The Effects of Competition and Entryin Multi-sided Markets*

Guofu Tan† Junjie Zhou‡

July 7, 2020

Forthcoming in the Review of Economic Studies

Abstract

We study price competition and entry of platforms in multi-sided markets. Uti-lizing the simplicity of the equilibrium pricing formula in our setting with hetero-geneity of customers’ membership benefits, we demonstrate that in the presence ofexternalities, the standard effects of competition can be reversed: as platform compe-tition increases, prices and platform profits can go up and consumer surplus can godown. We identify economic forces that jointly determine the social inefficiency ofthe free-entry equilibrium and provide conditions under which free entry is sociallyexcessive as well as an example in which free entry is socially insufficient.JEL classification: L13, L4.Keywords: multi-sided markets, cross-side externalities, cross subsidization, platformcompetition, discrete choice models, free entry.

*The paper was previously circulated under the title “Price Competition in Multi-sided Markets". Wethank the editor, Adam Szeidl, and three anonymous reviewers for their insightful comments and sug-gestions. We are also grateful for comments from Simon Anderson, Mark Armstrong, Andrei Hagiu,Matthew Kahn, Jonathan Libgober, Dennis Lu, Volker Nocke, Alessandro Pavan, Andrew Rhodes, AlexWhite, Adam Wong, Julian Wright, Jidong Zhou, and seminar participants at a number of universities andconferences.

†Department of Economics, University of Southern California. guofutan@usc.edu‡Department of Economics, National University of Singapore. zhoujj03001@gmail.com

1

1 Introduction

Multi-sided platforms generate enormous economic value by enabling customers fromdifferent sides to interact. For example, matchmakers such as Amazon and Alibabafacilitate transactions between buyers and sellers; Uber, Lyft, and other local companiesconnect riders and drivers through their competing ride-hailing platforms; there aremany advertising-supported media platforms such as newspapers, magazines, and TVnetworks that connect viewers to advertisers.1 In the era of digitalization such platformcompanies play an increasingly important role in the modern economy, as witnessedby the fact that seven of the ten most valuable companies in the world are now basedaround platforms.2

Multi-sided platforms run very different business models compared to traditionalfirms.3 To balance the interests of customers from different sides and to get multiplesides on board, typically, platforms charge a fee on one side of the market that canbe below marginal cost (or even free in many cases) while gaining most of the profitsfrom the other sides. This pricing strategy, called cross-subsidization, is driven by akey feature of the multi-sided markets: the existence of cross-side externalities, i.e., onegroup’s benefit from joining a platform depends on the sizes of the other groups thatjoin the same platform.

How do platforms compete for multiple groups (or sides) of potential customers byadopting cross-subsidization strategies? Does platform competition reduce fees chargedto users and increase welfare in the presence of cross-subsidization? Is the equilibriummarket structure (the number of platforms) efficient, resulting in socially optimal levelof welfare? There is by now a large literature on multi-sided platforms, which we willbriefly review later. However, the existing literature does not provide thorough answersto the above questions. This paper tries to fill the gap, complementing the rest of theliterature.

1Other examples that are relevant to our study include: Ad Exchanges including Google’s DoubleClick,Microsoft’s AppNexus, AOL’s Marketplace, and Facebook’s FBX; dating websites such as E-harmony andmatch.com; computer operating systems such as Windows, Mac OS, and Linux; shopping malls such asRoppongi Hills and Midtown in Tokyo (see Elberse et al. 2007); home services websites including Angie’sList, Yelp and HomeAdvisor.

2These seven companies are Microsoft, Apple, Amazon, Alphabet (Google’s parent company), Face-book, Alibaba, and Tencent.

3For a comprehensive discussion of what makes a multi-sided platform business model different fromsome traditional alternatives such as vertically integrated firms, resellers or input suppliers, see Hagiuand Wright (2015).

2

We follow this literature and provide a model of price competition with symmetricplatforms. We focus on customer heterogeneity (or platform product differentiation)by membership benefits and on homogeneous network effects (both within-side andcross-side externalities). Based on the assumptions of single-homing and full marketcoverage, we establish the existence of a symmetric pricing equilibrium in which theprice on each side follows a simple formula: the price equals a mark-up, due to marketpower associated with product differentiation, minus a subsidy, due to cross-side (andpossibly within-side) externalities. The market-power mark-up equals the semi-inverseprice elasticity of the demand system that comes from customer membership benefitheterogeneity, and the subsidy summarizes the aggregate marginal externality acrossall sides. Each of the two terms is separately determined by the primitives of the modeland depends on the number of platforms. The characterization of the pricing equilibriumallows us to evaluate several key questions regarding the effects of competition on prices,consumer surplus, and platform profits, and the social inefficiency associated with freeentry.

We show that, due to cross-subsidization associated with cross-side and/or within-side externalities in multi-sided markets, a perverse pattern between prices and competi-tion emerges: the prices charged to customers can increase as competition among plat-forms goes up, which is in contrast to the conventional wisdom about the competition-price relations in single-sided markets. We identify the main economic forces behind thisperverse competition-price effect and provide conditions on the primitives of the modelunder which this pattern does and does not occurs, respectively.

Intuitively, given the additively decomposable pricing formula in our setting, the neteffect of competition on the price for each side (as the difference between the mark-upassociated with product differentiation and the subsidy associated with externalities)depends on how fast each term decreases with the number of platforms n, relative tothe other term.4 When n is sufficiently large, we identify two cases which are describedby the behavior of the distributions of customer heterogeneity and externality functions.In the first case, the limiting elasticity of the mark-up is smaller in absolute value thanthe limiting elasticity of the subsidy, and hence the mark-up decreases slower than thesubsidy does. Consequently, the equilibrium price decreases with n, similar to the casein the standard single-sided markets. On the other hand, when the limiting elasticity

4As it is well known in the literature, under the log-concavity of the distribution, the market-powermark-up associated with product differentiation decreases with n. We show that the subsidy to each side,which accounts for the aggregate marginal externality of a group on the other groups, decreases generallywith n as well.

3

of the subsidy is smaller in absolute value than the limiting elasticity of the mark-up,the mark-up decreases faster than the subsidy does. In this second case, the equilibriumprice increases with n in settings with positive aggregate marginal externality, and theperverse competition-price effect persists in multi-sided markets with a sufficiently largenumber of competing platforms.

Moreover, such a perverse competition-price effect can reverse the conventional re-lationship between competition and consumer surplus and between competition andplatform profits. We decompose the effect of competition on consumer surplus intothe product variety effect, the price effect, and the network consolidation effect. The productvariety effect is positive due to customers’ preferences for variety, and the price effectcan be negative especially in the presence of the perverse price pattern. The networkconsolidation effect is a joint product of the aggregate marginal externality enjoyed bycustomers and the business-stealing effect (or decreased market share), and can be posi-tive or negative, depending on the sign of the aggregate marginal externality. Therefore,the net effect on consumer surplus can generally go in either direction. In an example,we illustrate that the consumer surplus as a function of n is U-shaped, i.e., it decreasesfor small n, then increases for large n. Likewise, in the same example, the platformprofit initially increases for small n, then decreases for large enough n, and eventuallyconverges to zero, generating an inverted U-shaped profit curve.

We further utilize the simplicity of the equilibrium prices to study whether free entryleads to the socially efficient level of platform entry. Intuitively, the wedge between themarginal welfare of an additional entry and the profit of the additional platform drivesthe inefficiency of a free entry equilibrium. In our setting we decompose such a wedgeinto the sum of the product variety effect, the business-stealing effect without externalities,and the aggregate marginal externality adjusted by network fragmentation, all of whichare aggregated over all sides. Each effect represents a distinct economic force: the va-riety effect is positive due to customers’ preferences for a variety of platform services;the business-stealing effect is negative due to a reduction in incumbents’ market sharesupon an additional entrant; the externality effect can be positive or negative dependingon the sign of the aggregate marginal externality. Each effect is either responsible for ex-cessive or insufficient entry, and collectively they determine whether there is too muchor too little platform entry. We provide sufficient conditions under which the overalleffect favors excessive entry. Through an example, we show it is possible that the freeentry equilibrium is socially insufficient with the presence of cross-side externalities butsocially excessive without. Therefore, multi-sidedness and externalities across sides may

4

help provide a plausible explanation for industry concentration in markets involvingplatforms.

As mentioned above, there is now a large literature on two-sided markets, whichstarts with the pioneer works by Rochet and Tirole (2003, 2006), Caillaud and Jullien(2003) and Armstrong (2006). These early papers provide basic frameworks for mod-eling externalities (or network effects) among customers between the two sides and forstudying pricing schemes of both monopolistic and duopolistic platforms. Several im-portant recent contributions to this growing literature include Weyl (2010), White andWeyl (2010, 2016), Jullien and Pavan (2019), among others.5 We follow this literature tostudy the effects of platform competition and entry on prices and social welfare. Weyl(2010) and White and Weyl (2010, 2016) consider platforms adopting insulating tariffs ina general model of customer preferences with heterogeneous membership benefits andnetwork effects in monopolistic and oligopolistic settings, respectively.6 They study sev-eral important issues, including comparing monopoly profit-maximization prices andinsulated equilibrium with the socially optimal ones, respectively. We focus on symmet-ric platforms competing in membership prices in a model of customer preferences withheterogeneous membership benefits but homogeneous and possibly nonlinear cross-sideexternalities. Jullien and Pavan (2019) study the impact of correlated preferences of cus-tomers across sides on prices and welfare in a duopolistic setting with linear cross-sideexternalities, while customer preferences across sides are independent in our setting.The equilibrium prices in these papers share a similarity with ours, but they also containdistinct features reflecting the specific setting in each paper. We defer further detaileddiscussions of these papers and comparison with ours to Section 3.3 and other parts ofour paper.

Platforms’ strategies in two-sided markets and corresponding equilibrium conceptshave been carefully discussed in a recent paper by Correia-da Silva et al. (2019). A largepart of the platform literature adopts the standard Bertrand-Nash equilibrium eitherin pure-membership pricing or in pure-usage pricing.7 White and Weyl (2016) pro-pose a novel equilibrium concept called insulated equilibrium. One of the advantagesof platforms using insulating tariffs is to avoid coordination failure among customers.

5Carrillo and Tan (2015) study the roles of complementors in a model of platform competition with twofirms providing horizontally differentiated platforms and two sets of complementors offering productsthat are complementary to each platform respectively. They analyze the impact of integration, contractualexclusivity, and technological compatibility between platforms and complementors.

6White and Weyl (2010) allow multiple platforms and within-side externalities.7See Rochet and Tirole (2006) for a review of the early literature.

5

Assuming homogeneous platforms and homogeneous network effects and using theexternality-adjusted price, Correia-da Silva et al. (2019) study a Cournot model of plat-form competition in two-sided markets.8 In the symmetric setting with a general de-mand, they provide conditions under which a reduction in the number of platformsharms (or benefits, respectively) consumers on both sides. They illustrate that whenthe externality across two sides is relatively strong, a merger can benefit both groupsof consumers, an insight that is similar to the perverse competition-price effect in ourpaper. Moreover, a combination of prices and quantities has been used to model mediaplatforms in which platforms set advertising intensities/quantities on the advertiser sidewhile setting prices on the viewer/reader side.9

Our study is related to a recent empirical literature on the impact of mergers andindustry concentration in different industries with two-sidedness features; see, amongothers, Chandra and Collard-Wexler (2009) and Fan (2013) on daily newspapers, Sweet-ing (2010) and Jeziorski (2014) on radio, and Song (2020) on TV magazines. For instance,based on data from the Canadian newspaper market in the late 1990s, Chandra andCollard-Wexler (2009) find that greater concentration did not lead to higher prices foreither subscribers or advertisers. Using data from the 1996-2006 merger wave in the USradio industry, Jeziorski (2014) finds that the merger wave resulted in a decline in adquantity but an increase in ad prices, implying a welfare gain for listeners and a loss foradvertisers. Using data from TV magazines in Germany, Song (2020) finds that followinga change in market concentration, the prices on the two sides can go up or down.

The early literature that investigates the relationship between free entry and socialefficiency includes Spence (1976) and Dixit and Stiglitz (1977), among others. Theydemonstrate that in a monopolistically competitive market, free entry can lead to toolittle entry relative to the social optimum. Mankiw and Whinston (1986) provide a

8The externality-adjusted price in the case of homogeneous network effects is like the “hedonic price”in Katz and Shapiro (1985) and also similar to fixed fee of insulating tariffs in White and Weyl (2016).Gabszewicz and Wauthy (2014) study a model of quantity competition with vertical differentiation intwo-sided markets and compare the equilibrium outcome with the one in a price competition model.

9See, for instance, Anderson and Coate (2005) and Anderson and Jullien (2015) for such a class ofmodels on two-sided media markets. As Weyl (2010) noted in footnote 14, the advantage of using quantityon just one side is to remove expectations from the decision making of users on the other side. Andersonand Jullien (2015) further illustrate this point that the equilibrium with platforms choosing advertisinglevels and subscription prices (as in Anderson and Coate 2005) is indeed an insulated equilibrium in thesense of White and Weyl (2016). Recently, using the aggregative game approach, Anderson and Peitz(2020) extend the framework of Anderson and Coate (2005) to allow for any number of (asymmetric)platforms in media markets. Among other things, they evaluate surplus effects of platform entry.

6

systematic analysis of free entry and social inefficiency in oligopolistic settings. Theyshow that the interaction between the business-stealing effect and the market-power ef-fect contributes to excessive entry. On the other hand, in the market with differentiatedproducts, an additional product diversity effect is present, and free entry can be exces-sive, insufficient, or optimal. Anderson et al. (1995) analyze an oligopoly model withdifferentiated products and price competition and provide sufficient conditions underwhich free entry leads to an excessive number of firms as compared to the second-bestsocial optimum.10 We provide a theoretical analysis on the relationship between freeentry and social inefficiency in markets in which platforms compete.

Since we incorporate network effects into the standard discrete choice model of ran-dom utility maximization by customers to study platforms price competition, our paperalso contributes to the literature on oligopoly theory with discrete choice demand. Ex-tending the Perloff and Salop (1985) model of symmetric oligopoly, Caplin and Nalebuff(1991) study a general framework with customer preferences for substitute productsand asymmetric single-product firms and identify sufficient conditions (weaker thanlog-concavity) on the distributions for existence of Nash equilibrium in prices. A com-prehensive survey of the early theoretical development of this literature is in Andersonet al. (1992). Some examples of empirical studies of price competition under consumers’discrete choice with random utility maximization include well-known papers by Berry(1994), Berry et al. (1995), and the large literature that follows. More recently, Quint(2014) further extends the existence of the price equilibrium to allow for products withsubstitutes and certain forms of complements, and Nocke and Schutz (2018) establishexistence and uniqueness of the price equilibrium of multi-product firms under nestedlogit and nested CES demand systems. Weyl and Fabinger (2013) and Zhou (2017) showthat, among many other findings, the log-concavity(-convexity) of the distributions ofheterogeneity shapes the comparative statics of equilibrium prices with respect to thenumber of firms. Gabaix et al. (2016) further study the impact of competition on priceswith numerous firms. By incorporating cross-side externalities into the oligopoly the-ory with discrete choice demand, our paper enriches the understanding of the effects ofcompetition and entry in platform markets.

The remainder of this paper is organized as follows. Section 2 introduces our modelwith assumptions. Section 3 characterizes the equilibrium outcome of the two-stage

10Berry and Waldfogel (1999) provide an empirical study to quantify the extent of inefficiency associatedwith free entry in radio broadcasting markets with listeners and advertisers. Berry et al. (2016) refine theanalysis of Berry and Waldfogel (1999), explicitly allowing for both horizontal and vertical differentiation,among other issues.

7

game. The effects of platform competition on prices, profit, and consumer surplus arepresented in Section 4. Section 5 compares free entry equilibrium with the socially opti-mal entry of platforms. Section 6 concludes. All the proofs are presented in Appendix.

2 Model

We consider n platforms competing for customers from s sides by charging prices (ormembership fees), n ≥ 2 and s ≥ 1. Let pk = (pk

1, · · · , pks) denote the prices charged

by platform k ∈ N := 1, 2, · · · , n, and xk = (xk1, · · · , xk

s) denote the participationprofile on platform k, representing the aggregate share of customers who select k. Letεi = (ε1

i , · · · , εni ) denote the matching values (or membership benefits) of a customer

on side i ∈ S := 1, 2, · · · , s with n platforms. Given εi, pk, and xk, the utility of acustomer on side i ∈ S from joining platform k ∈ N is

φi(xk) + εki − pk

i . (1)

Here we follow the recent platform literature (see Armstrong (2006), Weyl (2010),White and Weyl (2010, 2016), Jullien and Pavan (2019), among others) and model cus-tomer heterogeneity by membership benefits or matching values between customers andplatforms.11 We assume that there is a continuum of heterogenous customers (of mea-sure 1) and each customer knows her matching values εi and the matching values ofother customers provided by n platforms. From the perspectives of the platforms, εi

can be viewed as IID draws from a distribution Gi(εi). The function φi is a mappingfrom [0, 1]s to R, and φi(xk) captures the externalities (or network benefits) enjoyed bycustomers on side i from all sides in platform k.

Given the price profile P = (p1, · · · , pn) and participation profile X = (x1, · · · , xn),platform k’s profit is

∑i∈S

(pki − ci)xk

i (2)

where ci is the marginal cost of serving a customer on side i, which is assumed to beidentical across platforms. We normalize ci to be zero, and thus for some of our analysis

11Two types of preference heterogeneity are assumed in the literature. For example, Armstrong (2006),Correia-da Silva, Jullien, Lefouili, and Pinho (2019), and Jullien and Pavan (2019), like our paper, focus onheterogeneity by membership benefits, while Rochet and Tirole (2003) emphasize customers’ heterogeneityby their network benefits. Rochet and Tirole (2006),Weyl (2010), and White and Weyl (2010, 2016) allowboth forms of heterogeneity by membership benefits and network benefits.

8

later, the mark-up pki − ci is the same as the price. Moreover, we focus on the price as a

membership fee that is not conditional on the participation of customers on any side.12

We further make the following assumptions. First, we assume single-homing and fullmarket coverage each customer participates in one and only one platform. If there is anoutside option which yields utility level v0

i + ε0i on each side i, but v0

i is sufficiently smallso that in equilibrium customers always opt out of the outside option.

Second, for our analysis in the next section, we make two important assumptions onthe heterogeneity of customers’ matching values:

Cross-side Independence: Across different sides i 6= j ∈ S , εi are independent of εj.13

Cross-platform Symmetry: For each i, the joint distribution Gi(·) is symmetric acrossn platforms.

In addition, we assume that Gi is continuously differentiable. Similar to White andWeyl (2010, 2016), Gi is general to permit several special specifications. A well-knownspecial case of Gi with negative correlation of membership benefits between platforms isthe Hotelling specification, which is adopted in Armstrong (2006) and many subsequentpapers.14 Another special case is independently and identically distributed (IID) match-ing values across platforms.15 For characterization of equilibrium prices in Section 3 weimpose cross-platform symmetry which allows for possible correlations of matching val-ues across platforms. For comparative static analysis in Section 4 and the study of entryin Section 5, we further restrict Gi to the case of IID matching values across platforms,with distribution Fi(·) (side-specific), which is assumed to be continuously differentiableon support [θi, θi] with density function fi.

Third, we assume that φi(x) is continuously differentiable in x ∈ [0, 1]s with φi(0) = 0(normalization). The most common specification of φi is linear, i.e., φi(x) = ∑j∈S γijxj,

12Armstrong (2006), in an extension of his main model in Section 4, considers a platform’s two-parttariff on each side conditional on the participation of the other side on the same platform. White and Weyl(2016) allow general nonlinear tariffs that are potentially conditional on the participations of customers ofboth sides on all platforms, which we will discuss in more detail in Section 3.3.

13See Jullien and Pavan (2019) for a model of platform competition in two-sided markets with correlatedcustomer preferences between the two sides. The models of Chandra and Collard-Wexler (2009) and Whiteand Weyl (2016), among other features, also allow correlated preferences across different sides.

14To name a few, see, among others, Chandra and Collard-Wexler (2009), Belleflamme and Peitz(2019a,b), and Bakos and Halaburda (2020) for recent contributions.

15In addition, our setup can accommodate some discrete choice models with correlated alternatives suchas the multinomial probit model (i.e., Gi(·) is a n-variate normal distribution N(0, Ω)) with the restrictionthat for any permutation matrix B of order n, B−1ΩB = Ω).

9

where parameter γij is not necessarily nonnegative, i.e., negative externalities (conges-tion effect) are also allowed. Our model can incorporate general forms of externalities φi,including non-linear externalities and/or within-side externalities (network benefits).16

Moreover, for z ∈ [0, 1], define

σi(z) ≡ ∑j∈S

∂φj(x)∂xi

|x=z1s (3)

as the aggregate marginal externality provided by customers from side i to all sides atthe symmetric allocation x = z1s, where 1s = (1, 1, · · · , 1)′.

We study the subgame perfect equilibrium of the following two-stage game: Plat-forms simultaneously choose their prices first, followed by customers simultaneouslydeciding which platform to participate in.

3 Equilibrium analysis

3.1 Participation Equilibrium

We first study the equilibrium in the second stage, characterizing customers’ simultane-ous participation decisions. Define X(P) (the participation rate profile associated withP) as a participation equilibrium (PE) if and only if the following system holds:

xki (P) =

∫εi : k∈arg maxt∈N φi(xt(P))+εt

i−pti

dGi(εi) (4)

for any i ∈ S , k ∈ N . In other words, given a price profile P, each customer on side ijoins the platform that yields the highest utility, as specified in (1), while taking as giventhe participation decisions of all other customers. Furthermore, the participation rate onany side must equal the aggregation of participation choices among all the customers onthat side.17

Our definition of participation equilibrium and the way it is constructed is similar tothat in the literature (see Armstrong (2006), Rochet and Tirole (2003, 2006), Weyl (2010),

16See, among others, Miyao (1978), White and Weyl (2010, 2016) featuring nonlinear network benefits.See, among others, Miyao (1978), Anderson et al. (1992), Brock and Durlauf (2001, 2002), Bayer andTimmins (2005), and White and Weyl (2010) studying within-side(-group) network benefits.

17The same equilibrium definition is adopted by Bayer and Timmins (2005) in a location sorting gamewith spillovers.

10

White and Weyl (2010, 2016), Jullien and Pavan (2019), among others).18 In the presenceof own-side and cross-side externalities, the demand system X(P) is interdependentacross sides and across platforms, and it is implicitly determined by the above system ofequations. Define Bε as the upper bound of the slopes of the demand functions withoutexternalities, which is completely determined by the distributions of matching values.Similarly, define Bφ as the upper bound of the marginal externalities.19 The followingProposition provides a simple sufficient condition for the uniqueness of PE.

Proposition 1. For any price profile P, there exists a participation equilibrium. Moreover, theparticipation equilibrium is unique if Bφ < 1/Bε.

The bound Bφ measures the degree of within-side and cross-side externalities. Forinstance, scaling all externalities φi by 10 leads to a tenfold increase of Bφ. Similarly, 1/Bε

measures the dispersion of matching values, since scaling all εki by 10 leads to a tenfold

decrease of Bε. When the degree of externalities is small, relatively to the dispersionof customers’ heterogeneity, system (4) forms a contraction mapping, which leads to aunique equilibrium in participation, and there is no coordination problem.

Suppose that for i ∈ S , εki for k ∈ N are IID according to Gumbel distribution

with parameter βi, and that externalities take the linear form with coefficients γij fori, j ∈ S . Then, the sufficient condition for uniqueness in Proposition 1 reduces tomaxi∈S∑j∈S |γij| < 2 mini∈S βi, which can be easily checked.20

18Our definition of PE is related to rational expectations equilibrium adopted in Anderson et al. (1992)and self-consistent equilibrium in Brock and Durlauf (2001, 2002). Under their approaches, when makingparticipation/purchase decisions, customers hold conjectures about the participation rates of the firms,denoted by xk

i . The aggregate participation decisions are determined by

xki (P) =

∫εi : k∈arg maxt∈N φi(xt(P))+εt

i−pti

dGi(εi), (5)

which is similar (4) with the exception that the participation rates are xt, not xt. Rational expectations (orself-consistency) of customers require that in equilibrium these conjectures are correct, i.e., xk

i = xki , which,

together with (5), is equivalent to (4).19The formal definition of Bε and Bφ is given in the Appendix.20We provide an example, illustrating that multiple PE can arise when the above condition is violated.

Suppose s = 2, n = 2, γ11 = γ22 = 0, γ12 = γ21 = 2.1, and β1 = β2 = 1. For any symmetric priceprofile between the two platforms, there are three PE in which the demands for platform 1 are given by(0.5, 0.5), (0.315, 0.315), (0.685, 0.685), respectively. The non-uniqueness of PE for large externalities hasalso been observed by Caillaud and Jullien (2001, 2003), among others.

11

3.2 Pricing Equilibrium

We now characterize the price equilibrium in the first stage. For each i ∈ S , let Hi(·; n)and hi(·; n) denote the CDF and PDF of ε1

i −max(ε2i , · · · , εn

i ) (the difference betweenthe matching values from platform 1 and the maximum matching values from othern− 1 platforms), respectively. Due to cross-platform symmetry, we write both Hi and hi

dependent of n. Define

Mi(n) :=1− Hi(0; n)

hi(0; n)

as the inverse hazard rate of Hi evaluated at 0, and

ηi(n) :=1

n− 1σi(

1n)

where σi(z), defined in (3), is the aggregate marginal externality from side i to all sidesat the symmetric allocation.

Assumption 1. Every stationary point of R(·) on [0, 1]s is a global maximum point, where

R(z) := ∑i∈S

zi

p∗i + H−1

i (1− zi; n) +[

φi(z)− φi

(1s − zn− 1

)], z ∈ [0, 1]s (6)

and p∗i , i ∈ S in R(z) is given by (7).21

Proposition 2. Under Assumption 1, there exists a subgame perfect equilibrium with the out-come that all platforms charge the same prices p∗ = (p∗1 , · · · , p∗s ) in the first stage and thedemand for each platform in the second stage is x∗ = 1

n 1s, where

p∗i (n) = Mi(n)− ηi(n), i ∈ S . (7)

Proposition 2 shows that the equilibrium price, charged to each side, follows a sim-ple formula, which consists of two additively separable terms, and each represents adistinct economic factor and is explicitly and separately determined by the primitivesof the model.22 The first term Mi(n) is the standard measure of market power of the

21Assumption 1 imposes mild restrictions on revenue R as a function in quantities, which is weakerthan concavity. Without externalities, R is concave under log-concavity of 1− Hi (see, for instance, Caplinand Nalebuff 1991). Tan and Zhou (2019) provide simpler and stronger conditions on the distributionsand externality functions to guarantee the concavity of R in z ∈ [0, 1]s, which implies Assumption 1.

22Without normalizing marginal costs to zero, the equilibrium prices would be p∗i = ci + Mi(n)− ηi(n).Such a decomposition of the prices shares a similarity with many platform papers in the literature; see,among others, Armstrong (2006), Weyl (2010), White and Weyl (2010, 2016), and Jullien and Pavan (2019),which we will discuss in detail in the next subsection.

12

oligopolistic firms offering differentiated products and is determined by the heterogene-ity of customer membership benefits (see Perloff and Salop 1985). What is interestinghere is the determination of the subsidy term. Note that the participation by a side icustomer generates externalities to all sides (including her own side). When a platform,say platform 1, attracts ∆i additional users from side i, the externalities enjoyed by any

other side j from joining platform 1 is enhanced by ∆i∂φj∂xi

, while the externalities from

joining any competing platform is reduced by ∆in−1

∂φj∂xi

, since each of the remaining n− 1

platforms would lose ∆in−1 on side i due to a business-stealing effect. The difference in

willingness to pay by each side j customer equals (1 + 1n−1)∆i

∂φj∂xi

. The additional profitthat the platform could capture from attracting additional user ∆i is equal to the sum ofthe marginal externalities that side i imposed on all sides, multiplied by the equilibriummarket share x∗1j . In other words, to encourage customer from side i to participate, inequilibrium platform 1 needs to lower its price by an amount equal to the aggregate

marginal profits: ∑j∈S(1 + 1n−1)x∗1j

∂φj∂xi

, which is the subsidy in (7), where x∗1j = 1/n isthe equilibrium market share of platform 1. We call the factor n/(n− 1) as the loop effect,which will be useful in subsequent analysis.

3.3 Discussion

We discuss several closely related studies in the literature and draw some comparisonswith our paper in terms of model setups, equilibrium price formulae, and the maineconomic insights.

Example 1. In his model of platform competition (n = 2) in two-sided markets (s = 2),Armstrong (2006) uses a Hotelling specification with uniform distribution of consumerlocation. The equilibrium prices in his setting are given by

p∗1 = t1 − α2, p∗2 = t2 − α1,

where t1 and t2 are unit transport costs, and α1 and α2 are the degrees of cross-groupexternalities enjoyed by two sides, respectively. This pricing formula is the same as oursin (7). Indeed, in this Hotelling specification, the market power effect is just ti.23

Example 2. Consider a one-sided market with linear form of within-side linear exter-nalities and Gumbel distribution of matching values studied in Anderson et al. (1992).

23In this setting, R is a quadratic function of z, and Assumption 1 holds if and only if 4t1t2 ≥ (α1 + α2)2,

which is what Armstrong (2006) assumes.

13

Under the assumption that β > 827

nn−1 γ, the symmetric equilibrium price is given by

p∗ =n

n− 1β− γ

n− 1,

where β is the scale parameter of Gumbel distribution and γ is the constant networkeffect parameter. This pricing formula is the same as ours in (7).24

The above examples and our model share one common outcome: each of the twoterms in the pricing formula is determined exogenously and separately by the primitivesof the model. We extend these studies by allowing general distributions of customerheterogeneity of membership benefits and general functional forms of externalities.

In a general model of a monopolistic multi-sided platform facing both heteroge-neous customer membership benefits and heterogeneous network effects across sides,Weyl (2010) compares the monopoly profit-maximizing prices with the social welfaremaximizing prices. The pricing formula (e.g., equation (7) and (11) in his paper) has asimilar decomposition of classical market power distortion and Spence distortion relativeto the socially optimal price. He identifies the price distortions due to a discrepancy ofnetwork effects perceived by marginal users versus average users, in the spirit of Spence(1975). One of the important contributions of his study is to develop a novel concept,called insulating tariffs, for the platform to avoid possible coordination failure amongcustomers’ participation decisions.25

White and Weyl (2016) further explore insulating tariffs in a general duopoly model ofcompeting platforms. Insulated equilibrium (IE) that they propose is a Nash equilibriumof the two-stage game in which, after platforms have first moved, each side’s represen-tative user has a dominant strategy. They provide a characterization of IE and comparethe IE outcome with the socially efficient one. While both focusing on an oligopolisticsetting, White and Weyl (2016) is more general than ours in that their model accom-modates both single-homing and multihoming of customers, heterogeneity of both usermembership benefits and network effects, and asymmetric platforms, while we focus onsingle-homing, heterogeneous customer membership benefits and symmetric platforms,in order for us to evaluate explicitly the effects of competition and entry on prices andwelfare. Another major difference is with regard to platforms’ strategy spaces (and as-sociated equilibrium concepts): we follow Armstrong (2006) and related literature to

24 It can be shown that the stated assumption on parameters is sufficient for Assumption 1.25Relatedly, in the context of a one-sided market with homogeneous network effects, Dybvig and Spatt

(1983) proposed contingent prices as a way to implement the desired equilibrium in dominant strategies.

14

focus on membership prices (or flat prices) and adopt Bertrand-Nash equilibrium (FPfor short) while they adopt insulating tariffs and IE.26

In general settings a comparison between IE and FP is rather challenging.27 In aspecial case of Armstrong (2006)’s model of Hotelling product differentiation and linearexternality, White and Weyl (2016) compare IE and FP and conclude that the predictionsof equilibrium prices under IE and FP are qualitatively similar.28 In our setting with ho-mogeneous network effects and full market coverage, we further confirm and strengthentheir observation. To see that, we first show that a symmetric IE exists in our setting, inwhich the total price that each platform charges for side i ∈ S is

T∗IEi (n) = Mi(n)−

(n− 1

n

)ηi(n), (8)

where Mi and ηi are defined as before. As pointed out in White and Weyl (2016), with ho-mogenous network effects, the insulating tariff gives every user, not just a representativeuser, a dominant strategy, and therefore IE is strategically equivalent to the equilibriumoutcome of platforms competing in utilities.29 Comparing (8) with (7) in Proposition 2,we observe that (i) the market power effect is the same, and (ii) the subsidy term un-der IE is adjusted by the loop effect factor of (n − 1)/n which appears in our settingwith flat price competition but does not in IE. Such a discrepancy in subsidy reflectsa difference in how the network feedback effect is handled under IE and FP. Since theloop effect, n/(n− 1), approaches one for large n, the difference in equilibrium pricesunder IE and FP is quantitatively small in platform markets with numerous platforms.Moreover, our Theorem 1 in the next Section regarding the competition-price effect holdsunder IE. Thus, our message about the perverse pricing effect is robust to this alternativeequilibrium concept.

26Correia-da Silva et al. (2019) provide illuminating discussions about the way that different conductassumptions (Cournot, Bertrand, and Insulation) influence network feedback effects and thus platforms’pricing incentives. See our discussion in Section 1.

27One of the challenges is the lack of tractability of both FP and IE in general settings. Although theequilibrium prices in IE (Proposition 2 and Proposition 8 in White and Weyl (2016)) consist of a marketpower term and an externality term, these two terms are implicitly determined in a system of nonlinearequations together with the implicit demand system. Similarly, it is difficult to characterize FP equilibriumsince the demand elasticities feature complicated feedback effects.

28Proposition 3 in White and Weyl (2016) covers both IE and FP. For IE, T∗IE1 = t1 − (1/2)α2, T∗IE

2 =

t2 − (1/2)α1; for FP, p∗1 = t1 − α2, p∗2 = t2 − α1. These results are consistent with (7) and (8) in our setting.29The main gist of (8) follows from the pricing formulae for IE in White and Weyl (2010, 2016) for the

special case of homogenous network effects. A self-contained proof of (8), following a similar logic to theproof of our Proposition 2, is available upon request.

15

More recently, Jullien and Pavan (2019) study the effects of information dispersion(instead of competition) on prices, consumer surplus, and total welfare in two-sidedmarkets with dispersed information about users’ preferences. Compared with our set-ting, their model is more general in terms of preference correlation across sides, but fo-cuses on duopoly and linear forms of externalities. Since information dispersion affectsthe elasticity of the demand on each side, the equilibrium prices in their analysis (e.g.,Proposition 4 and Corollary 1) exhibit a novel extra distortion which is different fromthe one in the case where information about users’ preferences are commonly known byusers and platforms. Without correlation, this new distortion vanishes, and thus theirpricing formula becomes the same as ours in (7). Intuitively, the novel distortion (drivenby dispersed information) accounts for the discrepancy between the participation ratesexpected by the marginal agent on each side and the participation rates expected by theplatforms. If incorporating preference correlations in our oligopolistic setting, we con-jecture that a similar pricing formula to the one in Proposition 2 is likely to sustain, afteradjusting for the novel effect of correlation identified in Jullien and Pavan (2019).

The tractability of our model, and the simplicity of the pricing formula, is a jointproduct of symmetry, single-homing, full market coverage, and the additive separabilityof the customer utility, and does not rely on the functional forms of externalities andthe distributions of customer membership benefits. By extending the model in Arm-strong (2006) to an oligopolistic setting, our paper (together with other papers discussedabove and in Section 1) also demonstrates the boundary of applicability of the simplepricing formulae identified in Armstrong (2006). Allowing either asymmetric platforms,multi-homing, explicit outside option, or heterogeneous network effects would make theequilibrium prices implicitly determined rather explicitly expressed, which would makeit intractable to study the effects of competition and entry on prices and welfare.

4 The effects of competition

In this section we utilize the pricing equilibrium explicitly characterized in Proposition 2

to address several key questions regarding the effects of competition on prices, consumersurplus and platform profits. We show that, with the presence of externalities in multi-sided markets, the standard effects of competition can be reversed. For instance, asplatform competition increases, prices and platform profits can go up and consumersurplus can go down. Therefore, promoting competition in platform markets may haveunintended consequences, depending on the market environments.

16

4.1 The effects of competition on prices

Proposition 2 reveals an additively decomposable formula for symmetric equilibrium priceon each side which is equal to the mark-up (due to the product differentiation effect)minus the cross-subsidy. Each of the two terms varies with the degree of competitionand is exogenously and separately determined by the primitives of the model. We cananalyze these two terms separately and then evaluate the net effect.

To simplify our analysis and stress the impact of multi-sidedness, in this section wefocus on the case of IID matching values, as in such a case the effect of competition onmarket power mark-up in the context of the Perloff and Salop model is generally knownin the literature. For instance, Zhou (2017) shows that under the log-concavity of fi, themark-up (due to the product differentiation effect), Mi(n), is monotonically decreasingin n.30 On the other hand, in the literature on multi-sided markets, not much attentionhas been paid to the impact of platform competition on the extent of cross-subsidies andhence on platform prices. We proceed with the following Lemma.

Recall that cross-subsidy ηi(n) = 1n−1 σi(

1n ) and σi(z), defined in (3), is the aggregate

marginal externality from side i to all sides at the symmetric allocation x = z1s.

Lemma 1. If z1−z σi(z) increases (decreases) in z for z ∈ (0, 1), the equilibrium cross-subsidy

ηi(n) decreases (increases) with n. Moreover, if limz→0 zσi(z) = 0, then limn→∞ ηi(n) = 0.

Lemma 1 identifies sufficient conditions for the monotonic impact of platform com-petition on the cross-subsidy. Note that the cross-subsidy is determined by the multipli-cation of three effects: aggregate marginal externality σi(z), the market share z, and theloop effect 1/(1− z), all evaluated at the symmetric allocation z = 1/n. Here the loopeffect is equal to 1/(1− 1/n) = 1 + 1

n−1 , as each additional customer on one platforminduces 1/(n− 1) amount of losses to each of the other platforms. It is clear that boththe equilibrium market share and the loop effect decrease with n. On the other hand, theaggregate marginal externality evaluated at the equilibrium allocation may be positiveor negative, and increasing or decreasing with n. Thus, the net of the above three effectscritically depends on the bahavior of the aggregate marginal externality.

Consider a linear form of externalities for each side, φi(x) = ∑j∈S γijxj, implying thatthe aggregate marginal externality is constant (i.e., σi = γi ≡ ∑j∈S γji). Consequently,

30A slightly weaker assumption identified in Zhou (2017) is the log-concavity of 1− Fi, which is impliedby log-concavity of fi. Weyl and Fabinger (2013) and Quint (2014) have also provided related results anddiscussions. Note that the log-concavity of the density function is commonly assumed in the literature.See, for instance, Caplin and Nalebuff (1991), Anderson et al. (1995), and Zhou (2017).

17

the cross-subsidy decreases with n if and only if γi > 0. For a more general class ofexternalities allowing for variable returns to scale, φi(x) = ∑j∈S γijρ(xj) with ρ′ > 0,the aggregate marginal externality varies with market share, i.e., σi(z) = γiρ

′(z). Whenthe combined effects of the aggregate marginal externality and market share are positive(i.e., zρ′(z) increases with z for z ∈ (0, 1)), together with the decreasing loop effect, thecross-subsidy decreases with n if and only if γi > 0.31

As noted above, the mark-up due to product differentiation typically decreases withn. If the cross-subsidy also increases with n (under the condition in Lemma 1), then theprice decreases with n, as in the standard single-sided market setting. However, Lemma1 and the above examples illustrate that the cross-subsidy typically decreases with n aswell, which can easily offset the monotonicity of the product differentiation effect. Toillustrate the net effect of competition on prices, we consider two examples.

Example 3. Consider an exponential distribution Fi(θ) = 1 − e−λiθ and λi > 0 (seePerloff and Salop 1985), and linear forms of externalities with γi = ∑j∈S γji. The equi-librium prices are given by

p∗i (n) =1λi− 1

n− 1γi, i ∈ S .

We have an intriguing observation that the equilibrium price on each side increases withthe number of platforms whenever there are positive aggregate marginal externality.

Example 4. Consider the Gumbel distribution with parameter βi > 0, and linear formsof externalities with γi = ∑j∈S γji. The equilibrium prices are given by32

p∗i (n) =n

n− 1βi −

1n− 1

γi, i ∈ S (9)

which decrease (increase) with n if and only if βi > (<)γi. It is feasible to have theprice on one side (say i) decreases with n (when βi > γi) while the price on another side

31Moreover, ηi converges to 0 as n→ ∞, if limz→0 zρ′(z) = 0. The conditions [zρ′(z)]′ = zρ′′(z)+ ρ′(z) >0 and limz→0 zρ′(z) = 0 are satisfied when ρ is convex, or when ρ is a power function (ρ(z) = zr, 0 < r < ∞)or a logarithmic function (ρ(z) = log(1 + tz), t > 0). However, there exist specifications of ρ under whichthe cross-subsidy could increase with n even if γi > 0. This could happen for sufficiently concave ρ sothat zρ′(z) decreases with z. For example, consider ρ(z) = 10e−1/(10z), ηi(n) increases with n for n = 2 to9, then decreases in n afterward. For such ρ, zρ′(z) decreases with z for an open range of z.

32White and Weyl (2016) (in Section 2.4) consider Gumbel distribution and logit demand with outsideoption, where the equilibrium prices are implicitly determined. The full market coverage in our settingleads to the explicit pricing formula in (9).

18

(say j) increases with n (when β j < γj). The latter inequalities can hold for several sideswithout contradicting with the conditions for uniqueness of participation equilibriumand sufficiency for price equilibrium. Moreover, suppose βi = β j, then p∗i − p∗j = (γj −γi)/(n − 1). Hence p∗i > p∗j if and only if γi < γj. That is, each platform subsidizescustomers on side j by charging more on side i. As competition increases, the magnitudeof cross-subsidies decreases.

From Lemma 1 and Examples 3 and 4, we observe that due to cross-side externali-ties and cross-subsidization in multi-sided markets, the prices charged to certain sidescan easily increase as competition among platforms intensifies, a perverse relationshipbetween competition and price.33We have also illustrated that the extent of this perversecompetition-price effect depends on the distribution of the customer heterogeneity andthe behavior of the aggregate marginal externality function. In what follows, we utilizethe asymptotic properties of the product differentiation effect and cross-subsidy effectto analyze the price trend for markets with sufficiently large number of platforms.34

We provide a sufficient condition under which the perverse competition-price relationpersists when the number of platforms is large.

To state the conditions in Theorem 1, we introduce two indices. The first is the tailindex of a CDF F, with density f and support [θ, θ], as the minus of the limit of the slopeof the inverse hazard rate at the upper support, i.e.,

τ ≡ − limθ→θ

ddθ

(1− F(θ)

f (θ)

),

which measures the "fatness" of the right tail of the distribution.35 Under log-concavity33In a model of two-sided markets for newspapers with readers and advertisers, based on Armstrong

(2006)’s framework with a Hotelling specification, Chandra and Collard-Wexler (2009) provide numericalexamples to illustrate that a merger of the two newspapers may not necessarily increase the prices foreither side of the market, crucially depending on whether each platform could make positive profits fromthe marginal reader in the pre-merger equilibrium. Two key assumptions in their setting are that the ad-vertisers’ willingness to pay is specific to a reader’s location, with more loyal readers being more valuableto the advertisers, and that each platform could perfectly extract all surpluses from the advertisers. Ina model of trading platforms with buyers on one side and sellers on the other side, Karle et al. (2020)find that market concentration on the selling side and listing fees (prices charged to the sellers by theplatforms) can be negatively correlated. In a model of on-demand platforms with customers on one sideand workers on the other side, Nikzad (2020) shows the adverse effect of competition on the customers’side (i.e., the price under a duopoly can be greater than the one under monopoly) but not on the workers’side.

34As n goes to infinity, Mi(n) converges (e.g., under the log-concavity of the distributions), the cross-subsidy vanishes (e.g., under the assumption in Lemma 1), and hence the price generally converges.

35Our definition of the tail index is the same as in Gabaix et al. (2016), except for the minus sign.

19

of f , the inverse hazard rate, (1− F(θ))/ f (θ), is monotonically decreasing in θ, and thusτ ≥ 0. The second index ri is the elasticity of z

1−z σi(z) with respect to z at z = 0, which is

equivalent to ri = 1 + limz→0 z σ′i (z)σi(z)

. For example, ri = 1 for linear forms of externalities.The asymptotic behavior of Mi for large n can be described by its limiting elasticity

with n, which is the negative of the tail index τi of Fi (see Gabaix et al. 2016). The largerthe τi, the higher the rate of convergence for Mi to zero. Similarly, the limiting elasticityof cross-subsidy ηi is the negative of ri, which captures the rate of convergence for ηi.For large n the trend of the equilibrium price in each side (as the difference of two terms,Mi and ηi) is ultimately shaped by the dominant term between the two, i.e., the one witha smaller index and hence a lower rate of convergence. We summarize this comparisonin the following Theorem.

Theorem 1 (Competition-price effects). For each i ∈ S , assume fi is log-concave. Then thefollowing hold:

(i) If 0 < τi < ri, there exists a positive n0 such that p∗i (n) decreases with n for any n ≥ n0;

(ii) If 0 < ri < τi, there exists a positive n0 such that, for any n ≥ n0, p∗i (n) increases(decreases) with n whenever ηi(n) > (<)0.

Theorem 1 shows that the patterns of the monotonicity of the equilibrium prices forlarge n hold for general distributions and general forms of externalities. With a largenumber of platforms, the relative magnitudes of two simple primitive indices succinctlydetermine which of the two effects dominates the equilibrium prices, and whether theequilibrium prices increase or decrease with competition. When the tail index is smallerthan the externality index, the product differentiation effect dominates, and hence theprice decreases with n for large n. On the other hand, when the externality index issmaller, for large n the cross-subsidy effect dominates and the price trend is determinedby an additional factor, i.e., the sign of cross-subsidy: under negative aggregate marginalexternality, the price is decreasing in n; under positive aggregate marginal externality,the price is increasing in n. Thus, the perverse competition-price effect is persistent inmarkets with numerous platfirms, when the aggregate marginal externality is positiveand when the externality is relatively significant (i.e., the externality index is smallerthan the tail index).36

They derived the tail indices for several classes of distributions. For example, the tail index (using ourdefinition) is zero for exponential, Gumbel, and normal distributions, and it is 1 for uniform distribution.

36Moreover, we can use these two indices to succinctly demonstrate which effect dominates in the pricedecomposition: limn→∞ Mi(n)/p∗i (n) = 1 in case (i), and limn→∞ ηi(n)/p∗i (n) = −1 in case (ii).

20

To illustrate the above results, consider a uniform distribution, which has tail indexτi = 1,37 and φi(x) = ∑j∈S γij(xj)

r for r > 0, with the corresponding externality in-dex r.38 When r > 1, the cross-subsidy approaches zero faster than 1/n (the order ofthe product differentiation term), implying the dominance of the product differentiationeffect over the externality effect. Moreover, for large n, p∗

′i (n) is asymptotically propor-

tional to −1/n2 and p∗i (n) decreases with n. This corresponds to case (i) in Theorem1. On the other hand, when 0 < r < 1, the cross-subsidy approaches zero slower than1/n, implying the dominance of the externality effect in the price. In the latter case,for positive γi, the price (or precisely the price-cost mark-up) on side i is negative butincreases with n for sufficiently large n.

The perverse competition-price effect in Theorem 1 echoes some of the findings ina recent empirical literature on the impact of mergers and industry concentration, forinstance, Chandra and Collard-Wexler (2009) and Song (2020), which are discussed inSection 1. By decomposing the price into two components in Proposition 2, Theorem1 identifies the main economic forces behind this perverse competition-price effect, andprovides simple conditions under which it does and does not occur, respectively.39

4.2 Consumer surplus and platform profit

The perverse competition-price effect has immediate implications regarding the effectsof competition on the welfare of the customers and platform profits. We demonstratethat the presence of externalities could reverse the conventional relationship betweencompetition and consumer surplus and the one between competition and profit.

Under the symmetric equilibrium, the consumer surplus for side i can be written as

CSi(n) = δi(n)− p∗i (n) + φi(1sq∗(n)), i ∈ S , (10)

where the first term, δi(n) = E[maxk∈N εki ], denotes the expected maximum match value

from n platforms enjoyed by customers on side i, the second term is the price thatcustomers pay, the third term, φi(1sq∗(n)), represents network benefit from externalities,and q∗(n) = 1/n is the equilibrium per-platform market share.

We can decompose the impact of competition on consumer surplus into three ef-fects: the platform/product variety effect, ∂δi(n)/∂n, the price effect, −∂p∗i (n)/∂n, and

37If fi is the uniform distribution on [0, Li], Mi(n) = Li/n and τi = 1.38ηi(n) = rγi(

nn−1 )n

−r, where γi = ∑j∈S γji. We have ri = r as ηi(n) ≈ rγin−r for large n.39Since the asymptotic elasticity of ηi(n) with respect to n coincides with that of n−1

n ηi(n), Theorem 1

holds for the price under insulated equilibrium (IE) as presented in (8).

21

the network consolidation effect, ∂φi(1sq∗(n))/∂n. The product variety effect is positive,reflecting customers’ preferences for variety.40 In absence of any externalities, the priceeffect of competition is usually positive and hence customers generally benefit from anincreased competition. The presence of within-side and/or cross-side externalities af-fects consumer surplus in two ways: First, the price effect can now be negative, dueto the perverse competition-price pattern as discussed in Section 4.1 (e.g., see Theorem1). Second, the network consolidation effect is a joint product of the aggregate marginalexternality enjoyed customers on side i and the business-stealing effect (or decreasedmarket share).41 When the aggregate marginal externality enjoyed by customers on sidei is positive, which is a reasonable description of many multi-sided markets, competitionreduces the customer participation to each individual platform, which in turn reducesthe network benefits to customers. In such a situation the network consolidation effect isnegative and will partly offset the above positive benefits of competition. Consequently,the net effect of competition on consumer surplus in multi-sided markets can go in eitherdirection. Our example below illustrates the tradeoff of the above effects and presents aU-shaped relationship between competition and consumer surplus.

Likewise, under the symmetric equilibrium, the profit per platform is given by

Π(n) = q∗(n) ∑i∈S

p∗i (n), (11)

where the market share q∗(n) decreases with n. Thus, profit decreases with n wheneverthe prices decrease with n. Interestingly, when the perverse competition-price effect isstrong enough to offset the diminishing market share,42per platform profit could increasewith n, reversing the usual link between competition and firm profit. The example belowdepicts an inverted U-shaped relationship between competition and platform profit.

Example 4 (Continued). Each platform’s profit is

Π(n) =nβ− γ

n(n− 1)(12)

where β = ∑i∈S βi and γ = ∑i∈S γi. As competition increases, platform profit decreaseswhen γ/β < n2/(2n − 1), which holds for large n but is violated for small n if theexternality is significant. For each i ∈ S , the consumer surplus is

CSi(n) = βi

(ln(n) + κ − n

n− 1

)+

1n ∑

j∈Sγij +

1n− 1

γi

40It can be shown that δi(n) always increases with n (see (34) in the proof of Lemma 2).41Note that ∂φi(1sq∗(n))/∂n = −(1/n2)∑j∈S ∂φi(1s/n)/∂xj as q∗(n) = 1/n.42This holds if, for each side i, the elasticity of the price with respect to n exceeds one.

22

where κ ≈ 0.5772 is the Euler-Marcheroni constant. As competition increases, the con-sumer surplus first decrease and then increase with n.43

We use a numerical example to illustrate the behavior of the consumer surplus andplatform profit. Consider s = 2, β1 = β2 = 0.6, and γ11 = γ22 = 0, γ12 = γ21 = 1.Since the two sides are symmetric, the prices are identical between the two sides, and soare consumer surplus. The price on each side increases with n for any n, and thus theperverse competition-price effect pattern always holds. As a function of n, the consumersurplus is U-shaped (see Figure 1), while the platform profit is inverted U-shaped (seeFigure 2).

3 4 5 6n

0.60

0.65

0.70

0.75

CSHnL

Figure 1: The consumer surplus curve

3 4 5 6n

0.06

0.08

0.10

0.12

PHnL

Figure 2: The per-platform profit curve

Lastly, we briefly discuss the asymptotic impact of competition on consumer surplusand platform profits. For large n, the product variety term in (10) is unbounded for adistribution with unbounded support, while the other two terms regarding the price andnetwork benefit are bounded under fairly weak conditions. When n is sufficiently large,the variety effect dominates the two other effects, implying the ultimate monotonicity ofconsumer surplus. As n goes to infinite, market share diminishes to zero while the pricegenerally is bounded (under the log-concavity of the distributions and the assumptionin Lemma 1), thus the platform profit converges to zero, so it cannot increase with npersistently. Consequently, platform profit is most likely to be decreasing when n issufficiently large.

In sum, the presence of externalities in multi-sided markets can reverse the conven-tional wisdom about effects of competition when the markets are relatively concentratedwith a few platforms. In contrast, in very competitive markets with a large number ofplatforms, the impact of externalities regarding consumer surplus and platform profits

43CSi(n) increases with n if and only if n(n2 − n + 1)βi > (n− 1)2(∑j∈S γij) + n2γi.

23

is limited, thus standard effects of competition dominate. Yet, in either case, the perversecompetition-price effect can occur.44

5 Entry

In this section, we study endogenous entry by multi-sided platforms and particularlycompare free entry equilibrium with the socially optimal outcome. For simplicity, wefocus on the case of IID matching values.

Under the symmetric equilibrium, the total welfare with n platforms, as the sum oftotal platform profits and aggregate consumer surplus, equals

W(n) = nΠ(n) + ∑i∈S

CSi(n), (13)

where CSi is given by (10), Π(n) is given by (11), and both are functions of n.To facilitate our analysis, we treat n as a continuous variable. Let K > 0 be the fixed

cost of entry by a platform into the markets. Define n∗ = arg maxn≥2 (W(n)− nK) asthe socially optimal (i.e., the second-best) number of platforms,45 and ne denote the free-entry equilibrium number of platforms, i.e., the maximum n(≥ 2) such that Π(n)− K ≥0. Following the literature on free entry and social efficiency, we call entry excessive(insufficient) if the equilibrium number of platforms under free entry ne is at least aslarge (small) as the socially optimal (second-best) number n∗.

First note that the wedge between the marginal welfare of an additional entry andthe profit of the additional platform drives the inefficiency of free entry equilibrium. Forinstance, if the marginal welfare W ′(n) is less than or equal to the profit Π(n), for anynumber of platforms, then the profit with n∗ platforms minus entry cost is nonnegativeand hence excessive entry occurs (i.e., ne ≥ n∗), since at the welfare maximizing n∗, themarginal welfare must equal entry cost.

In our setting, we can write the marginal contribution of an additional platform on

44The extent of concentration in multi-sided markets is determined by many factors, including the fixedcost of investing and getting multiple sides on board, first move advantages, network effects, and so on.We will study some of these factors in the next section.

45If multiple maximizers for maxn≥2 W(n) − nK exist, for concreteness, we choose n∗ as the largestmaximizer. The excessive entry in Theorem 3 below clearly holds if any other maximizer, instead of thelargest maximizer, is selected in the definition of n∗.

24

total welfare as

W ′(n) = Π∗(n) + ∑i∈S

[δ′i(n)︸ ︷︷ ︸

variety effect

+∂qn

∂n(np∗i + σi(qn))︸ ︷︷ ︸

business-stealing effect

](14)

where the first term in the bracket describes the platform/product variety effect, which ispositive due to customers’ preferences for a variety of platform services, and the sec-ond term is the business-stealing effect, due to a reduction in incumbents’ market sharesupon an additional entrant. The above decomposition bears some similarity to the onein Mankiw and Whinston (1986).46 Unlike Mankiw and Whinston (1986), however, thebusiness-stealing effect in our setting with multi-sided platforms consists of two dis-tinct components: the first part is the standard multiplication of the mark-up and thereduction in market shares, and the second is the network consolidation effect, i.e., thecomposite effect of a reduction in market share on the aggregate marginal externalitygenerated by customers on side i. Several novel features arise. Since the mark-up inmulti-sided markets can be negative, the first business-stealing effect in each side canbe positive, in contrast to the negative business-stealing effect in the single-sided marketin Mankiw and Whinston (1986).47 The network consolidation effect, which vanisheswithout externalities, can generally be positive or negative when the aggregate marginalexternality is negative or positive.

Utilizing our characterization result of the symmetric equilibrium prices in Proposi-tion 2, we further decompose these two types of the business-stealing effect as follows:

∂qn

∂n[np∗i + σi(qn)] =

∂qn

∂n[nMi(n)−

1n− 1

σi(qn)],

which is proportional to the total mark-ups without externality minus the cross-subsidycaused by cross-side externalities, where 1

n−1 comes from the difference between thedirect network benefit and the indirect loop effect, n/(n − 1), in the price. In otherwords, the business-stealing effect can be decomposed into the same effect in absenceof externalities plus an additional effect solely caused by cross-side externalities. Wesummarize the above analysis in the following Theorem.

46Mankiw and Whinston (1986) show that, in a homogeneous product market with quantity competi-tion, the negative business-stealing effect leads to excessive entry. In the presence of consumer’s preferencefor variety, they illustrate that the product variety effect can mitigate the business-stealing effect.

47Nevertheless, the sum of the mark-ups over all sides, multiplied by the symmetric market share, isthe platform profit, which should be nonnegative in equilibrium. What matters in (14) is the sum of themark-ups over all sides.

25

Theorem 2 (Entry effects). The difference between the marginal welfare and per-platform profitequals

W ′(n)−Π(n) = ∑i∈S

[δ′i(n)−

Mi(n)n

+σi(1/n)(n− 1)n2

]. (15)

Theorem 2 shows that the difference between the welfare impact of an additional (ormarginal) entry and the profit of the entering platform can be expressed as the sum ofthe product variety effect, the business-stealing effect without externalities, and the aggregatemarginal externality adjusted by network fragmentation, all of which are aggregated overall sides. Each term of the decomposition in (15) represents a separate economic forceunder which entry can be inefficiently excessive or insufficient.

Understanding these new features enables us to identify sufficient conditions underwhich entry can be inefficiently excessive or insufficient. In the Perloff and Salop (1985)model without externalities (i.e., σi ≡ 0 in our setting), Anderson et al. (1995) show thatwhen fi is log-concave, the product variety effect is always dominated by the business-stealing effect (i.e., δ′i(n) ≤ Mi(n)/n), implying W ′(n)−Π∗(n) ≤ 0 and hence excessiveentry. In the presence of cross-side externalities, the above discussion suggests that theexcessive entry result also holds as long as the aggregate marginal externality acrossall sides is either zero or negative. On the other hand, when the aggregate marginalexternality is positive, which happens in many applications, we derive sufficient condi-tions under which there is an excessive entry, relatively to the socially optimal numberof platforms. Our results are summarized in the following Theorem.

Theorem 3 (Excessive entry). Assume that for each i ∈ S , fi is log-concave. Excessive entryoccurs when either one of the following three conditions holds:

(i) ∑i∈S σi(z) ≤ 0, for any z ∈ [0, 1],

(ii) for each i ∈ S , φi is linear or convex in x ∈ [0, 1]s, and

(iii) for each i ∈ S , fi satisfies the following property:48(n

n− 1

)δ′i(n) ≤

Mi(n)n

, for any n ≥ 2. (16)

48In the Perloff and Salop (1985) model without externalities, condition (iii) implies that the productvariety effect is dominated by the business-stealing effect by a factor of (1− 1/n). Hence, condition (iii)is slightly stronger than δ′i(n) ≤

Mi(n)n , which holds for any log-concave fi as shown in Anderson et al.

(1995). Nevertheless, the significance of condition (iii) is that it imposes no restriction on externalities.

26

We impose conditions on the functional forms of externalities in the first two casesand on the distributions of matching values in the third case. Condition (i) states that theaggregate marginal externality over all sides is nonpositive, which means that overall thenegative externalities outweigh the positive externalities. Condition (ii) states that themarginal externalities are weakly increasing, including the linear form of the external-ities. The case without any externalities (φi ≡ 0) obviously satisfies both conditions (i)and (ii), and hence Theorem 3 extends the well-known result on excessive entry in An-derson et al. (1995) (the case without externalities) to our setting of multi-sided marketswith externalities.

The first sufficient condition immediately follows from the discussion above. To un-derstand the intuition behind the second sufficient condition, we compare the extra wel-fare gained from one additional entry, W(n+ 1)−W(n), with the profit of the additionalplatform, Π(n + 1). If the difference is zero or negative, i.e.,

[W(n + 1)−W(n)]−Π(n + 1) ≤ 0,

for any n, then excessive entry occurs.49 Note that the total welfare is simply the sum ofthe consumer benefits from consuming a variety of platform services and the networkbenefits from cross-side externalities, and that by our characterization in Proposition 2,the platform profit simply consists of the product differentiation effect minus the cross-subsidy effect, multiplied by the market share. The additivity of both the extra welfareand the platform profit allows us to separate the impact of the distributions and thecross-side externalities on the difference, [W(n + 1)−W(n)]−Π(n + 1). Again, withoutexternalities, we know from Anderson et al. (1995) that the above difference is non-positive, leading to excessive entry. In the presence of externalities, the extra differencecaused by externalities equals

∑i

φi(1

n + 11s)−∑

iφi(

1n

1s) + ∑i,j

1n(n + 1)

∂φi(x)∂xj

|x= 1n+1 1s

where the combination of the first two terms together is the reduction in network benefitsfrom an additional entry and the third term represents the total subsidy (multiplied bymarket share 1/(n + 1) by the additional platform. If an additional entrant causes a

49Indeed, the maximization of W(n)− nK at n∗ implies

W(n∗)− n∗K ≥W(n∗ − 1)− (n∗ − 1)K,

or equivalently W(n∗)−W(n∗ − 1) ≥ K. It follows that Π(n∗) ≥ W(n∗)−W(n∗ − 1) ≥ K, which impliesimmediately from the definition of ne that ne ≥ n∗.

27

reduction in network benefits more than the aggregate subsidy paid by the entrant, thenthe presence of cross-side externalities amplifies excessive entry, since the additionalentrant does not take into the extra negative externality imposed on consumers andother platforms. This is true when the externality functions are convex.50

To understand the intuition behind our third sufficient condition in Theorem 3, weobserve that the difference between a scalar multiplication of the marginal welfare ofentry and the equilibrium profit has the following simple expression:(

nn− 1

)W ′(n)−Π(n) = ∑

i∈S

(n

n− 1

)δ′i(n)−

Mi(n)n

, (17)

where the scalar, n/(n− 1), is the loop effect of the platform competition that we dis-cussed before. What is interesting about the above equation is that the right-hand-sidedoes not rely on externalities, although both W ′ and Π depend on the externalities. Un-der condition (iii) in Theorem 3, the above observation implies that n

n−1W ′(n)−Π(n) ≤0, which in turn implies excessive entry.51 The slightly stronger requirement of distribu-tion in (iii) than log-concavity guarantees that the variety effect, which is responsible forinsufficient entry, is dominated by the business-stealing effect, which is responsible forexcessive entry.

We now provide an equivalent condition to (iii) in Theorem 3, which can be moreconveniently verified.

Lemma 2. For any i ∈ S , fi satisfies (16) if and only if the corresponding quantile densityfunction L(z) = fi(F−1

i (z)) satisfies the following:

n3(∫ 1

0zn−2L(z)dz

)(∫ 1

0

zn ln(1/z)L(z)

dz)≤ 1, for any n ≥ 2. (18)

The inequality (18) is satisfied by a large class of distributions, including commonlyused ones such as the Gumbel, uniform and reversed Weibull distributions.52

Theorem 3 demonstrates the joint forces in determining the nature of entry in multi-sided markets by the functional forms of externalities and the distributions of matching

50Indeed, by Jensen’s inequality, the convexity of φi implies

φi(1

n + 11s) + 〈∇φi(

1n + 1

1s),1n

1s −1

n + 11s〉 ≤ φi(

1n

1s),

where the left-hand-side is the linear approximation of φi at 1n 1s using the tangent plane at z = 1

n+1 1s.51It follows from W ′(n∗) = K that Π(n∗) ≥ n

n−1 W ′(n∗) > K and hence ne ≥ n∗.52Tan and Zhou (2019) characterize the class of distributions that satisfy (18). Note that log-concavity

does not necessarily imply (iii) in Theorem 3, as the exponential distribution in Example 5 violates (18).

28

values. Notably, all these three conditions in Theorem 3 are only sufficient, but notnecessary for excessive entry. Next, we provide an example to illustrate that free entryinto multi-sided markets can easily be socially insufficient.

Example 5 (Insufficient entry). Consider a two-sided market with distribution Fi(θi) =

1− exp (−θi) on both sides and externality function φ1(x1, x2) = α1xρ2 and φ2(x1, x2) =

α2xρ1. Under parameter values α1 = α2 = 11.9015, ρ = 0.05515, and entry cost K = 0.0978,

the free entry equilibrium number is shown to be 6, while the socially optimal numberis 7.53 The combination of the first two effects in (15) of Theorem 2 is negative, whichis consistent with Anderson et al. (1995), but it is dominated by the third positive termgenerated by the concave externality functions (i.e., decreasing returns to scale).

In the example, none of the sufficient conditions in Theorem 3 is applicable, as φi ismonotonically increasing and strictly concave, and the exponential distribution violatesthe inequality in Lemma 2. Under the exponential distribution (which has a log-concavedensity) and without externalities, there is excessive entry. However, in the presence ofcross-side externalities with decreasing returns of scale, free entry is socially insufficient.This example indicates that multi-sidedness and the nature of externalities may helpprovide a plausible explanation for industry concentration in many high-technologymarkets involving platforms.

To summarize, we have identified three distinct economic forces that jointly deter-mine the nature of free-entry and efficiency in multi-sided markets, and have providedinsights into which forces are responsible for excessive entry and which for insufficiententry, as compared to the socially optimal outcome. Unambiguous predictions can bemade when certain primitive sufficient conditions are satisfied.

6 Conclusion

In this paper, we provide a framework to study platform price competition in oligopolis-tic markets. The tractability of our framework enables us to address several importantissues. Among other things, we show that a perverse relationship between platform com-petition and price can generally occur. We provide primitive conditions under which the

53Under the given parametric values, we have Π(6) − K ≈ 0.0007 > 0, Π(7) − K ≈ −0.0056 < 0,Π(n) < K for any n ≥ 7, and hence ne is 6. Moreover, W(n) := W(n) − nK is single-peaked, W(7) ≈25.9029 > W(8) ≈ 25.9013 > W(9) ≈ 25.8913, W(6) ≈ 25.8946 < W(7), and hence n∗ = 7. Treating n as acontinuous variable, we find that ne ≈ 6.1 and n∗ ≈ 7.3, and that the difference between n∗ and ne exceeds1. The consumer surplus is U-shaped and platform profit is inverted U-shaped with n.

29

number of free-entry platforms is socially excessive and an example in which free-entryis socially insufficient.

Our analysis is based on a number of important assumptions. One of the assumptionsis single-homing. It would be interesting to study the effects of competition and entrywhen multi-homing by customers on one or more sides is feasible.54 Moreover, theregulation of big tech companies that are built around multi-sided platforms, such asAmazon, Apple, Facebook, and Google has attracted a huge amount of policy interestsin recent years. Our analysis on entry suggests that promoting competition in suchmarkets may be subtle and have unintended consequences. Yet richer models that allowfor asymmetry of platforms, along with customers’ choices of multi-homing and outsideoption, are necessary to provide a better guideline for regulation. We leave these andother important research questions for the future.

54There is a growing literature on the impact of multi-homing; see Armstrong (2006), Kaiser and Wright(2006), Armstrong and Wright (2007), White and Weyl (2010), Belleflamme and Peitz (2019a), Liu et al.(2019), Bakos and Halaburda (2020), and Jeitschko and Tremblay (2020), among others.

30

Appendix

Proof of Proposition 1: First, for given P define a mapping G with component

gki (X) = Qk

i (φi(x1)− p1i , · · · , φi(xn)− pn

i ), i ∈ S , k ∈ N (19)

on Ω, where

Ω := X = (xki , i ∈ S , k ∈ N ) ∈ Rsn|xk

i ≥ 0, ∀i, k; and ∑k∈N

xki = 1, ∀i ∈ S, (20)

Qki (u) :=

∫εi : k∈arg maxt∈N εt

i+utdGi(εi),

for u = (u1, · · · , un). Then, system (4) is equivalent to G(X) = X.Clearly, Ω is compact and convex, and for any X ∈ Ω, G(X) ∈ Ω. By the continuity

of Qki and φi, i ∈ S , k ∈ N , G is also continuous on Ω. The existence of PE follows from

Brouwer’s fixed-point theorem.Next, we apply contracting mapping theorem to show uniqueness. Recall the L∞

norm on vector X as ||X||∞ = maxi∈S,k∈N |xki |. We plan to show that for any X, Y,

||G(X)− G(Y)||∞ ≤ (BεBφ)||X− Y||∞, ∀X, Y, (21)

where55

Bε := maxi∈S ,k∈N

supu∈Rn

∑k′∈N|∂Qk

i (u)∂uk′ |, Bφ := max

i∈S(∑

j∈Ssup

x∈[0,1]s|∂φi(x)

∂xj|).

Since BεBφ < 1 by assumption, G has a unique fixed point by the contracting mappingtheorem. As a result, for any price profile there is a unique PE.

The following Lemma is repeatedly used in our proof of (21).Lemma C: For any continuously differentiable function ϕ(x1, · · · , xm) on a convex do-main Ξ ⊂ Rm, and for any x = (x1, · · · , xm) and y = (y1, · · · , ym), the following holds

|ϕ(x)− ϕ(y)| ≤ (maxi|xi − yi|)× (sup

z∈Ξ

m

∑j=1|∂ϕ(z)

∂xj|). (22)

55Using the properties of the demand system (i.e., the Roy’s identity, Slutsky symmetry and full market

coverage), Bε can be simplified as 2 maxi∈S ,k∈N supu∈Rn∂Qk

i (u)∂uk . For example, suppose for each i ∈ S , εk

ifor k = 1, · · · , n are IID according to Gumbel distribution with parameter βi, then Qk

i takes the standardlogit form, and Bε = maxi∈S 1/(2βi). For linear externalities, Bφ = maxi∈S∑j∈S |γij|.

31

Note that (maxi |xi − yi|) is L∞ norm of the vecor x− y, and (∑mi=1 supz∈Ξ |

∂ϕ(z)∂xi|) is the

supremum of the L1 norm of the gradient ∇ϕ over Ξ. Lemma C is a direct consequenceof the mean value theorem, and hence the proof is omitted.

We apply Lemma C to φi to show

|φi(xt)− φi(yt)| ≤ ||xt − yt|||∞(∑j∈S

supx∈[0,1]n

|Ψij(x)|) ≤ Bφ||X− Y||∞,

which is true for any i ∈ S , t ∈ N . Applying Lemma C to Qki (·), we obtain

|gki (X)− gk

i (Y)|= |Qk

i (φi(x1)− p1i , · · · , φi(xn)− pn

i )−Qki (φi(y1)− p1

i , · · · , φi(yn)− pni )|

≤ (maxk∈N|φi(xk)− φi(yk)|)

(supu∈Rn

∑t∈N|∂Qk

i (u)∂ut |

)

≤(

Bφ||X− Y||∞) (

supu∈Rn

∑t∈N|∂Qk

i (u)∂ut |

)≤ Bφ||X− Y||∞Bε.

Combing these inequalities, we have

|gki (X)− gk

i (Y)| ≤ (BεBφ)||X− Y||∞.

for any X, Y, any i ∈ S , k ∈ N . Therefore, (21) holds and the claim follows.

Proof of Proposition 2: To support the price profile P∗ = (p∗, · · · , p∗) in (7) and x∗ = 1n 1s

as the subgame perfect equilibrium outcome, we need to specify a PE for any priceprofile, which is described as follows:

(i) On the equilibrium path where every platform chooses p∗, pick the PE that giveseach platform x∗ = 1

n 1s.

(ii) If only one platform, say platform 1, deviates from p∗ to p1, we consider the fol-lowing semi-symmetric demand profile(

q1,1s − q1

n− 1, · · · ,

1s − q1

n− 1

),

which is a PE at (p1, p∗, · · · , p∗) if and only if q1 is a solution to the following

32

system of equations (it always exists by Brouwer’s fixed-point theorem):

q1i = Pr(φi(q1)− p1

i + ε1i ≥ max

t 6=1φi(

= 1s−q1n−1︷︸︸︷qt )−

:=p∗i︷︸︸︷pt

i +εti)

= Pr(ε1i −max(ε2

i , · · · , εni ) ≥ p1

i − p∗i − φi(q1) + φi(1s − q1

n− 1))

= 1− Hi

(p1

i − p∗i − φi(q1) + φi(1s − q1

n− 1); n)

, i ∈ S . (23)

To be consistent with item (i) above, if p1 happens to be exactly p∗, we pick q1 =1n 1s, which solves (23) using Hi(0, n) = 1− 1/n.

(iii) For any P involving more than two platforms deviating from p∗, pick any PE at P,which always exists by Proposition 1.

We now check that there is no profitable deviation by any platform from the proposedequilibrium candidate. Note that the deviating profit of platform 1 at price p1 is

Π1(p1) = 〈p1, q1(p1)〉

where q1(p1) is a solution to (23). By inverting (23), we observe that p1 must satisfy thefollowing inverse demand system:

p1i (q

1) = H−1i (1− q1

i ; n) + p∗i + φi(q1)− φi(1s − q1

n− 1), i = 1, 2, · · · , s. (24)

at quantity q1(p1). By changing variables, we can rewrite the profit of platform 1 as

Π1(p1) = ∑i∈S

q1i

(H−1

i (1− q1i ; n) + p∗i + φi(q1)− φi(

1s − q1

n− 1)− ci

)= R(q1(p1))

which holds by the definition of R.Notice that R( 1

n 1s) = 1/n ∑i∈S p∗i is the profit of platform 1 on the equilibrium path.Deviating from p∗ to p1 is not profitable if 1

n 1s maximizes R(q1) over q1 ∈ [0, 1]s, whichis indeed true by the following two observations: At 1

n 1s, the first-order conditions hold

∂R(q1)

∂q1i|q1= 1

n 1s=

1n−1

hi(0; n)+

1n ∑

j∈S(1 +

1n− 1

)∂φj(z)

∂zi|z= 1

n 1s+ p∗i − c = 0

by the price p∗ given by (7) in Proposition 2. Here we have used the following identities:

Hi(0; n) = 1− 1n

, H−1i (1− 1

n; n) = 0,

(φi(q1)− φi(

1s − q1

n− 1)

)|q1= 1

n 1s= 0, ∀i.

33

So 1n 1s is a stationary point of R(·), and moreover it is a global maximizer by Assumption

1. The claim follows.

Remark 1. Our argument also implies that, under uniqueness of PE, there is no othersymmetric equilibrium, except for p∗ defined in (7).

Proof of Lemma 1: Note that ηi(n) = 1/n1−1/n σi(

1n ). Clearly, ηi decreases (increases) with

n when z1−z σi(z) increases (decreases) in z. Moreover, limn→∞ ηi(n) = limz→0

z1−z σi(z) =

limz→0 zσi(z), so limn→∞ ηi(n) = 0 when limz→0 zσi(z) = 0.

Proof of the Results in Example 4: For IID Gumbel distribution with parameter βi, thefollow holds (see Anderson et al. (1992)):

Hi(θ; n) =n− 1

e−θ/βi + (n− 1), hi(θ; n) = Hi(θ; n)(1− Hi(θ; n))/βi, (25)

Mi(n) =1− Hi(0; n)

hi(0; n)=

nn− 1

βi, δi = E[ maxk=1,2,···n

εki ] = βi(ln(n) + κ).

With linear externalities, Proposition 2 implies56

p∗i (n) =n

n− 1βi −

1n− 1

γi, i ∈ S

where γi = ∑j∈S γji. Moreover p∗′

i (n) = (γi − βi)/(n− 1)2, so p∗i (n) decreases with n ifand only if βi > γi.

The equilibrium profit is given by

Π(n) =1n ∑

i∈Sp∗i =

1n ∑

i∈S(

nn− 1

βi −1

n− 1γi) =

nβ− γ

n(n− 1),

which proves (12). And Π′(n) = γ(2n−1)−βn2

n(n−1)2 < 0 if and only if γ/β < n2/(2n− 1).For each i ∈ S , the consumer surplus is

CSi(n) = βi

(ln(n) + κ − n

n− 1

)+

1n ∑

j∈Sγij +

1n− 1

γi,

which increases with n if and only if CS′i(n) = βi

(1n + 1

(n−1)2

)− 1

n2 ∑j∈S γij− 1(n−1)2 γi >

0, which is equivalent to n(n2 − n + 1)βi > (n− 1)2(∑j∈S γij) + n2γi.

56In this case, a sufficient condition for Assumption 1 is: 274 Diag(β1, · · · , βs)− n

n−1 (Γ + Γ′) is positivedefinite, where Γ = (γij)s×s.

34

Proof of Theorem 1: The proof works for each i ∈ S , and hence we omit subscript i.First note that η(n) = z

1−z σ(z) at z = 1/n, so

limn→∞

nη′(n)η(n)

= − limz→0

z[ z

1−z σ(z)]′[ z

1−z σ(z)] = −(1 + lim

z→0z

σ′(z)σ(z)

) = −r. (26)

Moreover, we have

limn→∞

nM′(n)M(n)

= limθ→θ+

ddθ

(1− F(θ)

f (θ)

)= −τ, (27)

where the first equality follows from Proposition 1 in Gabaix et al. (2016).Consider case (i) with 0 < τ < r. Combining (26) and (27) yields

limn→∞

n[η(n)/M(n)]′

[η(n)/M(n)]= lim

n→∞

nη′(n)η(n)

− limn→∞

nM′(n)M(n)

= τ − r, (28)

which is negative in this case. By (28) and Lemma D below, limn→∞η(n)M(n) = 0 57 and

limn→∞

p∗′(n)

M′(n)= 1− lim

n→∞

η′(n)M′(n)

= 1− limn→∞

(nη′(n)η(n)

)(

nM′(n)M(n)

) ( η(n)M(n)

)= 1− (

−r−τ

)(0) = 1. (29)

Thus, for sufficiently large n, p∗′(n) has the same sign as M′(n). Since −τ < 0, by (27),

Lemma D and the fact that M(n) is positive, we have M′(n) < 0 for sufficiently large n.The claim follows.

Next consider case (ii) with 0 < r < τ. Similar to (28), we obtain limn→∞ n [M(n)/η(n)]′

[M(n)/η(n)] =

−τ+ r, which is negative in this case. Using a similar argument, we obtain limn→∞M(n)η(n) =

058 and limn→∞p∗′(n)

η′(n) = −1. So, for sufficiently large n, p∗′(n) has the opposite sign as

η′(n), which, by (26) and Lemma D, has the opposite sign as η(n). In sum, for sufficientlylarge n, p∗

′(n) has the same sign as η(n) (or σ(1/n)).

Remark 2. Although Theorem 1 is not directly applicable to the case with τi = 0, thesign of p∗

′i (n) in this case can be easily determined by combining our results about ηi(n)

and the order of convergence for M′i(n).59 For instance, the sign of p∗

′i (n) in Examples 3

57This also implies that limn→∞M(n)p∗(n) = 1 in case (i).

58This also implies that limn→∞η(n)p∗(n) = −1 in case (ii).

59For many distributions with zero tail index, such as normal, Gumbel and exponential distributions,the orders of convergence for Mi(n) and M′i(n) are available (see, for instance, Gabaix et al. 2016).

35

and 4 is established along this line of argument. Similarly, when τi = ri > 0, both Mi(n)and ηi(n) have the asymptotic elasticities with respect to n for large n. In this case, thedominance argument in the proof of Theorem 1 does not apply, and further informationabout Mi(n) and ηi(n) is required to determine the sign of p∗

′i (n).

Lemma D: Suppose g(n) satisfies limn→∞ng′(n)g(n) = b < 0. Then, (i) there exists n0 > 0

such that, for any n ≥ n0, g′(n) has the opposite sign as g(n); (ii) limn→∞ g(n) = 0.

Proof of Lemma D: Take ε = −b/2 > 0, there exists n0 > 0 such that, for any n ≥ n0,

ng′(n)g(n)

∈ [b− ε, b + ε] = [3b2

,b2].

Since b < 0, n > 0, we have g′(n)g(n) < 0, showing part (i). For part (ii), note that for any

n ≥ n0,g′(n)g(n)

≤ b2n

.

Integrating from n0 to n on both sides yields

ln(|g(n)|/|g(n0)|) ≤b2

ln(n/n0),

implying limn→ ln |g(n)| = −∞. Hence limn→∞ |g(n)| = 0 and limn→∞ g(n) = 0.

Proof of Theorem 2: We first present some properties of Π(n) and W(n), which will beuseful for the proof of Theorem 2 and Theorem 3. From (11) and Proposition 2,

Π(n) =1n ∑

i∈Sp∗i =

1n ∑

i∈S

[Mi(n)−

1(n− 1)

σi(1n)

]. (30)

Using CSi as given by (10), Π(n) as given by (30), and simplifying, we obtain

W(n) = nΠ(n) + ∑i∈S

CSi(n) = ∑i∈S

(δi(n) + φi(

1n

1s))

, (31)

and

W ′(n) = ∑i∈S

δ′i(n)−1n2 ∑

i,j∈S

∂φi(x)∂xj

|x= 1n 1s

= ∑i∈S

[δ′i(n)−

1n2 σi(

1n)]. (32)

Theorem 2 is obtained by combining (30) and (32).

36

Proof of Theorem 3: Case (i): From Theorem 2, we have

W ′(n)−Π(n) = ∑i∈S

[δ′i(n)−

Mi(n)n

]+

∑i∈S σi(1/n)(n− 1)n2 .

The concavity of δi implies δ′i(n) ≤ δi(n)− δi(n− 1).60 Moreover, by Theorem 1 inAnderson et al. (1995), under log-concavity of fi, for any n ≥ 2,

δi(n)− δi(n− 1) ≤ Mi(n)n

, (33)

implying δ′i(n) −Mi(n)

n ≤ 0. Moreover, ∑i∈S σi(1/n) ≤ 0 by Assumption (i). As aconsequence, W ′(n)−Π(n) ≤ 0 for any n ≥ 2.

At n = n∗, W ′(n∗) = K, it follows that Π(n∗) ≥ W ′(n∗) = K. As ne is the maximal nsuch that Π(n) ≥ K, we must have n∗ ≤ ne, i.e., excessive entry occurs.

Case (ii): Using (30) and (31), we obtain

(W(n + 1)−W(n))−Π(n + 1) = I + I I

where

I = ∑i∈S

(δi(n + 1)− δi(n)−

1(n + 1)

Mi(n + 1))

,

I I = ∑i∈S

φi(1

n + 11s)− ∑

i∈Sφi(

1n

1s) + ∑i,j∈S

1n(n + 1)

∂φi(x)∂xj

|x= 1n+1 1s

.

By (33) the first term I is nonnegative. By Jensen’s inequality, the convexity of φi

implies

φi(1

n + 11s) + 〈∇φi(

1n + 1

1s),1n

1s −1

n + 11s〉 ≤ φi(

1n

1s),

where the term on the left-hand-side is the linear approximation of φi at 1n 1s using the

tangent plane at z = 1n+1 1s. Therefore, the second term I I is also nonnegative. In sum,

for any n ≥ 2,(W(n + 1)−W(n))−Π(n + 1) ≤ 0.

The maximization of W(n)− nK at n∗ implies

W(n∗)− n∗K ≥W(n∗ − 1)− (n∗ − 1)K,

60By (34), δ′i(n) decreases in n, which shows the concavity of δi(n) in n.

37

or equivalently W(n∗)−W(n∗ − 1) ≥ K. It follows that Π(n∗) ≥ W(n∗)−W(n∗ − 1) ≥K, which implies immediately from the definition of ne that ne ≥ n∗.

Case (iii): Direct calculation shows that(n

n− 1

)W ′(n)−Π(n) = ∑

i∈S

(n

n− 1

)δ′i(n)−

Mi(n)n

,

which is nonnegative under assumption (iii).At n = n∗, welfare maximization implies that W ′(n∗) = K, and therefore

Π(n∗) ≥ nn− 1

TW ′(n∗) =n

n− 1K > K,

implying ne ≥ n∗.

Proof of Lemma 2: First, note that

1/Mi(n) = n(n− 1)∫ θi

θi

f 2i (θ)Fn−2

i (θ)dθ.

Let z = Fi(θ), and thus θ = F−1i (z), fi(θ) = fi(F−1

i (z)) = L(z), and dθ = dz/ fi(θ) =

dz/L(z). Therefore,

1/Mi(n) = n(n− 1)(∫ 1

0zn−2L(z)dz

).

Recall that δi(n) =∫ θi

θiθdFn

i (θ). For any ∆ > 0, using integration by part, we obtain

δi(n + ∆)− δi(n) =∫ θi

θi

θdFn+∆i (θ)−

∫ θi

θi

θdFni (θ) = −

∫ θi

θi

(Fn+∆i (θ)− Fn

i (θ))dθ.

Therefore,

δ′i(n) = − lim∆→0

∫ θiθi(Fn+∆

i (θ)− Fni (θ))dθ

∆= −

∫ θi

θi

Fni (θ) ln Fi(θ)dθ. (34)

Changing variable z = Fi(θ) yields

δ′i(n) = −∫ θi

θi

Fni (θ) ln Fi(θ)dθ =

∫ 1

0

zn ln(1/z)L(z)

dz.

Using these expressions of Mi and δ′i , (16) reduces to (18).

38

References

Anderson, S. P. and S. Coate (2005). Market provision of broadcasting: A welfare analy-sis. The Review of Economic Studies 72(4), 947–972.

Anderson, S. P., A. De Palma, and Y. Nesterov (1995). Oligopolistic competition and theoptimal provision of products. Econometrica 63(6), 1281–1301.

Anderson, S. P., A. De Palma, and J. F. Thisse (1992). Discrete choice theory of productdifferentiation. MIT press.

Anderson, S. P. and B. Jullien (2015). The advertising-financed business model in two-sided media markets. In Handbook of Media Economics, Volume 1, pp. 41–90. Elsevier.

Anderson, S. P. and M. Peitz (2020). Media see-saws: Winners and losers in platformmarkets. Journal of Economic Theory 186, 104990.

Armstrong, M. (2006). Competition in two-sided markets. The RAND Journal of Eco-nomics 37(3), 668–691.

Armstrong, M. and J. Wright (2007). Two-sided markets, competitive bottlenecks andexclusive contracts. Economic Theory 32(2), 353–380.

Bakos, Y. and H. Halaburda (2020). Platform competition with multi-homing on bothsides: Subsidize or not? forthcoming in Management Science.

Bayer, P. and C. Timmins (2005). On the equilibrium properties of locational sortingmodels. Journal of Urban Economics 57(3), 462–477.

Belleflamme, P. and M. Peitz (2019a). Platform competition: Who benefits from multi-homing? International Journal of Industrial Organization 64, 1–26.

Belleflamme, P. and M. Peitz (2019b). Price disclosure by two-sided platforms. Interna-tional Journal of Industrial Organization 67, 102529.

Berry, S., A. Eizenberg, and J. Waldfogel (2016). Optimal product variety in radio mar-kets. The RAND Journal of Economics 47(3), 463–497.

Berry, S., J. Levinsohn, and A. Pakes (1995). Automobile prices in market equilibrium.Econometrica 63(4), 841–890.

39

Berry, S. T. (1994). Estimating discrete-choice models of product differentiation. TheRAND Journal of Economics 25(2), 242–262.

Berry, S. T. and J. Waldfogel (1999). Free entry and social inefficiency in radio broadcast-ing. The Rand Journal of Economics 30(3), 397–420.

Brock, W. A. and S. N. Durlauf (2001). Discrete choice with social interactions. TheReview of Economic Studies 68(2), 235–260.

Brock, W. A. and S. N. Durlauf (2002). A multinomial-choice model of neighborhoodeffects. American Economic Review 92(2), 298–303.

Caillaud, B. and B. Jullien (2001). Competing cybermediaries. European Economic Re-view 45(4-6), 797–808.

Caillaud, B. and B. Jullien (2003). Chicken & egg: Competition among intermediationservice providers. The RAND Journal of Economics 34(2), 309–328.

Caplin, A. and B. Nalebuff (1991). Aggregation and imperfect competition: On theexistence of equilibrium. Econometrica 59(1), 25–59.

Carrillo, J. D. and G. Tan (2015). Platform competition with complementary products.Available at http://www.jdcarrillo.org/PDFpapers/wp-platform.pdf.

Chandra, A. and A. Collard-Wexler (2009). Mergers in two-sided markets: An appli-cation to the canadian newspaper industry. Journal of Economics & Management Strat-egy 18(4), 1045–1070.

Correia-da Silva, J., B. Jullien, Y. Lefouili, and J. Pinho (2019). Horizontal mergers be-tween multisided platforms: Insights from cournot competition. Journal of Economics& Management Strategy 28(1), 109–124.

Dixit, A. K. and J. E. Stiglitz (1977). Monopolistic competition and optimum productdiversity. The American Economic Review 67(3), 297–308.

Dybvig, P. H. and C. S. Spatt (1983). Adoption externalities as public goods. Journal ofPublic Economics 20(2), 231–247.

Elberse, A., A. Hagiu, and M. Egawa (2007). Roppongi hills: city within a city. HarvardBusiness School Case 707-431.

40

Fan, Y. (2013). Ownership consolidation and product characteristics: A study of the usdaily newspaper market. The American Economic Review 103(5), 1598–1628.

Gabaix, X., D. Laibson, D. Li, H. Li, S. Resnick, and C. G. de Vries (2016). The impact ofcompetition on prices with numerous firms. Journal of Economic Theory 165, 1–24.

Gabszewicz, J. J. and X. Y. Wauthy (2014). Vertical product differentiation and two-sidedmarkets. Economics Letters 123(1), 58–61.

Hagiu, A. and J. Wright (2015). Multi-sided platforms. International Journal of IndustrialOrganization 43, 162–174.

Jeitschko, T. D. and M. J. Tremblay (2020). Platform competition with endogenous hom-ing. forthcoming in International Economic Review.

Jeziorski, P. (2014). Effects of mergers in two-sided markets: The US radio industry.American Economic Journal: Microeconomics 6(4), 35–73.

Jullien, B. and A. Pavan (2019). Information management and pricing in platform mar-kets. The Review of Economic Studies 86(4), 1666–1703.

Kaiser, U. and J. Wright (2006). Price structure in two-sided markets: Evidence from themagazine industry. International Journal of Industrial Organization 24(1), 1–28.

Karle, H., M. Peitz, and M. Reisinger (2020). Segmentation versus agglomeration: com-petition between platforms with competitive sellers. Journal of Political Economy 128(6),2329–2374.

Katz, M. L. and C. Shapiro (1985). Network externalities, competition, and compatibility.The American Economic Review 75(3), 424–440.

Liu, C., T.-H. Teh, J. Wright, and J. Zhou (2019). Multihoming and oligopolistic platformcompetition. mimeo, National University of Singapore.

Mankiw, N. G. and M. D. Whinston (1986). Free entry and social inefficiency. The RANDJournal of Economics 17(1), 48–58.

Miyao, T. (1978). A probabilistic model of location choice with neighborhood effects.Journal of Economic Theory 19(2), 347–358.

Nikzad, A. (2020). Size of the labor pool in on-demand service platforms. Available athttp://afshin-nikzad.com/pdfs/on-demand.pdf .

41

Nocke, V. and N. Schutz (2018). Multiproduct-firm oligopoly: An aggregative gamesapproach. Econometrica 86(2), 523–557.

Perloff, J. M. and S. C. Salop (1985). Equilibrium with product differentiation. The Reviewof Economic Studies 52(1), 107–120.

Quint, D. (2014). Imperfect competition with complements and substitutes. Journal ofEconomic Theory 152, 266–290.

Rochet, J.-C. and J. Tirole (2003). Platform competition in two-sided markets. Journal ofthe European Economic Association 1(4), 990–1029.

Rochet, J.-C. and J. Tirole (2006). Two-sided markets: a progress report. The RANDJournal of Economics 37(3), 645–667.

Song, M. (2020). Estimating platform market power in two-sided markets with an ap-plication to magazine advertising. forthcoming in American Economic Journal: Microe-conomics.

Spence, A. M. (1975). Monopoly, quality, and regulation. The Bell Journal of Economics 6(2),417–429.

Spence, M. (1976). Product selection, fixed costs, and monopolistic competition. TheReview of Economic Studies 43(2), 217–235.

Sweeting, A. (2010). The effects of mergers on product positioning: evidence from themusic radio industry. The RAND Journal of Economics 41(2), 372–397.

Tan, G. and J. Zhou (2019). Price competition in multi-sided markets. Available at SSRN3029134.

Weyl, E. G. (2010). A price theory of multi-sided platforms. The American EconomicReview 100(4), 1642–1672.

Weyl, E. G. and M. Fabinger (2013). Pass-through as an economic tool: Principles ofincidence under imperfect competition. Journal of Political Economy 121(3), 528–583.

White, A. and E. G. Weyl (2010). Imperfect platform competition: A general framework.NET Institute Working Paper #10-17.

White, A. and E. G. Weyl (2016). Insulated platform competition. Available at SSRN1694317.

42

Zhou, J. (2017). Competitive bundling. Econometrica 85(1), 145–172.

43