Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons: A Smplfed Approach Bn Zhang, Kemal Guler Intellgent Enterprse Technologes Laboratory HP Laboratores Palo Alto HPL-2002-86(R.) November 23, 2004 frst prce auctons, asymmetrc auctons, structural econometrcs, nonparametrc estmaton We present an approach to non-parametrc estmaton of pseudo-values n asymmetrc afflated prvate values (APV) models for frst-prce aucto ns that elmnates the 'curse of dmensonalty.' We show that pseudo-values can be estmated by usng two-dmensonal functons of bds for APV models regardless of the form asymmetry. Internal Accesson Date Only Copyrght Hewlett-Packard Company 2004 Approved for External Publcaton
Zhang & Guler, Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons: A Smplfed Approach Bn Zhang and Kemal Guler Decson Technologes Department Solutons and Servces Research Center Hewlett-Packard Laboratores, MS 40 50 Page Mll Rd Palo Alto, CA 94304 Abstract We present an approach to non-parametrc estmaton of pseudo-values n asymmetrc afflated prvate values (APV) models for frst-prce auctons that elmnates the curse of dmensonalty. We show that pseudo-values can be estmated by usng two-dmensonal functons of bds for APV models regardless of the formasymmetry. Keywords: Afflated Prvate Values, Frst Prce Auctons, Asymmetrc Auctons, Structural Econometrcs, Nonparametrc Estmaton JEL codes: C4, D44 Ths report s a substantal revson of an earler techncal report Unform Treatment of Symmetrc and Asymmetrc APV Models of Frst Prce Sealed Bd Auctons, HPL-2002-86. We gratefully acknowledge the comments and suggestons of Professor Quang Vuong that greatly mproved the paper. Errors, of course, reman our own.
Zhang & Guler, Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons: A Smplfed Approach Bn Zhang and Kemal Guler Hewlett-Packard Laboratores, Palo Alto, CA. Introducton Structural econometrcs analyss of auctons treats observed bd data as the Bayesan-Nash equlbrum of a game-theoretc aucton model. An mportant breakthrough n the structural econometrcs of frst prce auctons was Guerre, Perrgne & Vuong (2000) s (GPV n the sequel) procedure that uses the equlbrum strateges to express a bdder s unobserved valuaton n terms of hs bd and the dstrbuton and densty functons of observed bds. (See Perrgne and Vuong (999) for a survey of recent contrbutons to the structural econometrcs of frst prce auctons.) GPV procedure s a two-step estmaton procedure. In the frst step, a sample of pseudo-values s constructed usng the equlbrum nverse bd functon. In the second step, the sample of pseudo-values s used to estmate the underlyng densty of bdders valuatons. More recently, Campo, Perrgne & Vuong (2003) appled the same method to estmate asymmetrc afflated prvate values (APV) model of frst-prce auctons. Campo et al. s approach requres estmaton of hgh dmensonal functons of bds to construct the pseudo-values. As s well known sample szes requred by the kernel methods to acheve a gven qualty of estmaton (e.g. mean-square error) ncrease dramatcally as the dmensonalty of the functon to be estmated ncreases. Moreover, as the number of bdder segments ncreases, the computatonal complexty of functons to be estmated ncreases consderably. For example, Slverman (998) shows that the sample szes needed to estmate a multvarate normal densty at the orgn wth a mean-square error less than 0. for, 2, 3, 4, and 7-dmensonal denstes are 4, 9, 67, 223, and 0700, respectvely. 2
Zhang & Guler, Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons In ths paper, we show that one only needs to estmate two-dmensonal functons of bds to generate a sample of pseudo-values for the APV models regardless of the form of asymmetry. The proposed approach reduces the sample sze requrements consderably and t s much smpler computatonally. In the next secton we ntroduce the afflated prvate values model and the results of Campo et al. In secton 3 we present our smplfed approach that reduces the dmensonalty of the functons to be estmated to at most two. 2. An asymmetrc APV model of frst prce auctons In ths secton we ntroduce the asymmetrc APV model and the method of constructon of pseudovalues used n Campo et al. A sngle and ndvsble tem s auctoned to n rsk neutral bdders. The afflated prvate value model wth rsk neutral bdders s defned by an n-dmensonal cumulatve dstrbuton functon F(). The vector of prvate valuatons ( v, K, v n ) s a realzaton of a random vector wth jont dstrbuton F(). For,..., n, denote by b s ( v ) and φ ( b) s ( b) the equlbrum bddng strategy of bdder, and ts nverse, respectvely. In equlbrum, the jont dstrbuton of valuatons, F(), and that of bds, G(), are related by Gb (,..., bn) F( φ( b ),..., φn( bn )). Ex ante asymmetres among bdders are represented by approprate restrctons on the dstrbuton F(). Campo et al. use a model of asymmetry wth two bdder segments wth symmetrc bdders n each segment. It s assumed that segment 0 contans n0 bdders and segment contans n bdders, wth n0 + n n. The jont dstrbuton of bdder valuatons, Fv (,..., v, v,..., v ), s exchangeable 0, 0, n0, n wthn the frst n0 and the last n varables. For k 0, and j,..., nk, denote by bk, j sk( vk, j ) and φ s ( b ) the equlbrum bddng strategy of bdder j n segment k, and ts nverse, respectvely. k, j k k, j The th bdder n segment maxmzes hs expected payoff max Π ( b, v ) max[( v b )Pr( y < φ ( b ) and y < φ ( b ) v )],,, b, b,,,,, 0, 0,, 3
Zhang & Guler, Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons where y max{ v : j,..., n & j} and,, j y max{ v : j,2,..., n }. The objectve 0, 0, j 0 functon of a typcal bdder n segment 0 s defned smlarly. We refer the reader to Campo et al. (2003) for the detals of the dervatons and state ther results. Campo et al. show that the equlbrum nverse bd functons satsfy: G ( b,,,, 0, b b ) B B b v, φ( b, ) b, +, () dg ( b, b b )/ dx 0 B, B b,,, where B max{ b : j,..., n & j} and,, j B max{ b : j,2,..., n }. 0, 0, j 0 The expresson () nvolves functons of bds only and s the bass of estmatng the latent valuatons from observed bds. Campo et al. elmnate the condtonng on b n the second term by notng that ths rato can be nterpreted as Pr( B b, B b, b b) G ( b, b, b ) Pr( B b, B b, b b) Pr( B b, B b, b b) D ( b, b, b) D ( b, b, b) 0 B, B0, b 0 + 0 + 2. (2) The rato n (2) s then estmated by substtutng kernel estmates for the correspondng terms. The complexty of expressons n (2) and ther emprcal counterparts ncrease consderably for cases wth more than two bdder segments. When there are K bdder segments, for each of the segments the denomnator n the rato would have K functons each wth K+ arguments, requrng (K+)-dmensonal kernel estmaton. 3. A Smplfed Approach Our smplfed approach s based on workng wth the th bdder s payoff functon expressed n terms of equlbrum dstrbuton of rval bds condtoned on the th bdder s valuaton. Let B max( b ), the maxmum rval bd for bdder. The th bdder s expected payoff from partcpatng n the aucton wth a bd b s gven by Π ( b, v ) ( v b)prob( B < b v ). j j 4
Zhang & Guler, Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons The th bdder s optmal bd b should be the soluton of the followng equaton dprob( B < b v ) dπ ( b, v) db Prob( B < b v) + ( v b ) 0. (3) d b b b Rearrangng the terms n (3), we get a relaton between the valuaton and the correspondng optmal bd: v Prob( B < b v ) B b v db b b b + [ dprob( < ) ]. (4) Usng the notaton H ( b v ) Prob( B < b v ) () Bv and h ( b v ) dprob( B < b v ) db, () Bv we can rewrte (4) as v b + H ( b v ) h ( b v ). (5) ( ) () Bv Bv Gven the assumpton that bd functons are strctly monotone, the condtonng on valuatons can be replaced by condtonng on bds. Defnng G ( b b) H ( b φ ( b)) () () Bb B v and g ( b b) h ( b φ ( b)), () () Bb B v one can rewrte (5) as v b + G ( b b ) g ( b b ). (6) ( ) ( ) Bb Bb The expresson (6) wthout the bdder superscrpt s the same as the formula derved by L et al. (2002) for the symmetrc APV model. An mportant dfference s that now (6) apples wthout the symmetry assumpton. 2 For estmaton we assume that a sample of L ndependent auctons from the structure F s avalable. As n L et al. (2002) we estmate G ( b b )/ g ( b b ) by () () Bb Bb Gˆ ( b, b )/ gˆ ( b, b ), where () () Bb Bb 2 Note that although ndependent prvate values models can be estmated usng one-dmensonal kernels regardless of bdder asymmetres, APV models requre at least two-dmensonal functonals of bds. 5
Zhang & Guler, Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons L ˆ () b b, l GBb ( B, b) ( B, ) l BKG LhG l h G (7) L () B B, l b b, l gˆ Bb ( B, b), 2 Kg, Lh g l hg h g (8) KG s a one-dmensonal kernel wth bandwdth h G, and K g s a two dmensonal kernel wth bandwdth h g. 3 In addton to greatly smplfyng the estmaton of asymmetrc APV models, the proposed approach has several advantages 4. Frst, one need not mpose a pror restrctons on the form of asymmetry among the bdders. Valuaton dstrbutons can be estmated separately for each bdder usng (6). Furthermore, the estmated dstrbutons can be used to test the exstence and the nature of asymmetry among the bdders. Fnally, to the extent that the set of partcpatng bdders s not endogenous, one need not assume a fxed set of bdders n the sample. One can construct pseudo-values for each bdder n the sample by replacng L n (7) and (8) wth L - the subsample of auctons n whch bdder submts a bd. 4. Conclusons We have shown that one can avod the curse of dmensonalty n estmaton of pseudo-values for asymmetrc APV model of frst prce auctons. Regardless of the number of asymmetrc bdder segments, one needs no more than a two-dmensonal functon of bds to estmate the pseudo-values n the frst step of the two-step GPV procedure. Acknowledgements We are grateful to Quang Vuong and an anonymous referee for helpful comments and suggestons. 3 The detals on how to choose the kernels and the bandwdths are avalable n general densty estmaton lterature and n L et al. (2002). 4 We thank an anonymous referee for brngng these ponts to our attenton. 6
Zhang & Guler, Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons References. Campo, Sandra, Isabelle M. Perrgne, and Quang H. Vuong, 2003, Asymmetry n Frst-Prce Auctons Wth Afflated Prvate Values, Journal of Appled Econometrcs, 8, 79-207. 2. Guerre, Emmanuel, Isabelle M. Perrgne, and Quang H. Vuong, 2000, Optmal Nonparametrc Estmaton of Frst-Prce Auctons, Econometrca, 68, 525-574. 3. L, Tong, Isabelle M. Perrgne, and Quang H. Vuong, 2002, Structural Estmaton of the Afflated Prvate Value Aucton Model, Rand Journal of Economcs, 33, 7-93. 4. Perrgne, Isabelle M., and Quang H. Vuong, 999, Structural Econometrcs of Frst-Prce Auctons: A Survey of Methods, Canadan Journal of Agrcultural Economcs, 47, 203-223. 5. Slverman, B.W., 998, Densty Estmaton for Statstcs and Data Analyss, (Chapman & Hall/CRC, Boca Raton). 6. Zhang, Bn, and Kemal Guler, 2002, Unform Treatment of Symmetrc and Asymmetrc APV Models of Frst Prce Sealed Bd Auctons, Hewlett-Packard Laboratores Techncal Report HPL- 2002-86 7