Paul Belle
email: paul.belle‡amme@univ-amu.fr.

Martin Peitz

Managing competition on a two-sided platform

Keywords: Network e¤ects, two-sided markets, platform competition, intermediation, pricing, imperfect competition JEL-Classi…cation: D43, L13, L86

On many two-sided platforms, users on one side not only care about user participation and usage levels on the other side, but they also care about participation and usage of fellow users on the same side. Most prominent is the degree of seller competition on a platform catering to buyers and sellers. In this paper, we address how seller competition a¤ects platform pricing, product variety, and the number of platforms that carry trade.

Introduction

Many platforms enable or facilitate trade between buyers and sellers. Probably since the Stone Age, people gather at central places to exchange goods and provide services. Medieval trade fairs brought buyers and sellers into contact. The o-ine world features many platforms including stock exchanges, currencies, ‡ea markets, shopping malls, newspapers, magazines, and broadcasters. Electronic payment systems, software platforms, and digital marketplaces are more recent prominent platforms allowing two distinct groups of participants to interact and exchange goods or services.

Starting with the seminal papers by Caillaud and Jullien (2003), Rochet and Tirole (2003), and Armstrong (2006), the economic literature on two-sided platforms focuses on cross-group external e¤ects. Such focus seems natural as cross-group e¤ects directly stem from the desire of the two groups of agents to interact and, thereby, give their raison d'être to two-sided platforms. However, in many economic environments, platforms also have to factor in the fact that the attractiveness of a platform for the members of one group also depends on the participation of the members of the very same group. That is, there exist within-group external e¤ects, which platforms have to take into account when choosing their strategies. Let us describe a number of settings in which such e¤ects are present.

Negative within-group e¤ects appear when the members of one group compete with one another to interact with the other group. For instance, given a set of buyers on the platform, the expected pro…ts of sellers on Ebay decrease in response to the entry of competing sellers. Similarly, if an additional competing shop opens in a shopping mall, the expected pro…ts of existing shops decrease given a set of buyers in the mall. Another example is dating apps. These are characterized by positive cross-group external e¤ects, as the app becomes more attractive the more it is used by people of the opposite gender. However, they are also characterized by negative within-group external e¤ects, as more people of the same gender make it less likely that a match materializes for a particular person. Di¤erent from a buyer-seller context, there is typically no monetary transaction between users on a dating app.

Negative within-group e¤ects may also arise because of congestion problems-for instance, sellers may compete for buyer attention, which is scarce. Consumption externalities are another instance of negative within-group e¤ects. For instance, referring to Airbnb visitors, Slee (2016) reports that "as their numbers grow, they erode the very atmosphere in which they bask and threaten the livability of the city for residents." This arguably also applied to fellow visitors. Congestion e¤ects may also be present on digital platforms with limited bandwidth such that, e.g., the delivery of research result is slowed down-this would imply that the stand-alone utility of the platform su¤ers from a lot of buyer participation. In the o-ine world, congestion problems appear when the platform's physical venue is too crowded; for instance, shoppers may get stuck in a crowded shopping mall and, as a result, make fewer purchase attempts. [START_REF] Ambrus | Asymmetric Networks in Two-sided Markets[END_REF] In this paper, we focus on competition in a buyer-seller context and we examine how such competition a¤ects outcomes in platform markets. We consider a two-sided platform that enables trade between buyers and sellers-these sellers make non-discriminatory take-it-or-leave-it o¤ers to buyers. Imperfect competition between sellers has the standard property that an additional seller on the platform leads to lower per-buyer pro…t for each seller already on the platform-a negative within-group external e¤ect. It also often leads to lower prices and more variety, which buyers like. Thus, the additional seller may generate more participation on the buyer side, which in turn will bene…t all sellers-the combination of two positive cross-group external e¤ects. Our objective is to analyze how platforms take these con ‡icting e¤ects into account when choosing their price-and non-price strategies.

One way to carry out such analysis is to augment standard two-sided platform models by adding a parameter that measures a negative e¤ect within the group of sellers, and to study the allocative e¤ects in response to a change of this parameter; we review below some papers that follow this path (e.g., Belle ‡amme and Toulemonde, 2009). However, in light of the argument above, such comparative statics can be misconceived, as seller competition also a¤ects how buyers and sellers value the interaction with each other. For instance, an increase in seller competition drives down per-consumer pro…ts and, thus, increases the strength of the negative within-group external e¤ect on the seller side. At the same time, the per-seller consumer bene…t is a¤ected and, thus, the strength of the cross-group external e¤ect exerted by sellers on consumers changes as well. Hence, an increase in seller competition a¤ects two parameters in the reduced form model at the same time. This implies that these parameters should not be treated as primitives of the model. We need instead to couple standard two-sided platform models with micro-foundations of seller competition and buyer-seller relationships. This is the route that we take in this paper. [START_REF] Armstrong | Competition in Two-sided Markets[END_REF] This paper is organized as follows. First, we consider seller competition on a monopoly platform and its e¤ect on prices and product variety. Second, we study platform competition; in particular, we generalize the two-sided singlehoming model of Armstrong (2006) to allow for seller competition. Third, we examine the in ‡uence of seller competition on the number of platforms carrying positive volume of trade. Our objective is to synthesize existing literature, which will be referenced in the main text.

Prices and product variety on a monopoly platform

In this section, we focus on a single platform that facilitates the interaction between buyers and competing sellers. We …rst develop a baseline model to derive the main intuitions. We then extend the analysis by reviewing the extant literature.

probability that any project will be realized, which bene…ts all funders. Another instance is learning from others in the same group on a platform. [START_REF] Armstrong | Competition in Two-sided Markets[END_REF] Lin, Wu and Zhou (2016) also consider seller competition, and show that a monopoly platform bases its pricing on characteristics that a¤ect both sides of the market.

A baseline model

Let us de…ne the seller and buyer net surplus of visiting the platform (gross of any opportunity cost) respectively as

v s = r s + (n b ; n s ) m s and v b = r b + u (n b ; n s ) m b .
Here, r g is the stand-alone utility on side g 2 fb; sg and m g is the …xed membership or subscription fee charged by the platform to side g. The general functions (n b ; n s ) and u (n b ; n s ) represent the net gains from trade for any seller and any buyer on the platform. They both potentially depend on the number of buyers and on the number of sellers who are present on the platform, meaning that any form of cross-group and within-group external e¤ects are permitted. We assume that both functions are twice continuously di¤erentiable in their two arguments. In this paper, we focus on …xed fees per participant and do not allow the platform to charge usage fees. This is reasonable in situations in which monitoring transactions is prohibitively costly or in which consumers can bypass the platform at negligible cost. [START_REF] Ater | Vertical Foreclosure Using Exclusivity Clauses: Evidence from Shopping Malls[END_REF] We consider a model with three stages: …rst, the monopoly platform sets the …xed fees; second, buyers and sellers simultaneously decide to enter (alternatively, the platform admits a certain number of buyers of sellers on the platform and fees then clear the market); and, third, participating sellers play an oligopoly game. We analyze properties of the subgame-perfect Nash equilibrium.

In this monopoly model, we assume that a mass of Z buyers and of Z sellers have an outside option that is uniformly distributed on [0; Z], with Z large. [START_REF] Augereau | Coordination versus Di¤erentiation in a Standards War: 56K Modems[END_REF] Hence, for given m s and m b , the equilibrium number of buyers and sellers is implicitly characterized by n s = r

+ u (n b ; n s ) n b f b ] + n s [r s + (n b ; n s ) n s f s ].
The pro…t-maximizing number of sellers and buyers on a platform satis…es the system of …rst-order conditions 8 > < > :

@ @n b = [r b + u (n b ; n s ) 2n b f b ] + @u (n b ; n s ) @n b n b + @ (n b ; n s ) @n b n s = 0; @ @n s = [r s + (n b ; n s ) 2n s f s ] + @u (n b ; n s ) @n s n b + @ (n b ; n s ) @n s n s = 0:
The second-order conditions require

@ 2 @n 2 b < 0, @ 2 @n 2 s < 0 and @ 2 @n 2 b @ 2 @n 2 s @ 2 @n b @n s 2 > 0:
3 Otherwise, a platform may want to impose usage charges if participants are heterogeneous with respect to the bene…t they derive from usage. As discussed by Rochet and Tirole (2006) for the case in which sellers'products are independent (neither substitutes nor complements), membership fees are the"right"instrument to use if platforms only face heterogeneity in opportunity costs from participating. Seller competition does not a¤ect this insight. [START_REF] Augereau | Coordination versus Di¤erentiation in a Standards War: 56K Modems[END_REF] Due to seller heterogeneity inframarginal sellers will make a positive net surplus. If sellers were homogeneous, the platform would extract all rents on the seller side. For monopoly pricing with more general distributions, see Hagiu (2009). In models with a discrete number of sellers, we assume that the opportunity cost of the nth seller is n. For expositional convenience, with a …nite number of sellers, we treat the number of sellers at stage 1 as a real number.

We now focus on situations in which within-group external e¤ects are negative for sellers (re ‡ecting seller competition) and absent for buyers. As for cross-group external e¤ects, they are positive in both directions. We simplify the surplus functions as follows: (n b ; n s ) n b ~ (n s ) and u (n b ; n s ) n s ũ(n s ); both functions will be parametrized by a parameter that measures the intensity of competition among sellers. We now provide two examples with these properties. [START_REF] Belle ‡amme | The Economics of Crowdfunding Platforms[END_REF] Example 1 Cournot competition with horizontally di¤ erentiated products Suppose that sellers are Cournot competitors with horizontally di¤erentiated products and constant marginal costs c; the demand for variety k, produced by seller k (with k = 1 : : : n s ) is given by p k = a q k q k (with q k = P g6 =k q g ) for strictly positive prices and zero otherwise. [START_REF] Belle ‡amme | Platform Competition and Seller Investment Incentives[END_REF] Solving for the Cournot equilibrium and setting, without loss of generality, a c = 1, we can compute the equilibrium pro…t for each seller and the consumer surplus as (see the details in Appendix 6.1):

(n b ; n s ) = n b ~ (n s ) = n b 1 (2 + (n s 1)) 2 and u (n b ; n s ) = n s ũ(n s ) = n s (1 + (n s 1)) 2 (2 + (n s 1)) 2 :
Example 2 Monopolistic competition with CES demand Suppose that products are di¤erentiated and a continuum of single-product sellers simultaneously set prices. Conditional on visiting the platform, consumers have CES preferences over the di¤erentiated products. [START_REF] Belle ‡amme | Platform Competition: Who Bene…ts from Multihoming?[END_REF] Demand for product j is q(j) = p(j)

1 1 E R ns 0 p(i) 1 di ;
where p(j) is the price of variant j, E is income spent on the di¤erentiated goods industry, and 2 (0; 1) is inversely related to the degree of product di¤erentiation. Seller j maximizes his gross pro…t (per unit mass of buyers) ~ = (p c)p

1 1
A, where c is the constant marginal cost of production, and A E= R ns 0 p(j) 1 dj. Using symmetry, the …rst-order conditions of pro…t maximization yield the equilibrium price p = c= . In this example, an increase in product variety has no price e¤ect, so @p=@n s = 0. In equilibrium, the pro…t per unit mass of buyers and the utility per unit mass of sellers are respectively given by

~ (n s ) = (1 ) E n s and ũ(n s ) = E c n 1 2 s :
A special case is = 1=2, as in this case ũ(n s ) is constant in n s ; this describes a situation in which seller competition generates a negative within-group external e¤ect, whereas seller competition has no e¤ect on a consumer's per-seller gross bene…t.

In both examples, the equilibrium numbers of buyers and sellers is determined by the solution to n b = r b + n s ũ(n s ) m b and n s = r s + n b ~ (n s ) m s . Substituting the …rst equation into the second, we obtain n s = r s + [r b + n s ũ(n s ) m b ]~ (n s ) m s ; which implicitly de…nes the number of sellers as a function of the platform fees m b and m s . The positive indirect network e¤ect is captured by the square bracket-ũ(n s ) is always increasing in n s , as more buyers are attracted by a platform hosting more sellers. However, this interacts with a negative direct network e¤ect stemming from the fact that ~ (n s ) is decreasing in n s . [START_REF] Belle ‡amme | Negative Intra-Group Externalities in Twosided Markets[END_REF] The platform's pro…t maximization problem can be rewritten respectively as

max n b ;ns n b [r b n b f b ] + n s [r s n s f s ] + [ũ(n s ) + ~ (n s )]n s n b ; 8 > < > : @ @n b = r b 2n b f b + [ũ(n s ) + ~ (n s )] n s = 0; @ @n s = r s 2n s f s + [ũ(n s ) + ~ (n s )] n b + [ũ 0 (n s ) + ~ 0 (n s )] n b n s = 0: (1)
The system (1) implicitly de…nes the optimal numbers of buyers and sellers that the monopoly platform attracts, n M b and n M s and the associated prices that can implement this allocation. We now perform two exercises to evaluate how seller competition a¤ects the platform's equilibrium decisions and pro…ts. First, we observe in expression (1) that the term [ũ 0 (n s ) + ~ 0 (n s )] n b n s on the bottom line only appears when seller competition is present. We then make the following thought experiment. Upon observing n M b , n M s and the resulting ũ(n M s ) and ~ (n M s ), what happens if one wrongly believes that the platform maximized its pro…t while ignoring seller competition, and thus the term [ũ 0 (n s ) + ~ 0 (n s )] n b n s ? This hypothetical platform would solve the following system of equations (

r b 2n b f b + ũ(n M s ) + ~ (n M s ) n s = 0; r s 2n s f s + ũ(n M s ) + ~ (n M s ) n b = 0:
Graphically, the solution to this system is the intersection of the two linear functions depicting the pro…ting-maximizing n b for given n s and the pro…t-maximizing n s for given n b . Let us now account for the dependence of ũ(n s ) and ~ (n s ) on n s . This implies that the pro…t-maximizing n s as a function of n b is shifted inward if and only if ũ0 (n s ) + ~ 0 (n s ) < 0. In this case, the platform's solution must feature fewer sellers and fewer buyers. If the number of buyers is rather insensitive to the number of sellers the e¤ect on the number of buyers is weak; otherwise, the e¤ect is strong. The message that emerges from this simple comparison is that seller competition tends to have a negative e¤ect on product variety on a monopoly platform if ũ0 (n s ) + ~ 0 (n s ) < 0. We note that this comparison is arti…cial, since the absence of seller competition will have an impact on the level of ũ and ~ (as we note in our second exercise below).

It is of interest to see on which side a platform makes most or even all of its pro…ts. Denote pro…t on the buyer side by

b = n b (m b f b ) = n b [r b + n s ũ(n s ) n b f b ] and on the seller side by s = n s (m s f s ) = n s [r s + n b ~ (n s ) n s f s ].
Using the …rst-order conditions of pro…t maximization, we obtain

b = n b [n b ~ (n s )n s ] s = n s [n s ũ(n s )n b ] (ũ 0 (n s ) + ~ 0 (n s ))n b n 2 s :
Thus, it is less likely that sellers are subsidized if ũ0 (n s ) + ~ 0 (n s ) < 0 compared to a model in which cross-group external e¤ects are constant and …xed at the equilibrium value of the present model. [START_REF] Belle ‡amme | Who Bene…ts from Increased Competition among Sellers on B2C Platforms?[END_REF] Returning to our two examples, we observe the following:

ũ0 (n s ) + ~ 0 (n s ) = 8 < : 4+ (ns 1) 2(2+ (ns 1)) 3 in Example 1, E (1 2 )n 1= s cns(1 ) cn 3 s in Example 2.
It is immediate that ũ0 (n s )+ ~ 0 (n s ) < 0 in Example 1. In Example 2, we have that ũ0 (n s )+ ~ 0 (n s ) is locally positive or negative depending on the parameter values; it is necessarily negative if 1=2.

Our second exercise consists in assessing directly how the monopoly platform is a¤ected when seller competition becomes stronger; i.e., in our examples, when products are closer substitutes: when (Example 1) or (Example 2) increases. As noted in the introduction, a change in these parameters also a¤ects directly the consumer surplus. Letting 2 f ; g, we use implicit di¤erentiation to evaluate how a change in a¤ects the platform's optimal number of buyers and sellers, and pro…t. We summarize here our main results (the details can be found in Appendix 6.2). Using the envelope theorem, we compute the e¤ect of a change in on the platform's maximal interior's pro…t as

d d = @ ũ(n s ; ) @ + @ ~ (n s ; ) @ n s n b :
In Examples 1 and 2, where the degree of product substitutability measures the intensity of competition, both ũ and ~ decrease when sellers compete more …ercely. That more competition hurts sellers is obvious. To understand that more competition also reduces the surplus that buyers obtain from each seller, we note that in these two models, an increase in or has two opposite e¤ects: it decreases prices (because of more seller competition) but it also decreases utility (because buyers have an intrinsic preference for variety); it turns out that the latter e¤ect dominates the former. As a result, the bene…ts that both sellers and buyers obtain per transaction are reduced when seller competition increases, which reduces the platform's pro…t. By totally di¤erentiating the system of …rst-order conditions (1) with respect to n b , n s and , we obtain

dn b d = 1 K @ 2 @n b @ @ 2 @n 2 s + @ 2 @n s @ @ 2 @n b @n s ; dn s d = 1 K @ 2 @n s @ @ 2 @n 2 b + @ 2 @n b @ @ 2 @n b @n s ; where K @ 2 @n 2 b @ 2 @n 2 s @ 2 @n b @n s 2 > 0 (by S.O.C.).
Evaluating these expressions for the speci…c surplus functions derived above, we show in Appendix 6.2 that in both examples, the platform chooses to attract fewer sellers and fewer buyers when seller competition intensi…es. Noting that in our model, v b = n b and v s = n s , the previous results also mean that in both examples, sellers and buyers are worse o¤ when competition among sellers intensi…es.

Additional issues with monopoly platforms

In the previous section, we studied how a change in the degree of competition among sellers may a¤ect the platform's pricing strategy and, with it, the well-being of the various players (platform, sellers and buyers). In particular, we showed that if the total value generated in a buyerseller transaction decreases with the intensity of seller competition, then so does the platform's maximal pro…t. In this section, we explore a number of additional issues that have been studied in the literature. First, as a natural extension of our simple setting, we continue to focus on price strategies and examine whether pricing is e¢ cient, and how the results change when the set of strategies is either restricted (one-sided pricing) or expanded (price discrimination). Next, we ask how seller competition may lead platforms to modify their choices of non-price strategies, such as helping buyers search products or controlling sellers'quality.

Platform' s price strategies

Pricing e¢ ciency. Galeotti and Moraga (2009) consider a monopoly platform catering to a …xed discrete number of horizontally di¤erentiated sellers and a continuum of buyers. Their microfoundation of the buyer-seller interaction goes as follows: Sellers choose a probability to participate (this is interpreted as their decision to inform consumers about their product) and a price. Buyers on the platform draw match values and buy the product that maximizes the di¤erence between match value and price. Thus, a seller who increases his participation probability intensi…es competition. The authors assume that, in the second stage, sellers set price and make participation decisions, and buyers make participation decisions. Their decisions are guided by the membership fees set by the platform in the …rst stage.

At the participation stage, buyers and sellers are homogeneous. Therefore, the platform can extract the full surplus generated by intermediation. Galeotti and Moraga show that platform pricing is second-best e¢ cient-that is, a planner with the same instruments would implement the same allocation: The platform sets prices on both sides so as to ensure full participation by buyers and sellers, and this implements the social planner's second-best allocation.

One-sided pricing. Nocke, Peitz and Stahl (2007) analyze the product variety on a monopoly platform that charges only sellers a participation fee. [START_REF] Caillaud | Chicken and Egg: Competition among Intermediation Service Providers[END_REF] Thus, the platform has a single instrument to steer the degree of competition on the platform.

Since buyers can access the platform for free, their participation increases with the number of sellers hosted by the platform. As for sellers, their pro…ts are the ones derived in Example 2. As we showed above, the per-buyer pro…t is decreasing in the number of sellers. Yet, a countervailing e¤ect is the market-expansion e¤ect that arises since more sellers attract more buyers. When deciding about how much to charge sellers, a platform has to take these e¤ects into account-it maximizes n s m s C(n s ), where n s depends on m s and C(n s ) is the platform's weakly convex cost.

As is well-known from the literature on product variety, the fact that sellers engage in business-stealing tends to lead to socially excessive product variety. However, a monopoly platform takes this e¤ect into account. A pro…t-maximizing platform sets a high fee such that there is always a socially insu¢ cient number of sellers n s . Thus, product variety is less than in the second best, in which a social planner can pick the number of sellers hosted by the platform. By contrast, if a platform is open and, thus, does not restrict access by charging fees above marginal costs, product variety may well be socially excessive.

The lack of a pricing instrument on the buyer side suggests that the platform is overly concerned with the gross surplus generated on the seller side. To take an extreme example, suppose that sellers are ex ante homogeneous. A monopoly platform will then extract the full seller surplus. [START_REF] Cennamo | Value Co-creation and Capture in Platform Markets: Evidence from a Creative Industry[END_REF] This often means that the platform will set a high membership fee and e¤ectively limit competition on its platform. If sellers o¤er su¢ ciently close substitutes this implies that only a single seller will be active and the platform will extract the monopoly rent of this single seller. A platform can create such a commitment by o¤ering a contract that grants exclusivity to the seller on the platform-this has been documented in the case of shopping malls in Ater (2015). By contrast, if the platform can also charge buyers, a lower membership fee on the seller side leads to a larger number of participating sellers. This generates a larger gross surplus on the consumer side, which can be extracted by the platform through the membership fee on the buyer side. Thus, with the additional price instruments the platform allows for more product variety and generates a larger total surplus. 1 0 Such a situation arises if a platform cannot monitor participation decision by buyers or if it optimally payed buyers for participation but that such negative fees are not feasible or prohibited. With two-sided pricing, the platform has an additional price instrument that allows it to obtain higher pro…t. It may do so by paying buyers for participation or by charging them a positive price, depending on the parameters and functions.

Making discriminatory o¤ers. More generally, a platform may not only decide on how many sellers to host, but, taking into account that markets are often not symmetric and products di¤er by the degree of substitutability, actively pick sellers with particular characteristics by making discriminatory o¤ers. For instance, in the case of shopping malls, the empirical literature has established that a shopping mall manages its portfolio of shops to internalize externalities-see Pashigian and Gould (1998). In particular, a shopping mall provides better terms to those shops which serve as magnets and generate business for other stores (such as, traditionally, anchor stores). Translated into the online world, this means that a digital platform may be better at internalizing externalities by setting discriminatory membership or access fees on the seller side. Similarly, to internalize externalities, advertising-…nanced platforms may o¤er better terms to advertiser who posts ads that are less annoying or more attractive to buyers.

Platform' s non-price strategies

Product visibility. Search engines have an interest to guide consumers to products they like. [START_REF] Chen | Paid Placement: Advertising and Search on the Internet[END_REF] If the rents that accrue to buyer and sellers are correlated, a platform operating as a search engine, is interested in establishing such a match, as it may extract part of the rent on the seller and possibly also on the buyer side.

Suppose that the search engine charges only sellers. Then, as Chen and He (2011) and Eliaz and Spiegler (2011) show, [START_REF] Church | Network E¤ects, Software Provision, and Standardization[END_REF] a monopoly search engine may bias its search results when sellers compete. It is in its best interest that sellers with high value are highly ranked. A seller's value increases, as product market competition with other sellers is relaxed. Therefore, the search engine may distort search results to relax product market competition between sellers. In Chen and He (2011) and Eliaz and Spiegler (2011), the search engine has an incentive to decrease the relevance of its search results and, thus, discourages buyers from searching extensively. This degrades the quality of the platform. The platform faces a trade-o¤ between fewer buyers using the search engine and higher pro…ts on a per-buyer base, which it obtains from fees charged to sellers. The monopoly distortion introduced by the platform consists in fewer buyers on the platform who have to pay higher product prices than absent the distortion of the search results.

Quality control by platforms. Platforms may control the quality of sellers and remove underperforming sellers from the platform. [START_REF] Damiano | Competing Matchmaking[END_REF] In the presence of seller competition this may come at the cost of reducing competitive pressure. However, since quality control is of particular importance under asymmetric information (here, buyers being less informed about the seller's 1 2 The following discussion is similar to Peitz and Reisinger (2016). They discuss also other work on the strategies used by search engines, in particular, when they are partially vertically integrated.

1 3 Xu, Chen, and Whinston (2010, 2011) and Hagiu and Jullien (2011) analyze related models. 1 4 Selling through a for-pro…t platform may also a¤ect sellers' investment incentives. As Hagiu (2009) shows in a model in which sellers make their participation and non-contractible investment decision prior to buyers' participation decision, a platform who cannot commit to a membership fee on the user side before sellers make their investment decision wants to use a strictly positive royalty rate as an additional instrument.

quality and e¤ort than the seller himself), to investigate the e¤ect of minimum quality standards and other measures invoked by the platform to increase seller quality, one needs to study the interplay of asymmetric information, network e¤ects, and market power. Absent asymmetric information, reducing seller competition makes the platform less attractive for buyers, everything else given. However, with asymmetric information, removing underperforming sellers may actually make the platform more attractive for buyers, since the expected quality that will be consumed is increased. Thus, for example, security checks for apps on GooglePlay or certi…cation by Apple may be in the mutual interest of platform and buyers.

Buyers are often heterogeneous with respect to seller quality. For instance, some buyers may require fast delivery, whereas others care less about delivery speed. In this case, a platform may establish two market segments for sellers, in one segment they have to promise a certain quality, while in the other they do not. For example, Amazon hosts premium sellers with guaranteed fast delivery and standard sellers without that guarantee.

Price structure and product variety on oligopolistic platforms

We now consider situations in which several platforms compete to attract buyers and sellers. As in the previous section, we …rst draw a number of insights from a baseline model (which directly extends our previous model). We then enlarge the analysis by reviewing the existing literature.

Extension 1 of the baseline model

We de…ne the seller and buyer net surplus of visiting platform i (gross of any opportunity cost) as we did before:

v i s = r s + n i b ; n i s m i s and v i b = r b + u n i b ; n i s m i b
(the superscript refers to the platform). The general functions n i b ; n i s and u n i b ; n i s , which represent the net gains from trade for any seller and any buyer on platform i, are supposed to be twice continuously di¤erentiable in their two arguments. Buyers and sellers are uniformly distributed on the unit interval. Following Belle ‡amme and Toulemonde (2016), we consider the two-sided singlehoming model and identify the indi¤erent seller and buyer in the standard Hotelling fashion. A buyer of type x b incurs a disutility of b x b when visiting platform 1 and of b (1 x b ) when visiting platform 2; similarly, on the seller side with parameter s applied to a seller of type x s . The numbers of sellers and buyers at platform i can be expressed as:

n i b = 1 2 + 1 2 b u n i b ; n i s 1 2 b (m i b m j b ); n i s = 1 2 + 1 2 s n i b ; n i s 1 2 s (m i s m j s ); (2)
where

u n i b ; n i s u n i b ; n i s u 1 n i b ; 1 n i s ; n i b ; n i s n i b ; n i s 1 n i b ; 1 n i s :
Let us introduce the following notation:

u b @[ u n i b ; n i s ] @n i b ; u s @[ u n i b ; n i s ] @n i s ; b @[ n i b ; n i s ] @n i b ; s @[ n i b ; n i s ] @n i s :
In words, the function u n i b ; n i s measures the di¤erential in buyers'net gains from trade between platforms i and j when there are n i b buyers and n i s sellers on platform i.

= n 2 s = n 1 b = n 2 b = 1 2
at the symmetric equilibrium, it is then possible to show that the platforms set the following membership fees at the symmetric equilibrium of the game:

m s = f s + s 1 2 u s ( 1 2 ; 1 2 ) + s ( 1 2 ; 1 2 ) ; m b = f b + b 1 2 u b ( 1 2 ; 1 2 ) + b ( 1 2 ; 1 2 ) :
We observe that the equilibrium membership fees depend on the nature and strength of the within-and cross-group external e¤ects. In the complete absence of external e¤ects within and across groups, fees would be as in the Hotelling model. The presence of positive cross-group external e¤ects from, say, sellers to buyers leads platforms to lower the membership fee for sellers below the level that would prevail absent any external e¤ect. By contrast, the presence of negative cross-group external e¤ects from, say, sellers to buyers leads platforms to raise the membership fee for sellers above the level that would prevail absent any external e¤ect. This is the standard result of Armstrong (2006).

We add here a result related to the presence of external e¤ects within groups. Positive external e¤ects within groups leads platforms to lower the membership fee for the group below the level that would prevail absent any external e¤ect. Negative external e¤ects within groups leads platforms to raise the membership fee for the group above the level that would prevail absent any external e¤ect. [START_REF] Dukes | Negotiations and Exclusivity Contracts for Advertising[END_REF] 1 5 It is assumed that the functions u and are such that the system (2) leads to a unique solution n i b ; n i s 2 (0; 1) 2 , which is well-behaved in the sense that both n i b and n i s are decreasing functions of (m i b m j b ) and (m i s m j s ). To make use of examples 1 and 3, n i s should be interpreted as the fraction of the total number of sellers; that is, in the examples we have to scale up n i s . 1 6 To con…rm these statements, we recover the results of Armstrong (2006) We return to the special case in which (n b ; n s ) = n b ~ (n s ) and u (n b ; n s ) = n s ũ(n s ). Then, u b (1=2; 1=2) = 0, u s (1=2; 1=2) = 2ũ(1=2) + ũ0 (1=2), b (1=2; 1=2) = 2~ (1=2), and s (1=2; 1=2) = ~ 0 (1=2). Hence,

m s = f s + s ~ (1=2) 1 2 ~ 0 (1=2); m b = f b + b ũ(1=2) 1 2 ũ0 (1=2):
Since an additional seller reduces the pro…t per seller ( 0 < 0), one may think that increased seller competition (captured by less taste for variety in Example 2) has a mitigating e¤ect on the membership fee set on the seller side. However, one has to be careful with such reasoning. Take Example 2, which features that product prices are not a¤ected by the number of sellers. Here, ~ (1=2) + 1 2 ~ 0 (1=2) = 0 and, thus, a change in the taste for variety (in the example, measured by parameter ) does not a¤ect the membership fee on the seller side-it does a¤ect the membership fee on the consumer side. [START_REF] Edelman | Price Coherence and Excessive Intermediation[END_REF]

Additional issues with competing platforms

We now examine other issues by reviewing a number of papers that consider competition both between platforms and between sellers on the platforms.

Competitive bottlenecks. Hagiu (2009) considers competition among sellers in the competitive bottleneck world-more speci…cally, sellers multihome and buyers singlehome. Consumers have a preference for variety, which turns out to be a key factor determining the optimal pricing structure of competing platforms. Seller competition adds another force for lower prices on the consumer side in the seminal competitive bottleneck model of Armstrong (2006). [START_REF] Eliaz | A Simple Model of Search Engine Pricing[END_REF] Lowering the membership fee on the consumer side, makes also some sellers to withdraw from the competing platform. This further reduces the attractiveness of the competing platform (as u is increasing in n s ).

Consider now a decrease in the intensity of seller competition in the form of consumers' stronger taste for variety (e.g., as in Examples 1 and 2). Then, a given reduction of a platform's membership fee becomes less e¤ective in steering sellers away from the competing platform. This reduces the pressure to reduce the price on the buyer side. 1 7 Example 1 needs to be slightly modi…ed to …t with the present setting (we take here n i b and n i s as continuous values in the [0; 1] interval whereas we took them before as discrete values ranging from 0 to in…nity). Belle ‡amme and Toulemonde (2016) propose the following modi…cation. They assume that before choosing which platform to visit, buyers and sellers draw independently their location from a uniform distribution on the unit interval. Consequently, they consider n i b and n i s as the expected shares of buyers and sellers that decide to interact on platform i. They show that increased competition (captured by an increase in the degree of product substitutability ) reduces the buyers'equilibrium surplus, has ambiguous e¤ects on the sellers'equilibrium surplus, and reduces the platforms'equilibrium pro…ts. 1 8 Belle ‡amme and Peitz (2018) show under which conditions platforms do indeed set lower prices for the singlehoming side (here buyers) when sellers are allowed to multihome.

Competing search engines. As pointed out in the previous section, search engines trade o¤ participation with rent extraction from sellers (they do not set prices on the buyer side). An increased bias reduces the attractiveness of the search engine and, hence, reduces buyer participation, but it increases the expected rent of the preferred seller (gross of the payment to the platform) and, thus, allows the platform to make higher pro…t per unit mass of buyers. If competition between platforms increases the sensitivity by which buyers do not use a particular search engine, the quality reduction as the result of search engine bias becomes more "costly" for the platform. Therefore, we would expect that under competition platforms have weaker incentives to bias their search results. Eliaz and Spiegler (2011) show indeed that, compared to a monopoly search engine, competing search engines set a price-per-click that induces only the highest quality …rms to enter the search pools; as a result, search quality is maximized, which clearly bene…ts buyers.

Bargaining between sellers and platforms. Quite di¤erent from other works, Dukes and Gal-Or (2003) propose a bargaining model between platforms and participants on one side of the market. This requires that the number of participants on one side of the market is small. They consider competition between for-pro…t ad-…nanced media platforms, which sign exclusivity contracts with advertisers. More speci…cally, two sellers with horizontally di¤erentiated products post ads on di¤erentiated platforms to inform consumers about the existence of their productsprice and advertising competition between sellers is modeled as in Grossman and Shapiro (1984). Consumer preferences with respect to products are independent of their preferences with respect to media platform content. [START_REF] Ellison | Knife-Edge or Plateau: When Do Market Models Tip?[END_REF] Platforms obtain revenues only from advertisers. Advertising rates are set following a Nash-in-Nash approach, and price and advertising decisions are made concurrently.

In the informative advertising setting of Grossman and Shapiro, higher equilibrium levels of advertising can reduce advertisers'gross pro…ts, since this increases the share of consumers informed about competing o¤ers and, thus, makes price competition in the product market more intense. As media platforms take a cut in the advertisers' pro…ts, their pro…ts decline as well. When platforms provide exclusive advertising (i.e., no-compete clauses in advertising), consumers are less informed about competing o¤ers in the product market, and price competition in the product market is relaxed.

The number of platforms with positive volumes of trade

An important issue is the impact that within-side external e¤ects may have on the coexistence of competing two-sided platforms. Positive cross-group e¤ects generate positive feedback loops that may lead to situations where only one platform survives at equilibrium ('winner-takes-all') unless competing platforms are su¢ ciently di¤erentiated. We may conjecture, however, that seller competition may contribute to break the feedback loop and, thereby, facilitate the coexistence of competing platforms, even in the absence of di¤erentiation.

A case in point are industry standards; for example, the modem standard for Internet standards in the 1990s: Augereau, Greenstein, and Rysman (2006) point out that two di¤erent, but functionally equivalent, modem standards were used by Internet Service Providers (ISPs, which would be the sellers in our model) despite positive e¤ects of opting for a single standard and that the market seemed to have settled on this outcome-the two modem standards obtained similar market shares. By adopting di¤erent standards, ISPs created switching costs for consumers and, thus, were better able to reduce competition among them.

In what follows, we …rst analyze the impact that seller competition has on the coexistence of platforms in a setting that builds on our previous baseline model. We then review other models proposed in the literature.

Extension 2 of the baseline model

We sketch here an extended (and modi…ed) version of our baseline model that allows us to discuss how sellers may relax competition among them by trading on di¤erent platforms, thereby allowing these platforms to coexist even though, di¤erent from the baseline model in the previous section, buyers are sellers are homogeneous. Recall the speci…cation of seller and buyer net surplus with one-sided pricing:

v i s = r s + n i b ~ (n i s ) m i s and v i b = r b + n i s ũ(n i s )
. Di¤erent from the previous exposition, we assume that sellers and buyers are homogeneous-i.e., from the buyers' and the sellers' perspective, there is no exogenous di¤erentiation between platforms. Karle, Peitz and Reisinger (2017) analyze the two-sided singlehoming model (that is, each buyer and each seller joins exactly one platform) in which buyers observe product o¤ering on a platform only after having visited the platform. Platforms …rst set access fees on the seller side, and, then, buyers and sellers simultaneously decide which platform to select as their home-sellers can only sell to buyers on the same platform. Subsequently sellers set product prices and buyers make purchase decisions.

If all sellers co-locate on the same platform then, in equilibrium, all buyers will be active on this platform. Thus, there is agglomeration in equilibrium and network e¤ects are fully exploited. In such an equilibrium, all platform pro…ts are competed away and both platforms make zero pro…ts. Such a situation will necessarily emerge absent competition between sellers.

However, imperfect competition between sellers may lead to the equilibrium in which both platforms have a positive number of users and make positive pro…ts in equilibrium. [START_REF] Ellison | Competing Auctions[END_REF] Suppose that there are two sellers that have to decide whether to join platform 1, join platform 2 or not to participate at all. If they both join the same platform, they obtain duopoly pro…t d per buyer, which is less than the monopoly pro…t m per buyer that they would obtain if they were the only seller on the platform. If d = m is su¢ ciently small, there is an equilibrium in which sellers list on di¤erent platforms. Buyers are indi¤erent between the two platforms-some will join platform 1 and the others platform 2. Pro…ts are not competed away: platforms can extract the full seller surplus in equilibrium (remember that sellers are homogeneous). [START_REF] Gabszewicz | Vertical Product Di¤erentiation and Two-Sided Markets[END_REF]

Within-group external e¤ects and platform coexistence

In the previous model (as in much of our analysis so far), platforms are operated by for-pro…t intermediaries, which choose price-and non-price strategies in view of internalizing (within-and cross-group) external e¤ects. However, there exist environments in which some or all of the marketplaces buyers and sellers can choose to join are open. Although trade on such marketplaces is not intermediated, it is nevertheless interesting to analyze how seller competition a¤ects the potential coexistence of these marketplaces. We examine this issue …rst when all marketplaces are open and next, when a 'sponsored'platform competes with an open marketplace. Finally, we return to the competition between 'sponsored'platforms and discuss how within-group external e¤ects may facilitate the coexistence of platforms by being a source of endogenous platform di¤erentiation.

Trade on multiple market places. The issue as to whether market activities agglomerate or segment connects to an older literature on location decisions of sellers, which has analyzed the bene…ts and costs of clustering. Here, marketplaces are not actively managed and neither buyers nor sellers are charged for participation and trade on the platform.

There are typically two opposite forces at work. On the one hand, competing sellers prefer to locate in di¤erent market places, so as to relax competition-a segmentation force. On the other hand, sellers realize that a marketplace where many of them locate will attract many buyers, as it will, in expectations, drive prices down and o¤er a better …t-an agglomeration force. Depending on the setting, either force may dominate. If the segmentation force is stronger, multiple marketplace carry positive volumes of sales. [START_REF] Galeotti | Platform Intermediation in a Market for Di¤erentiated Products[END_REF] If agglomeration prevails, all sellers and buyers trade on a unique market place. [START_REF] Gehrig | Competing Markets[END_REF] Platform entry. Belle ‡amme and Toulemonde (2009) examine the extent to which negative within-group e¤ects among sellers may help a new platform operator lure buyers and sellers away from an existing marketplace. In their model, only the new platform can set membership fees; 2 1 As an extreme example, consider pure Bertrand competition between sellers. In this case d is equal to zero, and sellers have an incentive to avoid competition by locating at di¤erent platforms as long as platforms charge weakly less than m =2. In equilibrium, sellers segment despite the fact that platforms are ex ante homogeneous for both buyers and sellers and make pro…t m =2 from each seller.

2 2 Ellison, Fudenberg, and Möbius (2004) consider competition between two auction sites. They derive conditions for sellers not to agglomerate on one site. The rationale is that segmentation lowers the seller-buyer ratio on each platform and, thus, leads to higher prices. Ellison and Fudenberg (2003) provide general conditions such that tipping does not occur and multiple marketplaces coexist at equilibrium. 2 3 For example, Gehrig (1998) shows that, under some conditions, buyers and sellers agglomerate on a single open marketplace even though buyers are heterogeneous with respect to the preferred location of the marketplace. See also Stahl (1982), Dudey (1990), and Church and Gandal (1992) for related analyses.

this is not a model of price competition between platform, but it is not a monopoly model either, as the existing marketplace provides buyers and sellers with an endogenous outside option. As in Caillaud and Jullien (2003), the new platform faces a 'chicken-and-egg'problem, which it tries to solve by using a divide-and-conquer pricing strategy; that is, the platform must subsidize the participation of one side (divide) and hope to recoup the loss through the membership fee it sets on the other side (conquer). The question is whether the platform can make any pro…t with such strategy. The answer is 'yes'when the interaction among buyers and sellers only generates (positive) cross-group external e¤ects. However, the presence of negative within-group e¤ects among sellers (e.g., because they o¤er substitutable products) blurs the picture. Competition among sellers turns out to be a mixed blessing for the new platform. The upside is that the sellers'willingness to pay to join the new platform increases if only a few of them make the move; as a consequence, sellers are less sensitive to buyers'participation to the new platform, which alleviates the 'chicken-and-egg'problem. Yet, the downside is that it will be more costly for the new platform to attract buyers if only a small subset of the sellers join. The balance between the two e¤ects depends on the relative strength of the within-group e¤ects (with respect to the cross-group e¤ects). There may be situations in which entry is not pro…table.

Within-group external e¤ects as endogenous platform di¤erentiation. On matching platforms (job, dating or real-estate platforms), the external e¤ects across groups are clearly positive, as users in one group are more likely to …nd an attractive match if the platform attracts more users of the other group. Yet, negative external e¤ects also exist within each group, as a user in one group is less likely to be accepted by her chosen match if there are more users in her own group. Absent these negative within-group e¤ects (and absent su¢ cient horizontal di¤erentiation), the market for matching platforms would tend to a winner-takes-all situation, as all users would be happy to join the platform that attracts the largest set of users of the other group-a self-reinforcing process. But, the existence of negative within-group e¤ects generates a form of endogenous vertical di¤erentiation, which may allow competing platforms to coexist. The idea is simple: one platform would choose to be more expensive, so as to reduce participation and the intensity of competition within each group; conversely, the other platform would choose to be cheaper, so as to raise participation and the intensity of the positive cross-group e¤ects. Clearly, for this situation to emerge at equilibrium, there must exist users who value su¢ ciently being isolated from competition in their own group, with respect to the prospect of …nding a large number of potential matches. This is the assumption that Halaburda, Piskorksi and Yilidrim (2017) make. They consider a matching market with heterogeneous agents who di¤er in the value of their outside option (i.e., the utility they receive if they remain unmatched). Agents with low outside options prefer a platform with restricted choice; because they fear more being unmatched, they su¤er more from the competition within their own group. The reverse applies to users with high outside options, who prefer then a platform where choice is not restricted. It follows that platforms of di¤erent size can be sustained in equilibrium, leading to endogenous di¤erentiation between platforms. In particular, the platform that restricts choices is able to charge a premium to its users, which compensates for the smaller participation. [START_REF] Goos | Platform Pricing in Matching Markets[END_REF]

Conclusion and future research

In this paper, we examined how two-sided platforms manage the external e¤ects that users exert on other users in their own group (so-called 'within-group external e¤ects'). In particular, we have focused on platforms that intermediate between sellers and buyers, with sellers competing with one another (such that a negative within-group external e¤ect is present). In such situations, platforms face a trade-o¤ when attracting an additional seller, as this contributes (other things being equal) to raise the buyers'willingness to participate but to reduce the sellers'willingness to participate. It is thus of interest to understand how competition within one group of users a¤ects platforms'decisions, as well as the structure of markets with platforms.

To get some intuition, we developed a simple model throughout the paper. In this model, we assume positive cross-group e¤ects between buyers and sellers, negative within-group e¤ects among sellers, and no external e¤ect among buyers. Starting with a monopoly platform, we showed that a key variable to assess how the intensity of seller competition a¤ects the platform is the total value generated in a buyer-seller transaction: if this value decreases with the intensity of seller competition, then so does the platform's maximal pro…t. This is the case, e.g., if sellers produce horizontally di¤erentiated products and consumers have a taste for variety; here, a lower degree of di¤erentiation intensi…es seller competition and decreases both the buyers'and the sellers'surplus per transaction. We then extended the model to two competing platforms and showed that the presence of negative external e¤ects within a particular group leads platforms to raise the membership fee for that group above the level that would prevail absent any external e¤ect. Finally, we used the extended model to show that negative within-group e¤ects in one group may facilitate the coexistence of platforms with positive volumes of trade.

We complemented the intuition drawn from our simple model with a review of existing literature on the topic. This allowed us to consider a number of additional issues (e.g., price discrimination, non price strategies, entry of platforms). In terms of policy, it is not easy to draw clear-cut lessons given the variety of issues that are studied and the speci…cities of the models that are used. However, two words of caution clearly emerge. First, negative within-group external e¤ects should not be neglected, as they may deeply impact the platforms'conduct and, thereby, the structure of markets with platforms. Second, the analysis needs to be done carefully and on a case-by-case basis; the reason is that seller competition a¤ects, in a potentially complex way, how sellers and buyers value the interaction with each other. [START_REF] Grossman | Informative Advertising with Di¤erentiated Products[END_REF] Another conclusion that emerges from the literature review is that a number of important issues still have to be addressed. Within-group external e¤ects are usually abstracted away in the literature on multi-sided platforms; yet, this literature informs policy (e.g., antitrust) decisions. It is thus worthwhile to check if the recommendations drawn from the literature that abstracts from within-group external e¤ects are still valid for environments in which negative within-group external e¤ects are present (as on most platforms that intermediate between buyers and sellers).

To do so, the present analysis should be extended in a number of directions. Arguably, in many markets with platforms, users di¤er not only in their opportunity costs of participating to a platform but also in their bene…ts from using the platform. One important direction for future research is to evaluate the impacts of seller competition (or more generally of negative withingroup external e¤ects) in settings that allow for heterogeneity in usage bene…ts and feature usage fees. [START_REF] Hagiu | Two-Sided Platforms: Product Variety and Pricing Structures[END_REF] Of particular interest, also, is the interplay between platforms'price and non-price strategies in the presence of seller competition. For instance, in a recent paper (Belle ‡amme and Peitz, 2018), we assess how platforms, buyers and sellers are a¤ected when (non-competing) sellers have the possibility to multihome. We show that the perceived wisdom (according to which platforms and buyers should bene…t from this possibility, but sellers should not) is not always correct. Platforms may prefer to prevent sellers from multihoming, which would then hurt buyers. We conjecture that this conclusion is even more likely in environments with competing sellers. Competing sellers are indeed more willing to accept exclusivity agreements, as this can serve as a coordination device for them to split across platforms and, thereby, relax competition among them. This would make exclusive agreements more pro…table for platforms (as sellers are more willing to accept them) and even less desirable for buyers (on top of paying higher fees to platforms, they would face sellers that are less numerous and have more market power). These are important insights for antitrust authorities that examine whether exclusivity agreements between platforms and sellers should be allowed or not.

Similarly, one may want to investigate platforms' incentives to provide buyers with …rstparty content, which is a substitute to the third-party content that sellers provide. In the existing literature (see, e.g., Hagiu and Spulber, 2013), third-party sellers are not competing. The main trade-o¤ for platforms is then the following: …rst-party content helps attracting buyers and, thereby, sellers; yet, for a given number of buyers, …rst-party content steals pro…ts from third-party sellers. Now, if sellers not only compete with the platform's integrated content but also among themselves (as is, e.g., the case for third-party sellers on Amazon MarketPlace), novel issues arise. For instance, platforms may want to limit competition by third parties so as to protect their …rst-party sales. An example could be Apple which has been restrictive to "certify"sellers of accessories. Other platforms appear to be happy to accept more sellers (e.g., Amazon)-this may be a strategy to keep existing or potential competitors at bay and may also be a way for Amazon to learn about demand. [START_REF] Hagiu | Why do Intermediaries Divert Search?[END_REF] Recall that even if platforms do not produce their own content, they may have incentives to reduce seller competition; we have indeed shown in Sections 2 and 3 that platforms may su¤er from increased seller competition. The additional insight here is that such incentives may even be reinforced when platforms compete themselves with the sellers. Platforms could then increase the horizontal di¤erentiation between their …rstparty content and the sellers'third-party content; other things being equal, such a move should make buyers better o¤ if they value product variety. Di¤erentiation may also be with respect to the quality of service; for instance, …rst-party content (physical goods) may be delivered more quickly or return policies may be more generous. Sellers may react to the provision of …rstparty content, possibly also by adjusting their investment. A pro…t-maximizing platform must consider such issues when deciding whether to produce …rst-party content. [START_REF] Hagiu | First-Party Content and Coordination in Two-Sided Markets[END_REF] Finally, the analysis could be extended by looking deeper into the decisions that competing sellers make and that are relevant for their interaction with buyers on a platform. In our microfoundations of the buyer-seller relationships, we used simple oligopoly models, in which sellers only choose the price or the quantity of their goods. It would be interesting to consider richer models with multi-dimensional and sequential strategies. For instance, in Belle ‡amme and Peitz (2010), we study seller's incentives to invest in cost-reducing (or quality-enhancing) technologies when their trade with buyers is intermediated by platforms. This analysis is performed under the assumption that sellers o¤er independent products. However, if sellers o¤er substitutes, their investments become strategic if they are made before prices or quantities are chosen. Compared to non-strategic sellers (for instance, non-competing ones), competing sellers tend to invest more in cost-reducing technologies when the subsequent competition is in quantities and less with subsequent price competition. Insofar as cost-reducing investments by sellers increase the surplus that both, sellers and buyers, obtain from transactions, and insofar as platforms compete more …ercely for buyers and sellers when transactions are more valuable, we conjecture that more seller competition should hurt competing platforms when sellers compete in quantities, but should bene…t platforms when sellers compete in prices.

As a …nal disclaimer, all the conjectures that we just made should be properly examined, as should other issues that we did not mention here.

Appendix

Cournot example

Suppose also that each buyer has the following quadratic utility function: U (q 0 ; q 1 ; q 2 ; : : : ; q ns ) = a ns X k=1 q k 1 2

0 @ ns X k=1 q 2 k + ns X k=1 X g6 =k q k q g 1 A + q 0 ;
where q 0 is the Hicksian composite commodity (with a price normalized to 1), n s is the number of sellers on the platform and, hence, the number of varieties of the di¤erentiated good that the buyer has access to, and 0 < 1 measures the strength of the substitutability among varieties (varieties are homogeneous for = 1 and tend to be independent for ! 0). The buyer maximizes her utility U (q 0 ; q 1 ; q 2 ; : : : ; q n ) subject to the budget constraint y = q 0 + P ns k=1 p k q k , which gives rise to the following inverse demand functions p k = a q k q k (with q k = P g6 =k q g ) for strictly positive prices and zero otherwise. [START_REF] Halaburda | Competing by Restricting Choice: The Case of Search Platforms[END_REF] Seller k chooses its quantity q k to maximize n b (a c q k q k ) q k . The …rst-order condition yields 2q k = a c q k . Summing over the n s sellers and writing Q for P ns k=1 q k , one has 2Q = n s (a c) (n s 1) Q. Solving for Q, one obtains Q = n s (a c) =(2 + (n s 1)). By symmetry, each seller produces the same quantity at the Cournot-Nash equilibrium: q = Q=n s . It is easily found that, at equilibrium, p k c = q. Setting, without loss of generality, a c = 1, we compute the equilibrium pro…t for each seller as The consumer surplus is found by plugging the equilibrium prices and quantities into U ( ) P ns k=1 p k q k : u (n b ; n s ) = n s ũ(n s ) = n s (1 + (n s 1)) 2 (2 + (n s 1))2 :

Comparative statics

We establish here the comparative statics results presented in Section 2. For the second exercise, we look for the signs of the following expressions:

sign dn b d = sign ( b ) with b @ 2
@n b @ @ 2 @n 2 s + @ 2 @n s @ @ 2 @n b @n s ; sign dn s d = sign ( s ) with s @ 2 @n s @ @ 2 @n 2 b + @ 2 @n b @ @ 2 @n b @n s :

First, we develop the various derivatives of the platform's pro…t (using the facts that in our examples, the buyer surplus is only a function of n s and the seller surplus is a linear function of n b ):

@ 2 @n b @ = @u (n b ; n s ; ) @ + @ 2 (n b ; n s ; ) @ @n b n s ; @ 2 @n s @ = @ (n b ; n s ; ) @ + @ 2 u (n b ; n s ; ) @ @n s n b + @ 2 (n b ; n s ; ) @ @n s n s ; @ 2 @n 2 b = 2;

@ 2 @n 2 s = 2 @ (n b ; n s ; ) @n s 2 + @ 2 u (n b ; n s ; ) @n 2 s n b + @ 2 (n b ; n s ; ) @n 2 s n s ; @ 2 @n b @n s = @u (n b ; n s ; ) @n s + @ (n b ; n s ; ) @n b + @ 2 (n b ; n s ; ) @n b @n s n s :

In Example 1, we compute b = (2 2 (3 )n 3 s + (42 33 +5 2 )n 2 s +4(18+10 2 26

3 )ns [START_REF] Belle ‡amme | Negative Intra-Group Externalities in Twosided Markets[END_REF](2 )(3 ))n b 4(2+ (ns 1)) 6 ns(ns 1)(4+ (ns 1)) (2+ (ns 1))

:

To show that b < 0, note that the numerator of the …rst fraction is an increasing function of n s ; it reaches thus its lowest value for n s = 1, where it is equal to 8 (3 2 ) > 0. As the second fraction is positive, the whole term is negative. We proceed in the exact same way to show that s < 0; here, the numerator of the …rst fraction evaluated at n s = 1 is equal to 128n b > 0.

In Example 2, we compute It is easily seen that both b and s are negative for n s 3 (as ln n s > in this case). For n s = 1; 2, a su¢ ciently large value of c guarantees that b ; s < 0.

In sum, we have shown that dn s =d < 0 and dn b =d < 0 in both examples.

  s + (n b ; n s ) m s and n b = r b + u (n b ; n s ) m b . We postulate that there is a unique solution to this system of equations, n s (m s ; m b ) and n b (m s ; m b ), which imposes further restrictions on u and . From the system of equations, we have m s = r s + (n b ; n s ) n s and m b = r b +u (n b ; n s ) n b . The platform's pro…t is = n b (m b f b )+ n s (m s f s ). In light of the buyers'and sellers'participation decisions, the platform's pro…t maximization problem can be written as max n b ;ns n b [r b

1 1

 1 The number of sellers is determined by ms = rs + ~ (ns) n b . With two-sided pricing, the platform maximizes maxn b ;ns n b [r b n b f b ] + ns[rs ns fs] + [ũ(ns) + ~ (ns)]nsn b . If the platform can only charge sellers the platform's problem is maxn s n b f b + ns[rs ns fs + ~ (ns)n b ] subject to n b = r b + ns ũ(ns).

i b 1 , u = u 2n i s 1 ,

 11 by setting n i b ; n i s = n i b and u n i b ; n i s = un i s (i.e., cross-group e¤ects are positive and linear and within-group e¤ects are nil). We have then = 2n b = 2 , u s = 2u, and s = u b = 0. It follows that m s = cs + s u and m b = c b + b .

  (n b ; n s ) = n b ~ (n s ) = n b 1 (2 + (n s 1)) 2 :

1 . 2 (n 1 2 sRecall also that 0 < 1 ,

 1221 Let us recall the surplus functions derived in the two examples: (Example 1) u (n s ) = n s (1 + (n s 1)) 2 (2 + (n s 1)) 2 and (n b ; n s ) = n b 1 (2 + (n s 1)) 2 : Hence, ũ(n s ; ) + ~ (n s ; ) = 3 + (n s 1) 2 (2 + (n s 1)) Example 2) u (n s ) = n s E c and (n b ; n s ) = n b (1 ) E n s Hence, ũ(n s ; ) + ~ (n s ; ) = and 0 < < 1. Our …rst exercise aims at identifying the sign of d d = n s n b @ @ (ũ(n s ; ) + ~ (n s ; )) , with = f ; ; g .

2 s(Enb( 1 )(ln ns+2 1)+2n 2 s 2 s (2n 2 s 2 s ( 1 )( 2 ( 1 )n b ln ns+ 2 ( 1 )E+2 2 n b) +2c 2 2 n b n 2 s c 2 2 n 3 s

 2122221212123 En b (1 )(1 +ln ns)+cn 1+ s (ln ns ))+c 2 n +E(ln ns )+cn 1+ s :

  which depend on the combination of the four fees.[START_REF] Dudey | Competition by Choice: The E¤ect of Consumer Search on Firm Location Decisions[END_REF] Using implicit di¤erentiation and taking advantage of the fact that n 1 s

		The derivatives
	u b and u s measures the sensitivity of this di¤erential to a change in the mass of, respec-
	tively, buyers or sellers on platform i; the function	n i b ; n i s and derivatives b n i b ; n i s and
	s n i b ; n i s are de…ned accordingly for sellers.
	The system of equations (2) implicitly determines the demand functions for platform i,
	n i b (m i b ; m i s ; m j b ; m j s ) and n i s (m i b ; m i s ; m j b ; m j s ),

By contrast, there also exist market environments with positive within-group e¤ects; for instance, Belle ‡amme, Omrani and[START_REF] Belle ‡amme | The Economics of Crowdfunding Platforms[END_REF] explain that a larger 'crowd' of funders on a crowdfunding platform increases the

Another example, in which sellers compete on a Salop circle, can be found in Lin, Wu and Zhou (2016).

We suppose here that there is a …nite number of discrete sellers and then ignore the integer constrains. For the corresponding version of this model with a continuum of sellers, see, e.g.,[START_REF] Ottaviano | Monopolistic Competition, Multiproduct Firms and Product Diversity[END_REF].

This example is taken from Nocke, Peitz, and[START_REF] Nocke | Platform Ownership[END_REF].

This has also been pointed out by[START_REF] Hagiu | Two-Sided Platforms: Product Variety and Pricing Structures[END_REF].

For an analysis with general distribution functions, see[START_REF] Hagiu | Two-Sided Platforms: Product Variety and Pricing Structures[END_REF]. See also Goos, Van Cayseele and Willekens (2013) for a related analysis of a monopoly platform matching two groups, in the presence of positive cross-group and negative within-group external e¤ects.

[START_REF] Belle ‡amme | Who Bene…ts from Increased Competition among Sellers on B2C Platforms?[END_REF] Consumers are implicitly assumed to be ignorant about surplus in the product market when deciding about how much time to spend on the two media platforms; they only experience advertising as a nuisance when consuming platform content.

0 As Karle,[START_REF] Karle | Segmentation versus Agglomeration: Competition between Platforms with Competitive Sellers[END_REF] show, under some conditions, such a segmentation equilibrium also arises when sellers can multihome and some but not all buyers multihome.

[START_REF] Augereau | Coordination versus Di¤erentiation in a Standards War: 56K Modems[END_REF] See also[START_REF] Damiano | Competing Matchmaking[END_REF],[START_REF] Ambrus | Asymmetric Networks in Two-sided Markets[END_REF], and Gabszewicz and Wauthy (2014) for contributions in which asymmetric platforms coexist and the asymmetry is endogenously determined by user choices.2 5 Only in special cases is the buyer surplus from a transaction not a¤ected by the degree of seller competition.Such a special case is Example 2 with = 1=2, as in this case u is independent of the number of participating sellers. Comparative statics, for example, in the sellers' …xed cost then has no bearing on the buyers' per-seller surplus.

[START_REF] Belle ‡amme | Platform Competition and Seller Investment Incentives[END_REF] See, e.g.,[START_REF] Rochet | Cooperation among Competitors: Some Economics of Payment Card Associations[END_REF] or[START_REF] Edelman | Price Coherence and Excessive Intermediation[END_REF] for such settings.2 7 For an empirical investigation, see[START_REF] Zhu | Competing with Complementors: An Empirical Look at Amazon.com[END_REF].

[START_REF] Belle ‡amme | Negative Intra-Group Externalities in Twosided Markets[END_REF] For an empirical investigation of the video game industry, see[START_REF] Cennamo | Value Co-creation and Capture in Platform Markets: Evidence from a Creative Industry[END_REF]. For a general discussion, see[START_REF] Zhu | Friends or Foes? Examining Platform Owners'Entry into Complementors' Spaces[END_REF].

[START_REF] Belle ‡amme | Who Bene…ts from Increased Competition among Sellers on B2C Platforms?[END_REF] Income y has to be su¢ ciently large relative to the maximal number of sellers Z that can be active since we must have that y > Zpq; y > Z(a c)(a + c + c (Z 1))=[2 + (Z 1)] 2 .

gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG) through CRC TR 224.