MultiProduct Price Optimization and Competition under the Nested Logit Model with ProductDifferentiated Price Sensitivities


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1 MultProduct Prce Optmzaton and Competton under the Nested Logt Model wth ProductDfferentated Prce Senstvtes Gullermo Gallego Department of Industral Engneerng and Operatons Research, Columba Unversty, New York, NY 0027, Ruxan Wang The Johns Hopkns Carey Busness School, Baltmore, MD 2202, We study frms that sell multple dfferentated substtutable products and customers whose purchase behavor follows a Nested Logt model, of whch the Multnomal Logt model s a specal case. Customers make purchasng decson sequentally under the Nested Logt model: they frst select a nest of products and subsequently purchase a product wthn the selected nest. We consder the general Nested Logt model wth productdfferentated prce senstvtes and general nest coeffcents. The problem s to prce the products to maxmze the expected total proft. We show that the adjusted markup, defned as prce mnus cost mnus the recprocal of prce senstvty, s constant for all products wthn a nest at optmalty. Ths reduces the problem s dmenson to a sngle varable per nest. We also show that each nest has an adjusted nestlevel markup that s nest nvarant, whch further reduces the problem to a sngle varable optmzaton of a contnuous functon over a bounded nterval. We provde condtons for ths functon to be unmodal. We also use ths result to smplfy the olgopolstc multproduct prce competton and characterze the Nash equlbrum. Furthermore, we extend to more general attracton functons ncludng the lnear utlty and the multplcatve compettve nteracton model, and show that the same technque can sgnfcantly smplfy the multproduct prcng problems under the Nested Attracton model. Key words: multproduct prcng; Attracton model; Nested Logt model; Multnomal Logt model; productdfferentated prce senstvty; substtutable products Hstory : frst verson n December 20; ths verson on August, 203. Introducton Frms offerng a menu of dfferentated substtutable products face the problem of prcng them to maxmze profts. Ths becomes more complcated wth rapd technology development as new products are constantly ntroduced nto the market and typcally have a short lfe cycle. In ths
2 2 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model paper we are concerned wth the problem of maxmzng expected profts when customers follow a nested choce model where they frst select a nest of products and then a product wthn the nest. The selecton of nests and products depend on brand, product features, qualty and prce. The Nested Logt NL model and ts specal case the Multnomal Logt MNL model are among the most popular models to study purchase behavor of customers who face multple substtutable products. The man contrbuton of ths paper s to fnd very effcent solutons for a very general class of NL models and to explore the mplcatons for olgopolstc competton and dynamc prcng. The MNL model has receved sgnfcant attenton by researchers from economcs, marketng, transportaton scence and operatons management, and has motvated tremendous theoretcal research and emprcal valdatons n a large range of applcatons snce t was frst proposed by McFadden 974, who was later awarded the 2000 Nobel Prze n Economcs. The MNL model has been derved from an underlyng random utlty model, whch s based on a probablstc model of ndvdual customer utlty. Probablstc choce can model customers wth nherently unpredctable behavor that shows probablstc tendency to prefer one alternatve to another. When there s a random component n a customer utlty or a frm has only probablstc nformaton on the utlty functon of any gven customer, the MNL model descrbes customers purchase behavor very well. The MNL model has been wdely used as a model of customer choce, but t severely restrcts the correlaton patterns among choce alternatves and may behave badly under certan condtons Wllams and Ortuzar 982, n partcular when alternatves are correlated. Ths restrctve property s known as the ndependence of rrelevant alternatves IIA property see Luce 959. If the choce set contans alternatves that can be grouped such that alternatves wthn a group are more smlar than alternatves outsde the group, the MNL model s not realstc because addng new alternatve reduces the probablty of choosng smlar alternatves more than dssmlar alternatves. Ths s often explaned wth the famous redbus/bluebus paradox see Debreu 952. The NL model has been developed to relax the assumpton of ndependence between all the alternatves, modelng the smlarty between nested alternatves through correlaton on utlty components, thus allowng dfferental substtuton patterns wthn and between nests. The NL model has become very useful on contexts where certan optons are more smlar than others, although the model lacks computatonal and theoretcal smplcty. Wllams 977 frst formulated the NL model and ntroduced structural condtons assocated wth ts nclusve value parameters, whch are necessary for the compatblty of the NL model wth utlty maxmzng theory. He formally derved the NL model as a descrptve behavoral model completely coherent wth basc
3 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model 3 mcroeconomc concepts. McFadden 980 generated the NL model as a partcular case of the generalzed extreme value GEV dscretechoce model famly and showed that t s numercally equvalent to Wllams 977. The NL model can also be derved from Gumbel margnal functons. Later on, Daganzo and Kusnc 993 ponted out that although the condtonal probablty may be derved from a logt form, t s not necessary that the condtonal error dstrbuton be Gumbel. To keep consstent wth mcroeconomc concepts, lke random utlty maxmzaton, certan restrctons on model parameters that control the correlaton among unobserved attrbutes have to be satsfed. One of the restrctons s that nest coeffcents are requred to le wthn the unt nterval. Multproduct prce optmzaton under the NL model and the MNL model has been the subject of actve research snce the models were frst developed. Hanson and Martn 996 show that the proft functon of multple dfferentated substtutable products under the MNL model s not jontly concave wth respect to the prce vector. Whle the objectve functon s not concave n prces, t turns out to be concave wth respect to the market share vector, whch s n onetoone correspondence wth the prce vector. To the best of our knowledge, ths result s frst establshed by Song and Xue 2007 and Dong et al n the MNL model and by L and Huh 20 n the NL model. In all of ther models, the prcesenstvty parameters are assumed dentcal for all the products wthn a nest and the nest coeffcents are restrcted to be n the unt nterval. Emprcal studes have shown that the productdfferentated prce senstvty may vary wdely and the mportance of allowng dfferent prce senstvtes n the MNL model see Berry et al. 995 and Erdem et al has been recognzed. BorschSupan 990 ponts out that the restrcton for nest coeffcents n the unt nterval leads too often rejecton of the NL model. Unfortunately, the concavty wth respect to the market share vector s lost when prcesenstvty parameters are productdfferentated or nest coeffcents are greater than one as shown through an example n Appendx A. Under the MNL model wth dentcal prcesenstvty parameters, t has been observed that the markup, defned as prce mnus cost, s constant across all the products of the frm at optmalty see Anderson and de Palma 992, Aydn and Ryan 2000, Hopp and Xu 2005 and Gallego and Stefanescu 20. The proft functon s unmodal and ts unque optmal soluton can be found by solvng the frst order condtons see Aydn and Porteus 2008, Akcay et al. 200 and Gallego and Stefanescu 20. In ths paper, we consder the general NL model wth productdfferentated prcesenstvty parameters and general nest coeffcents. We show that the adjusted markup, whch s defned as prce mnus cost mnus the recprocal of the prce senstvty, s constant across
4 4 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model all the products n each nest at optmal locally or globally prces. When optmzng multple nests of products, the adjusted nestlevel markup, whch s an adjusted average markup for all the products n the same nest, s also constant for each nest. By usng ths result, the multproduct and the multnest optmzatons can be reduced to a sngledmensonal problem of maxmzng a contnuous functon over a bounded nterval. We also provde mld condtons under whch the sngledmensonal problem s unmodal, further smplfyng the problem. In a gametheoretc decentralzed framework, the exstence and unqueness of a pure Nash equlbrum n a prce competton model depend fundamentally on the demand functons as well as the cost structure. Mlgrom and Roberts 990 dentfy a rch class of demand functons, ncludng the MNL model, and pont out that the prce competton game s supermodular, whch guarantees the exstence of a pure Nash equlbrum. Bernsten and Federgruen 2004 and Federgruen and Yang 2009 extend ths result for a generalzaton of the MNL model referred to as the attracton model. Gallego et al provde suffcent condtons for the exstence and unqueness of a Nash equlbrum under the cost structure that s ncreasng convex n the sale volume. Lu 2006, Cachon and Kok 2007 and Kok and Xu 20 consder the NL model wth dentcal prce senstvtes for the products of the same frm and have characterzed the Nash equlbrum. Moreover, L and Huh 20 study the same model wth nest coeffcents restrcted n the unt nterval and derve the unque equlbrum n a closedform expresson nvolvng the Lambert W functon see Corless et al In all these models, the prcesenstvty parameters for the products of the same frm are assumed dentcal. Ths paper consders competton under the general NL model and shows that the multproduct prce competton s equvalent to a logsupermodular game n a sngledmensonal strategy space. The remander of ths paper s organzed as follows. In Secton 2, we consder the general Nested Logt model and show that the adjusted markup s constant across all the products of a nest. Moreover, the adjusted nestlevel markup s also constant for each nest n a multnest optmzaton problem. In Secton 3, we nvestgate the olgopolstc prce competton problem, where each frm controls a nest of substtutable products. A Nash equlbrum exsts for the general NL model and suffcent condtons for the unqueness of the equlbrum are also provded. In Secton 4, we consder an extenson to other Nested Attracton models and conclude wth a summary of our man results and useful management nsghts for applcaton n busness. 2. Prce Optmzaton under the Nested Logt Model Suppose that the products n consderaton are substtutable, and they consttute n nests and nest has m products. Customers product selecton behavor follows the NL model: they frst select
5 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model 5 a nest and then choose a product wthn ther selected nest. Let Q p be the probablty that a customer selects nest at the upper stage; and let q j p denote the probablty that product j of nest s selected at the lower stage, gven that the customer selects nest at the upper stage, where p = p,p 2,...,p m s the prce vector for the products n nest, and p = p,...,p n s the prce matrx for all the products n the n nests. Followng Wllams 977, McFadden 980 and Greene 2007, Q p and q j p are defned as follows: a p γ Q p = + n l= al p l γ l, e α j β j p j q j p = m, s= eα s β s p s 2 where α s can be nterpreted as the qualty of product s n nest, β s 0 s the productdfferentated prce senstvty, a l p l = m l s= eα ls β ls p ls represents the attractveness of nest l Anderson et al. 992 show that the expected value of the maxmum utlty among all the products n nest l s equal to loga l p l, and nest coeffcent γ can be vewed as the degree of nternest heterogenety. When 0 < γ <, products are more smlar wthn nest than cross nests; when γ =, products n nest have the same degree of smlarty as products n other nests, and the NL model degenerates to the standard MNL model; when γ >, products are more smlar to the ones n other nests. The probablty that a customer wll select product j of nest s equal to π j p = Q p q j p. 3 Apparently, m j= q j p = and m j= π jp = Q p. Wthout loss of generalty, assume that the market sze s normalzed to. For the NL model, the monopolst s problem s to determne prces for all the products to maxmze the expected total proft Rp, whch s expressed as follows Rp = m n p j c j π j p. 4 = j= The proft functon Rp s hghdmensonal and s hard drectly to optmze. Hanson and Martn 996 provde an example showng that Rp s not quasconcave n p even under the MNL model, so other researchers, ncludng Song and Xue 2007, Dong et al. 2009, take another approach. They express the proft as a functon of the marketshare vector and show that t s jontly concave wth respect to market shares. L and Huh 20 extend to the NL model wth nest coeffcent γ and dentcal prcesenstvty parameters wthn each frm may dfferent across frms. However, the proft functon may not be jontly concave when the prce senstvtes
6 6 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model are allowed to be productdfferentated, as we do n ths paper, wthn each nest. Appendx A analyzes the problem and shows that the objectve functon fals to be jontly concave wth respect to the market shares through an example. We wll next take a dfferent approach to consder the multproduct prcng problem under the general NL model. Although the markup s no longer constant for the NL model wth productdfferentated prcesenstvty parameters, we can obtan a smlar result, whch s crucal n smplfyng the hghdmensonal prcng problem. Theorem The adjusted markup, defned as prce mnus cost mnus the recprocal of prce senstvty, s constant at optmalty for all the products n each gven nest. Aydn and Ryan 2000, Hopp and Xu 2005 and Gallego and Stefanescu 20 observe that the markup, defned as prce mnus cost, s constant for all the products under the standard MNL model wth dentcal prcesenstvty parameters. L and Huh 20 extend t to the NL model but the prce senstvtes are stll dentcal for all the products wthn the same nest although they may be dfferent across nests. Let θ denote the constant adjusted markup for all products n nest,.e., θ = p j c j /β j. 5 For the sake of notaton smplcty, let Q θ be the probablty that a customer selects nest at the upper stage, where θ s the vector of adjusted markups for all the nests,.e., θ = θ,...,θ n ; and let q j θ denote the probablty that product j of nest s selected at the lower stage, gven that the customer selects nest at the upper stage, where the prces n each nest satsfy the constant adjusted markup as shown n equaton 5. Pluggng equaton 5 nto the probabltes defned n equatons and 2 results n Q θ = q j θ = a θ γ + n l= al θ l γ l, e eα j β j θ m, s= eeα s β s θ where a l θ l = m l s= eeα ls β ls θ l and αls = α ls β ls c ls for each l and s. Then, the total probablty that a customer wll select product j of nest s equal to π j θ = Q θ q j θ. Note that the average proft of nest can be expressed by m j= θ + /β j q j θ = θ + w θ, where w θ = m j= /β j q j θ. Then, the total expected proft correspondng to prces such
7 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model 7 that the adjusted markup s equal to θ for all the products n each nest, can be rewrtten as follows Rθ = n Q θ θ +w θ. 6 = Then, hghdmensonal prce optmzaton problem s equvalent to determnng the adjusted markup for each nest, whch sgnfcantly reduces the dmenson of the search space. We remark that the optmal adjusted markup θ does not have to be postve n general, but t must be strctly postve when the nest coeffcent s less than one for nest.e., γ because the total proft can also be expressed by θ + /γ w θ as shown below. Consequently, when γ >, t may be optmal to nclude lossleaders as part of the optmal prcng strategy. More specfcally, t may be optmal to nclude products wth negatve adjusted markups or even negatve margns for the purpose of attractng attenton to the nest. Example. To demonstrate the lossleaders phenomenon, we construct a smple example wth a sngle nest contanng two products. The parameters n the NL models are: α = 0.822,0.4687, β = , and γ = The costs are c = 0,0. At the upper stage of the NL model, the customer chooses an opton between purchase and nonpurchase ; then she selects one of the two products f choosng the purchase opton at the prevous stage. The problem s to determne the prces for the two products to maxmze the total proft assumng customers purchase behavor follows the NL model. By Theorem, the twodmensonal problem can be smplfed to a sngledmensonal problem of maxmzng the total proft wth respect to the adjusted markup. It s easy to obtan the unque optmal adjusted markup θ = , whch s negatve. Then, the optmal prces are p = θ +c+/β = ,.03. The total proft s equal to Note that the markup of the second product s negatve, whch s surprsng at the frst glance. In contrast, all the products wll be sold at fnte prces wth postve margns under the MNL model. Next, what f we do not offer the second product or equvalently set ts prce nfnte? Consder the prcng problem only for the frst product under the NL model. It s straghtforward to fnd the optmal prce , whch s lower than ts optmal prce when the second product s also offered at a fnte prce. The proft s , whch s 6% lower than that of offerng the two products. If the second product s offered wth a negatve margn, the attracton of the nest contanng the two product s hgher and more customers wll select the purchase opton at the upper stage of the NL model. If the second product s not offered, the nest attracton s lower and more customers
8 8 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model wll decde not to purchase. Although the second product contrbutes negatve proft, the nest has hgher attracton and more customers wll select the purchase opton at the upper stage. It results that the market share s hgher, the total proft can be hgher as well f the addtonal proft of the frst product outperforms the loss from the sales of the second product. We now state our man condton for multproduct prce optmzaton under the general NL model: Condton For each nest, γ or maxs β s mn s β s γ. Both the standard MNL model γ = for each nest and the NL model wth dentcal prcesenstvty parameters and γ < satsfy Condton. When γ >, t corresponds to the scenaro where products are more smlar cross nests; when 0 < γ <, t refers to the case where products wthn the same nest are more smlar, so the prce coeffcents of the products n the same nest should not vary too much and t s reasonable to requre max s β s /mn s β s / γ. Condton wll be used later to establsh mportant structural results. We remark that when Condton s satsfed, t requres ether γ or max s β s /mn s β s / γ for nest. More specfcally, t may happen that γ for nest, and γ < and max s β s/mn s β s / γ for another nest. Recall that θ + w θ s the average markup for all the products n nest, so we call θ + /γ w θ the adjusted nestlevel markup for nest. Theorem 2 Under Condton, the adjusted nestlevel markup, defned as θ + /γ w θ for nest, s constant for each nest. Then, the multproduct prce optmzaton problem can be reduced to maxmzng Rφ wth respect to the adjusted nestlevel markup φ n a sngledmensonal space,.e., n Rφ = Q θ θ +w θ, 7 where for each nest the adjusted markup θ s the unquely determned by = θ + /γ w θ = φ. 8 Proft Rφ s an mplct functon expressed n terms of θ, as there s a onetoone mappng between θ and φ for each accordng to equaton 8 under Condton θ + /γ w θ = γ w θ v θ /γ > 0, where v θ = m j= β j q j θ. The nequalty holds because w θ v θ max s β s /mn s β s as shown n Lemma n the Appendx.
9 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model 9 Corollary Under Condton, Rφ s strctly unmodal n φ. Moreover, Rφ takes ts maxmum value at ts unque fxed pont, denoted by φ. Then, the optmal adjusted markup θ for nest s the unque correspondng soluton to equaton 8. Thus, the optmal prce for product j of nest s equal top j = θ +c +/β j. The multproduct prce optmzaton can also be transformed to an optmzaton problem wth respect to the total market share. Let Rρ be the maxmum achevable total expected proft gven that the aggregate market share s equal to ρ,.e., n = Q p = ρ. n m Rρ := max p p = j= j c j π j p s.t., n Q = p = ρ. 9 Although n general the total proft s not jontly concave wth respect to the marketshare matrx, t has a nce structure n the aggregate market share. Corollary 2 Under Condton, Rρ s strctly concave wth respect to the aggregate market share ρ. If Condton s satsfed, the proft functon Rφ s unmodal n φ and Rρ s concave n ρ, so the frst order condton s suffcent to determne the optmal prces, whch can be easly found by several well known algorthms for unmodal or concave functons, e.g., the bnary search method and golden secton search algorthm; f Condton s not satsfed, Rφ may not be unmodal as llustrated n the followng example. Example 2. Consder the NL model wth a sngle nest contanng fve products. The parameters for the NL model are α = 4.850, , 6.60, 6.906, , β = ,.249,.0247,0.7968,0.050 and the nest coeffcent γ = Note that maxs β s mn s β s =.249/0.050 = > / γ =.76, whch volates Condton. By Theorem, the fve products should be prced such that the adjusted markup s constant. We use Rθ to represent the total proft of the fve products correspondng to the adjusted markup θ. Fgure clearly shows that Rθ s not unmodal wth respect to θ and there are three statonary ponts n the nterval,0:.90, , 2.984, and 4.736, Observe that the relatve dfference of profts s very small: /0.4479= 0.04%. Ths suggests that Rθ s very flat at the peak and any soluton to the frst order condton gves a good approxmaton.
10 0 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model Fgure Nonunmodalty of Rθ , , , Rθ Olgopolstc Competton θ We wll next consder olgopolstc prce competton where each frm controls a nest of multple products. Ths s consstent wth an NL model where customers frst select a brand and then a product wthn a brand. The olgopolstc prce Bertrand competton wth multple products under the standard MNL model has been wdely examned and the exstence and unqueness of Nash equlbrum have been establshed see Gallego et al. 2006, Allon et al. 20. Lu 2006 and L and Huh 20 have studed prce competton under the NL model wth dentcal prce senstvtes for all the products of each frm. However, ther approach cannot easly extend to the general NL model wth productdfferentated prce senstvtes. To the best of our knowledge, our paper s the frst to study olgopolstc competton wth multple products under the general NL model wth product dependent prcesenstvtes and arbtrary nest coeffcents. In the prce competton game, the expected proft for frm s m Game I: R p,p = p k c k π k p,p where p s the prce vector of frm,.e., p = p,p 2,...,p m, and p s the prce vectors for all other frms except frm,.e., p = p,...,p,p +,...,p n. By Theorem, the multproduct prcng problem for each frm can be reduced to a problem wth a sngle decson varable of the adjusted markup as follows k= Game II: R θ,θ = Q θθ +w θ. We remark that R θ,θ s logseparable. Because the proft functon R θ,θ s unmodal wth respect to θ under Condton, then t s also quasconcave n θ because quasconcavty
11 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model and unmodalty are equvalent n a sngledmensonal space. The quasconcavty can guarantee the exstence of the Nash equlbrum see, e.g., Nash 95 and Anderson et al. 992, but there are some stronger results wthout requrng Condton because of the specal structure of the NL model. Theorem 3 a Game I s equvalent to Game II,.e., they have the same equlbra. b Game II s strctly logsupermodular; the equlbrum set s a nonempty complete lattce and, therefore, has the componentwse largest and smallest elements, denoted by θ and θ respectvely. Furthermore, the largest equlbrum θ s preferred by all the frms. The multproduct prce competton game has been reduced to an equvalent game wth a sngle adjusted markup for each frm. The exstence of Nash equlbrum has been guaranteed and the largest one s a Pareto mprovement among the equlbrum set, preferred by each frm. To examne the unqueness of the Nash equlbrum, we wll concentrate on Game II, whch s equvalent to Game I by Theorem 3. We frst consder a specal case: the symmetrc game, and leave the dscusson for the general case n the Appendx. Suppose that there are n frms and that all the parameters α,β,γ n the NL model ncludng the cost vector c are the same for each frm. Note that the frm ndex s omtted n ths symmetrc game but the parameters n the NL model ncludng costs may be productdependent wthn a frm. Some propertes about the equlbrum set can be further derved. Condton 2 γ n n or maxs βs mn s β s n n γ. We remark that Condton 2 s a bt stronger than Condton and they are closer for larger n they concde when n goes nfnte. Theorem 4 a Only symmetrc equlbra exst for the symmetrc game dscussed above. b The equlbrum s unque under Condton 2. Under Condton 2, the unque equlbrum s the soluton to the equaton θ + γ Q θ w θ = 0, 0 where Q θ s the probablty to select each nest gven that the adjusted markup s θ for each frm,.e., Q θ = m s= eeα s β s θ γ +n m s= eeα s β s θ γ. It refers to the proof of Theorem 4 n the Appendx for detals.
12 2 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model 4. Extenson: Nested Attracton Model The attracton models have receved ncreasng attenton n the marketng lterature, as t specfes that a market share of a frm s equal to ts attracton dvded by the total attracton of all the frms n the market, ncludng the nonpurchase opton, where a frm s attracton s a functon of the values of ts marketng nstruments, e.g., brand value, advertsng, product features and varety, etc. As an extenson, we wll consder the generalzed Nested Attracton models, of whch the NL model dscussed above s a specal case. In ths secton, we extend to the general Nested Attracton model: a p γ Q p = + n l= al p l γ l, a j p j q j p = m a s= sp s, π j p = Q p q j p, where a s p s s the attractveness of product s of nest at prce p s and s twcedfferentable wth respect to p s, and a l p l = m l s= a lsp ls s the total attractveness of nest l. Note that a s p s = e α s β s p s for the NL model dscussed above; for the lnear model a s p s = α s β s p s, α s,β s > 0; for the multplcatve compettve nteracton MCI model, a s p s = α s p β s s, α s > 0,β s >. Condton 3 a a j p j 0, 2a j p j 2 > a j p j a j p j j, p j. b That a j p j = 0 mples that p j c j a j p j = 0. That a j p j 0 says that each product s attractveness s decreasng n ts prce; that 2a j p j 2 > a j p j a j p j can be mpled by a stronger condton that a j p j s logconcave or concave n p j. Condton 3b requres that a j p j converges to zero at a faster rate than lnear functons when a jp j converges to zero. In other words, when a jp j = 0, product j of nest does not contrbute any proft so t can be elmnated from the proft functon. Theorem 5 Under Condton 3, the prces at optmalty satsfy that p j c j + a j p j /a jp j s constant for each product j n nest. It s straghtforward to verfy that the MNL model, the Nested lnear attracton model and the Nested MCI model all satsfy Condton 3. The Corollary follows mmedately for the specal cases. Corollary 3 The followng quanttes are constant at optmal prces for the Nested lnear attracton model and the Nested MCI model, respectvely: 2p j c j α j β j, β j p j c j.
13 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model 3 Denote the constant quantty by θ for each nest,.e., θ = p j c j +a j p j /a jp j for each product j of nest. There s a onetoone mappng between θ and p j for each product j under Condton 3 because t holds that p j c j + a jp j = 2a j p j 2 a j p j a j p j < 0. a jp j a jp j p j Then, the multproduct prcng problem under the general Nested Attracton model can be reduced to maxmzng the total proft Rθ wth respect to θ for each nest, where Rθ under the Nested Attracton model s defned as follows Rθ = m n p k c k π k p, = k= where prce p j for each product j of each nest s unquely determned by p j c j + a jp j a jp j = θ. 2 Furthermore, the results n prce competton can also be obtaned under the Nested Attracton model. 5. Concludng Remark Dscrete choce modelng has become a popular vehcle to study purchase behavor of customers who face multple substtutable products. The MNL dscrete choce model has been well studed and wdely used n marketng, economcs, transportaton scence and operatons management, but t suffers the IIA property, whch lmts ts applcaton and acceptance, especally n the scenaros wth correlated products. The generalzed NL model wth a twostage structure can allevate the IIA property. Emprcal studes have shown that the NL model works well n the envronment wth dfferentated substtutable products. Ths paper consders prce optmzaton and competton wth multple substtutable products under the general NL model wth productdfferentated prcesenstvty parameters and general nest coeffcents. Our analyss shows that the adjusted markup, defned as prce mnus cost mnus the recprocal of prce senstvty, s constant for all products of each nest at optmalty. In addton, the optmal adjusted nestlevel markup s also constant for each nest. By usng ths result, the multproduct and multnest optmzaton problems s reduced to a sngledmensonal maxmzaton of a contnuous functon over a a bounded nterval. Mld condtons are provded for ths functon to be unmodal. We also use ths result to characterze the Nash equlbrum for the prce competton under the NL model.
14 4 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model We also study the general Nested Attracton model, of whch the NL model and the MNL model are specal cases, and show how t can be transformed to an optmzaton problem n a sngledmensonal space. The twostage model can allevate the IIA property and derve hgh acceptance and wde use n practce. In the future, the research and practce on customers selecton behavor wth three or even hgher stages may attract more attenton because t may be closer to the ratonalty of the decson process. Another future research drecton may consder the heterogenety of customers and nvestgate the dscrete choce model n the context wth multple heterogenous market segments. Appendx A: Nonconcavty of Market Share Transformaton In ths secton, we wll frst express the proft n terms of the marketshare vector and then show t s no long jontly concave under the NL model wth productdfferentated prce senstvtes. From equaton, Combnng wth equaton 2 results n Q n Q = l= l m γ e α s β s p s. s= e α j β j p j = π j Q γ. Q Q Then, p j can be expressed n terms of π := π,π 2,...,π m as follows p j π = β j logq logπ j + β j γ log Q logq + α j β j. 3 The total proft can be rewrtten as a functon of the marketshare matrx: Rπ = m n = j= where Q = m s= π s and c k = c k α k β k. logq logπ j + log Q logq cj π j, 4 β j β j γ Consder an NL model wth a sngle nest consstng of two products wth productdfferentated prce coeffcents β = 0.9,0. and nest coeffcent γ = 0.. Fgure 2 demonstrates that Rπ s not jontly concave wth respect to the marketshare vector π.
15 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model 5 Fgure 2 Nonconcavty of Rπ w.r.t. the Marketshare Vector Appendx B: Proofs : Proof of Theorem. Consder the frst order condton of Rp wth respect to prce p j n a gven nest [ Rp = π j p β j p j c j +β j γ p j m s= p s c s q s p +β j γ m n ] p s c s π s p = 0. l= s= Roots of the above FOC can be found by ether settng π j p = 0, whch requres p j = or lettng the nner term of the square bracket equal 0, whch s equvalent to p j c j m = γ p s c s q s p +γ β j s= m n l p ls c ls π ls p. 5 l= s= The rght hand sde RHS s ndependent product ndex j n nest, so p j c j β j, the socalled adjusted markup, s constant for all the products wth fnte prces n nest. We wll next show that all the products should be charged fnte prces such that the adjusted markup s constant. Let F be the set of products n nest offered at fnte prces. Denote the adjusted markup of each product k F by θ. Suppose the prces of all products n other nests are fxed. Note that the total attractveness of all other nests but nest s l al p l γ l = l m l s= eα ls β ls p ls γ l. Let ρ be the total market share, whch can be expressed as follows n al p l l γ l γ + s F eα s β s θ ρ = Q k θ,p = + al p l l γ l γ. + s F eα s β s θ k= Then, the total attractveness of nest can be expressed n terms of ρ and a j, j, γ e αs βsθ = ρ + al p l γ l. 6 s F l
16 6 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model Gven that the total market share s ρ and the offered product set n nest s F, the adjusted markup s the unque soluton to equaton 6, denoted by θ F. The total proft can be expressed as follows R F ρ = Q θ F,p θ F + w θ F + Q j θ F j m j,p p js c js q s j p j = ρ + al p l γ l θ F + l / ρ + + ρ al p l m γ l j p js c js q s j p j. l s= s= s F e α s β s θ F /β s al p l γ l /γ Let H F θ be the RHS of the above equaton wth θ F replaced by a free varable θ,.e., H F θ = ρ + al p l γ l θ F s F + e α s β s θ F /β s l / ρ + al p l l γ l /γ + ρ al p l m γ l j p js c js q s j p j. 7 l Consderng ts dervatve at θ = θ F H F θ θ =θ F = s= results n ρ + al p l γ l l l s F e α s β s θ / ρ + l a lp l γ /γ l θ =θ F The second equalty holds because of equaton 6. Recallng that H F θ s convex n θ, R F ρ s the mnmum of H F θ wth respect to θ,.e., R F ρ = mnθ H F θ = H F θ F. Suppose that another product z s added to set F and denote F + = 0. := F {z}. We wll next show that R F+ ρ > R F ρ for any 0 < ρ <. Smlarly, we have R F + ρ = mn θ H F+ θ = H F+ θ F+, where H F+ θ s defned n functon 25 and θ F+ s the unque soluton to equaton 6 correspondng to offer set F +. It s apparent that H F+ θ > H F θ for any θ. Then, R F+ ρ = H F + θ F + > H F θ F+ > H F θ F = R F ρ. The second nequalty holds because θ F s the mnmzer of H F θ wth respect to θ. Therefore, R F ρ s strctly ncreasng n F for any 0 < ρ < and t s optmal to offer all the products at prces such that the adjusted markup s constant n each nest. Lemma Defne w θ = m k= β k q k θ and v θ = m β k= k q k θ. The followng monotonc propertes hold. a w θ s ncreasng n θ and max s β s w θ mn s β s. b v θ s decreasng n θ and mn s β s v θ max s β s. Furthermore, w θ v θ for any θ, and all the nequaltes become equaltes when β s s dentcal for all s F.
17 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model 7 Proof of Lemma. a Consder the frst order dervatve of w θ. Then That w θ m w θ k= = + eeα k β k θ /βk m β s= se eα s β s θ m s= eeα s β s θ 2. 0 can be shown by CauchySchwarz nequalty that s m x m = y = m 2 = x y for any x,y 0. Because m k= e eα k β k θ m e eα k β k θ max s β s β k k= m k= e eα k β k θ mn s β s then, max s β s w θ mn s β s and the nequaltes become equaltes when β s s constant for all s =,...,m. b Consder the frst order dervatve of v θ. Then v θ It can be shown that v θ w θ v θ = = m k= β2 ke eα k β k θ m s= eeα s β s θ m + s= β se eα s β s θ 2 m s= eeα s β s θ 2. 0 by a smlar argument to part a. m k= eeα k β k θ /βk m β s= se eα s β s θ m s= eeα s β s θ 2 = w θ + The nequalty holds because of part a. Proof of Theorem 2. For the general NL model wth productdfferentated prcesenstvty parameters, the FOC of the total proft Rθ s Rθ = γ Q θv θ [ n l= Q l θθ l + w l θ l θ + w θ ] = 0. γ Agan, because v θ mn s β s > 0, the solutons to the above FOC can be found by ether lettng Q θ = 0, whch requres θ = or settng the nner term of the square bracket equal to zero, whch s equvalent to θ + γ w θ = n Q l θθ l + w l θ l. 8 l= The RHS of equaton 8 s ndependent of nest ndex, so θ + γ w θ s nvarant for each nest, denoted by φ. We wll next show that each nest should be charged fnte adjusted markup and the socalled adjusted nestlevel markup θ + γ w θ s constant all the nests. Let E be the set of nests whose adjusted markup satsfes equaton 8. The total market share can be expressed as follows ρ = m E s= Q θ = eeα s β s θ γ + m E E s= eeα s β s θ γ, 9
18 8 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model Then, m E s= γ e eα s β s θ = ρ ρ, 20 θ + γ w θ = φ, E. 2 The soluton to 20 and 2 s unque, whch wll be shown later, denoted by φ E and θ E = θ E,...,θ E n. The total proft can be expressed R E ρ = E Q θ E θ E + w θ E = Q θ E φ E + w θ E E = ρφ E + E m s= eeα s β s θ E γ / ρ γ w θ E Defne a new functon H E φ wth φ E replaced by a free varable φ but keepng the relaton between φ and θ as shown n 2,.e., H E φ = ρφ+ E m, γ s= eeα s β s θ γ w θ. 22 γ / ρ where θ satsfes 2 for each E. We wll next show that H E φ s convex n φ under Condton for each. where G E φ = E H E φ φ = ρ + E G E φ/ φ/ P m s= e eα s β s θ γ w θ γ / ρ G E φ = ρ ρ E m γ e eα s β s θ, s=. The second equalty holds because = ρ γ γ w θ v θ φ = γ γ w θ v θ. m γ e eα s β s θ, s= Moreover, H E φ φ = 0. φ=φ E The above equalty holds because of equaton 20. The second order dervatve of H E φ s 2 H E φ φ 2 = E H E φ/ φ = γ 2 ρv θ m s= eeα s β s θ γ > 0. φ/ γ w θ v θ E The nequalty holds because γ w θ v θ > 0 under Condton. Thus, H E φ s convex n φ, and the soluton to 20 and 2 s unque. Moreover, R E ρ s the mnmum of H E φ wth respect to φ,.e., R E ρ = mn φ H E φ. Let E + be the new nest set f the adjusted markup of another nest satsfes equaton 8. We wll next show that R E+ ρ > R E ρ for any 0 < ρ <.
19 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model 9 It s apparent that H E+ φ > H E φ for any φ, where H E+ φ s the functon defned n 22 correspondng to offer set E +. Then, R E+ ρ = H E+ φ E+ > H E φ E+ > H E φ E = R E ρ. The second nequalty holds because φ E s the mnmzer of H E φ. Therefore, R E ρ s strctly ncreasng n E for any 0 < ρ < and t s optmal to offer all the products at prces such that the adjusted nestlevel markup s constant for all the nests. Proof of Coronary. Consder the FOC of Rφ, Rφ φ = n = Rθ/ φ/ = Rφ φ n = γ 2 Q θv θ = 0, 23 γ w θ v θ where θ s the soluton to equaton 2. Because n γ 2 Q θv θ = γ w θ v θ > 0 under Condton for each, then, Rφ s ncreasng decreasng n φ f and only f Rφ φ. Case I: there s only one soluton to equaton 23, denoted by φ. Apparently Rφ s ncreasng n φ for φ φ and s decreasng n φ for φ > φ. Case II: there are multple solutons to equaton 23. Suppose that there are two consecutve solutons φ = Rφ < φ 2 = Rφ 2 and there s no soluton to equaton 23 between φ and φ 2. It must hold that Rφ < Rφ 2 for any φ < φ < φ 2 ; otherwse, there must be another soluton to equaton 23 between φ and φ 2, whch contradcts that φ and φ 2 are two consecutve solutons. We clam that Rφ s ncreasng n φ for φ [φ,φ 2 ]. Assume there are two ponts φ < φ < φ 2 < φ 2 such that Rφ > Rφ 2. Then, there must be a soluton to equaton 23 between φ and φ 2, whch also contradcts that φ and φ 2 are two consecutve solutons. Thus, Rφ s ncreasng between any two solutons to equaton 23 and Rφ may be decreasng after the largest soluton. Therefore, Rφ s unmodual wth respect to φ under Condton. Proof of Coronary 2. Let ρφ = n = Q θ, where θ s the soluton to θ + γ w θ = φ, for each =,2,...,n. Then, ρφ φ Rφ ρ = n = ρφ/ φ/ = Q 0 θ = Rφ/ φ ρφ/ φ = Rθ φ ρ. n = γ 2 Q θv θ γ w θ v θ, We can easly show that Rρ s unmodual n ρ by a smlar argument to part c. Moreover, we consder the second order dervatve under Condton for all, 2 Rρ ρ 2 = ρ Rθ φ ρ = Rθ φ ρ 2 + ρ Rθ φ ρ φ
20 20 Gallego and Wang: Prce Optmzaton and Competton under Nested Logt Model = ρ 2 n = γ 2 Q θv θ γ w θ v θ The last equalty hold because γ w θ v θ < for all θ and each under Condton. Therefore, < 0. Rρ s concave n ρ under Condton. Proof of Theorem 3. a Suppose that p,p s an equlbrum of Game I. From Theorem, the adjusted markup s constant for all the products of each frm,.e., p j c j β j s constant for all j, denoted by θ. We wll argue that θ,θ must be the equlbrum of Game II. If frm s betteroff to devate to ˆθ, then frm wll also be betteroff to devate to ˆp n Game I, where p = ˆp,..., p m and p j = θ + c j + β j. It contradcts that p,p s an equlbrum of Game I. Suppose that θ,θ s an equlbrum of Game II. We wll argue that p,p s an equlbrum of Game I, where p j = θ +c j + β j for each j. If frm s betteroff to devate to p := p, p 2,..., p m n Game I, p j c j β j must be constant for each j by Theorem, denoted by θ. Then, frm must be betteroff to devate to ˆθ n Game II, whch contradcts that θ,θ s an equlbrum of Game II. b Consder the dervatves of logr θ,θ : logr θ,θ = γ Q θ,θ v θ + w θ v θ θ + w θ, logr θ,θ = γ j Q j θ j,θ j v j θ j 0, j, θ j 2 logr θ,θ = γ γ j Q θ,θ Q j θ j,θ j v θ v j θ j 0, j. θ j Then, Game II s a logsupermodular game. Note that the strategy space for each frm s the real lne. From Topks 998 and Vves 200, the equlbrum set s a nonempty complete lattce and, therefore, has the componentwse largest element θ and smallest element θ, respectvely. For any equlbrum θ, t holds that θ θ θ and logr θ,θ logr θ,θ logr θ,θ. The frst nequalty holds because logr θ,θ / θ j 0; the second nequalty holds because θ,θ s a Nash equlbrum. Because logarthm s a monotonc ncreasng transformaton, then R θ,θ R θ,θ R θ,θ. Therefore, the largest equlbrum θ s preferred by all the frms.
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