Maxzng proft usng recoender systes Aparna Das Brown Unversty rovdence, RI aparna@cs.brown.edu Clare Matheu Brown Unversty rovdence, RI clare@cs.brown.edu Danel Rcketts Brown Unversty rovdence, RI danel.bore.rcketts@gal.co ABSTRACT Tradtonal recoendaton systes ake recoendatons based solely on the custoer s past purchases, product ratngs and deographc data wthout consderng the proftablty of the tes beng recoended. In ths work we consder the queston of how a vendor can drectly ncorporate the proftablty of tes nto ts recoendaton syste so as to axze expected proft whle stll provdng accurate recoendatons. Our approach uses the output of any tradtonal recoender syste and adjusts t accordng to te proftablty. Our approach s paraetrzed so the vendor can control the aount of devaton between the recoendaton ncorporatng profts and the tradtonal recoendaton. We study our approach under two settngs and show that t can acheve sgnfcantly ore proft than tradtonal recoendatons. Categores and Subject Descrptors H.3.5 [Inforaton Systes]: Inforaton storage and retreval Onlne-nforaton servces, Coercal servces Keywords Recoender systes, roduct proftablty, Algorths, Electronc coerce. 1. INTRODUCTION Recoendaton Systes are portant tools for ajor copanes such as Aazon, Netflx and andora. Recoendaton Systes use a custoer s deographc data, past purchases and past product ratngs to predct how the custoer wll rate new products [1, 11, 9, 13]. They have been shown to help custoers becoe aware of new products, ncrease sales and encourage custoers to return to the busness for future purchases [6, 15]. Desgnng recoendaton systes that accurately predct custoer ratngs has generated uch research and nterest both n the acadec and busness countes. The Netflx prze was a anfestaton of ths nterest [12]. However, the ajorty of the work on recoender systes has not explctly consdered how the proftablty of products could be ncorporated Supported by NSF grant CCF-0728816. Supported by NSF grant CCF-0728816. Copyrght s held by the author/owner(s). WWW2010, Aprl 26-30, 2010, Ralegh, North Carolna.. nto the recoendatons. An artcle publshed n Knowledge@Wharton clas that the actual Netflx recoendaton syste odfes ts ratngs to encourage consuers to order the ore obscure oves whch are presuably cheaper for Netflx to supply than ajor blockbusters[18]. Whle Netflx does not, publcly reveal whether t uses such ethods, t sees natural for a busness to ncorporate the proftablty of products nto ts recoendatons. Fro the vewpont of the custoer, recoendatons are helpful suggestons, but fro the vewpont of the vendor they are extreely targeted advertseents, and the explct goal of advertsng s to ncrease proft. In ths paper, we study the queston of how a vendor ght ncorporate the proftablty of tes nto ts recoendatons. A nave approach s to gve the ost proftable tes the hghest recoendatons. Then these tes would presuably be bought ore often and the busness would ake ore oney. However ths tactc has soe obvous flaws. Whle the custoer ay ntally follow the vendor s recoendaton, she ay fnd that she does not lke the tes as uch as the vendor predcted. After only a few such experences, she would realze that the vendor s recoendatons do not accurately reflect her tastes. In the best case for the vendor, the custoer would gnore the vendor s recoendatons and contnue her natural purchasng behavor whle n the worst case she would lose trust not only n the vendor s recoendatons but also n the vendor as a whole and take her busness elsewhere. Thus ncorporatng the proftablty of tes nto recoendatons ust be done carefully so that custoer s trust s not coprosed. A reasonable assupton s that as long as the vendor consstently presents recoendatons that are slar enough to the custoer s own ratngs, the custoer wll antan a hgh level of trust n the accuracy of the vendor s recoendatons. For ths reason, we use an establshed slarty easure as a easure of trust. We assue that the vendor has access to a vector c gvng the consuer s true ratngs for tes. The vendor s objectve s to present a recoendaton vector r to the custoer whch s wthn a certan threshold of slarty to c, and axzes the vendor s expected proft (by ncorporatng the proftablty of tes nto the recoendaton). Secton 2 gves detals of the odel. One queston that ght arse s how the vendor can deterne the custoer s true ratngs (.e c ) f the ratngs are for products that the consuer has not yet rated herself. There s a large aount of research and actvty n developng hghly accurate recoendaton systes that solve ths proble and we assue the predctons fro these systes
for well establshed custoers are good approxatons to the custoer s true ratngs. The dea that the custoer wll antan hgh trust as long as the vendor s recoendatons are slar to her own ratngs has eprcal support. Hll et al. showed that users asked to rate the sae te at dfferent tes supply dfferent ratngs [8]. Aatran et al. report slar fndngs n ther study whch evaluates nose n user ratngs of oves [2]. Thus there s soe natural varablty n the ratngs that users supply, and so slght dfferences between the predcted ratngs and the actual custoer ratngs should be nsgnfcant to the custoer. Chen et al. also consdered usng the proftablty of tes n recoender systes [5], but they do not explctly requre a level of accuracy n ther syste. The outlne of our paper s: secton 2 defnes our odel and proble, secton 3 descrbes the slarty easure and gves justfcaton for ts use as a easure of custoer trust and secton 4 descrbes our approach for axzng expected proft n dfferent scenaros. 2. MODEL Let n be the nuber of tes beng sold by the vendor. Let c and r be the vectors of length n where the th coponents denoted c and r gves a ratng for te. All tes are rated usng nubers between zero and soe axu ratng. We focus on the scenaro where the vendor s nteractng wth an establshed custoer who the vendor would lke to contnue to do busness wth n the long ter. We assue the vendor uses a recoendaton syste whch akes good predctons about how such a custoer rates tes and that vector c gves these predctons 1. The vendor presents the custoer wth recoendaton vector r to help her decde whch te to purchase. In any real applcatons, the vendor only wants to provde the custoer wth the top n recoended tes. In ths case the vendor would dsplay the n hghest rated tes n r. The custoer has a certan level of trust n the accuracy of the vendor s recoendaton whch she has developed fro experence wth the vendor s past recoendatons. When choosng an te to purchase, the custoer consders r based on how uch trust she has n the vendor s recoendatons. The exact nfluence that a custoer s level of trust n the vendor s recoendatons has on her purchasng behavor s too coplex to odel precsely. We ake the followng splfyng assuptons to allow for a odel of that behavor: Frst we assue that the consuer s trust s closely ted to how slar the vendor s recoendatons are to the ratngs she would gve tes. Second we assue that as long the slarty constrant s consstently et by the vendor, the consuer wll use the recoendatons when decdng what tes to purchase. Specfcally we assue there s a functon T ( r) that assgns a scalar value to the noton of slarty between r and c where hgher values ndcate greater slarty. Second, we assue that f for every r that the vendor presents to the custoer, T ( r) eets or exceeds soe threshold value, then the custoer s trust n the recoendaton syste wll rean at a constant, sgnfcantly hgh level. Fnally, we assue that f the cus- 1 The custoer does not necessarly know all the entres of c herself as she has not purchased all tes. toer s trust reans at ths level, her purchasng decson wll be solely a functon of r. The custoer s level of trust and her subsequent purchasng behavor s unknown f the vendor presents r such that T ( r) <. The ntuton behnd these assuptons s that, as long as the vendor recoendaton consstently predcts ratngs that are slar enough to how the custoer would rate the tes, the custoer wll antan a certan level of trust n the accuracy of the recoendaton syste. As a result, she wll use the vendor s recoendaton as nforaton when decdng whch tes to purchase. However f r s too far fro c, the custoer loses trust the vendor s recoendatons and no longer consders the when decdng whch te to buy. In that case, her purchasng behavor s unclear. The vendor s an goal s to axze proft. However, to antan custoer trust he s requred to present vector r such that T ( r) for soe constant. Denote ϕ( r) be a vector valued functon whose th coponent gves the probablty that the custoer wll purchase te at a gven te step. The custoer, at any step, can purchase zero or ore tes, so the coponents of ϕ( r) need not su to one. Let p be the proft vector whose th coponent, p, gves the proft receved when te s purchased. The vendor s expected proft s gven by E p = p ϕ( r). Forally our proble s to axze the vendor s expected proft whle antanng a level of trust wth the custoer: Max p ϕ( r) s.t. T ( r) (1) 3. SIMILARITY MEASURES FOR TRUST In ths secton we argue that the Dce coeffcent, whch easures slarty between two vectors, s an approprate easure of consuer trust. We brefly dscuss why soe of the other coon sngularty easures are lackng. Dce coeffcent. We adopt the Dce coeffcent gven n equaton 2, to easure trust T ( r). The Dce coeffcent s a popular easure whch has prevously been used to easure recoendaton accuracy [5, 7, 14]. Dce( r) = 2 cr c2 + r2 = 2 c r c 2 + r 2 (2) Above x = p x2 denotes the length of vector x. Norally the Dce coeffcent s denoted as a functon of the two vectors whose slarty s beng easured but here we denote t as only a functon of r to ephasze the fact that c s constant known to the vendor. Let θ denote the angle between c and r. An equvalent defnton of the Dce coeffcent s, Dce( r) = cos(θ) 2 c r c 2 + r 2 (3) We now lst soe propertes of the Dce coeffcent that akes t a reasonable functon to easure trust. roperty 3.1. Dce( r) s always between zero and one. Dce( r) = 1, f and only f c = r for every te. Dce( r) = 0, f and only f r = 0 on all tes such that c > 0. Thus the Dce( r) s one only when r s n coplete agreeent wth c, and t s zero only when r dsagrees wth c on all relevant tes. roof. (roof of roperty 3.1)
Dce( r) 0 because no te s rated less than zero. Thus the nuerator of Equaton 2 s always postve. Now suppose that Dce( r) > 1. By Equaton 2, ths ples that 0 > c2 2r c + r 2 = (r c)2 whch s a contradcton. Thus Dce( r) 1. Suppose that r = c for every. Then usng Equaton 2 Dce( r) = 1. To prove the other drecton suppose that Dce( r) = 1. The forulaton of the Dce coeffcent gven n Equaton 3 can be used to show that r = c for all. Note that 0 cos(θ) 1 as no te s rated less than zero. The second ter of Equaton 3 s non-negatve by defnton and the followng proof by contradcton shows t s at ost 1: suppose 2 c r /( c 2 + r 2 ) > 1 then 2 c r > c 2 + r 2 plyng that 0 > ( c r ) 2 whch s false. Thus f Dce( r) = 1 both ters of Equaton 3 ust be 1. As cos(θ) = 1 the angle between c and r s zero and the second ter beng 1 ples that ( c r ) = 0.e that r and c have equal length. Together ths ples that r = c for all. Fnally note that when Dce( r) = 0, the nuerator of Equaton 2 s zero whch ples that r ust be zero for each such that c > 0. Jaccard easure. The Jaccard slarty easure, gven below, behaves slarly to the Dce coeffcent and s used wdely n nforaton retreval and data nng [17, 10, 3]. Jac( r) = ( c r)/( c 2 + r 2 c r) It s another approprate easure of consuer trust and all results extend to the settng where T ( r) = Jac( r). Cosne easure. The cosne slarty easure, gven n equaton 4, s equal to the cosne of θ. It s always between zero and one as no te s rated less than zero. Cos( r) = cos(θ) = p r2 cr c2 The cosne easure s not nfluenced by dfference n the lengths of r and c and ths s the an reason t sees unsutable for easurng trust as deonstrated n the followng exaple. Exaple. Suppose the tes are rated between 0 and 5. Consder a pcky custoer who rates all tes soewhat low and a vendor that recoends all tes hgh,.e c = 1 and r = 5 for all. The cosne easure wll be to one because θ = 0 ndcatng that the pcky custoer has hgh trust for the vendor s recoendaton, whch s surely not the case. Mean squared error and dstance easures. The ean squared error (MSE) easures the average dsslarty between r and c. Let denote the axu possble ratng. To easure slarty we could use 1-MSE, Equaton 5, whch s always between zero and one. MSE( r) = ((c/ r/))2 n The 1-MSE easure gves equal credt for agreeents on tes the custoer dslkes as on agreeents on tes she prefers and ths akes t unsutable for easurng the trust as deonstrated by the exaple below. Ths exaple also apples to other dstance based slarty easures such as Eucldean dstance and Manhattan dstance. (4) (5) Exaple. Suppose the tes are rated between 0 and 5. Let c = [5, 5, 5, 1, 1, 1,..., 1] whch represents a custoer who rates a few tes very hghly but dslkes ost tes. Consder the followng recoendaton r = [1, 1, 1,..., 1, 5, 5, 5] where the vendor gves the hghest ratngs to a few tes that the custoer dslkes and gves the low ratngs to all other tes ncludng the tes the custoer prefers. MSE s Θ(1/n) so that 1-MSE approaches 1 as the nuber of tes n gets large ndcatng that the custoer would have hgh trust such recoendatons whch s not the case. 4. ROFIT MAXIMIZATION Usng the Dce slarty coeffcent defned n Secton 3 as the trust functon the vendor s optzaton proble s, Max E p( r) = p ϕ( r) s.t. Dce( r) = 2 c r c2 + r2 (6) We outlne a general approach for solvng Equaton 6 and then apply our technque on two dfferent objectve functons obtaned by alternatve defntons of φ( r). We analyze how uch proft the vendor gans by presentng the custoer wth recoendaton r rather than c. 4.1 General Approach Calculus gves us a general approach for solvng the vendor s axzaton proble stated n Equaton 1. Addng 1/ to both sdes of the Dce constrant above and splfyng reveals that t s equvalent to, X r c 2 1 2 1 «X c 2 (7) As c s a constant for our settng, the feasble r for our proble le n a regon enclosed by an n-sphere wth radus q ( 1 2 1) c2 whch we refer to as the Dce sphere. The general approach to solvng axzaton proble of Equaton 6 nvolves two parts. The frst s to deterne f there are any local axa that le strctly nsde the Dce sphere.e. whch satsfy Dce( r, c) >. The second part s to fnd the vector r that axzes expected proft over all vectors on the surface of the Dce sphere,.e. whch satsfy Dce( r, c) =. The largest of the local axa n the sphere and the axu on the surface s the global axu. The gradent of the objectve functon s zero at each local axu nsde the Dce sphere. Thus all solutons to E p = 0 are canddate vectors. Let r 1, r 2,..., r k be the lst of all vectors that satsfy ths property. Whle these vectors could be local na or saddle ponts nstead of local axa, t s not necessary to dstngush between the. It s only necessary to fnd the greatest value of E p( r ) for all and copare t to the axu of all vectors on the surface of the sphere. Let r n denote the axa vector nsde the Dce sphere. The axu vector on the surface of the Dce sphere can be found usng the ethod of Lagrange ultplers. The axu valued vector r s wll satsfy Dce( r s, c) = λ E p( r s) and Dce( r s, c) =. The axu of r s and s the soluton to the optzaton proble [4, 16]. 4.2 Sple probablty functon Consder a scenaro where the custoer can purchase zero or ore tes at each te step where the probablty that r n
the custoer purchases te s ndependent of the vendor s ratngs for other tes. Here s a sple way to defne the probablty functon whch satsfes ths assupton, r1 ϕ( r) =, r2,..., rn (8) Recall that s the hghest possble ratng for an te. Wth ths defnton of ϕ, the probablty that a custoer purchases an te s lnearly proportonal to the vendor s ratng of that te and the vendor s expected proft s E p( r) = p ϕ( r) = 1 X p r (9) As prces are all greater than zero, E p = ( p 1,..., pn ) 0 so there are no local axa nsde the sphere. Thus we can proceed to fndng the axu be on the surface of the Dce sphere. Usng Equaton 7 we have Dce( r, c) = `2(r1 c 1 ),..., 2(r n cn ), so the axa on the Dce sphere surface ust satsfy the followng syste of equatons, and p = 2λ(r c/) (r c/)2 = ` 1 2 for all 1 c2 where the last equaton requres r to le wthn the Dce sphere. Solvng the frst set of equatons we obtan that r = p 2λ + c (10) Substtutng r n the value of r nto Equaton 7 we get λ = p2. The fnal soluton s obtaned by pluggng 1 2 ( 1 2 1) c2 λ nto Equaton 10. s ( 1 1) r = p 2 j c2 j + c j p2 j (11) roft Gans. By presentng the custoer wth recoendaton r derved n Equaton 11 the vendor earns expected proft E p( r) = 1 p2 q(1/ 2 1) j c2j / j p2j + pc/, whch s splfed va the Cauchy-Schwarz nequalty to 2, r! 1 E p( r) 1 + 1 pc 2. (12) The expected proft fro c s E p( c) = ( pc)/. The vendor s proft gan fro presentng r rather than c s, r E p( r) E p( c) 1 = E p( c) 1 + 1 1 2(1/ 1). 2 Exaple. If the vendor presents recoendaton vectors that are wthn slarty threshold =.9, allowng a 10% devaton to c, then n expectaton he earns at least 2(10/9 1) > 22% ore proft by presentng r rather than c. 4.3 Sple dstrbuton Now we consder the scenaro where at each te step the custoer purchases only one te but chooses whch te to purchase based on how ts ratng copares to ratngs of other tes. A sple way to odel ths s to set the 2 The Cauchy-Schwarz nequalty s ( pc)2 p2 c2 purchase probablty of te to be r / j rj. «r1 r 2 r ϕ( r) =,,..., n r r r Thus the custoer scales each te by the su of the vendor s ratngs, and then chooses unforly aong all offered tes. Wth ths defnton of ϕ the expected proft s, E p( r) = p ϕ( r) = X p r j rj (13) The local axa nsde the Dce sphere occur where the gradent of the expected proft s zero. The gradent s * + p1 j rj r1p1 E p = 2,..., pn j rj rn pn 2 j rj j rj For E p = 0, t ust be the case that j rj = r for all. Thus f there are at least two tes wth dfferent ratngs then there are no local axa nsde the Dce sphere. We proceed to fndng local axa on the surface of the Dce sphere. Applyng the ethod of Lagrange Multplers as before we end up havng to solve the followng syste of equatons for varables λ and r for all : p j rj r p 2 = 2λ(r c ) for all and Dce( r) (14) j rj Unfortunately we do not know how to fnd a soluton for ths as the frst set of equatons nvolves r for all. Recall that n secton 4.2 the correspondng equatons for fndng the axa on the Dce surface were each a functon of only one r. Ths lead us to seek an alternatve approach. We wll reduce solvng the optzaton proble under the defnton of E p gven n Equaton 13 to solvng a seres of spler optzaton probles on whch we can effortlessly apply the ethod of Lagrange Multpler. To do so, frst consder the decson verson of our proble: Does there exst a r such that E p( r) V and Dce( r)? Under Equaton 13, havng E p( r) V s equvalent to havng (p V )r 0. Thus the decson verson of our proble s equvalent to solvng the followng axzaton proble and checkng that ts soluton has r 0 for all, Max E p( r) = X (p V )r s.t. Dce( r) (15) Equaton 15 can be solved usng the general approach outlned n Secton 4.1. The gradent E p( r) = 0 ff p V = 0 for all. Thus as long soe te s not prced V, there are no local axa for Equaton 15 nsde the Dce sphere 3. To fnd the axa on the surface of the sphere, we solve the followng syste of equatons, p V = 2λ(r c /) for all and Dce( r) (16) Note that Equaton 16 dffer fro Equaton 10 only by constants so the soluton derved n Secton 4.2 shfted by constants s a soluton for Equaton 16. We get that, r = p V 2λ + c where λ = 1 2 s (p V )2 (1/ 2 1) c2 3 If all te are prced V, all r yeld expected proft V and we could pck any r whch les nsde the Dce sphere.
If for all, r 0, we return a yes for the soluton of the decson proble and otherwse we return no. To fnd an a soluton for the orgnal optzaton proble whch s arbtrarly close to optal, we can do bnary search along a bounded nterval of possble values of E p, checkng the exstences of solutons usng decson verson algorth descrbed above. Let V ax = ax p. The ntal bnary search nterval can be set to [0, V ax] snce E p( r) V ax as the custoer purchases one te per te step. Each bnary search step reduces the search nterval by half and dong ore and ore bnary search steps brngs us closer and closer to the optal soluton for the optzaton proble. Let δ < 1. A soluton whch s wthn dstance V axδ of the optal can be found by perforng log( Vax ) bnary search steps. For exaple, let r δ denote the optal soluton. Wth δ = 1/V ax, we can obtan an approxate soluton r a such that E p( r a) + 1 E p( r ) n log( Vax ) = log(v 2 δ ax) = O(log V ax) bnary search steps. roft Gans. Thus we are able to fnd a near optal soluton to the optzaton proble wth E p( r) as defned n Equaton 13 by solvng a seres of optzaton probles of the knd solved n Secton 4.2. The proft gans analyss fro secton 4.2 extends to the last yes soluton obtaned for a decson proble. However as ths yes soluton s near optal for the orgnal optzaton proble, the proft gans wll be close to that fro secton 4.2. 5. CONCLUSIONS AND FUTURE WORK Tradtonal recoendaton systes do not drectly ncorporate the proftablty of tes nto ts recoendatons and n ths work we propose one of the frst drect ethods to do so. Our approach axzes vendor proft whle provdng trustworthy recoendatons for the custoer. Whle our approach s sple, ts splcty allows t to be used n conjuncton wth any tradtonal recoendaton syste. Our ethod s also tunable and allows the vendor to control how uch the proft based recoendaton should devate fro the tradtonal recoendaton. Our work s a startng pont and we hope t wll sulate new research on ncorporatng profts nto recoendaton systes. The an future drecton for ths work s conduct user studes to verfy our assupton that the Dce coeffcent s a sutable easure of custoer trust. It would also be useful to analyze how the custoer s trust s affected sall devatons fro tradtonal recoendatons systes. 6. REFERENCES [1] Adoavcus, G., and Tuzhln, A. Toward the next generaton of recoender systes: a survey of the state-of-the-art and possble extensons. In IEEE Trans. Know. Data. Eng. (2005), IEEE, pp. 734 749. [2] Aatran, X., ujol, J., and Olver, N. I lke t... lke t not: Evaluatng user ratngs nose n recoender systes. In roceedngs of the Conference on User Modelng, Adaptaton, and ersonalzaton (2009). [3] Berry, M., and Browne, M. Lecture Notes n Data Mnng. World Scentfc, 2006. [4] Can, G., and Herod, J. Multvarable Calculus. www.ath.gatech.edu/ can/notes/calculus.htl, 1997. [5] Chen, L., Hsu, F., Chen, M., and Y., H. Developng recoender systes wth the consderaton of product proftablty for sellers. In Inforaton Scences (2008), Elsever, pp. 1032 1048. [6] Chen,., and Wu, S. How does collaboratve flterng technology pact sales? eprcal evdence fro aazon.co. workng paper. Avalable at SSRN: http://ssrn.co/abstract=1002698, 2007. [7] Cho, Y., and K, J. Applcatons of web usuage and product taxonoy to collaboratve recoendatons n e-coerce. In Expert systes wth Applcatons (2004), Elsever, pp. 233 246. [8] Hll, W., Stead, L., Rosensten, M., and Furnas, G. Recoendng and evaluatng choces n a vrtual county of use. In roceedngs of ACM CHI 95 Conference on Huan Factors n Coputng Systes (1995), ACM ress, pp. 194 201. [9] J.S., B., Heckeran, D., and C, K. Eprcal analyss of predctve algorths for collaboratve flterng. In roceedngs of the 14th Conference of Uncertanty n AI (1998), Morgan Kaufann ublshers, pp. 42 52. [10] Kaufan, L., and Rousseeuw,. Fndng Groups n Data, An Introducton to Cluster Analyss. Wley, New York, 1990. [11] Lnden, G., B., S., and York, J. Aazon.co recoendatons:te-to-te collaboratve flterng. In IEEE Internet Coputng (2003), IEEE, pp. 76 80. [12] Netflx, Inc. Netflx prze., Dec 2008. [13] Resnck,., and Varan, H. R. Recoender systes. In Councaton of ACM (1997), ACM ress, pp. 56 58. [14] Sarwar, B., Karyps, G., Konstan, J., and Redl, J. Analyss of recoendaton algorths for e-coerce. In roceedngs of the 3rd ACM Conference on Electronc Coerce (2000), ACM ress, pp. 158 167. [15] Schafer, J., Konstan, J., and Redl, J. Recoender systes n e-coerce. In roceedngs of the 2nd ACM Conference on Electronc Coerce (1999), ACM ress, pp. 158 166. [16] Stewart, J. Multvarable Calculus 5e (Ffth Edton). Thopson-Brooks/Cole, Canada, 2003. [17] Strehl, A., and Ghosh, J. Value-based custoer groupng fro large retal data-sets. In roceedngs of SIE Conference on Data Mnng and Knowledge Dscovery (2000), pp. 33 42. [18] Wharton. Renforcng the blockbuster nature of eda: The pact of onlne recoender, Dec 2008.