Exploring Bundling Sale Strategy in Online Service Markets with Network Effects

Transcription

1 Explorng Bundlng Sale Strategy n Onlne Servce Markets wth Network Effects Weje Wu Rchard T. B. Ma and John C. S. Lu Shangha Jao Tong Unversty Natonal Unversty of Sngapore The Chnese Unversty of Hong Kong Emal: wejewu@sjtu.edu.cn tbma@comp.nus.edu.sg cslu@cse.cuhk.edu.hk Abstract In recent years we have wtnessed a growng trend for onlne servce companes to offer bundlng sales to ncrease revenue. Bundlng sale means that a company groups a set of ts products/servces and charges ths bundle at a fxed prce whch s usually less than the total prce of ndvdual tems. In ths paper our goal s to understand the underlyng dynamcs of bundlng n partcular what s the optmal bundlng sale strategy and under what stuatons t wll be more attractve than the separate sales. We focus on onlne servce markets that exhbt network effect. We provde mathematcal models to capture the nteractons between buyers and sellers analyze the market equlbrum and ts stablty and formulate an optmzaton framework to determne the optmal sale strategy for the servce provder. We analyze the mpact of the key factors ncludng the network effects and operatng costs on the proftablty of bundlng. We show that bundlng s more proftable than separate sale n most cases; however the heterogenety of servces and the asymmetry of operatng costs reduce the advantage of bundlng. These fndngs provde mportant nsghts n desgnng proper sale strateges for onlne servces. I. Introducton As the economy becomes more global and compettve t s becomng more mportant for onlne servce companes to fnd new ways to ncrease revenue. One way s by bundlng servces. Bundlng servce (or bundlng sale) means that companes group a set of ther products/servces and propose a sngle prce for ths group. Usually the prce of ths bundlng servce s less than the total prce of ndvdual tems. In onlne servce markets servces are provded over the Internet nfrastructure and the servce contents may vary. Typcal servces nclude nstant messagng onlne socal networks onlne games onlne recommendaton systems etc. Companes usually want to expand the servce scale to be as large as possble so as to ncrease ther market share. Although most onlne servce provders do not charge ordnary servces they do charge users for premum servces e.g. the largest move recommendaton network IMDb has an IMDbPro sesson where premum servces ( Get nformed Get connected and Get dscovered ) are provded to pad-users only. These premum servces are often provded n a bundlng manner e.g. customers are not allowed to buy these three Get servces separately but they have to pay a unform prce so as to obtan the premum servces as a whole. Though ths s a common sale strategy n onlne servce markets researchers have lmted understandngs of the underlyng ratonales /4/$3. c 24 IEEE There are a number of reasons why servce provders offer bundlng sales. An appealng justfcaton for provders s cost savng. The onlne servces usually share the same network or storage nfrastructure; therefore the cost of provdng an extra servce on the same nfrastructure s often margnal. Another mportant reason s that by bundlng the servce provders can reduce the varance of customers reservaton prces on the servces thereby ncreasng the revenue of the product. In here a customer s reservaton prce refers to a value such that she wll purchase the servce f and only f ts prce s no hgher than ths value. For example f customer s reservaton prce on servce A s $5 and f the sale prce of servce A s less than or equal to $5 then customer wll subscrbe to such servce. Note that dfferent customers have dfferent reservaton prces towards each servce. In Table I we use a smple example to llustrate ths concept. Suppose a company sells two servces (A and B) to two customers. The second and thrd column depct both customers reservaton prces on the servces. Assume a customer s reservaton prce of the bundle s the sum of reservaton prces of ndvdual servces then the two customers have the same reservaton prce on the bundle. If the servces are sold separately they can be prced at $5 and attract both customers (hence the revenue s $2); or they can be prced at $ and attract only one customer for each servce (hence the revenue s $2 too). In contrast f servces are bundled and prced at $5 then both customers wll purchase the servce and the total revenue s $3. Ths shows that bundlng can reduce the varance of customers reservaton prces on these servces and thus the company can ncrease ts revenue. Servce A Servce B Bundle Customer $5 $ $5 Customer 2 $ $5 $5 TABLE I: An example of bundlng sales One mportant unque feature of onlne servces s the network effect. It refers to the market effect at the customer s sde where a partcular customer s nterest on a servce s nfluenced by other customers purchasng decsons. For example n onlne socal networks (e.g. Facebook LnkedIn Twtter IMDb etc.) when the number of membershp ncreases the beneft each member receves also ncreases due to a hgher /4/$3. 24 IEEE 442

2 degree of nteracton and effcency of nformaton spreadng and ths causes more users to subscrbe to ths servce. Ths s a prme example of an onlne servce market where a large populaton sze ndcates a postve nfluence on each customer s valuaton and we call ths the postve network effect. Ths effect has a major mpact on the choce of prcng strateges for onlne servce provders. A number of exstng research [] [] [] [3] dscussed bundlng strateges but most of them focused on non-dgtal goods or servces and were manly based on graphcal explanatons case studes or algorthmc approaches. Very lmted work has been focusng on formal mathematcal models to provde deeper nsghts for the companes. Furthermore most exstng work dd not consder the mpact of network effect so they can only provde lmted nsghts on the onlne servce market. We am to address the followng questons: Is t more proftable for onlne servce provders to bundle a number of servces and sell them at a sngle prce? What are the factors that mpact the optmal prcng strategy wth network effect? In ths paper we make the followng contrbutons: We buld a mathematcal model whch captures the onlne servce market wth network effect. We analyze the market equlbrum and formulate an optmzaton framework to capture the optmal sale strategy. We dscuss the mpact of dfferent factors on the proftablty of the bundlng strategy. We show that bundlng s more proftable than separate sale n most cases whle the heterogenety of servces and asymmetry of operatng costs reduce the advantage of bundlng. Our paper s organzed as follows. Sec. II presents a general model to capture customers purchasng decson and the servce provder s proft. Sec. III focuses on the onlne servce market analyzes the market equlbrum and ts stablty and presents an optmzaton framework to capture the optmal sale strateges. In Sec. IV and Sec. V we analyze the role of network effect and operatng cost on the proftablty of bundlng. Sec. VI states related work and Sec. VII concludes. II. General Model We present a general model to characterze the Internet servce market and to study how customers and the servce provder make ther purchasng/prcng decsons. Let us frst provde formal defntons on the sale strateges. Defnton : Separate sale s a strategy by whch a servce provder sells each servce S at prce p ndvdually. Customers can choose to purchase any servce or not. Defnton 2: Bundlng sale s a strategy by whch a servce provder offers to sell a set of servces as a sngle unt. The bundlng servce s prced at p b. Customers can only choose to purchase the whole bundlng servce or not. There s also a negatve network effect f a large number of users cause congeston. However congeston s a physcal level nfrastructural problem but s not the focus of our paper on the applcaton level prcng problem. A. Utlty functons of separate sales Customers utlty. A customer determnes whether to purchase the servce(s) provded by the servce provder. We consder a sngle servce provder and a contnuum of customers wth dfferent reservaton prces on the servce. The customers heterogenety n reservaton prces s represented by ther types.e. each nfntesmal customer s characterzed by a one-dmensonal type parameter 2 whch has a contnuous dstrbuton over. The customer s utlty functon descrbes her purchasng behavor: a customer subscrbes to a servce f and only f she acheves a non-negatve utlty. Ths utlty functon depends on ) the customer s reservaton prce on the servce and 2) the sale prce p of the servce. We assume customer s reservaton prce on S s v ( ) ( ) where v ( ) s her ntrnsc valuaton on S s the fracton of customers that subscrbe to S and ( ) s a non-decreasng functon n representng the network effect. The multplcaton form v ( ) ( ) s to characterze that a customer wth a hgher ntrnsc valuaton s more senstve to the network effect [6] and that the reservaton prce s zero f ( ) =. We defne u ( ) as customer s utlty on servce S : u ( ) =v ( ) ( ) p. () Customers of dfferent types have dfferent ntrnsc valuatons v ( ) on S and we assume v ( ) has a contnuous dstrbuton n over. We further denote f( ) as the densty functon of and defne Z H (x) {v( )applex}f( )d (2) 2 as the cumulatve dstrbuton functon of v ( ).e. gven any value x H (x) represents the fracton of customers whose ntrnsc valuaton on servce S s less than or equal to x. Servce provder s utlty. The servce provder determnes whether t should provde separate or bundlng sale and to propose the prce(s) for the servce(s). We model the servce provder s utlty as ts total proft and we wll use utlty and proft nterchangeably n later analyss. We consder two factors that mpact the servce provder s utlty: ) the servce fee receved from customers whch we model as p ; 2) the varable operatng cost 2 whch we model as m. In here m represents the per-unt varable cost 3 and we call t the unt operatng cost or the unt cost for short. We defne U =(p m ) (3) as the servce provder s utlty on servce S. Therefore the servce provder s utlty from all separate sale servces s IX U s = (p m ). (4) = 2 Varable cost and fxed cost consst of the total cost. We consder the varable cost only snce the fxed cost only represents a lnear shft on the utlty and does not affect our concluson. 3 Some exstng lterature uses the term margnal cost to represent ths concept. In fact f the margnal cost s a constant ts value s equal to the per-unt varable cost whch we defne here. 443

3 B. Utlty functons of the bundlng sale The prevous subsecton has set up the stage for expressng the utltes of customers and the servce provder when servces are sold ndvdually. We now consder the bundlng strategy whch bundles servces S S 2...S I. Often tmes customers vew the bundled servces as a whole. We use the notaton b to denote the bundlng servce S b. For consstency we stll assume that the network effect functon mpacts the utltes of the bundle n a multplcaton form. By substtutng b for we denote the correspondng notatons for S b. In partcular u b ( ) v b ( ) b and p b represent customer s utlty on purchasng S b her ntrnsc valuaton on S b the fracton of users purchasng S b and the prce charged for S b respectvely. We have the customer s and the servce provder s utlty functons as u b ( ) =v b ( ) b ( b ) p b and U b =(p b m b ) b (5) where m b denotes the unt cost for S b. In partcular we assume m b = P I = m where 2 [ ] denotes the scalng factors of the operatng cost. In fact apple mples that bundlng can save unt costs e.g. f we bundle a number of bandwdth-related functonaltes n onlne game servces then the servces can rely on the same nfrastructure and save cost. C. Market equlbrum Due to the network effect and that customers subscrbe to servces at dfferent tmes the above model s n fact a dynamc system. We use the followng defnton to descrbe the steady state of the system. Defnton 3: Gven prce p > s a market equlbrum f Z {v( ) ( ) p }f( )d = (6) 2 where f( ) s the densty functon of. Ths defnton states that for any gven customer s utlty as u ( ) =v ( ) ( ) p f exactly fracton of customers have a non-negatve utlty to purchase S then s a market equlbrum. It represents the fracton of customers who purchase the servce S when the system reaches a steady state.e. gven ths fracton no customer has an ncentve to change her decson. In the followng our analyss s based on ths equlbrum. We wll dscuss prcng strateges under such scenaro. Unless we state otherwse we wll use to denote the equlbrum n the remanng of ths paper. Note that when = t may also be a steady state wth no users. But n ths case the servce s closed and there s no real market. Thus we exclude =from the defnton of equlbrum. Now let us characterze the value of. Lemma : Assume ( ) > for any >. The value s an equlbrum f and only f t satsfes = H p ( ) where H ( ) s the cumulatve dstrbuton functon of v ( ). The above lemma gves an mplct form to characterze and compute the equlbrum. In later analyss t s more convenent to use the followng corollary. Corollary : Assume H ( ) s a strctly ncreasng functon n [V ] and ( ) > for any >. Gven any prce p f there exsts an equlbrum then t s a soluton to the followng equaton: p = ( )H ( ) (7) where H ( ) s the nverse functon of H ( ) defned n [ ]. Up tll now we have set up a general model to capture the customers and the servce provder s utltes. Based on ths model we wll proceed to analyze the propertes of the market. III. Onlne Servce Market: Equlbrum and Optmal Sale Strategy In ths secton we study an onlne servce market. We frst model the users valuaton dstrbuton and the network effect and then analyze the market equlbrum. Lastly we establsh a framework to determne the optmal sale strateges. A. Dstrbutons of users ntrnsc valuatons Let us frst state two basc assumptons on the users ntrnsc valuaton dstrbutons. Frst gven a customer s ntrnsc valuaton on each ndvdual servce S as v ( ) we assume P I that her valuaton on the bundle satsfes v b ( ) = v ( ). The ratonale s that the onlne servces are often complementary.e. usng them n conjuncton can gve customers extra utltes 4. Second we assume the customers ntrnsc valuaton on dfferent servces are ndependent. Therefore f we let H (x) and H B (x) be the cumulatve dstrbuton functons of v ( ) and P I = v ( ) respectvely we have H B (x) =H (x) H 2 (x) H I (x) where the convoluton operaton s defned by H (x) H j (x) = R H (x t)dh j (t). 5 Let H b (x) be the cumulatve dstrbuton of v b ( ). Snce v b ( ) s lower bounded by P I = v ( ).e. any customer s ntrnsc valuaton on the bundle s at least P I = v ( ) thus H b (x) s upper bounded by H B (x).e. gven any value x the fracton of users whose v b ( ) apple x s at most H B (x). In fact H b (x) equals H B (x) f v b ( ) = P I = v ( ). In what follows we use H B (x) for the baselne analyss. It s easy to see that f n ths baselne analyss bundlng acheves a proft gan over the separate sale under a certan crcumstance t must acheve an equally hgh or even hgher proft gan under the dstrbuton H b (x). We focus on bundlng two servces S and S 2. Ths model represents a wde range of bundlng strategy decsons snce any bundlng of multple servces can be consequentally constructed by bundlng two servces. We consder the unform dstrbuton 6 of customer s ntrnsc valuaton whch s wdely 4 For example n the IMDb premum servces Get nformed can help users to fully utlze the Get connected and Get dscovered functonaltes. 5 Ths s a standard result n probablty theory and we omt ts proof. 6 We use the unform dstrbuton for ease of mathematcal dervaton. However the underlyng ratonale of bundlng does not rely on ths specfc form so our framework and nsghts do hold for other dstrbutons. 444

4 HB(x) Fg. : Cumulatve dstrbuton H B (x) =2V 2 =3 adopted n economc lteratures (e.g. [2] []). Formally we have the cumulatve dstrbuton of v ( )= 2 as 8 < f x< H (x) = x/v f apple x apple V (8) : f x>v where V ( = 2) s the maxmal ntrnsc valuaton of S. Wthout loss of generalty we let apple V 2 and we have Lemma 2: The baselne dstrbuton functon H B (x) s 8 f x< >< x 2 /(2 V 2 ) f apple x apple H B (x) = (2x )/( ) f <xapplev 2 ( +V 2 x) >: 2 /(2 V 2 ) f V 2 <xapple +V 2 f x> + V 2. Proof: By takng the convoluton operatons on H (x) we can drectly reach the result. In Fg. we llustrate the shape of ths dstrbuton. It shows that H B (x) ncreases more rapdly n the mddle range of the nterval; n other words customers are more concentrated to have a moderate valuaton of the bundle comparng wth the unform dstrbuton of separate servces. Ths shows that bundlng can reduce the varance of customers valuatons and t s an mportant underlyng reason to make bundlng proftable: f the servce provder sets a relatvely low bundlng prce then t becomes easer for hm to attract more customers snce there are a lot of customers wth moderate valuatons and hence the servce provder can make more proft. However f the servce provder only targets at a small amount of customers wth hgh valuatons then bundlng may not have an advantage. Ths s because H B (x) ndcates fewer customers wth hgh valuatons comparng to the unform dstrbuton. B. Network effect and utlty functons We model the network effect n form of ( ) = where 2 ( +) represents the shape of the network effect. When the two servces wth apple 2 are bundled customers vew t as a new servce wth the network effect functon b ( b )= b b where apple b apple 2. We use the above form for a number of reasons. Frst u ( ) =f =.e. no customer has an ncentve to enter an empty market. Ths s a common fact n many nteractve applcatons e.g. onlne socal network or recommendaton systems; and ths also shows that t s mportant for a servce provder to promote the servce and have some ntal users x for startup. Second s ncreasng n so t represents a postve network effect. Last but not least ths s an soelastcty functon whch allows us to use a sngle parameter to model the elastcty or the shape of the network effect. Large (or a convex functon) means that gven a small s small and many users wll lose ther nterest so a large start-up populaton s necessary. On the contrary f s small (or a concave functon) then s large gven a moderate or small. Ths means a small amount of ntal users can be enough to nduce a large network effect and later on the servce can potentally attract many more customers. Note that our model generalzes the lnear network effect models n many exstng lteratures [6] [2]; n fact when = our model exactly represents the lnear network effect. Based on the above settngs servce S s unquely defned by ) the users maxmal ntrnsc valuaton V 2) the unt operatng cost m and 3) the network effect parameter. In later analyss we use a tuple S = hv m = 2 to denote a separate sale servce. Based on the above customer s utlty and the servce provder s utlty on servce S are: u ( ) =v ( ) p U =(p m ) ; (9) and the servce provder s utlty on all separate sales s 2X 2X U s = U = (p m ). () = = For the bundlng sale the utlty functons are! 2X u b ( ) =v b ( ) b b p b U b = p b m b. () C. Analyss of market equlbrum In ths subsecton we derve the condtons for the exstence of the market equlbrum (or equlbra). Theorem : Consder any separate or bundlng sale S 2 { 2b}. There exsts a threshold p such that for any gven servce prce p we have 8 < # of equlbrum (or equlbra) = : = f p > p f p =or p 2 f p 2 ( p ).. + In partcular for separate sale we have p = V + Ths theorem states that the condton for the exstence of equlbrum s that the servce prce s not too hgh otherwse no customer wll purchase the servce. We also note that f the exstence s guaranteed then almost surely there are two equlbra. We next dscuss the stablty property and explan why we are nterested n the larger equlbrum. Dscusson on stablty 7. We say that an equlbrum s stable f there exsts a postve such that f at any tme a nonequlbrum fracton 2 ( + ) of customers subscrbe to the servce then the dynamc market wll eventually reach the equlbrum. In fact f we consder the two equlba < 2 2 n the above theorem the only stable one s. If 7 The features of equlbra are qute smlar to dscussons n [4]. 445

5 the market s now wth fracton of customers then eventually all customers wll leave the market and the servce wll be closed; f the market s now wth + fracton of customers t wll not reach but wll reach 2. Hence s an unstable equlbrum. In contrast f we consder the market wth any fracton 2 2 ( ) of users the dynamc market wll eventually reach the equlbrum 2; hence 2 s a stable equlbrum. Due to the page lmt we omt the detals here; nterested readers may refer to [6] for a detaled dscusson. Due to ths stablty property n later analyss we can safely restrct our analyss n the larger equlbrum. We use max{ (p )} to denote the maxmal equlbrum for a gven p and defne max ; =to capture an empty market when the prce s too hgh and the equlbrum does not exst. D. An optmzaton framework for the sale strateges In ths secton we establsh an optmzaton framework to determne the optmal sale strateges. A natural way to model the optmal sales s that the servce provder ams to fnd a prce (or prces) for the separate or the bundlng sale whch maxmzes ts proft.e. the optmal separate or bundlng sale S 2 { 2b} can be modeled as max p U (p )=(p m ) max{ (p )} s.t. p. (2) However ths form s not easy for analyss and we opt to change the decson varable from p to. Accordng to Corollary we can transform (2) nto the followng problem 8 max U ( )=( ( )H ( ) m ) s.t. apple apple. (3) Snce the above optmzatons for 2 { 2b} have contnuous objectve functons over a compact set they are guaranteed to have optmal solutons. By solvng the above optmzatons we automatcally choose the largest equlbrum fracton for any gven prce whch s the stable one as we desre. 9 Up tll now we have establshed an optmzaton framework to determne the optmal sale strateges. In what follows we use Us and Ub to denote the maxmal proft of the servce provder under the separate and bundlng sales respectvely. If we could calculate Us and Ub then we can determne whether bundlng s more proftable than separate sale by comparng ther values. Formally we have the followng defnton to capture the proft gan of bundlng over the separate sale: Defnton 4: The proft gan rato s the dfference between the maxmal proft of the bundlng sale and that of the separate sale dvded by maxmal proft of the separate sale.e. =(U b U s )/U s. (4) If > t means the optmal bundlng sale s more proftable than the optmal separate sale and vce versa. A larger value of ndcates a larger proft gan by the bundlng sale. 8 Although =s excluded from Corollary we can verfy that f the soluton to (2) s p and max{ (p )} = then the soluton to (3) s. 9 To see ths note f there exst < 2 such that ( )H ( )= ( 2)H ( 2 ) m then obvously cannot be the soluton to (3). We beleve ths framework s crtcal for onlne servce provders to evaluate ther best sale strateges; but we should also pont out t may not be easy to have general results at ths stage n partcular when a general nduces dffculty n solvng the optmzatons. In the followng sectons we explore the mpact of varous factors.e. the network effect parameter and the unt cost m on the proftablty of bundlng. Before we proceed let us present the followng lemma whch reflects the scalng propertes of the sales. Lemma 3 (Scalng property): Let c be a postve number. () If the equlbrum and proft of the optmal sale S = hv m are and U then the equlbrum and proft of the optmal sale S = hcv cm are and cu. (2) If the proft gan rato for bundlng S = h m and S 2 = hv 2 m 2 2 s then the proft gan rato for bundlng S = hc cm and S2 = hcv 2 cm 2 2 s also. Applyng the optmzaton framework we can easly prove the above lemma. Ths lemma ponts out that f V and m ncreases (or decreases) by the same factor then t does not mpact the equlbrum or the proftablty of bundlng. Hence n later analyss we can normalze to be and vary V 2 so as to explore the whole desgn space. Ths smplfes our analyss and does not lose any generalty. IV. Impact of Network Effect Up to now we have formulated a framework to capture the prcng strateges and the market equlbrum. In ths and the next sectons we dscuss the mpact of key factors on the proftablty of bundlng. We frst focus on the network effect. Many onlne servce provders ncur a much larger fxed cost comparng to ther varable cost. For example onlne socal network servce needs to nvest a large amount of money to ntally set up the hardware and nfrastructure but the cost s mnmal to ncrease one user membershp. So n ths secton we set the per-unt varable cost m =and consder S = hv. The servce provder s utlty can be expressed as U s = p + p 2 2. (5) Ths smplfcaton captures the feature of a wde range of dgtal onlne servces and t allows us to solate dfferent factors so as to better understand the mpact of network effect. A. Homogeneous network effect ( = 2 = ) We start our dscusson wth the network effect functons ( ) = 2 ( ) =. Naturally we assume the bundlng servce S b also have b ( ) =. Such settng represents bundlng two servces wth smlar network effects. We have the followng theorem to show that bundlng s more proftable than separate sales under ths settng. Theorem 2: Consder S = h and S 2 = hv 2. ) The proft gan rato of the bundlng sale >. 2) In partcular when S = S 2 = hv we have a) The optmal separate sale s = + +2 p = V +2 b) The optmal bundlng sale s + +2 U s =2 p. 446

6 proft gan rato γ = V 2 = = V 2 =2 = V 2 = network effect parameter α proft gan rato γ η=.8 η=.5 η= network effect parameter α 2 (a) = V 2 proft gan rato γ = V 2 = = V 2 =2 = V 2 = network effect parameter α 2 (b) 6= V 2 Fg. 2: Impact of network effect = b = 2 +3 p b = r V U b = b p b. c) The proft gan rato of the bundlng sale s p 2 (2 + 4) +2 ( ) = 4 (2 + 3) +3/2 and t s an ncreasng functon n. Ths theorem states that when = 2 bundlng s always more proftable and large (.e. a convex network effect functon) ndcates a hgh proft gan. Let us use examples to show how bundlng acheves a hgher proft. In Fg. 2 we consder S = h and S 2 = hv 2 where we vary V 2 2 { 2 5} and 2 [. 3.]. We can see s always postve and ths valdates our results n Theorem 2. We can also observe that when there s a large gap between and V 2 the proft gan rato reduces. Ths s because the jont dstrbuton H B (x) becomes less concentrated n the mddle range so bundlng sale can attract fewer customers. To summarze we have the followng observaton: Observaton : The advantage of bundlng becomes more domnant when the network effect functon s more convex (.e. larger ); however the heterogenety n ntrnsc valuaton dstrbutons reduces the proft gan rato of bundlng. B. Heterogeneous network effect ( 6= 2 ) Now let us consder bundlng two servces wth dfferent network effects. We frst consder = V 2. It s natural to assume b the network effect parameter for the bundle s between and 2. Let b = +( ) 2 where 2 [ ]. We have the followng result. Theorem 3: Consder S = hv and S 2 = hv 2. Let b = +( ) 2. The proft gan rato s b + q = +2 2 b +2 2 b b (6) Proof: By notng the form of Us and Ub n Theorem 2 we can drectly reach the concluson. Let us show the mpact of network effect on. In Fg. 3(a) we consder S = h. and S 2 = h 2 where we Fg. 3: Impact of network effect 6= 2 vary 2 2 [..]. We plot three curves of when =.8.5 and.2 respectvely. We frst focus on a partcular curve e.g. =.5. When 2 ncreases from. also ncreases; ths s because a large network effect parameter has a postve mpact on bundlng. But when 2 s large begns to decrease and eventually becomes negatve; ths s because when and 2 dffer a lot the optmal equlbra and 2 also dffer a lot. In such cases t s not ratonal to bundle S and S 2 snce the bundlng sale needs to fnd a unque equlbrum b whch s ether far away from or far away from 2 so bundlng s not as proftable as separate sale. We also observe when s larger s also larger. Ths s because larger mples larger b or a more convex network effect functon whch makes bundlng even more proftable. Smlar to the prevous dscussons we also consder the servces wth dfferent V. In Fg. 3(b) we consder S = h. and S 2 = hv 2 2 where we vary 2 2 [. ] and V 2 2 { 2 5}. We set = V +V 2 to represent the relatve weght of each servce. We can observe the smlar feature as we have shown before: when 2 ncreases frst ncreases and then decreases. We also observe that the nflecton pont ncreases when V 2 ncreases. Ths s because when V 2 s large servce S 2 has a major mpact on the bundle so the postve mpact of 2 on bundlng can be effectve n a larger range. To summarze we have the followng observaton: Observaton 2: The heterogenety of network effect functons decreases the proftablty of bundlng. V. Impact of Operatng Cost In the prevous secton we have dscussed the mpact of network effect when the varable operatng cost equals zero. Although ths approxmaton apples to many exstng servces there mght be exceptons. For example n onlne storage systems (e.g. Dropbox) the unt cost of storng the data mght not be neglgble. In ths secton we dscuss how the operatng cost mpacts the prcng strateges. Our dscussons generalze the results we obtaned n the prevous secton. A. Impact of operatng cost when = 2 = We start our dscusson when both servces have the same network effect functon.e. ( )= = 2. We want to explore how our results n Theorem 2 can be generalzed wth non-zero unt operatng costs. Accordng to Lemma 3 we 447

7 proft gan U* α=2 α= α=.5 proft gan U* α=2 α= α=.5 proft gan rato γ γ : m = m 2 = γ 2 : m = m 2 =. γ 3 : m =. m 2 = unt cost m (a) m = m unt cost m 2 (b) m 6= m network effect parameter α 2 Fg. 4: Impact of unt cost = 2 can normalze V so that the effectveness of the unt cost s represented by m V. We wll dscuss when m : m 2 = : V 2 and m : m 2 6= : V 2.e. symmetrc and asymmetrc unt costs respectvely. We start from the symmetrc case. Theorem 4: If and m : m 2 = : V 2 then Ub Us. Ths theorem states that f S and S 2 have the same convex network effect functon and symmetrc unt costs then the proft of the optmal bundlng s no less than that of the optmal separate sale. The key reason s that under ths settng the optmal separate sale always obtans an equlbrum larger than /2 so bundlng can attract more customers. However f < and m s large then may be less than /2 and bundlng may not be always proftable. Let us use examples to show ths phenomenon. Snce Us may be zero whch leads = we opt to use the proft gan defned by U = Ub Us as the performance measure. In Fg. 4(a) we consder S = S 2 = h m and vary m 2 [.34] 2 {.5 2}. We observe for any gven U reduces wth respect to m. Ths ndcates that unt cost reduces the proft gan of bundlng. When =or 2 bundlng s always no worse than separate sale. Ths valdates our result n Theorem 4. When =.5 U can be negatve when m s large. Ths ndcates when the network effect functons are concave a large unt cost can make bundlng less proftable than separate sale. Now let us consder the mpact of asymmetrc unt costs.e. m : m 2 6= : V 2. The asymmetry nduces dfferent equlbra and bundlng s not always more proftable. It s not easy to quantfy the domnant doman of the bundlng or the separate sales. We have the followng theorem as a suffcent condton to guarantee the proftablty of bundlng. Theorem 5: Let = 2 be equlbra of optmal separates 2 sales S and S 2. If 2 2 apple 2 apple (where 2 s the scalng factor such that m b = m + 2 m 2 ) then Ub >U s. Ths theorem states that f the optmal equlbra of separate sales are close then bundlng s more proftable. The underlyng reason s smlar to the prevous analyss: f two servces are hghly asymmetrc and have very dfferent equlbra then t s not feasble to fnd a sutable servce prce for the bundle because the correspondng equlbrum b of the bundle s ether too far from or too far from 2; only when and Fg. 5: Impact of unt cost 6= 2 2 are close bundlng can be more proftable. Let us use examples to show how the asymmetry mpacts the proft gan. In Fg. 4(b) we consder S = h.4 S 2 = hm 2 and vary m 2 2 [.34] and 2 {.5 2}. We can observe that when m 2 ncreases the proft gan U decreases; when m 2 s greatly larger than m then U can be negatve for =or 2. Comparng wth Fg. 4(a) where U s always non-negatve for =or 2 we can see the asymmetry n unt costs further reduces the proftablty of bundlng. To summarze we have the followng observaton. Observaton 3. Under the symmetrc operatng costs and homogenous network effect the operatng costs reduce the proftablty of bundlng; n partcular when the network effect functon s concave bundlng may be less proftable than separate sales. When the operatng costs are asymmetrc the proftablty of bundlng s further reduced. B. Impact of operatng cost when 6= 2 In ths secton we observe the mpact of the unt operatng cost when the two servces have dfferent network effect functons. In partcular we show how our result n Fg. 3(a) changes wth consderaton of the unt cost. To keep consstency wth Fg. 3(a) we evaluate the proft gan rato. In Fg. 5 we consder bundlng S = hm. and S 2 = hm 2 2. We fx = 2 = =.5 and vary 2 2 [..]. We consder three cases of : = f m = m 2 = = 2 f m =m 2 =. and = 3 f m =.m 2 =. We frst note 3 <. Ths means a unt cost on S dscourages bundlng whch s the same as our fndng n the prevous subsecton. Next we focus on the curve of 2 and we have some nterestng observatons. We note 2 > when 2 s moderately large. Ths shows the unt cost of S 2 can sometmes ncrease the proftablty of bundlng. The reason s ths unt cost reduces 2 so that the gap between and 2 reduces. Therefore the unt cost reduces the asymmetry of S and S 2 so t ncreases the proftablty of bundlng. However when 2 s qute large the unt cost further reduces 2 and ts negatve mpact on the proft becomes domnant. To summarze we have the followng observaton: Observaton 4. The operatng costs play a complcated role when network effect functons are dfferent. In partcular a 448

8 moderate operatng cost on the servce wth larger may ncrease the proftablty of bundlng. VI. Related Work Bundlng strategy has been dscussed n economc communty. Early studes [] [] [] revealed basc understandngs and they were all based on non-dgtal goods. Authors n [2] [3] [8] dscussed bundlng strategy of dgtal goods wth zero margnal cost but there was no network effect. Network effect (or network externalty) has also been extensvely studed. Early works [7] [9] set up basc models to defne and analyze network effect and recent works [5] [4] [5] have dscussed varous applcatons under network effects. Although network effect and bundlng sale have been both extensvely studed there are very few works that combne them. We only fnd one recent work [2] closely related to our paper where the authors dscussed bundlng strategy of technologcal products wth network externalty. The paper presented nterestng fndngs but they manly rely on numercal and graphcal explanatons and ther analyss was restrcted to some specal cases. Meanwhle the lnear and addtve form of network externalty appled n ther work s n a specal form whch does not capture all mportant features of onlne servces. Hence we need a more accurate model on the network effect to capture today s onlne market. Our paper dffers from prevous works n that ) we buld a formal optmzaton framework that captures the optmal separate sale and bundlng strateges 2) we gve rgorous analytcal results based on the multplcaton form of network effect and 3) we analytcally show how network effects and operatng costs mpact on the proftablty of bundlng under varous scenaros. VII. Concluson In ths paper we dscuss the bundlng sale strategy for onlne servce markets whch exhbt network effects. In such market a customer s purchasng decson s nfluenced by other customers purchasng decsons. We formulate a formal optmzaton framework to characterze the optmal sale strateges whch allows the servce provders to determne ther best sale strateges. Based on ths we analyze and quantfy the mpact of the key factors. Our mportant fndngs nclude: ) when the network effect functon s more convex the proft gan of bundlng over separate sales becomes larger; 2) the operatng cost usually plays a negatve role towards bundlng; but when the two servces have dfferent network effects a moderate operatng cost on a partcular servce may ncrease the proftablty of bundlng; and 3) the asymmetry n operatng costs and the heterogenety n valuaton dstrbutons or network effects reduce the proftablty of bundlng and can even make bundlng less proftable than separate sales. We beleve these fndngs provde valuable nsghts for onlne servce provders to desgn effectve prcng schemes and we plan to better explore bundlng sales va real data analytcs. Acknowledgement: Weje Wu s supported n part by Natonal Key Basc Research Program of Chna (NKBRP) 23CB32963 and Natonal Natural Scence Foundaton of Chna Project Rchard T.B. Ma s supported n part by the Mnstry of Educaton of Sngapore under AcRF Grant R John C.S. Lu s supported n part by the RGC Grant (452). REFERENCES [] W. Adams and J. Yellen. Commodty bundlng and the burden of monopoly. The quarterly journal of economcs pages [2] Y. Bakos and E. Brynjolfsson. Bundlng nformaton goods: Prcng profts and effcency. Management Scence 45(2): [3] Y. Bakos and E. Brynjolfsson. Bundlng and competton on the nternet. Marketng scence 9(): [4] C. Buragohan D. Agrawal and S. Sur. A game theoretc framework for ncentves n p2p systems. In Proc. P2P pages [5] O. Candogan K. Bmpks and A. Ozdaglar. Optmal prcng n the presence of local network effects. In Proc. 6th nt l conference on Internet and network economcs pages Sprnger-Verlag 2. [6] D. Easley and J. Klenberg. Networks crowds and markets. Cambrdge Unv Press 2. [7] M. Katz and C. Shapro. Network externaltes competton and compatblty. The Amercan economc revew 75(3): [8] J. Kephart and S. Fay. Compettve bundlng of categorzed nformaton goods. In Proc. ACM EC pages ACM 2. [9] S. Lebowtz and S. Margols. Network externalty: An uncommon tragedy. The Journal of Economc Perspectves 8(2): [] C. Matutes and P. Regbeau. Compatblty and bundlng of complementary goods n a duopoly. The Journal of Industral Economcs pages [] R. McAfee J. McMllan and M. Whnston. Multproduct monopoly commodty bundlng and correlaton of values. The Quarterly Journal of Economcs 4(2): [2] A. Prasad R. Venkatesh and V. Mahajan. Optmal bundlng of technologcal products wth network externalty. Management Scence 56(2): [3] R. Schmalensee. Gaussan demand and commodty bundlng. Journal of Busness pages [4] C. Wang Y. Hsu and W. Fang. Acceptance of technology wth network externaltes. Journal of Informaton Technology Theory and Applcaton 6(4): [5] C. Wang S. Lo and W. Fang. Extendng the technology acceptance model to moble telecommuncaton nnovaton: The exstence of network externaltes. Journal of consumer Behavour 7(2): 28. APPENDIX Proof of Lemma : Snce for any > we have ( ) > so the condton v ( ) ( ) p s equvalent to p v ( ) ( ). Accordng to the defnton of equlbrum we have = R 2 n o p v ( ) f( )d. By notng the above ( ) equaton and recallng the defnton of H (x) n Eq. (2) we can prove the lemma. Proof of Theorem : Accordng to Corollary s an equlbra f and only f t satsfes p = H ( ) ( = 2) p b = b b H B ( b) where H ( ) =V ( ) and 8 p >< + V 2 2V V 2 b f apple b apple V H 2V B ( 2 b)= 2 >: V V 2 + V 2 ( b) f < b apple p V V 2V V 2 ( b) f < b apple. Let g (x) = x H ( x) = 2 and g b (x) = x b H B ( x). Then equlbrum s a soluton to p = g ( ). (7) 449

9 By lettng g (x) == 2 we see there s a unque soluton x = n ( ]. By lettng + g b (x) =we have 8 2 >< 2 b (V+V2)2 x (4 2 b +4 b+)v 2 f apple x apple V b = b (+) V 2( >: b +)V 2 f V <xapple 2 b V 2 b + f <xapple. b Snce 2 b + < 2 < to gb (x) =. If b (+) (2 b +)V 2 > V 2 If 2 b (V+V2)2 (4 2 b +4 b+)v 2 apple V then V2 apple b (+) 2 (2 b +)V 2 and 2 b (V+V2)2 t s not a soluton then V2 > 2 b. 2 b. Ths means (4 2 b +4 b+)v 2 cannot be both solutons to gb (x) =. So g b (x) =has at most one soluton n ( ]. Note that for 2 { 2b} g () = g () = g (x) > f <x< and that g (x) =has at most one soluton n (] we conclude g (x) has one and only one maxma when x 2 ( ). Let us denote ths value as p. Thus when x ncreases from to g (x) frst ncreases from to p and then decreases from p to. Therefore when p > p Eq. (7) has no soluton; when p = p or Eq. (7) has only one soluton; when <p < p Eq. (7) has two solutons. In partcular we have p = g ( = 2 whch completes the proof. + )= V + + for Proof of Theorem 2: We frst prove the second part of the theorem. By applyng the optmzaton framework we see the optmal separate sale s a soluton to max U ( )= 2X = ( + ) s.t. apple apple. (8) We can easly obtan the soluton as = and have p = V U s =2p. Smlarly we can solve the optmal bundlng sale. By notng the defnton of and takng the forms of Us and Ub nto Eq. (4) we can derve the proft gan rato ( ) as desred. By takng the dervatve of log ( ) wth respect to t s easy to show ( ) s ncreasng n. Now we prove the frst part of the theorem. For any gven we have = 2 = + +2 and we denote ths value as. Obvously > /2 and we have U s = ( )H ( )+ 2 ( )H 2 ( ). (9) We consder a bundlng sale wth prce p b such that the largest equlbrum s. The servce provder s utlty n the optmal bundlng sale satsfes Ub b ( )H B ( ). (2) Gven the form of H (x) and H B (x) one can easly verfy H ( ) V < H B ( ) that +V 2 for > /2. Therefore we have H ( )+H2 ( ) <H B ( ). Snce ( )= 2 ( )= b ( )= we have 2X ( )H ( ) < b ( )H B ( ). (2) = Combnng nequaltes (9) (2) and (2) we have U b >U s therefore the proft gan rato >. Proof of Theorem 4: Let us denote m = m2 V 2 = m. Accordng to Lemma 3 the equlbra of separate sales satsfy = 2 and we denote t as. Based on the optmzaton formulaton we know s a soluton to the followng optmzaton: max U( )= + +2 m s.t. apple apple. (22) We next show =or > /2. Note that U( )= g( ) where g( )= + m. Let us consder f = s not the unque soluton then g( ). By takng frst and second order dervatves we can derve g( ) acheves the unque maxmal value n [ ] when = +. Let us suppose < + then apple g( ) < g( + ) so g( ) < + g( + ) whch s U( ) < U( + ). Ths contradcts wth our assumpton that U( ) s the maxmal value n []. Therefore we have =or + > /2. If = then the optmal separate sale acheves a proft of zero whch s obvously no larger than the optmal bundlng sale; f > /2 then usng the same approach n the proof of Theorem 2 we can prove the optmal bundlng sale s more proftable than the optmal separate sale. Combnng the above two cases we prove the theorem. Proof of Theorem 5: We frst analyze the bundlng sale wth prce p b such that the equlbrum b = = 2. Gven the forms of H ( ) and H B ( ) we have H ( ) V < H B ( ) +V 2. Snce p b = H B ( ) and p = H ( ) we have p b >p V+V2 V so the servce provder s utlty under ths settng satsfes U b > (p ( + V 2 )/V m 2 m 2 ). (23) Snce the above settngs ( b b2) are two realzatons n the bundlng strategy we have the optmal bundlng utlty satsfes 2U b U b + U b2 > p ( + V 2 )/ + p 2 2( + V 2 )/V 2 ( + 2)m 2 ( + 2)m 2. (24) Therefore we have 2(U b U s ) > (V 2 )(p / p 2 2/V 2 ) (( 2) + 2 )m (( 2 2) )m 2.(25) Let us consder another servce S = h and assume ts optmal equlbrum s. Snce the ncrease of unt cost reduces the value of the optmal equlbrum and that 2 apple we have 2. Snce and 2 are also two sale strateges of S and that s nearer to the optmal separate sale we have p p 2 2 V 2. Recall V 2 we have (V 2 )(p / p 2 2/V 2 ). (26) Snce apple 2 apple 2 we have (( 2) + 2)m apple. Snce apple 2 we have (( 2 2) )m 2 apple. Therefore we have (( 2) + 2 )m +(( 2 2) )m 2 apple. (27) Combnng (25) (26) and (27) we conclude U b U s >. 45