Optmzng Merchant Revenue wth Rebates Rakesh Agrawal Search Labs Mcrosoft Research rakesha@mcrosoft.com Samuel Ieong Search Labs Mcrosoft Research saeong@mcrosoft.com Raja Velu School of Management Syracuse Unversty rpvelu@syr.edu ABSTRACT We study an onlne advertsng model n whch the merchant remburses a porton of the transacted amount to the customer n a form of rebate. The customer referral and the rebate transfer mght be medated by a search engne. We nvestgate how the merchants can set rebate rates across dfferent products to maxmze ther revenue. We consder two wdely used demand models n economcs lnear and log-lnear and explan how the effects of rebates can be ncorporated n these models. Treatng the parameters estmated as nputs to a revenue maxmzaton problem, we develop convex optmzaton formulatons of the problem and combnatoral algorthms for solvng them. We valdate our modelng assumptons usng real transacton data. We conduct an extensve smulaton study to evaluate the performance of our approach on maxmzng revenue, and found that t generates sgnfcantly hgher revenues for merchants compared to other rebate strateges. The rebate rates selected are extremely close to the optmal rates selected n hndsght. Categores and Subject Descrptors J.4 [Socal and Behavoral Scences]: Economcs; F.2.0 [Analyss of Algorthms and Problem Complexty]: General General Terms Algorthms, Economcs Keywords Internet advertsng, Rebates 1. INTRODUCTION Sponsored search, where merchants pay search engnes for dsplayng ther advertsements or for redrectng traffc Work done whle author was vstng Search Labs. Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. WSDM 11, February 9 12, 2011, Hong Kong, Chna. Copyrght 2011 ACM 978-1-4503-0493-1/11/02...$10.00. to ther webstes, has played a key role n makng nternet search ubqutouson the web. Underthepay per clck model of sponsored search, merchants pay for clcks that brng users to ther webste. An ncreasngly popular varant s the pay per acton model, where merchants pay the search engne only when users take some acton such as placng an order at the webste. See [10, 13, 16, 20] for concept, hstory, and analyss of sponsored search. Cashback, an expermental program ntroduced by Mcrosoft n May 2008, added a novel twst to the pay per acton model. Merchants frst select products that they would lke to advertse. They also select the referral commsson they are wllng to pay per transacton as a fracton of the sales prce. As n pay per acton, merchants only pay the search engne when a transacton occurs. But the search engne then returns the commsson to the consumer n the form of a cash rebate. Ths program can thus be vewed as one where merchants offer rebates on ther products, facltated by the search engne. In effect, when a user searches for a product and buys t through a partcpatng merchant, the user receves a percentage of the amount she pays as rebate. 1 There has been lttle work on analyzng Cashback as an advertsng program. An excepton s [6], where Chen et al. proposed several dfferent revenue sharng mechansms that are remnscent of Cashback, and compare ther revenue propertes from the search engne s perspectve. In ths paper, we adopt the merchant s perspectve, and examne the queston of how merchants can best use rebates to maxmze ther revenue. Assumng that the search engne passes on the whole of commsson receved from a merchant for a transacton to the customer as rebate, the problem of determnng how much commsson should a merchant pay for a transacton becomes the same as computng optmum rebate. 2 We thus address the problem of selectng rebate rates to maxmze revenue subject to a budget constrant, takng nto account that a merchant may carry more than one product lne, and that the effects of rebates on dfferent product lnes may dffer. Our approach s to tackle ths problem n two steps. Frst, 1 The program has been dscontnued as of July 2010. The study of the queston of how to select rebate rates remans nterestng nonetheless from a research vewpont. There are also other webstes where merchants can offer rebates to onlne shoppers, such as ebates.com and fatwallet.com. 2 It s straghtforward to extend our work to the case where the search engne charges a fxed amount of admnstratve fee for the rebate processng servce or charges a fee whch s some fracton of the amount of rebate handled.
for dfferent products, we estmate the senstvty of demand to changes n prces and rebates usng hstorcal data. Second, treatng the values estmated n the prevous step as parameters, we formulate an optmzaton program to fnd the best rebate rate for every product. We consder two mportant economc demand models [18] lnear demand and log-lnear demand and show how rebates can be ncorporated nto these models. We also address the algorthmc queston of revenue maxmzaton under both demand models. Whle the two demand models gve rse to dfferent optmzaton problems, we provde effcent algorthms to solve both problems. We evaluate our approach usng both real and synthetc transacton data. We frst valdate our modelng assumptons wth transacton data obtaned from Mcrosoft Cashback operatons. For evaluaton, as we do not have drect access to merchants, we conduct an extensve smulaton study usng synthetc transacton data, and compare our proposed soluton to a number of rebate selecton heurstcs on the amount of revenue each method generates. Our approach outperforms all tested heurstcs n almost all of the cases. The rest of the paper s organzed as follows. We dscuss related work from the economc and marketng lterature n Secton 2. We state our problem formulaton n Secton 3. In Secton 4, we explan how the lnear and the log-lnear demand model can be extended to estmate demand senstvty to rebate rates. In Secton 5, we show how to solve the rebate optmzaton problem effcently under both demand models. In Secton 6, we valdate our modelng assumptons usng real transacton data collected over a year. In Secton 7, we conduct an extensve smulaton study usng a synthetc data generaton process desgned to mmc real data, and evaluate the end-to-end performance of our approach to maxmzng revenue. We conclude wth the key fndngs and drectons for future work n Secton 8. 2. RELATED WORK Merchants often choose between prce cuts and rebates for stmulatng sales [9, 17, 19]. The key dfference between these two tactcs s that prce cuts offer dscounts up front whereas rebates offer dscounts after the product s purchased at the regular prce. Ths dfference leads to a phenomena known as slppage, where some rebates are not redeemed [11]. In the context of Cashback, the rebate credt to the customer collecton process s automatc, hence slppage n ts tradtonal sense s a non-ssue. Nonetheless, userswat tocollect therebatescreatngafloatforthesearch engne. There has been work on fndng rebate polces that maxmze profts by Arcelus et al. [1, 2, 3] and Khouja et al. [14, 15]. Ther work focuses on settngs where merchants can choose to set the sales prce, the rebate value, and the order sze (for managng nventory). Our work s dfferent n several ways. Frst, n our settng, the merchant carres multple products, and the optmal rebates may dffer for each product. Second, the sze of the rebate program s governed by a budget that plays a central role n determnng the sze of the advertsng program. On the other hand, we do not model nventory holdng costs and treat prces as gven, and focus on the selecton of rebate rates. Past work on modelng the relatonshp of demand and rebates s dscussed further n Secton 4. 3. PROBLEM SETUP Let the products a merchant sells be P = {1,2,...,n}, and the budget for rebates be b. The relatonshp among demand, prce, and rebates s product-dependent. For product, let ts prce be p, ts rebate rate be r, and ts demand, whch depends on p and r, be q (p,r ). We defne revenue as the net proceeds the merchant receves,.e., the gross revenue from the products sold less the rebates pad. Our objectve s to fnd rebate rates, one for each product, that maxmzes revenue wthout exceedng the budget. Prces are treated as nput parameter to the problem. The rebate optmzaton problem can be stated as follows. 3 max r subject to q (p,r )p (1 r ) q (p,r )p r b 0 r 1 P Budgets are central to the problem formulaton, as they are nstrumental n controllng exposure rsk n sponsored search [16]. Even though n the long run, a budget may be effectvely unlmted as long as a merchant s makng a proft, n the short run, t s needed to balance the allocaton of captal across dfferent operatons. Hence, we treat the budget as gven to the optmzaton. The presence of a budget presents a trade-off between offerng more rebates on one product versus another. It also requres careful plannng of rebates so as not to run out of budget before the end of an advertsng campagn. In the (unlkely) scenaro where a merchant s not constraned by a budget, selectng rebate rates can be smplfed to selectng the optmal rate for each product ndvdually. The above formulaton treats prces as nput parameters, and tactly assumes that prces stay unchanged durng the rebate program. Gven that rebate programs often run for short duraton, ths assumpton s realstc. In stuatons where prces may change, one can rerun the optmzaton and adjust the rebate rates accordngly. The formulaton also assumes that there s no constrant on supply for each product. Ths assumpton holds for dgtal goods [12], or when supply s large compared to demand. Lmts on supply can be modeled by addng constrants of the form q (p,r ) S to the optmzaton problem (1). Our results generalze to ths settng. Ths work assumes that the demand of a product can be estmated ndependently of prces and rebates offered by other products, and leaves the general case where prces and rebates of dfferent products may nteract to future work. To nstantate the optmzaton problem, the demand functon q (p,r ) s estmated usng hstorcal data. Ths ntroduces uncertanty n the underlyng problem. Our results drectly apply f one s nterested n maxmzng expected revenue whle satsfyng the constrant n expectaton. When the constrant has to be satsfed wth hgh probablty, one wll need to extend proposed technques employng deas from stochastc programmng [22]. 3 Our formulaton and optmzaton technque allows one to specfy lmts on the mnmum and maxmum rebate rate for each product; ths s useful when a merchant s runnng a targeted campagn. We keep the lmts to be 0 and 1 for ease of exposton. (1)
4. ESTIMATING DEMAND SENSITIVITY To determne the rebates to offer, we start wth estmatng a relatonshp among prces, rebates, and quantty of goods sold (demand). We have chosen to treat prces (p) and rebates (r) as two separate varables rather than treatng them as a net prce varable (p(1 r)) for two reasons. Frst, consumers often consder the value of a dollar rebate to be dfferent from a dollar dscount. Second, n our experments n Secton 6, models that treat the two varables separately ft the data much better. A smple yet wdely-used demand model s the lnear demand model [18]. It has been extended to treat prce and rebate separately n [3, 14, 15]. q = β 0 +β 1p+β 2pr+ɛ (2) where β 0, β 1, β 2 are parametersof themodel tobeestmated and ɛ s random nose. Gven hstorcal sales data of the form (q,p,r), we can estmate the parameters of the model usng lnear regresson. We expect the coeffcent β 1 to be negatve and β 2 to be postve, consstent wth the expected propertes of a demand curve. Another model we consder s the log-lnear demand model [18]. It s an mportant and well-studed model n economcs due to ts nterpretablty, though t has been examned less n the context of rebates, the excepton beng [3]. To treat rebates separate from prces, we extend the model as follows. logq = γ 0 +γ 1logp+γ 2log(1 r)+ɛ (3) where γ 0, γ 1, γ 2 are parameters of the model to be estmated and ɛ s random nose. As n the case of the lnear demand model, we can learn the parameters of the model from hstorcal data usng lnear regresson. In the economc lterature, the coeffcent γ 1 has been nterpreted as the prce elastcty of demand, and the coeffcent γ 2 as the rebate elastcty of demand. For our model, both of these coeffcents are expected to be negatve, correspondng to the expectaton that demand ncreases when prce decreases or rebate ncreases. The descrpton above makes the smplfyng assumpton that demand relatonshp remans unchanged for the duraton of the analyss. As rebate programs are often short, ths assumpton s reasonable. If demand relatonshp may shft over tme, one may use deas from tme seres analyss to model the tme-dependent effects [4]. The central dea s to augment the basc model wth tme, and express demand as a functon of prce, rebate rates, and past demand. Our technques also apply to such tme-varant models; the lmtng factor s whether one has suffcent data. 5. OPTIMIZING REBATE RATES Treatng the coeffcents of demand senstvty as parameters, we now dscuss the queston of how to select the optmal rebate rates. Dependng on the demand model used, the soluton to the optmzaton problem requres dfferent technques. In the followng, we are gong to consder the lnear demand model and the log-lnear demand model separately. 5.1 Lnear Demand Model Under the lnear demand model (Equaton (2)), the optmzaton program (1) becomes: max r subject to ( ) β 0, +β 1,p +β 2,(p r ) p (1 r ) ( ) β 0, +β 1,p +β 2,(p r ) p r b 0 r 1 P The decson varables n ths optmzaton are the rebate rates r. The prces p, the demand senstvtes (β 0,, β 1,, β 2,), and the budget b, are all nputs to the problem. To solve ths optmzaton problem, we start wth some preprocessng. Note that for any product for whch β 2, 0, one should not offer any rebates. We thus set r = 0 for these products and remove them from further consderaton. After ths step, remanng products have β 2, > 0. Our problem has both a quadratc objectve and a quadratc constrant. We can formulate the problem as a (convex) quadratcally constraned quadratc program (QCQP), whch can be solved n polynomal tme usng nteror pont methods [5]. There are also effcent off-the-shelf solvers for the problem [7]. Theorem 1. Selectng rebate rates that maxmze revenue under the lnear demand model can be solved n polynomal tme va a convex QCQP. Proof. The objectve s quadratc n r and the budget constrant s quadratc n r, hence we need to verfy that the optmzaton s a convex one,.e., the objectve s concave (snce t s maxmzed) and the constrants are convex [5]. Recall all products wth negatve β 2, have been removed durng preprocessng. The objectve can be expressed as a sum of quadratc and lnear terms n r s. The coeffcent to the quadratc terms are β 2,p 2, hence the ndvdual quadratc terms are concave, and the overall functon s concave. Smlarly, the budget constrant can be expressed as a sum of quadratc and lnear terms n r s. The coeffcents to the quadratc terms are β 2,p 2, hence they are convex and the constrant s convex. The other constrants are lnear and hence also convex. Next, we wll develop a combnatoral algorthm for solvng the optmzaton n the case of log-lnear demand model. The technque can be adapted to lnear demand model as well. However, gven the speed of exstng solvers for convex QCQP, a combnatoral algorthm for the lnear demand case mght not be needed. 5.2 Log-lnear Demand Model We start by re-wrtng the log-lnear demand model(equaton (3)) as follows. logq = γ 0 +γ 1logp+γ 2log(1 r) q = exp(γ 0 +γ 1logp)exp(log((1 r) γ 2 )) = c(1 r) γ 2 where c s a postve number ndependent of rebate rate r. As the coeffcents of the demand functon (γ 0, γ 1, γ 2) and prces p are nput toour optmzaton, we can compute c and treat t as part of the nput as well. Note that each product wll have ts own constant c. Rewrtng our optmzaton
program (1), we have: max c r (1 r ) γ 2, p (1 r ) subject to c (1 r ) γ 2, p r b (4) 0 r 1 P To solve ths optmzaton problem, smlar to the case for lnear demand model, we start by removng products for whch offerng rebates do not mprove revenue. In ths case, for any product, f γ 2, 1, we should set r = 0. We can verfy ths as follows. Denote the objectve functon of Eq. (4) by f(r). Takng the partal dervatve of f wth respect to r, f r = c p (γ 2, +1)(1 r ) γ 2,, (5) whch s non-postve when γ 2, 1. One arrves at the same concluson by nterpretng γ 2, as the rebate elastcty: when t s at least 1, a unt ncrease n rebates wll generate at most a unt return n gross revenue, hence offerng rebates do not mprove net revenue. After preprocessng, all remanng products have γ 2, < 1. Unlke the case for lnear demand model, however, after preprocessng, the resultng problem s a non-convex optmzaton problem. Ths s because we are maxmzng an objectve that s not necessarly concave. Indeed, evaluatng f the partal dervatve n Eq. (5), when γ 2, < 1, r > 0; the objectve s actually convex rather than concave! However, by a careful change of varables, one can fnd an equvalent optmzaton problem that s convex. Theorem 2. Selectng rebate rates that maxmze revenue under the log-lnear demand model can be (re)formulated as a convex optmzaton problem. Proof. For all P, let x = (1 r ) γ 2,+1. Rewrtng rebate rate r n terms of x, r = 1 x 1/(γ 2,+1). Changng the optmzaton varables n the problem from r to x, and makng the substtuton to Eq. (4), we have max p x c x ( ) subject to p c x γ 2,/(γ 2, +1) x b (6) x 1 P To verfy that ths problem s convex, frst, note that the objectve functon s lnear n x, hence t s concave. Denote the budget constrant by g(x). Takng the partal dervatve of g wth respect to x, g x = p c ( 1 γ 2, γ 2, +1 x γ 2, +1 1 g When γ 2, < 1, x > 0, hence g s a convex functon, and the problem s a convex optmzaton problem. Snce the optmzaton problem can be formulated as a convex program, n theory t can be solved n polynomal tme usng nteror pont method. However, unlke convex ) Algorthm 1: Prmal-dual algorthm for (6) forall the P do x 1 S {} Sort(h (x )) // Ensures h 1(x 1) h P (x P ) for 1 to P do Add to S forall the j S do Increase all x j whle mantanng the nvarant h 1(x 1) = = h (x ) = λ untl (C1) λ = h +1(x +1), or (C2) S g(x) = b. f (C1) then Break f (C2) then Termnate end end QCQP, off-the-shelf solvers are slow for general convex programs. Hence, we develop a prmal-dual-based combnatoral algorthm for t. The detals are descrbed n Algorthm 1. 4 The correctness of the algorthm s based on two techncal lemmas. We state the two lemmas here. Let f (x ) = p c x, g (x ) = p c ( x γ 2,/(γ 2, +1) x ), h (x ) = f(y) x / g(y) x = 1 γ 2, γ 2, +1 x 1/(γ 2,+1) 1. Lemma 1. A feasble soluton x s optmal for (6) f and only f there exsts a constant c such that For P where x > 1, h (x ) = c; For j P where x j = 1, h j(x j) c. Lemma 2. The functon h (x ) s non-negatve and monotoncally decreasng n x. Theorem 3. Algorthm 1 fnds the optmal x to optmzaton problem (6) n O(n 2 ) + T tme, where T s the tme needed to solve for the root of a unvarate polynomal equaton, and s bounded by the parameters of the problem. Proof. One can verfy that g x > 0. Hence, g(x) ncreases as the x s ncrease. Condton (C2) wll eventually be satsfed and the algorthm always termnates. Because of Lemma 2, for each j, as x j ncrease, the value h j(x j) decreases, hence t s possble to mantan the nvarant. At termnaton, the condtons to Lemma 1 wll be satsfed, snce for all,j S, h (x ) = h j(x j) = λ, and for k S, x k = 1, and h (x ) h k (x k ) due to sortng. Hence the algorthm fnds the optmal soluton. In actual mplementaton, one does not ncrease x j contnuously; the descrpton s only for ntuton. Instead, one checks for condtons (C1) and (C2) dscretely. Let λ = h (x ). In the outer loop, whle the budget s not exceeded, we add to S, and solve for h 1(x 1) = = h (x ) = λ +1. 4 An alternatve approach, suggested by an anonymous revewer, stoperformthesubsttutonx = p c r (1 r ) γ 2, /b to Eq. (4), whch gves rse to optmzng an convex objectve over a smplex constrant wth respect to x. Off-the-shelf solvers run more effcently for ths type of problems.
The moment the budget s exceeded, we know that we cannot add another product to S. Note that x can be expressed as a functon of λ,.e., x = h 1 (λ). Ths nverse s well-defned snce h s monotoncally non-decreasng n x. Hence we solve a polynomal equaton n one varable, λ, S g(h 1 (λ)) = b. The LHS s monotonc n λ, hence ts root can be found by bnary search. The number of steps s bounded by parameters to the optmzaton problem. Note that only one equaton needs to be solved. The rest of the algorthm s bounded by O(n 2 ), as there s a maxmum of n loops, each of whch takes O(n) tme. 6. MODEL VALIDATION A central assumpton n our demand model s that potental customers do not value prce and rebates equally, and hence the effects of these varables on demand should be treated separately. We valdate ths assumpton usng real transacton data n ths secton. 6.1 Transacton Data We obtaned transacton data from Mcrosoft Cashback operatons over a year. We grouped the transactons frst by merchant, and then by product. We randomly sampled 40 thousand such groups of merchant-product pars, constranng to pars for whch the product was sold at least 5 tmes, and for whch the unts sold were not dentcal for all days. We retreved all transactons for the selected pars. Ths samplng process results n about 3 mllon transactons for evaluaton. Each row of data descrbes one transacton, and ncludes nformaton such as the merchant, the product, the date of sales, the prce, the rebate rate, and the number of unts sold. Ths provdes the nput to our demand estmaton experments dscussed next. 6.2 Demand Estmaton Evaluaton For each merchant-product par, we compute for each day the average prce, the average rebate, and the number of unts sold for each group of transactons, and run regresson to estmate the parameters to four models. The frst two models are the lnear and the log-lnear models presented n Secton 4. They model prce and rebates separately. The other two models, the lnear net-prce (Lnear-NP) and the log-lnear net-prce (Log-lnear-NP), are the control models that only use the net prces (computed as prces tmes one mnus rebate rates) to model demand. If our assumpton s vald, we should see an ncrease n the explanatory powers of the frst two models over the latter two. To measure the explanatory powers of the models, we use the coeffcent of determnaton (commonly known as R 2 ), whch measures the proporton of the varablty n the observatons accounted for by the statstcal model [21]. The value les between 0 and 1, and a hgher value suggests a better ft. Snce the models have dfferent number of ndependent varables, we also compute the adjusted R 2 of the models. 5 Adjusted R 2 takes nto account the dfference n the number of ndependent varables of the models, and a hgher value suggests that the mprovement of explanatory power due to the addtonal varable(s) cannot be explaned by chance alone. The results are reported n Table 1. (n 1) 5 Adjusted R 2 = 1 (1 R 2 ), where n s the number of samples and m s the number of ndependent var- (n m 1) ables [21]. Model R 2 Adjusted R 2 Lnear 0.4379 0.2292 Log-lnear 0.4420 0.2348 Lnear-NP 0.2827 0.1730 Log-lnear-NP 0.2898 0.1810 Table 1: R 2 and adjusted R 2 on transacton data. Both the R 2 and the adjusted R 2 values of the models are sgnfcantly hgher when prces and rebates are modeled as two separate varables (Lnear and Log-lnear) than when they are modeled as net prces (Lnear-NP and Loglnear-NP). Ths ndcates that potental customers under real transacton condtons do not value prce and rebates equally. It valdates our demand modelng assumptons, and renforces the mportance of treatng rebate selecton as a dfferent problem from prce selecton. 7. SIMULATION STUDY Our objectve s to develop an approach that the merchants can use to maxmze ther revenue. We conducted two sets of experments. In the frst set of experments, we consder the case of a sngle product, and evaluate f our approach fnds good rebate rates that both satsfy the budget and maxmze revenue, and whether t s better than the alternatves consdered. We also study the robustness of our approach under varous parameter settngs. In the second set of experments, we consder the case of multple products. We nvestgate whether our approach can dscover the product that s more senstve to rebates, and whether t can fnd the rght trade-off between rebates offered on one and the other. Ideally we would lke to evaluate our approach wth real operatons, but we do not have drect access to merchants. Hstorcal data cannot help wth evaluatng the effcacy of rebate selecton as such evaluaton requres counterfactual changes to the rebates offered. To crcumvent ths dffculty, we desgn a synthetc data generator that ams to mmc real transactons, and evaluate our approach usng smulaton. 7.1 Synthetc Transacton Generator To better understand the characterstcs of real transactons, we examned the sample transactons used for model valdaton n Secton 6. We observed that n most transactons, only a sngle unt of product s sold. Ths suggests that most potental customers face a dscrete choce gven the prce and rebate, whether to purchase the product or not. Hence, we adopt the followng process for transacton generaton. For each day, the number of potental customers of a merchant for a gven product, referred to as traffc henceforth, s drawn accordng to a Posson dstrbuton, parameterzed by µ, the average traffc per day. We note that the traffc to a webste has also been modeled as a Posson dstrbuton n [23]. Each potental customer faces a bnary choce of whether to purchase the product. Followng the dscrete choce lterature [24], the decson s modeled usng a bnary logt functon; for prce p and rebate rate r, the probablty t that the potental customer s gong to purchase the product
s gven by t = 1 1+exp( (α 0 +α 1p+fα 1pr)), (7) where α 0, α 1, and f are parameters specfc to the product. The probablty t can be nterpreted as the converson rate of the merchant. The parameter α 1 captures how senstve the potental customers are to prce changes. The parameter f captures the relatve value of rebate to prce. The parameter α 0 can be vewed as an offset that helps determne the converson rate; n our experments, we vary the mnmum converson rate of the merchant and compute the correspondng value for α 0. To determne f a potental customer makes a purchase, a number s drawn unformly at random between 0 and 1. If ts value s less than t, the product s bought. We refer ths sample to be the potental customer s deal-seekng atttude, as a hgher value means the person s seekng for a better deal (lower prces or hgher rebates). Fnally, the number of unts sold per day s obtaned by aggregatng over all traffc. Ths synthetc transacton generator s desgned to mmc how transactons take place for merchants. The key assumpton s based on the dscrete choce process, supported n the economc lterature [18, 24]. We beleve t does not create a bas that favors our proposed approach; our approach uses only transacton data, and t s unaware of how traffc s generated, as well as the dscrete choce process underlyng the decsons of the potental customers. 7.2 Expermental Setup We want to conduct an end-to-end evaluaton of our approach, startng from demand estmaton and endng wth measurng the revenues generated based on rebate optmzaton. Therefore, n each tral, we frst generate transacton data over some pre-determned prce and rebate ranges, correspondng to a perod durng whch a merchant s learnng the demand relatonshp. The data s then ftted to the lnear and the log-lnear models and the parameters are estmated. Then, over a of 12-week evaluaton perod, gven a budget parameter b, the merchant fxes the prce of the product at prce p and selects a rebate rate ether based on our optmzaton routnes or some other heurstcs. The merchant offers the sad rebate untl budget runs out, after whch zero rebates are offered for the remanng perod. We measure the revenue generated both over the duraton of the rebate program, and over the entre 12 weeks, averaged over 500 trals for each experment. The former whle rebate lasts (WRL) scenaro s approprate when a merchant s requred to offer rebates to partcpate n the program, whereas the latter entre duraton (ED) scenaro s approprate when that s not the case. The revenue under ED s at least as hgh as the revenue under WRL, and strctly hgher when budget s exhausted due to potental customers that purchase at zero rebates. Revenue s senstve to both traffc and the deal-seekng atttudes of the potental customers. To control for ths varablty, nstead of runnng the smulaton ndependently for each rebate program, we couple the smulatons together: for each tral, we sample one set of potental customers along wth ther atttudes, and evaluate all rebate programs wth respect to them. Any dfference n revenue s therefore due only to the choce of rebates, but not due to dfferences n traffc or the atttudes of the potental customers. Gven a set of potental customers, we can compute the optmal rebate rate (n hndsght) that would maxmze revenue for ths specfc nstance. Of course, ths revenue cannot be acheved n realty as t requres foreknowledge of the number of potental customers and ther atttudes, but t can serve as an nstance upper bound for each tral. We refer ths upper bound as the optmal revenue. 7.2.1 Alternatve Approaches to Selectng Rebates We compare our approach to the followng heurstcs. The frst two heurstcs are selected due to ther popularty n Cashback data. The last heurstc, motvated by feedback control, tres to adapt to demand patterns, and consttute a compettve baselne for comparson. 1. Fxed rates (Fx-r). Fx rebate rate at r for the entre perod, untl budget runs out. In our experment, we try three popular rates 5%, 10%, and 15%. Note that our approach also selects a fxed rate, although the rate s determned algorthmcally to optmze revenue. 2. H-Lo. A merchant alternates between offerng hgh rebate rates and low ones. In our experment, the hgh rate s set at 15% and the low at 5%, and the merchant changes the rebate rate every week. 3. Adaptve. A merchant changes the rebate rates dependng on the remanng budget. When the remanng budget s hgher than expected, the rates are ncreased; f lower, they are decreased. In our experment, we start wth a rate of 5% and adjust the rates multplcatvely by a factor of 1.5 when the budget fals to track by more than 10%. These parameters were chosen after some basc tunng and appear to do well n smulaton. 7.2.2 Performance Metrc Our metrc for evaluatng the dfferent approaches to selectng rebates s the % of optmal revenue acheved. The revenues acheved under both WRL and ED scenaros are measured. Naturally, the hgher the value for ths metrc, the better the approach. When we evaluate our proposed approach, we wll also examne the rebate rates selected accordng to the lnear and the log-lnear models for each tral, and compare them to the optmal rebate rates. Ths helps to measure how close we are to the optmal choce. 7.2.3 Smulaton Parameters There are altogether fve parameters that we vary n our smulaton study. Four of these parameters, µ, α 0, α 1, and f, govern the synthetc transacton generaton process. As mentoned, we do not explctly select α 0, but rather determne α 0 based on the mnmum converson rate, t mn, whch we vary n our experments. The parameter t mn corresponds to the expected converson rate when a merchant selects the default prce and offers zero rebate for the product. The ffth parameter s the budget b, whch determnes how much rebates are avalable durng the evaluaton perod. These parameters, along wth ther default values and the ranges wth whch we have expermented, are summarzed n Table 2. Throughout the study, prce p s set at 100. Ths s wthout loss of generalty, as an ncrease n prce can be mapped to a correspondng decrease n α 1 and n budget b. Hence, varaton n prce s mplctly tested when we vary the parameters α 1 and b.
Parameter Default Mn Max µ (traffc) 100 10 400 α 1 (prce senstvty) 0.08 0.06 0.10 f (relatve value of rebate) 0.8 0.4 1.2 t mn (mn converson rate) 0.04 0.005 0.08 b (budget) 5, 000 2, 500 10, 000 Table 2: Summary of smulaton parameters. Program % of optmal revenue (standard devaton) whle rebate lasts (WRL) entre duraton (ED) Fx-5% 83.1 (2.0) 82.7 (2.0) 6 Fx-10% 86.4 (2.5) 98.0 (1.0) Fx-15% 54.4 (1.6) 92.5 (1.8) H-Lo 74.7 (2.7) 94.4 (1.7) Adaptve 92.6 (2.4) 92.4 (2.4) Lnear 97.8 (1.5) 97.6 (1.6) Log-lnear 93.3 (4.2) 98.8 (0.9) Table 3: % of optmal revenue acheved by dfferent rebate programs under WRL and ED. The default values for the parameters α 1, f, and t mn are chosen to gve rse to realstc converson rates. Under the range of parameter values consdered, dependng on the rebates offered, the converson rates for the product fall n the range of 0.5% to 20%. These values are wthn ranges observed n onlne marketng[8]. The default value for traffc s set to 100. Ths selecton s based on an educated guess, snce ths number could not be relably estmated from the Cashback data. Consequently, we vary ts value n a wde range as we test for senstvty. Whenwe varytraffc, we also vary budget at the same tme to keep the average amount spent per potental customer constant. The default value for budget s set to be 5,000, correspondng to spendng about 50 cents per potental customer, or $5 per converson, assumng a 10% converson rate (for a product sold at $100). We beleve ths value s a realstc estmate of the amount merchants are wllng to pay per transacton. 7.3 Experment 1: Sngle-Product Case Under the default parameter values to the smulaton, the average fractons of optmal revenue acheved (and ther standard devatons) for the dfferent rebate approaches are presented n Table 3. Optmzaton under the lnear and the log-lnear models are respectvely the best methods under WRL and ED, and acheves close to 100% of the optmal revenue. These are very strong performance numbers, especally consderng that the optmal revenues are determned wth the beneft of hndsght. The dfferences compared to other rebate programs are statstcally sgnfcant n both cases under pared t-tests (wth p-values < 0.0001). Examnng the results closer, we fnd two trends. On the one hand, rebate programs that use up more of the budget generally acheve hgher revenue. On the other hand, f bud- 6 Note that the optmal revenue depends on the scenaro, and s hgher under ED. Hence, we see that the fracton of optmal revenue acheved for Fxed-5% s lower under ED than under WRL even though ts revenue stays the same. get s exhausted too early, less revenue s acheved. Indeed, the strong performance of our approach can be attrbuted to the ablty to balance the two factors. The mportance of balancng these factors can be llustrated usng the three fxed-rate programs, Fx-5%, Fx-10%, and Fx-15%. At a rebate rate of 5%, there s leftover budget, and one can mprove revenue by settng a hgher rate. At rebate rates of 10% and 15%, budget s fully exhausted. Comparng the two, we see that a rebate rate of 10% does better as t exhausts the budget later than a rebate rate of 15%. At the default parameter values, the average optmal rebate rate s 8.7%. The average rebate rates computed usng the lnear model and the log-lnear models are 8.4% and 9.3% respectvely. They are both close to the optmal rates. The lnear model performs better than the log-lnear model under WRL as t does not exhaust the budget before the end. On the other hand, under ED, as revenue stll counts after budget s exhausted and the merchant starts offerng zero rebates, the log-lnear model performs better as t puts the entre budget to work. 7.3.1 Senstvty Analyss We next vary each of smulaton parameters (holdng others to ther default values) to understand ther nfluence on the performance of our approach. The results are presented n Fgure 1. In ths fgure, the panels on left rght show the rebate rates computed by our approach usng the lnear and the log-lnear demand models, along wth the optmal rates, as a functon of the parameter value. The panels on the left show the % of optmal revenue acheved. We have shown revenue plots only for the lnear model under the WRL scenaro and for the log-lnear model under the ED scenaro. It s because the lnear (resp. log-lnear) model consstently outperformed the log-lnear (resp. lnear) model under the WRL (resp. ED) scenaro. Smlarly, we only show plots for our approach snce t outperformed the alternatves n more than 90% of cases. We observe the followng wth respect to the rebate rates: 1. Overall, the rebate rates selected by our approach are very close to the optmal rates, wth an average dfference of less than 0.5%. Comparng the rebate rates chosen, the lnear model tends to pck rates slghtly below optmal, whereas the log-lnear model pcks ones slghtly above. 2. When the traffc s very small (µ =10), the probablty that there s no transacton for an entre day s 67%. Ths mssng data problem manfests tself n adversely affectng demand estmaton. The resultant error n the regresson coeffcents lead to the suboptmal values of rebate rate. Computed rebate rates start trackng the optmal wth moderate ncrease n traffc (Fgure 1(a)). 3. As expected, the computed rebate rates decrease as the senstvty of the demand to the prce of the product (parameter α 1) ncreases (Fgure 1(c)). Smlarly, rebate rates decrease as the relatve value of rebate (parameter f) ncreases (Fgure 1(e)). 4. Asmnmumconversonrate(parametert mn)ncreases, the average number of transactons ncreases, and rebate rate has to decrease to match the budget (Fgure 1(g)). The decrease s more sgnfcant for smaller values of t mn due to the nverse proportonal relatonshp between converson rate and rebate rate, whch n turn s due to the budget constrant (number of transactons tmes rebate per transacton must be less than a fxed budget).
Rebate rate senstvty Revenue senstvty (a) Senstvty to µ (b) Senstvty to µ (c) Senstvty to α 1 (d) Senstvty to α 1 (e) Senstvty to f (f) Senstvty to f (g) Senstvty to t mn (h) Senstvty to t mn () Senstvty to b (j) Senstvty to b Fgure 1: Senstvty analyss.
5. As budget ncreases (parameter b) ncreases, the model pcks up larger rebate rates as there s no ncentve to leave the budget unspent (Fgure 1()). We observe the followng wth respect to the revenues: 1. Followng the strategy of usng the lnear model when optmzng revenue under WRL, and adoptng the log-lnear model under ED, one does extremely well, achevng average % of optmal revenue of at least 95% n most cases. Only when the traffc s very small (µ = 10) or when transactons are very rare (t mn = 0.5%), our approach does not acheve 95% of maxmum possble revenue (Fgures 1(b) and (h)). The reasons are due to estmaton errors because of mssng values, and that when transactons are rare, the relatve value of each transacton ncreases, and so mssng out a few transactons becomes more costly. But once we have suffcent data, and transactons are not rare, our approach starts performng at a very hgh level. 2. For the parameters α 1 and f, the performance of our approach s very consstent, wth lttle varaton n % of optmal revenue acheved across all choces of parameters (Fgures 1(d) and (f)). Ths s due to the data-drven nature of our approach. As these parameters vary, the rebate senstvty of the product changes. By leveragng transacton data, our approach dentfes these changes durng demand estmaton and optmze accordngly. 3. The performance of our approach s also very consstent for the budget parameter b. As can be seen n Fgure 1(), despte large varatons n the optmal rates, our approach found rates that are close to the optmal ones for dfferent budget values. Here, the reason s due to the algorthmc nature of our approach. As our approach takes budget as n nput parameter and selects rebate rates through optmzaton, t can adapt well to dfferent budgets well. 7.3.2 Summary Our approaches for selectng rebate rates acheves close to the best possble revenue, and ther performances are consstent across almost all choces of smulaton parameters. Based on the expermental results, the lnear model works better under WRL, and the log-lnear model works better under ED. Both models manage to select rebate rates that are very close to the optmal ones. 7.4 Experment 2: Mult-Product Case In ths experment, we nvestgate whether our approach can dentfy products that are more senstve to rebates and fnd the rght trade-offs amongst the rebates offered for dfferent products. We smulate a stuaton where a merchant carres twoproducts. Thefrstproduct, P1, hasthesame default parameter values as the ones used n the sngle product case. The parameters for the second product, P2, s dentcal n all aspects except for f, ts rebate senstvty. Holdng f for the frst product constant, we vary f of the second product. For the base case, f of P2 s set at 0.4. Ths s smaller than f of P1, whch s 0.8, and hence P2 s less responsve to rebates than P1. The revenues acheved under dfferent approaches are shown n Table 4. 7 When there are two or more products, we do not know how to compute the optmal revenue under ED effcently. Based on our experments on sngle products, the optmal revenue s < 1% hgher than under WRL. Hence we use the optmal revenue under WRL as the approxmate benchmark. Program % of optmal revenue (standard devaton) whle rebate lasts (WRL) entre duraton (ED) 7 Fx-5% 81.4 (1.5) 81.4 (1.5) Fx-10% 91.0 (2.2) 95.5 (1.0) Fx-15% 57.3 (1.4) 91.7 (1.3) H-Lo 80.9 (2.4) 93.1 (1.2) Adaptve 88.9 (3.7) 93.4 (1.5) Lnear 97.1 (1.3) 97.2 (1.4) Log-lnear 92.8 (3.7) 98.6 (0.9) Table 4: % of upper bound on revenue acheved by dfferent rebate programs wth two products. Fgure 2: Rebate rates selected under the two demand models compared to optmal, as the rebate senstvty f of P2 vares. Lke the case of a sngle product, our approach usng the lnear and the log-lnear models acheves the hghest % of optmal revenue respectvely under WRL and ED. The dfferences n performance are statstcally sgnfcant under a pared t-test (wth p-value < 0.0001). The dfferences are larger n ths experment compared to the case wth only a sngle product. Ths s due to the mportance of offerng dfferent rebate rates on products wth dfferent rebate senstvtes. For the current settng, the optmal rebate rate s about 12.7% for P1 and 1.2% for P2. A sngle rebate rate wll fal to take nto account these dfferences. To complete the experment, we vary the rebate senstvty f of P2 and compare the computed rebate rates usng coeffcents produced by the lnear and the log-lnear models to the optmal rates. The result s presented n Fgure 2. As P2 becomes more senstve to rebates, both approaches correctly ncrease the rebates rates for P2, and they closely mrror the optmal rates for both products. As a santy check, when the rebate senstvty of both products are equal, both the optmal rebate rate and the rates selected by our approach are roughly equal as well. 7.4.1 Summary Our approach performs relatvely even better when there are multple products. The demand estmaton step correctly dentfes the product more senstve to rebates, and the optmzaton selects correspondngly hgher rebate rate for the more senstve product. The approach s robust to changes n rebate senstvtes, and strkes a good balance among the rates selected for dfferent products. 8. CONCLUDING REMARKS We studed the problem of how onlne merchants can best
use rebates to maxmze ther revenue. Our soluton conssts of two steps an estmaton step and an optmzaton step. Our estmaton routne bulds on classcal demand models n economcs, and extends them to model the effect of rebates separately from that of prces. We develop effcent solutons to the optmzaton problem under both the lnear and the log-lnear demand model, drawng upon deas from convex optmzaton. We valdated our modelng assumptons usng transacton data obtaned from Mcrosoft Cashback operatons, and conducted an extensve smulaton study to evaluate the performance of our proposed approach. We found that across a wde range of parameters, our approach consstently generates hgher revenue than other approaches, and acheves close to the maxmum possble revenue. Through these smulaton studes, we found that selectng good rebate rates requres carefully balancng two factors puttng the entre budget to use and spreadng the budget over the entre perod. Our approach does well n balancng these factors, and hence performs better than other approaches. Between the lnear and the log-lnear model, the former s more suted for the scenaro where revenues are measured whle rebate lasts (WRL), whereas the latter s more suted for the scenaro where revenues are measured over the entre duraton (ED). The rebate rates selected are often wthn 1% of the optmal rates. We also note that the performances of the lnear and the log-lnear models are very close both n real and synthetc data. Ths s an nterestng and somewhat surprsng fndng, as economsts have often favored the log-lnear model over the lnear model for demand, and merts further nvestgaton. The optmzaton approach we presented n ths paper can be extended n several ways. For example, t can be used to maxmze profts nstead of revenue by takng nto account the cost of producton. It can also be used to solve more sophstcated problems that nclude addtonal constrants such as mnmum and maxmum rebate rate per product or lmts on the supply of each product. There are mportant future drectons to explore. One drecton s technques for optmzng rebates when demand of one product may be affected by prces and rebates of others. From the estmaton standpont, ths presents a challenge due to ts requrement for large volume of data. The optmzaton problem can no longer be formulated as a convex program and new technques wll be needed. In ths work, we do not consder the revenue generated durng the perod when a merchant s learnng a demand model for the product. If ths perod s consdered as part of the evaluaton, merchants face a new problem that may requre nterleavng exploraton for addtonal data and explotng the demand model. Solvng ths problem optmally (or approxmately optmally) wll requre new technques. 9. 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