Firms nowadays are increasingly proactive in trying to strategically capitalize on consumer networks and

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1 Information Sytem Reearh Vol. 24, No. 1, Marh 2013, pp ISSN (print) ISSN (online) INFORMS Engineering Optimal Network Effet via Soial Media Feature and Seeding in Market for Digital Good and Servie Yifan Dou Shool of Eonomi and Management, Beihang Univerity, Beijing , China, Mariu F. Niuleu, D. J. Wu Sheller College of Buine, Georgia Intitute of Tehnology, Atlanta, Georgia 30308, Firm nowaday are inreaingly proative in trying to trategially apitalize on onumer network and oial interation. In thi paper, we omplement an emerging body of reearh on the engineering of wordof-mouth effet by exploring a different angle through whih firm an trategially exploit the value-generation potential of the uer network. Namely, we onider how oftware firm hould optimize the trength of network effet at utility level by adjuting the level of embedded oial media feature in tandem with the right market eeding and priing trategie in the preene of eeding diutility. We explore two oppoing eeding ot model where eeding-indued diutility an be either poitively or negatively orrelated with utomer type. We onider both omplete and inomplete information enario for the firm. Under omplete information, we unover a omplementarity relationhip between eeding and building oial media feature that hold for both diutility model. When the ot of any of thee ation inreae, rather than ompenating by a tronger ation on the other dimenion to retore the overall level of network effet, the firm will atually ale bak on the other initiative a well. Under inomplete information, thi omplementarity hold when eeding diutility i negatively orrelated with utomer type but may not alway hold in the other diutility model, potentially leading to fundamentally different optimal trategie. We alo diu how our inight apply to aymmetri network. Key word: oial ommere and oial media; network effet; oial interation; eeding; adoption proe; digital good and ervie Hitory: David Gode, Senior Editor; Zolt Katona, Aoiate Editor. Thi paper wa reeived on January 15, 2012, and wa with the author 4 month for 2 reviion. Publihed online in Artile in Advane Deember 20, Introdution For many ategorie of produt, it ha been widely known in the indutry and doumented in a rih reearh literature that the exiting network of uer an impat in many way the adoption proe. Firt, it may indue word-of-mouth (WOM) effet, leading to fater or more effiient propagation of information about the produt, helping onumer in the valuation learning proe. Seond, inreaed adoption within ertain oial group, even in the abene of informative interperonal ommuniation, may lead to herding behavior, i.e., imitation effet, epeially when the intended market exhibit homophili tendenie. Third, if the produt i ueptible to network effet at individual utility level, a larger network may boot the value of the produt to eah uer and, impliitly, inreae the willingne-to-pay (WTP) of potential adopter. The rapid expanion of the Internet uer bae (with over 2.26 billion individual onneted aording to Internet World Stat 2012) and the relatively reent advent of oial media tool provided firm with unpreedented abilitie to reah and apitalize on the network of uer. Many buinee with an online torefront (e.g., Amazon, Apple App Store, Beah Camera, Bet Buy, Newegg, Target, et.) introdued feature and hannel to allow uer to rate produt. Similarly, many pratie and buinee (from ar dealer to medial dotor) are now rated online by utomer. Two-ided platform provider (e.g., Airbnb, ebay, elane) introdued feedbak mehanim through whih partiipant an build reputation. Many provider of ontent reation and produtivity oftware (e.g., Adobe, Google Do, Mirooft Word, wiki) introdued ollaboration tool that made thee produt or ervie more appealing to variou uer. Along the ame line, everal loud torage ervie (e.g., Dropbox, Mozy) allow uer to hare doument. Some ompanie alo upport online forum through whih utomer an 164

2 Information Sytem Reearh 24(1), pp , 2013 INFORMS 165 interat with eah other and tart diuion thread about produt-related topi (e.g., Dell, Amazon Web Servie). Moreover, the value of oial and profeional network (e.g., Faebook, LinkedIn, Salefore Chatter); maive multiplayer online game (e.g., Blizzard World of Warraft); video, voie, and text hatting tool (e.g., Google Talk, Skype, Yahoo Meenger); or blogging and miroblogging tool (e.g., Twitter) i prediated on oial interation. Many other example an be mentioned where oial media feature have failitated growth in value or aeleration of information diemination in aoiation with produt by enhaning the potential benefit of uer interation. Mot of the literature apturing the influene of onumer network on the produt adoption traditionally onider the manifetation, trength, and impat of uh effet outide the influene reah of the firm (e.g., Ba 1969, Robinon and Lakhani 1975, Kalih 1983). In other word, although oial interation do our and have been hown to influene onumer behavior, very few tudie atually explored how the firm hould manage and trategially influene thee interation (Gode et al. 2005). Reent tudie began opening thi path, fouing primarily on firm trategie and opportunitie to engineer WOM effet. Biyalogorky et al. (2001) tudy how onumer referral ation hould be inentivized. Dellaroa (2006) inpet how trategi manipulation of online forum an hift the information value of online review for utomer. Chen and Xie (2008) explore the firm benefit from etablihing an online ommunity where onumer an pot review. Forman et al. (2008) empirially how that the prevalene of reviewer diloure of identity information an be aoiated with an inreae in onumer trut in the review, with impat on ubequent online produt ale. Thu, firm might benefit from trategially reating online review platform and inentive mehanim that enourage reviewer to hare more information. Gode and Mayzlin (2009) empirially tudy how firm hould trategially reruit utomer for WOM ampaign baed on loyalty onideration to drive ale. Aral and Walker (2011) highlight the effetivene of viral produt feature in generating oial ontagion. Aral et al. (2011) analyze the performane of eeding and referral inentive program a two popular method to engineer oial ontagion. We extend thi literature by onidering how a monopoliti firm an trategially engineer the trength of network effet at utility level via oial media. On thi dimenion, we fou on deriving the optimal level of oial media funtionality that inreae the value of oial interation to eah uer. Suh funtionality inlude feature and environment that failitate ommuniation between uer (e.g., hat apabilitie, virtual reality environment where avatar an interat, reen haring), ollaboration on and oreation of ontent (e.g., wiki, ontent editing, and tagging), peer endorement or referral (e.g., on profeional network uh a LinkedIn), building of reputation, et. In a different ontext, Bako and Katamaka (2008) explore how an intermediary would optimally engineer the trength of ro-ide network effet when deigning a twoided Internet platform. In ontrat, we onider a vendor that endogenize the trength of diret (ameide) network effet experiened by the buyer in parallel with eeding the market, when there are ynergie between the two ation, a will be further diued in the paper. Our tudy foue on paid digital good and ervie where the value i motly indued by the network. Some example inlude (but are not retrited to) maive multiplayer online game (e.g., World of Warraft), oial dating ite (e.g., eharmony, Math.om), profeional oial network (Salefore Chatter Plu), and peialized online forum with paid memberhip (e.g., Angie Lit, Natureape.net). We point out that our reult go beyond digital good and ervie, and apply to other produt and ervie (e.g., voie ommuniation ervie) where marginal ot are negligible and the bulk of value i derived from the network. When network effet trongly dominate tand-alone benefit from the produt (i.e., benefit in the abene of the network), firm may find it profitable to park adoption by giving away ome onumption for free. In thi paper, we fou on eeding trategie, whereby the firm give the produt with full funtionality and perpetual liene to a few utomer to boot the WTP of other utomer and atalyze adoption (Lehmann and Eteban-Bravo 2006, Jiang and Sarkar 2009, Galeotti and Goyal 2009). Alternative trategie that are alo employed in the indutry to park adoption involve freemium approahe (limited-time free trial or free verion with tripped-down funtionality Niuleu and Wu 2012). In addition to potential demand annibalization and boot in network effet, in the ontext of paid produt, eeding may indue a eparate effet on the paying utomer. If ome utomer are harged for the produt, then eeding implie prie dirimination in the market: eeded utomer pay le (zero) ompared to uneeded individual who end up buying the produt. If eeding i extenive and paying utomer oberve it, then they might onider the prie heme unfair. Extant empirial tudie illutrate that prie dirimination ould potentially lower utomer WTP. Oliver and Shor (2003) how evidene of trong negative effet on fairne pereption, atifation, and purhae ompletion among

3 166 Information Sytem Reearh 24(1), pp , 2013 INFORMS online hopper that are prompted to enter a oupon ode toward the onluion of the hekout proe for uh utomer that did not reeive a ode in advane. Novemky and Shweitzer (2004) explore the role of internal oial omparion (between buyer and eller) and external oial omparion (between buyer and buyer) in negotiator atifation. They find that buyer atifation will be inreaed if other buyer have a maller urplu. Xia et al. (2004) provide a literature review and oneptual framework to undertand the fairne of priing. When prie omparion are pereived a unfavorable for imilar tranation, they predit that utomer will have an advere repone to the eller trategy, whih may involve negative emotion (uh a anger or outrage), negative WOM, and redued demand. Aording to Hinz et al. (2011), when prie dirimination i oberved, it i often the ae that utomer feel unhappy about the unfair priing. Alo, it i not unommon in the indutry for adopter to quetion priing pratie of the firm and exert preure on them. For example, Amazon offered a publi apology and refund to over 6,000 utomer in repone to publi baklah over a erie of prie tet through whih different online hopper were quoted different prie for variou DVD (?). In another example, after introduing the firt generation iphone in June 2007, within jut three month Apple dropped the prie by $200 for the entire market. Faed with a flood of omplaint from early adopter, Apple deided to refund eah of them $100 in the form of tore redit (Wingfield 2007). Building on prior literature and indutry obervation, in the ontext of digital good and ervie with an aoiated prie tag, we formalize the negative effet of eeding aoiated with prie dirimination via a diutility inurred by paying utomer. In our model, the more utomer are eeded, the greater i the baklah and valuation downgrading from paying utomer who quetion the fairne of the priing heme. To apture variou potential market enario, we onider two ontrating eeding diutility model. Under the firt eeding diutility model, SDU +, for eah utomer, the eeding-indued diutility i poitively orrelated with her type. Thu, the highet-type utomer are experiening the highet eeding-indued diutility. Under the eond model, SDU, every utomer experiene a eeding diutility that i negatively orrelated to her type. In thi ae, high-type utomer do not experiene muh diutility beaue of eeding. Variou example jutifying eah etup are inluded in 2. In thi paper we explore the trade-off between benefit and ot aoiated with eeding and building oial media feature into the digital produt. On one hand eeding boot WTP of potential utomer beaue it lead to larger uer network. On the other hand, eeding indue diutility for paying utomer and, ontingent on eed alloation, an annibalize demand. Similarly, building oial media feature that boot the trength of network effet at the utility level (i.e., by allowing more value extration from oial interation) would lead to inreaed WTP for the utomer but involve building effort. Taking thee trade-off into onideration, we eek to find out what are the optimal eeding, priing, and oial media trategie for the firm. Depending on firm ae to market information, we explore two enario: omplete information on firm ide (the firm know enough about the utomer uh that it an perform targeted eeding) and inomplete information (the firm doe not know muh about the utomer other than the onumer ditribution and, thu, annot reort to targeted eeding). Firt, under omplete information, for eah of the diutility model, we olve ompletely the market equilibrium, diu market overage, and invetigate the interation effet between eeding and building oial media feature. Under both diutility enario, we find that if the marginal ot/penalty aoiated with one of thee initiative inreae, the firm will ale bak (or ometime leave unhanged) it effort on the other dimenion a well. Thi i intereting beaue, at the utility level, an upward hange in eah of thee two dimenion would inreae the impat of the network effet. However, if the invetment required to build more oial media feature in the produt i higher, while the firm ale bak on uh feature, it will not try to ompenate by eeding more. Similarly, if the eeding diutility rate i higher, while the firm ale bak on eeding, it doe not try to boot the trength of network effet. Under inomplete information, while the firm deide trategially on the eeding volume, we aume the eed end up being pread uniformly in the market. Under diutility model SDU, omplementarity between eeding and building oial media feature ontinue to hold. However, under SDU +, ome new pattern emerge. When the eeding penalty i mall, it may be atually poible for both level of eeding and oial media feature to be inreaing in the eeding penalty. When eeding penalty i intermediate, the two ation at a ubtitute to eah other with repet to hange in penalty rate. When the eeding penalty inreae, the firm redue eeding and build at the ame time tronger network effet. One eeding diutility i large, we enounter the ame omplementarity effet a in all the other etting. Alo, for mall oial media building ot, the ation at a ubtitute to eah other with repet to hange in oial media ot. If it i more expenive to build oial media feature but not too expenive, the firm inreae the eeding ratio and dereae the level of

4 Information Sytem Reearh 24(1), pp , 2013 INFORMS 167 oial media feature. However, one the oial media ot beome large enough, the outome revert to the previouly unovered omplementarity in ation. We alo explore how our framework and reult extend to aymmetri network where onnetion between uer are ative only in one diretion. We how that ome of the previou inight ontinue to remain robut, and we alo inpet variou eeding pattern indued by the network truture. Our paper ha important pratial impliation. Firt, in the indutry, firm reognize the value of network effet and put a lot of effort into harneing them. Thi paper and the framework and reult within provide a hot of pratial and ationable inight into how to optimally ue in tandem two lever (eeding and building oial media feature) aimed at booting network effet taking into aount how they an jointly impat the market outome. In partiular, in addition to development ot onideration, firm hould alo onider the magnitude and form of eeding diutility when deiding the optimal oial media trategy. For example, a diued above, higher eeding diutility may inentivize firm to ale bak on the level of oial media feature embedded in their produt. In parallel, we alo ugget priing trategie that effetively jumptart and utain adoption. Our analyi further indiate that barrier to entry are higher for firm in environment haraterized by SDU + eeding diutility model. Another pratial impliation of our reult i toward meauring the value of information. Unpreedented amount of onumer data are olleted in the market and many player are now engaging in big data analyti. Firm an hooe to invet internally in uh apabilitie or proure uh information from an inreaing number of third-party provider who peialize in olleting, analyzing, and elling market information. By deriving the optimal trategie and profit under omplete and inomplete information, thi paper take an important tep toward meauring how muh value firm hould plae on identifying utomer harateriti. We further diu the value of information in the onluion. The ret of the paper i trutured a follow. In 2, we introdue our general modeling framework. In 3, we preent the analyi of the omplete information ae. In 4, we extend our diuion to inomplete information etting. In 5, we extend our analyi to aymmetri network and provide further diuion on robutne of our key finding. We onlude in 6. All proof of our reult an be found in the appendix. 2. General Model Conider a oftware market with a monopoliti firm and a heterogeneou and tationary pool of potential utomer with ma normalized to 1 and type ditributed uniformly in the interval 0 1. For impliity, in the main part of the paper we fou on a ymmetri, fully onneted onumer network and aume the oftware exhibit heterogeneou network effet. We relax the ymmetry aumption in 5. Thu, if the urrent intalled bae ha ize, then a utomer of type will get a diret benefit b from the oftware, where b apture the trength of network effet. Apart from the network-generated value, we aume the produt arrie negligible tand-alone value. Our model i onitent with etup in Dhebar and Oren (1985, 1986). To boot paying utomer produt valuation, the oftware firm eed a fration of the market. While the eeding proe in itelf jumptart adoption, it alo generate diutility at the individual level for paying utomer a diued in the Introdution. For eah paying utomer, we model thi diutility a a funtion that i nonnegative, onvex and inreaing in, and apture heterogeneity of thi effet a experiened by eah utomer. The nonlinear dependene on the ize of the eeded pool apture the fat that diutility i very limited for mall eeded pool but rapidly inreae a more utomer are eeded. In thi paper, we onider a etup where utomer progreively join the network toward a market equilibrium. Seeding our immediately before the produt i releaed for ale. Cutomer do not know the overall type ditribution in the market and at in a myopi fahion, making their adoption deiion baed on the pereived utility from the produt omputed uing the urrent oberved intalled bae. In general, in the oftware indutry (and other), there are many ae where firm make publi the information regarding intalled bae or uh information i etimated and reported with regularity by market reearh firm. 1 Conitent with uh obervation, we aume in our model that utomer have ae to thi information. If at a given moment the intalled bae i of ize (inluding eeded utomer), then a paying utomer of type would momentarily pereive the utility from buying the produt a u b p = b p (1) A utomer of type adopt a oon a he pereive u 0. Thi etup i onitent with Rohlf (1974) and Dhebar and Oren (1985), among other. 1 For example VG Chartz ( report indutry-wide weekly ale number (in term of number of unit old) for both oftware game and onole hardware. Alo, in Japan, all mobile teleom arrier make publi on a monthly bai their mot urrent ubriber number on their orporate webite and through the Japanee Teleommuniation Carrier Aoiation (Niuleu and Whang 2012).

5 168 Information Sytem Reearh 24(1), pp , 2013 INFORMS The firm i proative in managing the trength of network effet via hooing the right amount of oial media feature that boot the value of oial interation to eah uer. A uh, we aume that the firm will inur a onvex ot b 2 to indue network effet at marginal trength b, with > 0. If we denote by N b p the ma of paying utomer 2 at the onluion of the adoption proe (in equilibrium), then the firm optimization problem beome max b p = pn b p b 2 (2) 0 1 b>0 p>0 We onider all development effort with the exeption of the building of oial media feature unk. Thi reflet a realiti enario where the firm i reevaluating it trategy loer to market releae when more aurate information i available. 3 Moreover, a we are fouing on digital good and ervie, the reprodution ot are aumed negligible. Although, a diued in the Introdution, it ha been doumented that prie dirimination an lead to a dereae in WTP beaue of pereived unfairne, there i very little reearh onneting thi diutility to onumer harateriti. Related literature offer variou inight a to how variou utomer group reat to negative firm ation (e.g., prie inreae or ervie failure). For example, under ondition of high prie inequality, onumer that hop with higher frequeny pereive prie inreae a le fair ompared to utomer that hop with lower frequeny (Huppertz et al. 1978). A different tudy by Martin et al. (2009) take a omewhat oppoite tane by howing that loyal utomer do not neearily pereive major prie inreae le fair than nonloyal utomer (and, in the ae of mall prie inreae, atually the oppoite might our). The ame tudy propoe that, under ondition of a prie inreae, pot utomer loyalty i greater for previouly loyal utomer than nonloyal utomer. Suh finding are alo onitent with He et al. (2003), whereby it i argued that more loyal utomer invet in maintaining the relationhip with the vendor and, thu, are more forgiving toward minor negative ation ompared to nonloyal utomer. However, ome utomer may ome to expet ertain relational benefit in exhange for their loyalty. For example, utomer ervie quality expetation may be poitively orrelated to the longevity of the utomer-firm relationhip duration (Heilman et al. 2000). 2 Whih, in the ontext of the market ize being normalized to one, i the ame a the fration of the market that ontitute paying intalled bae. 3 Our reult till hold under a fixed ot aoiated with reearh and development not diretly related to oial media feature. The only differene would be that the region where the firm hooe to enter the market would hrink. Thu, the above literature ugget that diutility from firm ation may be different for ditint utomer group. However, in the abene of a lear onenu regarding how eeding-indued diutility i related to utomer type (or WTP), for ompletene of the analyi we hooe to explore two oppoing model to aount for variou market peuliaritie. The eeding diutility funtion under the two model are parameterized a follow: model SDU + model SDU = 2 = 2 1 where 0. Under model SDU +, for eah utomer, the eeding-indued diutility i poitively orrelated with her type. Thu, the highet-type utomer are experiening the highet eeding-indued diutility. Under model SDU, every utomer experiene a eeding diutility that i negatively orrelated to her type. In thi ae, high-type utomer do not experiene muh diutility beaue of eeding. If adoption tart, then, at any ubequent moment, the intantaneou utility i inreaing in type under both SDU + and SDU. 4 Conequently, our model i onitent with the extant literature on vertial differentiation in the ene that if a utomer adopt, all higher-type utomer mut adopt a well. A uh, utomer type i poitively orrelated with intantaneou onumer WTP. If there are different type that pereive at a given time nonnegative utility from the produt and they have not adopted yet, for impliity we aume the higher-type move firt. In that ene, we aume that type (hene, utility) i poitively orrelated with the urgeny to ue the produt for whatever miion-ritial need that utomer ha. 3. Complete Information We firt onider the ae where the firm ha omplete information about utomer type and, thu, an perform eeding targeted toward peifi individual type. For example, uh enario may orrepond to market where onumer leave a oniderable and relevant informational footprint after (online) ativitie, whih i made available to the vendor. Suh information may be olleted perhap in aoiation with the onumption of a related produt/ervie offered by the ame vendor or by a partner of the vendor. In other ae, uer have to atify ertain ondition to qualify for the free offer, and they mut reveal thi information to the provider prior to reeiv- 4 Under SDU +, at a given point along the adoption proe, if intalled bae i, then u = b 2 p. If the adoption tart, it mean that b 2 > 0 uh that the firm an harge a poitive prie. Then, at later tage, beaue intalled bae inlude both eeded and paying utomer. Under SDU, u = b p. Thu, for any, if adoption tart, then at any given point intantaneou utility i inreaing in utomer type.

6 Information Sytem Reearh 24(1), pp , 2013 INFORMS 169 ing the produt. For example, through it Dreampark global initiative, Mirooft i making developer grade oftware available for free to tudent in many ountrie around the world (Mirooft 2008). To qualify, tudent have to prove their affiliation with an aademi intitution (whih, in the United State, an be done by providing a.edu addre). In another example, in 2009, in a $15 million initiative, Autodek eeded 100 early-tage lean teh ompanie with free oftware bundle eah worth approximately $150,000 (Autodek 2009). In thee example, Mirooft and Autodek managed to target a partiular egment of the market with their free offer Optimal Strategy Under Model SDU + We tart by exploring neeary ondition for optimality: Lemma 1. Under SDU + and omplete information, if the firm tay in the market (i.e., it an make profit) then it optimal trategy b p and optimal eed alloation mut atify the following: (i) b 2 p (ii) all utomer with type 0 are eeded; (iii) all utomer with type 1 purhae the produt. Part (i) of Lemma 1 tate that eeding annot be effetive at parking adoption unle it i oupled with trong enough network effet. In other word, the optimal trategy ha to be hoen in uh a way that at leat the highet-type utomer want to adopt at the very beginning. Part (ii) and (iii) baially apture the fat that, under optimal eed alloation, there i no egment of the market left without a produt. Although it eem more or le intuitive that eed hould go to the low-type utomer to prevent ale annibalization, what i intereting i that, under the optimal trategy, if any uneeded utomer purhae the produt, then all uneeded utomer purhae the produt. Seeding more indue two oppoing effet at the intantaneou utility level: it inreae the network benefit and it alo inreae the eeding diutility. Neverthele, the firm will hooe to manipulate the three ontrol (eeding, level of oial media feature, and priing) in uh a way a to eed right up to the lowet-type paying utomer. Paid adoption our in dereaing order of type for utomer with type 1. The following reult haraterize the optimal trategy of the firm. Propoition 1. Under SDU + and omplete information, if 1/4, then the firm exit the market. Otherwie, if < 1/4, then the firm enter the market and it optimal trategy b p i given by = b = 1 p = b 2 2 A it turn out, the individual rationality (IR) ontraint at adoption time i binding for the highet type. Thu, under an optimal trategy, the firm will hooe a prie uh that eeding jut jumptart adoption and thi minimal puh i enough for the adoption to gain momentum and not tall until every uneeded onumer purhae the produt. We emphaize that thi equilibrium trategy i not dependant on adoption equene. Suppoe utomer exhibit variou degree of inertia in reating to market hange (uh a intalled bae growth), but they all eventually reat to uh hange. Firt, for any given eeding ratio, priing uh that IR ontraint i binding for the highet-type utomer repreent the highet prie the firm an harge uh that adoption an tart (one the highet-type utomer deide to make a move). One adoption tart, if it evolve in dereaing order of type, it an be hown that IR ontraint at adoption time will not be binding for any of the other paying utomer exept for the highet-type paying utomer. Even if adoption doe not our in dereaing equene of type, a long a it doe not tall, we reah the ame full market overage and, thu, an idential optimal trategy. Otherwie, if at any moment adoption tall momentarily (perhap beaue of ome higher-type utomer being low in reating to market evolution ompared to lower type utomer), it will pik up eventually a there will alway be ome uneeded type for whom adoption yield nonnegative utility. A uh, full market overage i reahed again. In uh ae, priing lower uh that IR ontraint i not binding for the highet type i uboptimal. Thu, the trategy derived in Propoition 1 remain optimal even under different adoption equene. Next, we fou our attention on the interation between building more oial media feature to inreae the trength of network effet and eeding the market. Tehnially eah of thee ation on the firm behalf i aimed at booting WTP but they both ome at a ot. So a natural quetion arie: are thee ation omplementary or in ubtitution of eah other? In other word, if the ot/penalty aoiated with being more proative on one of thee two dimenion inreae, would the firm inreae or dereae it ativity on the other dimenion? The following reult addree thi quetion: Propoition 2. Under SDU + and omplete information, when the firm tay in the market < 1/4, eeding and embedding more oial media feature are omplementary ation. If any of the ot aoiated with thee ation or inreae, the firm ale bak on both dimenion i.e., / 0, / 0, b / 0, b / 0. The fat that the firm ale bak on a partiular ation if the aoiated ot/penalty with that repetive ation inreae (i.e., / 0, b / 0) i

7 170 Information Sytem Reearh 24(1), pp , 2013 INFORMS to be expeted. However, the intereting reult in Propoition 2 haraterize the interation between the two ation. At firt glane, one might expet that a the ot of booting the trength of network effet via more ophitiated oial media feature inreae, the firm might turn to the other lever it ha ae to, eeding, in order to inreae WTP (and vie vera). However, eeding i valuable to the firm a long a it doe not annibalize too muh demand and doe not indue paying utomer to downgrade their valuation of the produt too muh. When it i more otly to generate trong marginal network effet via oial media, the firm redue it invetment along that dimenion. In turn, if it were to ompenate uh an ation by a boot in eeding, to reah the ame overall level of network effet the firm would have to eed more utomer. Thu, for imilar network effet, the firm would atually ee both an inreae in eeding penalty and, beaue of full market overage, a dereae in the number of paying utomer. A it turn out, thee two effet dominate the benefit from the boot in overall network benefit from eeding, and, onequently, the firm prefer to downize the eeding pool a well. A imilar argument goe in the other diretion. If the eeding-indued downgrading of WTP of paying utomer i more intene, then the firm firt ale bak on eeding. Again, at firt glane, it might eem like a good idea in uh a ae to imultaneouly boot the trength of network effet o that the firm doe not have to rely on eeding that muh. Neverthele, one operating at optimal network effet trength level, it i otly to further upgrade b, and thi ot i not reovered by the benefit of an upward hift in WTP beaue of tronger network effet. When eeding doe not indue any diutility for the paying utomer ( = 0), then = 1/2, b = 1/8, and p = 1/16. In uh ae, the optimal eeding ratio and, impliitly, the ratio of paying utomer are independent from the trength of network effet embedded in the produt. If the ot of adding oial media feature i inreaing, the firm ettle for a lower trength of network effet and, at the ame time, harge utomer le uh that IR ontraint remain binding for the highet type while the eeding ratio i kept unaltered. A een from Propoition 1, the upper bound for i 1/2. Thu, under the optimal trategy, the oftware firm prefer an outome where the majority of utomer are paying utomer whoe WTP i influened by a well-balaned ombination of eeding and oial media feature that boot network effet. Lat, we mention that the reult in thi etion an be extended to more general type ditribution funtion and utility truture. Thi diuion ha been inluded in Appendix B Optimal Strategy Under Model SDU In thi etion, we explore a etting where utomer type i negatively orrelated with eeding-indued diutility. A argued in 2, in ome intane, more loyal utomer may pereive le diutility beaue of eeding proedure ompared to le loyal utomer. It may be the ae that more loyal utomer alo have higher WTP. Cutomer with higher WTP may be big lient uh a orporation who developed a relationhip with the vendor over time and for whom withing ot would be too high. Suh lient might be le likely to fret muh over ome other utomer reeiving the produt for free. At the other end of the type petrum, utomer who do not derive muh value from the produt and might operate on a tight budget might be more upet if other got it for free. The following lemma haraterize the market egmentation under the vendor optimal trategy. Lemma 2. Under SDU and omplete information, if the firm tay in the market (i.e., it an make profit) then it optimal trategy b p and optimal eed alloation mut atify the following: (i) b p (ii) there exit a marginal type m > uh that all utomer with type m 1 purhae the produt, no utomer of type < m purhae the produt, and all eed ome from the interval 0 m (though they may not need to be grouped at the very low end). Condition (i) illutrate the fat that adoption ha to tart with the highet-type utomer. Condition (ii) tate that, imilar to the SDU + ae, paid adoption our among the top-tier utomer. Neverthele, unlike in the ae of SDU +, full market overage may not be optimal. We will reviit thi point later. Firm optimal trategy i preented below. Propoition 3. Under SDU and omplete information, the firm alway enter the market. Let m be the lowet type among paying utomer under optimality. Then firm optimal trategy i a follow: Region 1: 0 1/8. Then = 1 2 b = 1 8 m = 1 2 Region 2: 1/8 < < /16. Then = 3 m 1 b = 1 m 2 m m = x where x i defined a the unique real olution to the equation 2 6x + x 3 = 0 over the interval ; Region 3: /16. Then = 3 m 1

8 Information Sytem Reearh 24(1), pp , 2013 INFORMS 171 b = m = b b b Under all region, p = b 1 m + m 2 1 m. One immediate differene from SDU + i that, under SDU, the firm will alway prefer to enter the market. That i beaue the highet-type utomer experiene very mall eeding diutility and thu their WTP i more or le ditated by network-generated value. If i very high, low b an indue poitive WTP at top tier, whih, oupled with low (but not too low) prie would indue revenue that would dominate aoiated oial media ot. Another differene from SDU + i that, under SDU, the IR ontraint at adoption time i binding for the marginal-type paying utomer rather than the highet paying utomer. Thu, at the very beginning, many utomer may be willing to adopt olely baed on the network value generated by the eed. One of the reaon leading to thi outome i the fat that eeding diutility i low for top-tier utomer under model SDU. One intereting apet of the optimal trategy under SDU i that, when eeding penaltie and ot aoiated with oial media feature are relatively low ( 1/8), then the firm repond to the diutility olely through adjuting the prie downward by the bigget eeding diutility a paying utomer an experiene (i.e., the one experiened by the lowet paying type m ). It eeding and oial media engineering trategie do not hange under mall flutuation in eeding penalty. Alo, in uh region, eeding i not influened by hange in the ot of adding more oial media feature. However, in market haraterized by higher ot ( > 1/8), the optimal eeding and oial media engineering trategie will depend on both and. A it turn out, differently from SDU +, under SDU, full market overage doe not alway hold. The following orollary to Propoition 3 apture thi: Corollary 1. Under SDU and omplete information, full market overage = m i attained only when 1/8. When > 1/8, then < m, and there are alway utomer that are not purhaing the produt and are not eeded. When the ot aoiated with eeding and/or building more oial media feature into the produt are relatively high, it i not optimal to eed more or lower the prie to a level where all uneeded utomer adopt. Uneeded utomer of low type atually have high diutility from eeding o their WTP would be rather low. The firm i better off keeping the prie higher and extrating more onumer urplu from the high tier. Seeding the aforementioned uneeded utomer would alo generate a dereae in WTP for utomer in the mid-type range (where there i ignifiant eeding diutility) that would hrink or ompletely eliminate any benefit from the inreaed WTP at the high end (where there i little eeding diutility). In region 2 and 3, a < m and = 3m 1, it follow immediately that < 1/2. Thu, again, it i never optimal to eed more than half of the market. In pite of the differene in both model and optimal trategy, we find that the omplementarity reult between eeding and building oial media feature for model SDU + (Propoition 2) extend to SDU. Propoition 4. Under SDU and omplete information, optimal eeding ratio and optimal trength of network effet b are both noninreaing in and. For region 1, b i independent of and i independent of both and. Strit monotoniity i experiened in region 2 and 3. A diued above, under SDU, under optimality, IR ontraint i binding for the lowet paying type beaue highet type have negligible eeding diutility. If adding more oial media feature beome otlier, the firm will redue it invetment in oial media and thu, dereae the marginal network-generated value of the produt. Thi, in turn, would lower WTP for all utomer. If the firm would repond by inreaing eeding to boot network effet, atually the WTP of the low end of the paying group would dereae a lot and that ditate prie. Thu, the firm would ee either a maller paying egment or would have to harge a lower prie, and thoe ation would lead to a lower profit. A a reult, it i better for the firm to dereae eeding a well. Alternatively, if inreae, the low end get affeted mot. To ompenate for that dereae, the firm would have to invet a lot in oial media feature. Granted that at the top tier thi inreae WTP for the utomer, however, in region 2 and 3, where wa alo relatively high, aoiated oial media ot would inreae teeply for uh a proe and wipe out other benefit. Therefore, the firm find it more profitable to atually dereae oial media feature and manipulate demand more through prie. 4. Inomplete Information When the firm ha inomplete market information, i.e., it know the type ditribution but not the exat type of eah utomer, it may not be able to target individual utomer by type. In thi enario, when a firm attempt to eed the market, one of the inherent downide i the potential to annibalize ome of the demand from high-type utomer beaue the firm annot enure that the eed go to the lowet

9 172 Information Sytem Reearh 24(1), pp , 2013 INFORMS end (Niuleu and Wu 2012). For example, in the app market for ios and Android devie, it i not unommon for ome of the developer to offer their app for free for a limited period of time to boot adoption in the market. In uh ae, any utomer who ome aro the app during thi limited time window an download it for free. We onider the ae where the firm deide trategially on the eeding volume but the eed end up being pread uniformly in the market. We again explore both model SDU + and SDU for a omplete piture of how the firm adjut eeding and the level of oial media feature in produt in repone to flutuation in ot depending on market peifi. Before we diu eah model, we would like to point out ome other differene between uniform eeding and eeding under omplete information. Under omplete information, we aw in 3 that full market overage alway our under SDU + and it may our alo for SDU ontingent on mall ot. However, under uniform eeding, full market overage never our, regardle of the eeding diutility model. Thi i beaue for any poitive prie, there will alway be uneeded utomer with low type for whom benefit fall below that prie. Moreover, for both eeding diutility model, the IR ontraint at the adoption time i binding for the lowet-type paying utomer. Thi i not the ae under omplete information for SDU Uniform Seeding Under Model SDU + In thi ae, firm optimal olution i a follow: Propoition 5. Under SDU + and uniform eeding, when the firm deide to enter the market, it optimal trategy b p i given by [ = zb ] 2 17 b = arg max {b1 2zbzb 2 zb 4 b 0 b zb 2 1 3zb b 2 } p = b where zb i defined a the unique olution to the equation b 2 1 4z 4 2 z 3 + bz 2 + 9z = 0 over the interval /17. The marginal paying type i m = b /b 1. Alo, b exit and i unique. Moreover, there exit a bound 1/4 uh that when > then the firm doe not enter the market. Firt, we oberve that the upper bound on eeding ha dereaed dramatially / ompared to the omplete information ae 1/2. The fat that uniform eeding effetively redue the available market at the high end deter the firm from puruing overly aggreive eeding ampaign. Reviiting the app market example, thi obervation i onitent with the pratie to limit the free promotion to a hort window to avoid exeive eeding. Moreover, profit under omplete information eeding dominate profit under uniform eeding. Thu, when the firm would not enter the market under the former ae, it will alo not do o under the latter. When = 0, the optimal trategy i given by = 1/4, b = 1/16, and p = 1/64. Given the omplexity of the olution, we perform a numerial enitivity analyi on the optimal eeding ratio and level of oial media feature b with repet to their aoiated ot. Thi analyi yield very intereting and, in ertain region, different reult, ompared to the one under omplete information. We apture thee reult in Figure 1, where panel (a) () illutrate how and b hange with repet to, and panel (d) (f) explore enitivity with repet to. We ee that the omplementarity reult now hold only when ot are uffiiently large ( high enough). In uh a ae, imilar inight ompared to the one in 3.1 apply and we omit thi diuion for brevity. When i mall, we an ee that it i optimal to build a high level of oial media feature in the produt, but uh a pratie would not be optimal for high. Thu, imilar to the omplete information ae and quite intuitive in nature, b will tend to dereae in. However, thi i where imilaritie top. Firt, under mall we notie that i inreaing in both and. Moreover, b tend to alo inreae in when i mall. To get a better undertanding of the enitivity with repet to, we illutrate in Figure 2 everal propertie of the equilibrium outome orreponding to panel (a) in Figure 1. Note that under omplete information we have full market overage and an inreae in would alway reult in a hrinking of the ize of the paying group. When new eed are given away, all of them are atually annibalizing paying utomer. However, a it turn out, under uniform eeding, beaue we do not have full market overage (beaue of uneeded low type not adopting), new eed only annibalize a fration of the paying utomer a they get ditributed uniformly. Thu, the firm an atually inreae at the ame time both the eeding ratio and the ize of the paying group 1 1 m, a an be een from panel (a) and (d) of Figure 2. Given that inreaing eeding involve more eed to the high type a well, the high-type paying group hrink. However, the firm repond by lowering prie and expanding adoption toward the lower end of the market, a an be een from panel (b) and (). When i mall, inreaing eeding ome at a low penalty and thu, it reult motly in inreaed WTP. Additional inreae in b alo

10 Information Sytem Reearh 24(1), pp , 2013 INFORMS 173 Figure 1 Optimal and b Under Uniform Seeding and Seeding Diutility Model SDU (a) = (b) = () = * () * () * () * () b * () * () b * () b * () b * () b * () b * () * () (d) = (e) = (f) = * () * () * () * () * () * () b * () b * () b * () b * () b * () b * () Note. The yy -plot have b on the left-hand y -axi and on the right-hand y -axi. Panel (a) () onider enitivity with repet to, wherea panel (d) (e) onider enitivity with repet to. further add to the inreae in WTP. In uh a ae, the newly added revenue from the low end of the market (beaue of lower prie and higher WTP, both of whih indue more utomer to join) may atually ut the double loe at the high end (beaue of fewer uneeded utomer and lower prie). A inreae, for intermediate range we ee that the firm will with toward uing oial media engineering a a ubtitute for eeding. One i not too mall, the benefit from extra eeding vanih a the WTP annot be booted that high without ubtantial invetment in booting b beaue of the inreaing diutility. Thi effet, together with the redued pool of paying high-type utomer, make inreaing the eeding ratio uboptimal. In uh a region, in parallel with a dereae in we ee an inreae in b. The firm find it optimal to ontinue to expand the market into lower type by dereaing prie, thu aborbing the inreae in the eeding diutility. At the ame time, the dereae in alo expand the group of high-type paying utomer. When the eeding diutility i not too high, the firm will till keep at relatively high level, adjuting upward b in parallel with the dereae in to revere a dereae in network value. In thi ae, maintaining b at high level i alo worth it given that inreae in WTP of Figure 2 Detail of Equilibrium When = 03 Under Uniform Seeding and Model SDU (a) * and b * (b) * m () p * 0.70 (d) (1 * m ) (1 * ) * () b * () 0.21 b * () * ()

11 174 Information Sytem Reearh 24(1), pp , 2013 INFORMS high type. Neverthele, one beome too high, it i too otly for the firm to ue thi approah beaue it would be neeary to invet a lot in b to maintain WTP at high level. Thu, in uh region, the firm will reort to dereaing all three ontrol b p. For, we ee in panel (d) (f) of Figure 1 that under mall value, eeding ratio i inreaing in, whih, again, indiate that eeding and oial media engineering are ubtitute to eah other in generating profit. An inreae in will indue a dereae in the level of oial media feature embedded in the produt. Similar evolution of p, m, and 1 1 m with repet to are oberved a in the ae of hange in (plot are omitted for brevity). For mall range of, even if b dereae, it remain moderate in value, providing the potential for a ubtantial network value if intalled bae i robut. Small upward adjutment in eeding ratio might be profitable a they help retain network value and allow the firm to profitably expand in the lower type egment of the market by lowering the prie. However, the boot in eeding i taken advantage of exatly through network effet and, thu, one get really high, the firm would not find it optimal to further invet in b whih, in turn, would expoe utomer to a potentially high eeding diutility without high benefit from the network. To ompenate for thi, the firm will redue the eeding ratio, thu inreaing the number of paying utomer among the higher type and alo reduing the diutility at all level. The aoiated prie dereae alo allow the firm to trategially extend the group of paying utomer toward more of the lower-type utomer Uniform Seeding Under Model SDU Under uniform eeding and model SDU, firm optimal trategy i a follow: Propoition 6. Under SDU and uniform eeding, the firm alway enter the market and it optimal trategy b p i given by [ 2b + 4b 2 + 3b 0 5 ] 5 > = 1 4 = 0 b = p = 2b + b The marginal paying type i m = 1/3. Under SDU and uniform eeding, the marginal paying type i alway the ame (m = 1/3), in ontrat with mot of the ae previouly tudied, with the exeption of the low-ot region 1 under SDU and omplete information. Even ompared to the latter, there i one important differene, namely, that in the urrent ae, while m i ontant, the firm adjut all it ontrol b p in repone to flutuation in and. Moreover, when > 0, a an be rewritten a = 1/3 2b / + b /b / + 3, given that b / i a funtion of, it follow that i a funtion of, and oppoite-diretion ot flutuation that leave unhanged alo leave the eeding ratio unhanged. Moreover, we note that, ompared to the full information ae, the upper bound on the eeding ratio alo dereae ignifiantly from 1/2 to 5 5/ Similar to the diuion in 4.1, given that eeding trategie redue the top-tier paying group under uniform eeding, the firm will not engage in overly ubidizing adoption. We next explore the enitivity of and b with repet to ot. An inreae in would reult in a dereae in WTP pronouned more ignifiantly at lower type level. A uh, unlike under SDU +, the firm doe not ee it benefiial to expand into the lower market by induing lower m and more paid adoption. In turn, it hift toward inreaing the paying group toward the top tier by lowering. In tandem, it alo lower b a there i le preure to indue trong network effet to ompenate for eeding diutility and thi allow the firm to eae down on ot of building a high level of oial media feature into the produt. On the other hand, when inreae, given onvexity of oial media ot, inight remain robut behind b being dereaing. One the networkindued value dereae for all type, if the firm would try to ompenate with an inreae in eeding, it will indue a higher diutility and redue the number of paying utomer at the top tier. The firm would then have to operate at uboptimal prie level to utain a robut adoption. A uh, the firm alo prefer to redue the eeding ratio, preventing a trong drop in WTP and, at the ame time, inreaing the ize of the paying group. Thu, the omplementarity reult unovered under full information ontinue to hold. The above inight are formalized in the next reult. Propoition 7. Under SDU and uniform eeding, optimal eeding ratio and optimal trength of network effet b are both noninreaing in and. Unlike in the ae of omplete information, the two eeding diutility model may lead to optimal trategie that are different not only in peifi value but in more fundamental way. A uh, when there i eeding diutility in the market, it i very important for firm to aount for it proper form when detailed onumer information i laking. Returning to the ios example at the beginning of 4, app developer an ue uh inight to better time the length

12 Information Sytem Reearh 24(1), pp , 2013 INFORMS 175 of the promotional ampaign (i.e., ontrol the ize of the eeded pool of utomer) ontingent on market harateriti. 5. Extenion: Aymmetri Network In thi etion, we preent an illutration of how our reult an be extended to aymmetri network. Among other, network aymmetry may be indued by lak of full onnetivity (e.g., Zubek and Sarvary 2011) and/or by ingle-diretion link (e.g., Lehmann and Eteban-Bravo 2006). A uh, there are a vat number of poible aymmetri network enario. We will explore one ae that pertain to the latter ategory. Suppoe the market ha two dijoint onumer egment: low-value egment L ( 0 r) and highvalue egment H ( r 1). Suppoe the network i fully onneted but aymmetri in nature. Within eah egment, all link are bidiretional. Aro egment, link are unidiretional, going from highvalue egment to low-value egment, but not vie vera. In a ene, the high-value and low-value egment orrepond to the innovator and imitator in Lehmann and Eteban-Bravo (2006). We aume that the network-indued value for eah utomer i given by the volume of inoming link. For eah egment, the eeding ratio i alway upper bounded by the ize of the egment. In thi extenion, we fou on SDU + eeding diutility model. 5 We further aume that eeding diutility i only manifeting within the high valuation egment (and only with repet to eed in that egment), wherea in the low-value egment it i negligible ( L = 0, H = > 0). Of oure, thi i jut one example of how eeding diutility an manifet in the market. We onider the ae of full information. Although the market i egmented, we till aume that the eller only approahe it with a unique prie. Seller optimization problem onit of how to hooe p, b, and eeding level { L H orreponding to eah egment. It i traightforward to etablih the following propertie in equilibrium under optimal trategy. Firt, all eed hould go to the lowet end in eah egment. Seond, both egment will be fully overed. Let u fou on the high-end egment firt. If the egment H i not ompletely eeded (that might be an option), and prie i low enough uh that paid adoption an tart in that egment, given that egment H an be treated in iolation (it i not influened by egment L), imilar to our previou reult it turn out that adoption goe all the way until the high egment i overed without talling. It i irrelevant if at 5 Similar inight hold under SDU and their diuion ha been omitted for brevity. ome point along the proe adoption alo tarted at the low level. Note that if there i any paid adoption in egment H, the highet-type utomer mut alway be willing to adopt from the tart. It annot be optimal for ome utomer to be left without eed in the high-end egment and the prie be et above the WTP of the highet type. Thi i beaue lowegment adoption annot inreae WTP at the high end (beaue of unidiretional link) and thu, thoe uneeded utomer in egment H will never buy. However, eeding them would inreae WTP in the low-end egment even further (beaue egment L doe not exhibit eeding diutility). A uh, in optimality, either the entire egment H i eeded or all uneeded utomer purhae the produt. Next, in term of equilibrium, we point out that it i irrelevant whether paid adoption tart in egment L before being omplete in egment H. If it tart after, baially the WTP of the firt adopter in egment L will be higher. Neverthele, if it tart before, if at any point there i any talling in egment L, adoption will pik up again one adoption from above pik up. A uh, any equilibrium outome where paid adoption in the low egment i tarting before paid adoption i omplete in the high egment an be repliated under a trategy where the low-egment market i opened after high-egment market i fully overed. If it i optimal to et the prie very high uh that there i no adoption in the low-value egment, then it i irrelevant whether utomer are eeded in that egment or not beaue of aymmetry. A uh, for impliity, we an aume in thee etting full eeding of the low-value egment. However, if it i optimal to have paid adoption in the low-end egment a well, then, one adoption tart, it an again be hown that it will not tall until everyone i overed (beaue of the onavity of the intantaneou WTP, via a imilar argument a in the ae of ymmetri network). Under optimality, it an be eaily hown that when r i loe to one, it i optimal to eed the entire high-end egment, and when r i loe to zero, it i optimal to prie in uh a way that there will be no paid adoption anyway in the low-end egment. When r i in an intermediate range, it i optimal to eed a fration of eah egment and have paid adoption in eah egment a well. We fou the remaining part of our diuion on the intereting region for r,, and, where it i optimal to have paid adoption in both egment. Figure 3 and 4 illutrate enitivity of L, H, L + H, and b with repet to ot parameter and. In the plot, parameter r i hoen at 04 and we onider 01 04, range that enure optimality of paid adoption in both egment. Firt, a it an be een from panel (a) () in Figure 3, aggregate eeding ratio = L + H i dereaing in. Thu, we unover a imilar pattern a in

13 176 Information Sytem Reearh 24(1), pp , 2013 INFORMS Figure 3 Senitivity of Optimal Strategie with Repet to Under Aymmetri Network Struture and Seeding Diutility Model SDU + When r = (a) = (b) = () =0.4 * H + * L * H + * L * H + * L (d) = (e) = (f) = * L () * H * H () * L () * L * H * H () * L () * L * H * H () * L (g) = (h) = (i) = 0.4 b * () b * () b * () the ae of the ymmetri network. For low range of, if inreae, the eeding diutility in egment H inreae but there i no impat on egment L. All ele equal, the dereae in WTP in egment H put downward preure on prie. Thi impat not only revenue from high-end egment but alo that from low-end egment. To prevent too muh revenue lo, the firm will reat firt by releaing ome of the preure on prie in the high-end by booting b. When i low, a in panel (d) and (g) or (e) and (h), the firm will embed a relatively higher level of oial media feature in the produt. A b i puhed even higher, while i till mall, it may be benefiial to inreae H in repone to an inreae in beaue in both egment eeding i taken advantage of via network effet. In the low-end egment, the boot in b eliminate ome of the need for a high level of eeding and, thu, the firm an atually dereae L without affeting too muh the WTP of onumer in that egment. For high, a in panel (f) and (i), in mall range of, given that it i too expenive to operate at high level of b, the boot in b allow the firm to redue it reliane on eeding in the low-end egment. It will alo reult to the ame pratie in the high-end egment for different reaon. Beaue b i mall, for mall, a inreae the mall boot in b annot ompenate for the inreae in eeding diutility. A uh, we ee in panel (e) and (f) that H i dereaing in. Thi reaoning applie pretty muh in all ae one get large enough beaue eeding diutility imply grow too big and the firm will urb eeding in the high-end egment, beaue it annot boot b too muh beaue of the aoiated onvex ot. When i high, thi i alo aompanied by an expeted dereae in b, a an be een in panel (i) of Figure 3. Figure 4 apture enitivity with repet to. Again, from panel (a) () we ee that aggregate eeding ratio i dereaing in. At the ame time, onvexity of oial media ot indue b to dereae in. Thu, inight from the ymmetri network ae arry through. A inreae, b dereae, and, a a reult, in both egment, the network value of the produt dereae. When i low and i not too high, to avoid preure on prie, the firm might try to boot H. Thi an be een in panel (d) and (e). However, one eeding diutility i too high, thi i not optimal anymore, a an be een in panel (h). One i

14 Information Sytem Reearh 24(1), pp , 2013 INFORMS 177 Figure 4 Senitivity of Optimal Strategie with Repet to Under Aymmetri Network Struture and Seeding Diutility Model SDU + When r = (a) = (b) = () = 0.4 * H + * L * H + * L * H + * L (d) = (e) = (f) = * H * H * H * L () * H () * L () * L * H () * L () * L * H () * L (g) = (h) = (i) = 0.4 b * () b * () b * () too high, b i puhed too low and eeding penalty would be too high if the firm would try to puh eeding further. A uh, it will dereae H. Looking at the low-end egment, a the firm make adjutment in the high-end egment to prevent too muh preure on prie, it will expand it ale in the low-end egment by reduing eeding ratio, thu having more paying utomer. 6. Conluion Firm nowaday are inreaingly proative in trying to trategially apitalize on onumer network and oial interation. In thi paper, we omplement an emerging body of reearh on the engineering of WOM effet by exploring a different angle through whih firm an trategially exploit the value-generation potential of the uer network. Namely, we onider how oftware firm hould optimize the trength of network effet at the utility level in tandem with the right eeding and priing trategie in the preene of eeding diutilitie. To the bet of our knowledge, our tudy i one of the firt exploration of thi reearh path. Our reult have important managerial impliation for pratitioner in the indutry who are trying to apitalize on network effet in a more effetive way. Moreover, the appliability of our reult i augmented by the fat that we onider two potential eeding diutility enario, thu overing multiple plauible onumer reation in the market. Under omplete information when the firm an target individual utomer type, after deriving the optimal firm trategy and the aoiated market truture, we unover ounterintuitive omplementaritie between eeding and the building of oial media feature in the preene of diutility aoiated with eeding. Although both initiative ontribute to a diret inreae in WTP of onumer, eah of them ome at a ot. The inherent trade-off indue the firm to ale bak on any of thee initiative if the ot aoiated with the other inreae. Thu, market with high eeding diutility beaue of prie dirimination are alo exhibiting low network effet embedded in the produt. Alternatively, market for produt where ignifiant invetment in oial media feature i neeary to make an impat on the network value alo experiene low level of market

15 178 Information Sytem Reearh 24(1), pp , 2013 INFORMS eeding. We further how that our reult are robut to both eeding diutility model. Under inomplete information, although omplementarity between eeding and building oial media feature ontinue to hold for SDU everywhere, under diutility model SDU + we unover a hot of new inight. In partiular, a more peuliar omplementarity an our a well, where, for mall eeding penaltie, it may be optimal for both eeding ratio and level of oial media feature to be inreaing in the eeding penalty. Moreover, for intermediate eeding penaltie or low ot we unover the potential for the two ation to at a ubtitute. Thu, different diutility model may atually lead to very different optimal trategie under inomplete information. Thi highlight the importane of not overlooking the form of the eeding diutility in variou market, epeially in the abene of onumer information. In an extenion, we alo explore how our framework and inight an be extended to ertain lae of aymmetri network. At a high level, ome of the previou reult remain robut, while new inight alo emerge. Moreover, we explore how the firm jointly approahe the eeding proe for different egment in the market by alloating eed in eah egment. For example, when both ot are high and the market i almot evenly plit between the high and the low egment, omplementarity between eeding and building oial media feature ontinue to hold. On the other hand, when eeding penalty i low, thi omplementarity relationhip break a the firm may find it optimal to adjut overall eeding ratio and oial media feature in oppoite diretion in repone to an inreae in eeding penalty. A expeted, the value of information omputed by taking the differene (or perent differene) between profit under omplete and inomplete information i dereaing in both and. We only briefly mention here thi point beaue it i intuitive. A it i beoming inreaingly otly to indue high WTP, firm are le willing to pay high premium for low return on additional information. Although our paper i among the firt to addre in an analytial etting the interation between oial media and eeding trategie, it doe have it own limitation that preent variou intereting opportunitie for future reearh. Firt, we aumed utomer have limited information about the market truture (type ditribution) and at myopially. In that ene, our tudy add to the rih literature that aume a more or le pronouned form of bounded rationality on the demand ide. In future tudie, thi aumption an be relaxed toward an analyi of market in the preene of trategi utomer. We expet many of our inight to arry through. Seond, one ould look into extending our modeling framework to explore the impliation of ompetition on how firm would jointly adjut eeding and oial media feature. Third, alternative form of network effet (e.g., additive) an be onidered in aoiation with produt that alo arry an intrini value that i independent of the uer network. Fourth, a an alternative to eeding, firm an onider freemium trategie to jump-tart adoption. Fifth, it would be intereting to dive even deeper into exploring aymmetri network, going beyond our illutration in 5. Although we foued motly on the exitene and diretion of link between uer, an intereting etting to onider would be one where link alo have weight. A uh, firm might fou eeding trategie toward opinion leader. Alo, it would be intereting to explore firm trategie in aymmetri network where ome of the uer might onider a produt to be a tatu good, deriving additional utility from being aoiated with the repetive brand image (Kukov 2007, Kukov and Xie 2012). Sixth, for impliity and, in ome ae, analytial tratability, we aumed a reaonable equene of adoption whereby higher type move firt in ae when multiple utomer who did not adopt yet would derive poitive utility at a given moment. A diued in the paper, the adoption equene doe not influene the reult in 3.1 and 5. However, it would be inightful to relax thi aumption and explore how it affet optimal eeding and oial media trategie under SDU. Lat but not leat, it would be a very informative exerie to empirially tet our model predition. Aknowledgment The author thank the enior editor, aoiate editor, and the two anonymou reviewer for their ontrutive feedbak on the manuript. The author alo extend their gratitude to Terrene Augut, Hemant Bhargava, Rahel Chen, Vidyanand Choudhary, Debabrata Dey, Vijay Mookerjee, Barrie Nault, Yong Tan, Yong-Pin Zhou, Feng Zhu; the partiipant at the Information Sytem Reearh Speial Iue Workhop on Soial Media and Buine Tranformation, Univerity of Maryland, June 2 3, 2012; and the partiipant at the Theory in Eonomi of Information Sytem Workhop, Laguna Beah, June 9 10, 2012, for their inightful omment. The author aknowledge the finanial upport from the NET Intitute ( [Summer Reearh Grant #11-07] and the Center for International Buine Eduation and Reearh at Georgia Intitute of Tehnology. Yifan Dou alo aknowledge partial finanial upport from the National Natural Siene Foundation of China (NSFC) [Grant and ] and from Tinghua Univerity Initiative Sientifi Reearh Program [Grant ]. Appendix A. Proof Proof of Lemma 1. (i) If b 2 < p, then paid adoption annot tart regardle of how eed are aigned. Immediately after eeding, there i a ma of onumer of

16 Information Sytem Reearh 24(1), pp , 2013 INFORMS 179 ize in the market. Given that we aume utomer are myopi, even the highet-type utomer ( = 1) annot pereive a poitive momentary utility at that tage. Thu, no uneeded utomer i willing to be the firt to pay for the oftware. (ii, iii) We prove (ii) and (iii) imultaneouly. Note that if the firm hooe to enter the market (i.e., it an make profit), then > 0 and b > 0. Via imple interhange argument, it an be eaily hown that the lowet-type paying utomer mut have type greater or equal to the highettype eeded utomer. Baially all that we have to how i that there are no uneeded utomer that are left without a produt. Let be the highet type among eeded utomer and p be the lowet type among paying utomer. Then 0 p 1. It i trivial to ee that the highet-type utomer (with = 1) will be purhaing utomer under any optimal eeding poliy and eed alloation. Conider the funtion h = b p Then h i onave in. If all utomer with type above adopt, then h would repreent the pereived utility for type at the moment thi type i onidering adoption. Beaue onumer of type = 1 are adopter, it mut be the ae that h1 0. Moreover, the following hold: Conequently, b 2 = b 3 b 2 h h1 0 Beaue h i onave in, thu, h 0 for all 1. Beaue, then it immediately follow that adoption annot tall before adopt. Conequently, we have = p and all utomer above adopt. The only thing left to prove i that =, i.e., there are no uneeded utomer below. We prove thi argument by ontradition. Suppoe that under an optimal eeding alloation and optimal parameter value, >. Then, by eeding in interval 0, the utomer in the interval will alo adopt given that h 0 for, and thu profit will inreae. Contradition. Suh a eeding alloation annot be optimal. Thu, =. Cutomer in the interval 0 are eeded and all other utomer end up buying the produt. Proof of Propoition 1. Aording to Lemma 1, the firm will only onider enario where b 2 p. Then, utomer with type 0 are eeded and utomer with type 1 are purhaing the produt. For eah paying utomer, at purhae time the intalled bae i = 1 + beaue, in addition to all eeded utomer, all utomer with higher type would have already purhaed the produt and no utomer with lower type i moving ahead of urrent type. Therefore, the utility of the utomer of type at purhae time i u b p = b1 + 2 p (A1) The utility funtion i onave in. Thu, for adoption to tart and not to tall, it i neeary and uffiient that the utilitie of the firt paying utomer and lat paying utomer are nonnegative: u = 1 b p 1 = b 2 p 0 u = b p = b 2 p 0 Given that 0 1 we have b 2 b 3. Beaue the firm i profit maximizing, the IR ontraint will be binding for the highet type and thu Conequently p b = b 2 b = b 2 1 b 2 (A2) b i quadrati and onave in b. Note that for the ontraint b 2 p to hold, it i neeary that b. Solving thi ontrained optimization problem, it immediately follow that { b = max } 1 2 Note that Then 1 = if < 1 2 b 2 otherwie Thu, if 1 2 the firm annot make any profit and would exit the market. Therefore, to make a profit, the firm would hooe < 1 2. In thi ae, note that = 0 ha four root 1 = 2 = 0, 3 = 1, and 4 = 1 4. We ditinguih two ae: (i) 1/4. Then 4 1 = 2 = 0 < 3. Then i dereaing on 4, dereaing and then inreaing on 4 0, bouning off at zero beaue that i a double root, dereaing and then inreaing on 0 1, and inreaing on 1. Given that the firm onider onequently we mut have = 0. The optimal prie and optimal network effet are zero and the firm exit the market beaue it annot make any profit. (ii) 1/4 >. In thi ae, 1 = 2 = 0 < 4 = 1 4 < 3 = 1. Then i dereaing on 0, bouning off at zero beaue that i a double root, inreaing and then dereaing on 0 1 4, dereaing and inreaing on 1 4 1, and inreaing on 1. Therefore, i the unique root of the firt order ondition (FOC) that fall in the interval That root i not zero (whih i one of the root of the FOC). Computing the FOC, we get = The FOC give optimal eeding ratio 31 2 = 4 It i trivial to verify that 1/ Proof of Propoition 2. Follow diretly from omputing the derivative of the optimal quantitie and b derived in Propoition 1 with repet to and.

17 180 Information Sytem Reearh 24(1), pp , 2013 INFORMS Proof of Lemma 2. (i) At the very beginning, right after the eeding proe and before any paid adoption ha ourred, a utomer of type pereive an intantaneou utility u = b 2 1 p = b p. Thu, at the very beginning (and atually alo at every moment afterward), the intantaneou utility i inreaing in type. Therefore, at leat the highet-type utomer mut want to adopt. Thu, we need u1 0, or b p. (ii) Given that at any moment utility will be inreaing in type (diutility i the ame one eeding ha ourred and the network benefit inreae more for the higher type), it an be eaily oberved (via an interhange argument) that eeding hould not be at the high end. No eeded utomer hould have a higher type than a paying utomer. Moreover, given monotoniity of utility in type at any given time, everyone with type above the marginal type hould adopt a well. Proof of Propoition 3. We preent here a keth of the proof. Some portion are omitted for brevity but available from the author upon requet. Firt, for the firm to make any profit, it i neeary to harge a poitive prie. That mean that the highet-type adopter ( = 1) mut have poitive benefit from the produt before any paid adoption our. Beaue b > 0, highet-type adopter alway have a poitive WTP for the produt. For a paying utomer of type, at the moment of purhae (after all the higher type have already adopted and before any other lower type adopt), her pereived utility i u = b p The utility funtion i onave in. Given that u0 = 2 p 0, adoption top at a ertain marginal type m, where u m = 0 u1, or, equivalently, m + 2 /b. Optimal prie i given by p = b1 m + m 2 1 m. Alo, m. Then the profit i given by m b = p1 m b 2 = b1 m + m 2 1 m 1 m b 2 (A3) We onider two different ae: (i) = 0 and (ii) > 0. (i) = 0. In thi ae, the optimal trategy under SDU i idential to the optimal trategy under SDU +, whih ha been derived in Propoition 1. (ii) > 0. In thi ae, profit funtion in (A3) i quadrati in. We optimize firt in under the ontraint 0 m min + 2 /b 1. Moreover, we verify within eah region that the profit i poitive. Note that the profit funtion i onave in. Solving unontrained / = 0, we obtain root = b m /21 m. Moreover, given that 0, it an be hown that + 2 /b m i equivalent to b + bb + 4 m /2. Thu, we have { { }} b + bb + 4m b = max min 2 m m 21 m After onidering and omparing all the feaible ae (analyi omitted for brevity but available from the author), it an be hown that the optimal olution i the following: Region 1: 0 < 1/8. In thi ae = m = 1/2, b = 1/8, = 1 4/64. Region 2: 1/8 < < /16. In thi ae, = b m /21 m, m i the unique real olution to the equation 2 6 m + m 3 = 0 over the interval , b = 21 m 2 m /4 2 m, and = 1 m 4 m 2 /4 m 2. Replaing = m 3 /6m 2, it an be hown that = 3m 1, b = 1 m m2 /, and = 1 m 3 3m 1. Region 3: /16. Then = b + b b + 4m 2 m = b b b and b = and = b b + m Replaing m into and, we obtain = b + b b and = b 6 b b + 3 9b + b 9 Then, we ee that b b b + 3 = 3. Replaing in m, we obtain m = 1 + /3, or = 3m 1. It an be hown that the optimal profit i poitive in all region. Proof of Corollary 1. Follow immediately from the proof of Propoition 3. Proof of Propoition 4. A Propoition 3 tate, the optimal trategy i haraterized over three ditint region for the value of. We explore eah region eparately. Region 1: 0 1/8. Thi ae i traightforward. Reult hold in weak form with repet to hange in and only b repond to hange in. Region 2: 1/8 < < /16. Note firt that the optimal value and b depend on m. Thu, we need to firt undertand the monotoniity of m with repet to and ; m i defined in impliit form a the unique olution to equation 2 6 m + m 3 = 0 over the interval Let u denote = and define funtion m = 2 6 m +m 3. Then m i the unique funtion uh that m = 0 for every 1/ /16. Therefore, over thi interval, m = / = 2 6 m / m 3m 2 2 A diued in the proof of Propoition 3, in region 2, m < 2. Moreover, m > 1/3. Thu, learly, m / < 0. Conequently, ine =, m i dereaing in both and. Beaue = 3m 1, it immediately follow that / < 0 and / < 0. Next, we onider the derivative of b with repet to and : b = m 2 3 m m / 1 m m and 2 b = m 2 3 m m It an be eaily een that m < 1/2. Given that 2 3 m > 0, m / < 0, m / < 0 then it immediately follow that b / < 0 and b / < 0

18 Information Sytem Reearh 24(1), pp , 2013 INFORMS 181 Region 3: /16. Beaue we have the formula for b only in term of and, it an be eaily verified that b / < 0 and b / < 0. From the proof of Propoition 3, in region 3, we know that = b + b b + 3/3. It immediately follow that = b + 3 b 2 6 b b + 3 = b and b + 3 b 2 b b / 6 2 b b + 3 It an be eaily een that / < 0. Furthermore, note that b =, where x = 1 24x 36x x x + 36x 2 /16x. It an be hown that x i dereaing when x /16. Then b b / = 2 > 0. Then, it immediately follow that / < 0. Proof of Propoition 5. Note that the profit in thi model annot exeed the profit under full information, where the eed are alloated optimally. Sine, aording to Propoition 1, under full information the firm doe not enter the market when > 1/4, thi hold true a well in the uniform eeding ae. Thu, there exit 1/4, uh that, when > then the firm hooe not to enter the market. Firt, for the firm to make any profit, it i neeary to harge a poitive prie. That mean that the highet-type adopter ( = 1) mut have poitive benefit from the produt before any paid adoption our. Thu, it i neeary to have b. For any (paying or nonpaying) individual with type, there are eeded utomer with type maller than. Therefore, when a paying utomer of type deide to purhae the produt, at the moment of purhae (after all the higher type already adopted and before any other lower type adopt) her pereived utility i u = b1 + 2 p The utility funtion i onave in. Given that u0 = p 0, adoption top at a ertain marginal type m, where u m = 0 u1, or, equivalently, m b /b1. 6 Then, it follow that p = b1 m + m 2 m, and the profit i given by m b = p1 1 m b 2 = b1 m + m 2 m 1 1 m b 2 (A4) We optimize firt in m under ontraint m minb /b1 1. Note that the profit i ubi in m with a poitive oeffiient for m 3. Solving for / m = 0, we obtain the following two loal extreme: m1 = 2b b 2 b 2 b 2 +b 2 2 b 2 b b1 m2 = 2b b 2 + b 2 b 2 +b 2 2 b 2 b b1 6 We dimi the ae m = 1 beaue in that ae the firm annot make profit. It an be hown that 0 m 1 1 m 2. Thu, the profit i inreaing in m over 0 m 1 and dereaing over m 1 1. Thu m = min { m 1 } b b1 (A5) We will plit the analyi into 10 ae, baed on when m 1 < b /b1. Cae 1. 1/2 < < 1 and b 2 /2 1. Cae /10 < < 1 and / b < 2 /2 1. Cae /10 < < 1 and b < / Cae 4. = 5 + 5/10 and 3 + 5/5 5 1 b < 2 /2 1. Cae 5. = 5 + 5/10 and b < 3 + 5/ Cae 6. 1/2 < < 5 + 5/10 and / b < 2 /2 1. Cae 7. 1/2 < < 5 + 5/10 and b < / Cae /10 < 1/2 and / b. Cae /10 < 1/2 and b < / Cae < 5 5/10. For Cae 1, 2, 4, 7, and 9, we have m = m 1 < b /b1. For all the other Cae (3, 5, 6, 8, 10), we have m = b /b1 m 1. Replaing m in (A4), we obtain an expreion for profit in term of b and. It an be hown that when > 5 5/10, i.e., in ae 1 9, for any given feaible b, profit i dereaing in. Thu, none of thee ae i poible under optimality. Conequently, under optimality, the firm will hooe and b uh that ae 10 our. Thu, under optimality m = b b 1 Replaing m in (A4), we obtain 0< 5 5 b (A6) 10 b = b b 2 b (A7) It an be hown that when < 5 5/10, then 2 b/ 2 0 for any feaible b, i.e., profit i onave in the eeding ratio. Alo, b/ =0 > 0, b/ =5 5/10 < 0. Moreover, when b <, it an alo be hown that b/ =b/ < 0. Thu, the optimal eeding ratio i interior, unique, and atifie FOC. Furthermore, b < b/ for any b, o the only ontraint on b that we impoed will be atified. For any b, the eeding ratio i given in impliit form a the unique olution to b/ = 0 over the interval 0 5 5/10. Simplifying FOC, for any b, b atifie b b 4 2 b 3 + b b b = 0 b < and (A8) For any general value /17 5 5/10, we have b b < 0 for any

19 182 Information Sytem Reearh 24(1), pp , 2013 INFORMS b 0, 0. Thu, it mut be the ae that b /17. In region where the firm enter the market, we need to prove exitene and uniquene of b maximizing b b in (A7). Firt, when = 0, it immediately follow that the unique olution i b = 1/16. For the remaining part of the proof we fou on the ae when > 0. Uing (A8) to implify (A7), we obtain b = b b 2b + b + 11b 8 b 2 16 (A9) From (A8) we have lim b 0 b = 0; hene from (A9) we have lim b 0 b = 0. Moreover, when b = 0, the firm annot make profit a no onumer will buy the produt for a poitive prie. A uh 0 = 0. It an be hown that lim b b = 1/4. From Equation (A7), we ee that, in thi ae, b 2 will be the dominant term a b get large. Hene, lim b b =. It follow immediately that for large b, the profit will be dereaing in b. Given that b /17, there exit b > 0 uh that b < 0 for all b > b, 0 = 0, and b i ontinuouly differentiable on ompat 0 b, the exitene of a maximum i guaranteed for b over 0. We next move to prove uniquene of b in ae where > 0 and the firm hooe to enter the market, i.e., in ae when b > 0. Sine 0 = 0, b mut be interior and atify FOC of (A9), i.e., b/b b=b = 0. We will prove below that FOC of (A9) an have a unique olution when b /17, b atifie (A8), and b > 0. We an kip ae = 0 beaue in that ae the firm will et b = 0 beaue it annot make any profit. Differentiating (A8) with repet to b and uing the impliit funtion theorem, we obtain d b db = 2b b2b + 9b (A10) It an be eaily hown that b2b + 9b > 0 for any b > 0, > 0, /17. Thu, the derivative in (A10) i properly defined. Taking total derivative in Equation (A9) with repet to b, we obtain d/db = /b + / d /db. Inerting (A10), we obtain d db = 32b b b b2b + 9b (A11) A diued above, the denominator i alway poitive. Thu, for FOC to be atified, the numerator mut be zero. We diu two ae: (i) = 1/4. Then, from (A8) it follow that b =. Replaing b in (A11), we have d/db = 0 if and only if = 33/512 (ii) 0 1/4 1/ /17. Then, from (A8) we have b 2 = b (A12) Repeatedly uing (A12) to redue the degree of b, we an tranform the numerator in the left-hand ide of (A11) into a linear equation in b. It follow that a maximizing b value mut atify b = (A13) A profit-maximizing pair b b mut atify both (A8) and (A13). Inerting (A13) into (A8), we obtain where = 0 1 = = = (A14) It an be immediately een that 1 < 0 and 2 > 0 for all 0 1/4 1/ /17. Thu, (A14) an hold if and only if 3 = 0. Thu, for b to be unique, it i neeary and uffiient for 3 to have a unique olution over 0 1/4 1/ /17 when < 1/4. It an be hown that d 3 /d > 0 on /17 when < 1/4. Moreover, 3 0 = < 0 and /17 = / / Thu, 3 = 0 ha a unique olution in the deired region. Moreover, given that 3 1/4 = 1/ , we ee that a unique optimal olution under ae (ii) alway exit when 33/512. When = 33/512, no olution exit under ae (ii) but a unique olution exit under ae (i). Proof of Propoition 6. Note that in thi model, for any b > 0 and > 0, beaue b > 0, the firm an get the adoption tarted under a poitive prie beaue the highet-type utomer have no eeding-indued diutility. The firm will never hooe = 0 or b = 0. For any (paying or nonpaying) individual with type, there i a ma of ize of eeded utomer with type maller than. Therefore, when a paying utomer of type deide to purhae the produt, at the moment of purhae (after all the higher type already adopted and before any other lower type adopt) her pereived utility i u = b p The utility funtion i onave in. Given that u0 = 2 p 0, adoption top at a ertain marginal type m, where u m = 0 u1, or, equivalently, m b + /b1. 7 Then, it follow that the optimal 7 We dimi the ae m = 1 beaue in that ae the firm annot make profit.

20 Information Sytem Reearh 24(1), pp , 2013 INFORMS 183 prie i p = b1 m + m m 2 1 m, and the profit i given by m b = p1 1 m b 2 = b1 m + m m 2 1 m 1 1 m b 2 (A15) We optimize firt in m under ontraint m minb + /b1 1. Note that the profit i ubi in m with a poitive oeffiient for 3 m. Solving for / m = 0, we obtain the following two loal extreme: m1 = 2b b+2 b 2 b 2 +b 2 2 2b 2 +4b b1 m2 = 2b b+2 + b 2 b 2 +b 2 2 2b 2 +4b b1 It an be hown that 0 m 1 1 m 2. Therefore, the profit i inreaing in m over 0 m 1 and dereaing over m 1 1. Thu m = min { m 1 } b + b1 (A16) We will plit the analyi into five ae, baed on when m 1 b + /b1. Cae 1. 1/2. Cae 2. 0 < < 1/2 and b 2 /1 2. Cae /10 < 1/2 and 2 /1 2 < b. Cae 4. 0 < < 5 5/10 and 2 /1 2 < b / Cae 5. 0 < < 5 5/10 and / < b. For Cae 1 4, we have m = m 1 b + / b1. For Cae 5, we have m = b + / b1 m 1. Replaing m in (A4), we obtain an expreion for profit in term of b and. It an be hown that in all Cae 1 4, for any given feaible b, profit i dereaing in. Thu, none of thee ae i poible under optimality. Cae = 5 5/10 an be eaily ruled out a well (to make ure that the optimal olution i not at the boundary of Cae 3). Conequently, under optimality, the firm will hooe and b uh that Cae 5 our. Thu, under optimality, m = b + b 1 0 < < b > Replaing m in (A15), we obtain b = b1 2 3 b 2 (A17) (A18) We onider two ae: (i) = 0 and (ii) > 0. (i) = 0. In thi ae, the optimal trategy under SDU with uniform eeding i idential to the optimal trategy under SDU +, whih ha been derived in Propoition 5. (ii) > 0. In thi ae, from A18, for any b, we ee that profit i ubi in with negative oeffiient for the degree 3 term. The olution to b/ = 0 are 1 = 2b 4b 2 +3b 3 <0< 2 = 2b+ 4b 2 +3b 3 < It immediately follow that It an be verified that b = 2b + 4b 2 + 3b 3 b b + b1 b 1 5 b + 5 b 2 < b (A19) for any b > 0. Thu, the ontraint in Cae 5 hold for any b one the firm eed a fration b of the market. Then, inerting b into (A18), we obtain b = b8b + 6 b4b b 2 9b (A20) Then, it an be hown that b/b = 0 ha three root b 1 = < b 2 = 0 < b 3 = and b/b < 0 on b 1 0 b 3 and b/b > 0 on b 1 0 b 3. Conequently, b =b 3 = (A21) Then, it immediately follow that m = 1/3. Plugging in all the optimal parameter in the profit funtion, it an be hown that the optimal profit i poitive for any > 0 and > 0. Proof of Propoition 7. Follow diretly from omputing the derivative of the equilibrium and b derived in Propoition 6 with repet to and. Appendix B. Generalization of Model SDU + Under Complete Information The analyi of firm trategie under eeding diutility model SDU + and full information (on the eller ide) an be extended to general onumer utility truture and type ditribution a detailed in the next two ubetion. B.1. General Utility Struture In thi etion we explore a more general form of the utility funtion, where the link between utomer type and WTP i moderated by a funtion w that i twie differentiable, with w/ > 0, w0 = 0, and w1 = 1. Thu, if at the urrent moment the intalled bae ha ize, the utility pereived momentarily by a paying utomer of type i u b p = b 2 w p (B1) The following reult apture firm optimal trategie when w i onave. Propoition B1. Under SDU + and omplete information, when w i onave, then the optimal trategy i the ame a in Propoition 1. Proof of Propoition B1. When w i onave, it an be hown that Lemma 1 till applie. Beaue the proof of thi tatement follow imilar tep a the proof of Lemma 1, we