Platform Pricing with Strategic Buyers

Transcription

1 0 45th Hawaii Intrnational Confrnc on ytm cinc Platform Pricing with tratgic uyr Yifan Dou Tinghua Univrity D J Wu Gorgia Intitut of Tchnology djwu@mgtgatchdu Jian Chn Tinghua Univrity chnj@mtinghuaducn Abtract Digital latform ar ubiquitou Whil thr ha bn a growing acadmic litratur on latform tratgi, littl i known about latform ricing whn buyr ar tratgic In contrat to myoic buyr who mak dciion olly bad on thir currnt riod utility, tratgic buyr tak into account futur riod utility in thir dciion making Thy may wait-and-, ntrarly, or fr-rid, in ronding to ky latformcific factor uch a cro-id ntwork ffct, and witching cot Uing a two-riod analytical modl, w driv otimal buyr-id ricing tratgi for a monoolitic latform ownr undr thr rvalnt ricing modl: ubcrition-bad, licn-bad, and tim-limitd frmium (TF Comard with myoic buyr, tratgic buyr do not affct otimal ricing tratgi undr th ubcrition modl or undr TF with no witching cot, but ignificantly chang otimal ricing tratgi and adotion dynamic undr th licn modl and undr TF with oitiv witching cot Th thr modl ar alo comard numrically Motivation Elctronic markt bad on latform -- multiidd ntwork that bring togthr buyr and llr -- rrnt a ignificant and growing har of th global conomy [, 3] Platform tratgi hav bn ubiquitou Thy ar of incraing imortanc for ractitionr Examl includ traditional oftwar firm lik Adob and ky, a wll a mrging gnration of cloud-bad innovator lik Al, Amazon and alforc In buin ractic, thr ar thr rvalnt ricing modl ud for digital latform ownr: ( th ubcrition f modl, whr th latform charg a r-riod fixd ric to a buyr; ( th rtual licn modl, whr th latform charg a on-tim fixd licn f to a buyr; and (3 timlimitd frmium (TF modl, whr th latform do not charg until latr riod [, 6, 7] Th latform may alo u othr ricing modl uch a a hybrid among th thr Th multi-riod natur of latform u giv ri to oibl tratgic buyr bhavior In contrat to nontratgic or myoic buyr who mak dciion olly bad on thir currnt riod utility [4], tratgic buyr tak into account futur riod utility into thir dciion making A tratgic buyr may ( oton th adotion dciion until th numbr of llr or r adotr i larg nough (wait-and-, ( adot arly at a lo whn h for that arly lo can b rcovrd latr (arly ntry; and myoic buyr would not bhavior lik thi, or (3 imly njoy th fr u riod but lav th latform onc th ownr tart charging (fr rid uch tratgic bhavior ar rfctly rational in latform adotion du mainly to two latformcific factor Firt and formot, latform growth oftn go hand in hand with ntwork ffct [3] Thi mak it oibl for om non-adotr to adot latr whn ntwork ffct lvat thir utility from ngativ to oitiv Early ntry i anothr oibility du to ntwork ffct cond, thr ar oftn witching cot for a buyr to lav a latform [7, 0] Whn th witching cot i high, tratgic buyr may not want to tak th fr offr in arly riod bcau thy for th f charg latr Whil thr ha bn a growing acadmic litratur on latform tratgi, to th bt of our knowldg, littl i known about latform ricing whn buyr ar tratgic W aim to mak th firt t toward filling thi ga by dvloing a tworiod analytical framwork to tudy ricing modl for a monoolitic latform ownr incororating tratgic buyr bhavior Our framwork allow u to addr a cntral rarch qution: nchmarkd with th ca whn buyr ar myoic, will (and how tratgic buyr bhavior affct th latform otimal ricing tratgi? W tudy thr ricing modl: th ubcrition modl, th rtual licn modl, and TF W find that tratgic buyr bhavior do not affct th latform otimal ricing tratgi undr th ubcrition modl or undr TF with no witching cot, but ignificantly chang th otimal ricing tratgi and adotion dynamic undr th licn modl and undr TF with oitiv witching cot W alo illutrat with numrical xaml how our thortical rult may b ud to hl th latform to choo among th abov thr modl Th rt of th ar i organizd a follow W brifly rviw rlvant litratur in ction Th modl tu i introducd in ction 3 W / $600 0 IEEE DOI 009/HIC

2 invtigat thr ricing chm rctivly in ction 4, 5 and 6 ction 7 rovid om numrical rult and ction 8 conclud itratur Rviw Two tram of rarch ar rlatd to our: latform ricing and tratgic urchaing bhavior Dit growing acadmic intrt in ricing information good [6, 6], th imact of tratgic buyr bhavior rmain largly unxlord in th oftwar or latform ricing litratur Whil thr hav bn a fw latform ricing modl in th litratur [8, 9, 3], a w ummariz in Tabl, th majority of th modl focu on comaring ricing modl with and without ntwork ffct Mot of th xtant litratur rtrict to a ingl-riod tting [], which mak it infaibl to xamin forward-looking buyr bhavior tratgic urchaing bhavior ar motly tudid in th oration managmnt (OM litratur A ky finding of thi litratur i that tratgic buyr oftn lad to lowr ric, bcau th rtailr comt with itlf acro diffrnt riod whn buyr hav th otion to wait until a latr riod to urcha whn ric i lowr [, 3, 4, 4] In thi litratur, tratgic buyr ar dfind a tho who tak th utility of both riod into conidration whil myoic on ar tho who only concrn about th utility of th currnt riod [4] Our modl of buyr bhavior (tratgic v myoic i conitnt with thi tram of litratur Our ar intnd to contribut to both tram of litratur by tudying latform ricing with tratgic buyr Whil tratgic buyr wait in anticiation of oibl futur ric markdown in invntory modl in th OM litratur [5], in latform tting, thir bhavior ar richr du to latform-cific factor A dicud arlir, thy may: wait-and- du to otntial WTP incra rathr than ric cut, or tak a lo by adoting arly, or only tak th fr offr uch tratgic buyr bhavior ar rarly modld in th latform (or OM litratur Our modl i caabl of caturing uch latform adotion bhavior drivn by ky latformcific factor uch a cro-id ntwork ffct, and witching cot Fixd f icn f undararajan (004 [6] Uag-bad f Rocht and Fixd f Tirol (006 [3] Uag-bad f Choudhary (007 [7] aa Parkr and Van Altyn(005 [] Tabl Poition of thi ar to th rlvant litratur Platform Providr Quality Pricing Modl Diffrnc icn f Frmium Farrll and Klmrr (007 [7] Dynamic ricing Pntrating ricing Niculcu and Wu (0 [] Thi ar Adding fixd f to uagbad f i rofitimroving Find th otimal ricing tratgy aa modl lad to highr rofit Offring fr roduct on on id could b rofit imroving witching cot hav ignificant imact on th ricing tratgi Find th otimal ricing tratgy and how that Frmium modl i conditionally bttr than for-f modl Charg-forvrything Frmium ubcrition f icn f Frmium 3 Th Modl tu Multiidd Ntwork Effct uyr witching Cot Forward-looking havior Major Finding tratgic bhavior will chang th otimal ricing tratgi of licn and TF modl Conidr a two-idd oftwar latform that connct buyr and llr W aum th latform ha a lifcycl of two riod Th latform ownr wih to maximiz it total rofit ovr two riod by charging buyr and llr acc f inc w ar rimarily intrtd in latform ricing with tratgic buyr (and in comarion with nontratgic buyr, w aum a fixd r-riod f A for ach llr throughout thi ar W aum a unit ma of buyr with ty v uniformly ditributd on [ k,] Ty v rrnt th buyr initial r-riod willingn-to-ay (WTP to acc th latform W aum k i ufficintly Thi imli that th numbr of buyr on [a, bi ] ( b a /( + k For imlicity, w cal to b a and omit th contant cal factor ( + k throughout thi ar, a it do affct any of our rult

3 larg uch that thr ar alway om cutomr who would not buy W dnot Ui( i, v a a buyr utility function in ach riod i {, }; i a th latform rannouncd acc f; > 0 ( > 0 a th trngth of oitiv cro-id llr (buyr ntwork ffct to th buyr (llr; Ni a th numbr of buyr (aftr caling by a factor of + k at th nd of riod i W aum thr ar N = I (aftr caling by a factor of + k llr at th bginning of riod At th bginning of riod, mor llr ar attractd to join th latform du to incrad intalld ba of buyr uch that N = I + N om riod non-adotr may adot in riod du to incrad WTP v+ N Thrfor, th buyr utility function ar: U, v = v ; U, v = v+ N = v+ I+ N W aum a witching cot c if a riod buyr ot out in riod W dnot λ [ 0,] a a buyr tratgicn, atinc or dicount factor for futur utility A buyr with λ = 0 i non-tratgic, myoic or xtrmly imatint, a hi adotion dciion at th bginning of ach riod dnd olly on hi currnt riod utility A myoic buyr adot if and only if hi currnt riod utility i nonngativ In contrat, a buyr with λ = i tratgic, or xtrmly atint, a h comut hi total utility from th currnt riod to th nd, taking into account futur riod utiliti whn making a dciion at th bginning of ach riod A tratgic buyr adot if and only if hi total utility at ach riod i non-ngativ Figur dict a buyr' dciion tr (choic and ayoff in our two-riod framwork At th bginning of ach riod, th buyr dcid whthr to adot { a} or not { n} A buyr ha four oibl tratgi:{ aa, },{ an, },{ na, } and { nn, } W aum that th ayoff of nvr-adot tratgy{ nn, } i zro W ar intrtd in th imlication of tratgic buyr to latform ricing modl, in comarion with non-tratgic (or myoic buyr For thi raon, w hall rtrict our attntion to th buyr id W comar and contrat thr latform ricing modl: th ubcrition modl ( = =, th licn modl, and a oular form of low-high R ( = 0, = ntrating ricing modl whn th latform offr fr acc in th firt riod and charg in th cond riod (TF W hav ut all th roof in th andix Priod Priod Adotion Dciion Figur uyr' dciion tr 4 ubcrition Modl Undr th ubcrition ricing modl, th latform charg ach buyr a fixd acc f r riod Thi ricing modl i ud by many Intrnt rvic latform rovidr who uually charg ach buyr a flat monthly f For xaml, Ntflix, th lading Intrnt traming mdia rovidr, offr buyr unlimitd TV tram for a monthly flat f of $799 Economit magazin offr unlimitd acc to it lctronic vrion for an annual f of $0 Prooition 4 Undr th ubcrition modl,{ an,} i a dominatd tratgy for any buyr Undr th ubcrition modl, cond riod utility of any firt riod adotr i non-dcraing, bcau ( ( U v = v + N v = U v,, If a buyr follow tratgy{ an, } thn (, λ 0 aa bcau U v c, which i in turn dominatd by {,} a n Adotion Dciion Adotion Dciion U (, v + λu (, v U, v U, v λc > 0 > c Prooition 4 imli that th numbr of adotr in nithr riod i affctd by th witching cot c, bcau witching i ruld out a an otion Givn thi, a can b n from th buyr dciion tr (Figur, adotion at th bginning of riod i drivn olly by cond riod utility Uing backward induction, at th bginning of riod, adotion i drivn olly by utility function Patinc or dicount factor λ do not lay a rol in ithr riod dciion making Put it diffrntly, undr th ubcrition modl, a myoic and a tratgic buyr nd u making idntical dciion, thu th latform do not nd to ric diffrntiat btwn thm a two riod n U(, v λc firt riod only a λu (, v cond riod only n 0 non adotion

4 Prooition 4 Undr th ubcrition modl, otimal latform ricing i + I + A * = ( + Givn Prooition 4, w rov th following comarativ tatic ( Figur for illutration: ( N N N N < 0; > 0; > 0 Thi giv th following inight: undr th ubcrition modl, cro-id llr ntwork ffct ( driv buyr-id adotion from firt riod toward cond riod, whthr buyr ar tratgic or myoic Figur Adotion undr ubcrition modl ( =, I = 0, A= 5 icn Modl ow N N Undr th licn modl, th latform charg ach buyr a on-tim fixd rtual licn acc f Th buyr nd to ay only onc icn modl ha bn ud by many oftwar latform rovidr Microoft, for xaml, charg a rtual licn f for it orating ytm Window and dvlomnt latform Viual tudio Prooition 5 Undr th licn modl, { an,} i a dominatd tratgy for any buyr Thi can b hown uing a imilar argumnt for Prooition 4 Givn Prooition 5, which ay that no buyr would conidr adoting only in th firt riod, thr ar only two oibl ricing tratgi, rlativ to a common thrhold Abov thi thrhold, i a high ric tratgy A that focu on rofiting from th firt riod, blow which i a low ric tratgy that balanc rofit from both riod (i tratgy A (High Pric: λ A > ( I + It can b hown that, undr tratgy A, adotion follow: A λ N N = N = + λ (iitratgy (ow Pric: λ ( I + Undr tratgy, adotion occur in both riod: N = λ, N = + N N Prooition 5 Undr th licn modl, if buyr ar myoic ( λ = 0, otimal latform ricing i + ( I + A * = ( + Givn Prooition 5, for myoic buyr undr th licn modl, w rov th following comarativ tatic ( Figur 4a for illutration: N ( N N N < 0; > 0; > 0 Prooition 53 Undr th licn modl, if th buyr ar tratgic ( λ =, otimal latform ricing tratgy i ithr tratgy A or givn blow, whichvr giv th highr rofit tratgy A (High Pric: ( I +, if ( I + + A, A = + I A, othrwi tratgy (ow Pric: + I +, if ( I + >, = ( I +, othrwi tratgy A tratgy Figur 3 Otimal tratgy undr licn modl: tratgic buyr ( λ =, I = 0, A= It follow dirctly from Prooition 53 that: ( I + + A, tratgy i otimal; ( if if ( I + + A < and ( I, + < tratgy A i otimal; (3 othrwi both tratgy A and tratgy could b otimal, dnding on th rofit comarion W u Figur 3 to illutrat thortical rult from Prooition 53 tratgy (tratgy A i otimal whn, or thir combination i ufficintly larg (mall Again, cro-id ntwork ffct lay a cntral rol in th latform choic of

5 ricing tratgi (high ric tratgy A v low ric tratgy, and conquntly adotion dynamic Givn th otimal ricing tratgy a cifid in Prooition 53, for tratgic buyr undr th licn modl, w rov th following comarativ tatic ( Figur 4b for illutration: N N ( N N 0, 0, and 0 Figur 4 comar myoic buyr adotion dynamic with tho of tratgic buyr Whn buyr ar myoic, adotion dynamic undr th licn modl ar imilar to tho undr th ubcrition modl Whn buyr ar tratgic, firt riod adotion incra (until aturation a incra Nw adotion in th cond riod do not kick in (du to latform high ricd tratgy A until xcd a crtain thrhold (du to latform witch to th low ricd tratgy High N N ow ( a Myoic uyr ( b tratgic uyr Figur 4 Adotion undr licn modl ( =, I = 0, A= Prooition 54 Undr th licn modl, otimal ricing whn buyr ar tratgic i highr comard with th ca whn buyr ar myoic Th intuition bhind Prooition 54 i a follow Whn buyr ar tratgic, at th bginning of riod, thir total utility i highr than th utility whn buyr ar myoic, bcau a tratgic buyr for that h can borrow from hi cond riod incrad utility (du to incra WTP and no nd to ay Taking thi into account, th latform rai th ric whn buyr ar tratgic 6 Tim-limitd Frmium Tim-limitd frmium (TF i a oular form of low-high{0, R } ntrating ricing tratgy whn th latform offr fr acc in th firt riod but charg in th cond riod Examl abound (g, [3], uch a alforccom Prooition 6 Undr th TF modl, if thr i no witching cot c = 0, otimal latform ricing i not affctd by buyr atinc factor λ N = N N N Whn thr i no witching cot, c = 0, w from th dciion tr in Figur that a buyr adotion dciion at ach riod do not dnd on λ Put it diffrntly, undr TF with zro witching cot, a myoic and a tratgic buyr nd u making idntical dciion, thu th latform do not nd to ric dicriminat btwn thm, othr thing qual Thi intuition can b hown by comaring th following Prooition 6 and Prooition 63, which ar idntical if c = 0 Prooition 6 If buyr ar myoic ( λ = 0, otimal latform ricing tratgy i ithr tratgy A or givn blow, whichvr giv th highr rofit tratgy A (High Pric: + ( I + + c, if ( I + + c, = ( I + + c, othrwi tratgy (ow Pric: + ( I + *, ( R if I + >, = ( I +, othrwi Givn Prooition 6, for myoic buyr undr TF, w rov th following comarativ tatic ( Figur 7a for illutration: N N ( N N = 0, 0, and 0 W u Figur 5 to illutrat th inight from Prooition 6 Whn buyr ar myoic, both tratgy A and tratgy can b otimal, dnding on th trngth of th ntwork ffct tratgy (tratgy A i otimal if, or thir combination i ufficintly larg (mall Thi contrat with th licn modl in th rnc of myoic buyr, whr th high ric tratgy A i ruld out rgardl of ntwork ffct; Hr th high ric tratgy A i otimal whn ntwork ffct ar low Clarly, th diffrnc i drivn by th witching cot, a { an, } now bcom a viabl otion for th buyr Rcall that it i dominatd in th licn (and th ubcrition modl A witching cot incra from c = 0 in Figur 5a to c = 08 in Figur 5b, maning it bcom mor cotly for tho firt-riod adotr to lav th latform, a high ric tratgy A bcom vn mor dirabl to quz mor rofit from th cativ arly buyr Prooition 63 If buyr ar tratgic ( λ =, otimal latform ricing tratgy i among tratgy

6 A, or C givn blow, whichvr giv th hight rofit tratgy A (High Pric: + ( I + + ( c, if ( I + + ( 3 c = ( I + + ( c, othrwi tratgy (ow Pric: + I +, if ( I + > = ( I +, othrwi tratgy C (Mdium Pric: ( I +, if ( I+ + A = ( I+ + ( cif, ( I+ + A + ( c< + ( I A /, othrwi Figur 5 Otimal tratgy undr TF modl: myoic buyr ( λ = 0, I = 0, A= W u Figur 6 to illutrat th inight from Prooition 63Whn buyr ar tratgic, tratgy A,, C all can b otimal, dnding on th trngth of th ntwork ffct Th low ric tratgy i otimal whn ntwork ffct ar larg Th high ric tratgy A i otimal whn ntwork ffct ar low In-btwn i th mdium ric tratgy C, which i otimal whn ntwork ffct ar mdium A witching cot incra from c = 0 in Figur 6a to c = 04 in Figur 6b, th mdium ric tratgy C bcom mor dirabl A tratgy A ( a c= 0 tratgy C ( a c= 0 ( b c = 04 Figur 6 Otimal tratgy undr TF modl: tratgic buyr ( λ =, I = 0, A= Thi i o bcau a tratgic buyr will not tak th fr offr in th firt riod whn th witching cot i high, if h for a high ric in th cond riod To mitigat thi concrn of tratgic buyr and boot riod adotion, th latform lowr th ric A tratgy A ( b c = 08 C trat Givn Prooition 63, for tratgic buyr undr TF, w rov th following comarativ tatic ( Figur 7b for illutration: N N ( N N 0, 0, and 0 Figur 7 illutrat adotion dynamic undr TF Th dottd lin rrnt N Whn buyr ar myoic (Figur 7a, N = Whn ntwork ffct ar high, w an incrad intalld ba in th cond riod du to th low ric tratgy ( Figur 5a Whn ntwork ffct ar low, w a dcrad intalld bad in th cond riod du to th high ric tratgy A ( Figur 5a om fr ridr ar ricd to lav in riod N N ( a Myoic uyr ( b tratgic uyr Figur 7 Adotion undr TF modl ( c = 0, =, I = 0, A= In contrat, whn buyr ar tratgic, if ntwork ffct ar low, om tratgic buyr will not tak th fr offr, bcau thy for th high ric tratgy A in th cond riod, rulting in N < Only whn ntwork ffct ar larg nough, w adotion growth in both riod du to lowr ric (th mdium ric tratgy C and low ric tratgy, Figur 6a 7 Modl Comarion N N W illutrat comarion among thr ricing modl with numrical xaml Whn buyr ar myoic, a w illutrat in Figur 8a, th ubcrition modl i rfrrd ovr th licn modl ovrall Hr i th intuition Givn th am ric charg, th numbr of firt-riod adotr undr ach modl i th am but th ubcrition modl can doubl di (in contrat, th licn modl cannot, a w know from Prooition 4 that firt-riod adotr would not lav Th licn modl, howvr, may win whn i ufficintly larg and i vry mall In thi ca, cond riod adotion outnumbr ignificantly that undr th ubcrition modl Whn thi han, th licn modl i in turn dominatd by th TF modl, a w hall now xlain

7 Myoic uyr λ = 0 tratgic uyr λ = Tabl ummary of otimal ricing tratgi ubcrition Prtual icn Tim imitd Frmium Condition Otimal ric + I + A Condition ( + + I + A ( + Otimal ric + ( ( I + ( ( I + (3 (: ( I + + A and ( I + I A (4 + ( I + (5 + I + + c (6 I + + c (7 + ( I + + > ; (: ( I + + A and ( I + ; (3: ( and ( I + + A < I + ; (4: c < [ ( I + ] / 4 and ( I + > ; (5: ( I + + c < ; (6: ( I + + c > and ( I (7: ( I + + A and ( I + <, or c [ ( I + ] /4and ( I + > ; + > TF TF ( a ubcrition ( v icn ( ( b Comarion among Thr tratgi Figur 8 Modl comarion: myoic buyr ( λ = 0, I = 0, A= Undr both th licn modl and TF, th total numbr of aying adotr qual to th numbr of cond-riod adotr Howvr, th intalld ba of TF modl i largt among th thr modl du to firt riod giv-away For th am ric, thi giv TF an dg ovr th licn modl In hort, th licn modl i lat favord with myoic buyr Whn th ntwork ffct ar mall, th ubcrition modl win ovr TF bcau th numbr of aying adotr undr TF i mallr (du to otimality of th high ric tratgy A Othrwi, TF i favord mot Th inight ar illutratd in Figur 8b Whn buyr ar tratgic, th comarion of th ubcrition modl with th licn modl fli If ntwork ffct ar low, th high ric tratgy A of th licn modl driv all adotion in th firt riod ignoring adotion from riod Whn ntwork ffct ar high, th low ric tratgy mak th ubcrition modl mor aaling a it balanc rofit ovr two riod ( a ubcrition ( v icn ( Figur 9 Modl comarion: tratgic buyr ( λ =, I = 0, A= W illutrat thi trad-off in Figur 9a Whn thi han, th ubcrition modl i in turn dominatd by th TF modl TF win again du to it largt intalld ba Howvr, whn ntwork ffct ar low, tratgic buyr may fr rid, hurting TF In thi ca, th licn modl i favord mot Th inight ar illutratd in Figur 9b 8 Concluion ( b Comarion among Thr tratgi Dit a growing litratur on latform tratgi, whn buyr ar tratgic, otimal latform ricing rmain largly unxlord, to th bt of our knowldg In thi ar, w intnd to fill thi ga by dvloing an analytical framwork of latform ricing whn buyr ar tratgic In th roc, w comar with th ca whn buyr ar non-tratgic (or myoic On contribution of thi ar i an intgrativ framwork that nabl a had-to-had comarion of thr oular latform ricing modl ud in

8 buin ractic: ubcrition, licn, and tim limitd frmium Our modl i caabl of caturing vral ditinct tratgic bhavior obrvd in latform adotion: Wait-and-, ntr-arly and fr-rid W invtigat imlication of uch tratgic buyr bhavior on latform ricing W find that th latform do not nd to ric dicriminat tratgic v myoic buyr, undr th ubcrition modl or undr TF whn thr i no witching cot, bcau tratgic and myoic buyr bhav xactly th am In har contrat, thy bhav rathr diffrntly undr th licn modl and undr TF with oitiv witching cot A a conqunc, th latform hould ric dicriminat btwn thm W ummariz our ky finding in Tabl Finally, w illutrat with numrical xaml on how our thortical framwork mayb ud to hl ractitionr to comar and lct among th thr ricing modl For futur rarch, w intnd to xtnd our finding to altrnativ form of cro-id ntwork and information tructur Anothr intrting xtnion i to if and how th otimal ricing on th llr id i affctd by th buyr' tratgic bhavior Whil th lattr roblm will quickly bcom analytical intractabl, numrical aroach may b viabl Acknowldgmnt Th work wa artly uortd by th National Natural cinc Foundation of China, undr grant and 70800, and Tinghua Univrity cintific Rarch Initiativ Program grant Rfrnc [] Aviv, Y and A Pazgal, "Otimal ricing of aonal roduct in th rnc of forward-looking conumr," Manufacturing & rvic Oration Managmnt, vol 0, , 008 [] ako, Y and E rynjolfon, "undling information good: Pricing, rofit, and fficincy," Managmnt cinc, vol 45, , 999 [3] anko, D and W Winton, "Otimal ric kimming by a monoolit facing rational conumr," Managmnt cinc, vol 36, , 990 [4] Cachon, G and R winny, "Purchaing, ricing, and quick ron in th rnc of tratgic conumr, "Managmnt cinc, vol 55, 497-5, 009 [5] Choudhary, V, "oftwar a a rvic: Imlication for invtmnt in oftwar dvlomnt," Procding of th 40th Hawaii Intrnational Confrnc on ytm cinc, 007 [6] Choudhary, V, "U of ricing chm for diffrntiating information good," Information ytm Rarch, vol, 78-9, 00 [7] Farrll, J and P Klmrr, "Coordination and lockin: Comtition with witching cot and ntwork ffct," Handbook of Indutrial Organization (d Armtrong, M and R Portr, vol 3, Elvir, , 007 [8] Jain, and P Kannan, "Pricing of information roduct on onlin rvr: Iu, modl, and analyi," Managmnt cinc, 3-4, 00 [9] Kauffman, R J and E A Waldn, "Economic and lctronic commrc: urvy and dirction for rarch," Intrnational Journal of Elctronic Commrc, vol 5, 5-6, 00 [0] Klmrr, P, "Comtition whn conumr hav witching cot: An ovrviw with alication to indutrial organization, macroconomic, and intrnational trad," Th Rviw of Economic tudi, vol 6, 55, 995 [] Niculcu, M F and D J Wu, "Whn hould oftwar firm commrcializ nw roduct via frmium buin modl?," Working ar, Gorgia Intitut of Tchnology, 0htt://arrncom/ol3/arcfm?abtract _id= [] Parkr, G G and M W Van Altyn, "Two-idd ntwork ffct: A thory of information roduct dign," Managmnt cinc, vol 5, , 005 [3] Rocht, J C and J Tirol, "Two-idd markt: a rogr rort," Th RAND Journal of Economic, vol 35, , 006 [4] u, X, "Intrtmoral ricing with tratgic cutomr bhavior," Managmnt cinc, vol 53, [5] u, X and F Zhang, "tratgic cutomr bhavior, commitmnt, and uly chain rformanc," Managmnt cinc, vol 54, , 008 [6] undararajan, A, "Nonlinar ricing of information good," Managmnt cinc, vol 50, , 004 [7] Varian, H R, "Pricing information good," ar rntd at th Rarch ibrari Grou ymoium on cholarhi in th Nw Information Environmnt hld at Harvard aw chool, May 3, 995 (wwwimbrklydu/~hal/par/ricinfo-gooddf Andix Proof of Prooition 4 th txt immdiat aftr thi rooition Q E D Proof of Prooition 4 Givn any, w hav N =, N = I +, N = + I + Th latform' total rofit i, ( N + N + A( N + N = ( + A N + N + AI Not that, throughout thi ar, w hall dro th

9 contant trm AI from th rofit function inc it do not affct th otimal olution Th latform roblm i quivalnt to: ( max ( A ( [ I ] olving, giv th otimal olution in th txt Th following comarativ tatic ar immdiat: * I A( N + N I + I = = < 0; = + > 0; ( + ( + ( N N A I = I ( + + ( + + I > I + 0 QED + + Proof of Prooition 5 Undr th licn modl, cond riod utility of any firt riod adotr i not dcraing, U( 0, v = v+ N v = U(, v If a buyr follow tratgy { an, } thn U (, v λc 0, which i in turn dominatd by {,} aa bcau U( 0, v U(, v λc > c QED Proof of Prooition 5 If λ = 0, only tratgy i faibl W hav N = ( λ =, N = + N Platform rofit i N + A N + N = N + A N + AI Platform roblm bcom max ( + N + A ( tn = I+ ( olving, giv th otimal olution in th txt It follow immdiatly that, + A( ( + * I N N I + = = < 0; = > 0; ( N N A I = I ( + + ( + + I > I + 0 QED ( + + Proof of Prooition 53 (i tratgy A: ( I + Undr tratgy A, adotion follow N N = N = Platform roblm i N ( + A ( + I max ( + A ( = t ( I + olving, yild th otimal olution in th txt It follow immdiatly that, if * N = ( I +, N = N = = ; othrwi if * + I A + A + I =, N = N = (ii tratgy : < ( I + Undr tratgy, adotion follow N =, N = + N Platform roblm i max [ + ( I + ] + A t < ( I + olving, yild th otimal olution in th txt It follow immdiatly that, if * = I +, N = N = ; othrwi if * + I + I + + =, N = Combin Ca (i and (ii, w hav N N ( N N 0, 0, and 0 Q E D W now laborat condition whn ach tratgy (A v i otimal Thr ar four ca Ca (i: ( I + + A and ( I + Corronding otimal ricing for tratgy A and ar + ( I + and ( I + [ + ( I + ] A 4 ( + + I A tratgy rofit + i biggr than tratgy A rofit Ca (ii: ( I + + A and ( I + < oth tratgi ndd u chooing th am otimal ric ( I +, thu obtaining th am rofit Combin Ca (i and (ii,tratgy wakly I + + A dominat A if Ca (iii: ( I + + A < and ( I + < Corronding otimal ricing for tratgy A and ar + I A and ( I + tratgy A rofit ( + I + A i biggr than tratgy rofit 4( I + + A Ca (iv: ( I + + A < and ( I + Corronding otimal ricing for tratgy A & + I A + I + &, rofit for A & ar A + ( I + oth can win ar ( + I + [ ] & 4( 4 + A QED Proof of Prooition 54 Conidr th four ca in th roof of Prooition

10 Ca (i: ( I + + A and ( I + ( I + + ( I + A + = > = ( + * * Ca (ii: ( I + + A and ( I ( I + + ( I + A * * = ( I + > = ( + ( + + < Firt inquality i trivial cond inquality i du to A <, a dirct conqunc of ( + + and ( I + < Ca (iii: ( I + + A < and ( I + < + I A + ( I + A I A = > = ( + * * A Th inquality hold a long a < which i a dirct conqunc of ( I + < Ca (iv: ( I + + A < and ( I + ( I + ( I + ( + > = ( I + = + A ( + ( + ( ( + * I I A + = ( + ( + ( + > = I + hold du to both condition A cond du to ( I + QED Proof of Prooition 6 th txt immdiat aftr thi rooition Q E D Proof of Prooition 6 Whn 0, R R + c+ N, if c+ N, N = R R + N, if < N R Ca (i: c+ N Th latform roblm i R R max R ( + c+ N + A t N = I +, R c+ ( I + olving, yild th otimal olution in th txt It follow immdiatly that, = I + + c, N = N = othrwi If + I + + c I + + c+ =, N = R If Ca (ii: < N Th rofit maximization roblm i R R max R ( + N + A t N = I +, R < ( I + Th olution giv th otimal ricing in th txt It follow immdiatly that, if = I +, N = N = ; othrwi if + I + I + + =, N = Combin ca (i and (ii, w hav N N ( N N = 0, 0, and 0 Q E D Proof of Prooition 63 Ca (i: R c+ N Adotion follow, N = c, N = R + c+ N Th latform roblm i R R max R ( + c+ N + A ( c t N = I + ( c, R ( I + + ( c olving, yild th otimal olution in th txt It follow immdiatly that, if = ( I + + ( c, N = N = c; othrwi if + * ( I + + ( c + ( I + + ( R c =, N = R Ca (ii: < N Adotion follow, R N =, N = + N Th latform roblm i R R max R [ + ( I + ] + A t R < [ I + ] olving, yild th olution in th txt It follow * R = I +, N = N =, immdiatly that, if + othrwi if * ( I + + R ( I + =, N = R Ca (iii: N < N + c Adotion R ( N follow N = N = Th latform roblm i R R ( N max R ( + A ( R t ( N N = I + (, R N < N + c olving, yild th otimal olution in th txt It follow immdiatly that, if = ( I +, N = N = ; othrwi if = I + + ( c, N = N = c; othrwi if I A + I + A = +, N = N = ( Combin Ca (i, (ii and (iii, w hav N N ( N N 0, 0, and 0 Q ED