Inference of Bidders Risk Attitudes in Ascending Auctions with Endogenous Entry

Transcription

1 Infeence of Biddes Risk Attitudes in Ascending Auctions with Endogenous Enty Hanming Fang y Xun Tang z Fist Vesion: May 28, 2. This Vesion: Apil 7, 22 Abstact Biddes isk attitudes have impotant implications fo selles seeking to maximize expected evenues. In ascending auctions, auction theoy pedicts bid distibutions in Bayesian Nash equilibium does not convey any infomation about biddes isk pefeence. We popose a new appoach fo infeence of biddes isk attitudes when they make endogenous paticipation decisions. Ou appoach is based on the idea that biddes isk pemium the di eence between ex ante expected po ts fom enty and the cetainty equivalent equied fo enty into the auction is stictly positive if and only if biddes ae isk avese. We show biddes expected po ts fom enty into auctions is nonpaametically ecoveable, if a eseache obseves the distibution of tansaction pices, biddes enty decisions and some noisy measues of enty costs. We popose a nonpaametic test which attains the coect level asymptotically unde the null of isk-neutality, and is consistent unde xed altenatives. We povide Monte Calo evidence of the nite sample pefomance of the test. We also establish identi cation of isk attitudes in moe geneal auction models, whee in the enty stage biddes eceive signals that ae coelated with pivate values to be dawn in the bidding stage. Keywods: Ascending auctions, Risk attitudes, Endogenous enty, Nonpaametic Test, Bootstap JEL Classi cation Codes: D44, C2, C4 We ae gateful to Fedeico Bugni, Xu Cheng, Flavio Cunha, Ken Hendicks, Tong Li, Fank Schofheide and Peta Todd fo helpful discussions. We also thank semina paticipants at Nothwesten, Bekeley, U Penn, Whaton, NAES Summe Meeting (St. Louis 2), and SED Annual Meeting (Ghent 2) fo comments. Any eos ae ou own. y Depatment of Economics, Univesity of Pennsylvania, 378 Locust Walk, Philadelphia, PA 94; and the NBER. hanming.fang@econ.upenn.edu z Depatment of Economics, Univesity of Pennsylvania, 378 Locust Walk, Philadelphia, PA xuntang@econ.upenn.edu

2 Intoduction We popose a nonpaametic test to infe biddes isk attitudes in auctions with endogenous enty of potential biddes. In these auctions, potential biddes obseve some (possibly idiosyncatic) enty costs, such as bid pepaation/submission costs o infomation acquisition costs that need to be incued befoe leaning about pivate values, and decide whethe to pay the costs to be active in the bidding stage. In any Bayesian Nash Equilibium (BNE), biddes make ational enty decisions by compaing expected utility fom enty with that fom staying out, based on thei knowledge of enty costs o peliminay signals of pivate values to be ealized in the subsequent bidding stage. Infeence of biddes isk attitudes have impotant implications fo selles choice of evenue-maximizing auction fomat. When paticipation of biddes is exogenously given and xed, the Revenue Equivalence Theoem states that expected evenues fom st-pice and ascending auctions ae the same if biddes ae isk-neutal with symmetic, independent pivate values (IPV). On the othe hand, Matthews (987) showed that, if biddes ae isk-avese in such models, then st-pice auctions yield highe expected evenues than ascending auctions. Biddes isk attitudes also a ect evenue ankings among symmetic IPV auctions when paticipation decisions ae endogenous. Fo isk-neutal biddes, Levin and Smith (994) implied any given enty cost induces the same enty pobabilities in st-pice auctions (with entants obseving the numbe of othe entants) and in ascending auctions. Thus the Revenue Equivalence Theoem implies expected evenues must be the same fom both st-pice and ascending fomats unde endogenous enty. On the othe hand, Smith and Levin (996) established the evenue anking of st-pice ove ascending auctions unde endogenous enty fo isk-avese biddes, except fo the case with deceasing absolute isk avesions (DARA). While some ealie papes had studied the identi cation and estimation of biddes isk attitudes in st-pice auctions (e.g. Bajai and Hotascu (26), Campo, Guee, Peigne and Vuong (27) and Guee, Peigne and Vuong (29)), infeence of isk attitudes in ascending auctions emains an open question. Athey and Haile (27) pointed out biddes isk attitudes cannot be identi ed fom bids alone in ascending auctions whee paticipation is given exogenously. This is because bidding one s tue values is a weakly dominant Even in the case with DARA, st-pice fomat can yield highe expected evenues than ascending fomats when enty costs ae low enough. To see this, conside a simple case whee enty costs ae low enough so that the di eence between enty pobabilities in st-pice and ascending auctions ae su ciently small. In such a case, these two pobabilities ae both close to and only di e by some " >. By Matthews (987), conditioning on any given numbe of entants, ascending auctions have smalle expected evenues than st-pice auctions. With di eence between the two enty pobabilities " being small enough, such a evenue anking esult will be peseved.

3 stategy in ascending auctions, egadless of biddes isk attitudes. Thus, biddes with vaious isk attitudes could geneate the same distibution of bids in Bayesian Nash equilibia. Consequently, the distibution of bids fom entants is not su cient fo infeing biddes isk attitudes. Futhemoe, we show in Section 4 that isk attitudes cannot be ecoveed fom tansaction pices and enty decisions of a given set of potential biddes, when nothing is known about enty costs. It follows that some knowledge of enty costs is necessay fo ecoveing isk attitudes. We popose a non-paametic test fo biddes isk attitudes when eseaches obseve the tansaction pices and biddes enty decisions in ascending auctions. Ou appoach only equies data to contain some noisy measues of biddes enty costs. This is motivated by the fact that enty costs ae often measuable (at least up to some noises) in applications even when isk attitudes ae unknown. Fo example, enty costs may consist of bid pepaation costs (such as mailing costs), admission fees o othe infomation acquisition expenses, which ae usually obseved with noises in data. The main insight fo ou test can be illustated using the mixed-stategy enty model (which is analogous to that consideed in Levin and Smith (994) fo st-pice auctions). In the enty stage, all potential biddes obseve some common enty cost and decide whethe to pay the cost and ente an ascending auction in the bidding stage. In a Bayesian Nash equilibium, potential biddes paticipation in the auction will be in mixed stategies with the mixing pobability detemined to ensue that a bidde s expected utility fom enty equals that fom staying out. Hence biddes isk attitudes can be identi ed by compaing the expected po ts fom enty and the cetainty equivalent. As long as the expectation of enty costs can be identi ed fom data, the distibution of tansaction pices and enty decisions alone can be used to make such a compaison. Building on this intuition, we show that identi cation of isk attitudes can also be achieved in a elated model whee biddes enty costs ae idiosyncatic and enty decisions follow a pue-stategy. Pehaps moe inteestingly, we extend the idea to ecove isk attitudes in moe geneal models whee pivate values in the bidding stage ae a liated with peliminay signals obseved by potential biddes in the enty stage. We apply the analog pinciple to constuct a non-paametic test statistic, using data on tansaction pices and enty decisions as well as estimates of the mean of enty costs. We chaacteize the limiting distibution of this statistic, and popose a bootstap test that attains coect asymptotic level and is consistent unde any xed altenative of isk-avesion o isk-loving. We povide evidence fo its decent nite sample pefomance though Monte Calo simulations. The emainde of the pape is stuctued as follows. In Section 2 we discuss the elated liteatue; in Section 3 we pesent two basic models of auction enty and bidding; in Section 4 we descibe the theoetical esult undelying ou test fo biddes isk attitudes unde 2

4 the two basic models; in Section 5 we popose the test statistic and deive its asymptotic distibution; in Section 6 we pesent Monte Calo evidence fo the small sample pefomance of ou test statistics; in Section 7 we extend ou test to an auction model with selective enty; and in Section 8 we conclude. The poofs ae collected in the appendices. 2 Related Liteatue This pape ts in and contibutes to two banches of the liteatue on stuctual analyses of auction data. Some ealie papes analyzed the equilibium and its empiical implications in auctions with endogenous enty and isk-neutal biddes. These include Levin and Smith (994), Li (2), Ye (27) and Li and Zheng (29). Mame, Shneyeov and Xu (2) study a model of st-pice auctions between isk-neutal biddes with selective enty, and discuss testable implications of vaious nested enty models. Robets and Sweeting (2) estimated a model of ascending auctions with selective enty and isk-neutal biddes, assuming identi cation is attained fo a model with paametized stuctue. Othe papes studied the identi cation and estimation of biddes utility functions along with the distibution of pivate values in st-pice auctions without endogenous enty. Campo, Guee, Peigne and Vuong (29) showed how to estimate a semipaametic model of st-pice auctions with isk-avese biddes when the identi cation of a paametic utility function is assumed. Bajai and Hotascu (27) used exogenous vaiations in the numbe of biddes in st-pice auctions to semi-paametically estimate the utility function while leaving the distibution of biddes pivate values unesticted. Guee, Peigne and Vuong (29) used exogenous vaiations in the numbe of potential biddes to non-paametically identify biddes utility functions along with the distibution of pivate values in st-pice auctions. Lu and Peigne (28) consideed a context whee data contain bids fom both st-pice and ascending auctions that involve biddes with the same undelying utility function and the distibution of pivate values. Thei idea is to st use bids fom ascending auctions to ecove the distibution of pivate values, and then use bids fom st-pice auctions to ecove the utility function. Ou wok in this pape contibutes to these two banches of empiical auction liteatue by studying a model which endogenizes biddes enty decisions and elaxes the isk-neutality assumption at the same time. To the best of ou knowledge, ou pape is the st e ot to non-paametically infe biddes isk attitudes in ascending auctions with endogenous enty. Levin and Smith (996) pesented some esults on the evenue anking of auction fomats in tems of selle evenues when auctions ae known to involve isk-avese biddes who make endogenous enty decisions. Thei focus is not on the identi cation of biddes isk attitudes. Ackebeg, Hiano and Shahia (2) studied a class of e-bay auctions whee a typical online ascending auction is combined with an option of paying the buy-out pice posted by 3

5 the selle in ode to puchase the object immediately. They showed how to identify the biddes utility functions and the distibution of pivate values using exogenous vaiations in the buy-out pices and othe auction chaacteistics. The fomat of auctions they conside is qualitatively di eent fom the one we conside in this pape, which is a standad ascending fomat with endogenous enty. We do not embak on a full identi cation of the utility function in this pape, and theefoe equie fewe souces of exogenous vaiations to pefom the test. (With exogenous vaiations in enty costs, identi cation of the utility function may be possible in ou model as well.) Ou appoach does not ely on vaiations in enty costs, fo ou test can be pefomed fo any given level of enty costs. Anothe di eence is that, we also goes beyond identi cation and popose a method of obust infeence. We popose a non-paametic test statistic, deive its limiting distibution, and pesent evidence fo good pefomance in nite samples. Ou pape ts in a categoy of empiical auction liteatue on nonpaametic tests of the empiical implications/pedictions of auction theoy. Ealie woks in this categoy included tests of biddes ationality in st-pice auctions with common values in Hendicks, Pinkse and Pote (23), tests fo pesence of intedependent values in Haile, Hong and Shum (24), and test fo a liations between biddes pivate values in Li and Zhang (2) and Jun, Pinkse and Wan (2). 2 3 Ascending Auctions with Endogenous Enty Conside an empiical context whee eseaches obseve data fom a lage numbe of independent single-unit ascending auctions. Each of these auctions involve N potential biddes who have symmetic independent pivate values and make endogenous enty decisions. In the enty stage, each potential bidde decides whethe to incu an enty cost K i so as to become active. Following thei enty, active biddes see thei pivate values V i, and then compete in an open out-cy (ascending) auction in the bidding stage. A binding eseve pice may be implemented in an auction, and is obseved by all potential biddes in the enty stage. In each auction, pivate values and enty costs ae independent daws fom some distibution F (V ; :; V N ; K ; :; K N ), which is common knowledge among all potential biddes befoe making enty decision. Upon enty, each bidde may o may not be awae of the total numbe of active entants (denoted A). All biddes in data shae the same bounded von Neumann-Mogenstein utility function u : R +! R with u > and the sign of u is the same ove R +. A winne who has value V i and pays a pice at P i eceives a payo of u(v i P i K i ). Simila to Li and Zheng (29), we conside two elated enty models that di e in whethe enty costs K i vay acoss potential biddes. With a slight abuse of notation, we 2 See Athey and Haile (27) and Hendicks and Pote (27) fo ecent suveys. 4

6 use N and A to denote espectively both the numbe and the set of potential and active biddes. Let F ; 2, F 2 j denote espectively the joint and conditional distibutions of geneic andom vectos ; 2. We use uppe cases to denote andom vaiables and lowe cases to denote thei ealizations. Fo notational simplicity, we dop the efeence to N when thee is no ambiguity. Model A (Identical Enty Costs). In each auction, biddes shae the same costs ( K i = K fo all i) in the enty stage. Pivate values V i ae independently and identically distibuted ove bounded suppot [v; v] given K. Acoss auctions, enty costs ae dawn independently fom the same distibution F K ove [k; k]. That is, P(V v ; :; V N v N jk = k) = Q i2n F V jk(v i ). Model B (Heteogeneous Enty Costs). Idiosyncatic enty costs K i ae pivately known to bidde i, and ae i.i.d. daws fom some continuous, inceasing distibution F K ove [k; k]. Pivate values ae independent fom enty costs, and ae i.i.d. daws fom some distibution F V. (That is, the joint distibution of pivate values and enty costs is i2n F K (k i )F V (v i ).) In Models A and B, we assume F V is continuous, atomless and inceasing ove [v; v]. We focus on cases whee biddes pivate values ae symmetically distibuted in Sections 4-5. In both models, in Bayesian Nash equilibium (BNE), each entant i in bidding stages follows a dominant stategy to dop out at his tue value V i if A 2. When A = in the bidding stage, the lone entant wins and pays the eseve pice. Yet enty stategies in BNE di e acoss these two models. In Model A, thee is no pivate infomation available to potential biddes in the enty stage. Playes adopt mixed stategies and make independent decisions to ente with cetain pobabilities. In Model B, potential biddes have pivate infomation about thei own enty costs, and we focus on pue stategies whee potential biddes decide to ente if and only if thei pivate enty costs ae lowe than cetain cuto s. We chaacteize the BNE in both models fo the est of this section. Let A i denote the set of active entants that bidde i competes with if he decides to ente. Let be a binding eseve pice ( > v). De ne P i max j2a i fmaxfv j ; gg as i s payment if he entes and wins while othe biddes in A i follow weakly dominant bidding stategies. If A i =?, then de ne P i. Then i s po t unde dominant stategies is (V i P i ) + k, whee (:) + maxf:; g. Let! A (k; i ) denote the expected utility fo a bidde i in Model A conditional on paying enty cost k and on potential competitos enteing with pobabilities i f j g j2nnfig. Unde assumptions of Model A,! A (k; i ) u( k)f V jk () + h(v; k; i )df V jk (v), 5

7 whee fo all v >, h(v; k; i ) u(v k)f Pi (jk; i ) + u(v p k)df Pi (pjk; i ) +u( k)[ F P i (vjk; i )] () with F Pi (:jk; i ) being the distibutions of P i in Model A when K = k, and i s potential competitos ente with pobabilities i. Similaly, in Model B, let! B (k i ; k i ) denote the expected utility fo a bidde i conditional on paying k i to ente and potential competitos enteing when thei costs ae lowe than k i fk j g j6=i. Let F Pi (:jk i ) denote the distibution of P i conditional on i s competitos enteing when thei pivate enty costs ae lowe than k i. Unde assumptions of Model B,! B (k i ; k i ) = u( k i )F V () + ~h(v; k i ; k i )df V (v) whee h(v; ~ k i ; k i ) is de ned by eplacing k and F P i (:jk; i ) in () espectively with k i and F P i (:jk i ) in Model B. Due to symmety in distibutions of pivate values acoss biddes in both models, both F P i (:jk; i ) in Model A and F Pi (:jk i ) in Model B do not change with the bidde identity i. Consequently,! A ;! B ae also independent fom the bidde identity i. Given ou speci cation of Model A,! A is deceasing in i fo any given k. This is due to the following two obsevations: Fist, the distibution of active entants competing with a bidde i is stochastically inceasing in i. (A highe i leads to a highe pobability of competing with a geate numbe of ivals.) Second, the distibution of u((v i P i ) + k) conditional on enty is stochastically deceasing in the numbe of active competitos. By simila easoning, we can show in Model B that! B is deceasing in k i and k i. Using these popeties, enty stategies in Bayesian Nash equilibium is chaacteized in the following lemma. Its poof is included in Appendix A. Lemma (a) Suppose the common enty cost k is such that! A (k; (; :; )) < u() <! A (k; (; :; )) in Model A. Then thee is a unique symmetic BNE in which all biddes ente with pobability k, whee k is the unique solution to! A (k; ( k; :; k)) = u(). (b) Suppose! B (k; (k; :; k)) < u() <! B (k; (k; :; k)) in Model B. Then exists a unique symmetic BNE in which all biddes ente i k i k whee k is the unique solution to! B (k ; (k ; :; k )) = u(). In Model A, wheneve! A (k; (; :; )) u() (o! A (k; (; :; )) u()), the equilibium enty pobabilities must be (o espectively). Thus the assumption! A (k; (; :; )) < u() <! A (k; (; :; )) can be tested as long as enty decisions ae obseved in data. Likewise, enty stategies in Model B will be chaacteized by k (o k espectively) if u() >! B (k; (k; :; k)) (o u() <! B (k; (k; :; k))). The equilibium enty pocesses in Models A and 6

8 B ae both non-selective, in the sense that the potential biddes enty decisions ae not based on any infomational vaiables that ae coelated with pivate values to be dawn in the bidding stage. 4 Identi cation of Biddes Risk Attitudes In this section, we assume the numbe of potential biddes is xed at N and known to the eseache. We stat by assuming that eseaches have complete knowledge of enty costs. In Model A, this means eseaches obseve enty costs. In Model B, this means the eseache knows the distibution of enty costs F K. In pactice, the auctionee sometimes chages a xed admission fee to entants (such as at auctions). This ts in Model A povided the admission fees ae obseved in data. Ou identi cation agument builds on the simple intuition that the cetainty equivalent fo isk-avese biddes must be stictly smalle than the expected po ts fom enty. The di eence between these two can be ecoveed fom biddes enty decisions and the distibution of tansaction pices alone. In Section 4.2, we genealize this agument to allow fo the case when the distibution of enty costs is only impefectly known to the eseache. 4. Complete knowledge of enty costs Let k denote enty pobabilities in symmetic BNE of Model A when the common enty cost is k; and let k denote the cuto that chaacteizes the enty stategies in symmetic BNE of Model B. Let A ( k; k) denote expected po ts fo a bidde i if he entes in Model A, conditional on the enty cost k and that each of his potential competitos also entes with pobability k. That is, A (k) E[(V i P i ) + kj K = k]. [We suppess j = k 8j 6= i in the event conditioned on in ode to simplify notations]. Similaly, let B (k ) denote i s expected po ts fom enty in Model B, given his idiosyncatic cost k, and that competitos ente if and only if thei costs ae lowe than k. That is, B (k ) E[(V i P i ) + k j k ], whee k denotes the event A i = fj 6= i : k j k g. Both A and B ae independent fom bidde identities due to the symmety in pivate value distibutions. Lemma 2 (a) Suppose k is such that < k < in Model A. Then A (k) = i biddes ae isk-neutal, and A (k) > (o < ) i biddes ae isk-avese (o espectively, iskloving). (b) Suppose k < k < k in Model B. Then B (k ) = i biddes ae isk-neutal, and B (k ) > (o < ) i biddes ae isk-avese (o espectively, isk-loving). Poof. Conside Model A. By Lemma, in any symmetic BNE, biddes decide to ente with pobability k with! A (k; k) E[u((V i P i ) + k)jk; j = k 8j 6= i ] = u(). Thus, is the cetainty equivalent associated with u and the distibution of (V i P i ) + k given 7

9 enty cost k and pivate value distibution F V jk. Theefoe, A (k) > if u < (biddes ae isk-avese). Likewise, it can be shown that A (k) = (o, espectively, A (k) < ) if biddes ae isk-neutal (o isk-loving). Simila aguments poves pat (b) fo Model B. Lemma 2 suggests biddes isk attitudes can be identi ed in Model A if A (k) can be ecoveed at least fo some k with < k <. Likewise, isk attitudes can be identi ed in Model B with k< k < k, povided B (k ) can be constucted fom enty decisions and the distibution of tansaction pices. Poposition states this can be done when k is obseved in data. Let F V (s:m) jk denotes the s-th smallest out of m independent daws fom F V jk fo all s m. Poposition (a) Fo any enty cost k such that < k < in Model A, A (k) is identi ed fom biddes enty decisions and the distibution of tansaction pices, povided the enty cost k is obseved in data. (b) If k < k < k in Model B, then B (k ) is identi ed fom enty decisions and the distibution of tansaction pices, povided the enty cost distibution F K is known. Poof. Poof of (a). By de nition, A (k) = E[(V i P i ) + Kj K = k]. Conditional on k, enty decisions ae independent acoss biddes, and jointly independent fom pivate values in Model A. Futhemoe, pivate values ae i.i.d. acoss biddes given k. Hence, once conditional on k and A i (the numbe of active competitos fo i in the bidding stage), (V i ; P i ) ae independent fom mixed stategies adopted by potential competitos. Using the Law of Iteated Expectations, E[(V i P i ) + jk] = P N a= E[(V i P i ) + jk; A i = a] P(A i = ajk). (2) With common cost k and enty decisions obseved fom data, k is diectly identi ed as the pobability that a bidde entes unde cost k. Consequently, P(A i = ajk) is identi ed as a binomial distibution with paametes N and k. Conditional on enteing with cost k, pivate values ae independent daws fom F V jk. Let f:g denote the indicato function. By the Law of Iteated Expectations, E[(V i = P i ) + jk, A i = a] is E[(V i P i )fv i > P i > gjk, A i = a] + E[(V i )fv i > gfp i = gjk, A i = a] (s v) df V jk (s) df V jk (v) a + F V jk () a (v ) df V jk (v) (3) v fo all A i = a, due to independence between V i and P i given k. Applying integation by pats to the st tem in (3), we have E[(V i P i ) + ja i = a; k] = F V jk (v) a F V jk (v) a+ dv fo a. Besides, E[(V i P i ) + j A i = ; k] = E[(V i ) + jk] = R v FV jk (v) dv. Since k is obseved in data, A (k) can be ecoveed as long as F V jk (v) is identi ed fo v. 8

10 Let W denote tansaction pices obseved in data. If no entants bid above, then de ne W <. The symmetic IPV assumption implies fo any m 2, P(W < ja = m; k) = P(V (m:m) < jk) = F V jk () m and P(W = ja = m; k) = mf V jk () m [ F V jk ()]. Hence fo any m 2 and t, P(W tja = m; k) = P(W < jm; k) + P(W = jm; k) + P( < W tjm; k) = F V jk (t) m + mf V jk (t) m [ F V jk (t)] = F V (m :m) jk(t). Fo any m 2, de ne m (t) t m + mt m ( t) so that F V (m :m) jk(t) = m (F V jk (t)). Since m (t) is one-to-one fo any m 2 ove t 2 [; ], F V jk (t) is (ove-)identi ed fo each t fom the distibutions of W conditional on k and A = m. Poof of pat (b) uses simila aguments and is included in the appendix. Remak If bids fom those who lose (i.e. pices at which they dop out) ae obseved in data, the distibution of the othe ode statistics V (s:m) with s m 2 can also be used fo identifying A (k) and B (k ). This is because a one-to-one mappings between F V jk and the distibutions of these smalle ode statistics in Model A still exists. Likewise fo Model B. Such an ove-identi cation can be exploited to impove e ciency in estimation. Remak 2 In pactice, auction data may be tuncated in the sense that only those involving at least one entants ae obseved. In such a case, a positive enty pobability k in Model A is (ove-)identi ed using atios of tuncated distibutions fo all n N P(A=njA;k) = CN n ( k) n ( k )N n P(A=n+jA;k) Cn+( N k) = (n+)( k ) n+ ( k )N n (N n) k. Simila aguments show F K (k ) is identi ed fom tuncated data in Model B. Same aguments in Poposition show A (k) and B (k ) can be ecoveed. 4.2 Impefect obsevation of enty costs We now conside cases whee eseaches only have impefect knowledge about enty costs. Fist, conside an extension of Model A whee enty costs K common to potential biddes vay acoss auctions, but the eseache only obseves noisy measues of costs K ~ = K + with E() =. Then K E(K) = E( K) ~ is diectly identi able fom data. In such a case, the test fo biddes isk attitudes can still be conducted, povided enty costs vay independently fom biddes values. Coollay Let [k; k] denote the suppot of enty costs K in Model A. (a) Suppose < k < fo all k 2 [k; k]. Then E[ A (K)] = when biddes ae isk-neutal, and E[ A (K)] > (o < ) when biddes ae isk-avese (o espectively, isk-loving). (b) If K is independent fom (V i ) i2n, then E[ A (K)] is identi ed fom biddes enty decisions, the distibution of tansaction pices and noisy cost measues K. ~ 9

11 Poof. Pat (a). By Poposition and the suppot condition in pat (a), A (k) = fo all k 2 [k; k] if biddes ae isk-neutal, and A (k) > (o A (k) < ) fo all k if biddes ae isk-avese (o, espectively, isk-loving). Integating out k using F K poves (a). Pat (b). That K? (V i ) i2n implies, given A i = a, the vecto of ode statistics (V (s:a+) ) sa+ is independent fom K. Thus '(a) E[(V i P i ) + ja i = a; k] does not depend on k fo all a (ecall P i when A i =?). By (2), E[ A (K)] is: Z k k k + P N a= '(a) P(A i = ajk)df K (k) = P N a= '(a) P(A i = a) K. (4) To identify P(A i = a) (o R k P(A k i = ajk)df K ), note that given any k and N, A i is binomial (N ; k) while A is binomial (N; k). By constuction, P(A i = ajk) = N a N P(A = ajk) + a+ N P(A = a + jk) (5) fo all k and a N. Integating out k on both sides of (5) implies P(A i = a) = N a a+ P(A = a) + P(A = a + ). Since the unconditional distibution of A is diectly N N identi ed, so is the distibution of A i. As fo K, it is identi ed as E( K). ~ Obseving some noisy measues of enty costs is su cient but not necessay fo identifying isk attitudes in Model A. As suggested by the poof of Coollay, only K needs to be known (o ecoveable fom data). Thee is an altenative appoach fo ecoveing the unconditional distibution of A i. Because enty decisions in data ae ationalized by a symmetic BNE, the distibution of A i can be identi ed as the distibution of the numbe of entants fom a andom subset of N potential biddes. Such subsets can be fomed by emoving a andomly-selected bidde i fom the set of N potential biddes. We adopt this altenative appoach while constucting the test statistic. Now conside a special case of Model B whee eseaches have impefect knowledge of enty costs. Suppose the data contain noisy measue of idiosyncatic costs K i = K i + i. The measuement eos i s ae independently dawn fom the same distibution F, which is independent fom K i. Even with K i now unobsevable, expected po ts fom enty E[(V i P i ) + j k ] is ecoveable as in Poposition. Thus, biddes isk attitudes can be identi ed as long as k can be ecoveed using the distibution of K i. If the distibution F is known to eseaches and has a non-vanishing chaacteistic function, then the chaacteistic function of K i is ecoveed as E(e itk i )=E(e it i ). With the equilibium enty pobability identi ed fom data, the equilibium cuto k can be ecoveed by inveting F Ki at the enty pobability. Even with F unknown to eseaches, the cuto k can be ecoveed povided eseaches have multiple noisy measues of idiosyncatic costs K i. Suppose fo each K i, data contain two noisy measues (K i;; K i;2) (K i + i; ; K i + i;2 ), whee (K i ; i; ; i;2 ) ae mutually independent with non-vanishing chaacteistic functions,

12 and the mean o some quantile of eithe i; o i;2 is known. 3 The Kotlaski Theoem (see Kotlaski (967) and Rao (992)) then applies to identify the maginal distibutions of K i and i;. Inveting the distibution of K i at the enty pobability then gives k. 4.3 Futhe discussions about enty costs Ou test fo biddes isk attitudes equies that the eseache have at least some patial knowledge about enty costs (such as its expectation). We now elaboate on theoetical and empiical justi cations of such an assumption. We also discuss possibilities fo constucting a test without such knowledge by exploiting the exogenous vaiations in the numbe of potential biddes. Fist, with the numbe of potential biddes xed and eseaches having no infomation about enty costs, it is impossible to nonpaametically infe biddes isk attitudes fom obseved enty decisions and tansaction pices alone. To see this, conside a simpli ed vesion of Model A whee all auctions in data have the same xed enty cost k, which is not epoted in data. The distibution of pivate values is still identi ed fom the distibution of tansaction pices and the numbe of entants; and the equilibium enty pobability is also identi ed fom enty decisions. Nonetheless, even in this simpli ed case, the biddes utility u(:) and the xed cost k cannot be jointly identi ed. To see this, suppose biddes ae isk-neutal with utility function u(:) and k 2 (; ). Then one can always eplace u with a slightly concave ~u 6= u, the obseved enty behavios ( ) would still be ationalized by some level of enty cost k ~ 6= k that equals the expected utility fom enty to that fom cetainty equivalent (i.e. E[~u((V i P i ) + k)j ~ ; k] = u()). Thus, ou hope fo infeing isk attitudes fom enty and bidding behavios (when the numbe of potential biddes is xed) must utilize at least some patial knowledge of enty costs. To ou knowledge, ou wok in this pape is the st attempt to exploit possible infomation fom enty costs to ecove isk pefeences. Second, depending on the empiical application consideed, enty costs may well be measuable (at least up to andom noises) though additional suveys o data collection wok. Fo example, in timbe auctions held by US Foest Sevice, the enty costs fo potential biddes (i.e. that milles and logges located in a neaby geogaphic egion) consists lagely of infomation acquisition costs. These costs ae the pices fo conducting a cuise on foest tacts in ode to plot the distibution of the diamete and height of tees, etc. (See Athey, Levin and Seia (2) fo details.) Such pivate cuises ae institutionalized and standad pactices on the maket. Thus it is plausible that thei costs vay little acoss potential biddes, and can be leaned though suveys o maket obsevations up to andom measuement 3 The two measuement eos i;, i;2 could be dawn fom di eent maginal distibutions, in which case the identi cation would equie eseaches to know how to distinguish Ki; and K i;2 in data.

13 eos. Thus the assumption of having an unbiased estimato fo the common enty costs seems plausible in such contexts. Othe well-known examples whee this assumption might be expected to hold include the high-way pocuement auctions consideed in Li and Zheng (29) and Kasnokutskaya and Seim (2). If the numbe of potential biddes can be expected to vay exogenously in data (in the sense that maginal distibution of V i is invaiant to N), then thee is hope fo constucting a non-paametic test fo biddes isk attitudes fom ascending auctions with IPV without elying on any infomation about enty costs. To see this, conside the scenaio whee the data contains two sets of auctions, with each obseved to involve eithe N o N > N potential biddes espectively. The enty cost k is not obseved in data, but known to emain the same acoss all auctions. Then by the same aguments above, the null of isk neutality implies E[(V i P i ) + j N;k; N] = k and E[(V i P i ) + j N ;k; N ] = k, whee N;k denotes the equilibium enty pobability when enty cost is k and the numbe of potential biddes is N. Thus the null yields a testable implication E[(V i P i ) + j N;k; N] = E[(V i P i ) + j N ;k; N ], (6) whee both sides ae identi able fom data and do not equie obsevations of enty costs. Howeve, the main challenge is to deive a fomal desciption of the powe of a test unde the altenatives (of isk-avesion o isk-loving). We conjectue (6) should fail in geneal when the null of isk-neutality is false. Nevetheless, it is not clea how the diection of inequality would be elated to the types of altenatives (i.e. avesion o loving). We leave this fo futue eseach. 5 Infeence of Biddes Risk Attitudes We now constuct a test fo biddes isk attitudes using the analog pinciple. We focus on the extended vesion of Model A, whee enty costs K vay acoss auctions independently fom (V i ) i2n. Reseaches only get to obseve noisy measues K ~ = K +, whee? (K, (V i ) i2n ) and E() =. Futhemoe, < k < fo all k 2 [k; k]. Heeinafte we efe to these assumptions as the Conditions of Model A. Independence of fom (K; (V i ) i2n ) is not necessay fo ou infeence pocedue in Section 5.2 to be valid. Nonetheless it simpli es deivation of the limiting distibution of ou test statistic. To simplify exposition, we x the numbe of potential biddes N, and dop it fom the notations fo obsevable distibutions in data when thee is no ambiguity. 2

14 5. Asymptotic popety of the test statistic Recall ou goal is to daw a conclusion as to which of the following thee competing hypotheses is suppoted by data: H : Biddes ae isk-neutal, = ; H A : Biddes ae isk-avese, > ; and H L : Biddes ae isk-loving, <. whee E[ A (K)]. Ou data contain T independent auctions, each of which is indexed by t, involves N potential biddes and has at least one active entants. Let A t denote the numbe of entants in auction t. By de nition, W t < if and only if thee is no tansaction. On the othe hand, when thee is tansaction (W t ), the tansaction pice in auction t is W t = maxf; V (At :At) g when A t 2 and W t = when A t =. Ou test statistic amounts to estimating the ight-hand side of (4) using the analog pinciple. Constuction of the Test Statistic ^ T Step Fo m 2, calculate the empiical distibution of tansaction pices ^F W;m;T (s) N P tn fw P t s and A t = mg= N tn fa t = mg fo any s. Step 2 Fo any m 2, estimate the distibution of pivate values at s by ^F V;m;T (s) m ( ^F W;m;T (s)), whee m (t) t m + mt m ( t). Aggegate these estimates by ^F V;T (s) P N ^F N m=2 V;m;T (s) fo any s. Step 3 Estimate '(a) fo a using sample analogs: 4 ^' T (a) h a h i ^FV;T (s)i ^FV;T (s) ds. The integal is calculated using mid-point appoximations. Step 4 In each auction t, select a potential bidde i andomly. Let ~ At denote the numbe of entants fom the othe N potential biddes (excluding i). Estimate ((a)) an 4 As shown while poving pat (a) of Poposition, '(a) = [F V (s)] a [ F V (s)] ds fo each a. The distibution of pivate values (V i ) i2n Model A in Coollay. ae independent fom K unde Conditions of 3

15 by ^ T (^ T (a)) an, whee ^ T (a) ~ T K t. Finally, estimate by T P tt P tt f ~ A t = ag. 5 Then estimate K by ^ T ^ T P N a= ^' T (a)^ T (a) ^ T. We now deive the limiting distibution of p G (^ T ). Let denote weak convegence of stochastic pocesses in a nomed space. (Fo Euclidean spaces R N, this is educed to convegence in distibution, denoted!.) d Let F ~V be a shot-hand fo the section of F V ove the domain [; v) and ^F ~V ;T be a shot-hand fo its estimato as de ned in Step 2. Denote the limiting distibution of p T (^ T ) and p T (^ T K ) by N and N espectively. We chaacteize the covaiance between N and N in Lemma B4 in the appendix. We also show in Lemma B5 in the appendix that unde mild conditions p T ^F ~V ;T F ~V G V, whee G V is a zeo-mean Gaussian Pocess indexed by [; v). Lemma B5 chaacteizes the covaiance kenel of G V as well as its covaiance with N and N. The poof of these esults builds on the fact that ^ T, ^ T ae simple sample aveages, while ^F V;T is a known function of sample aveages. Let S [;v) denote the set of functions de ned ove the domain [; v) that ae stictly positive, bounded, integable, ight-continuous and have limits fom the left. Unde the supnom, S [;v) is a nomed linea space with a non-degeneate inteio. De ne ' : S [;v) 7! R N + as '(F ) ('(a; F )) a= N, whee '(a; F ) F (s) a F (s) a+ ds. By de nition, '( ^F ~V ;T ) = (^' T (a)) N a= ^' T and '(F ~V ) = ('(a; F ~V )) N a= '. The mapping ' is Hadamad di eentiable at F ~V tangentially to S [;v) (see Appendix B). Fo any h 2 S [;v), the Hadamad deivative D ';F ~V D ';F ~V (h)(a) : S [;v)! R N + is fo a N ; and D ';F ~V (h)() R v h(s)ds. af ~V (s) a (a + )F ~V (s) a h(s)ds Poposition 2 Suppose the eseve pice is binding and F V is continuously distibuted with positive densities ove [; v). Unde the Conditions in Model A, p T (^ T ) N whee N D ';F ~V (G V ) + 'N N follows a univaiate nomal distibution with zeo mean. 5 An altenative is to estimate (a) by T PtT N a N fa t = ag+ T PtT a+ N fa t = a + g. (See poof of Coollay fo details.) 4

16 The poposition uses the Functional Delta Method (Theoem in van de Vaat and Wellne 996), and its poof is included in Appendix B. The poof builds on the fact that both ^ T ; ^ T ae simple sample aveages while ^' T is a known non-linea functional involving sample aveages. This allows us to st apply the Functional Delta Method to show that p T (^'T ') D ';F ~V (G V ), with the covaiance kenel of G V and its covaiance with N ; N completely chaacteized in tems of population distibution in the data-geneating pocess (DGP). The Jacobian of the ight-hand side of (4) with espect to ('; ; K ) at the tue DGP is the -by-(2n + ) vecto [; '; ]. Thus anothe application of the multivaiate delta method delives the esult. The Gaussian pocess G V and N ; N ae Boel-measuable and tight. Besides, by constuction the Hadamad deivative D ';F ~V is a linea mapping ove S [;v). Thus the limiting distibution D ';F ~V (G V )+ 'N N is a zeo-mean univaiate nomal (by Lemma of van de Vaat and Wellne (996)). 5.2 Bootstap Infeence Pocedue Ou goal is to test the null H : = against two diectional altenatives H A : > (isk-avese) and H L : < (isk-loving). Set the level fo ou test to be. In pinciple, we can estimate the standad deivation of the limiting distibution in Poposition 2 using the analog pinciple. Then the asymptotic plug-in appoach can be applied to estimate the citical value fo testing H : =. In this section, we adopt an altenative appoach using bootstap pocedues to test H. This avoids the explicitly estimating the standad deviation of the limiting distibution of p T (^ T ). Building on esults in Poposition 2, we show the bootstap test is consistent against xed altenatives (of isk-avesion o isk-loving), and attains the coect level asymptotically. Bootstap Pocedue fo Testing H : = Step : Calculate ^ T using the oiginal sample. Step 2: Daw a bootstap sample with size T fom the oiginal sample with eplacement. Estimate using this bootstap sample and denote the estimate by ^ T;. Step 3: Repeat Step 2 fo B times and denote the bootstap estimates by f^ T;b g bb. Find the quantile of the empiical distibution of the bootstap estimates f p T j^ T;b ^ T jg bb (denoted by ^c =2;T ). Step 4: Do not eject H if ^c =2;T p T ^ T ^c =2;T. Reject the null in favo of H A (o H L ) if p T ^ T > ^c =2;T (o espectively if p T ^ T < ^c =2;T ). Poposition 3 establishes consistency and asymptotic validity of the test. Let P(^ T : j ) denote the distibution of ^ T given tue value of in the data geneating pocess. 5

17 Poposition 3 Suppose the Conditions in Model A hold, and data contains T independent auctions, each involving N potential biddes and p lim P T ^ T ^c =2;T j = c = 8c > ; (7) T!+ p lim P T ^ T ^c =2;T j = c = 8c < ; (8) T!+ p lim P T ^ T > ^c =2;T o p T ^ T < ^c =2;T j = =. (9) T!+ These esults ae due to the fact that the empiical distibution of p T (^ T;b ^ T ) calculated fom bootstap samples povides a consistent estimato fo the nite sample distibution of p T (^ T ) unde mild conditions. Such conditions ae stated in Bean and Duchame (99) and vei ed fo ou context hee in Appendix C. (Hee consistency means the distibution of p T (^ T asymptotic distibution of p T (^ T ) as estimated fom bootstap samples gets unifomly close to the ) as sample size inceases.see Hoowitz (2) fo a fomal de nition of bootstap consistency.) The esults then follow fom the fact that p T is zeo unde the null but diveges to positive (o negative) in nity unde the altenative. Ou bootstap infeence uses an asymptotically non-pivotal statistic p T (^ T ). One could constuct asymptotically pivotal statistics using the pe-pivoting appoach. This would help attain asymptotic e nements in the appoximation of test statistic distibution elative to st-ode asymptotic appoximation o bootstap using asymptotically non-pivotal statistics. This is computationally intensive due to bootstap iteations and theefoe we do not pusue this appoach hee.. 6 Monte Calo Expeiments This section pesents evidence fo the pefomance of ou test in simulated nite samples. We conside the following data-geneating pocess (DGP): Each auction involves N potential biddes who face the same enty cost K. Upon enty, biddes daw pivate values fom a unifom distibution with suppot [v; v] [; ]. The eseve pice is R = 3. The data epot pices paid by the winne and the numbe of entants in each auction. When thee is no tansaction (when all entants ealized pivates values ae lowe than R), the tansaction pices is set to an abitay level lowe than R. The data only epot noisy measues of the enty costs K ~ = K + whee is dawn fom a unifom distibution [ ; ]. Biddes von Neumann-Mogensten utility is speci ed as u(t) t+5 That is, biddes ae isk-neutal (and espectively, isk-avese o isk-loving) if = (and < o > ).Fo a xed level of enty cost and, the enty pobability deceases in the numbe of potential biddes. This con ms that the patten as stated fo the case with isk-neutal biddes in Li and Zheng (29) is caied ove into the geneal case with non-isk-neutal biddes. Besides, once 6

18 contolling fo the enty costs and the numbe of potential biddes, the enty pobability inceases as biddes become moe isk-loving ( inceases). [Inset Figue (a), (b), (c), (d) hee.] To illustate how pefomance of the test depends on biddes isk pefeences, we st focus on a simple design with the enty cost xed at K = :7 o K = :9 espectively. We also set N = 4 in the data-geneating pocess. We epot test pefomance unde vaious in the DGP in Figue. Fo each gidpoints between :8 and :2 (with gidwidths being :2), we simulate S = 3 data sets, each containing T = 5; o T = ; independent auctions. In evey single auction, data contain K ~ = :7 + whee is unifom on [ =2; =2]. Fo each simulated sample, we calculate the statistic ^ T, and then pefom the test by dawing B = 3 bootstap samples with eplacement fom that estimation sample. We expeiment with = 5% and % espectively. The solid cuves in Figue show the pecentage of these S simulated samples whee the test fails to eject the null of isk-neutality (H : = ). The dashed (and dotted) cuve plots the popotion of these samples in which the test ejects the null in favo of the altenative H A : < (and H L : > ) espectively. Each panel of Figue 2 epots these popotions fo a given pai of sample size T and enty costs K. In all panels of Figue, the test attains appoximately the tageted level fo both = 5% and = %, when the null is tue in the DGP. All panels show that unde the altenative the powe of the test inceases faily quickly to as moves away fom the isk-neutal value. 6 A compaison of panels (a) and (c) with panels (b) and (d) suggests an incease in sample size impoves test pefomance both in tems of eos in ejection pobabilities unde the null, and powe unde the altenative. Fo a given sample size T, the test pefoms bette when K = :9. This can be patly ascibed to the fact that the di eence between enty pobabilities unde the null and the altenatives ae slightly moe ponounced when K = :9 than when K = :7. Next, we pesent pefomance of the test when the DGP contains andom enty costs that ae obseved with noises unde the Conditions of Model A (which ae speci ed at the beginning of Section 5). We let K be dawn fom a unifom multinomial distibution with suppot [:7; :8; :9], and K ~ = K + as befoe. We expeiment with DGP with 2 f; :9; :g and N 2 f4; 5g. Fo each DGP, we pefom ou test in S = 3 simulated data sets, each containing T = 5; o ; independent auctions. As befoe, fo each simulated sample, we conduct ou test by dawing B = 3 bootstap samples. The test esults ae summaized in Table. Table (a): Test Pefomance unde Random Costs (N = 4) 6 In the pesence of diectional altenatives (i.e. H A and H L ), we de ne the powe of the test as the pobability of ejecting the null in favo of the tue altenative that undelies the DGP. 7

19 T = 5; = 5% = % = 5% = [5.%, 93.67%,.33%] [4.%, 93.%, 3.%] [5.33%, 89.67%, 5.%] = :9 [.%, 36.%, 64.%] [.%, 25.%, 75.%] [.%, 2.33%, 79.67%] = : [77.%, 23.%,.%] [86.%, 4.%,.%] [9.67%, 9.33%,.%] T = ; = [3.33%, 95.33%,.33%] [8.67%, 87.67%, 3.67%] [.67%, 84.%, 4.33%] = :9 [.%, 9.67%, 9.33%] [.%, 3.%, 97.%] [.%, 2.33%, 97.67%] = : [95.33%, 4.67%,.%] [97.33%, 2.67%,.%] [99.%,.%,.%] The ows in Table epesent di eent DGPs and sample sizes, each coesponding to a pai (N; ). The column heads ae tageted levels. We test the null of isk-neutality (H ) against two altenatives isk-avesion (H A ) and isk-loving (H L ). Fo each cell in Table, we epot (fom the left to the ight) the popotions of S simulated samples whee the test ejects H in favo of H L, whee the test does not eject H, and whee H is ejected in favo of H A espectively. Table (a) shows the esult fo N = 4 and T = 5; o ;. In such cases, ou test attains a ejection pobability close to the tageted level when the null of isk-neutality is tue ( = ). When the biddes ae not isk-neutal, the test yields easonably high chances fo ejecting the null in favo of the coect altenative. When biddes ae not isk-neutal, the pecentage of samples in which the null is ejected in favo of an incoect altenative (also known as the Type-thee eo) is zeo acoss all speci cations and sample sizes. Table (a) also shows the pefomance of the test impoves as the sample size T inceases. Fo a xed speci cation, the pobability fo ejecting the null when biddes ae isk-neutal is close to the tageted level when T = ;. Thee is also a quite substantial incease in the powe of the test (i.e. pobability fo ejecting H in favo of the coect altenative) unde both altenatives as T inceases. T = 5; Table (b): Test pefomance unde Random Costs (N = 5) = 5% = % = 5% = [3.67%, 94.67%,.67%] [6.%, 9.%, 3.%] [8.%, 86.33%, 5.67%] = :9 [.%, 45.67%, 54.33%] [.%, 32.33%, 67.67%] [.%, 27.%, 73.%] = : [73.%, 27.%,.%] [8.%, 2.%,.%] [88.%, 2.%,.%] T = ; = [3.67%, 95.67%,.67%] [7.33%, 9.33%, 2.33%] [.33%, 84.33%, 5.33%] = :9 [.%, 2.67%, 87.33%] [.%, 6.33%, 93.67%] [.%, 3.67%, 96.33%] = : [93.67%, 6.33%,.%] [98.33%,.67%,.%] [98.67%,.33%,.%] 8

20 Table (b) epots simila esults fo N = 5. A compaison between Table (a) and (b) shows the impact of a lage numbe of potential biddes on the elative pefomance of the test is ambiguous, depending on the tue DGP. With = in DGP and fo a xed sample size T, the test yields smalle eos in the ejection pobability when N = 5 in most cases. (The only exception takes place when T = ; and = 5%, in which case eos in ejection pobabilities ae both small when N = 4 and N = 5.) By constuction, inceasing the numbe of potential biddes a ects the test pefomance though two channels that have o setting impacts. Fist, with T xed, the numbe of auctions involving a given numbe of entants m N may decease as N gets lage. Hence fo each m, the empiical pice distibution ^F W;m becomes a wose estimato fo F W;m in the population as its vaiance is inceased. On the othe hand, the numbe of empiical distibutions f ^F W;m g mn available fo estimating F V incease as N inceases, which might help incease pefomance of ^F V;T as an estimato fo F V. The impovement of test pefomance unde a lage N when = appeas to be evidence that the second impact has dominated the st when the DGP involves isk-neutal biddes only. On the othe hand, the impact of a lage N on the powe of the test when 6= is ambiguous. Fo example, conside the case with lage sample size T = ;. When biddes ae isk-avese with = :9, the powe of the test is highe fo any given when N = 4. Meanwhile, the powe appeas to be highe when N = 5 if = : and = %. We conjectue that such pattens ae due to combined impacts of the shape of u(:) and the fact that the enty pobability deceases as N inceases (which in tun implies the distibution of payo s (V i P i ) + fom enty is stochastically deceasing). 7 Extension: Selective Enty with Infomative Signals So fa we have consideed models whee biddes infomation in the enty stage is uncoelated with thei pivate values to be dawn in the bidding stage. In this section, we elax this assumption and conside models whee biddes enty decisions ae based on infomative signals coelated with pivate values in the bidding stage. Nonpaametic identi cation of biddes isk attitudes can be attained in this case, povided data is ich enough to contain auctions with continuous vaiations in obsevable enty costs. The model is speci ed as follows. In the enty stage, each bidde i eceives a peliminay signal S i and decides whethe to become active by paying the enty cost k (which is the same fo and known to all biddes). The costs ae incued only fo entants. Upon enty, each bidde sees his tue value V i, and bids in an ascending auction with a eseve pice. The joint distibution F (S ; :; S N ; V ; :; V N ) and ae common knowledge among all potential biddes in the enty stage. Each entant may o may not be awae of the numbe of active competitos A. SAF (i) Peliminay signals and pivate values (S i ; V i ) ae i.i.d. acoss biddes ( F (S ; :; S N ; 9