Online Appendix for Supporting New Product or Service Introductions: Location, Marketing, and Word of Mouth

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1 Onlne Appendx for Supportng New Product or Servce Introductons: Locaton, Maretng, and Word of Mouth Vahdeh Sadat Abed, Oded Berman, mtry Krass A1 - Customer Purchase Process; Faclty Choce & Assgnment Compatblty The formulaton n Secton 3 s of the drected assgnment type:.e., t s assumed that the assgnment of customers to facltes can be controlled by the frm and s selected so as to mze the overall proft. In ths secton, we consder the alternatve customer choce case, where customers choose the faclty that mzes ther own utlty. In order to do ths, we frst defne the customer utlty and the resultng customer choce model, and then compare t to the drected assgnment case. We follow and extend the lne of argument employed by Chatterjee and Elashberg 1990 to defne a customer choce model that gves rse to our GPHR demand functon. The prevous wor s extended n two ways: frst we nclude the dstance nconvenence varable n the customer utlty functon, allowng us to model the customers choce of faclty for potental purchase and to derve dstance dependent aggregate demand dffuson. Second, we ntroduce the dependence of the total potental maret sze on the dstance nconvenence varable, allowng the potental sze to be less than 100% of the avalable demand. The choce process for a typcal customer n maret N observng the set of open facltes can be descrbed axomatcally as follows: 1 True product qualty and customer percepton: The true qualty or performance of the product, µ, s unnown to the customer. At any tme t, the customer assocates a normally dstrbuted random varable to the true product qualty. The parameters of the dstrbuton are affected by the cumulatve nformaton receved about the product by that tme from maretng and word-of-mouth. Wth I unts of nformaton accumulated, the mean and standard devaton of ths dstrbuton are gven by mi and si, respectvely. Thus, the customer perceves the true product qualty to be xi N mi, si 2. 2 Customer choce: the customer s rs averse, wth the utlty functon followng the von Neumann- Morgenstern framewor. That s, the customer values any realzaton of product qualty accordng to the utlty functon u x x I = 1 exp cx I where c s the coeffcent of rs averson. In adon, we consder a lnear dsutlty wth respect to prce and dstance nconvenence and assume that A1

2 the customer s perceptons of dstance and faclty attractveness are ndependent of product qualty. The customer wll select the faclty that mzes her overall utlty.e., the dfference between the utlty and the dsutlty functons. Thus, gven the prce P, the customer from maret decdes to purchase f: { j } E u x xi 1 P + 2 j = 1 exp cm I + c2 si P + 2 mn j > 0, j where 1 represents the shadow prce of the ndvdual customer s budget when she solves her ndvdual ncome-constraned expenure problem; t s a measure of the perceved dsutlty of prce and dstance to the customer. The non-decreasng functon 2. represents the cost of dstance nconvenence. Note that the above equaton mples that the utlty mzng customer wll always select the faclty wth the least dstance nconvenence. 3 Percepton updatng: Each new unt of nformaton comes wth an mpled product qualty level z. However, the customer does not rely on the new nformaton receved to completely reveal the true product qualty. Instead, she treats z as the realzaton of a random varable from a normal dstrbuton, Nµ, σ 2, where σ reflects the perceved level of nformaton relablty. The customer who has not yet purchased the product and has just receved the I + 1-st unt of nformaton updates her percepton of product qualty accordng to Bayes rule as follows: m I m I + 1 = si 2 + z 1 σ 2 / si σ 2 ; si = 1/ si σ 2 Ths update rule mples that the percepton of the customer of product qualty s dependent on the realzaton of a stream of mpled product qualty conveyed by an nformaton process. To aggregate the purchase process over all customers n maret, we consder customers n maret to be heterogeneous n ther ntal percepton of product qualty m0 and s0, level of rs averson c, the personal budget whch determnes cost percepton 1, the cost functon of dstance nconvenence 2., and the perceved relablty of nformaton σ. Note that we only assume the homogenety of customers wth respect to dstance nconvenence; customers can stll have dfferent cost of dstance nconvenence as a functon of the level of dstance nconvenence. The followng observaton shows that t s optmal for all customers n maret to choose the same faclty. Observaton 17 All customers n maret choose a faclty for purchase from the set arg mn j j whch s a tme-ndependent set. If the set has only one element, then all customers n maret choose the same A2

3 faclty over the tme horzon; otherwse the customers are ndfferent between any of the facltes n that set. Therefore, the choce of a statonary common faclty s optmal for all customers n maret. As tme passes, all customers n maret accumulate nformaton about the product qualty from maretng and word-of-mouth of other customers who have already adopted the product. In order to formulate the aggregate adopton level n contnuous tme, Chatterjee and Elashberg consder nformaton to be accumulated contnuously over tme. Then they assume the mpled product qualty by an nformaton stream to follow a contnuous Wener process and show that a determnstc nformaton process serves as a very good approxmaton. Let represent the dstance nconvenence perceved by customers n maret by ther choce of faclty. Followng the lne of wor of Chatterjee and Elashberg 1990 wth a determnstc mpled product qualty process, the ndvdual characterstcs of a customer can be summarzed nto two functons of as follows: cs0 2 α = 1 σ 2 2 c ln P m 0 s0 2, θ = 1 c ln P Here α s bascally a measure of how much nformaton a customer needs n order to adopt whle µ θ represents the speed at whch a customer becomes convnced to adopt by the arrval of new nformaton. Therefore, we can conclude that I = α µ θ n some sense represents the crtcal level of nformaton a customer needs n order to adopt. Note that dstance nconvenence ncreases the level of persuason a potental customer needs n order to adopt α, and the level of her resstance to new nformaton receved µ θ. The followng proposton shows how these functons along wth the proper choce of nformaton rate and dstrbutonal assumpton of customers along ther heterogenety characterstcs can be used to yeld the demand dffuson accordng to our GPHR model. We let the soluton of the GPHR model be gven by F t,, representng the fracton of maret adopted by tme t when t realzes dstance nconvenence. Proposton 18 Let the cumulatve nformaton level perceved by customers n maret, It, change over tme accordng to I t = 1 r λ t, F t,, a t, where r s an arbtrary postve functon of. In adon, let α > 0 for all customers n maret and for all dstance nconvenence values achevable. If the dstrbuton of customers along I = α µ θ follows a shfted exponental dstrbuton, G I = 1 exp r I h for I h r,, then the resultng aggregate demand dffuson exactly reproduces our GPHR model 1. A3

4 Proof of Proposton 18: Based on defntons of α and θ, customers at maret, wth populaton m, can be parttoned nto three groups when s ept fxed: Ω 1 = {α, θ α < 0}, Ω 2 = {α, θ α 0, θ < µ}, Ω 3 = {α, θ α 0, θ µ}. Ω 1 represents the customers who wll purchase mmedately at tme 0, Ω 2 represents customers who wll fnally adopt over tme, and Ω 3 represents customers who wll never purchase. Usng the result of the wor of Chatterjee and Elashberg 1990, t s straght forward to show that, for customers n Ω 2, the crtcal level of nformaton needed for a customer to adopt s I = α µ θ. Note that for customers n Ω 1 and Ω 3, I < 0. Let G 2. represent the cumulatve dstrbuton of customers n Ω 2 along the crtcal level of nformaton, I. Combnng the nformaton process It wth G 2., the aggregate demand fracton adopted by tme t at maret, F t,, can be wrtten as follows: F t, = Ω 1 m + Ω 2 m G 2 I t. Snce α > 0 for all customers, Ω 1 =. Therefore no customer wll purchase mmedately at tme 0 and Ω 2 + Ω 3 = m. Also I < 0 for a customer f and only f the customer belongs to Ω 3 Snce α > 0, I wll never tae the value of 0. Therefore, Ω 2 m = Pr I > 0 = 1 G 0 = exp h. To fnd G 2., note that t s defned only over Ω 2 and hence represents the cononal dstrbuton of G I on Ω 2. For I > 0 we have, G 2 I = G I G 0 1 G 0 = 1 exp r I To obtan the temporal behavour of demand, we substtute I wth the gven nformaton process I t = t 0 1 r λ s, F s,, a s, ds, whch results: F t, = e h G 2 I t = e h 1 exp Therefore we have ln 1 e h F t, = t 0 t 0 λ s, F s,, a s, ds. λ s, F s,, a s, ds. fferentatng the above equaton wth respect to t yelds the dfferental equaton for F t, of equaton 1. The shfted exponental dstrbuton assumpton of Proposton 18 s smlar to the exponental dstrbuton assumpton n Chatterjee and Elashberg 1990 please refer there for further dscusson. In fact, our framewor s more flexble snce we allow r and α to depend on, nclude the h functon n the A4

5 dstrbuton of customers along I, and allow for the nter-maret word of mouth. A2 - Proofs Proof of Proposton 3: Let and maretng functons a t be fxed, and defne Λ t, F t to represent the rght-hand sde of 1. Theorem 10.VIII of Walter 1998 provdes conons for the exstence and unqueness of a C-soluton that we show are met as follows: The frst exstence conon requres Λ to satsfy the Caratheodory conon, whch states that Λ should be contnuous n F for fxed t and Lebesgue measurable as a functon of t for fxed F. These conons are satsfed as λ s contnuous n t, F and a, and a t s pecewse contnuous. The second exstence conon of a C-soluton requres the exstence of a Lebesgue-measurable and ntegrable functon z 1 t such that Λ t, F ; N z 1 t. We show that z 1 t exsts by showng tme dependent bounds for λ and e h F exst, whch results n the exstence of Lebesguemeasurable and ntegrable functons z 1 t for each N, such that Λ t, F z 1 t. Then z 1t = z 1 t; N. To show the bounds, note that as F s defned to be the fracton of adopted customers, we can restrct the doman of defnton of Λ wth respect to F 0, 1. Snce λ s ncreasng n F, λ t, 0, a t, λ t, F, a t, λ t, 1, a t, wth both bounds beng measurable and ntegrable. Also e h F s bounded snce t s lnear n F whch can change only n 0, 1. Multplyng the two bounds, we set z1 t = λ t, 1, a t,. Therefore the conon on exstence of z 1 t s met and a C-soluton exsts. In adon to the exstence conons, the unqueness of a C-soluton requres a generalzed Lpschtz conon to hold for Λ t, F ; N for some Lebesgue-measurable and ntegrable functon z 2 t, that s, we should have Λ t, v Λ t, ν ; N z 2 t ν ν for every ν and ν n 0, 1 N. Ths conon s automatcally satsfed as for each, Λ s dfferentable wth respect to F wth bounded dervatve. Therefore a unque C-soluton exsts. To show that F t e h for all t 0, T, assume the contrary that there are ponts of tme that ths expresson s not vald. Let t 0 0, T be the nfmum of such tmes. The contnuty of F t and t 0 beng the nfmum of such tmes requre F t 0 = e h F τ > e h we should have df τ and the exstence of ɛ > 0 such that for all τ t0, t 0 + ɛ. Based on dfferental equaton 1 and non-negatvty of λ 0 for almost all τ t 0, t 0 + ɛ. The absolute contnuty of F t allows us to employ the fundamental theorem of calculus for F t wth Lebesgue ntegraton, whch establshes the A5

6 contradcton: t0 +ɛ e h < F t 0 + ɛ = F t 0 + t 0 df τ dτ F t 0 = e h. The non-decreasng property of F t s now straght forward: based on dfferental equaton 1, nonnegatvty of λ, and boundedness property proved above, we have df t 0. To show the least upper bound property, let the demand dffuson be allowed to run over 0, and lm nf t λ t, 0, a t, > 0. Monotoncty of λ n F results n df t e h F t λ t, 0, a t, for almost every t 0,. Integratng both sdes over 0, t we have: t 0 df τ e h F τ = F t F 0 whch results n the followng, for almost everyt: e h As t, 0 λ τ, 0, a τ, e h. F t exp e h df F t 0 t 0 λ τ, 0, a τ, λ τ, 0, a τ, dτ. dτ dτ = based on the lm nf conon, forcng lmt F t = Proof of Theorem 4: Based on the dfferentablty assumptons on λ and Theorem 9.2 of Amann 1990, F t, exsts and s contnuous over the ntervals of contnuty of {a r t; r R}, whch conssts of all t except a fnte number of ponts. Let Λ t, F t,, represent the rght hand sde of 1 for maret and defne g t, = F t, system 1 can be represented as, over the ponts of contnuty of F t,. Therefore, the demand F t, t = Λ t, F t,,, wth F, 0 = 0. For a fxed, dfferentatng ths system wth respect to gves us the followng system of dfferental equatons for g t, ; N that concdes wth F t, for all t 0, T except a fnte number of ponts. g t, t = l N F l Λ t, F, g l t, + 1 = Λ t, F, ; g 0, = 0 6 In dervng the above dfferental equaton, we nterchange the order of dfferentaton to obtan the left hand sde and apply the chan rule to Λ to obtan the rght hand sde. These are justfed by the contnuty of the partal dervatves of Λ. Also snce F 0, = 0 and s ndependent of, the ntal conon of the above dfferental equaton becomes g 0, = 0. Snce we now the functon F t, satsfes ts defnng dfferental equaton, we can treat t as a nown functon of t,. Therefore the system of dfferental equatons 6 s a lnear dfferental equaton n gl t, wth tme-dependent coeffcents. Also A6

7 smlar to the argument n the proof of Proposton 3, g t, satsfyng 6 s contnuous n tme. We would le to show that g t, 0 for all, N. Assume the contrary that there s a neghbourhood of tme such that for some N, g t, > 0. Let tme t 0 be the nfmum of all such tmes. The contnuty of g t, mples that there s a set of marets N 0 N such that for all N 0, g t 0, = 0 and that there s a neghbourhood of tme after tme t 0 such that g t, remans non-negatve and s postve for at least some N 0. Let t 1 > t 0 be a tme such that g t, does not change sgn over t 0, t 1 for all N. Therefore for all t t 0, t 1, g t, s non-negatve for N 0, and t s non-postve for all other marets. Wrtng the system 6 for marets N 0, we can arrve at a system of dfferental nequaltes below. g t, t l N 0 F l Λ t, F, g l t, 7 e h F λ t, F, gl F t,, N 0, t t 0, t 1 l l N 0 In dervng the above system, note that F F l λ t, F, 1l = λ, meanng that Λ t, F, 0, and F l Λ t, F, = e h F l Λ t, F, 0 for l. As N 0 whch mples that t does not belong to N N 0, we have F l Λ t, F, g l t, 0 for l N N 0. Now defne G t, = N 0 g t, and let M be a fnte upper bound on e h F F l λ t, F, for all l N 0. Such a bound exsts as λ s have bounded dervatves. Summng the equatons n the system of dfferental nequaltes above for all N 0, we arrve at the followng dfferental nequalty for G t, : G t, t M N 0 G t, Based on the defnton of set N 0, G t, s non-negatve and G t 0, = 0 by defnton of t 0. Therefore we can apply the Gronwall nequalty whch states that we have G t, G t1 t 0, exp M N 0 = 0 for almost all t t 0, t 1 and all. Further, as G t, s the sum of a number of non-negatve terms for t t 0, t 1, g t, should be 0 for all N 0 and t t 0, t 1 as well. Ths establshes a contradcton wth the exstence of a neghbourhood of tme for some N 0 that g t, s postve. Proof of Theorem 9: Assume the set of faclty locatons are fxed and all marets are self-motvated. We frst show that lm T F T, a,t, = e h for all N: The second self-motvated assumpton mples that a contnuous maretng polcy â and an nterval t 0, t 1 exsts such that over ths nterval λ t, 0, εâ, â = a 0. Also defne demand adoptons F ε t, a, > 0 for any 1 > ε > 0 and outsde ths nterval for any maretng polcy a that satsfes the followng t 0 A7

8 system: df ε t = e h F ε t λ t, F ε t, a r t + εâ r t 1 N r r R, F ε 0 = 0 N ; Recall that a 0 s the maretng polcy that s zero over tme. The defnton of F ε t, a, F ε t 1, a 0, > 0. Therefore, the frst self-motvated assumpton holds wth lm nf t λ t, F ε t 1, a 0,, a 0, > 0 whch means that lm t F ε t, a 0, = e h means that. But note that as the optmal maretng polcy for system 1 s non-negatve and λ s ncreasng n maretng, F ε T, a 0, F ε T, a,t, e h resultng n lm T F ε T, a,t, = e h. However, note that based on the conons on λ n Secton 3 n adon to â beng contnuous n t, the hazard rate of the system for F ε Walter 1998, p148 mples that F ε T, a, F t, a,. Therefore, s contnuous. The theorem on contnous dependence n parameters of s contnuous n ε and T, and as ε 0, F ε t, a, lm F T, a,t, = lm lm F ε T, a,t, = lm lm F ε T, a,t, = e h T T ε 0 ε 0 T the followng concludes the clam as the contnuty of F ε T, a, above. allows for the nterchange of the lmts When all marets are self-motvated at level, they are self-motvated f they are assgned to any faclty wthn the canddate set of locatons and for any faclty locaton, lm T F T, a,t, = e h. The lmt of the objectve functon becomes: lm Π = lm m P F T, a,t, T T N r R = m P e h m lm T N r R N 0 r m N r T T lm T 0 C r a r,t t. C r a r,t t Note that the frst term s ndependent of the maretng strateges used, whch means that n order to mze proft we should have lm T 0 C r T a r,t t = 0. Ths conon s satsfed when lm T a r,t t = 0 for almost all t 0,. Note that the saturated level of demand adopton from any maret s ndependent of demand adopton at all other marets and therefore the problem can be transformed nto an nstance of the UFLP problem wth b j = m P e h. Proof of Theorem 10: Frst, f for a fxed set of facltes, maret s assgned to faclty j, we have lm T 0 F T = P e h j λ 0, 0, a r 01 N r ; r R, j whch s ndependent of demand adopton from all other marets. Therefore, we can use problem P for formulaton of the optmzaton problem. A8

9 The objectve functon for mzaton of the rate of change n proft s: a r 0, j,y j m P e hj λ 0, 0, a r 01 N r ; r R, j 1 N r C r a r 0 r R j L N j L f R j j Y j Note that n the above optmzaton problem, we can optmze for a r 0 for every tool r, ndependent of other tools snce all tools are perfectly targetable. In adon, the optmal ntal maretng support of tool r, targetable at maret, would depend on the dstance nconvenence perceved by the assgnment of maret. Therefore, the optmzaton for tool r, targeted at maret that s assgned to a faclty wth dstance nconvenence s as follows, and we denote the optmal soluton as â r : m a r P e h λ 0, 0, a r 01 N r ; r R, C r a r 0 0 Snce the optmal soluton for tool r should belong to the nterval 0, ā r and the above functon s contnuous n a r 0, a mzer exsts and equals to ether of the end ponts 0 or a or s at the soluton of the FOC that can be obtaned from the followng equaton f a soluton exsts: P e h λ 0, 0, a r 01 N r ; r R, a r C a a r = 0. ervaton of optmal maretng necessary conons for GPHRL model: Let F t = F t, N and ar t be the optmal demand adopton and maretng strateges, wth a t = ar t 1 N r, r R, and denote the correspondng costate varables by θt = θ t, N. The Hamltonan functon for the GPHRL problem s: H F, a r, θ, t = N θ e h F λ t, F, a t, The frst Hamltonan conon requres F t and a r t to satsfy system 1: df t = e h F t λ t, F t, a t, Σ N m C r a r. r R ; F 0 = 0 N The second Hamltonan conon defnes a system of dfferental equatons for the costate varables θ H t = F, N whch results n the followng system: θ t = θ λ t, F t, a t, θ e h λ t, F, a t, F, N F N The ntal conon for the above system s θ T = F P N m F T = P m. The last conon corresponds to mzaton of the Hamltonan functon wth respect to maretng A9

10 strateges, that s a r t = arg a r 0,ā H F t, a r, θt, t. The optmal soluton for maretng tool r at tme t, a r t, s ether 0, ā or satsfes the FOC for the Hamltonan functon: 0 = a r H F t, a r, θt, t = θ t e h F λ t, F t, a t, t a r N r N r m C ra r ; r R Proof of Theorem 11: Frst note that the frst two constrants of T-GPHRL defne proper assgnment varables, that s, a maret can be assgned to at most one open faclty. Also the rght hand sde of the thrd constrant defnng dscretzed demand adopton s decreasng n j and ncreasng n F t+1, therefore, t can be easly shown wth nducton that t s optmal to set y j = 1for every maret for only the faclty wth least dstance nconvenence, whch s the same assgnment as the GPHRL problem. If no faclty s open, all y j = 0 for all and j whch maes demand from all marets equal to zero throughout the tme horzon. Also the non-negatvty of the rght hand sde mples that F t s ncreasng n, smlar to Proposton 3. To show the upper boundng property, consder the problem T-GPHRL-2 whch s the same as GPHRL except that demand ˆF s pecewse constant over tme and changes only at the grd ponts of Γ. ˆF at the grd ponts are defned as follows: ˆF t+1 = ˆF t + e t +1 h ˆFt λ τ, ˆF t+1, a r t1 N r, t τ t,t +1 N, = 0,..., Γ 1 ; ˆF0 = 0 We establsh the clam n two steps: Frst we let the faclty locatons and maretng strateges be fxed. We show that ˆF t F t for all N and t Γ, that s ˆF t over-approxmates demand of system 1. Snce ths s true for all faclty locatons and maretng strateges, we can conclude that the optmal proft of T-GPHRL-2 s at least as great as the optmal proft of GPHRL. Secondly, we show that pecewse constant maretng strateges, wth dscontnutes happenng at most at grd-ponts of Γ, are optmal n T-GPHRL-2 problem. Note that T-GPHRL s the same as T-GPHRL-2 except that only the former type of maretng strateges are permssble. Ths means that proft of T-GPHRL-2 s the same as the optmal proft of T-GPHRL and we can conclude that the optmal proft of T-GPHRL s an upper bound on optmal proft of GPHRL. Below we establsh the two clams made here: ˆF t over-approxmates F t for fxed faclty locatons and maretng strateges: We show ths property by nducton on : At tme 0, the clam s true as ˆF t0 = F t 0 = 0. Now assume that ˆF t F t, we would le to show that ˆF t+1 F t +1. Frst note the followng resultng from the monotoncty of A10

11 λ n F : F t +1 = F t + Hence: F t + t+1 t t+1 t e h F t λ t, F t, a r t1 N r, e h F t τ t,t +1 λ τ, F t+1, a r t1 N r, ˆF t+1 F t +1 ˆF t F t + e t +1 h ˆFt e h F t = ˆFt F t 1 t t +1 t t+1 t + e t +1 h ˆFt t λ τ, ˆF t+1, a r t1 N r, τ t,t +1 λ τ, F t+1, a r t1 N r, τ t,t +1 e h F t τ t,t +1 λ τ t,t +1 λ τ, ˆF t+1, a r t1 N r, τ, F t+1, a r t1 N r, τ t,t +1 λ τ, F t+1, a r t1 N r, The last equalty s obtaned by addng and subtractng the followng term, e h ˆFt t +1 t λ τ, F t+1, a r t1 N r, τ t,t +1 Now based on the C 1 -Mean Value Theorem for functons on a mult-dmensonal space See Pugh 2000,. p. 278 we have the followng : λ τ, ˆF t+1, a r t1 N r, τ t,t +1 = φ t ˆFt+1 F t +1, τ t,t +1 λ τ, F t+1, a r t1 N r, where φ t = 1 0 F τ t,t +1 λ τ, F, a r t1 N r, F =F t+1+α ˆF t+1 F t +1 dα. Ths A11

12 allows us to smplfy the lower bound on ˆF t+1 F t +1 : ˆF t+1 F t +1 ˆFt F t 1 t+1 t + e t +1 h ˆFt ˆFt F t t 1 + e t +1 h ˆFt t τ e h F t φ t ˆFt+1 F t +1 τ t,t +1 λ λ τ, 1; N, 1 N r ā; r R, 0 φ t ˆFt+1 F t +1 τ, F t+1, a r t1 N r, Now the frst term of the nequalty above s non-negatve based on the assumpton on and the nducton assumpton, whch allows us to smplfy: ˆF t+1 F t +1 e t +1 h ˆFt t φ t ˆFt+1 F t +1. In order to show ˆF t+1 F t +1 0 for all, assume the contrary that a non-empty subset N 0 of N exsts such that ˆF t+1 F t +1 < 0 for all N 0. Now for N 0, as ˆF t+1 F t +1 < 0 we can reduce the rght hand sde of the above nequalty by replacng φ t and e h ˆFt wth ther respectve mum values and wrte: ˆF t+1 F t τ λ τ, F, a r, ˆFst+1 s N 0 F,a r, F F st +1 s τ λ τ, F, a r, ˆFst+1 N F,a r, F F st +1 If we sum the above nequaltes over all N 0, we get ˆFst+1 F st +1 1 N 0 s N 0 F,a r, s N 0 τ λ τ, F, a r, 0. F But based on the assumpton on the second term n the summaton above s non-negatve whch establshes a contradcton as s N ˆFst+1 0 F s t +1 < 0 by defnton of set N 0. Therefore, we should have ˆF t+1 F t +1 0 for all. Optmal maretng strateges of T-GPHRL-2 problem: Let ˆF t be the optmal demand adoptons for T-GPHRL-2 and be the correspondng optmal set of faclty locatons. Consder mzng proft over the perod t, t +1 by optmzng over the maretng strateges but constranng the optmzaton to the set of maretng strateges that eep the level of demand adopton constant at the optmal levels at tmes n Γ.e. ˆF t. Snce ˆF t s are fxed, optmzaton of proft n one perod does not affect the level of proft that can be obtaned at tmes not n t, t +1. Ths optmzaton problem s as follows where we treat ˆF t s A12

13 as fxed: a r t,r R t t,t +1 s.t. t+1 m N ˆF t +1 = ˆF t + t e h P e h ˆF t t+1 t ˆF t τ t,t +1 λ 1 N r r R C ra r t λ τ t,t +1 τ, ˆF t +1, a r t1 N r, τ, ˆF t +1, a r t1 N r, ; N The optmzaton of ths system s equvalent to optmzaton of the followng Hamltonan functon where the costate varables, θ, are tme-nvarant based on the ntegral form of the objectve functon and the constrants. a r,r R N P m + θ e h 1 N r m C r a r r R ˆF t τ t,t +1 λ τ, ˆF t +1, a r 1 N r, Snce the Hamltonan mzaton problem above s ndependent of tme and ˆF t s represent optmal demand adopton, t should be that constant maretng strateges over t, t +1 are optmal and hence the optmal maretng strateges are pecewse constant wth ther dscontnuty ponts belongng to Γ. Proof of Theorem 13: When there are no ntermaret communcatons and maretng actvtes are perfectly targetable, the demand adopton n each maret becomes ndependent of adopton at the others. Therefore, we can redefne the proft earned from each maret as Π = P F T Subject to: df t a r t 0, N r={} m = e h r R, N r T F t λ t, F t, a t, 0 C a a r t, F 0 = 0 efne N j for j / to be the set of { N; j < }. It represents the set of marets that wll swtch to faclty j f t s added to the set of open facltes. Note that the adon of faclty j to set does not change the dstance nconvenence perceved for marets n N N j, and hence ther assgnment to a faclty remans unchanged. Also snce the maretng decsons for all marets are ndependent and no nter-maret word of mouth s present, ntroducton of faclty j does not change the optmal maretng expenure for all marets n N N j, and the proft obtaned from them. Therefore, we can remove these marets from proft consderatons n ρ j expresson and smplfy t as follows, A13

14 notng that Π S {j} ρ j = N j = Π Π {j} {j} for all N j : Π f j = N j Π {j} Π f j. Now for any sets S T L and j / T, N j T N j S snce there are more facltes n set T than S. Consequently we have: ρ j S ρ j T = = N j T N j S Π {j} Π S Π T Π S + N j T N j S N j T Π Π {j} S {j} Π T Π S The two summatons above nvolve dfferental proft from a maret when more facltes are open and we show that these terms are postve. Let a, t be the optmal maretng polcy that targets maret when facltes n set are open and denote Π, a be the proft obtaned from maret when maretng polcy a t that targets maret s employed. Hence Π = Π, a,. Now we can wrte: Π T Π S = Π T, a, Π T, a,s + Π T, a,s Π S, a,s 0 The rght hand sde of the above s non-negatve: the frst term s postve as a,t s the optmal level of maretng wth faclty set T ; the second term s also postve snce the dstance nconvenence realzed by faclty T s less than or equal to that of S, and proft s decreasng n dstance nconvenence by Theorem 4. We can use a smlar argument to show Π S {j} Π S and conclude that ρj S ρ j T 0, showng that the proft functon s submodular. Proof of Theorem 14: Based on the dfferentablty assumptons on λ and Theorem 9.2 of Amann 1990, 2 F t, l exsts and s contnuous over the ntervals of contnuty of {a r t; r R}, whch conssts of all t except a fnte number of ponts. Let h l t, = 2 F t, l = g t, l = gl t, over the ponts of contnuty of 2 F t, l. Equaton 6 provdes a dfferental equaton for the frst dervatve of F wth respect to dstance nconvenence, g t,. To arrve at a dfferental equaton for hl t,, we tae dervatve of 6 wth respect to l, whch s justfable because of proper smoothness of Λ t, F, explaned n A14

15 Secton 3. t hl t, = t hl t, = s N 2 g t, t { l Λ t, F, = s N l + 1 = l g s t, + F s Λ t, F, F s Λ t, F, } gs t, l 2 Λ t, F, gq l t, + 1l = 2 Λ t, F, gs t, F s F q F s q N + Λ } t, F, h l s t, F s + 1 = 2 Λ t, F, gs l t, + 1 = l = 2 Λ t, F, F s 2 s N = 2 Λ t, F, gq l t, gs t, + 1l = F s F q s N q N s N + 1 = s N + 1 = l = 2 Λ t, F, 2 2 Λ t, F, gs l t, + F s s N Λ t, F, h l s t, F s 2 Λ t, F, F s g s t, Snce we now the functon F t, and g t, satsfy ther defnng dfferental equatons, we can treat them as nown functons of t,. Therefore the system of dfferental equatons 8 s a system of lnear dfferental equatons for h l t,, N, wth tme-dependent coeffcents. Also snce g 0, = 0 and s ndependent of, the ntal conon of the above dfferental equaton becomes h l 0, = 0. Smlar to the argument n the proof of Proposton 3, h l t, as the soluton of 8 s contnuous n tme. Now we would le to show that h l t, 0 for all,, l N when l N. Assume the contrary that there s a neghbourhood of tme such that for some N, h l t, > 0. Let tme t 0 be the nfmum of all such tmes. The contnuty of h l t, and the defnton of t 0 mply that there s a set of marets N 0 N such that for all N 0, h l t 0, = 0 and that there s a neghbourhood of tme after tme t 0 such that h l t, remans non-negatve and s postve for at least some N 0. Let t 1 > t 0 be a tme such that h l t, does not change sgn over t 0, t 1 for all N. Therefore for all t t 0, t 1, h l t, s non-negatve for N 0, and t s non-postve for all other marets. Wrtng the system 8 for marets N 0, we can arrve at a system of dfferental nequaltes below. In dervng the system, note that the frst three terms on the rght hand sde of 8 are non-postve based on the assumptons and Theorem 4. Snce l, the last term also vanshes. 8 A15

16 h l t, t Λ t, F, h l s t, F s s N 0 e h F λ t, F, F s s N 0 h l s t,, N 0, t t 0, t 1 Ths system of dfferental nequaltes s smlar to the system derved n the proof of Theorem 4. Usng a smlar argument as n that proof, we can show that h l t, = 0 for all N 0, t t 0, t 1 whch s a contradcton wth the exstence of a neghbourhood of tme for some N 0 that h l t, s postve. Proof of Lemma 15: Let the maretng strateges be fxed, and defne F and Π represent the total demand adopton and total proft from openng facltes by the end of the tme horzon. efne ρ j S = Π S {j} Π S, for S T and j / T. The submodularty requres us to show that ρ j S ρ j T 0. In adon, let N j for j / be the set { N; j < }. It represents the set of marets that wll swtch to faclty j f t s added to the set of open facltes. By defnton, we have N j S N j T. ρ j S ρ j T = Π S {j} Π S = N P m { F S {j} F S Π T {j} Π T F T {j} F } T Let represent the th term n the bracet of the above summaton. Then we have: = F S {j} F S F T {j} F T N { = F S {j} 1,..., S {j}, +1 S,..., S N F S {j} 1,..., S {j} 1, S,..., S N =1 F T {j} 1,..., T {j}, +1 T,..., T N F } T {j} 1,..., T {j} 1, T,..., T N 9 We show that the th term n the summaton above s non-negatve. In the dervatons below, we use the decreasng dfferences property of F n dstance nconvenence as a result of Theorem 14 whch states that for representng the vector of dstance nconvenences except, and for vector of dstances 1 2 we have: F 2, 2 F 1, 2 F 2, 1 F 1,. 1 We consder followng three cases. 1 N j T, S {j} = T {j} = {j} : Usng the decreasng dfferences property, the th term of A16

17 9 becomes: F S {j} 1,..., {j}, S +1,..., S N F S {j} 1,..., S {j} 1, S,..., S N F T {j} 1,..., {j}, T +1,..., T N F T {j} 1,..., T {j} 1, T,..., T N F T {j} 1,..., {j}, T +1,..., T N F T {j} 1,..., T {j} 1, S,..., T N F T {j} 1,..., {j}, T +1,..., T N F T {j} 1,..., T {j} 1, T,..., T N = F T {j} 1,..., T {j}, T,..., T N F T {j} 1,..., T {j} 1, S,..., T N N j S N j T, S {j} = {j} and T {j} = T : In ths case, the last two expressons nsde the th term of 9 cancel out and t summarzes to the followng based on the decreasng property of F n. F S {j} 1,..., {j}, S +1,..., S N F S {j} 1,..., S {j} 1, S,..., S N 0, 3 N N j S, S {j} = S and T {j} = T : The th term of 9 reduces to 0. Snce n all of the cases above, the th term of the summaton 9 s non-negatve, 0, whch results n ρ j S ρ j T = N P m 0. Proof of Theorem 16: Wth a lttle abuse of notaton, defne Π, a to be the total proft obtaned from openng faclty set and employng maretng strategy a. Then we have the followng for S T and j / T : ρ j S ρ j T = Π S {j} Π S Π T {j} Π T = Π S {j}, a S {j} Π S, a S Π T {j}, a T {j} Π T, a T Π S {j}, a S Π S, a S Π T {j}, a T {j} Π T, a T {j} Π T {j}, a S Π T, a S Π T {j}, a T {j} Π T, a T {j} = Π T, a T {j} Π T, a S Π T {j}, a T {j} Π T {j}, a S. The frst nequalty holds because of optmalty of a S {j} and a T, and the second s due to the submodularty of Π, a S based on Lemma 15. The conon 2 F T,a, l 0 means that Π, a T {j} Π, a S s ncreasng n. As T T {j}, we have ρ j S ρ j T 0 whch establshes the submodularty of Π. In order to see whether 2 F T,a, 0 holds wth the added assumptons stated n the theorem, we l let A l t, a, = 2 F t,a, over the ponts of the contnuty of 2 F t,a, l whch s almost all t, and l A17

18 derve a dfferental equaton for A l t, a,. Ths dfferental equaton s obtaned by tang dervatve of 6 calculated at maretng strateges a wth respect to l : A l t, a, t = s N r R = g T, a, l ar 1 N r l 2 Λ t, F t, a,,, a + Λ t, F t, a,,, a A l s t, a, F s So based on the assumptons and Theorem 4, we have: A l t, a, t s N r R F s a r gs T, a, + 1 = 2 Λ t, F t, a,,, a a r r R Λ t, F t, a,,, a F s a r l A l s t, a, a r l Ths s a dfferental nequalty wth ntal conon A l 0, a, = 0 smlar to 7. Based on a smlar argument we can show that the soluton A l t, a, 0 for all t, and, whch concludes the proof. Adonal References for the Appendx Amann, H Ordnary dfferental equatons, an ntroducton to nonlnear analyss, Walter de Gruyter Pugh, C.C Real Mathematcal Analyss. Sprnger-Verlag. Walter, W Ordnary dfferental equatons. Sprnger, 121. A18