Bid-Response Models for Customized Pricing

Transcription

1 Bd-Response Models for Customzed Prcng Vshal Agrawal * Mark Ferguson + June 2007 Abstract: In ths paper, we study prcng stuatons where a frm provdes a prce quote n the presence of uncertanty n the preferences of the buyer and the compettve landscape. We ntroduce two customzed-prcng bd-response models used n practce, whch can be developed from the hstorcal nformaton avalable to the frm based on prevous bddng opportuntes. We show how these models may be used to explot the dfferences n the market segments to generate optmal prce quotes gven the characterstcs of the current bd opportunty. We also descrbe the process of evaluatng competng models usng an ndustry dataset as a test bed to measure the model ft. Fnally, we test the models on the ndustry dataset to compare ther performance and estmate the percent mprovement n expected profts that may be possble from ther use. Keywords: bd-response functons, customzed prcng, prce optmzaton, bd-prcng. * Vshal Agrawal, School of Industral & Systems Engneerng, Georga Insttute of Technology, Atlanta GA Emal: vshalagrawal@gatech.edu. + Mark Ferguson, College of Management, Georga Insttute of Technology, Atlanta GA Emal: mark.ferguson@mgt.gatech.edu. 1

2 1. Introducton Whle the maorty of the prevous lterature n the prce-optmzaton area focuses on the prcng of consumer goods or the optmal desgn of auctons, a large percentage of frms face prcng decsons n a busness-to-busness settng where a customer requests bds from a small set of competng frms and the frms vyng for the customer s busness respond wth a sngle prce quote for the product or servce. When the total annual sales to the frm requestng the bd does not ustfy a dedcated sales person on behalf of the frm respondng to the bd, many frms have started usng bd-response models to provde customzed prcng recommendatons on what prce to offer for the busness beng bd upon. Customzed-prcng bd-response models (CPBRMs) provde a probablty of wnnng for every possble prce response, allowng a frm to balance a decreasng margn wth an ncreasng wn probablty needed n a prce optmzaton model. Examples of frms usng CPBRMs nclude Unted Parcel Servce (UPS) when respondng to bds for from ther small to medum sze customers (Knple, 2006), and BlueLnx, the largest buldng products dstrbutor n the U.S, respondng to requests for products from constructon companes (Dudzak, 2006). (Phllps, 2005a) descrbed a prevalent use of these models n the fnancal servces ndustry when frms determne what nterest rate to offer when respondng to request for mortgages (prme, home equty, sub prme), credt cards, and auto-loans. The fnancal mpact from usng CPBRMs can be sgnfcant. UPS reported an ncrease n profts of over $100 mllon per year by optmzng ther prce offerngs usng CPBRMs (Boyd et al. 2005). In determnng the wnnng bd probablty, CPBRMs effectvely determne the prce segment the current bd falls n. Prce segments are defned as sets of transactons, classfed by customer, product, and transacton attrbutes, whch exhbt smlar prce senstvtes. Customer attrbutes may nclude customer locaton, sze of the market the customer s n, type of busness the customer s n, the way the customer uses the product, customer purchase frequency, customer sze, and customer purchasng sophstcaton. Product attrbutes may nclude product type, lfecycle stage, and the degree of commodtzaton. Transacton attrbutes may nclude order sze, 2

3 other products on the order, channel, specfc compettor, when the order s placed, and what the urgency s of the bdder. In addton, some models assume knowledge of the hstorcal and current bd-prce of competng frms partcpatng n the bd. A common characterstc of stuatons where frms employ CPBRMs s when the bdder wth the lowest prce does not always wn the bd. Thus, markets are characterzed by product dfferentaton where a gven frm may command a postve prce-premum over ts compettors; dependent upon the partcular customer offerng the bd. Even assumng a frm collects enough hstorcal data to perfectly derve ts prce premum for a gven customer, there may stll be some nherent amount of uncertanty n the bd wnnng probablty due to the bd-requestng frm randomly allocatng ts busness to dfferent compettors to ensure a compettve market for future bds. Therefore, a frm wll never be able to remove all uncertanty from the bd-prce response process and must work wth probablstc models. Another common characterstc of stuatons where frms have used CPBRMs s when the sze of the bd opportuntes s not large enough to ustfy a dedcated sales person for each bd opportunty. Thus, the most common alternatves to usng CPBRMs s ether to charge a fxed prce to all customers or to have a sales agent respond to each separate bd opportunty wth a customzed prce. Chargng a fxed prce leads to mssed opportuntes to prce dscrmnate between dfferent customer segments, a practce that has been well publczed for sgnfcantly ncreasng a frm s proft n many dfferent ndustres. The other alternatve, relyng on a sales agent to respond to multple bd opportuntes, s also problematc. Theoretcally, the sales agent should have knowledge of the market, based on a hstory of former bd-responses wth the customer requestng the bd, allowng the sales agent to respond wth a customzed prce that optmzes ths nherent trade-off between decreasng margns, due to lowerng the prce, and ncreasng probabltes of wnnng the bd. In realty, sales agents often do not make good tradeoff decsons n these stuatons, ether because of a lack of hstorcal knowledge, the nablty to process ths hstorcal knowledge nto probablty dstrbutons, or ms-algned 3

4 ncentves (Garrow et al., 2006). The udcous use of CPBRMs allows frms to capture hstorcal bd nformaton, process t, and present non-based prce recommendatons to bddng opportuntes. If there s addtonal nformaton avalable regardng the bddng opportunty that can not be captured n the CPBRMs, the CPBRM s recommended prce may serve as one of possbly many nputs to the person responsble for makng the bd-response decson. To summarze, CPBRMs apply to stuatons where a frm sellng a non-commodty product must respond to frequent request for small to medum szed bds from a number of dfferent customers where the bd-wnnng crtera s not always the lowest prce. To use a CPBRM, a frm must have access to ther hstorcal bd hstory that ncludes, as a mnmum, the prce the frm bd at each opportunty and the correspondng bd result (wn or loss). Other useful hstorcal nformaton used n developng CPBRMs s, for each hstorcal bd opportunty, the customer, the length of the relatonshp wth the customer, the sze of the order, delvery date requrements, compettors bds, and any other pertnent nformaton useful for market segmentaton. When CPBRMs are used as an nput to a prce optmzaton model, there s also the mplct assumpton that the actons of the compettors can be determned probablstcally and ndependently of the decson maker s acton. If all compettors have smlar analytc capabltes and ontly optmze aganst each other, compettve response modelng technques such as game theory must be used. In ths paper, we evaluate two CPBRMs, namely the Logt and Power functons, whch model the response of the buyer subect to the segmentaton crtera descrbed above. We demonstrate, on an ndustry dataset, how each model may be developed and expected mprovements n profts may be estmated. By assesses two goodness-of-ft crtera for each model, we fnd the Logt functon provdes a better ft when there s lmted data avalable on the hstorcal bd opportuntes for determnng customer segments. When detaled nformaton s avalable about each former bddng opportunty such as the compettors prces and the sze of the order, the Power functon s a better ft on our test dataset. We demonstrate how to modfy the 4

5 functons to ncorporate varous degrees of segmentaton data avalable to the frm. We then test both functons on the ndustry dataset to analyze the relaton between the nature of the segmentaton nformaton avalable to the frm and the potental mprovements n proft generated by our approach. Fnally, we observe that the model provdng the better ft to the data also results n hgher expected proft mprovements. The rest of the paper s organzed as follows. In 2 we revew the academc lterature and ndustry practces related to the modelng of bd-prce responses. In 3 we present two CPBRMs that are used n practce and show how they can be modfed to be used under three dfferent levels of avalablty of hstorcal and compettve nformaton. We dscuss a step-by-step procedure for developng CPBRMs and usng them for quotng customzed prces. We also dscuss dfferent dagnostc measures and segmentaton methods whch can be used, based on the nature of data avalable. In 4 we present the results from applyng the two CPBRMs to an ndustry dataset and assess ther ft for un-segmented and segmented data. We then compare ther performance by measurng the percent mprovement n expected profts under dfferent nformaton levels. In 5 we summarze our observatons from the numercal comparson and conclude wth some lmtatons and manageral mplcatons of usng CPBRMs. 2. Lterature Revew In ths secton, we dscuss the academc lterature on bd-prce response models and how CPBRMs are unque. We also dscuss the motvaton from ndustry practces related to such compettve prcng settngs. Several papers develop bd-prce response models where prce s the only attrbute of the model. Fredman (1956) and Gates (1967) both develop models whch use the hstorcal bd nformaton avalable. Morn and Clough (1969) buld on ther work by dentfyng key compettors and capturng temporal senstvty to changes n strategy by gvng recent data more mportance. However, these models consder prce as the sole crteron for wnnng a bd and only consder the obectve of maxmzng profts. Chapter 4 n Llen et al. (1992) provdes an 5

6 overvew of competton orented prcng where the frm makes a trade-off between margn and probablty of wnnng the bd. Ths s the same trade-off the frm makes n our models. The dfference, however, occurs n the estmaton of the wnnng probabltes. In ther model, the lowest bd always wns, so the probabltes are based on the number of compettors and each compettor s estmated bd-to-cost rato. Kng and Mercer (1991) dscuss estmaton methods for determnng the dstrbutons for these ratos. The models we revew are more general; they nclude non-prce factors such as order sze and contnue to hold when factors other than ust prce are ncluded n the buyer s decson. Papaoannou and Cassagne (2000) provde a detaled revew of bd-prce response models and develop a ServPrce model whch, lke CPBRMs, accommodates several frm obectves and accounts for both prce and non-prce attrbutes. However, ther model reles only on the sales or prcng agents to nternally make tradeoffs and analyze the hstorcal nformaton wthout provdng any analytcal tools for dong so. In contrast, CPBRMs help the frm obtan a non-based nput to the bd-response decson by processng the relevant hstorcal nformaton statstcally. Lawrence (2003) develops an analytcal model for provdng bd-response quotes that predcts the outcome of a bd as a functon of ts attrbutes. Hs model requres a more extensve bd hstory (number of hstorcal bds) than a typcal CPBRM and uses a machnelearnng approach. In addton, t doesn t explot any addtonal nformaton that s avalable to the frm such as order sze, length of relatonshp, etc. CPBRMs, n contrast, can explot ths addtonal nformaton to determne partcular market segments. We also study the dfference n mprovements when CPBRMs are used wth dfferent levels of ths nformaton. Ths paper s most closely related to the work presented n Chapter 11 of Phllps (2005b) and the U.S. patent of Boyd et al. (2005), who dscuss the use of CPBRMs n ndustry and develop models usng a Logt functon as a bd-response functon. These models capture the nherent preference uncertanty and non-prce factors whch play a crtcal role n wnnng a bd. Fnally, Elmaghraby (2006) provdes a bref revew of CPBRMs n relaton to busness-to- 6

7 busness auctons but does not descrbe the models or how to mplement them n practce. The contrbutons of our work over the methods suggested by Phllps and Boyd et al. are as follows: We extend the Logt bd-response functon to nclude the compettor s prce whch helps to capture the compettve dynamcs. We also present another CPBRM, the Power functon (sometmes found n practce) whch ncludes a parameter of the rato of the bdder s prce to the expected bd prce of the bdder s compettor(s). We numercally test both models on an ndustry dataset and, based on the ft and performance of each model, we provde observatons on when each model may be preferred. To the best of our knowledge, ths s the frst academc paper to present such models and dscuss ther use for customzed prce optmzaton. 3. Customzed-Prce Bd-Response Models In ths secton we descrbe what CPBRMs are, present two CPBRMs used n practce, and dscuss how they may be developed to be used n a prce optmzaton model. CPBRMs calculate the probablty of wnnng a bd opportunty for each possble prce response gven the market characterstcs and compettve dynamcs for a partcular customer segment. The parameter values for these models are statstcally estmated from hstorcal bd nformaton and nclude, at a mnmum, the bddng frm s responses to prevous bd opportuntes and the outcome from each bddng opportunty (wn or loss). Intutvely, f the prce quoted by a frm s very low compared to ts compettor s prce, the probablty of wnnng the bd should be close to 100%. If t s very hgh by comparson, the probablty of wnnng should be close to zero. Ths probablty of wnnng the bd should monotoncally decrease wth an ncrease n prce (or prce rato). Also, the slope of the response curve should be steeper for prces close to the compettor s prce as compared to prces far hgher or lower than the compettor s prce. Hence, the bd response curve s generally S-shaped n nature. In a sngle compettor settng wth no prce premum enoyed by ether frm, prcng equal to the compettor s prce should result n a 50% chance of wnnng the bd opportunty. In practce however, one of the frms usually enoys some 7

8 prce-premum over the other. CPBRMs can be used to dentfy what ths prce-premum s for each customer segment. 1.0 Prob. Of Wn , Prce Rato Fgure 1 Bd-Response Curve Fgure 1 shows a CPBRM curve appled to one of our test case datasets. The prce rato on the x-axs s the rato of the frm s prce relatve to ts compettor s prce. For the partcular frm correspondng to ths bd-response curve, a prce equal to ts compettors prce (prce rato = 1) results n a probablty of wnnng the bd of 49% (a prce rato =.99 equates to a 50% wn probablty for ths frm). Thus, ths frm has a negatve prce-premum and must prce below ts compettor s prce for an equal opportunty of wnnng the busness of the frm offerng the bd. Before presentng the two CPBRM functons revewed n ths paper, we frst ntroduce some notaton. p Table 1: Notaton Unt prce quoted by the frm for bd opportunty (our decson varable) ρ ( p ) Bd-Response Functon,.e. probablty of wnnng bd opportunty gven a prce of p a, α b, γ Index for bd opportuntes, = 1, 2 Parameters related to non-prce factors for a segment Parameters related to prce factors for a segment Index for segments, = 1, 2 8

9 c c Coeffcent for the prce quote of the compettor p c, Unt prce quoted by the compettor(s) for bd opportunty c q Q Coeffcent for the order sze Order sze for bd opportunty r( p ) Prce rato = p pc, for bd opportunty x Indcator varable for segment (bnary) ε ( p ) Elastcty of the bd- response functon c p W y Margnal costs for the frm Wn/Loss ndcator varable for a bd opportunty ( bnary) Actual Outcome for the th bd. 3.1 Two Common CPBRMs In ths secton we descrbe the two CPBRMs compared n ths paper and dscuss how they can be adusted to nclude segmentaton and compettve prcng nformaton; whch have been conectured to sgnfcantly enhance the predctve power of a CPBRM. Bd-responses may dffer based on customers, channels, or product attrbutes such as warranty or payment terms. We capture these possble aspects n our models through a sngle countng varable, where = 1, 2 representng the number of dstnct, dscrete customer segments. Other factors such as the sze of the order or the compettve prce can often be modeled (dependng on the CPBRM) on a contnuous scale and may sometmes be treated separately. When dstnct clusters or segments exst n the data, a dscrete approach should be used. Logt Bd-Response Functon Phllps (2005b) & Boyd et al. (2005) both present the Logt functon as ther representaton of a CPBRM. As dscussed n Phllps (2005b, pg. 289), for a dataset wth dstnct segments, the general form for the Logt functon s: 1 ρ( ) = 1+ e p a + b p. 9

10 One of the man advantages of the Logt functon s the ease of addng addtonal segmentaton factors such as the sze of the order, Q and the compettor s prce quote, p c,. If the segmentng varables can be used as contnuous varables, the model may be adusted to nclude these segmentatons by addng coeffcents such as c q to measure the effect of order quantty segmentaton and c c to measure the effect of the compettor s prce segmentaton. These coeffcents are multpled by Q and c, p respectvely: ρ( p Q, p ) = 1+ e 1 c, a + b p + c Q + c p q c c,. Note that a relatve prce rato may also be used n the Logt functon by replacng c c p c, wth c p n the equaton above. In our performance test, we found lttle dfference between c( p /, ) c these two representatons so we only present the smpler form wth ust the compettor s prce. Usng the smplest form of the Logt functon: 1 ( p) =, the slope ρ ( p) and elastcty a bp 1 + e ρ + ε ( p) of the Logt functon s (Phllps 2005b pg. 284): Power Bd-Response Functon ρ ( p) = bρ( p)(1 ρ( p)) and ε ( p) = bp(1 ρ( p)). An alternate CPBRM sometmes used n practce s the Power functon, defned n ts general form as: ρ( p ) = α + r( p ) γ α. The man advantage of the Power functon s that, compared to the Logt functon, compettve prce dynamcs are explctly captured. The man dsadvantage s that t s more cumbersome to adust the model for non-prce, contnuous varable attrbutes. Segmentaton parameters can be added to the Power functon but only through a dscrete characterzaton. Thus, a varable such as order sze must be broken nto dscrete ntervals and captured through the parameterγ, where the subscrpt now represents the dscrete 10

11 ntervals of the order sze. Usng the smplest form of the Power model wth no segmentaton, α ρ( p) =, the slope and elastcty of the Power functon s α + r( p) γ γ ρ ( p) = ρ( p)(1 ρ( p)) and ε ( p) = γ(1 ρ( p)). p For a CPBRM to be a strctly decreasng functon n p, the prce dependent parameters must be strctly greater than zero. More specfcally, for the Power functon: γ >0. The parameter γ s a measure of the prce senstvty of the buyer where hgher values of γ mply greater prce senstvty. The effect of the parameter γ on the probablty of wnnng s shown n Fgure Prob. of Wn Gamma=5 Gamma =15 Gamma = Prce Rato Fgure 2: Effect of γ (Prce Senstvty) on the probablty of wnnng The parameter α s a measure of the prce premum the frm enoys, wth a hgher value of α mplyng a larger prce premum on the market. Thus, an ncrease n the value of α allows the frm to charge a hgher prce for the same probablty of wnnng. The effect of the parameter α on the probablty of wnnng s shown n Fgure 3. 11

12 1 Prob. of Wn Alpha=0.1 Alpha =1 Alpha= Prce Rato Fgure 3: Effect of α (Prce Premum) on the probablty of wnnng 3.2 Estmaton of Parameter Values The parameter values of a CPBRM can be estmated statstcally by fttng a curve to the avalable bd-hstory data based on mnmzng the squared errors or usng maxmum-lkelhood estmates. Before estmatng the parameter values of the models however, t s mportant to dvde the dataset nto two segments; one for estmatng the parameter values and the other for measurng the ft. Smlar to tme-seres forecastng models, measurng the goodness-of-ft on the same data as the parameter values are estmated on may result n a msleadngly close ft as compared to testng the model on a holdout sample. We brefly descrbe two estmaton methods below, usng the Power functon as the CPBRM of reference. (Phllps 2005b, pg. 285) descrbes how each estmaton method s appled to the Logt functon. The two methods are: a. Mnmze the squared error resduals: Mnmze [ ρ( p α, γ) W ] αγ, 2 b. Maxmze lkelhood estmates: Maxmze ln[ ρ( p α, γ) W + [1 ρ( p α, γ)](1 W)]. αγ, 3.3 Segmentaton Methods Many dfferent approaches to segment data exst. The number and type of segments can be determned n advance (a-pror) or can be determned on the bass of data analyses (post-hoc). 12

13 Predctve methods where one set conssts of dependent varables to be predcted by the set of ndependent varables can also be used. Some of the more popular methods are non-overlappng and overlappng clusterng methods, classfcaton and regresson trees, and Expectaton Maxmzaton algorthms. A detaled analyss of these methods s beyond the scope of ths paper but we refer the reader to Wedel & Kamakura (1998) for a detaled overvew. In general, the number of bd attrbutes (segments) that can be accurately estmated depends on the amount of hstorcal bd-nformaton avalable. If extensve nformaton s avalable, greater degrees of segmentaton can be acheved wthout compromsng the accuracy and robustness of the statstcal estmaton of the parameter values. 3.4 Dagnostc Measures After the segments have been determned and the parameters of the model have been estmated, the goodness-of-ft of the response functon should be assessed usng the holdout sample of the dataset. Varous dagnostc measures are avalable to check the ft of the model to the data. Some of the more common nclude the Hosmer & Lemeshow tests (H & L tests), the very smlar Pearson Ch-Squares tests, and the Akake Informaton Crtera (AIC). Hosmer and Lemeshow (1989) dscuss the H & L tests and the Pearson Ch-Squares tests and, Burnham and Anderson (1998) dscuss the AICc estmates n detal. For our analyss, we use H & L tests and AICc estmates to assess the goodness-of-ft of our models to the holdout sample. H & L tests are a popular dagnostc measure for logstc regresson models. The observatons are parttoned nto 10 equal segments based on the estmated probabltes and the dagnostc measure s then calculated usng X ( Ok Ek) =, E (1 E / n ) k = 1 k k k where O = k y = Sum of the actual outcomes for each segment k, k k n s the number of records n segment k, and E = k k ρ k = Sum of the estmated probabltes for each segment k. If 13

14 the ftted model s approprate, the dstrbuton of X 2 s well approxmated by the ch-square dstrbuton wth (k-2) degrees of freedom. Thus, a p-value can be calculated from the ch-square dstrbuton usng k degrees of freedom to test the ft of the model to the data. If the p-value from the H & L tests s 0.05 or less, we can accept the null hypothess at the 95% confdence level that the model does not predct values consstent wth the observed values. However, f the p-value s greater than 0.05 we cant reect the null hypothess that there s no dfference between the observed values and the model predcted values, mplyng that the model s estmates ft the data at an acceptable level. Pearson Ch-Square estmates can be calculated n a smlar way, but the groupng s based on the predctor varables. AIC estmates compare two alternatve systems usng the number of parameters estmated (K), the number of observatons (N) and the resdual sum of squares (SS). The estmate s calculated as follows: SS A IC = N ln + 2 K N. If the number of observatons s small, a correcton factor s often added to the Akake s Informaton Crtera: 2 K ( K + 1) AIC c = AIC + N + K 1. If N s much greater than K (as n our dataset), then corrected the AIC c s approxmately equal to the AIC estmate. A model wth a lower AIC c estmate s more lkely to be a better ft for the data. After estmatng the parameter values of a CPBRM usng hstorcal bd data and determnng the CPBRM fts the data well usng the holdout sample, the CPBRM can now be used to determne the optmal bd-response prce for an upcomng bd opportunty. Ths process s descrbed n the next secton. 3.5 Use of a CPBRM n Prce Optmzaton We now look at how CPBRM curves can be used n prce optmzaton. For the followng dscusson, we use the obectve of maxmzng expected profts. However, other strategc or operatonal obectves can be easly accommodated such as ncreasng market shares or ncludng constrants on capacty, nventory, prce or margn. The prce optmzaton problem for bd opportunty s 14

15 Max π( p ) = ρ( p ) ( p c ) Q. p p Note, the margn ( p c ) s strctly ncreasng n prce (Fgure 4) but the probablty of wnnng p the bd s strctly decreasng n prce (Fgure 5). Therefore, the expected proft s a unmodal functon as shown n Fgure 6. Margnal Deal Cont. Unt Prce Prob. Of Wn Unt Prce Fgure 4: Margnal Deal Contrbuton vs. Unt Prce Fgure 5 Prob. Of Wn vs. Unt Prce Expected Proft Unt Prce Fgure 6 Expected Proft vs. Unt Prce Determnng the optmal prce nvolves fndng a global maxma for the expected proft whch s unmodal n nature. The proft-maxmzng prce occurs where the elastcty of the expected proft functon s equal to the nverse of the margnal contrbuton rato, p ε ( p ) = p c p. The dervaton s avalable from the authors by request. 15

16 We have descrbed two CPBRMs and explaned how they can be used to fnd an optmal prce response for a specfc bd opportunty. In the next secton we demonstrate how to apply the CPBRMs to hstorcal bd data and test them on two ndustry datasets correspondng to two extremes of hstorcal nformaton avalable to the user. 4. Numercal Comparsons of CPBRMs on Industry Data In ths secton, we compare the performance of the two CPBRMs descrbed n the prevous secton usng a bd-hstory ndustry dataset. The dataset contans a sngle-compettor settng where extensve bd hstory s avalable ncludng the compettor s prce at each bd opportunty. We test the two CPBRMs under a wde set of scenaros pertanng to: 1) the amount of knowledge of the compettors prce response to the current bd request, and 2) the amount of segmentaton ncluded n the models based on the sze of the order n each bd opportunty. 4.1 Test Case Scenaros The frm provdng our dataset manufactures and sells medcal testng equpment to laboratores at hosptals, clncs, and unverstes across North Amerca. One of ther popular products s a gas chromatograph refll cartrdge that has a lst prce of $ The margnal cost assocated wth each unt s $6.00. The refll cartrdges are ordered n batches rangng n sze from 100 to over Orders for fewer than 200 unts are handled through the company s webste or through resellers wth no assocated dscount from the lst prce. At the other extreme, the company receves about 100 orders per year for more than 1000 unts. These large deals are negotated by a natonal account manager, usually as part of a much larger sale. Orders for unts are handled by a regonal sales staff that has consderable leeway wth regard to dscountng. We only look at ths mddle-sze segment to apply the CPBRMs. The requested sze of the order for each bd opportunty s also recorded, allowng us to test both segmented and unsegmented versons of the CPBRMs. Because of the specalzed nature of the product, the frm has only one sgnfcant compettor and they are able to capture ther compettor s prce after each bd 16

17 opportunty. Ther bd hstory nformaton s exhaustve, wth approxmately 2400 records of prevous bd opportuntes. A snapshot of ths dataset s shown n Table 2. Table 2: Bd Hstory for a Medcal Devce Company Bd Number Wn Frm s Bd Compettor s Bd Order Sze 1 Y $8.44 $ N $11.88 $ N $11.29 $ Y $9.78 $ Y $9.28 $ For ths applcaton, the probablty of wnnng the bd at a prce equal to the compettor s prce (.e. prce rato s one) s 51%. Ths percentage mples the frm doesn t enoy any sgnfcant postve or negatve prce premum compared to ts compettor. Knowledge Level of Compettors Prcng We tested the two CPBRMs under three dfferent levels of knowledge a frm may posses regardng ts compettors prcng,.e. worst, medum, and best cases. Hstorcal compettve bdprce nformaton s often avalable n many B2B applcatons through ether formal or nformal channels, dependng on the relatonshp the bdder shares wth the buyer. UPS, for example, obtans compettors bds n approxmately 40% of the parcel shppng bd opportuntes they partcpate n (Knple, 2006). In some busness-to-busness scenaros, a frm may even be provded wth the compettors bds and asked to respond wth a quote of ther own (note that for reasons explaned earler, provdng the lowest bd does not always guarantee a wn n these stuatons). In many B2C markets such as loan and nsurance quotes, nformaton about the compettor s prce may be avalable from a smple web-page search. Worst Case: No Prce Informaton Case: In ths case, the frm has no hstorcal prce nformaton on ts compettors, nor does t have any nformaton about how ts compettors wll prce for the current bd opportunty. Ths scenaro s rare n practce but, for our analyss, serves 17

18 as a lower bound on the knowledge of compettors prcng. Wth no compettor prce nformaton, the Logt functon s the only CPBRM avalable, as the Power functon requres an estmate of the compettor s prce n the current perod (va the prce rato). Medum Case: Naïve Prce Estmaton Case: In ths medum case, the frm has no nformaton about how ts compettors wll prce n the current perod except for the prce hstory of ts compettors on past bddng opportuntes. Thus, the frm can estmate ts compettors prces for the current perod through some type of forecastng or regresson model. In our analyss, we use a smple 10-perod movng average to predct the compettor s prce n the current perod. We expermented wth movng averages of dfferent numbers of perods but found the 10-perod movng average resulted n the most accurate and least based estmates for the future compettor s bd. By estmatng the compettor s prce response, we can now test both the Logt and Power functons. Best Case: Perfect Compettve Prce Knowledge: In ths best case, the frm knows exactly what ts compettors bds wll be n the current perod. Ths can be consdered an upper bound on the frm s forecastng capabltes. It also apples to cases where the buyer provdes compettors bds before requestng a bd from the frm or n applcatons where a frm can check ts compettors prces (possbly va ther web pages) before respondng wth ts own prce quote. The chart below summarzes whch knowledge levels were tested for each CPBRM. Logt Power Worst Case Medum Case Best Case The next secton descrbes the procedure we used to develop and test the two CPBRMs on the dataset and scenaros descrbed above. 4.2 Procedure for Testng CPBRMs 1. We dvded the dataset nto two dstnct sets; the frst for estmaton of the model parameter values and the second for performance evaluaton (the holdout sample). We used the frst 90% of the hstorcal bd records as our estmaton data and the remanng 18

19 10% as our performance test data. Whle the choce of 10% for the performance test may seem arbtrary, t s a common choce for holdout samples n forecast methods evaluatons. Senstvty test wth dfferent percentages of the hstorcal data used for measurng performance were also performed. The changes n the parameter estmates and performance results on the dataset were nsgnfcant when tested over a range of 10% - 20% for the performance test dataset. 2. Usng the estmaton data, we calculated the parameter values for both the Logt and Power functons usng ordnary least squares and maxmum lkelhood estmators. We found lttle dfference n the ft of the models between the two estmaton methods so we present the values found usng least squares. The parameter values from the unsegmented analyss of the dataset are: Table 3: Parameter Estmates for Unsegmented Analyss Logt Model Power Model c α γ a b c Worst Case NA NA NA Medum Case Best Case Because the Power functon requres an estmate of the compettor s prce, t could not be used under the Worst nformaton case. The reason the parameter values are the same for the Medum and Best nformaton cases s because past bd opportuntes are used for estmatng the parameter values when the compettor s prce s known wth certanty (note the nformaton cases pertan to knowledge of the compettors prce n the current perod; the past prces are assumed to be known wth certanty). In the next secton, we descrbe how we also used the order sze as a segmentaton varable. 3. After estmatng the parameter values for each model, we measured the goodness-of-ft of the models usng both the H & L tests and the AIC c. 4. After selectng the model that provded the best ft for the holdout sample data, we used the CPBRM to optmze the bd-prces for all the bds n the performance test data subset. 19

20 5. Fnally, we computed the percent mprovements over expected profts and over actual profts as explaned n secton 4.3. Ths provded us wth two metrcs of performance. The ncorporaton of the compettor s prce s only one possble nput to CPBRMs (although for the Power functon t s a requred nput). Another common nput s the sze of the order request. It s reasonable to assume the prce senstvty of customers only orderng a few unts wll dffer sgnfcantly from customers orderng large quanttes. Thus, we descrbe how we segmented the bds based on the sze of the order below. Segmentaton Based on Order Sze In our dataset, order quanttes range between 200 and For segmentaton based on the order sze usng the Logt functon, we used dscrete segmentaton based on the order sze and estmated the model parameter values by fttng the followng model to the estmaton data: ρ( p Q, p ) = 1+ e 1 c a+ p bx+ ccpc For dscrete order sze segmentaton, a separate parameter must be estmated for each segment whle for contnuous order sze segmentaton, the estmaton of only one parameter s requred. Therefore, t s easer to use a contnuous varable f the model allows t. For segmentng based on the order sze usng the Power model however, a dscrete approach s the only opton. The bd response for the Power functon was calculated by estmatng a dfferent value of γ (our estmates for α dd not change and was thus, held constant) for each order sze segment q by fttng the followng model to the estmaton data: ρ( p q) = α α + r( p ) γ x. The postve correlaton between the sze of the order and the estmate of γ (a measure of prce senstvty) ndcates that wth an ncrease n order sze, the prce senstvty ncreases. To test f there were any nherent order sze segments n our dataset, we tred varous clusterng algorthms 20

21 and segmentaton technques. These efforts dd not expose any naturally occurrng order sze segments, so we used an a-pror approach wth the followng segmentaton: Order Sze Between Segment To estmate the parameter value for each segment, we used a bnary ndcator varable x, = 2, 3.9, whch was assgned a value of one for the order sze segment a specfc bd fell under. Ths classfcaton scheme s demonstrated n the table below: Bd Wn Frm's Compettor's Order Indcator Varables for Order Sze Segments Number Bd Bd Sze $8.44 $ $11.88 $ Based on least squares fts, Table 4 provdes the estmated parameter values for each segment. Table 4: Parameter Estmates for Segmented Analyss Knowledge of Comp. Prce Logt c q, a c c Worst Case NA Medum Case Best case Power α Medum Case Best Case γ Although each order sze segment had approxmately the same number of bds, the prce senstvty parameters ( c, and γ ) dd not ncrease monotoncally wth the order sze. Ths s q an nterestng observaton as our ntuton led us to predct otherwse (demand becomes more senstve to prce, on a contnuous scale, as the sze of the order ncreases). It s possble however, that ths s ust an artfact of our dataset. In summary, we bult models for the Logt and Power CPBRMs usng an estmaton data subset, three levels of knowledge of the compettors prces, two levels of segmentaton on the 21

22 order sze, and measured performance usng two performance measures. Thus, we had a total of 12 scenaros to base our observatons. Table 5 summarzes the varous scenaros. 4.3 Dagnostc Measures: Table 5: Summary of Test Scenaros Knowledge of Compettors Prce Segmentaton on Order Sze Performance Measure Worst Case Segmented Actual Profts Medum Case Unsegmented Expected Profts Best Case For each model and case scenaro, we measured the goodness-of-ft for the dataset and dentfed the better fttng model. We analyzed the models usng H & L tests and AIC c estmates. For the worst case of nformaton.e. (no nformaton about the compettor s prce), only the Logt model could be used. Therefore, for ths case, we dd not use the AIC c estmates as they can only be used to compare two competng models. The H & L tests however, were used to assess the ft of the Logt model to the data. The results from ths test for the worst nformaton case scenaros are summarzed n Table 6. Table 6: H & L Test Results for Worst Case Scenaros Worst Case: No Compettve Informaton Logt Model p-value Unsegmented Segmented The p-values are large enough to ndcate the model s a good ft for the dataset n these scenaros. The models developed for the medum and the best nformaton cases only dffer n the compettor prce estmates used durng the prce optmzaton sub-problem. Therefore, the model developed for both cases are dentcal. For reportng purposes, we refer to both cases together as Compettve Informaton Cases. The results for the Logt and the Power model for the unsegmented and the segmented cases usng H & L tests are n Table 7. Note that larger p-values ndcate a better ft for the H & L tests. X 2 22

23 Table 7: H & L Test Results and AIC c Estmates for Medum & Best Case Scenaros Compettve Inf. H & L tests: AIC c Cases X 2 p-value Estmates Logt Unsegmented Segmented Power Unsegmented Segmented Accordng to the results from the H & L tests, for the unsegmented scenaro, the Logt model provdes a better ft. Also, the AIC c estmate s smaller than the estmate for the Power model. Therefore, the Logt s a better fttng model for ths scenaro. Smlarly, for the segmented scenaro, both the dagnostc tests ndcate the Power model provdes a better ft to the data. One fnal observaton s that the segmented models provde a better ft for the data than the unsegmented models. In the next two sectons, we determne f a better ft also leads to better performance. 4.4 Measures of Percent Improvement n Actual and Expected Profts To test the mpact of usng CPBRMs on the ndustry datasets, we used two performance metrcs: percent mprovement n profts over un-optmzed actual profts and percent mprovement n profts over un-optmzed expected profts. To understand the dfference between the two performance metrcs, consder the followng numercal example from our dataset: Bd Wn Order Sze Orgnal Bd Optmal Bd Probablty of Wn Probablty of Wn at Orgnal Bd at Optmzed Bd 1 Y 353 $ 8.44 $ Applyng the unsegmented, worst nformaton case, Logt CPBRM wth the parameter values 1 obtaned through the procedure outlned n secton 4.2 we get: ρ( p) = p 1 + e Substtutng n the orgnal bd prce of $8.44, we calculate the probablty of wnnng for the unoptmzed bd = *$8.44 1/(1 e + ) + = Applyng the optmzaton procedure 23

24 descrbed n secton 3.3, we calculate that the optmal bd prce for ths bd opportunty should have been $9.35. Substtutng n ths prce results n a probablty of wnnng for the optmzed e *$9.35 bd = 1/(1 + ) = The actual proft from ths bd opportunty s = (Orgnal Unt Prce- Margnal Cost)* Order Sze* Wn/Loss Indcator Varable = $(8.44-6)*353*1 = $ If the orgnal bd had resulted n a loss, the actual proft would be zero. The orgnal bd expected proft = (Orgnal Unt Prce-Margnal Cost)* Order Sze * Probablty of Wn at the Orgnal Bd Prce = $(8.44-6)*353* = $ Note that the expected proft s always smaller than the actual proft when the bd was won, and s always larger when the bd was lost. The optmzed bd expected proft = (Optmzed Prce- Margnal Cost)*Order Sze* Probablty of Wn at the Optmzed Bd Prce = $ (9.35-6)*353*0.64= $ We now compare the percent mprovement of the latter case over the frst two: Percent Improvement n Optmzed Bd Expected Profts over Un-Optmzed Bd Actual Profts = ($ $861.32)/$ = = % Percent Improvement n Optmzed Bd Expected Profts over Un-Optmzed Bd Expected Profts = ($ $676.37)/$ = = 11.01%. The calculatons above were performed for every bd opportunty n the performance test data subset and the average of each measure (over each bd opportunty n the performance test data subset) was used as the performance metrcs presented n the next secton. 4.5 Model Performance For our dataset, the un-optmzed bd actual and expected profts over the performance test subset of our dataset were exactly the same. Therefore, we only present the percent mprovements n actual profts for each scenaro (segmented & unsegmented and three levels of nformaton). The percent mprovements are summarzed n Fgure 7. For the unsegmented case, the results are from the Logt model only. For the segmented case, we used the Logt model for 24

25 the worst case and the Power model for the medum and best nformaton cases, where the frm has some knowledge about the compettor s prcng strategy. % Improvements 30% 20% 10% 0% Worst Case Medum Case Best Case Segmented Unsegmented Knowledge Level Fgure 7: Percent Improvement n Actual Profts 5. Observatons from Numercal Comparsons and Conclusons In ths secton, we summarze our observatons based on our numercal comparsons and attempt to answer the queston: Gven a partcular set of condtons, whch CPBRM should a frm use to optmze prces? We then summarze our work and provde areas for future research. Our observatons come wth the followng caveats; they are based purely on the performance of the models on our avalable dataset and may not be generalzable to applcatons dfferent than the ones tested. Thus, a frm should rgorously test the models usng ther own bd hstory data before drawng conclusons on the sutablty of a partcular model for ther specfc applcaton. Based on the performance on our two ndustry datasets, we provde four man observatons: Observaton 1. If enough hstorcal bd data s avalable to segment based on the order sze, the segmented Power functon, adusted for each dscrete customer segment, provdes a better ft to the performance data than the segmented Logt functon. Ths observaton s evdent from the results of the goodness-of-ft tests presented n Table 7. If the frm has no knowledge about the compettors prces for the current bd opportunty however, the Power functon can not be used. 25

26 Observaton 2. If the frm does not segment based on the order sze, the Logt functon provdes a better ft to the performance data than the Power functon. Ths observaton s agan evdent from the results of the dagnostc measures n Table 7. Observaton 3. The model whch provdes a better ft for the data also provdes hgher mprovements n profts. Also, the order sze segmented models ft the data better and resulted n larger proft mprovements than the unsegmented models. Table 8 summarzes the best fttng model and the correspondng percent mprovement n expected profts for both models. One would expect a better fttng model to provde better performance and ths s confrmed by the percent mprovements n Table 8. It s also evdent that the segmented models outperformed the unsegmented models. Table 8: Comparson of % Improvements Model wth Better Ft % Improvements Unsegmented Analyss Logt Power Medum Case Logt 18.45% 16.90% Best Case Logt 29.36% 28.76% Segmented Analyss Logt Power Medum Case Power 16.94% 20.21% Best Case Power 27.46% 29.98% Observaton 4. Incorporatng hstorcal compettor prces nto a CPBRM does not ensure better performance. Ths observaton s evdent n Fgure 7. In both the unsegmented and segmented results, the ablty to perfectly forecast nformaton about the compettor s prcng results n the largest mprovements. Havng access to hstorcal data on the compettor s bds can sometmes make a company worse off however, as evdenced by the worst nformaton case (no hstorcal compettors prce data) outperformng the medum nformaton case (10 perod movng average of compettors prces). A quck check of our 10-perod movng average forecast (used to provde the compettor s prce estmate) showed the forecast was unbased but had a large varance n the forecast error. Thus, for ths dataset, the past hstorcal compettor s bds were poor ndcators of how the compettor would bd n the future. The use of these poor estmates 26

27 led to worse performance usng the CPBRMs than f no estmate of the compettor s prce was used at all. Conclusons A frm adoptng CPBRMs for prce optmzaton needs to be aware of ther lmtatons. CPBRMs assume the bd opportuntes are exogenous and are not affected by the bd responses suggested through the optmzaton model. In realty, a frm s prcng strategy may have a sgnfcant mpact on customer retenton, especally f the optmzaton model recommends consstently prcng hgher than the competton for a partcular customer class. Also, CPBRMs, and ther correspondng optmzaton models, do not assume any response from the frm s compettors. Instead, they assume the actons of compettors can be determned probablstcally and ndependently of the decson maker s actons. In realty, compettors may react to a frm s new prcng strategy causng the hstorcal bd opportunty data to be unrepresentatve of future bd prce responses. To help detect these possbltes, mechansms should be put n place to montor and evaluate the performance of the CPBRMs over tme. If compettors change ther bd-prcng behavor due to the mplementaton of a CPBRM, more nvolved models usng concepts from game theory should be employed. In summary, we present two CPBRMs used n practce and explan how they may be used to calculate optmal bd-response prces, dscuss how they may be adusted to accommodate segmentaton based on the dfferent levels of nformaton avalable, and descrbe how to measure ther ft and performance. We llustrate the procedure through a numercal analyss on an ndustry dataset and, based on these results, offer a set of recommendatons about the type of CPBRM a frm should use dependng on the avalablty of past nformaton and the level of compettve knowledge avalable to a frm. 27

28 Acknowledgments: The authors wsh to thank Loren Wllams for hs nsghtful comments, and Robert Phllps for provdng the ndustry data set and for nformng us of the Power model. We also thank the partcpants from presentatons at Unversty of Notre Dame and the Unversty of Maryland for ther helpful suggestons. References Aldrch, J. and D. Nelson Lnear Probablty, Logt, and Probt Models. Seres: Quanttatve Applcatons n Socal Scences. Boyd. D, M. Gordon, J. Anderson, C. Ta, F. Yang, A. Kolamala, G. Cook, T. Guardno, M. Purang, P. Krshnamurthy, M. Cooke, R. Nandwada, B. Montero and S. Haas Manugstcs. Target Prcng System. PatentNo: US B1 Burnham, K. P., and D. R. Anderson, Model Selecton and Multmodel Inference: A Practcal- Theoretc Approach, 2nd ed. Sprnger-Verlag. Bussey, P, N. Cassagne, and M. Sngh Bd Prcng - Calculatng the Possblty of Wnnng, IEEE SMC, Orlando USA, Dudzak, Bll Senor manager n the plannng and analyss group of BlueLnx. Panelst n nontradtonal ndustres, Georga Tech 2 nd annual workshop on Prce Optmzaton and Revenue Management, May 18 th. Edelman, F Art and Scence of Compettve Bddng, Harvard Busness Revew (July/August), Elmaghraby, W. J. (2006) "Prcng and Auctons n EMarketplaces," forthcomng n the Handbook of Quanttatve Supply Chan Analyss: Modelng n the E-Busness Era, edted by Davd Smch- Lev, S. Davd Wu, and Z. Max Shen, Internatonal Seres n Operatons Research and Management Scence, Kluwer Academc Publshers, Norwell, MA. Fredman, L A Compettve - Bddng Strategy. Operatons Research 4: Gates, M Bddng strateges and probabltes. The Journal of Constructon Dvson 93:

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