ascendng aucton nduces buyers to bd truthfully, and returns the economcally effcent bass. Unlke other ascendng auctons for ths envronment, ours runs n pseudo-polynomal or polynomal tme. Furthermore we prove the mpossblty of an ascendng aucton for nonmatrodal ndependence set-systems. Keywords Matrod, Vckrey, mult-tem, combnatoral, aucton, polymatrod. An extended abstract of ths paper appeared n SODA 2008; see Bkhchandan, de Vres, Schummer, and Vohra, Anderson Graduate School of Management, UCLA, Los Angeles, CA FB IV - Mathematk, Unverstät Trer, Trer, Germany The second author gratefully acknowledges support through the Alexander von Humboldt foundaton, and s grateful to the Yale Economcs Department for an nsprng 2-year vstng appontment, durng whch ths work was completed. Department of Manageral Economcs and Decson Scences, Kellogg School of Management, Northwestern Unversty, Evanston IL Department of Manageral Economcs and Decson Scences, Kellogg School of Management, Northwestern Unversty, Evanston IL

2 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 2 1 Introducton Consder sellng bundles of ndvsble objects to buyers wth utlty whch s addtvely separable n money and ndvdual tems. In partcular, we analyze such stuatons where the seller s constraned to sell a subset of objects that forms a bass wth respect to some underlyng matrod structure (descrbed below) on the objects. Whle ths may seem lke an abstract class of problems n the mechansm desgn lterature, t turns out to contan varous nterestng specal cases. Examples (see Secton 5) nclude: schedulng matrods (Demange et al., 1986), the allocaton of homogeneous objects (Ausubel, 2004), parwse kdney exchange (Roth, Sönmez, and Ünver, 2005), spatally dstrbuted markets (Babaoff et al., 2004), bandwdth markets (Tse and Hanly, 1998), and multclass queung systems (Shanthkumar and Yao, 1992). The prmary objectve s to allocate the objects effcently;.e. to allocate the objects whch form a maxmum-weght bass, where the weghts are the agents valuatons for the objects. However t s a partcular lst of addtonal desgn constrants that motvates us. In ths paper we present an ascendng aucton that both meets these constrants and allocates objects to form a maxmum-weght bass. One desgn constrant nvolves ncentves. If the agents valuatons were known to the seller, t would be easy to determne the economcally effcent outcome by usng the greedy algorthm. When valuatons are prvate nformaton, however, the greedy algorthm does not apply. Ths constrans us to desgn a method that gves agents an ncentve to reveal accurate nformaton. There s a well-known sealed-bd aucton whch can acheve both the ncentves and effcency objectves: the VCG mechansm (Vckrey, 1961, Clarke, 1971, Groves, 1973). It operates by askng agents for ther valuatons of all objects, computng an effcent allocaton, and chargng Vckrey prces (descrbed below) to bdders. Nevertheless, there are reasons for eschewng sealed-bd auctons n favor of ascendng auctons. Informally, an ascendng aucton has the auctoneer announcng prces and bdders reportng ther demands at the announced prces. If the reported demands can be feasbly satsfed, the aucton ends. Otherwse, the prces are adjusted upwards. An agent who reduces demand s not allowed to ncrease t later. Ascendng auctons dffer accordng to the knds of prces that are used (e.g. lnear, non-anonymous, or non-lnear) and the rules used to adjust prces. Ausubel (2004) and Cramton (1998) lst some of the advantages

3 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 3 of ascendng auctons, ncludng that they are more transparent, bdders reveal less prvate nformaton, communcaton and computaton costs for bdders may be lower, and they may be better even wth mld nterdependences among bdders valuatons. Ths motvates our second desgn constrant: to search for an ascendng aucton n our envronment. Ths constrant, too, has been addressed n the lterature. Prevous work descrbed n Secton 1.1 has developed ascendng auctons for envronments more general than ours. However, that generalty comes at the prce of complexty. The auctons dscussed n Secton 1.1 rely on a number of prces that s exponental n the number of objects. Our thrd concern, therefore, s to desgn an aucton wth low complexty. Unlke other combnatoral envronments, we show that the matrod envronment admts an ascendng aucton that runs n polynomal tme. Furthermore, we prove the mpossblty of an ascendng aucton for nonmatrodal ndependence set-systems. We meet our objectves n ths paper by descrbng an ascendng aucton for the matrod envronment that mplements the outcome of a sealed-bd VCG aucton. Specfcally, 1. Bddng truthfully s an ex-post equlbrum of the aucton. It should be noted that ths result reles nether on proxy bddng schemes that restrct the bds that bdders make nor on consstency checks on bds. 2. The truthful equlbrum results n the effcent outcome. 3. The complexty of the aucton scales polynomally n the number of elements and unts. The sealed-bd VCG aucton mentoned above gves bdders the ncentve to report ther valuatons truthfully because t charges Vckrey prces. To descrbe ths aucton for general envronments, let A be a set of abstract alternatves. Each agent j N has a valuaton v j (a) for each alternatve a A. Economc effcency requres choosng an alternatve a that yelds total value V (N) := max a A j N vj (a). The VCG aucton fnds an allocaton that s effcent wth respect to the reported valuatons. Each bdder s nduced to truthfully report hs valuatons, however, because he s charged the net effect hs presence has on

4 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 4 the other bdders, n the followng sense. If bdder j N were absent, economc effcency would requre fndng an alternatve that provdes value V (N \j) := max a A k N\j vk (a). Therefore, the net effect that j s presence has on the other bdders s V (N \ j) k N\j vk (a ) whch defnes bdder j s Vckrey payment. Bdder j s net Vckrey payoff s therefore [ v j (a ) V (N \ j) ] k N\j vk (a ) = V (N) V (N \ j). That s, hs net payoff equals hs net contrbuton to attanable socal surplus, whch s why ths amount s also called bdder j s margnal product (see Makowsk and Ostroy (1987) for the connecton between margnal product, perfect competton, and the Vckrey mechansm). The payments n a sealedbd VCG aucton can be found by solvng n + 1 optmzaton problems: one to fnd V (N) (and a ) and n more to fnd each V (N \ j). 1.1 Related Lterature There s a substantal lterature on mult-object, ascendng-prce auctons. We descrbe some of ths prevous work. Models n whch bdders have gross substtute preferences over heterogeneous objects have receved partcular attenton. Kelso and Crawford (1982) and Gul and Stacchett (2000) present ascendng auctons n ths settng. Whle these auctons return the effcent outcome when bdders bd truthfully, truthful bddng s not ncentve compatble. The dynamc aucton of Ausubel (2006) returns the effcent outcome n equlbrum and s ncentve compatble. It works by runnng several dummy auctons (each of whch s an nstance of the one n Gul and Stacchett 2000) and then uses the prces from each of these runs to determne the Vckrey payments. In contrast, our paper acheves the same outcome wthout the necessty of runnng dummy auctons. These dummy auctons not only ncrease the duraton, but also substantally ncrease the potental for collusve behavor (see de Vres, Schummer, and Vohra, 2007). Snce preferences n our model also satsfy the gross substtutes condton, the Agents are Substtutes condton (Bkhchandan and Ostroy, 2002)

5 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 5 holds. The ascendng auctons of Parkes and Ungar (2000a), Ausubel and Mlgrom (2002), de Vres et al. (2007), and Mshra and Parkes (2007) rely on ths condton. These auctons attan (wth the approprate consstency checks) an effcent outcome n equlbrum but they rely on an exponental (n the number of goods) number of prces. Ths would make the complexty of runnng such auctons exponental n the number of goods. Our aucton n contrast produces the same outcome wthout relyng on a proxy bddng scheme (e.g. Parkes and Ungar (2000b), Ausubel and Mlgrom (2002)) and wth a complexty that s polynomal n the number of elements and unts E + ρ(e). 1.2 Motvatng Example: Sellng a Tree from a Graph Prmarly to provde ntuton for our analyss, we descrbe an nstance of a smple matrod envronment: sellng a spannng tree of a graph. Whle ths example may appear abstract, a procurement verson of t has applcatons to constructng communcaton networks. Other applcatons are dscussed n Secton 5. Let G = (V, E) be a complete graph wth vertex set V and edge set E. For smplcty, suppose each agent j N s nterested only n a sngle edge e E, so that we can dentfy edges wth agents. Let v e be agent e s value from obtanng edge e. Our goal s to derve an ascendng Vckrey aucton to sell off edges that form a maxmum weght spannng tree. 1 Snce G s complete, no sngle agent s ntally n a poston to prevent a spannng tree from formng. Whle there s no competton for edges between agents, there s competton between dfferent bdders to cover the same mnmal cut (and mnmal cuts are cocrcuts of the underlyng graphcal matrod). To see ths let us compute the margnal product of an agent e that s part of a maxmum weght spannng tree T. To determne agent e s margnal product we dentfy the reducton n weght of the spannng tree when we remove agent e and replace her wth a (next best) edge. If f T s the largest weght edge such that 1 Though we speak n terms of sellng edges, one nterpretaton for ths problem nvolves a procurement settng, where the auctoneer wants to purchase the rght to use an edge and the bdder ncurs some cost ( v e ) when t s used (e.g. constructng a complete communcatons network at mnmal total socal cost). For consstency wth the rest of the paper, we avod procurement termnology and speak of sellng.

6 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 6 T f contans a cycle through e, then the maxmum weght spannng tree that excludes e s (T \ {e}) f. Thus agent e s margnal product s v e v f. Not all algorthms for fndng a maxmum weght spannng tree lend themselves to an ascendng aucton nterpretaton or generate Vckrey prces. The greedy out algorthm does: Startng wth the complete set of edges, delete edges n order of ncreasng weght, whlst keepng the graph connected. An edge s spared deleton when all smaller weght edges that could cover the same cut have been deleted. Ths algorthm can be nterpreted as an aucton whch begns wth a prce p = 0 on each edge. Throughout the aucton, ths prce s ncreased. At each pont n tme, each agent announces whether he s wllng to purchase hs edge at the current prce. As the prce ncreases, agents drop out of the aucton when the prce exceeds ther value v e for the edge, reducng the connectvty of the graph. At some pont an agent becomes crtcal: removng the agent dsconnects the graph. The edge of the crtcal agent s a brdge of the subgraph of remanng edges. At ths pont the auctoneer mmedately sells the edge to the crtcal agent at the current prce. Ths edge s to be part of the fnal maxmum weght spannng tree and does not drop out. The aucton contnues wth other agents droppng out or becomng crtcal and ends when the last crtcal agent s awarded an edge and a tree s formed. A crtcal agent acqures hs edge at the prce where another bdder dropped out of the aucton. That bdder s edge s the best alternatve to the crtcal edge. The Vckrey aucton would charge just the prce of ths best alternatve. It s clear that ths should generalze from trees to matrods. However, care s necessary regardng detals of the mplementaton: Can an arbtrary cut n the graph be pcked and ts prce ncreased untl t contans only one element? What procedure should be followed f multple agents become crtcal at the same tme? Suppose an agent s nterested n multple elements, say b and b, that are part of a maxmum weght bass. Suppose also that an element f of nterest to a second agent s a second-best replacement for both b and b. Can the value the second agent assgns to f be used to set the Vckrey prce for b and b?

7 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 7 These ssues are addressed n the next secton. 2 Sellng Bases of a Matrod We begn wth a revew of basc matrod concepts. Subsequently we descrbe the model and dscuss smplfyng assumptons that are wthout loss of generalty. In Subsecton 2.3, we recall a nonstandard algorthm for fndng the maxmum weght bass of a matrod due to Dawson (1980). Ths algorthm selects a sequence of cocrcut and element pars that certfes feasblty and optmalty smultaneously. We specalze ths algorthm to select VCG-sequences that also certfy optmalty for all margnal allocaton problems obtaned after removng one buyer. The margnal allocaton problems are essental to mplementaton of Vckrey payments and hence to effcency and desrable ncentve propertes of the aucton. After establshng that these sequences yeld Vckrey prces, we present Aucton 3 whch s an ascendng-prce mplementaton of the sealed-bd VCG aucton. In Subsecton 2.9 we nvestgate ts runtme and prove n Subsecton 2.10 that truthful bddng s an ex post equlbrum n ths aucton. 2.1 Matrod Bascs We use the standard notons of matrod theory; see Oxley (1992). A matrod M s an ordered par (E, I) of a fnte ground set E and a set I of subsets of E satsfyng the axoms: (I1) I, (I2) f I I and I I then I I, and (I3) f I 1, I 2 I and I 1 < I 2, then there s an element e I 2 I 1 such that I 1 e I. Subsets of E that belong to I are called ndependent; all other sets are called dependent. Mnmal dependent sets of a matrod M are called crcuts; the set of all crcuts of M s denoted C(M). Crcuts consstng of a sngle element are called loops. A set C s the set of crcuts of a matrod f and only f t satsfes these three propertes: (C1) / C. (C2) If C 1, C 2 C and C 1 C 2 then C 1 = C 2. (C3) If C 1, C 2 C, e C 1 C 2, f C 1 \ C 2, then there exsts a C 3 C such that f C 3 (C 1 C 2 ) e (strong crcut elmnaton). A maxmal ndependent set s called a bass of the matrod; the set of all bases of M s denoted B(M). All B B(M) have the same cardnalty. The rank r(s) of S E s the sze of a largest ndependent set contaned

8 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 8 n S. Matrods are characterzed amongst all ndependence systems by the property that for any addtve weghtng w : E R, the greedy algorthm fnds an optmal (maxmum-weght) bass. Let M be a matrod, E(M) ts ground set, and B (M) := {E(M) B : B B(M)}. Then B (M) s the set of bases of a matrod on E(M), called the dual matrod of M, denoted M. 2 Independent sets and crcuts of M are called condependent sets and cocrcuts of M respectvely. A wellknown fact we use later s that any cocrcut of a matrod ntersects each of ts bases, and each element of a bass belongs to at least one cocrcut. For M = (E, I) and X E the deleton of X from M s the matrod defned by M \ X := (E \ X, {I E \ X : I I}). The contracton of X n M s defned by M/X := (M \ X). A matrod M that s derved by contractons and deletons from a matrod M s called a mnor of M. Some propertes of mnors follow. Proposton 1 (Oxley, 1992, Prop ). The crcuts of M/T consst of the mnmal non-empty members of {C T : C C(M)}. Proposton 2 (Oxley, 1992, Cor ). M \ e = M/e f and only f e s a loop or coloop of M. Proposton 3 (Oxley, 1992, ). For every subset T of a matrod M: C(M \ T ) = {C E \ T : C C(M)}. Lemma 4. Let C be a cocrcut of the matrod M and e E(M) : () If e / C then C s a unon of cocrcuts of M \ e. () If e C and {e} s a not a coloop of M, then C e s a cocrcut of M \ e. Lemma 4 s the dual of Exercse 2 n Secton 3.1 of Oxley (1992). It s proven n the onlne e-companon to ths artcle and mples the followng. Corollary 5. Let C be a cocrcut of the matrod M. If T E(M) contans no coloops then C T s a unon of cocrcuts of M \ T. 2 See Oxley (1992) for ths and other results.

9 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod The Economc Model We consder a fnte matrod M = (E, I) and set of agents N. The ground set E of M s parttoned nto sets (E j ) j N. Interpret E j as the set of elements that agent j N may feasbly purchase from the auctoneer. Let o(e) denote the prospectve buyer (owner) of element e,.e. the unque ndex such that e E o(e). Snce the E j s partton E, o(e) denotes the ndex j wth e E j. If agent j acqures some element e E j t would provde hm a nonnegatve value v e Q +. Acqurng a set S E j provdes value v(s) := e S v e. The auctoneer s constraned to sell combnatons of elements that form a bass of the matrod; for an extenson see Secton 4. Socal surplus s maxmzed by dentfyng an optmal (maxmum weght) bass of M wth respect to the values v e. Untl Subsecton 2.6 we assume that all values are dstnct. We assume as a generalzaton of the brdge-condton of the motvatng example, the followng no-monopoly condton, whch states that t s possble to form a bass wthout allocatng any elements to any gven bdder j: r(m) = r(m \ E j ) for all j N. Ths s equvalent to requrng that no cocrcut C of M belongs to any bdder j: C E j for all j N. If ths condton were volated for some j, then every bass of M would contan elements of E j, and V (N \ j) would be undefned ( ). Bdder j s Vckrey payment would also be undefned ( ). The assumpton that agents are nterested n dstnct tems s wthout loss of generalty. In any envronment where two agents are nterested n the same tem e, one can add an element e parallel to e, and mpose that the frst agent can only acqure e, and the second only e. Settng I = I {I {e }: I {e} I}; I fulflls the matrod axoms, whle any ndependent set contans at most one of {e, e }. Our aucton also apples to settngs where the seller s permtted to sell any ndependent set (when agents can have negatve values). To handle ths case we can ntroduce a dummy bdder and, for every element of M, a parallel element that the dummy bdder values at Computng the Optmal Bass For any cocrcut C of M let b C := arg max{v e : e C } be ts hghestvalued element. Algorthm 1 due to Dawson (1980) fnds an optmal bass for a matrod,.e. one that maxmzes v e. As values are dstnct, M and

10 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 10 ts mnors have unque optmal bases. Algorthm 1 Optmal bass for a matrod (Dawson, 1980) Requre: Matrod M on ground set E wth dstnct values. 1: 0 2: whle M has a cocrcut that s dsjont from {b 1,..., b } do 3: + 1 4: Let C be such a cocrcut 5: Let b = b C 6: end whle 7: r Proposton 6 (Dawson, 1980, Thm. 1). For any matrod M wth dstnct values, Algorthm 1 determnes ts optmal bass B and ts rank r = r(m). The way n whch cocrcuts C are chosen n Lne 4 of Algorthm 1 s arbtrary (when more than one cocrcut s dsjont from {b 1,..., b }) accordng to Dawson s Theorem. Intutvely, the algorthm works despte ths because any cocrcut of a matrod ntersects each of ts bases and each element of a bass belongs to at least one cocrcut. From ths t can be shown that f a cocrcut s dsjont from a subset of the optmal bass, then the hghest-valued element n the cocrcut s part of the optmal bass. We show that cocrcuts C must be chosen n a partcular order so that we obtan an algorthm that can be nterpreted as an ascendng aucton wth Vckrey prces. As we formalze next, ths order s monotonc wth respect to the hghest-valued element of C \E o(bc ), that s, the hghest-valued element of C gnorng elements whch can be purchased by the owner of b C. 2.4 Sequences and Certfcates For any cocrcut C of M, we defne b C to be the hghest-valued element n C. Excludng the elements whch can be owned by agent o(b C ), denote the hghest-valued remanng element as f C := arg max{v e : e C \ E o(bc )}. Ths s the best element n C \ E o(bc ); t s second-best n C n the sense

11 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 11 of beng the best element of C that s assocated wth a bdder dstnct from o(b C ). The element f C s well-defned because the no-monopoly condton ensures C \ E o(bc ). Notce that n the tree case n Subsecton 1.2, fc s just the element wth value equal to the current prce whose removal makes another agent crtcal. Smlarly, o(b C ) s the agent that becomes crtcal. Gven that agent o(b C ) mght be nterested n multple elements, t may not be, that removal of b C makes formng a maxmum weght bass mpossble (snce the rank drops) but that only removal of all hs elements (from C ) decreases the rank; nevertheless, of hs elements n C he want b C the most. A sequence of cocrcut-element pars ((C1, b 1 ), (C2, b 2 ),..., (Cj, b j )) s called sutable f t can be generated durng the executon of Algorthm 1. That s, such a sequence s sutable for M f for all (S1) C C (M) and C {b 1,..., b 1 } =, and (S2) b = arg max e C v e. (S1) ensures that C satsfes the condton n Lne 2 of Algorthm 1. It s easy to see that a lst of elements {b 1,..., b r(m) } s an optmal bass f and only f there s a sequence of r(m) cocrcuts wth whch t s sutable. Although there s consderable flexblty n choosng cocrcuts n a sutable sequence, our choce s constraned by the requrement that the sequence should also yeld Vckrey prces for elements as they are allocated to buyers. It wll be shown that the values of second-best elements of selected cocrcuts, v fc, = 1, 2,..., r determne Vckrey prces. The ascendng aucton wll satsfy: () selected cocrcuts, together wth ther best elements, form a sutable sequence, () the value of second-best elements n the selected cocrcuts s non-decreasng, and () the ncrease n v fc along the sequence s mnmal. That s, n any sutable sequence ((C1, b 1 ), (C2, b 2 ),..., (Cj, b j )) selected by the aucton, v fc v fc and for each there does 1 not exst C whch satsfes (S1) and v fc > v fc v fc 1. In addton, f at any stage v fc > v fc 1 then t wll turn out that elements e E such that v fc > v e v fc 1 can be removed from consderaton; these elements wll never be ether best or second-best elements of cocrcuts n the remanng porton of the sutable sequence. We collect such elements n sets D n the defnton below; hence D collects the elements wthdrawn between the award of b 1 and b.

12 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 12 A sequence ((C1, b 1, D 1 ), (C2, b 2, D 2 ),..., (Cr, b r, D r ), (D r+1 )) (r = r(m)) s a VCG-sequence for M f for all, (V1) C arg mn {C C (M):v fc v f C and C {b 1,...,b 1 }= } v fc 1 (For ease of notaton, let v fc = ) 0 (V2) b = arg max e C v e, and (V3) D = {e E \ {b 1,..., b 1 }: v fc ( 1 r D r+1 = E \ {b 1,..., b r } ) j=1 D j v e < v fc }, 1 r and In addton to ensurng that the value of second-best elements of cocrcuts s mnmally non-decreasng along the sequence, (V1) mples (S1). Therefore, f we omt the component D of a VCG-sequence then ((C1, b 1 ), (C2, b 2 ),..., (Cr, b r )) forms a sutable sequence. The resultng set of elements B := {b 1,..., b r } forms the optmal bass, whle the D s partton the complement of B. Let p = v fc, the value of the second-best element n C. Call a cocrcut C of M feasble at p 0 f v fc = p. We say that C s feasble at p 0 va e f C s feasble at p and e = f C. A VCG-sequence can be dentfed by an algorthm whose th teraton s descrbed below. 3 1: Let {e 1, e 2,... } = {e E : v e v fc } \ 1 {b 1,..., b 1 } be n ncreasng order of v e. Let k 1 and e e k 2: whle every C C (M) wth f C = e ntersects {b 1,..., b 1 } do 3: D D {e}; k k + 1; e e k 4: end whle 5: Choose C C (M) so that e = f C and C {b 1,..., b 1 } =. 6: b b C It would be convenent to dscard elements of the matrod as we go along; for that we ntroduce the next defnton. A sequence ((C 1, b 1, D 1 ), (C 2, b 2, D 2 ), 3 We follow the conventon that the whle condton n step 2 below s satsfed f there exsts no C C (M) wth f C = e.

13 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 13 (..., (Cr, b r, D r ), (D r+1 )) s called a condensed VCG-sequence f D r+1 = E \ r {b 1,..., b r } ) for r = r(m) and (wth M 1 := M and M +1 := j=1 D j M /b \ D ) we have for all 1 r: (CV1) f = arg max {f E(M ):M \{e:v e<v f } fulflls the no monopoly condton} v f and D = {e E(M ): v e < v f }, (CV2) C s a cocrcut of M \ D wth f C = f, and (CV3) b = arg max e C v e. Clearly, a condensed VCG-sequence can be dentfed by an algorthm whose th teraton (startng wth D = ) s descrbed n Procedure 2. Procedure 2 Determnng one step of a condensed VCG sequence. 1: Let {e 1, e 2,..., } = E(M) be n ncreasng order of v. Let k 1 and e e k 2: whle M \ (D {e}) fulflls the no-monopoly condton do 3: D D {e}; k k + 1; e e k 4: end whle 5: Choose C C (M \ D ) wth e = f C. 6: b b C and p v e 7: M = M +1 M/b \ D Notce that f M \ D fulflls the no-monopoly condton, then the whle condton s equvalent to that every C C (M) wth fc = e ntersects {b 1,..., b 1 }. Our goal s to prove that for every condensed VCG sequence, t s possble to fnd a VCG sequence that has the same D, b, and second best element f C, and vce versa. Theorem 7. For every VCG-sequence ((C1, b 1, D 1 ), (C2, b 2, D 2 ),..., (Cr, b r, D r ), (D r+1 )) there exsts a condensed VCG-sequence ((C 1, b 1, D 1), (C 2, b 2, D 2),..., (C r, b r, D r), (D r+1)) wth (b k, D k, f C k ) = (b k, D k, f C k ) for 1 k r + 1. Also for a condensed VCG-sequence there exsts a correspondng VCG-sequence. The farly routne proof s gven for the reader s convenence n the e- Companon A.2.

14 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod Vckrey Prces We show that an element f C can be second-best for at most one cocrcut per bdder. Ths property s mportant for determnng Vckrey payments. If t dd not hold then one bdder mght be awarded two elements b, b, when another bdder, o(f C ), drops hs nterest n f C ; the removal of b, b would requre that f C be nserted twce nto the bass, whch s not possble. Lemma 8. If a VCG-sequence contans two dstnct cocrcuts C, Cj o(b ) = o(b j ) then f C f C j. wth Proof. Recall from Procedure 2, Step 6 that p = v fc. If p p j then f C f C j. Assume p = p j. Suppose that f C = f C j and o(b ) = o(b j ), and wthout loss of generalty that v b > v bj. By strong crcut elmnaton, there exsts a cocrcut C (C Cj ) f C that contans b. Snce b s the hghest valued element of C Cj, t follows that b C = b. Furthermore, snce f C s the hghest valued element of (C Cj ) \ E o(b ), t follows that v fc < v fc (because f C / C, and values are dstnct). Therefore C must have been selected n an earler step k < mn{, j} of the sequence, and b k = b. Ths contradcts C {b 1,..., b 1 } =. The next lemma shows that no element s best for one cocrcut and second-best for another cocrcut n the same VCG-sequence. Lemma 9. For a VCG-sequence and all, j holds b f C j. Proof. Snce C j s dsjont from {b 1,..., b j 1 }, < j mples f C j b. Clearly b f C (coverng the case = j). Fnally, > j mples v b > v fc v fc j, provng b f C j. To determne Vckrey payments for the optmal bass B = {b 1,..., b r } derved from a VCG-sequence, we need nformaton about the optmal bases for each margnal matrod M j := M\E j. To prove that the p s are Vckrey payments for the b s we show that the optmal bass of each M j s B j := (B \ E j ) {f C : b B E j }. (1) We argue that B 1 s an optmal bass of M 1, the proof of optmalty of the other B j s s analogous. Gven the VCG-sequence ((C 1, b 1, D 1 ),

15 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 15 (C2, b 2, D 2 ),..., (Cr, b r, D r ), (D r+1 )) consder the sequence B 1 = (b 1,..., b r): { b b f o(b ) 1 = f C f o(b ) = 1. Lemmas 8 and 9 mply that b 1,..., b r are dstnct. To prove that B 1 s optmal for M 1 we need to construct a sequence of cocrcuts that makes B 1 sutable for M 1. As M 1 s a deleton mnor of M and M has no coloops, t follows from Corollary 5 that for every cocrcut C of M, the set C \ E 1 s the unon of cocrcuts of M 1. For 1 r let C C be the cocrcut of M 1 that contans b. The sequence ((C 1, b 1), (C 2, b 2),..., (C r, b r)) has the property that b C and b = arg max e C v e by constructon. As all b are dstnct, the C are also dstnct. However, ths sequence need not be sutable, as there could be ndces < j wth b = f C C j. The next result establshes that ths dffculty can be overcome. Lemma 10. Suppose a sequence K := ((C 1, b 1),..., (C r, b r)) of M 1 s such that (1) all b are dfferent, (2) b = arg max e C v e for 1 r, (3) and the sequence ((C 1, b 1),..., (C j, b j)) s sutable for some 1 j < r. Then the cocrcut C j+1 can be modfed so that condtons (1) (3) hold for j + 1 and M 1. Proof. If ((C 1, b 1),..., (C j+1, b j+1)) s sutable we are done; otherwse consder the smallest < j + 1 wth b C j+1. Usng strong crcut elmnaton we can choose a cocrcut C of M 1 n (C C j+1) b that contans b j+1. We replace C j+1 by C. By assumpton, b k / (C C j+1) b for every k <. Therefore, {b 1,..., b } C =, and we can replace C j+1 by C. Let K = ((C 1, b 1),..., (C j, b j), (C, b j+1), (C j+2, b j+2),..., (C r, b r)). Now K fulflls (1) (3) for j and {b 1,..., b } C j+1 =. Ether K fulflls (3) for j + 1, or there exsts another ndex > wth b C j+1, n whch case we repeat the procedure untl K fulflls (1) (3) for j + 1.

16 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 16 The hypotheses of Lemma 10 are satsfed by a VCG-sequence and j = 1. Repeated applcaton of the lemma proves there are cocrcuts C 1, C 2,..., C r that make the sequence ((C 1, b 1), (C 2, b 2),..., (C r, b r)) sutable n M 1 ; hence B 1 s optmal for M 1 and B s optmal for M. Theorem 11. Gven a VCG-sequence. Then B, B 1,..., B n are the unque optmal bases of M, M 1,..., M n, respectvely. The Vckrey payoff of bdder j N s :b {b 1,...,b r} E j (v b v fc ), and hs Vckrey payment s :b {b 1,...,b r} E j v fc = :b {b 1,...,b r} E j p. The theorem shows that f bdder j s awarded element b, then hs fnal payment ncreases by p. Therefore, n a dynamc settng he could be charged p at the moment he s awarded element b. 2.6 Te Breakng Tes among valuatons presents a slght techncal complcaton whch can be resolved wthout changng the deas n our results. The reader wllng to gnore ths techncalty may skp ths subsecton. When values are not necessarly dstnct, we can break tes by choosng suffcently small postve δ and perturbng v e by ɛ e := δ e to v e := v e + ɛ e = v e + δ e. For suffcently small δ, the perturbed valuatons have the property that v (S) v (T ) mples v(s) v(t ). Clearly a bass that s optmal for v on a mnor of M s also optmal for v on the same mnor yeldng the followng generalzaton of Theorem 11. Lemma 12. Suppose a VCG-sequence for the perturbed valuaton v of a nomonopoly matrod M wth nonnegatve valuaton v. Then B, B 1,..., B n are the unque optmal bases wth respect to v and v of M, M 1,..., M n. The Vckrey payment (under v) by bdder j N s :b {b 1,...,b r} E j v fc. Lemma 12 allows us to extend the notons of feasble sequence, VCGsequence, and condensed VCG-sequence, to valuaton wth tes, by requrng that the sequence condtons are fulflled for a perturbed valuaton. To sdestep the ssue of how to determne suffcently small δ, one could choose an arbtrary strct order on E and use t to break tes wth respect to v. Henceforth we wll break tes n ths way and wll demonstrate how ths can be acheved wthout relyng on the cooperaton of the selfsh bdders.

17 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod The Ascendng Aucton Aucton 3 Effcent ascendng aucton (unt-step) Requre: No-monopoly matrod M wth nonnegatve nteger valuaton 1: 1, p 0, r r(m) 2: Determne an orderng of E (for te-breakng) 3: whle < r do 4: Ask bdders to determne F = {f 1,..., f k } {f E(M): v f = p} and label the elements of F so that they are ncreasng accordng to the tebreakng. 5: for l 1 to k do 6: whle there exsts a bdder j and a cocrcut C of M \ f l wth C E j do 7: Ask bdder j to determne arg max e C v e and let b be the most valuable element from ths set accordng to te-breakng. 8: Award b to j and charge hm p p. 9: M M/b 10: : end whle 12: M M \ f l 13: end for 14: M and p p : end whle 16: B {b 1,..., b r } s the optmal bass Aucton 3 s our man algorthm. Notce that the condton C E j n Lne 6 generalzes the condton we had n the motvatng example that an agent owns a brdge. Observe n addton that the te-breakng s carred out wthout the bdders cooperaton: In Lnes 4 and 7 the bdders are asked questons regardng ther valuatons; afterwards the auctoneer breaks tes n ther answers. In ascendng prce auctons, t s standard to ask bdders to reveal ther demand at current prces. Lnes 4 and 7 are to be nterpreted n ths sprt. These steps n Aucton 3 ask bdders to reveal ther demand correspondences

18 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 18 at current prces, and n partcular reveal those elements that they are just ndfferent between buyng or not at current prces. In mplementatons, ths latter nformaton can also be extracted by askng bdders to reveal demand at current prces at the next hgher prce ncrement. It s easy to show that Aucton 3 determnes a condensed VCG-sequence f bdders behave truthfully, yeldng our man result. Theorem 13. For every no-monopoly matrod wth nonnegatve nteger valuaton, f bdders behave truthfully, then Aucton 3 determnes an effcent allocaton and charges Vckrey prces. Proof (Sketch; full proof n e-companon A.3). Notce that for dstnct valuatons the aucton determnes a condensed VCG-sequence. Ths would become clear f we add the lne 12.5: f f l has not been awarded then D D {f l }. By Theorem 7 there s a correspondng VCG-sequence; hence (wth Subsecton 2.6) the effcent allocaton s found. Wth Theorem 11 t follows that the p lead to Vckrey prces. The classcal Englsh-aucton style of ncreasng p n Lne 14 s p p + 1, whch s why we refer to ths as the unt-step verson of our aucton. It s clear why ths works when valuatons are nteger. There s another varant, the long-step verson, that s faster and able to handle arbtrary (nonnteger) valuaton; t s ntroduced n Subsecton 2.9. The unt-step aucton can be descrbed smply n the followng way. Start wth the prce set at zero where, by assumpton, no bdder holds a monopoly. Then (Lne 4) ask the bdders whether they have any elements at current prce that they are ndfferent about (because the prce matches ther values); such a queston s not unusual for an aucton. Among the elements that the bdders announce, the auctoneer can break tes n the way he has predetermned to. Hence he can pck the smallest element f among them (and later larger ones). Now Lne 6 s a queston that requres only knowledge of the matrod and the partton of E nto sets of nterest E ; snce the auctoneer knows both, checkng whether there s a C of M \ f l and a bdder j wth C E j s easy for hm and requres no further knowledge. Therefore the auctoneer can determne whether the removal of f would gve another bdder j a monopoly. If t does, let bdder j determne n Lne 7 hs best elements from C ; among these best elements the auctoneer determnes

19 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 19 wth hs te-breakng plan a unque best element e C, awards t to j and charges j the current prce. (All of ths requres no advanced knowledge of bdders valuatons). Then contract e and check for any other bdders that mght own a cocrcut. Afterwards delete the unnecessary element f from the matrod. If at current prce p another element f was announced (but not yet awarded) then repeat the prevous step; otherwse ncrease p and contnue. In the next secton, an example llustrates the aucton. 2.8 An Example An example run of the aucton on a graphcal matrod M s descrbed n Fg. 1. As shown n Fg. 1(a), there are three bdders {a, b, c} and fve elements {a:5, a:4, b:3, b:2, c:1}; an element encodes the prospectve buyer and value. The matrod has rank 3. Snce Fgs. 1(b) 1(d) show that removal of any sngle agent does not decrease the rank of the matrod t fulflls the no-monopoly condton. At prce p = 1 the stuaton n Fg. 1(e) results: only bdder c announces that the value of hs element f = c :1 (n gray) s reached and that he s ndfferent between payng 1 for t or not gettng t. At ths moment t has to be checked whether deleton of f would gve bdder a a monopoly. As the cut (dashed lne) ndcates n Fg. 1(f), f f s removed then bdder a owns a monopoly, as removal of a:5, a:4 from M f would reduce ts rank. Consequentally bdder a s asked for hs most valuable element out of a:5, a:4, to whch he wll answer a:5. Now the auctoneer awards hm a:5 and charges 1 for t. To remove a:5 from consderaton he contracts a:5 yeldng Fg. 1(g). Bdder c s ndfference has another consequence. As can be seen n Fg. 1(h), removal of bdder b and c:1 decreases the rank of the matrod, hence b would have a monopoly. He s awarded the better of b:2, b:3 at a prce of 1. Agan the auctoneer contracts the awarded edge b:3 yeldng Fg. 1(). Snce no more monopoles occur at p = 1, element c:1 s removed and the prce ncreases to p = 2. Bdder b then announces that the value of hs element b:2 s reached; see Fg. 1(j). If b:2 were removed then bdder a would agan have a monopoly; see Fg. 1(k). So bdder a s awarded a:4 at a prce of p = 2, element a:4 s contracted, and b:2 s deleted; Fg. 1(l) results. As r(m) elements have been awarded, the aucton ends. In the sealed-bd VCG aucton, the elements a:5, a:4, b:3 would be awarded and bdder a would pay 3, bdder b would pay 1, and bdder c pays nothng. These payments match the outcome of our ascendng aucton carred out

20 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 20 above. a:5 a:5 a:5 a:5 b:2 b:3 b:2 b:3 b:2 b:3 b:2 b:3 c:1 a:4 c:1 a:4 c:1 a:4 c:1 a:4 (a) p = 0 (b) a has no monopoly (c) b has no monopoly (d) c has no monopoly a:5 a:5 b:2 b:3 b:2 b:3 b:2 b:3 b:2 b:3 c:1 a:4 c:1 a:4 c:1 a:4 c:1 a:4 (e) p = 1; c:1 becomes ndfferent (f) p = 1; a gets a monopoly (g) p = 1; a:5 s awarded and contracted (h) p = 1; b gets a monopoly b:2 b:2 b:2 c:1 a:4 a:4 a:4 () p = 1; b:3 s awarded and contracted (j) p = 2; b:2 becomes ndfferent (k) p = 2; a gets a monopoly (l) p = 2; a:4 s awarded and contracted Fgure 1: An example llustratng Aucton 3 on a graphcal matrod wth elements {a :5, a :4, b :3, b :2, c :1} (encodng bdder and value) and bdders a, b, c. 2.9 Polynomal Aucton Not much has been sad about the runtme of Aucton 3. The problem wth the unt-step verson s that we have to determne several numbers (the values of the second-best elements) but do so by queryng every smaller nteger. Clearly ths yelds an aucton that s only pseudo-polynomal (n the hghest

21 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 21 value). More fundamentally, the classcal Englsh ascendng aucton to sell a sngle unt of a sngle good has the same problem. From a computatonal standpont, bnary search seems to be one way to solve ths problem; however, under straght-forward bnary search t turns out that truth-tellng need not be a weakly domnant strategy. Recently Grgoreva, Herngs, Müller, and Vermeulen (2007) proposed the bsecton aucton for a sngle good that takes polynomal tme and n whch truth-tellng s agan a weakly domnant strategy. To handle arbtrary values or to be faster, however, the aucton can be easly modfed to the long-step verson by changng Lne 14 to read 14L: Ask each bdder j to determne u j = mn e Ej M v e > p; M p M and p mn j N u j. Ths step asks each bdder for the value of hs least valuable remanng object, then ncreases the prce to the lowest of these reported values. 4 Ths can be mplemented n a varety of ways n practce. For nstance, bdders can be asked for suggested prce ncreases, wth the assurance that only the lowest of these suggestons wll be mplemented n order to speedly move the aucton along. If the bdder wth the lowest reported prce ncrease suggests too low of a prce ncrease (.e. under reports hs u j ), the aucton s delayed as no bdder wthdraws at the resultng prce. The long-step verson of Aucton 3 runs n polynomal tme and, snce t avods unt-steps, can accommodate nonnegatve ratonal valuatons. The outer loop of the long-step aucton s executed at most E tmes. It makes only polynomally many calls to an oracle fndng a cocrcut of a mnor of M contaned n some E, and t performs only polynomally many steps. Theorem 14. For every no-monopoly matrod wth nonnegatve valuaton, f bdders behave truthfully, then the long-step verson of Aucton 3 determnes an effcent allocaton and charges Vckrey prces. The aucton goes through a polynomal number of steps (n E, N, and the largest encodng length of a value) and makes polynomal use of a cocrcut oracle (for M and ts mnors). 4 In the specal case of a sngle object aucton, the prce smply jumps from zero through all the bdders values from lowest to second-hghest and then awards the object to the hghest bdder wthout revealng hs bd. If there are more goods, however, then sgnfcantly less nformaton s revealed.

22 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 22 Proof. The whle loop of Lnes 3 15 s carred out at most r(m) n tmes (because n each teraton at least one element gets contracted as bdders are honest). The nner loops are carred out at most E and N tmes respectvely. Therefore the polynomal-tme result holds. Notce, that under our assumpton of sngle ownershp we have N = O( E ) and hence one could say that the aucton s polynomal n E and the largest encodng length of a value. On the other hand, clearly, the aucton s polynomal n the number of rounds t needs, but the latter s only endogenously determned; however, as already mentoned, t can be bounded above by E. Theorems 13 and 14 are computatonal results. That s, t s mplctly assumed that the bdders honestly report ther nformaton to the Aucton. Of course ths assumpton s not wthout loss of generalty when bdders are strategc. In the next secton, we consder ths ssue of ncentves Incentves We show that n Aucton 3, truthful bddng forms an ex post equlbrum. That s, no bdder can beneft by bddng untruthfully provded that all other bdders bd truthfully. Ths s a stronger equlbrum property than Bayesan Nash equlbrum. When Vckrey payments are mplemented through the use of a sealedbd aucton, each bdder maxmzes hs payoff by bddng truthfully (.e. truthfully reportng hs valuatons) regardless of the strategy used by other bdders. In other words, the Vckrey mechansm s strategy-proof. Therefore t should not be surprsng that an ascendng aucton (.e. extensve form game) that mplements Vckrey payments nherts good ncentve propertes. The argument follows the logc of the Revelaton Prncple. Suppose a bdder behaves n a way whch s consstent wth some valuaton functon dfferent from hs true one. Ths causes hm to receve elements and make payments that correspond to the Vckrey allocaton/payments for the false valuaton. By strategy-proofness of the sealed-bd (drect revelaton) mechansm, the bdder cannot be better off than f he had behaved truthfully. Ths yelds Lemma 15 below. Formally, for any possble lst of valuatons that a bdder could realze, a strategy dctates how a bdder should behave after any possble hstory throughout the aucton. Whle we omt a tedous mathematcal defnton,

23 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 23 ths means that each bdder must decde what nformaton to declare n Lnes 4 and 7 of Aucton 3, gven any possble aucton hstory up to that pont (e.g. the order and prce at whch elements already have been awarded or wthdrawn by other bdders). A strategy s a truthful bddng strategy f, regardless of the aucton hstory, the bdder truthfully declares whch of hs elements have value p (at Lne 4) and whch are n arg max e C v e (at Lne 7, f the bdder s bdder j). Lemma 15. For any profle of valuatons, suppose each bdder s usng a truthful bddng strategy. Then any bdder receves a payoff at least as great as the payoff he would receve usng any other strategy whch s truthful wth respect to some other valuatons. The result follows from the strategy-proofness of the VCG sealed-bd aucton, as stated above. Ths argument s not suffcent to prove a domnantstrategy result, however, as bdders could bd n a way whch s nconsstent wth any valuaton functon. It s concevable for a bdder to make declaratons n Lnes 4 and 7 of Aucton 3 that are nconsstent wth truthful bddng under any possble valuatons. For example, when a bdder s awarded a cocrcut {a, b, c} he mght declare that v a = v b > v c and be awarded, say, a. Later f he s awarded the cocrcut {b, c, d} but announces v c > v b, ths s nconsstent wth any valuaton. Aganst strateges that make such nconsstent announcements condtonal on the announcements of others, truthful bddng may not be a best response. 5 Ths s why truthful bddng s not a domnant strategy n dynamc, ascendng (.e. extensve-form) auctons. Nevertheless, t turns out that t s an equlbrum for all bdders to commt to truthful bddng at all tmes throughout the aucton. When all other bdders are behavng truthfully, a bdder can do no better than to bd truthfully. The key to provng ths s Lemma 16, whch makes nontruthful strateges partally rrelevant. Lemma 16. Fx a profle of valuatons and a bdder. Suppose the other bdders are usng truthful bddng strateges but bdder s not. Then there exsts a truthful bddng strategy for (that s truthful wth respect to some valuaton ṽ) that yelds the same aucton outcome as hs orgnal strategy. Proof. Under the assumptons of the lemma, suppose bdder s usng an arbtrary bddng strategy s. Usng s, suppose s awarded the set of elements 5 Smlar dffcultes appear n the lterature on extensve-form mplementaton. See Moore and Repullo (1988).

24 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 24 {e 1,..., e k }, n that order, at prces p e1,..., p ek, respectvely. For every e E \ {e 1,..., e k } that he s not awarded, denote by p e the prce at whch he dropped the element from the aucton (Lne 4). We defne a valuaton ṽ and show that the same outcome would be obtaned (for all agents) f bdder bds truthfully wth respect to ṽ. For all e E, let { (1 + p r ) + (k l) f e = e l {e 1,..., e k } ṽ e = p e f e E \ {e 1,..., e k } (where p r s the hghest charged prce). Therefore, ṽ e > ṽ f for all e {e 1,..., e k } and f E \ {e 1,..., e k } and ṽ el > ṽ ej f 1 l < j k. Let s be the strategy of bddng truthfully wth respect to ṽ. The dfferences between s and s are the followng. In Lne 4 of the aucton when the current prce s p, under s bdder wll announce the elements e that he wll not wn such that ṽ e = p. Under s t could be that he announces addtonal elements that he s actually awarded n an executon of Lne 7 (whle the prce s stll p, but before the element s deleted n Lne 12). Suppose that such an element (wth smallest ndex after te-breakng) s f m. Snce f m was awarded to hm later n Lne 7 under s, ths happens before the loop n Lne 5 gets to l = m. Later, at l = m, the whle condton n Lne 6 cannot be fulflled, hence hs announcng of f m has no nfluence on the course of the computaton. So we acheve the same outcome (allocaton and prces) for any addtonally announced element. In Lne 7, under s, bdder may announce multple elements of C. However, only one element s awarded to hm, and the remanng elements are rrelevant n the computaton. Under s, he announces that same element only; hence the computatons n both cases are dentcal. A profle of strateges s an ex post equlbrum f, for any realzaton of valuatons, no bdder has an ncentve to change hs strategy even f he could fully learn the others valuatons. We have mposed no assumpton on the nformaton that bdders possess. At one extreme, bdders may know a lot about each other. At another extreme, bdders may only have probablstc belefs about others valuatons; these belefs may not even be commonly known. The strength of the ex post equlbrum concept s that such assumptons become rrelevant. For any possble realzaton of valuatons, no bdder

25 An Ascendng Vckrey Aucton for Sellng Bases of a Matrod 25 regrets havng used the strategy he dd, all else beng fxed. Whle ths s a very strong (hence desrable) concept, we should also pont out that t s not qute as strong as the concept of domnant strateges. 6 Theorem 17. For any profle of valuatons, the profle of truthful bddng strateges s an ex post equlbrum of Aucton 3. Proof. Suppose all bdders are bddng truthfully. Lemma 16 mples that each bdder has a best response among the set of strateges whch are truthful wth respect to some (possbly false) valuaton ṽ. Lemma 15 mples that the truthful one (wth respect to true valuatons) s one best response. Therefore bdder cannot do better than truthful bddng. The long-step verson of Aucton 3 also possesses strong ncentves propertes. It s straghtforward to show that Lemma 15 also apples to the long-step aucton (see the e-companon). Slght care needs to be taken wth Lemma 16, however. For example, consder Lne 14. Even f all other bdders are truthful, an untruthful bdder could announce u = p + 2 t n every round t. The aucton could then contnue ndefntely f the bdders announcements yeld F = n Lne 4, n each round. It s smple to fx ths by addng a consstency check to Lne 4 whch would requre the followng: f bdder j announces the mnmal utlty u j n Lne 14L of the long-step aucton, then he must announce at least one element n the next executon of Lne 4. Formally, Lne 4 could be changed to the followng. 4L: Ask bdders to determne F = {f 1,..., f k } {f E(M): v f = p} and label the elements of F so that they are ncreasng accordng to the te-breakng. Every bdder j wth p = u j must announce at least one element. Ths addton guarantees that f all bdders are bddng truthfully, no sngle bdder can devate to an arbtrary strategy n order to prolong the aucton ndefntely. Wth ths, the arguments behnd Lemma 16 and hence Theorem 17 would apply to the long-step aucton. 6 In fact domnant strategy equlbra are not obtanable n many (reasonable) dynamcaucton settngs. It s beyond the scope of ths paper to elaborate much on ths pont, but the essental reason for ths s that no strategy can be optmal (best response) when another bdder s behavng n a way talored to punsh players usng that gven strategy; hence none can be domnant.