A. Existence of Pure-strategy Bayesian Nash Equilibrium

Transcription

1 Web Suppement for The Roe of Quaity in On-ine Service Markets Eena Krasnokutskaya, Kyungchu Song and Xun Tang August, 4 This web suppement is organized as foows. Section A estabishes the existence of purestrategy Bayesian Nash Equiibrium in the mode considered in the paper. Section B provides further detais regarding the identification of mode eements that are not incuded in the main text. Section C discusses extension to modes where seers participation decisions are endogenous. Section D provides detais in the nonparametric cassification procedure. Section E eaborates on the technica detais in the GMM estimation of the mode (incuding the forms of moments used in the estimation). Section F provides additiona empirica resuts that are not incuded in the main text of the paper. A. Existence of Pure-strategy Bayesian Nash Equiibrium In this section, we show PSBNE exists in our mode. To simpify the notation and exposition, we abstract away from any auction or seer characteristics that are reported in the data. Our arguments can be extended to accommodate these observed heterogeneities by conditioning the game on them. Proposition A. Suppose that the foowing conditions hod. (i) The bids are commony chosen from a cosed interva of a rea ine. (ii) The conditiona CDF of ɛ given (α, U, Q) = (ᾱ, ū, q) are uniformy continuous for a (ᾱ, ū, q) in the support of (α, U, Q). (iii) The project costs C i are private, independent and identicay distributed, and have a density function bounded uniformy away from zero. (iv) The entry costs E i are private, independent, and identicay distributed, and have a density function bounded uniformy away from zero. (v) The entry costs and the private costs are independent from each other. Then there exists a PSBNE in the game. Proof of Proposition A. We first consider a subgame in the bidding stage where a subset A N of bidders have aready decided to enter. We first show that in each subgame with A, there exists a PSBNE. For this, we invoke Coroary. of Athey () by which it suffices to show that the payoff faced by each bidder is continuous in the bid profie, and the expected payoff satisfies singe crossing of incrementa returns. Given the other payers strategies b i = (b j ) j A\{i}, the interim expected payoff for bidder i is given by U i (b i, c i ; b i ) u i (b i, b i (c i ); c i )df (c i c i ),

2 where u i (b; c i ) is the payoff for bidder i conditiona on the vector of bids b and private cost c i, and independence among the private costs are used. In our mode, the payoff u i (b; c i ) is given as foows: u i (b; c i ) (b i c i ) Pr{i wins b}, where, with F ( ε ᾱ, ū, q) denoting the joint CDF of ɛ given (α, U, Q) = (ᾱ, ū, q), Pr{i wins b} = h(b; ᾱ, ū, q)df (ᾱ, ū, q), and h(b; ᾱ, ū, q) H i (b;ᾱ,ū, q) df ( ε ᾱ, ū, q) and H i (b; ᾱ, ū, q) = { ε R A : ᾱ q j,i + ε j,i b j b i and ū ᾱq i ε i b i, j N p α( q k q i ) + ε k,i b k b i, k N t }. Since F ( ᾱ, ū, q) is uniformy continuous for a (ᾱ, ū, q) in the support of (α, U, Q), we find that h( ; ᾱ, ū, q) is continuous. By the Bounded Convergence Theorem, Pr{i wins b} is continuous in b. As for singe crossing of incrementa returns, we consider the foowing. For each bidder i, for b i b i and c i c i u i (b i, b i, c i ) u i (b i, b i, c i ) = (b i c i ) Pr{i wins b i, b i } (b i c i ) Pr{i wins b i, b i } = (b i b i) Pr{i wins b i, b i } + (b i c i ) (Pr{i wins b i, b i } Pr{i wins b i, b i }). The ast term is bounded from beow by because (b i c i) (Pr{i wins b i, b i } Pr{i wins b i, b i }) Pr{i wins b i, b i } Pr{i wins b i, b i }. Therefore, the payoff u i (b i, b i, c i ) satisfies non-decreasing differences in (b i, c i ). This impies that its expected payoff U i (, ; b i ) satisfies singe crossing of incrementa returns. Hence by Coroary. of Athey (), each subgame with given A has a PSBNE. We now consider a PSBNE in the entry stage. For this, we use Theorem of Athey (). The action space is {, } with action denoted by d. We set d = to denote participation and d = to denote non-participation. The private type for each bidder i is entry cost E i. The interim expected payoff function for each bidder i with entry cost e i is given by ( d i ) (E [ U i (b i (C i ), C i ; (b j ) j A(d(e))\{i} ) E i = e i ] ei ) = ( d i ) (E [ U i (b i (C i ), C i ; (b j ) j A(d(e))\{i} ) ] e i ) where A(d) = {i N : d i = }, the bid strategy profie b(c) are given by the subgame PSBNE whose existence is previousy estabished, and d(e) = (d i (e i )) i N. (Reca that the private costs C i are reveaed ony after the bidder decides to enter and are independent of E i.) Certainy Assumption A and singe crossing of incrementa returns in Theorem of Athey () are satisfied. Thus, the existence proof of PSBNE foows from the theorem. Q.E.D.

3 3 B. Further Detais about Identification Resuts B. Quaity differences and buyers tastes We now provide more detais in the argument about the identification of the difference in quaity eves, the distribution of buyers weights for quaity α, and the match components ɛ. To do so, it is convenient to first iustrate the main idea using a simpe case where the set of participants consist of two permanent bidders i, j and a singe transitory bidder k. Under the maintained independence between (U, α, ɛ) and C i s (and hence B i s) and the assumption that participation decisions are non-strategic (in the sense that they do not depend on the reaization of (U, α, ɛ) and private costs), the probabiity that i wins conditiona on the reaized vector of bids (b i, b j, b k ) is Pr (α q j,i + ɛ j,i b i b j and Y i,k ɛ i b i ), () where random variabe Y i,k is defined as the maximum of U αq i and α(q k q i ) b k + ɛ k. First, suppose q i is identica to q j so that q i,j = (which is an event that can be conditioned on given that the quaity cassifications are identified before). Then a deconvoution argument based on the Kotarski Theorem impies the margina distributions of ɛ i, ɛ j and Y i,k are identified up to scae, provided the joint support of (B i B j, B i ) is arge reative to that of ( ɛ j,i, Y i,k ɛ i ) conditiona on the price from the transitory bidder b k. Next, consider a simiar winning probabiity except that q i q j. The joint distribution of (α q j,i + ɛ j,i, Y i,k ɛ i ) is recovered from () as ong as the support of (B i B j, B i ) is sufficienty arge given b k. Thus the independence between (α, Y i,k ) and (ɛ i, ɛ j ) impies the joint distribution of (α q j,i, Y i,k ) is recovered using knowedge of the distribution of ɛ i obtained in the previous step. A scae normaization such as E[α] = then impies the difference in quaity eves and the distribution of buyers taste for quaity are identified from the margina distribution of α q j,i. In addition, the distribution of Y i,k given α is aso recovered. This means, with a ocation normaization that q j =, the distribution of max{u, αq k B k + ɛ k } given α and b k is identified as that of Y i,k + αq i given α and b k. The support conditions invoved in the identification of the distribution of buyers s tastes is aso discussed in further detais in Section B4 beow. We now formaize the argument for identifying buyers tastes presented. For any pair of bidders i, j, et B i,j denote the vector of bids submitted by the set of the other entrants A\{i, j}, where A A p A t denote the set of entrants. Let Y i,j (B i,j ; a) denote the maximum of U αq i and α(q k q i ) b k + ɛ k for k A\{i, j}. The foowing assumption is needed. (A3) There exists a set a = a p a t, a pair i, j a p and a vector of bids b i,j {b k } k a\{i,j} such that (i) the characteristic function of the joint distribution of (α q j,i + ɛ j ɛ i, Y i,j (b i,j ; a) ɛ i ) does not vanish; and (ii) the joint support of (B j B i, B i ) incudes the joint support of (α q j,i + ɛ j ɛ i, Y i,j (b i,j ; a) ɛ i ) conditiona on b i,j. Proposition B. Suppose (A), (A) hod. (i) If (A3) hods for some i, j with q i = q j and some (a, b i,j ), then the margina distribution of ɛ i and the distribution of Y i,j given (a, b i,j ) are jointy identified up to a ocation normaization (e.g. E(ɛ i ) = ). (ii) If in addition (A3) aso hods for some i, j with q i q j and some (a, b i,j), then the quaity difference q i,j and the distribution of α are jointy identified up to a scae normaization (e.g. E(α) = ). Besides, the joint distribution of (α q j,i, Y i,j ) given (a, b i,j) is aso identified. Proof of Proposition B. Part (i). Consider i, j and a t, a p, b i,j satisfying Assumption (A3) and q i = q j. Under Assumptions (A) and (A), ɛ i, ɛ j, Y i,j are mutuay independent given any

4 4 (b i,j, a). Hence, for any b i, b j R, the probabiity that i wins conditiona on B i = b i, B j = b j and a, b i,j equas ϕ(b i, b j ; q i, q j ) Pr {α q j,i + ɛ j,i b j b i and Y i,j (b i,j ; a) ɛ i b i } where q j,i q j q i and ikewise for ɛ j,i. Note the equaity uses independence of private costs (and hence bids) from U, α, and ɛ. With B j and B i being independent of (U, α, ɛ), evauating ϕ(.,.; q i, q j ) at different reaizations of B i, B j ony amounts to evauating the same joint distribution of (ɛ j ɛ i, Y i,j (b i,j ; a) ɛ i ) (with b i,j, a fixed) at different points on the support. Thus, under Assumption A3-(ii), i.e. the arge support condition, this joint distribution is identified. Mutua independence between ɛ i, ɛ j and Y i,j (b i,j ; a) given (b i,j, a) impies their margina distributions are identified up to some ocation normaizations (e.g. E(ɛ) = ) by an appication of the Kotarski s Theorem, or Theorem.. in Rao (99). Part (ii). Without oss of generaity, suppose conditions for part (ii) hod for q i > q j. Repicating arguments in part (i) with i, j, a t, b i,j satisfying Assumption A3 and q i > q j shows that the joint distribution of (α q j,i + ɛ j,i, Y i,j (b i,j ; a ) ɛ i ) given (b i,j, a ) is identified. This impies the margina distribution of α q j,i + ɛ j,i and the distribution of Y i,j (b i,j ; a ) ɛ i given (b t, a ) are aso recovered. With the margina distribution of match components aready identified from part (i), so is the distribution of ɛ j,i. With α q j,i being independent from ɛ j,i, this means the distribution of α q j,i can be identified as ong as the characteristic-function of ɛ j,i is non-vanishing. It then foows that q j,i and the distribution of α are identified up to some scae normaization (such as E[α] = ). Finay, note the joint distribution (α q j,i, Y i,j (b i,j ; a )) given (b i,j ; a ) can be recovered from knowedge of the distributions of ( ɛ j,i, ɛ i) and the distribution of (α q j,i + ɛ j,i, Y i,j (b i,j ; a ) ɛ i ) given (b i,j ; a ), as ong as the characteristic function for ( ɛ j,i, ɛ i) is non-vanishing. This is because (A) and (A) impy ( ɛ j,i, ɛ i) is independent from (α q j,i, Y i,j (b i,j ; a )) with (b i,j ; a ) fixed. The proof of identification of the distribution of outside option and the distribution of quaities of transitory bidders conditiona on bids is as presented in the text. Q.E.D. B. Buyers taste for observed characteristics We now expain how to recover the distribution of β (buyers tastes for observed characteristics) in a context when a participants in the auction are permanent seers. Let their quaity eves be identified. Then Pr (i wins A = a, (b i ) i a ) ɛ j,i + α q j,i + x j,i β b j b i j a\{i} = Pr and U αq i ɛ i b i where x j,i x j x i. We need to introduce the foowing condition for identification of the distribution of buyers taste for observed heterogeneity. (B) There exist some i and a subset of permanent seers a with i a such that (i) the ( a )- by-j matrix ( x j,i ) j a\{i}, with J being the number of coordinates in x i, has fu rank; (ii) the support of (( ɛ ji + α q ji + x ji β) j a\{i}, U αq i ɛ i ) is a subset of the support of ((B j B i ) j a\{i}, B i ); and (iii) (( ɛ j,i + α q j,i ) j a\{i}, U αq i ) have non-vanishing characteristic functions.

5 5 Both (i) and (iii) in (B) are mid and standard assumptions in nonparametric identification. The support conditions in (ii) can be structuray justified aong simiar ines discussed in Section B4 beow. Proposition B Suppose the distribution of (α, ɛ, U ) and the quaity eves are identified. If the condition (B) hods, then the distribution of β is identified. We now sketch the proof of this proposition. Provided the joint support of (B j B i ) j a\i and B i is arge enough to cover the joint support of (( ɛ ji + α q ji + x ji β) j a\i, U αq i ɛ i ), we can recover the joint distribution of the atter. With F ɛi and F α identified as above, and under our assumption that (ɛ i ) i a are i.i.d. and jointy independent from α, the distribution of ɛ j,i + α q j,i is known. With β assumed to be independent from α and (ɛ i ) i N, we can identify the distribution of ( x j,i β) j a\i by taking the ratio of the characteristic functions of ( ɛ j,i + α q j,i + x j,i β) j a\i and ( ɛ j,i + α q j,i ) j a\i. As ong as the matrix ( x j,i ) j a\i is furank, then the joint density of β is identified as the product of the joint density of ( x j,i β) j a\i and the absoute vaue of the determinant of the square matrix ( x j,i ) j a\i by changing variabes. B3. Distribution of private costs We discuss how the distribution of project s costs can be identified. We consider a simpe case when bidders entry costs are independent of the project s costs. The genera resut obtains by combining steps presented beow with the identification strategy proposed by?. The identification of the distribution of project s costs. in a simpe case of signas independence foows an argument simiar to Guerre, Perrigne, and Vuong (). To see this, note that the quaity eves for permanent seers, the distribution of quaity eves of transitory seers, and buyer tastes can be considered known since they are identified in preceding sub-sections. The inverse bidding strategy can be recovered as foows. The first-order condition for bidder i choosing price b in equiibrium is: (b c i ) b Pr {ϖ(a) b} b=b = Pr {ϖ(a) b } () where ϖ(a) denotes the maximum of max j A\{i} (αq j αq i + x j,i β + ɛ j,i B j ) and U αq i. Thus, it suffices to show that the distribution on the right-hand side can be identified. It woud impy that the derivative on the eft-hand side woud aso be identified, in which case the inverse bidding strategy (and consequenty the distribution of private cost C i ) woud aso be recovered for every subpopuation of seers with (x, q). Note the right-hand side of () is: {a:a N} P {ϖ(a) b i A = a} Pr {A = a}. For those entrants in A\{i} who are transitory seers, their quaities are mutinomia random variabes whose distributions are recoverabe, with the distribution of quaities assumed to be the same across the two subpopuations of permanent seers and transitory seers (i.e. π p,k = π t,k ). Finay note that, given any fixed a, the distribution of ϖ(a) can be constructed from the joint distribution of α, ɛ, β, U identified earier and the distribution of submitted bids given a independence assumptions in (A) and (A).

6 6 B4. Discussion of Support Conditions Parts of our identification requires the supports of prices quoted by permanent seers be arge in the sense of condition (ii) in (A3). In this section we provide a heuristic argument showing that our mode is capabe of generating the variation in prices necessary for this condition to hod. We do so in the context of a simpe mode which abstracts away from the differences in unobserved quaities, stochastic participation, and the presence of transitory bidders but aows for the aocation rue to incude a stochastic match component. Adding these features compicates the agebra but does not require any additiona insight. In this simpe mode, the support condition for identifying the distribution of match components is reduced to: There exist i, j with q i = q j such that the support of (B j B i, B i ) incudes that of (ɛ j ɛ i, U ɛ i ). Note that, in a type-symmetric equiibrium which we consider here, the support of bids from i, j are identica, and denoted by [b i, b i ]. The respective supports of (B j B i, B i ) and (ɛ j ɛ i, U ɛ i ) are depicted in Figure??. It is cear from these figures that a set of sufficient conditions for the support restrictions above is: there exist i, j with q i = q j s.t. b > ε u and b < ε u, which can be satisfied if (a) b > ε u and (b) b b > (u u ) + (ε ε). Condition (a) hods provided b c i > ε u, where c i is the supreme of the support of private costs for i and j. Condition (b) essentiay requires the support of bids to be arge reative to that of outside utiity and match components. Intuitivey, (b) aso hods when the support of seers private costs is sufficienty arge. We now provide a simpe argument for this intuition. The idea is to show that the bidding strategy is continuous in the ength of the support of match component ε ε and the support of outside option. Under type-symmetric PSBNE the bidders strategies sove the maximization probem: ( ) σ i (c) arg max(b c) Pr (U ɛ i b) Pr b ε ε max { B j + ɛ j } ɛ i b j A\{i} The second probabiity in (3) represents i s beief, which is formed from i s knowedge of S, the distribution of private costs {F k } k K, and the distribution of quaities in the popuation of seers. Suppose ɛ i are i.i.d. uniform over [ε, ε]. Appying the Law of Tota Probabiity, we can write the objective function for seer i with costs c as ε [( ε ) ] F Bj (b ε i,j ) FU ( b + ε i ) (b c) dε j dε i. (4) ε ε ε ε Changing variabes between ε r and τ r εr ε ε ε ( (b c) for r = i, j, we can write (4) as F U (ε b + δτ i ) [ F Bj (b δ τ i,j ) ] dτ j ) dτ i (5) where δ ε ε is the ength of support of match components. Note (5) is continuous in both δ and the ength of support for U. It then foows from an appication of the Theorem of Maximum that the support of bids is continuous in the size of the support of match component and outside options. In this Figure, ε and ε denote the infinum and supremum of the support of ɛ i (and ɛ j ) and u, u denote the infinum and supremum for the support of U. (3)

7 7 Provided private costs vary sufficienty, the support of bids in a standard auction mode with no match components (i.e. ɛ is degenerate at ) and no outside option is an interva with non-degenerate interior. It then foows from the impication of the Theorem of Maximum that condition (b) hods whenever the support of ɛ and U is sma enough. By the same token, we can provide simiar structura justifications for the support conditions for recovering quaity eves using i, j with q i q j, as ong as variation in private costs and quaity differences are sufficienty arge reative to that of buyers tastes in (α, ɛ, U ). In the current exampe, the presence of transitory seers woud not entai any quaitative different arguments for the structura justification of the support conditions. C. Mode with Endogenous Entry This section provides further detais of the mode where seers decisions to participate in auctions are endogenous. We discuss how to extend the arguments for the identification from the text in this case. Existence of PSBNE in a mode with endogenous entry is shown in the preceding section. Throughout this section, we assume an anaog of (A) and (A) in the text hod for a set of potentia bidders. (A ) The private signa E i, and the private cost C i, are independent from each other, and the random vectors (E i,, C i, ) are independent across a i N and across auctions. For each i with q i = q k, these costs are independent draws from the continuous distributions Fk E and with a density positive over supports [e k, e k ] and [c k, c k ] respectivey. F C k (A ) The three random vectors (α, U, ), ɛ and (E i,, C i, ) i N are mutuay independent; match components ɛ i, are i.i.d. across i s; ɛ i, and (α, V, ) are continuousy distributed with a density positive over [ε, ε] and over [, α] [u, u ] respectivey. We focus on type-symmetric equiibria in which any pair of participants i, j who are ex ante identica (i.e. either i, j N p and q i = q j or i, j N t ) adopt the same strategies. The strategy of seer i from the quaity group q k, σ r,k, consists of a participation strategy σ r,k E (as defined above) and a bidding strategy, σ r,k B : [c k, c k ] Supp(I N, ) R +. Conditiona on participation, a seer i s expected profit from bidding b is given by Π r,k (b, c i, I N, ; σ i E, σ i B ) (b c i) Pr(i wins b, (r, k), I N, ; σ i E, σ i B ), where (σ i E, σ i B ) is the profie of strategies of the other participants. A PSBNE is a profie of strategies {(σ r,k E, σr,k B )} r {p,t}, k {,...,K} 3 such that and if and ony if σ r,k E (e i,, I N, ) =. σ r,k B (c i,, I N, ) = arg max b Π r,k (b, c i,, I N, ; σ i E, σ i B ) E[Π r,k (σ r,k B (C i,, I N, ), C i, I N, ) E i, = e i, ] e i, Our identification resuts coud be extended to the case when E i, and C i, are correated for each seer i in an auction, but the random vectors (E i,, C i, ) are independent across i N and across auctions. 3 We assume that σ r,k B = if σr,k E =.

8 8 As discussed above, PSBNE with monotone strategies exists. Given orthogonaity between E i, and C i, and independence of (E i,, C i, ) across bidders, the equiibrium participation strategies are monotone and are characterized by a threshod rue. That is, for any I N,, there exists e p,k and e t,k such that for a permanent potentia bidder i with quaity eve qk, σ r,k E (e i,, I N, ) = iff e i, e p,k ) whereas for each transitory seers with quaity eve q k, σ r,k E (e i,, I N, ) = iff e i, e t,k. It is easy to estabish that under Assumptions (A ), (A ) and (A3) and for any given I N, and in any pure-strategy Bayesian Nash equiibrium, the buyer s tastes {α, ɛ, U } are independent from {D i, } i N (and therefore A ); participation decisions {D i, } i N are mutuay independent across a i N ; and bids are mutuay independent within each reaized set of active bidders. The pairwise comparison approach for identification in Proposition?? remains unchanged in the presence of endogenous participation. Since participation decisions are independent across potentia seers, for given i and j the probabiities for various sets of competitors to reaize conditiona on i A and j A are sti identica to those conditiona on j A and i A. Thus, the proof of Proposition which reies on this property remains vaid. 4 Identification of quaity differences and the distribution of buyers tastes aso foow from the same arguments as before. In particuar, since entry decisions and bid distributions are orthogona to (α, ɛ, U ) the variation in bids by permanent seers is sti a vaid source of exogenous variation for identification. Further, these resuts aso rey on the independence of entry and bidding strategies across active bidders (and in particuar on independence of permanent bidders strategies from those of transitory bidders) which continue to hod under endogenous entry as mentioned above. Finay, note that with C i, being independent from E i, the standard arguments used for backing out the distribution of private costs remain vaid in the presence of endogenous entry. D. Detais of Cassification Procedure This section provides the detais of the nonparametric cassification methodoogy. A compete treatment of this methodoogy and forma resuts are found in Krasnokutskaya, Song, and Tang (4). D. Estimation of Cassifications with Known Number of Groups We consider the case where X i takes vaues from a finite set (x,, x Λ ). This impies that the set of the permanent seers can be partitioned into S p groups, S,, S Λ, such that for each λ =,, Λ, and any i, j S λ, x i = x j. The anaysis in this section is performed conditiona on x. For brevity, we omit conditioning on x in the exposition beow, so that we simpy write S instead of S λ. Define K to be the number of distinct quaity eves among the permanent seers. For ease of exposition, we first present the case with the number of quaity eves K equa to so that q i { q h, q } for a pair of unknown numbers q h and q. We expain how the agorithm generaizes to the case with K > ater. Let S h S be the coection of high quaity seers within group λ and S S be the coection of ow quaity seers within group λ. We estimate 4 Notice that in the mode with endogenous entry the pairwise index aso depends on the set or rather on reevant information about the set of potentia bidders, I N. In practice, the index is computed by pooing a the observations that are characterized by the same I N which may entai using auctions with distinct (in terms of identities) sets of potentia bidders.

9 9 an ordered partition (S h, S ) of S in three steps. First, for each i S, we estimate two ordered partitions: one partition consists of the group of the seers with higher or equa quaity than that of i (denoted by S (i)) and the rest (denoted by S\S (i)), and the other partition consists of the group of seers with ower or equa quaity than that of i (denoted by S (i)) and the rest. Second, among the two ordered partitions, we choose the one that is mosty ikey to coincide with (S h, S ). Third, we choose i such that the estimated partition associated with this i is most strongy supported by the data. Our method reies on the estimates of winning probabiities in Proposition. Let W p i, {, } be an indicator taking if the i-th seer that is permanent wins at auction and otherwise. Define ˆδ ij (b) ˆr ij (b) ˆr ji (b), where ˆr ij (b) L = W p i, K h(b i, b){j / A } L = K h(b i, b){j / A }, where K h (v) = K(v/h)/h for a univariate kerne function K. Then we construct test statistics: ˆτ + ij = max{ˆδ ij (b), }db, ˆτ ij = max{ ˆδ ij (b), }db, and ˆτ ij = ˆδ ij (b) db. We confine the integra domains to the intersection of bids submitted by i and j, and this restriction is omitted from the notation. We use a bootstrap method to estimate the finite sampe distributions of the test statistics. We first construct ˆr ij,s(b) and ˆδ ij,s(b) using the s-th bootstrap sampe, s =,, B, and consider the re-centered bootstrap test statistics: ˆτ + ij,s = max{ˆδ ij,s(b) ˆδ ij (b), }db ˆτ ij,s = max{ ˆδ ij,s(b) + ˆδ ij (b), }db, and ˆτ ij,s = ˆδ ij,s(b) ˆδ ij (b) db. Note that Lee, Song, and Whang (3) estabished asymptotic properties of these bootstrap test statistics in a more genera set-up. Using the bootstrap test statistics, we define the bootstrap p-vaues as foows: p z(i, j) = B B s= z {ˆτ ij,s > ˆτ ij} z with z {+,, }. We proceed in three steps as outined above. Step : Define Ŝ (i) = {j S\{i} : p +(i, j) p (i, j)} and Ŝ (i) = {j S\{i} : p +(i, j) > p (i, j)}. Step : We determine now whether seer i has the same quaity as those of S (i) or S (i),

10 i.e., ow quaity or high quaity. If both Ŝ(i) and Ŝ(i) are non-empty, 5 we cassify seer i as foows: Take Ŝ(i) = Ŝ(i) and Ŝh(i) = Ŝ(i) {i} if min og p (i, j) < min og p (i, j) j Ŝ(i) j Ŝ(i) Take Ŝ(i) = Ŝ(i) {i} and Ŝh(i) = Ŝ(i) if min og p (i, j) min og p (i, j). j Ŝ(i) j Ŝ(i) That is, we put seer i into the high-quaity group, if the evidence against the hypothesis that seer i has the same quaity as the seers in group Ŝ(i) is stronger than the evidence against the hypothesis that seer i has the same quaity as the seers in group Ŝ(i). Step 3: For each i S, we compute the foowing index: s (i) = { Ŝ(i) Ŝh(i) j Ŝ(i) og p +(i, j) if i Ŝh(i) j Ŝh(i) og p (i, j) if i Ŝ(i), where Ŝ(i) denotes the cardinaity of the set Ŝ(i). The quantity s (i) indicates the weakness of the ikeihood that i is cassified into her right quaity group. Then choose i that minimizes s (i) over i S, and et Ŝh = Ŝh(i ) and Ŝ = Ŝ(i ). We take Ĉ = (Ŝh(i ), Ŝ(i )) as an estimated cassification of payers into two quaity groups. 6 The generaization of the procedure to the case of K > with K known can proceed as foows. First, we spit S into Ŝh and Ŝ using the agorithm for K =. Then we find a minimum vaue (denoted by ˆp h ) of og p (i, j) among the pairs (i, j) such that i j, and i, j Ŝh, and a minimum vaue (denoted by ˆp ) of og p (i, j) among the pairs (i, j) such that i j, and i, j Ŝ. If ˆp h < ˆp, we spit Ŝh into Ŝhh and Ŝh using the same agorithm for K =, and otherwise, we spit Ŝ into Ŝh and Ŝ using the same agorithm for K =. We repeat the procedure. For exampe, suppose that we have cassifications Ŝ,, Ŝk obtained. For each r =,, k, we compute the minimum vaue (say, ˆp r ) of og p (i, j) among the pairs (i, j) such that i j and i, j Ŝr, and then seect its minimum (say, ˆp r ) over r =,, k. We spit Ŝr into Ŝr h and Ŝ r using the agorithm for K = to obtain a cassification of S into k groups. We continue unti the groups become as many as K. 5 If Ŝ(i) is empty, we pick some eve α =.5: Take Ŝ(i) = and Ŝh(i) = Ŝ(i) {i} if min j Ŝ(i) p (i, j) α. Take Ŝ(i) = {i} and Ŝh(i) = Ŝ(i) if min j Ŝ(i) p (i, j) < α. We proceed in a simiar way if Ŝ(i) is empty. 6 There may be aternative ways to obtain estimators of the quaity partition. One way is to fix i and appy hypothesis testing to the nu hypothesis that q i q j. This essentiay bois down to comparing the p- vaue p z(i, j) with a certain eve of the test. One unattractive feature of this aternative procedure is that the resut can be different depending on whether the nu hypothesis is taken to be q i q j or q i q j. This is because the conventiona hypothesis testing procedure treats the nu hypothesis and the aternative hypothesis asymmetricay. Hence in this paper, instead of comparing a p-vaue with a fixed eve of the test, we compare a p-vaue with an aternative p-vaue, to capture the symmetry of the comparison.

11 D. Seection of the Number of Groups The methodoogy outined above assumes that we know the exact number of groups. To accommodate the situation with rea ife data without knowedge of the number of the groups, we offer a method of consistent seection of the number of groups. We suggest that the number of groups shoud be seected to minimize the criterion function that baances a measure of goodness-offit that captures a misspecification bias versus a penaty term that penaizes overfitting. The goodness-of-fit measure is based on the variance test approach. Given an estimated cassification S = K k=ŝk with K groups, et ˆV k (K) = min og p (i, j), i,j Ŝk for each k =,, K. Suppose that K is the true number of groups. Let g(l) be such that g(l)/ L as L. Then, define ˆQ(K) K ˆV k (K) + Kg(L). K We seect K as foows: k= ˆK = argmin K N ˆQ(K). The idea of this seection is based on the foowing intuition. First, K K k= ˆV k (K) = O P (), as L, if K K, because there is no misspecification bias in this case. The imit O P () measures the asymptotic behavior of the goodness-of-fit of the mode when the cassification is weaky finer than the true cassification. Since Kg(L) as L, the minimization of ˆQ(K) over K eans toward a ower choice of K that is coser to K. On the other hand, if the cassification is stricty coarser than the true cassification, i.e., K < K, the quantity K K k= ˆV k (K) diverges at a rate faster than g(l), as L, due to misspecification. In this case, the minimization of ˆQ(K) over K excudes K such that K < K for arge sampes. Thus we obtain the consistency of ˆK under reguarity conditions. Detais in a more genera set-up are found in Krasnokutskaya, Song, and Tang (4). E. Detais of GMM Estimation E. Moment conditions Using the resut in Proposition, we derive the moment conditions as foows. For any vector vaued map h p (x,q), : R A R d h, and for each x X, q Qx, we can write the moment condition as: [ ] E h p (x,q), (B, I )W p (x,q), I [ ] = K A, E k= h p (x,q), (B, I )e p k,(x,q), (B, I ; θ) g k (B t, I,, I t, ; θ) I KA, d= g d(b t, I,, I t, ; θ).

12 where g k (B t, I,, I t, ; θ) is a parametric approximation of g k(b t, I,, I t, ), W p (x,q), = e p (x,q),k, (B ; θ) = A p (x,q), j A p (x,q), A p (x,q), j A p (x,q), W p j, and e (x,q),k, (B ; θ, j). This can be further re-written as: [ ) ] E h p (x,q), (B, I ) (W p(x,q), ep(x,q), (θ, g) I = with (6) K e p (x,q),(θ, g) = e p (k,x),q, (B ; θ) k= g k (B t, I,, I t, ; θ). KA, d= g d(b t, I,, I t, ; θ) As we discussed in the previous subsection, it may be usefu, especiay in the parametric setting, to separatey specify and estimate f(bi, t Qt A,i, = q A,i,k, I, ) and P (i A t x, Qt A,i, = q A,i,k, I, ) functions. In such a case, restrictions associated with (a) transitory seers bid distribution, (b) the transitory seers probabiity of participation, as we as (c) restrictions summarizing optima participation behavior may be additionay imposed. More detais are given ater. To construct a sampe version of the moment conditions, we first obtain a consistent estimator ˆω k, of ω k, via the sampe anaog principe, and construct ) ˆρ (x,q), (θ) = h p (x,q), (Bp, I ) (W p(x,q), êp(x,q), (θ), where ê p (x,q), (θ) is equa to ep (x,q), (θ) except that ω k, is repaced by ˆω k, and ˆω k, is the sampe anaogue of ω k,. We define ˆρ (θ) to be a coumn vector with ˆρ (x,q), (θ) stacked up with (x, q) running in X Q. Hence the dimension of ˆρ (θ) is d h,p X Q, where X Q is the cardinaity of the set X Q. Then define a genera method of moment estimator as foows: ˆθ GMM = argmin θ Θ ˆQ GMM (θ), where ˆQ GMM (θ) = ˆΣ = L ( L ) L ˆρ (θ) = L ˆρ ( θ)ˆρ ( θ), = ˆΣ ( L ) L ˆρ (θ), and = and θ is the first step estimator of θ which is a minimizer of ˆQ GMM (θ) ony with ˆΣ repaced by the identity matrix. Under reguarity conditions, the estimator is known to be asymptoticay norma with a positive definite covariance matrix. Note that the estimation error due to ˆω k, does not affect the asymptotic variance matrix because the components P {Q t A,i, = q A,i,k I } of ω k, take vaues ony from a finite set and hence has a convergence rate that is arbitrariy fast as L.

13 3 Next, we expain some additiona restrictions used in estimation. (a) The restriction associated with transitory seers bid distribution: f(b t i I ) = k f(b t i Q t A,i, = q A,i,k, I, ) Pr(Q t A, = q A,k, I ) = (7) K A k= ωa,k, t {f(b j Āt j Q t A,j, = q A,j,k, I, ) Pr(j A t Qt A,j, = q A,j,k, I, )} Pr(j A t j Āt Qt A,j, = q. A,j,d, I, ) K d= ωt A,d, Moment conditions associated with this restriction woud reate the empirica moments of the f(b t i I ) distribution to the theoretica moments computed using (7). (b) The restriction associated with the transitory seers probabiity of participation: Pr( A t x, = m x, I, ) = K A, k= j Āt K A, k= Pr( A t x, = m x, and Q t A, = q A,k I, ) = (8) { Pr(j A t Q t A,j, = q A,j,k, I, ) Pr(Q t N,j, = q j x t j,) } q N A Q N A Ω i N t Āt { Pr(i N t A t Q t N,i, = q i, I, ) Pr(Q t N,i, = q i x t i,) }. The derivation for this expression is provided in the proof of Proposition. Moment conditions associated with this restriction woud reate the transitory seers empirica probabiity of participation and expected x characteristics of entrants conditiona on I, to their theoretica counterparts using (8). (c) The restriction reated to the expected profit condition. This restriction summarizes optima participation behavior. It is summarized by the threshod strategy where potentia bidders with entry cost draws beow the ex-ante expected profit participate in the auctions and those with higher draws stay out. This impies that in equiibrium Pr(j A t x, Q t j, = q k,x, I, ) = F E (E[π t (j, Q t A,j, = q k,x, I, )]), where F E (.) is the distribution function of entry costs E and q k,x Q x. E. Semiparametric Estimation The estimation method in the previous section empoys parametrization of nonparametric functions g k. The functions g k invove the density of the transitory seers bids and participation probabiities. Since the bids and participations are equiibrium objects, one might prefer to use a more fexibe specification for the nonparametric functions g k. In this section, we expain how this extension can be done in practice. Reca our definition e p x,q, (θ, g) in (6) and define ˆρ x,q,s (θ, g) = h p x,q,(b p, I ) ( W p x,q, êp x,q, (θ, g)),

14 4 making its dependence on the nonparametric function g expicit. Let ˆm x,q, (θ, g) = L s= ˆρ x,q,s(θ, g) {I s = I } L s= {I s = I } and define ˆm (θ, g) to be a coumn vector with ˆm x,q, (θ, g) stacked up with (x, q) running in X Q. Hence the dimension of ˆm (θ, g) is d h,p X Q, where X Q is the cardinaity of the set X Q. In the semiparametric estimation, we regard the nonparametric function g as an infinite dimensiona nuisance parameter, and empoy the sieve minimum distance estimation method of Ai and Chen (3). As argued previousy, the functions g are nonparametricay identified under our set-up. Now et G L be the space of finite dimensiona sieves whose dimension increases as the number of auctions L grows. Detais about the appropriate sieves are found in the appendix. We construct the estimator as foows: where (ˆθ CMD, ĝ) = argmin (θ,g) Θ GL ˆQ CMD (θ, g), ˆQ CMD (θ, g) = L L = ˆm (θ, g)ˆσ ˆm (θ, g), and ˆΣ is a consistent estimator of a non-singuar matrix Σ. Ai and Chen (3) showed that under reguarity conditions, we have n (ˆθCMD θ ) d N(, V ), where V is a non-singuar covariance matrix. We turn to the form of the covariance matrix formua which foows Ai and Chen (3). For each i =,, N, we et for α = (θ, g) A L = Θ G L, and r () (α) = [r () x,q, (α) ] q Q x,x X r () x,q, (α) = E [ h p x,q,(b, I ){W p x,q, ep x,q, (α)} I ] r () x,q, (α) = E [ h t x,q,(b, I ){Wx, t et x,q, (α)} I ] and r() (α) = [r () x,q, (α) ] q Q x,x X. In other words, r() (α) is a coumn vector with vectors r () x,q, (α), q Q x, x X, stacked up and r () (α) = [r () x,q, (α) ] q Q x,x X is a coumn vector with vectors r () x,q, (α), q Q x, x X, stacked up. We define m (α) = For each j =,, d θ, we define for w G, m (θ, g ) g [ r () r () ] (α). (α) [w g ] = m (θ, τw + ( τ)g ) τ=. τ The eft-hand side term represents the pathwise derivative of m (θ, g ) aong the direction w g.

15 5 Let w j be the soution to the minimization probem: where ( m (θ, g ) inf m ) ( (θ, g ) m (θ, g ) [w j g ] Σ m ) (θ, g ) [w j g ], w j G θ j g θ j g Then we take [ ] Σ = E [r () (α), r () (α)][r () (α), r () (α) ] I. D w, = m (θ, g ) m (θ, g ) [ ] w θ g g,, wd θ g. The asymptotic covariance matrix formua becomes V = E [ D w,σ D w,]. The estimation of the asymptotic covariance matrix formua can be done in a straightforward way. For exampe, we may foow Section 5 of Ai and Chen (3), except that instead of using the series estimation to obtain the sampe anaogue of the conditiona expectation E[ I ], we use the usua sampe anaogue of the conditiona expectation with discrete conditiona variabes because I is a discrete random vector. Detais are omitted. Now et us consider the choice of the sieve space G L. Let K > be an integer such that for a L, max =,,L K K. Now et us discuss the construction of the sieve space G L. For each k =,, K, and reaized vaue of (I,, I t, ), the function g k(, I,, I t, ) defined in (??) is of particuar form. First we take G L = G,L G K,L, where G k,l s are constructed as foows. For i =,, N, L, and k =,, K, et G k,l and F k,l be sieve spaces, where for each g k,l G k,l, g k,l : I [, ] and for each f k,l F k,l, f k,l : R I [, ) and for each I, I, fk,l (b, I, )db =. Then we construct a sieve space G k,l as the coection of maps g k,l (b, I, ), where g k,l (b, I, ) = x X Āt x, j= f k,j,l (b, I, )v x k,j,l(i, ), where vk,j,l x Vx k,j,l, and f k,j,l F k,j,l. For Vk,j,L x, we choose a sieve space Mx k,j,l of rea-vaued functions and define { } exp(m( )) Vk,j,L x = + exp(m( )) : m Mx k,j,l. As for M x k,j,l, we can take poynomia series. As for F k,l = K j= F k,j,l, we use a Hermite poynomia sieve, where F k,j,l = { fk,kl,k (x; σ k, ε,k, r,k, a k ) : ε >, σ >, r, a s,k R, s =,, K L,k fk,kl,k (x; σ k, ε,k, r,k, a k )dx = and f k,kl,k (x; σ k, ε,k, r,k, a k ) is defined to be ε,k + πσk K L,k s= a s,k ( x r,k σ k ) s exp ( (x r,k) σ k ). },

16 6 Observe that f k,kl,k (x; σ k, ε,k, r,k, a k )dx = ε,k + K L,k K L,k s= t= a s,k a t,k E ( Z s+t), where Z is a standard norma random variabe. The quantity E (Z s+t ) can be expicity computed from the moment generating function. F. Additiona Empirica Resuts Tabe : Estimated Quaity Structure for a Given Number of Groups Number of Groups K = K = K = 3 K = 4 K = 5 K = V Q(K) This tabe shows the estimated quaity group structures for the various numbers of quaity groups for Eastern European suppiers with the medium eves of average reputation score. Rows -6 record the number of suppiers estimated to beong to a respective group. Rows 7 and 8 record the vaue of the p vaue component of the criterion function and the vaue of the criterion function. The resuts are based on the penaty function g(l) = og(og(l)). Resuts indicate that the number of groups most supported by the data is equa to three.

17 7 Tabe : Participation Decision and Bid Distribution Score Q I(T) II(T) I(P) II(P) Mean Constant.55 (.9) -.5 (.9).585 (.5) -.73 (.9) No Ratings -.83 (.5) -.33 (.8) < Ratings (.6).5 (.) 3 < Ratings (.7).5 (.9) Number of Ratings.7 (.63).3 (.7) Average Score -.5 (.5) -.4 (.) Average Score.5 (.). (.3) North America, Low, -.8 (.3).3 (.) -.37 (.46).3 (.7) North America, Low, -.73 (.8) -.4 (.) -.7 (.44) -.53 (.5) North America, Medium, -.38 (.) -.39 (.3) -.6 (.46) -.5 (.6) North America, High, -.73 (.).34 (.3) -.4 (.4).65 (.7) North America, High, -.8 (.) -.65 (.3) -.66 (.39) -.93 (.3) Eastern Europe, Low, -.6 (.7).3 (.) -.83 (.43).5 (.7) Eastern Europe, Low, -.6 (.) -.94 (.3) -.76 (.38) -.8 (.) Eastern Europe, Medium, -.99 (.35).34 (.5) -.46 (.44).3 (.) Eastern Europe, Medium, -.9 (.4) -.45 (.) -.6 (.48) -.98 (.) Eastern Europe, Medium, (.3) -.57 (.7) -.67 (.5) -.3 (.9) Eastern Europe, High, -.9 (.4) -.3 (.3) -.48 (.38) -.57 (.34) Eastern Europe, High, -.78 (.43) -.9 (.3) -.49 (.5) -.99 (.) Eastern Europe, High, (.) -. (.) -.7 (.38) -.6 (.9) South-East Asia, Low, -.4 (.34).5 (.).86 (.43).88 (.3) South-East Asia, Low, -.46 (.34) -.3 (.) -.6 (.4) -.4 (.) South-East Asia, Low, (.44) -.75 (.) (.43) -.5 (.33) South-East Asia, Medium, -.8 (.63) -.75 (.) -.96 (.44) -.7 (.7) South-East Asia, Medium, -.83 (.8) -.7 (.3) -.3 (.38).33 (.) South-East Asia, Medium, (.35) -.74 (.3) ( (.7) South-East Asia, High, -. (.37) -.95 (.) -.78 (.38) -.5 (.) South-East Asia, High, -.95 (.4) -.99 (.4) -.65 (.53) -.8 (.34) Std Error.7 (.9).38 (.) This tabe reports the effects of the covariates and the group premiums on seers bid distribution and participation decisions for the specification presented in the paper. Coumns I(T), II(T), and I(P), II(P) report estimated coefficients for the bid distribution and probabiity of participation of the transitory and permanent seers respectivey. Average Score and Average Score denote interactions of the current average score variabe with the indicators for Ratings 3 and 3 Ratings 6. The stars,, indicate that a coefficient is significant at the 95% significance eve.

18 8 Tabe 3: Participation Decision and Bid Distributions: Competitive Effects I(T) II(T) I(P) II(P) North America, ow,. (.7). (.3).3 (.). (.5) North America, ow,. (.).5 (.) -.5 (.3). (.4) North America, medium, -.3 (.) -.89 (.) -. (.) -.4 (.4) North America, medium, -.5 (.8) -.3 (.5) -.4 (.) -.7 (.3) North America, high,.5 (.3). (.78) -.7 (.) -. (.3) North America, high, -.6 (.) -.85 (.) -.8 (.3) -.6 (.4) Eastern Europe, ow, -.5 (.). (.7) -. (.). (.5) Eastern Europe, ow, -.7 (.) -.5 (.) -.5 (.6) -.9 (.6) Eastern Europe, medium, -.5 (.3) -.7 (.4) -.3 (.) -.7 (.) Eastern Europe, medium,.34 (.3).6 (.). (.4). (.3) Eastern Europe, medium, 3 -. (.) -.6 (.5). (.) -.4 (.4) Eastern Europe, high, -. (.) -. (.7) -. (.3) -.3 (.5) Eastern Europe, high,.8 (.5).7 (.4). (.5).3 (.) Eastern Europe, high, (.4) -. (.9). (.3) -. (.3) South-East Asia, ow,. (.) -.7 (.3) -. (.) -. (.) South-East Asia, ow, -.3 (.4) -.4 (.8) -.8 (.3) -.9 (.) South-East Asia, ow, -.3 (.) -.6 (.) -. (.). (.3) South-East Asia, medium, -. (.) -.3 (.) -.4 (.3) -. (.5) South-East Asia, medium, -.3 (.) -. (.) -. (.) -.4 (.) South-East Asia, medium, 3.3 (.33).5 (.4) -. (.).7 (.5) South-East Asia, high, -.4 (.3) -.7 (.4) -. (.). (.) South-East Asia, high, -.4 (.) -.39 (.8) -.4 (.) -.7 (.3) This tabe reports the coefficients summarizing the impact of the various potentia competitors on seers bid distribution and participation decisions. Coumns I(T), II(T), and I(P), II(P) report estimated coefficients for the bid distribution and the probabiity of participation of transitory and permanent seers respectivey. The resuts are based on the data set consisting, 3 projects. The quaity eve for South and East Asia, ow score, Q =, is normaized to be equa to zero. The stars,, indicate that a coefficient is significant at the 95% significance eve.

19 9 Figure : Bid Functions Low Score Medium Score High Score North America Low Quaity High Quaity Diagona 3.5 ow quaity high quaity diagona 3.5 ow quaity high quaity diagona Bids.5 bids.5 bids costs costs Eastern Europe medium quaity high quaity diagona 3.5 Low Quaity Medium Quaity High Quaity Diagona 3.5 Low Quaity Medium Quaity High Quaity Diagona Bids.5 Bids.5 Bids South and East Asia Low Quaity Medium Quaity High Quaity Diagona 3.5 Low Quaity Medium Quaity High Quaity Diagona 3.5 Low Quaity High Quaity Diagona Bids.5 Bids.5 Bids The figure shows the equiibrium bidding strategies of permanent seers recovered from the first order conditions of bidders optimization program. The convexity at the upper end of the costs support arises due to presence of stochastic component in buyers tastes.

20 Figure : Density of Project s Cost Distribution Low Score Medium Score High Score North America 5 Low Quaity High Quaity Low Quaity High Quaity Density 6 Density Density Eastern Europe 8 6 Medium Quaity High Quaity Low Quaity Medium Quaity High Quaity Low Quaity Medium Quaity High Quaity Density Density 6 Density South and East Asia 4 5 Low Quaity Medium Quaity High Quaity 6 Low Quaity Medium Quaity High Quaity Medium Quaity High Quaity Density 6 Density 3 Density The figure shows the permanent seers distribution of costs recovered by combining estimated bidding strategies and bid distributions.

21 Using Standard as Opposed to Muti-Attribute Auctions In this section we evauate the wefare gains associated with using muti-attribute auction aocation mechanism rather than standard auction mechanism in an on-ine market. We abstract away from characteristics other than quaity and focus on Eastern European permanent seers with medium reputation scores as potentia auction participants. We set the number and the composition of the set of potentia bidders to match these statistics for a typica auction. The resuts are presented in the tabe beow. Coumn () and coumn () summarize the resuts of a muti-attribute and a standard auctions respectivey where a quaity eves are aowed to participate. In Coumns (3), in addition to using a standard auction mechanism, we aow the intermediary to imposes a pre-certification requirement that ensures that ony medium- and high-quaity bidders are aowed to participate in these auctions. We find that using a standard instead of a muti-attribute auction mechanism resuts in a substantia oss of vaue to buyers. Whie the prices paid by buyers decrease by 5.8% on average, this reduction is aso associated with 9.5% reduction in the average purchased quaity which in turn entais 6.5% reduction in buyer s vaue and 9.% reduction in the overa surpus. The reasons for these effects are cear. Our findings indicate that the provision of high-quaity service is substantiay more expensive than the provision of medium-quaity service. This makes highquaity seers reativey uncompetitive on price. Thus, they win a sma fraction of standard auctions and buyers end up mosty procuring the services of medium-quaity seers even if they woud be wiing to pay a premium for higher quaity eve. The standard auction mechanism aso does not aow to take into account seer s idiosyncratic suitabiity for a given project (ɛ component). Both effects contribute to the oss of vaue. The tota surpus is further reduced by ower profits earned in this market. Pre-certification requirement improves the average purchased quaity. However, this effect is off-set by the increase in price associated with the reduction in the number of participating seers. Overa, buyers in this market gain substantiay from using muti-attribute rather than standard auction as an aocation mechanism. These resuts importanty depend on the features of our environment. For exampe, in settings where owquaity seers are more competitive the effects are ikey to be even stronger.

22 Tabe 4: Standard Auction with Certification Muti-Attribute Standard Auction Auction Excude Low A Leves A Leves Quaity Average Price Average Net Expected Vaue Net Expected Vaue, τ α,9%..6. Net Expected Vaue, τ α,7% Net Expected Vaue, τ α,5% Net Expected Vaue, τ α,3% Net Expected Vaue, τ α,% Expected Tota Surpus Average Number of Entrants Market Shares High Quaity 3% 3% 7% Medium Quaity 5% 6% 66% Low Quaity 4% % Not Aocated % 5% 7% Note: In this tabe the expected vaue and the tota surpus are computed reative to the outside option. To faciitate the comparison across auction formats we assume that in a standard auction a buyer does not aocate the project if his net vaue from the owest bid is beow his outside option. References Ai, C., and X. Chen (3): Efficient Estimation of Modes with Conditiona Moment Restrictions Containing Unknown Functions, Econometrica, 7, Athey, S. (): Singe Crossing Properties and the Existence of Pure Strategy Equiibria in Games of Incompete Information, Econometrica, 69(5), Guerre, E., I. Perrigne, and Q. Vuong (): Optima Nonparametric Estimation of First-Price Auctions, Econometrica, 68(3), Krasnokutskaya, E., K. Song, and X. Tang (4): Uncovering Unobserved Agent Heterogeneity through Pairwise Comparisons, Working paper, University of Pennsyvania. Lee, S., K. Song, and Y.-J. Whang (3): Testing Functiona Inequaities, Journa of Econometrics, 7, 4 3.