Platform Competition: Who Benefits from Multihoming?

Transcription

1 Univerity of Mannheim / Department of Economic Working Paper Serie Platform Competition: Who Benefit from Multihoming? Paul Belleflamme Martin Peitz Working Paper Novemer 2017

2 Platform Competition: Who Benefit from Multihoming? Paul Belleflamme Aix-Mareille Univerity Martin Peitz Univerity of Mannheim Thi verion: Novemer 2017 Atract Competition etween two-ided platform i haped y the poiility of multihoming. If uer on oth ide inglehome, each platform provide uer on either ide excluive acce to it uer on the other ide. In contrat, if uer on one ide can multihome, platform exert monopoly power on that ide and compete on the inglehoming ide. Thi paper explore the allocative effect of uch a change from ingle- to multihoming. Our reult challenge the conventional widom, according to which the poiility of multihoming hurt the ide that can multihome, while enefiting the other ide. Thi i not alway true: the oppoite may happen or oth ide may enefit. Keyword: Network effect, two-ided market, platform competition, competitive ottleneck, multihoming JEL-Claification: D43, L13, L86 We thank Marku Reiinger and Julian Wright for helpful comment. Martin Peitz gratefully acknowledge financial upport from the Deutche Forchunggemeinchaft (PE 813/2-2). Aix-Mareille Univ., CNRS, EHESS, Centrale Mareille, AMSE; Paul.Belleflamme@univ-amu.fr. Other affiliation: KEDGE Buine School and CESifo. Department of Economic and MaCCI, Univerity of Mannheim, Mannheim, Germany, Martin.Peitz@gmail.com. Other affiliation: CEPR, CESifo, and ZEW.

3 1 Introduction Two-ided platform cater to the tate of two audience in many intance, uyer and eller. Deciion among thee audience are interdependent ecaue of poitive cro-group external effect. One of the principle achievement of the literature on two-ided market ha een to characterize the price tructure and aociated price ditortion in alternative market environment. Competing platform have to take into account the effect of a price change to participation level not only on the market ide directly affected, ut alo indirect effect ariing from altered participation on the other ide. A poile market environment i that oth ide inglehome. To reach a particular agent on one ide, an agent from the other ide ha to e on the ame platform. If a platform lure an agent from either ide away from a competitor onto it ite, thi platform ecome more attractive to agent on the other ide, a more tranaction partner ecome availale on the platform ite and fewer partner are availale on the competing ite. Another poile market environment i that agent on one ide can multihome and agent on the other can inglehome. Thi i the o-called competitive ottleneck, which ha een decried in thee term: Here, if it wihe to interact with an agent on the inglehoming ide, the multihoming ide ha no choice ut to deal with that agent choen platform. Thu, platform have monopoly power over providing acce to their inglehoming cutomer for the multihoming ide. Thi monopoly power naturally lead to high price eing charged to the multihoming ide, and there will e too few agent on thi ide eing erved from a ocial point of view [...]. By contrat, platform do have to compete for the inglehoming agent, and high profit generated from the multihoming ide are to a large extent paed on to the inglehoming ide in the form of low price (or even zero price). (Armtrong, 2006, pp ) Thi inight ha een appreciated and reproduced in variou policy document. For intance, in a recent report, the German Cartel Office write: 1 Armtrong analye a contellation which he decrie a competitive ottleneck with one ide applying inglehoming, the other one multihoming. In thi cenario, the platform were competing for uer on the inglehoming ide. Accordingly, on the multihoming ide, platform provided monopolitic acce to inglehoming uer who were memer of the platform. Regarding the framework of the model reviewed, thi led to a monopolitic price on the multihoming ide, while the price on the inglehoming ide would e fairly low a a reult of platform competing for uer on thi ide. In thi repect, thi may reult in an inefficient price tructure depite potentially intenive platform competition (on the inglehoming ide). (BKartA, 2016, p. 58) 1 Another intance i the tatement y the European Commiion in OECD (2009, p. 169). 1

4 In thi paper, we take a cloer look at the price and urplu effect of multihoming. We compare the competitive ottleneck to the two-ided inglehoming market environment. In the latter, platform compete on oth ide of the market, wherea on the former they compete on only one. One may therefore e tempted to conclude that an audience that otain the poiility to multihome face higher price and otain a lower urplu, while the other audience face lower price and otain a higher urplu. Alo, ince in the competitive ottleneck, platform compete on only one ide, one may expect that their profit are higher than in the market environment in which oth audience inglehome. Yet, the effect of making one ide multihome intead of inglehome i le traightforward than what may in general e perceived. While it i true that platform exert monopoly power over the multihoming ide, participant on thi ide may actually enefit from multihoming. A Evan and Schmalenee (2012, p. 16) oerve, 2 in oftware platform, for intance, the price tructure appear to e the oppoite of what the competitive ottleneck theory would predict. Mot peronal computer uer rely on a ingle oftware platform, while mot developer write for multiple platform. Yet peronal computer oftware provider generally make their platform availale for free, or at low cot, to application developer and earn profit from the ingle-homing uer ide. A our analyi will reveal, while thi oervation run counter the claim that the multihoming ide face high price, it i perfectly compatile with the competitive ottleneck model. For the ake of concretene, we aume that the eller ide i the ide of the market which potentially can multihome, while uyer alway inglehome. Our main finding are a follow. When going from inglehoming to multihoming on one ide, price on oth ide of the market alway move in oppoite direction. It i not necearily the cae that eller pay a higher fee and uyer, a lower fee; the oppoite may occur ecaue eller may pay a low price to tart with in the competitive ottleneck cae. Platform prefer to impoe excluivity to eller (i.e., to prevent them from multihoming) if the eller intrinic value (the difference etween their tand-alone enefit and the marginal cot of accommodating them) i not too large. There exit configuration of parameter for which thi condition i alway, or never, atified. Buyer tend to prefer the competitive ottleneck environment when they value a lot the preence of eller and eller find it profitale to multihome; they are then more likely to interact with a larger et of uyer and to e charged lower fee. However, it may alo happen that platform charge higher fee to uyer when eller multihome than when they inglehome, and that thi negative price effect outweigh the poitive participation effect, leading uyer to prefer the two-ided inglehoming environment. A for eller, if they perceive the two platform a more differentiated and if they exert weaker cro-group effect on uyer, then they are more likely to e etter off in the competitive ottleneck cae. Comining thee finding, we otain three important inight aout how the urplu effect play out for the three group. Firt, the reulting market outcome may have the feature that uyer, eller and platform are all etter off when eller are allowed to multihome. Second, 2 Evan, Hagiu, and Schmalenee (2006) made the ame point. 2

5 whenever platform enefit from impoing excluivity, they necearily hurt uyer and poily alo eller. Thu, in an environment with potential eller multihoming, an agency hould prohiit the ue of excluivity of the eller ide if it aim i to maximize uyer urplu. Third, whenever uyer uffer from eller multihoming, platform and eller enefit from it. While comparing a two-ided inglehoming model to a competitive ottleneck model i an intereting exercie, it may appear to e of little practical relevance ecaue inglehoming on oth ide i oerved in only few market environment. We want to challenge thi view. Firt, while multihoming may e feaile for ome uer on one ide, it may well e the cae that due to hait or other latent factor a fraction of uer doe not conider the poiility of multihoming. Then, a comparion etween two-ided multihoming and the competitive ottleneck i an extreme verion of a comparion etween market in which uch hait and other latent factor are preent and thoe in which they are not. Second, the preent comparion inform policy maker aout poile effect when taking action to enale multihoming or prohiiting excluive dealing for a fraction of eller (or, in the flip ide of the model, of uyer). The former may take the form of aggregator comining the functionalitie and liting y oth platform. There exit urpriingly little work that tudie the competitive effect of multihoming, depite the policy deate aout mean to encourage multihoming. In the eminal paper y Armtrong (2006) oth market environment that i, two-ided ingle-homing and competitive ottleneck are analyzed in detail, ut no comparion i undertaken. We follow hi approach of conidering platform that are horizontally differentiated on oth ide of the market and charge acce fee to each ide. A an alternative, platform may charge tranaction fee, a analyzed in Rochet and Tirole (2003). For an inightful dicuion of the ue of different price intrument, ee Rochet and Tirole (2006). While it would e intereting, to extend our analyi to other price intrument, we retrict attention to acce fee, which i the natural aumption to make when platform cannot monitor tranaction. Armtrong and Wright (2007) endogenize the multihoming deciion of uyer and eller. In their etting, the competitive ottleneck model emerge endogenouly a one ide decide not to multihome along the equilirium path. They, a well a the ret of the literature, do not look at the urplu effect of the poiility of multihoming. In general, there exit little work on urplu effect in market with two-ided platform. An exception i Anderon and Peitz (2017) they evaluate urplu effect of policy intervention in the competitive ottleneck world. The remainder of the paper i organized a follow. We firt lay out the model (Section 2) and olve it when oth ide inglehome (Section 3) and when one ide i allowed to multihome (Section 4). We are then in a poition to derive our main reult y comparing the two etting (Section 5). After howing that our reult are rout to a more general formulation of the utility of multihomer (Section 6), we propoe ome concluding remark (Section 7). 3

6 2 The model Two platform compete to facilitate the interaction etween a unit ma of eller and a unit ma of uyer, with thi interaction generating poitive cro-group external effect. Following Armtrong (2006), we aume that platform compete in memerhip fee and that uyer and eller perceive them a horizontally differentiated. Horizontal differentiation i modeled in the Hotelling fahion: platform are located at the extreme point of the unit interval and face contant cot c and c for each additional eller and uyer, repectively; eller and uyer are uniformly ditriuted on thi unit interval and incur an opportunity cot of viiting a platform that increae linearly in ditance at rate τ and τ, repectively. We aume the following form of interaction etween uyer and eller on a platform: uyer purchae one unit of the perfectly differentiated product offered y each eller who i active on the platform; each trade generate a enefit for the uyer and a profit for the eller. 3 Buyer and eller alo derive a tand-alone enefit from viiting a platform; we aume that thee enefit are equal acro platform and we note them r and r. Letting n i and ni denote the ma of uyer and eller active on platform i, and noting m i and mi the memerhip fee that platform i charge uyer and eller, we can expre uyer and eller urplue of viiting platform i (gro of any opportunity cot) a v i = r + n i m i and v i = r + n i m i. We conider a two-tage game in which platform i {1, 2} et memerhip fee on each ide of the market, m i, m i, imultaneouly in tage 1 and uyer and eller imultaneouly make their ucription deciion in tage 2. We compare different etting for tage 2 according to whether participant in one or the other group chooe at mot one platform (i.e., to inglehome ) or alo have the option to e active on oth platform (i.e., to multihome ). We olve for ugameperfect equiliria and will retrict attention to parameter contellation uch that oth platform will e active in equilirium (we will provide precie aumption elow). A a enchmark, conider the cae that cro-group external effect are zero i.e., = = 0. Thu, if participant on a particular ide are inglehoming, we olve the tandard Hotelling model and otain c g + τ g a equilirium price on ide g {, }, provided that there i full market coverage. Thi require that r g τ g /2 (c g + τ g ) > 0 or, equivalently, r g c g > (3/2)τ g hold. If participant are multihoming, their choice of uying from one platform i independent of the pricing of the other platform, and each platform olve a monopoly prolem. Solving the firt-order condition of profit maximization when the marginal conumer i in the interior of the [0, 1]-interval, we otain the price (r g + c g )/2. At thi price, the marginal conumer i indeed in the interior if r g τ g (r g + c g )/2 < 0, which i equivalent to r g c g < 2τ g. In thi cae, the price under multihoming i le than under inglehoming if and only if (r g + c g )/2 < c g + τ g or, equivalently, r g c g < 2τ g. To ummarize, for (3/2)τ g < r g c g < 2τ g, the price under 3 It i aumed, quite realitically, that in a eller uyer relationhip, price or term of tranaction are independent of the memerhip fee that applie to uyer and eller. 4

7 multihoming which coincide with the monopoly price in a ingle-product monopoly prolem i le than the price under inglehoming which coincide with the tandard Hotelling duopoly prolem. The reaon for thi counterintuitive reult i that the firm face a more elatic demand when the conumer outide option i a contant rather than the competitor offer (which ecome increaingly attractive when the firm want to reach conumer located further away) that i, the ame price cut lead to a larger increae in the demand in the monopoly than in the duopoly etting. 3 Two-ided inglehoming In thi ection we conider a market environment in which oth ide of the market inglehome. Our aim i to characterize the equilirium in which oth platform are active. 4 If uyer and eller inglehome, the eller and the uyer who are indifferent etween the two platform are repectively located at x and x uch that v 1 τ x = v 2 τ (1 x ) and v 1 τ x = v 2 τ (1 x ). It follow that n 1 = x, n 2 = 1 x, n 1 = x, and n 2 = 1 x and the total numer of each ide agent on the two platform add up to 1: n 1 + n 2 = n 1 + n2 = 1. Comining the indifference equation together with the expreion of v i and v i, we otain the following expreion for the numer of uyer and eller at the two platform: n i ( ) ) n i = τ ((2n i 1) (m i m j ), n i ( ) ( ) n i = τ (2n i 1) (m i mj ). Solving thi linear equation ytem, we derive the equilirium numer of uyer and eller at tage 2 a a function of the memerhip fee: n i (m i, m j, m i, mj ) = (m j mi ) + τ (m j m i ), 2(τ τ ) n i (mi, m j, m i, mj ) = (m j m i ) + τ (m j mi ). 2(τ τ ) Platform i chooe m i and m i to maximize Πi = ( m i c ) n i ( ) + ( m i c ) n i ( ). At the ymmetric equilirium (m 1 = m 2 m and m 1 = m2 m ), the firt-order condition can e written a { m = c + τ τ ( + m c ), m = c + τ τ ( + m c ). The equilirium memerhip fee for the eller i equal to marginal cot plu the productdifferentiation term a in the tandard Hotelling model, adjuted downward y the term τ ( + m c ). A pointed out y Armtrong (2006), to undertand thi term, note from expreion (1) that each additional eller attract /τ additional uyer. Thee additional uyer allow the intermediary to extract per eller without affecting the eller urplu. In addition, each of 4 Our analyi follow Armtrong (2006). A textook treatment can e found in Belleflamme and Peitz (2015). (1) 5

8 the additional /τ uyer generate a margin of m c to the platform. Thu τ ( + m c ) repreent the value of an additional uyer to the platform. The ame hold on the uyer ide. Solving the ytem of firt-order condition (with full participation) give explicit expreion for equilirium memerhip fee (where the upercript 2S denote two-ided inglehoming ): m 2S = c + τ and m 2S = c + τ. We oerve that the equilirium memerhip fee for one group i equal to the uual Hotelling formulation (marginal cot plu tranportation cot) adjuted downward y the cro-group external effect that thi group exert on the other group (ee Armtrong, 2006). A platform et the ame fee at equilirium, the indifferent participant of oth ide are located at 1/2, meaning that n 2S = n 2S = 1/2. It follow that in equilirium, a eller and a uyer otain, repectively, a urplu (gro of their tranport cot) given y v 2S = h τ + and v 2S = h τ +, where h g r g c g denote the difference etween the tand-alone enefit and the cot of accommodating a participant of ide g {, }. Thu eller and uyer aggregate urplue are calculated repectively a P S 2S = v 2S 2 CS 2S = v 2S τ xdx = h 5 4 τ , τ xdx = h 5 4 τ Finally, equilirium profit are the ame for oth platform and are computed a Π 2S = 1 2 (τ + τ ). They are increaing in the degree of product differentiation on oth ide of the market (a in the Hotelling model) and decreaing in the uyer and eller urplu for each tranaction, i.e., the magnitude of the cro-group external effect. The intuition for the latter reult i the following: a cro-group external effect increae, platform compete more fiercely to attract additional agent on each ide a they ecome more valuale. A erie of condition have to e met for the previou equilirium to e valid. Firt, the econd-order condition of the profit-maximization program are τ τ > and 4τ τ > ( + ) 2. Both condition require that the tranportation cot parameter τ and τ (which meaure the horizontal differentiation etween the two platform) are ufficiently large with repect to the gain from trade and (which meaure the cro-group external effect). Thee condition are alo ufficient to have a unique and tale equilirium in which oth platform are active. We check that the former condition make ure that the numer of memer of one group at one platform, n i ( ) or n i ( ), decreae not only with the memerhip fee that they have to pay ut alo with the memerhip fee that the other group ha to pay on thi platform. 5 We alo 5 For tronger cro-group external effect and/or weaker horizontal differentiation (i.e., for > τ τ ), the numer of agent on one platform would e an increaing function of their memerhip fee and the market would tip; i.e., all uyer and eller would chooe the ame platform. 6

9 oerve that the latter condition i more retrictive than the former. Furthermore, we need to guarantee full participation on the two ide; that i, the indifferent participant on each ide (located at 1/2) mut have a poitive net urplu at equilirium: on the eller ide, v 2S 1 2 τ > 0 or 2h > 3τ 2 ; on the uyer ide v 2S 1 2 τ > 0 or 2h > 3τ 2. In um, we make the following et of aumption for the two-ided inglehoming cae. Aumption 1 In the two-ided inglehoming cae, parameter atify 4τ τ > ( + ) 2 2h > 3τ 2 2h > 3τ 2 (Soc2S) (FP2S) (FP2S) 4 Multihoming on one ide (competitive ottleneck) Suppoe now that eller have the poiility to multihome (i.e., to e active on oth platform at the ame time), while uyer continue to inglehome. We aume that the deciion whether to ucrie to one platform i independent of the deciion whether to ucrie to the other platform. In particular, if a eller x i ucried to oth platform hi urplu i r + n 1 m 1 τ x + r + n 2 m 2 τ (1 x) = 2r + τ (m 1 + m 2 ), which i independent of hi location. According to our aumption, a multihoming eller enjoy the tand-alone enefit on oth platform i.e., platform are ymmetric ut provide different ervice leading to tandalone enefit that can e comined when multihoming. 6 Seller can e divided into three uinterval on the unit interval: thoe eller located on the left regiter with platform 1 only, thoe located around the middle regiter with oth platform, and thoe located on the right regiter with platform 2 only. At the oundarie etween thee interval, x i0, we find the eller who are indifferent etween viiting platform i (i {1, 2}) and not viiting thi platform. Their location are found a, repectively, x 10 uch that r + n 1 m 1 = τ x 10, and x 20 uch that r + n 2 m 2 = τ (1 x 20 ). We aume for now that 0 < x 20 < x 10 < 1 (we provide neceary and ufficient condition elow), o that n 1 = x 10 and n 2 = 1 x 20, with the multihoming eller eing located etween x 20 and x 10. A far a uyer are concerned, we have the ame ituation a in the previou ection. The numer of uyer and eller viiting each platform are thu repectively given y n i = (n i n j ) (m i mj ) and n i = r + n i m i. 2τ τ Solving thi ytem of four equation in four unknown, we otain uyer and eller participation a a function of ucription fee and the parameter of the model, n i = (m j m i ) + τ (m j mi ), ( 2 (τ τ ) n i = 1 τ 2 + (m j m i ) + τ (m j mi ) ) + r m i. 2 (τ τ ) τ 6 Thi aumption differ form Armtrong and Wright (2007) who aume that the ervice that give rie to the tand-alone utility are the ame on oth platform. We dicu the implication of our aumption in Section 6. 7

10 The maximization prolem of the two platform are the ame a aove. Platform 1 et repone are implicitly defined y the firt-order condition, which can e expreed a m 1 = ( + ) m 1 + m 2 + τ m 2 ( c ) + τ (τ + c ) 2τ, m 1 = ( + ) τ m 1 + m 2 + τ m 2 ( + c + 2r ) + τ c + ( + 2c + 2r ) τ τ. 2 (2τ τ ) Solving the previou ytem of equation, we find the equilirium memerhip fee, which are equivalent for oth platform (with the upercript CB tanding for competitive ottleneck ): m CB = 1 2 (r + c ) ( ), m CB = c + τ 4τ ( r 2c ). On the eller ide, platform have monopoly power. If the platform focued only on eller, it would charge a monopoly price equal to (r + c ) /2+ /4 (auming that each eller would have acce to half of the uyer and, therefore, would have a gro willingne to pay equal to /2). We oerve that thi price i adjuted downward y /4 when the cro-group effect that eller exert on the uyer ide i taken into account. Similarly, on the uyer ide, platform charge the Hotelling price, c + τ, le a term that depend on the ize of the cro-group effect and on the parameter characterizing the eller ide (r, c, and τ ). It i ueful to compare price change in the competitive ottleneck model to thoe in the twoided inglehoming model. We oerve that the equilirium memerhip fee for eller i increaing in the trength of the cro-group effect in the competitive ottleneck model ( m CB / > 0), wherea it i contant in the two-ided inglehoming model. Thi i due to the monopoly pricing feature on the multihoming ide. Everything ele equal, if eller are multihoming, the platform operator directly appropriate part of the rent generated on the multihoming ide y etting higher memerhip fee. Thi i not the cae in the inglehoming world, where the memerhip fee doe not react to the trength of the network effect on the ame ide ince platform compete for eller (and uyer). At equilirium, eller and uyer participation i n CB = 1 2 and ncb = 1 4τ ( + + 2h ). Thi allow u to compute the equilirium net urplu of eller and uyer (gro of tranportation cot and for one platform) a: v CB = 1 4 ( + + 2h ), v CB = 1 4τ ( ( + ) h ) + h τ. Note that v CB i the per-platform eller urplu (gro of tranport cot). 7 We oerve that v CB and v CB are increaing in the net gain of the other ide and in the net gain of the own ide. 7 Seller located etween 1 n CB hand, v CB and n CB i the urplu earned y the eller located etween 0 and 1 n CB only, and y the eller located etween n CB multihome and, therefore, earn a urplu of 2v CB. On the other, who chooe to viit platform 1 and 1, who chooe to viit platform 2 only. 8

11 Aggregated over all uyer, in equilirium, uyer urplu i CS CB = v CB 2 1/2 0 τ xdx = 1 4τ (( + ) ( + + 2h ) + 2 ) + h 5 4 τ. Aggregated over all eller, in equilirium, eller urplu i computed a (uing the definition of n CB and v CB ): P S CB = 1 n CB 0 n CB (v CB τ x)dx + 1 n CB = 1 ( ) τ v CB 2 = 1 16τ ( + + 2h ) 2. 1 (2v CB τ )dx + n CB (v CB τ (1 x))dx We oerve that the aggregated eller urplu i decreaing in the degree of platform differentiation on the eller ide, increaing in the tand-alone enefit on the eller ide, and increaing in cro-group external effect on oth ide. The platform equilirium profit are Π CB = 1 16τ (8τ τ ( + ) h 2 ). A aove, a et of condition need to e atified for thi equilirium to hold. The econdorder condition are here τ τ > and 8τ τ > ( + ) 2 + 4, the econd condition eing more tringent than the firt. Note that the latter condition implie that even if platform do not offer tand-alone utilitie (h = r c = 0), equilirium profit are trictly poitive. Thee condition alo guarantee that a unique and tale equilirium exit in which oth platform hare the market in hort, haring equilirium. We alo impoe that ome (ut not all) eller multihome at equilirium (if ome eller multihome, thi alo implie that all eller participate). Thi i the cae if 1/2 < n CB < 1, which i equivalent to 2τ < + + 2h < 4τ. Finally, all uyer mut e willing to participate; i.e., r + n CB m CB 1 2 τ > 0 which i equivalent to 4τ h + 2 ( + ) h > 6 (τ τ ) ( ) 2. We collect all thee condition in the following aumption. Aumption 2 In the competitive ottleneck cae (with multihoming eller), parameter atify 8τ τ > ( + ) h > 2τ 2h < 4τ 4τ h + 2 ( + ) h > 6 (τ τ ) ( ) 2 (SocCB) (FPCB) (ShCB) (FPCB) Comparing condition in Aumption 1 and 2, we note the following. Firt, the econdorder condition are le demanding in the competitive ottleneck cae than in the two-ided inglehoming cae; that i, if (Soc2S) i atified, o i (SocCB). Second, for τ >, full participation of eller i guaranteed in the competitive ottleneck cae if full participation i guaranteed in the two-ided inglehoming cae; that i, if (FP2S) i met, then o i (FPCB). 9

12 5 Singlehoming v. multihoming In thi ection, we compare the haring equilirium of the two previou environment. Thu, the et of aumption 1 and 2 have to hold. Regrouping them, we impoe: 8 4τ τ > ( { + ) 2 ( )} h > max 1 2 (3τ 1 2 ), 4τ 6 (τ τ ) ( ) 2 2 ( + ) h h min max { 1 2 (2τ ), 1 2 (3τ 2 ), 0 } < h < h max 1 2 (4τ ) To tart with, we ak when a haring equilirium can e upported in the two environment. The relevant aumption regarding the relationhip etween platform differentiation and cro-group external effect i (Soc2S) with two-ided inglehoming market and (SocCB) in the competitive ottleneck market. A mentioned aove, the former implie the latter. Thi confirm the claim that a haring equilirium i more likely to arie in a competitive ottleneck environment than in a two-ided inglehoming environment. We now examine in turn the price, the platform profit, and the urplue of the participant. 5.1 Price We recall that in the model in which eller multihome, platform hold an excluive acce to their et of inglehoming uyer (the ottleneck ), which make uyer valuale to extract profit on the eller ide. Platform then et monopoly price on the multihoming ide and low (and poily even negative price) on the inglehoming ide, a ha een pointed out y Armtrong (2006). In other word, we expect platform to compete fiercely for uyer (inglehomer) and, in return, to milk eller (multihomer). Hence, we may expect lower price on the uyer ide and higher price on the eller ide when compared to the two-ided inglehoming model. We call thi the ottleneck effect. However, price charged to eller in the competitive ottleneck model may e low to tart with and competition may well afford poitive margin in the two-ided inglehoming environment. A tated in the following lemma, it depend on the parameter whether eller pay a lower price in the competitive ottleneck model. Comparing price, the ottleneck effect doe not necearily dominate (there are parameter configuration that atify all our aumption for either cae). In addition, when moving from inglehoming to multihoming on one ide, price on oth ide of the market alway move in oppoite direction. Lemma 1 Allowing eller to multihome increae the fee paid y eller and decreae the fee paid y uyer, m CB > m 2S and m CB < m 2S, if and only if 2h + + > 4τ 2. The oppoite happen that i, m CB > m 2S and m CB < m 2S if and only if the left-hand and the right-hand ide of the inequality are revered that i, 2h + + < 4τ 2. Both cae are compatile with the condition impoed on the parameter in Aumption 1 and 2. 8 It i the third line linking the value of h, τ,, and that will e crucial to derive our reult. Given the value of thee parameter, it i alway poile to find value of h and τ that atify the firt two line. 10

13 Proof. The proof follow directly from computing the difference etween the eller and the uyer fee in the two cae: m CB m CB m 2S = 1 2 (r + c ) ( ) (c + τ ) = 1 4 [(2h + + ) (4τ 2 )], m 2S = c + τ 4τ ( r 2c ) (c + τ ) = 4τ [(4τ 2 ) (2h + + )]. A for the compatiility with Aumption 1 and 2, we recall that condition (FP2S), (FPCB) and (ShCB) impoe that max {2τ, 3τ } < 2h + + < 4τ. Clearly, 4τ 2 < 4τ. Moreover, if τ >, then 4τ 2 > max {2τ, 3τ }. Baed on the intuition that eller have to pay monopoly price in the competitive ottleneck model and that platform compete in thi cae fiercely on the multihoming ide, we would conider m CB > m 2S and m CB < m 2S a the natural outcome. However, we have hown aove, in the enchmark cae with no cro-group effect, that the monopoly price (correponding to the competitive ottleneck model) i alway lower than the duopoly price (correponding to the two-ided inglehoming model). Thi i ecaue a drop in the fee on the eller ide i more effective to expand the numer of eller when they are multihoming (in which cae, the offer of a platform compete againt the outide option, jut a in the monopoly prolem) intead of inglehoming (in which cae, the offer of a platform compete againt the rival offer, jut a in the tandard duopoly prolem). Thi reult till hold in the orderline cae in which only eller are uject to a poitive cro-group external effect ( > 0 and = 0): we indeed check that m CB m 2S = 1 4 (2h + 4τ ) < 0 y virtue of condition (ShCB), which ecome 2h < 4τ in thi particular cae. It follow that the natural outcome can occur only if the uyer utility increae with the numer of eller ( > 0). 5.2 Platform incentive What are the platform incentive regarding ingle- v multihoming? Thi i not a rhetorical quetion, a platform may e ale to ue non-price trategie to prevent participant from multihoming. For intance, a platform may impoe excluivity on eller and, thu, force them to ecome inglehomer. How do platform profit depend on excluivity? To anwer thi quetion, we compare equilirium profit: Π CB Π 2S = [( m CB ) c n CB + ( m CB ( m CB m 2S = 1 2 ) ] [( c n CB m 2S ) ( 1 τ ) + ( n CB 1 2 c ) n 2S ) ( m CB c ). + ( m 2S c ) n 2S ] where the econd line ue our previou reult, namely n CB = n 2S = n 2S = 1 2, ncb > 1/2 and m CB m 2S = ( /τ ) ( m CB m 2S ). We ee that if, for intance, eller pay a higher fee in the competitive ottleneck cae and thi fee i larger than marginal cot (m CB > m 2S and m CB > c ) while τ >, then all term are poitive, meaning that platform make higher 11

14 profit when eller can multihome. Converely, till in the cae where τ >, if platform uidize eller in the competitive ottleneck cae (m CB two-ided inglehoming cae (m CB prevent eller from multihoming. < m 2S < c ) and et a lower fee than in the ), then all term are negative and platform prefer to To formalize thi intuition, we ue the value of m CB, m 2S and n CB to compute Π CB Π 2S = 4h2 + 8 ( + ) τ 8τ 2 ( ). 16τ A ufficient condition for Π CB > Π 2S i 8 ( + ) τ 8τ 2 ( ) > 0 (a we aume h > 0). Thi polynomial in τ ha two poitive root, ( + ) /2 ± 2 ( ) /4, and i poitive if τ i compried etween the two root. Otherwie, we have that Π CB < Π 2S h < 1 2 8τ 2 8 ( + ) τ + ( ) 2 h Π. Recalling that 2h < 4τ according to condition (ShCB), we oerve that the latter inequality i alway atified if τ < /2. 9 We record our reult in the following lemma. Lemma 2 If 2 < 4τ 2 2 < 2, then platform are alway willing to allow eller to multihome. In contrat, if τ < /2, then platform are alway willing to prevent eller from multihoming. Outide thi region of parameter, platform prefer to allow eller to multihome if and only if h > h Π. 5.3 Participant urplue In thi uection we compare the aggregate urplu of uyer and eller in the two environment. In particular, we want to know to whether uyer and eller preference aligned or mialigned regarding the multihoming of eller. Our previou dicuion aout equilirium fee point at a major ource of mialignment, a fee move in oppoite direction: when eller pay lower fee in the competitive ottleneck cae, uyer pay lower fee in the two-ided inglehoming cae, and vice vera. However, participant alo care aout the numer of agent of the other group they can interact with, and thee numer alo differ in the two environment. Firt, a n CB > 1/2, there are more eller active on a platform under multihoming than under inglehoming, thu adding value to participation on the uyer ide. Second, multihoming eller have acce to all uyer, which may poitively affect their urplu (even if they pay twice the fee and the tranportation cot). Thu, we need to examine how the effect of price and participation alance one another. Buyer. For uyer, we have v CB v 2S = (n CB 1/2) ( m CB m 2S ). The firt term i the participation effect and i clearly poitive, a multihoming ring more eller on each 9 It i readily checked that τ < /2 < min { ( + ) /2 2 ( ) /4, ( + ) /2 + 2 ( ) /4 }. Recall alo that parameter mut atify condition (Soc2S), i.e., 4τ τ > ( + ) 2, which can e rewritten a 2τ 2 > ( ( + ) 2 / (2τ ) )2. Thi condition i compatile with τ < /2 a long a τ > ( + ) 2 /(2 ). 12

15 platform; the econd i the price effect and, a we have een aove, it can e either negative (if m CB > m 2S ) or poitive. At the aggregate level, CSCB = v CB τ /4 and CS 2S = v 2S τ /4 imply that CS CB CS 2S = v CB v 2S. Uing the definition of ncb and the expreion derived aove for m CB m 2S, we can write [ ] CS CB CS 2S = 1 4τ (2h + + ) 1 2 4τ [(4τ 2 ) (2h + + )]. It i clear that the participation effect (the firt term) increae with the trength of the crogroup effect that eller exert on uyer ( ), and with the equilirium numer of eller in the competitive ottleneck cae, which i itelf an increaing function of h and a decreaing function of τ (more eller decide to multihome when their intrinic enefit are larger and when tranportation cot are maller). It can e checked that an increae in or h, or a decreae in τ alo reduce the importance of the price effect (the econd term), therey making it unamiguouly more likely that CS CB > CS 2S. 10 Developing the previou expreion, we find CS CB > CS 2S h > 4τ 2 + τ + h. It i readily checked that the latter inequality i more likely to e atified when increae or when τ decreae, which confirm our previou intuition. Recall that condition (FP2S), (FPCB) and (ShCB), together with h > 0, impoe h min max { 1 2 (2τ ), 1 2 (3τ 2 ), 0 } < h < h max 1 2 (4τ ). It i clear that h i le than the upper ound. Hence, there alway exit admiile value of h that are ufficiently large for conumer to prefer that eller e allowed to multihome. However, parameter may e uch that h i alo le than the lower ound, implying that uyer prefer eller to multihome for any admiile value of h. Thi i o, for intance, if >, i.e., if uyer value more the interaction with eller than vice vera. We formalize thee finding in the next lemma (which i proved in Appendix 8.1). Lemma 3 If > or if > and τ < ( ) / (2 ( + 2 )) τ min, then uyer are alway etter off when eller are allowed to multihome (CS CB > CS 2S ). Otherwie (i.e., for > and τ > τ min ), they may prefer that eller e forced to inglehome (CS 2S > CS CB ) for mall value of h = r c (i.e., for h min < h < h ). In ummary, Lemma 3 how that we can expect conumer to welcome the poiility for eller to multihome, ecaue it give them acce to a larger et of eller with whom they can interact, and it may even lead platform to charge them lower fee. Thi i all the more likely that they value a lot the preence of eller and that eller find it profitale to multihome. We provide ufficient condition for thi to e only poile outcome. However, it may happen that platform charge higher fee to uyer when eller multihome than when they inglehome, and that thi negative price effect outweigh the poitive participation effect if thi i the cae, uyer prefer that platform actually prevent eller multihoming. 10 The net effect of a change in i amiguou. 13

16 Seller. For inglehoming eller, we have v CB v 2S = m 2S m CB ; here, there i no participation effect, a uyer equally plit etween the two platform in oth environment; and we have een aove that the price difference can go oth way. A for multihoming eller, we focu on the one located at the middle of the Hotelling line for whom the urplu difference i equal to 2v CB τ (v 2S τ /2) = r + ( τ ) /2 ( 2m CB m 2S ). Developing the latter expreion, we find that thi eller i etter off in the competitive ottleneck environment than in the inglehoming environment if and only if τ >. 11 At the aggregate level, we recall the expreion derived in Section 3 and 4: It follow that P S CB = 1 ( ) τ v CB 2 = 1 16τ ( + + 2h ) 2, P S 2S = v 2S 1 4 τ = h 5 4 τ P S CB > P S 2S 4h 2 4 (4τ ) h + 20τ 2 8 (2 + ) τ + ( + ) 2 > 0. If τ > 2, thi polynomial in h ha no real root in which cae it can e hown that it i poitive everywhere, meaning that P S CB > P S 2S. Otherwie, for τ < 2, the polynomial ha two real root. The larger root i equal to 1 2 (4τ ) + τ (2 τ ). Recalling that our parameter retriction impoe that h < h max = 1 2 (4τ ), we immediately ee that thi root lie aove the upper ound of the admiile range. A for the lower root, we denote it h 1 2 (4τ ) τ (2 τ ). A etalihed in the following lemma (ee Appendix 8.2 for the formal proof), the inequality h > h min mut hold unle τ i ufficiently mall. Then, we have that P S CB > P S 2S for h < h and P S CB < P S 2S otherwie. In the cae in which h < h min, we have that P S CB < P S 2S for all admiile parameter configuration. Lemma 4 If τ > 2, then eller are alway etter off when they are allowed to multihome (P S CB > P S 2S ). In contrat, for ufficiently mall value of τ, eller are alway etter off when they are prevented from multihoming (P S CB < P S 2S ). For intermediate value of τ, eller prefer to multihome if h < h, and to inglehome otherwie. According to Lemma 4, eller preference regarding multihoming crucially depend on the ratio etween the degree of platform differentiation, τ, and the trength of cro-group effect that eller exert on uyer, : the larger thi ratio that i, the larger τ and/or the maller the more likely it i that eller are etter off in the competitive ottleneck cae. Thi tand in harp contrat with what we oerved for uyer, who are more likely to prefer that eller inglehome when τ increae and/or decreae. Thu, we have identified here an important 11 Recall that under thi condition, full participation of eller i guaranteed in the competitive ottleneck cae if full participation i guaranteed in the two-ided inglehoming cae i.e., condition (FP2S) implie condition (FBCP). 14

17 ource of divergence etween the preference of uyer and eller. However, thi doe not exclude the exitence of parameter configuration uch that uyer and eller agree, a we dicu in the following uection. 5.4 Surplu comparion: The complete picture To cloe thi ection, we uperimpoe the reult of Lemma 2 4, o a to meaure the extent to which platform, uyer and eller agree or diagree with repect to the enefit from eller multihoming. There are a priori eight poile cenario. In the next propoition, we eliminate three of them y proving a clear divergence etween uyer on one ide, and eller and platform on the other ide (the proof i relegated to Appendix 8.3). Propoition 1 Whenever uyer prefer that eller e forced to inglehome (CS 2S > CS CB ), oth eller and platform prefer the oppoite (P S CB > P S 2S and Π CB > Π 2S ). Propoition 1 leave u with four other cenario. The firt two cenario correpond to the conventional widom: CS CB > CS 2S and P S 2S > P S CB ; that i, the poiility of multihoming i eneficial for the ide that continue to inglehome, ut harmful for the ide that i allowed to multihome. Then, platform may have higher profit in either environment: they will pleae eller if they chooe to impoe excluivity, and pleae uyer otherwie. In the lat two cenario, oth uyer and eller are etter off in the competitive ottleneck environment: CS CB > CS 2S and P S CB > P S 2S. Again, platform may prefer one or the other environment. Here, if they impoe excluivity (which can only occur if > ), they hurt oth group; otherwie, the poiility of multihoming for eller i welfare improving a it make all partie etter off. We ummarize our finding in the next propoition. Propoition 2 (1) Whenever platform find it preferale to impoe excluivity, they necearily hurt at leat one group of participant. (2) It i poile that uyer, eller and platform are all etter off when eller are allowed to multihome. Proof. The firt part directly follow from Propoition 1, a uyer and eller never agree that excluivity would make them all etter off. To how the exitence of the other configuration, we uild numerical example. Firt, take > and τ > 2, o that repectively uyer and eller alway prefer the competitive ottleneck environment. Set = 40, = 10 and τ = 85. Then, we have h Π = while h min = 82.5 and h max = 145. For h = 83, we check that Π CB Π 2S = 43/170 < 0, in which cae platform would impoe excluivity, therey hurting oth uyer and eller. In contrat, for h = 85, we have Π CB Π 2S = 25/34 > 0, in which cae all partie agree that the competitive ottleneck environment i preferale. We ummarize our reult in Tale 1 and Figure 1. Tale 1 how the five poile comination of preference for the three partie uyer, eller, and platform ( CP indicate a preference for the competitive ottleneck cae, 2S for the two-ided inglehoming cae). In the 15

18 lat column of the tale, we indicate the zone of Figure 1 that correpond to the variou comination (the left panel correpond to the cae in which > and the right panel, to the cae in which > ). Buyer Seller Platform Zone in Figure 1 CB 2S 2S 1, 5 CB 2S CB 2, 6 CB CB CB 3, 7 CB CB 2S 4 2S CB CB 8 Tale 1. Preferred market environment h > h h max h > h h max h h h h min h h h min Figure 1: Surplu effect of eller multihoming We conclude that without further information, a competition authority or regulator cannot know whether allowing multihoming on one ide (with the other ide inglehoming) lead to higher or lower net urplue on either ide. It i, therefore, a priori not poile to ay whether the ide that change it ehavior from inglehoming to multihoming (or revere) enefit or uffer from thi change of ehavior. 6 Extenion: Stand-alone enefit when multihoming We propoe here a more general formulation of the competitive ottleneck prolem y auming that a multihoming eller enjoy a total tand-alone enefit equal to (1 + ρ) r. We let the parameter ρ take any value etween 0 and 1 to cover any ituation regarding the ervice that give rie to the tand-alone utility. At one extreme (ρ = 1), we have the cae conidered o far in thi paper: platform provide completely differentiated ervice. At the other extreme (ρ = 0), we have the cae analyzed y Armtrong and Wright (2007): platform provide exactly the ame ervice, o that joining a econd platform doe not generate any extra tand-alone 16

19 enefit. In thi ection, we conider the whole pectrum etween thee two extreme. Our goal i to how that our reult till hold under thi more general formulation and that our main meage i even reinforced that i, the conventional widom aout the effect of multihoming cannot e truted. A the analyi follow the ame tep a in Section 4, we kip mot of the development. Derivation of equilirium. The eller who are indifferent etween viiting platform i (i {1, 2}) and viiting oth platform are now identified y x 1m = 1 1 ( ρr + n 2 τ m 2 ) and x2m = 1 ( ρr + n 1 τ m 1 ). A aove, we aume that 0 < x 1m < x 2m < 1, o that n 1 = x 2m and n 2 = 1 x 1m. Noting that nothing change for uyer, we can proceed in the ame way a in Section 4: we olve for uyer and eller participation level and then ue thee expreion to derive the platform profit a function of the four fee. Solving the ytem of the four firt-order condition, we find the following equilirium fee and equilirium participation, where h,ρ tand for ρr c (note that h,ρ = h = r c for ρ = 1): m c = c + τ ( h,ρ ), 4τ m c = c h,ρ ( ), n c = 1 2 and nc = 1 4τ ( + + 2h,ρ ). From there, we can compute the equilirium net urplu of eller and uyer (gro of tranportation cot) a: where v h v h = 1 4 ( + ) ((2 ρ) r c ), v mh = 1 2 ( + ) + (r c ), v c = 1 4τ ( ( + ) h,ρ ) + h τ, denote the urplu for inglehoming eller (i.e., eller located etween 0 and 1 n c who chooe to viit platform 1 only, and eller located etween n c platform 2 only) and v mh, and 1, who chooe to viit the urplu for multihoming eller (i.e., eller located etween 1 n c = 2v h (1 ρ) r that i multihoming eller earn le than twice and n c ). Note that v mh the urplu of inglehoming eller ecaue of the poile duplication etween the tand-alone enefit provided y the two platform. For the pecial cae ρ = 1 (no duplication), v mh a potulated in Section 4. Aggregated over all uyer, in equilirium, uyer urplu i CS c = 1 4τ (( + ) ( + + 2h,ρ ) + 2 ) + h 5 4 τ. Aggregated over all eller, in equilirium, eller urplu i P S c = x c 1m 0 (v h τ x)dx + x c 2m x c 1m = 1 16τ ( + + 2h,ρ ) 2 + (1 ρ) r. (v mh τ )dx (v h x c 2m τ (1 x))dx = 2v h,

20 Finally, the platform equilirium profit are computed a Π c = 1 16τ (8τ τ ( ) + 4h 2,ρ ). It i eaily een that the condition of Aumption 2 guarantee that thi equilirium hold for any value of ρ (the condition ecome more tringent a ρ increae). Multihoming v. inglehoming. We oerve that the expreion for the equilirium price, participation, uyer urplu and platform profit are iomorphic to the one otained in Section 4: we jut need to replace h y h,ρ. A h,ρ h, we can immediately conclude that m c mcb, m c m CB, n c n CB, CS c CS CB, and Π c Π CB. Thu, the more imilar the ervice provided y the platform (i.e., the maller ρ), the higher the fee charged to uyer and the lower the fee charged to eller; there are alo fewer multihoming eller, which reult in a lower urplu for uyer and lower profit for platform. The impact on eller i le oviou, ince an additional term appear in P S c. We compute P S c P S CB = 1 4τ ( + + h,ρ + h ) (h,ρ h ) + (1 ρ) r ) = (1 ρ) r (1 1 4τ ( + + h,ρ + h ) ( = (1 ρ) r 1 n CB + (1 ρ) r ) 0. 4τ Hence, eller otain a higher urplu a ρ decreae. The previou reult ugget than when comparing the competitive ottleneck and the twoided inglehoming environment, it i more likely, a ρ decreae, that eller prefer the former, while uyer and platform prefer the latter. Hence, a larger duplication of tand-alone enefit (i.e., a lower ρ) further undermine the conventional widom, according to which the poiility of multihoming hould hurt the ide that can multihome (here, eller) while enefiting the other ide. 7 Concluion In thi paper, we have reconidered the claic Armtrong (2006) two-ided platform etting with the aim to analyze the impact of multihoming on one ide on price, platform profit, and uyer and eller urplu. The competitive ottleneck world i decried a a world in which the multihoming ide ha to pay monopoly price and platform compete on the inglehoming ide. However, thi doe not imply that the multihoming ide were to pay lower price if it could not multihome. We recall the three main inight regarding the urplu platform, uyer, and eller otain. (i) Whenever uyer prefer that eller e forced to inglehome, oth eller and platform prefer the oppoite. (ii) Whenever platform find it preferale to impoe excluivity, they necearily hurt at leat one group of participant. (iii) It i poile that uyer, eller 18

21 and platform are all etter off when eller are allowed to multihome. All our finding are eaily reformulated if the uyer intead of the eller ide i the ide that may e ale to multihome. Future work may look into alternative etting to addre the effect of multihoming. Thi appear to e a worthwhile endeavour given the inclination of ome competition authoritie and regulator to encourage and facilitate multihoming. 8 Appendix 8.1 Proof of Lemma 3 We firt note that h > 1 2 (2τ ) τ >, h > 1 2 (3τ 2 ) (τ ) ( ) > 0, h > 0 τ > ( + 2 ) τ min. (1) Take >. (1a) If τ >, then h min line aove that h < h min = 1 2 (3τ 2 ) and we ee from the middle. (1) If ( + ) /2 < τ <, then h min = 1 2 (2τ ) and we ee from the top line aove that h < h min. (1c) If τ < ( + ) /2, then h min > ( + ) /2 when >, we have again, from the ottom line aove, that h < h min τ min = 0 and a. (2) Take >. (2a) If τ > ( + 2 ) /3, then h min = 1 2 (3τ 2 ) and a ( + 2 ) /3 > when >, we have that h < h min from the middle line aove. (2) If τ < ( + 2 ) /3, h min = 0 and a τ min 8.2 Proof of Lemma 4 < ( + 2 ) /3, we have that h > h min for τ < τ min. We firt etalih the ign of C 20τ 2 8 (2 + ) τ +( + ) 2. Thi quadratic form in τ ha real root provided that > 0, which i guaranteed if < If > 7.45, C > 0 for all τ. If < 7.45, then the two root, which we note τ < τ + = 1 5 (2 + ) < τ = 1 5 (2 + ) 1 10 and τ +, are uch that < 2. So, C i poitive for τ < τ or τ > τ +, and negative otherwie. We now rewrite the condition P S CB > P S 2S a 4h 2 4 (4τ ) h +C > 0. For thi quadratic form to have real root, we need 4 (4τ ) 2 4C = 16τ (2 τ ) > 0, or τ < 2. If τ > 2, then C > 0, implying that P S CB > P S 2S. Conider now the cae where τ < 2. There are then two poitive root: h 1 2 (4τ ) τ (2 τ ) and 1 2 (4τ ) + τ (2 τ ) > h max. Clearly, h > 0 if and only if C > 0. So, in cae h > h min, then P S CB > P S 2S if h < h, wherea P S CB < P S 2S otherwie. If h < 0, then P S CB < P S 2S for all admiile h > h. We till need to compare h to h min. We have that h > 1 2 (2τ ) τ >, h > 1 2 (3τ 2 ) (τ ) (5τ ) > 0. 19