A NON-PARAMETRIC ESTIMATOR FOR RESERVE PRICES IN PROCUREMENT AUCTIONS

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1 A NON-PARAMETRIC ESTIMATOR FOR RESERVE PRICES IN PROCUREMENT AUCTIONS MARTIN BICHLER and JAYANT KALAGNANAM IBM T. J. Watson Researc Center Yorktown Heigts, NY 0598, USA {bicler, Abstract. Electronic auction markets collect large amounts of auction field data. Tis enables a structural estimation of te bid distributions and te possibility to derive optimal reserve prices. In tis paper we propose a new approac to setting reserve prices. In contrast to traditional auction teory we use te buyer s risk statement for getting a winning bid as a key criterion to set an optimal reserve price. Te reserve price for a given probability can ten be derived from te distribution function of te observed drop -out bids. In order to get an accurate model of tis function, we propose a nonparametric tecnique based on kernel distribution function estimators and te use of order statistics. We improve our estimatior by additional information, wic can be observed about bidders and qualitative differences of goods in past auctions rounds (e.g. different delivery times). Tis makes te tecnique applicable to RFQs and multi-attribute auctions, wit qualitatively differentiated offers. Keywords: Reserve prices, auction teory, non-parametric estimation. Introduction During te past few years, electronic reverse auctions ave become a very popular economic institution for automating procurement negotiations. Te competitive process of tese auctions serves to aggregate te scattered information about supplier s costs and to dynamically set a price. Te literature on optimal auction design tries to find te auction mecanism tat provides te greatest revenue/profit for te seller, or te lowest cost for te buyer in a procurement auction, respectively. Suc a question is of considerable practical value. In te well-studied independent private value model wit risk neutral bidders any auction mecanism suc as first-price, second-price, Englis, and Dutc auction, generates te same revenue for te seller. Terefore, te problem of optima l auction design reduces to determining an optimal reserve price []. Te latter is expressed as a functional of te distribution of private values and its corresponding density function [, 3]. Wit te advent of large-scale electronic auction markets, te access to large amounts of transaction data as become considerably easier. Tis enables a structural estimation of te empirical distributions relevant to setting an optimal reserve price, and as led to an increased interest in empirical applications of optimal auction teory (see section 5). Tis paper is motivated by our work on a large-scale electronic procurement platform for te retail industry. On tis platform retail companies conduct repeated purcases of retail

2 commodities using classic reverse auctions, mostly in an open-cry or Englis format. Tese commodities are purcased from an establised set of suppliers for te respective commodities. Providing good decision support for setting appropriate reserve prices is an important feature for purcasing managers on tis platform. Te results of optimal auction teory, owever, ave been criticized because tey seem to be of teoretical rater tan practical significance so far. Te optimal auctions are usually quite complex, and tere is no evidence for teir use in practice. [4] As argued by McAfee and Vincent [5], a major difficulty in implementing te optimal reserve price is te use of unobservables suc as te distribution of supplier s costs and te bid taker s own valuation for te good (i.e. er opportunity cost). First, tere are metodological difficulties in estimating te latent supplier costs from te observed bids in te transaction data, wic essentially assume knowledge of a bidder s strategy. Tis is also called te identification problem. Te accuracy of tis estimate, owever, as a big impact on te reserve price. Second, also assessing te buyer s own opportunity cost can be non-trivial. Often, tere is no alternative market value for te good. Additionally, a procurement auction is not necessarily a one-time event, and it is common procurement practice to repeat unsuccessful auctions wit varied reserve prices. Were sould te optimal reserve price be set, if tere is a possibility to repeat te auction wit a different reserve price? In tis paper we propose a new approac to setting reserve prices in suc a procurement environment. Our contribution is two-fold: In contrast to traditional auction teory we use te buyer s risk statement for getting a winning bid as a key criterion to set an optimal reserve price. Te reserve price for a given probability can ten be derived from te distribution function of te observed drop-out bids. In order to get an accurate model of tis function, we propose a nonparametric tecnique based on kernel distribution function estimators and te use of order statistics. We improve our estimatior by additional information, wic can be observed about bidders and qualitative differences of goods in past auctions rounds (e.g. different delivery times). Tis makes te tecnique applicable to RFQs and multi-attribute auctions, wit qualitatively differentiated offers. Te paper is structured as follows. Te next section summarizes te relevant results of optimal auction teory. We will focus on two important aspects, namely asymmetry of bidders costs and correlation. We will ten describe a univariate and a multivariate estimator for bid prices in section 3. Section 4 will present some first evidence from a Monte Carlo study. We ten point to related literature in Section 5 and provide a brief summary and conclusions in section 6.. Teory of Optimal Auctions Tere is a considerable academic literature on te effects of reserve prices in auctions. Te basic teory as been developed by Vickrey [6] and extended by Riley and Samuelson [], Levin and Smit [7] and Monderer [8]. Many laboratory experiments ave tested different predictions of auction teory [9]. Empirical work using field data is summarized in Hendricks and Paarsc [0], or more recently in a so called field experiment by Lucking-Reiley []. Lucking-Reiley s analysis sows tat implementing reserve prices () reduces te number of bidders, () increases te frequency wit wic goods go

3 unsold, and (3) increases te revenues received on te goods conditional on teir aving been sold. McAfee and McMillan state tat te IPV model applies to (government) contract bidding wen eac bidder knows wat is own production cost will be if e wins te contract []. Terefore, in te following section we will introduce te basic IPV model and derive te main results wit respect to setting reserve prices. We will also discuss relevant factors suc as te number of bidders, te asymmetry of bidders costs, and correlation among te bidders costs. Altoug relevant in many instances we will ignore oter aspects suc as collusion among bidders, teir risk attitudes, or royalities for te sake of brevity... Optimal Auctions in te Independent Private Values Model Te most torougly researced auction model is te symmetric independent private values (IPV) model. In tis model applied to reverse auctions: A single indivisible object or task is put up for auction to one of several bidders. Eac bidder i knows er true cost, c i, and can revise er signal wen tat of rival bidders are disclosed. If c i is lower tan te bid b i, ten te bidder makes a profit of b i - c i. All bidders are symmetric/indistinguisable, i.e., te costs are drawn from a common distribution G( ) wit support [c, c] n, wic is known to all bidders. Te unknown costs c i are statistically independent, identically distributed, and continuous random variables. Te bidders are risk neutral concerning teir cance of winning te auction, and so is te seller. Te vector (c,, c n ) is a realization of a random vector wose n-dimensional cumulative distribution function is G(c). Denoting c (i) as te i t smallest order statistic for a sample of size n from te distribution of c, te bidder of te auction will be te player wit te lowest cost c (). Tat is, c () is te first order statistic, and c () is te second order statistic [3]. In an Englis auction te second-last bidder will drop out of te bidding as soon as te price is below er own cost of te item. From te point of view of te winning bidder, er expected rent is te expected difference between c () and c (), wic is te difference between te first order statistic and second order statistic, given by G(c)/g(c), were c is te bidder s production cost, and G and g are te probability distribution function and density function of bidders costs. Consequently, te expected buyer s payment is te winning supplier s cost plus te winning supplier s rent: G( c() ) J ( c() ) = c() + () g ( c ) () Riley and Samuleson [] reduce te optimal auction problem to te optimal coice of te reservation price. Tat is, an optimal auction requires te buyer to set a reserve price, r, above wic se will not buy te item and make it public (i.e., a maximum bid). Tis price is set to mimic te expected bid of te second lowest bidder and is lower tan te buyer s valuation, i.e. cost for not getting te good, c 0, namely,

4 r J c0 ) < = ( c () 0 Tis reserve price minimizes te expected cost of te buyer, based on te distribution of costs in te market. For te IPV model, any of te Englis, Dutc, first-price sealed-bid, and second-price auctions is optimal, provided te reserve price is set optimally as in (). Remarkably, tis optimal reserve price is independent of te number of bidders n. Tis is a powerful result, as no restrictions ave been placed on te types of policies te seller can use. For instance, te seller can ave several rounds of bidding, or carge entry fees, or allow only a limited time for te submission of bids. None of tese procedures would increase te expected price for te seller. Tis optimal level of te reserve price is determined by a trade-off. Te disadvantage of setting a reserve price is tat it is possible for te remaining bidder to ave a valuation tat lies between te sellers valuation and te reserve price, c 0 > c > r. In tis case, te buyer doesn t find a supplier even toug te bidder would ave been willing to carge a lower price tan wat te buyer was willing to pay. On te oter and, if te reserve price is below te second-lowest bidder s cost, te bidder carges less tan se would ave in absence of te reserve price. In summary, te buyer imposes te reserve price in order to capture some of te informational profits tat would oterwise ave gone to te winner. However, tis can ave a negative impact on te efficiency of te auction. Bulow and Roberts [4] point out te relationsip of optimal auction teory to te teory of price differentiation in monopolies... Number of Bidders and Distribution of Costs Te number of bidders and te particular type of cost distribution ave a considerable impact on te outcome of an auction. Increasing te number of bidders increases te revenue of te bid taker on average. Tis is because te second-lowest cost approaces te lowest possible cost. In addition to te number of bidders, te variance of te distribution of costs as an impact. Te larger te variance, te larger on average is te difference between te lowest cost and te second lowest cost. Tis means te average revenue of te buyer and te winning supplier are increased. As can be seen in () and () te distribution of bidder s costs as well as te buyer s cost for not getting te good, c 0, are te two parameters impacting te reserve price. In te following we will analyze te impact of tese state variables for two common types of distributions, te uniform and te normal distribution. For a uniform distribution te reserve price, r, is c0 + r = J ( c0 ) = lo (4) were lo is te lower bound of te uniform distribution. Te reserve price calculation for te case of a normal distribution is more comp licated. Te distribution function of te normal distribution can be computed based on te error function, wic can be integrated numerically, or approximated as in [5]. Tis leads to a J-function as in equation (5). Function J is continuous and monotonically increasing. Terefore, te inverse to function J - for te optimal reserve price can be found troug bisection or oter root finding metods.

5 J( r) = r + r e x µ σ e r µ σ dx = r e r µ σ π µ r σ Erf σ Figure sows te sapes of J - for different levels of c 0 using tree different distributions. Te solid line illustrates te reserve price for a uniform distribution wit a lower bound of. Te dased line sows a normal distribution wit a of 50 and a of only 4, wereas te dotted line sows a normal distribution wit a of 30 and a of 0. Te intuition is tat wit only a low variance of cost, it does not make sense to raise te reserve price way beyond te mean of te normal distribution, even wit a ig opportunity cost c 0. r c0 Figure : Sapes of J - functions in reverse auctions for different distributions Te figure illustrates te strong impact te buyer s opportunity cost c 0 on te coice of te optimal reserve price. A bias in c 0 leads to a considerable bias in r, wic makes te application of tis formula difficult in many real-world cases. (5).3. Asymmetry of Bidders Altoug, IPV assumptions are similar to te conditions one can find in typical procurement negotiations, tey need to be carefully evaluated before applying te model to auctions in te field. One IPV assumption, wic is often violated in practice, is te symmetry of bidders. In many procurement situations bidders fall into recognizably different classes, i.e., bidders are asymmetric. It would terefore not be appropriate to represent all bidders as drawing teir valuations from te same probability distribution F. An example migt be suppliers from different countries, were tere are systematic cost differences between domestic and foreign firms. Asymmetry of bidders leads to a breakdown of te well-known revenue equivalence teorem and terefore impacts te coice of te auction format. In absence of a reserve price, an Englis auction yields an efficient outcome, wereas a sealed-bid auction may yield an inefficient outcome.

6 In an optimal asymmetric auction, te seller sets a different reserve price for eac type of bidder, computed as in (). Tis means, tat te optimal auction is discriminatory between te types. By setting k different reserve prices, eac type as an incentive to bid for te contract. However, tere is a possibility tat one bidder wins despite anoter bidder s aving a iger valuation. Tis is true because asymmetry implies tat te probability distributions F are different, so tat it is possible tat te buyer s expected payment J k (c k ()) > J k+ (c k+ ()) even toug c k () < c k+ (), wit k being te different types of bidders. Because tis optimal policy leaves a positive probability of te item being awarded to someone oter tan te bidder wit te lowest cost, te policy is not Pareto efficient. If te distributions of valuations are identical except for teir means, ten te class of bidders wit te iger average cost receives preferential treatment in te optimal auction. Because it migt appen tat te iger type bidder is below er reserve price, wereas te lower type bidder is not. In general, te IPV assumes tat te distributions are observable by all of te bidders, wic migt not be given in many procurement auctions. For example, in many privatesector sealed-bid auctions bidders do not even know wo as been invited to bid. In addition, empirical distributions migt be complex mixture distributions, wic cannot be assumed known by te bidders..4. Correlated Costs Yet anoter assumption, wic is sometimes violated in procurement negotiations is te independence of bidders costs. Often te bidders estimates about te cost of executing a contract are somewat correlated. Two oter models ave been discussed in te auction literature, namely te common-value model and te affiliated-values model [6]. Te common-value model describes situations were te good as a single objective value to all bidders, and te bidders ave different guesses about ow muc te item is objectively wort. Tis model is particularly apt to situations were goods ave a resale value, suc as securities, antiques or te amount of gold in a mine. Wile most procurement auctions, wic can be observed in practice do not exibit te caracteristic of a single objective value of te good in question, it is likely tat te valuation for a contract is somewat dependent on te bids of oter bidders. For example, if tere is a common element of tecnological uncertainty (e.g., in long-term contracts), ten te appropriate assumption is a degree of affiliation among te bidder s bids. Te affiliated-values model accounts for tis influence. Wit n bidders, let x={x,, x n } represent te private signals about te item s value observed by bidders, < i < n; Let s={s,, s m } be a vector of variables tat measure te quality of te item for sale. Te bidder s valuation may depend not only upon is own signal, but also upon te oter bidders private signals and te true quality of te item, v i (s, x). If variables are affiliated, ten tey are positively correlated. In oter words, affiliation means tat large values for some of te components make te oter components more likely to be large tan small. Altoug, affiliation plays a role in te auctioning of contracts wit uncertain estimates about te actual costs incurred for te supplier, we will focus on te simpler IPV model in te following, assuming for example te purcase of direct or indirect materials, were te costs are known by te bidders and affiliation can be ignored. For a teory of reserve prices in an auction wit affiliated values, see [7].

7 3. Estimation As sown in equation (), te optimal reserve price for an auction is determined by te latent distribution of costs in te auction and te buyer s cost for not getting te good, c 0. Te price of te good on a secondary market, or te loss incurred troug not getting te good in te subsequent production step, migt serve as an estimate for c 0. Neverteless, tis variable is often difficult to set, in particular, since in many settings te procurement manager can initiate a second or tird round of auctions, if te first round was unsuccessful. In contrast to traditional auction teory we use te buyer s risk statement for getting a winning bid as a key criterion for setting a reserve price. For example, a buyer wants to find te best reserve price given a probability Pr = 30% of getting a winner. Te key tecnique to deriving suc a reserve price is a good fitting estimation for te distribution of te winning bid. In particular, in situations wit only a few bidders te accuracy of te estimate can ave a big impact. In te following, we will describe a non-parametric estimator for te probability distribution function of prices in a new auction. Te tecnique is based on well-known kernel density estimators [7] and te teory of order statistics. We will first describe te one-dimensional case and ten extend it to a mu ltivariate case, wic considers te impact of qualitative differences in te goods and services put up for auction. Due to teir popularity in procurement, we will focus on Englis auctions, were te bidding strategy is simple and te drop out bids equal te true costs of te suppliers. We will in general make IPV assumptions, wit te notable exception of symmetry in cases. A basic assumption in tis section is tat te bid prices in te transaction data do not contain any systematic seasonal or long-term trends. 3.. Univariate Bid Price Estimators In a first step, we will estimate te distribution of bids witout considering qualitative differences in previous auction rounds. Tere are two approaces to estimating bid distributions. Te parametric approac assumes a particular functional form wit some unknown parameters for te valuation distributions. Te non-parametric approac does not assume te valuation distributions to be part of a specified parametric family. Since little is known about te actual sape of te empirical bid distribution, kernel estimation as been cosen as a non-parametric approac to estimating te bid distribution. Besides te missing prior knowledge about te sape of te bid distribution, kernel estimation as a number of additional advantages over parametric estimations wit respect to outliers or missing values [7]. A goal of density estimation is to approximate te probability density function (pdf) g( ) of a random variable C, wic describes te bidders cost or drop-out bid in case of an Englis auction. Assume we ave n independent observations c,, c n from te random variable C. Te kernel density estimator gˆ ( c) for te estimation of te density value g(c) at point c is defined as n K ( c Ci ) i= gˆ ( c) = n (6)

8 wic is also called te Rosenblatt-Parzen kernel density estimator [8, 9] of C, were ( u) = K( u / ) is te kernel wit scale factor. Te sape of te kernel K weigts is determined by K, wereas te size of te weigts is parametrized by, wic is called bandwidt. Commonly used kernel functions are te Epanecnikov kernel (7) K ( u) = 0.75( u ) I ( u ) (7) wic as a parabolic sape, or te Gaussian function (8) wit its bell sape. K( u) exp u π = It is easy to see tat for estimating te density at point c, te relative frequency of all observations c i falling in an interval around c is counted. Te factor /(n) in (9) is needed to ensure tat te resulting density estimate as integral ˆ ( z) dz =. For more detailed information on te coice of kernel functions and appropriate bandwidt, we refer to [7]. n gˆ ( c) = /( n) 0.75( ( u / ) ) I ( u ) (9) i= Instead of a kernel density estimator in (9), we propose a kernel distribution function estimator Gˆ ( c) (KDF) for G(c). For c <, te KDF of G is given by n K ( c Ci ) i = G ˆ ( c) = n (0) were for u < K ( u) = K ( v) dv. () ], u] K ( u) = K ( u / ), wit K ( u) = ] K( v) dv being te probability distribution, u] function. Te KDF wit an Epanecnikov kernel (9) leads to n 3 ˆ u G ( c) = 0.75 u ( < ) 3 I u () n i = Te probability Gˆ ( c) in () is a continuously and monotonically increasing function of te bid price c. Using bisection or te Newton-Rapson metod one can find te reserve price to a particular probability of getting a winner. Following IPV assumptions, we suppose tat te drop-out bids are n independent variates C, C,, C n, eac wit te cdf G(c), i.e. te bids are independent, identically distributed (iid). Terefore, we are particularly interested in te distribution function of te smallest order statistic C (), wic appens to be te lowest bid. Te KDF can ten be rewritten as KDF (). g (8)

9 ˆ n c Gˆ ( ) ( ) = [ ( c (3) G )] 3.. Multivariate Estimators Field data from large-scale procurement auction platforms offer te possibility of getting transaction data from repeated auctions on te same good and service wit essentially te same supplier pool. Neverteless, tese auctions are not completely omo geneous over time. For example, tey migt ave differing qualitative attributes suc as delivery time or bio degradability. In addition, buyers migt invite different numbers of bidders to te auction. Multivariate estimators are a possibility to take tis additional information into account. Te kernel density estimator can be generalized to te multivariate case in a straigtforward way. Suppose we now ave observations c,, c n were eac of te observations is a d-dimensional vector c i =(c i,, c id ) T. Te multivariate kernel density estimator at a point c is defined as n c c c c i id d gˆ ( c ) = Μ,..., (4) n i=... d d wit M denoting a multivariate kernel function, i.e. a function working on d- dimensional arguments. Note, tat (4) assumes tat te bandwidt is a vector of bandwidts. A possibility for a multivariate kernel is te radial symmetric Epanecnikov kernel T T M ( u) ( u u) I( u u ) (3) Radial symmetric kernels can be obtained from univariate by defining T M ( u) K ( u ), were u = u u denotes te Euclidean norm of te vector u. Te indicates tat te appropriate constant as to be multiplied. Radial symmetric kernels use observations from a ball around c to estimate te density at c. For te bivariate case wit one qualitative attribute and price, te function (9) wit an Epanecnikov kernel can be rewritten as n ci c ci c gˆ ( c ) = n = (4) i Similar to (0) and (), te integral of te kernel function leads to a multivariate kernel distribution function estimator (MKDF). Note tat te arguments of te MKDF G ˆ ( c), (c ij -c i ) are constrained between and. Figure illustrates te sape of a bivariate MKDF. For a given quality of a newly auctioned good and probability of getting a winner, te reserve price can be derived troug slicing te landscape at te appropriate quality level. Te resulting function needs to be normalized by dividing troug its asymptotic probability value. Te estimator for te first-order statistic MKDF () can ten

10 be derived as in (3). Tis procedure can also be used for multi-attribute bid data as can be found in RFQs and multi-attribute auctions. quality Pr(b<x) price - Figure : Sape of MKDF 4. Some Monte Carlo Evidence In tis section, we use Monte Carlo metods to compare different estimation tecniques. We analyze te results of a naïve approac, wic fits a Gaussian cdf to te data based on empirical sample moments, wit te univariate and te multivariate kernel estimator described in te previous section. Since data are often scarce, we considered te effect upon te estimators of small to medium sized samples of 0 training auctions. 4.. Experimental Design and Data Generating Process In all of our simulation experiments, we assumed tat te latent distribution of costs c follows a mixture distribution, similar to an example wit asymmetric bidders. Allowing c to ave a diffuse distribution also mimics some of te empirical evidence, wic we ave encountered in field data. We ave conducted 0 training auction rounds wit 4 bidders eac, from wic we estimate G ˆ ( c). In a consecutive set of 0 test auction rounds, we ave set a reserve price wit a probability Pr = 30% for getting a winner, using four different estimators: no reserve price at all te naïve Gauss estimator te KDF () te MKDF (). After te simulation we analyzed te total cost for te buyer, and te number of successful auctions. In te following sections we will describe te results of two treatments:

11 . Treatment A: We assume to ave auction data from completely identical auction rounds (exactly te same quality, number of bidders and identity of bidders).. Treatment B: We assume differentiated quality in te field auction data (e.g. different delivery times) 4.. Discussion of Simulation Results Te first simulation wit treatment A ad 4 bidders wit different costs. Te bids in eac auction round were drawn from normal distributions wit a standard deviation of 4 and a mean of 0, 3, 38, and 09, respectively. After 0 training runs reserve prices were set for te two estimators at a probability of 30% for getting a winner. Naïve Gauss-based estimator: 8. KDF:.6 We ave evaluated te results for 0 additional auction rounds and acieved te following results: No Reserve Price Total cost for 0 auction rounds witout reserve price Naïve Gauss-based estimate for Pr = 30% 0 Auctions wit savings 0 Successful auctions 0 Total cost using te estimator 0 Total cost witout using te estimator 0 KDF () for Pr = 30% 6.34 Auctions wit savings 4 Successful auctions 5 (5%) Total cost using te estimator Total cost witout using te estimator 99.4 Table : Results of Simulation wit Treatment A and 4 Bidders Te Naïve Gauss estimator was too low and as a consequence all auctions were unsuccessful. For te KDF (), only 5 or 5% of all te test auctions ad a winner, and 4 of tem ad savings. Figure 3 sows bot estimators as a function of te bid prices. KDF () 40 sows te KDF () after te 0 training auction rounds plus te 0 test auction rounds. Te results illustrate tat goodness-of-fit of te estimator is important for setting a reserve price at a certain level. Wit an increasing number of bidders, te difference between te first-order and te second-order statistic decreases and also te savings will are less significant.

12 Naïve Gauss KDF() 0 KDF() 40 Figure 3: Estimators as a function of bid price A second simulation used treatment B wit differentiated qualities of te goods to be purcased. Trougout te 0 test auction rounds witout reserve price, we introduced 3 different qualities L(ow), M(edium), and H(ig). Depending on te quality level demanded by te buyer, we introduced increased costs for quality M (+0) and H (+0) for all bidders. In te 0 subsequent auction rounds, te buyer purcased goods of quality L. No Reserve Price Total cost for 0 auction rounds witout reserve price Naïve Gauss-based estimate for Pr = 30% 8.40 Auctions wit savings 0 Successful auctions 0 Total cost using te estimator 0 Total cost witout using te estimator 0 KDF () for Pr=30% 3.80 Auctions wit savings 0 Successful auctions 9 (95%) Total cost using te estimator Total cost witout using te estimator MKDF () for Pr=30% and quality L 0.60 Auctions wit savings 4 Successful auctions 7 (85%) Total cost using te estimator Total cost witout using te estimator Table : Results of Simulation wit Treatment B and 4 Bidders Tis time, we used tree different estimators, te naïve Gauss estimator, te univariate, and te multivariate estimator for a probability of 30% for getting a winner. MKDF takes te different quality levels into account and provides an estimate for a particular quality in question (L), wereas te oter estimators ignore tese qualitative differences. Te simulation illustrates tat ignoring qualitative differences is penalized. Te Gauss-based estimator is again too optimistic. Wit KDF () 95% of all auctions ave a winner, and 0 acieve savings, wereas wit te MKDF () for quality L 85% of te auctions ave a

13 winner and 4 acieve savings. Figure 4 sows all estimators as a function of te bid prices, including te tree MKDF for different quality levels Naïve Gauss KDF() MKDF() L MKDF() M MKDF() H Figure 4: Estimators as a function of bid price 5. Related Literature Recently, tere ave been a number of approaces in te econometrics literature focusing on te empirical analysis of auctions. Hendricks and Paarsc (995) classify empirical work in auctions into two categories, structural and non-structural/reduced-form approaces. Te non-structural approac tests necessary conditions of auction teory using reduced form econometric models. Te reduced form provides a data admissible statistical representation of te economic system, wereas te structural form can be seen as a reformulation of te reduced form in order to impose a particular view suggested by economic teory [0]. An example for non-structural analysis is te detection of bid rigging in Porter and Zona []. A key question in te structural analysis of auctions is te estimation of te latent distributions tat generate bidder valuations in te auction from observed bids. Te strategy is to estimate te distribution of bids and ten to retrieve te distribution of costs. An issue tat all structural estimations ave to address is te issue of identification, tat is, te question of te extent to wic te unobservable cost distributions can be recovered from te observed bid distributions. Tis approac relies upon te ypotesis tat observed bids are te equilibrium bids of te auction model under consideration. In te first price sealed bid auction te focus lays on Bayesian-Nas bidding strategies. Some of tese structural estimation procedures are parametric and assume a particular functional form wit some unknown parameters for te valuation distributions. Leading examples tat analyze te first price sealed bid auction wit private values include Donald and Paarsc [], and Laffont, Ossard, and Vuong [3]. Donald and Paarsc [4] present metods, wic consist of finding estimators maximizing te likeliood function for te symmetric IPV, were te valuation distributions are identical across bidders. Muc of te

14 literature is concerned wit te identification problem under various conditions suc as asymmetry [5], or affiliations among te valuations [6]. Some newer approaces also use non-parametric estimators, wic do not make assumptions on te sape of te distribution function (see [7], [8] and [9]). Note, tat also in non-parametric estimations, some assumptions are made, suc as tat te distributions are identical across bidders and continuous. In addition, structural estimations make assumptions about te type of equilibrium in an auction. In contrast to existing analysis, our approac relies on te bid taker s risk statement. Terefore, we do not necessarily need to know te latent valuations/costs of suppliers. More relevant is te goodness of fit of our estimator. For tis reason we coose a non-parametric tecnique and take additional information into account, suc as qualitative differences in te auctioned goods, te number, and identity of te bidders. Anoter relevant stream of literature deals wit multi-attribute auctions and RFQs [30]. Multi-attribute reverse auctions allow negotiation over price and qualitative attributes suc as color, weigt, or delivery time. A toroug analysis of te design of multi-attribute auctions as been provided by Ce [3]. He derived a two-dimensional version of te revenue equivalence teorem. Ce also designs an optimal scoring rule based on te assumption tat te buyer knows te probability distribution of te supplier s cost parameter, and proves tat using tis scoring function is in fact an optimal mecanism. More recently, Beil and Wein [3] suggested an inverse-optimization based approac tat allows te buyer via several canges in te announced scoring rule, to determine an optimal scoring rule. Wile elegant, tese approaces assume a number of prerequisites (e.g., knowledge of te parametric sape of te supplier s cost functions), wic are ardly given in practice. An alternative approac to increase te buyer s revenue/utility is to set reserve prices, based on te attribute values a supplier as specified. Tis is applicable to bot, multi-attribute auctions and RFQs, wic do not even use public scoring functions. 6. Summary and Conclusions Game-teoretic auction teory is based on te assumption tat te distribution of valuations, or costs respectively, is known among te participants in an auction. In many procurement auctions, tis assumption is not given. Knowledge about empirical distribution enables a bid taker to fine-tune reserve prices. We ave proposed a multivariate estimator, wic is useful in estimating empirical bid distributions in auctions. Te estimator allows us to set reserve prices, based on te risk statement provided by a buyer. We plan to incorporate te estimator into a tool, wic takes new auction rounds into account and suggests reserve prices based on past auction rounds. Essentially, te user determines a description of te good in question, te invited suppliers and te probability for getting a winner, and te software suggests a reserve price. Tere are multiple ways ow we intend to improve te estimator and make it more applicable to different environments. In many settings, we will ave past auction rounds wit reserve prices. It is important in tese settings to take into account te fact tat certain bidders did not submit bids. Anoter issue is te consideration of underlying trends in te bid data, in particular, if te data is collected over a longer time period, were trends and seasonal deviations play a role. A general problem in multivariate prediction is

15 also called te curse of dimensionality. Te basic element of nonparametric smooting averaging over neigboroods will often be applied to a relatively meager set of points since even large samples are surprisingly sparsely distributed in te iger dimensional Euclidean space. We plan to investigate additive models suc as te projection pursuit regression [33], or stocastic gradient boosting [34] for tese cases. Setting reserve prices is of course not te only application of suc an estimator. It can as well be used for bid pricing, in order to elp a bidder in estimating te likeliood of winning, or to compare te attractiveness of different auction markets for buying or selling goods. 7. References [] E. Wolfstetter, "Auctions: An Introduction," Journal of Economic Surveys, vol. 0, pp , 996. [] J. G. Riley and J. G. Samuleson, "Optimal auctions," American Economic Review, vol. 7, pp , 98. [3] R. B. Myerson, "Optimal auction design," Matematics of Operations Researc, vol. 6, pp , 98. [4] M. H. Rotkopf and R. M. Harstad, "Modeling Competitive Bidding: A Critical Essay," Management Science, vol. 40, pp , 994. [5] R. P. McAfee and D. Vincent, "Updating te Reserve Price in Common-Value Auctions," American Economic Review, vol. 8, pp. 5-58, 99. [6] W. Vickrey, "Counterspeculation, Auctions, and Competitive Sealed Tenders," Journal of Finance, pp. 8-37, 96. [7] D. Levin and J. L. Smit, "Optimal Reservation Prices in Auctions," Economic Journal, vol. 06, pp. 7-8, 996. [8] D. Monderer and M. Tennenoltz, "Optimal Auctions Revisited," presented at AAAI 98, Madison, Wisconsin, 998. [9] J. H. Kagel, "Auctions: A Survey of Experimental Researc," in Te Handbook of Experimental Economics, J. H. Kagel and A. E. Rot, Eds. Princeton: Princeton University Press, 995, pp [0] K. Hendricks and H. J. Paarsc, "A Survey of Recent Empirical Work Concerning Auctions," Canadian Journal of Economics, vol. 8, pp , 995. [] D. Lucking-Reiley, "Auctions on te Internet: Wat's Being Auctioned, and How?,", vol. 000, 999. [] R. McAfee and P. J. McMillan, "Auctions and Bidding," Journal of Economic Literature, vol. 5, pp , 987. [3] S. M. Ross, Probability Models. San Diego, et al.: Harcourt Academic Press, 000. [4] J. I. Bulow and D. J. Roberts, "Te Simple Economics of Optimal Auctions," Journal of Political Economy, vol. 97, 989. [5] M. Abramovitz and A. Stegun, Handbook of Matematical Functions. Dover, 964. [6] P. R. Milgrom and R. J. Weber, "A Teory of Auctions and Competitive Bidding," Econometrica, vol. 50, pp. 089-, 98.

16 [7] W. Haerdle, Smooting Tecniques wit Implementation in S. New York: Springer Verlag, 99. [8] E. Parzen, "On estimation of a probability density function and mode," Ann. Mat Statist., vol. 33, pp , 96. [9] M. Rosenblatt, "Remarks on some non-parametric estimates of a density function," Ann. Mat.Statist., vol. 7, pp , 956. [0] T. Haavelmo, "Te Probability Approac in Econometrics," Econometrica, 944. [] R. H. Porter and D. J. Zona, "Detection of Bid Rigging in Procurement Auctions," Journal of Political Economy, vol. 0, pp , 993. [] S. G. Donald and H. J. Paarsc, "Piecwise Pseudo-Maximum Likeliood Estimation in Empirical Models of Auctions," International Economic Review, vol. 34, pp. -48, 993. [3] J.-J. Laffont, H. Ossard, and Q. Vuong, "Econometrics of First-Price Auctions," Econometrica, vol. 63, pp , 995. [4] S. G. Donald and H. J. Paarsc, "Maximum Likeliood Estimation wen te Support of te Distribution depends upon Some or All of te Unknown Parameters," Department of Economics, University of Western Ontario, Ontario 99. [5] P. Bajari, "Econometrics of te First Price Auction wit Asymmetric Bidders," Harvard University, Boston, MA, USA February [6] T. Li, I. Perrigne, and Q. Vuong, "Structural Estimation of te Affiliated Private Value Auction Model," University of Soutern California, San Diego, Working Paper 000. [7] E. Guerre, I. Perrigne, and Q. Vuong, "Nonparametric Estimation of First-Price Auctions," Economie et Sociologie Rurales, Toulouse 95-4D, 995. [8] B. Elyakime, J.-J. Laffont, P. Loisel, and Q. Vuong, "First-Price Sealed-Bid Auctions wit Secret Reserve Prices," Annales d'economie et Statistiques, vol. 34, pp. 5-4, 994. [9] P. A. Haile and E. Tamer, "Inference wit an Incomplete Model of Englis Auctions," Journal of Political Economy, vol. fortcoming, 00. [30] M. Bicler, Te Future of emarkets: Multi-Dimensional Market Mecanisms. Cambridge, UK: Cambridge University Press, 00. [3] Y.-K. Ce, "Design Competition troug Multidimensional Auctions," RAND Journal of Economics, vol. 4, pp , 993. [3] D. R. Beil and L. M. Wein, "An Inverse-Optimization-Based Auction Mecanism To Support a Multi-Attribute RFQ Process," MIT, Boston, MA, USA, Researc Report December 00. [33] J. Friedman and W. Stuetzle, "Projection Pursuit Regression," Journal of te American Statistical Association, vol. 76, 98. [34] J. Friedman, "Stocastic Gradient Boosting," Stanford University, Stanford, Tecnical Report 999.