Jan 18, 2016 - When Apple launched its first iPhone in 2007, the market responded with ... outlet, a video-games console, or a new web-site for e-commerce) ... of the platform's product on the opposite side (for example, a developer with.
Platform Pricing under Dispersed Information Bruno Jullien
Alessandro Pavan
Toulouse School of Economics (CNRS)
Northwestern University and CEPR
January 2016
Abstract We study platform pricing in a two-sided market with dispersed information about users’ preferences. First, we show how the dispersion of information introduces idiosyncratic uncertainty about participation rates and how the latter shapes the elasticity of the demands and thereby the equilibrium prices. We then study informative advertising campaigns a¤ecting the agents’ ability to estimate their own as well as other agents’valuations, and platform design a¤ecting the distribution of valuations on the two sides of the market. JEL classi…cation: D82 Keywords: two-sided markets, dispersed information, platform competition, global-games, informative advertising
An earlier version circulated under the title "Platform Competition under Dispersed Information." For useful comments and suggestions, we thank Philipp Kircher, anonymous referees, Mark Armstrong, Emilio Calvano, Marco Ottaviani, Michael Riordan, Patrick Rey, Jean Tirole, Glen Weyl and seminar participants at Berkeley Haas, Bocconi University, the 2013 CSIO-IDEI Conference, Universidad Carlos III of Madrid, NYU Stern, the 2013 SAET Conference, the 2013 Berlin IO Day, the 2014 Econometric Society European Meetings, and the 2015 Shanghai Microeconomics workshop. Pavan also thanks the National Science Foundation (grant SES-60031281) and Nokia for …nancial support and the University of California at Berkeley for hospitality during the 2011-1012 academic year. Emil Temnyalov and Jonas Mishara Blomberger provided excellent research assistance.
1
Introduction
When Apple launched its …rst iPhone in 2007, the market responded with hesitation: developers were uncertain about the attractiveness for end-users of the phone’s touch-screen feature, while consumers were uncertain about the potential for new applications. While subsequent versions of the iPhone were better received, new versions of Windows smart phones continue to be welcome with skepticism.1 More generally, the launch of new platforms (e.g., a new smart-phone operating system, a media outlet, a video-games console, or a new web-site for e-commerce) comes with uncertainty about the platform’s success in attracting agents from various sides of the market. For instance, Lucking-Reiley and Spulber (2001) report that only 15% of online trading exchanges funded by venture capital prior to 2000 were active by mid-2000.2 In the face of such uncertainty, market participants typically hold heterogenous beliefs about the success of a new platform, an issue that has been ignored by the literature on two-sided markets.3 In fact, the assumption commonly made in the analysis of such markets is that preferences on each side are commonly known, so that each agent from each side can perfectly predict the participation decisions of all other agents. This property, while a convenient shortcut, does not square well with most markets, especially those in which the product, or service, o¤ered by one (or multiple) platforms is relatively new. In this paper, we develop a tractable, yet rich, model of platforms’pricing that explicitly accounts for the heterogeneity of agents’beliefs about the success of a new platform. We then use the model to study how such dispersion of beliefs shapes the equilibrium prices on each side. Finally, we examine the platforms’incentives to change the information available to each side through informative advertising and/or marketing campaigns, as well as their incentives to invest in platform design so as to change the way their product is likely to be perceived relative to those o¤ered by the competitors. In a two-sided market, agents’participation decisions depend on two dimensions. First, agents typically derive a direct utility from a platform’s product, often referred to as "stand-alone valuation".4 This utility may originate from the one-sided goods or services that the platform bundles with its two-sided services (see e.g., Edelman 2015), or from preassigned agents on the opposite side (e.g., vertically integrated suppliers –see Lee, 2013; or marquee buyers, see Rochet and Tirole, 2003). 1
An issue faced by Microsoft in the smart-phone market is the self-defeating nature of agents’ expectations (see,
e.g., "Microsoft Banks on Mobile Apps", by Shira Ovide and Ian Sherr, April 5, 2012, The Wall-Street Journal). 2 While uncertainty is perhaps most pronounced for certain Internet platforms, it also surrounds other platform industries. For instance, in the market for personal computers, many of Microsoft new OS releases were received with signi…cant hesitations (e.g., Windows 3, Windows 95, Windows 7). 3 See Rochet and Tirole (2006), Rysman (2009), and Weyl (2012) for excellent overviews of the developments of this literature. 4 Other expressions favored in the literature are "intrinsic bene…t" — e.g., Armstrong and Wright (2007) — and "membership bene…t" — e.g., Weyl (2010). In the baseline version of the model, we assume agents know their own stand-alone valuations. However, in an extension, we relax such assumption to capture the possibility that agents may be uncertain about which products and services best serve their needs.
1
In addition, agents derive an indirect utility from interacting with other agents from the opposite side of the market, often referred to as "network e¤ ects".5 To capture the uncertainty mentioned above, we consider a model in which agents are uncertain about the distribution of stand-alone valuations in the population at the time they must choose which platform to join.6 Each agent then uses his own stand-alone valuations to form beliefs about the distribution of valuations in the population. For example, a developer of a new application may use his or her own appreciation for the features of a new smart-phone, or video game console, to form beliefs about the success of the platform in attracting end-users. Likewise, a consumer may use his or her appreciation for the features of the new smart phone to form beliefs about the potential for new applications or games. Naturally, such beliefs may depend on past experiences as well as other individual factors. For this reason, we do not impose that beliefs be derived from a common prior. We allow for the possibility that agents with a higher stand-alone valuation may expect a higher or a lower appreciation of the platform’s product on the opposite side (for example, a developer with a high appreciation for the sophistication of a new smart-phone operating system may expect fewer end-users to appreciate the new system when these latter users are known to favor dimensions such as the simplicity of the core tasks over the sophistication of the code). For simplicity, but also to isolate the novel e¤ects brought in by the dispersion of information, we consider markets in which all agents from the same side attach the same value to interacting with agents from the opposite side. However, because di¤erent agents hold di¤erent beliefs about the platform’s success on the opposite side of the market, the model de facto accommodates for a particular form of heterogeneity in the estimated network e¤ects. To understand the consequence of this heterogeneity, we build on the global-game literature7 by assuming that the dispersion of beliefs is such that, under regularity conditions, the equilibrium demand functions are unique (thus avoiding the usual "chicken and egg" problem of many two-sided markets emphasized in Caillaud and Jullien (2003)).8 Insights for pricing. When information is dispersed, a platform must anticipate, when choosing its prices, both the stand-alone valuation of the marginal agents on each side (the marginal agents 5
Other expressions favored in the literature are "usage value" — e.g., Rochet and Tirole (2006) — "cross-side
externality" — e.g., Armstrong (2006) — and "interaction bene…t" — e.g., Weyl, (2010). Importantly, while in the baseline version of the model, we abstract from within-side externalities, in one of the extensions, we consider the possibility that agents on one or multiple sides of the market may also experience a network externality from their own side’s participation, as in the case of advertisers competing for viewers’ attention (negative externality) or software developers gaining from the popularity of an operating system in the IT community (positive externality). 6 The baseline version of the model abstracts from the possibility that agents revise their choices over time. In one of the extensions, however, we consider the possibility that agents change platforms as their information becomes more precise. 7 See, among others, Carlsson and Van Damme (1993), and Morris and Shin (2003). 8 See Weyl (2010) and White and Weyl (2012) for a discussion of how platforms can control for coordination issues among sides by o¤ering two-part "insulating tari¤s" where the price on each side is a function of the participation rate on the other side. As we show in one of the extensions, equilibria in insulating tari¤s do not exist in our model.
2
are those who are indi¤erent between joining one platform or the other) and the marginal agents’ beliefs about the other side’s participation. This leads to two major di¤erences with respect to the complete information case, both related to the fact that the beliefs of the marginal agents regarding the participation of the opposite side di¤er from the platform’s own beliefs. The …rst di¤erence concerns the platform’s ability to leverage on a higher participation on one side by inducing either a higher participation, or a higher price, on the opposite side. Under complete information, this e¤ect is known to contribute to lower prices (see, e.g., Rochet and Tirole, 2003, 2006, or Armstrong, 2006). Under dispersed information, a platform’s ability to leverage on a higher participation on side i to raise its price on side j naturally depends on the side-i’s price elasticity expected by the side j’s marginal consumer vis-a-vis the elasticity expected by the platform. The second di¤erence is that, as the platform changes its price on side i, the side i’s marginal agent, and hence the beliefs of the marginal agent about the side j’s participation, also change. Dispersed information thus induces a speci…c form of correlation between the agents’ stand-alone valuations and their estimated network e¤ects. Importantly, the endogeneity of such correlation has novel implications for the elasticity of the demands expected by the platforms that di¤er from those obtained by assuming an exogenous correlation structure within each side. To illustrate, suppose that agents with a higher stand-alone valuation for a platform’s product expect, on average, a higher appreciation from the opposite side. Then suppose that a platform were to raise its price on, say, side i. Because the marginal agent from side i who is excluded is the most "pessimistic" about the side-j’s participation, among those who would join the platform without the price change, the drop in the side-i’s demand expected by the platform is smaller than in a market where all agents share the same beliefs about side-j’s participation (as is necessarily the case under complete information). In this case, the dispersion of information thus reduces the own-price elasticity of the demand functions. As a result, the equilibrium price on each side increases with the intensity of that side’s network e¤ects, an e¤ect that is absent under complete information.9 Despite the complexity of the strategic interactions, the model yields a relatively simple and intuitive formula for the equilibrium prices when competition between platforms is symmetric (in the sense that platforms set equal prices and expect equal market shares). Implications for advertising, marketing, and platform design. The results have natural implications for the platforms’incentives to change the information available to the agents, possibly through advertising and/or marketing campaigns, but also other information disclosures aimed at a¤ecting the agents’ beliefs, such as forums10 and trial periods11 . We study such implications by specializing the analysis to markets in which stand-alone valuations are drawn from a common prior 9
In fact, under complete-information, the equilibrium price on each side is invariant in that side’s intensity of network
e¤ects, as longs as the importance that each agent assigns to such network e¤ects is independent of the agent’s own stand-alone valuation, as assumed in this paper as well as in Armstrong (2006) and Rochet and Tirole (2006). 10 See Godes & al. (2005) for a discussion of the role of social interactions in marketing. 11 See Haruvy and Prasad (2005) and Gaudeul (2010) for analysis of freewares and sharewares.
3
and then enriching the model by allowing agents to face uncertainty about their own stand-alone valuations in addition to the uncertainty they face about the distribution of valuations on the opposite side of the market. While most marketing campaigns operate simultaneously over various dimensions and alter both the agents’perception of their own stand-alone valuations as well as the agents’beliefs about other agents’participation, we …nd it useful to isolate the various e¤ects by distinguishing between campaigns that alter the agents’ability to estimate their own stand-alone valuations and campaigns that alter the agents’ability to predict the participation by the other side. We show that campaigns that increase the agents’ability to estimate their own stand-alone valuations always increase pro…ts. This is because such campaigns, by increasing the sensitivity of the demands to the agents’idiosyncratic information increase the ex-ante degree of di¤erentiation between the platforms (equivalently, they reduce the elasticity of the residual demands), thus softening competition.12 According to this result, a platform entering the market with a new product should promote online discussion forums aimed at facilitating consumers re…ne their idiosyncratic valuations for the platform’s product. Whether such forums should also be opened to the other side is more subtle. We show that helping each side predict the participation decisions on the opposite side of the market increases pro…ts if and only if stand-alone valuations are positively a¢ liated across the two sides. In this case, the own-price elasticity of each side’s demand decreases with either side’s ability to estimate the stand-alone valuations on the opposite side. Because the e¤ects of information on the own-price elasticities prevail over their e¤ects on the cross-price elasticities, improving the ability of each side to predict the participation decisions of the opposite side contributes positively to pro…ts. In contrast, the reverse conclusion holds when stand-alone valuations are negatively a¢ liated across the two sides for, in this case, the own-price elasticity of each side’s demand increases with either side’s ability to forecast the stand-alone valuations on the opposite side. We also investigate how equilibrium pro…ts change with variations in the prior distribution from which stand-alone valuations are drawn. These comparative statics, contrary to the ones pertaining to the quality of information, are meant to shed light on a platform’s incentives to di¤erentiate its product from the competitors’, without knowing the exact distribution of preferences on either side of the market. For instance, we show that raising the similarity with an opponent’s product always reduces the equilibrium pro…ts by intensifying competition. Similarly, aligning the preferences of the two sides by favoring dimensions that are expected to be appealing to both sides increases pro…ts as long as the sum of the network e¤ects is positive. For instance, it may be more pro…table for a new e-commerce platform to align the interests of buyers and sellers by focusing on a niche in which agents from either side share similar preferences than to aim at getting on board a broader base of undi¤erentiated users on each side of the market. 12
A related result can be found in Anderson and Renault (2010) in the contest of an ex-ante symmetric one-sided
market.
4
Outline. The rest of the paper is organized as follows. We wrap up the Introduction with a brief discussion of the contribution of the paper to the pertinent literature. Section 2 then presents the model. Section 3 introduces some benchmarks that facilitate the subsequent analysis. Section 4 characterizes optimal prices under dispersed information. Section 5 contains implications for advertising and platform design. Section 6 examines a few extensions (dynamics, within-side externalities, and competition in insulating tari¤s). Section 7 o¤ers a few concluding remarks. All proofs are in the Appendix. Some material omitted in the main text can be found in the paper’s Supplement. (Most) pertinent literature. Our model is the incomplete-information analog of the models of di¤erentiated platforms with homogenous network e¤ects studied in Rochet and Tirole (2003, 2006), and Armstrong (2006). Weyl (2010) and White and Weyl (2012) extend this model by allowing for heterogenous network e¤ects, focusing respectively on a monopolistic platform and a duopoly platform (see also Veiga and Weyl (2011) and Ambrus and Argenziano (2009) who were the …rst to introduce heterogenous network e¤ects and show how the latter can lead to asymmetric networks). These papers, however, do not consider the possibility that preferences are correlated across sides, which is one of the key forces behind the mechanism we consider in the present paper. We thus identify a new channel by which the dispersion of information a¤ects the elasticity of the demands on the two sides and thereby the equilibrium prices. This in turn permits us to uncover novel e¤ects. For example, under complete information, it is the discrepancy between the importance assigned to network e¤ects by the marginal user and by the average user that is responsible for distortions in prices and in network allocations, along the lines of those identi…ed in Spence (1975) and Sheshinski (1976) – see Weyl (2010). In contrast, under dispersed information, it is the discrepancy between the participation rates expected by the marginal agent on each side and the participation rates expected by the platforms that is responsible for novel distortions. Our paper focuses on dispersed information at the subscription stage pertaining preferences in the population. In contrast, Halaburda and Yehezkel (2013) examine optimal tari¤s when agents privately learn their valuations only after joining a platform but before transacting with the other side. Hagiu and Halaburda (2014) focus on information about prices, showing that unlike in the monopoly case, in a duopoly market, platforms may prefer agents to remain uninformed about the prices charged on the opposite side. While these papers address very di¤erent questions, they both point to the importance of asymmetric information for platforms’pricing decisions. The paper also contributes to the literature on coordination under incomplete information, and in particular the global-games literature. A di¤erence though is that, in our work, the dispersion of information is a central part of the phenomenon under examination, as opposed to a convenient tool to arrive at equilibrium selection. To the best of our knowledge, this is the …rst paper to examine a global game in which two distinct populations (the two sides) coordinate under dispersed information and where the outcome of such coordination is shaped by two competing "big players" 5
(the platforms). The paper in this literature closest to ours is Argenziano (2008) which uses a globalgame approach to study one-sided network competition. The questions addressed in that paper relate to allocative e¢ ciency under product di¤erentiation and di¤ers from those addressed in the present paper which are largely motivated by the two-sideness of the problem under examination. Finally, the paper contributes to the literature that studies informative advertising and marketing campaigns (for recent contributions see, among others, Anderson and Renault (2006, 2009), Johnson and Myatt (2006) and the references therein). Our results about the e¤ects on pro…ts of campaigns that help the agents understand their own needs and preferences are in the same spirit as those established in this literature. The main contribution is to analyze the e¤ects of campaigns that help agents predict other agents’preferences and behavior, which is new and brings novel implications.
2
Model
Players. Two platforms, indexed by k = A; B, compete on two sides, i = 1; 2: Each side is populated by a measure-one continuum of agents, indexed by l 2 [0; 1]:
Actions and payo¤s. Each agent l from each side i must choose which platform to join. The
payo¤ that each agent l from each side i derives from joining each platform k is given by k i mj
Uilk = vilk +
pki
where vilk is the agent’s stand-alone valuation for platform k (think of this as the direct utility the agent derives from the platform’s product), mkj 2 [0; 1] is the mass of agents from side j 6= i that
join the platform; side i; and
pki
i
2 R+ is a parameter that controls for the intensity of the network e¤ ects on
is the price (the access fee) charged by the platform to side i:13 As in most of the
literature, we abstract from price discrimination. By paying the fee pki , the agent is granted access to the platform’s product and to all agents joining the same platform from the opposite side of the market. The payo¤ that each agent l from each side i obtains from not joining any platform (that is, the agent’s outside option) is assumed to be equal to zero. Each platform’s payo¤ is the total revenue collected from the two sides:14 k
= pk1 mk1 + pk2 mk2 :
All players are risk-neutral expected-utility maximizers. 13
To ease the exposition, we assume positive network e¤ects on both sides. All results extend to markets in which
network e¤ects are negative on one side (as in certain advertsining models) provided that
1+ 2
> 0; which, empirically,
appears the most relevant case. The results in the …rst four sections extend to markets in which the results in Section 5 depend on whether
1+
2
> 0 or
1+
2
1
+
2
< 0. Instead,
< 0. We comment on the di¤erences in that section.
Also note that it is only to ease the exposition that we assume that the intensity of the network e¤ects is the same across platforms. The results extend to the case in which each ik is platform-speci…c. 14 All results extend to markets in which the platforms incur costs to provide access to the agents. Because these costs do not play any fundamental role, we disregard them to facilitate the exposition.
6
We focus on equilibria with full participation and singlehoming. To maintain the analysis as simple as possible, we then assume that each agent’s stand-alone valuations for the products of the two platforms are given by vilA = si
vil vil and vilB = si + all l 2 [0; 1]; i = 1; 2: 2 2
This speci…cation is chosen so that the "type" of each agent is unidimensional and coincides with the di¤erential in stand-alone valuations vil = vilB
vilA , i =; 1; 2, which simpli…es the analysis. The
scalar si 2 R, which we assume is commonly known to the platforms and to all agents, only serves
the role of guaranteeing that full participation is robust to the possibility that agents opt out of the market (see the discussion in the Supplementary Material). Beliefs. We assume that the distribution of the agents’stand-alone valuations on the two sides of the market (hereafter, "the aggregate state") is unknown both to the platforms and to the agents. Each agent’s di¤erential in stand-alone valuations vil thus parametrizes both the agent’s preferences and the agent’s beliefs about the aggregate state, which in turn he uses to form beliefs about the participation of the other side of the market. For simplicity, we assume that platforms do not possess private information. This assumption permits us to abstract from the signaling role of prices (and the associated multiplicity of equilibria), and focus on the novel e¤ects that come from the agents’ idiosyncratic uncertainty about the participation of the other side of the market. For any v 2 R; any i = 1; 2; any k = A; B; we denote by
k (v) i
the measure of agents from side
i that platform k expects to have a di¤erential in stand-alone valuations no greater than v. The function
k i
thus represent the cumulative distribution function of platform k’marginal beliefs over
the side-i cross-sectional distribution of stand-alone valuations. We assume that each function strictly increasing and di¤erentiable over the entire real line and then denote by
k i (v)
d
k i
is
k (v)=dv i
its density. Next, consider the agents. For any i; j = 1; 2; j 6= i; any (vi ; vj ) 2 R2 , we denote by Mji (vj jvi )
the measure of agents from side j with di¤erential in stand-alone valuations no greater than vj , as expected by an agent from side i with di¤erential in stand-alone valuations equal to vi . These
functions are thus the cumulative distribution functions of the agents’ marginal beliefs over the cross-sectional distribution of di¤erential in stand-alone valuations on the other side of the market. We again assume that, for any vi ; Mji (vj jvi ) is strictly increasing in vj and di¤erentiable in each
argument, i; j = 1; 2; j 6= i. As for the dependence of Mji (vj jvi ) on vi ; we will be primarily interested in the following two cases:
De…nition 1 We say that stand-alone valuations are positively a¢ liated between the two sides if, for all vj 2 R, Mji (vj jvi ) is decreasing in vi , i; j = 1; 2; j 6= i. Conversely, we say that stand-alone
valuations are negatively a¢ liated if, for all vj 2 R, Mji (vj jvi ) is increasing in vi , i; j = 1; 2; j 6= i.
7
In markets in which stand-alone valuations are positively a¢ liated, agents with a higher appreciation for a platform’s product thus also expect a higher appreciation by the agents from the opposite side, whereas the opposite is true in markets in which stand-alone valuations are negatively a¢ liated. Importantly, note that the de…nition does not presume that stand-alone valuations are drawn from a common prior. It simply establishes a monotone relationship between beliefs and stand-alone valuations. Finally note that, while the above speci…cation assumes that agents face uncertainty about the aggregate distribution of stand-alone valuations for each of the two platforms’products, all the results extend to markets in which the stand-alone valuations for one of the two platforms’products (say, an incumbent) are commonly known. Common prior. As mentioned already, the above speci…cation does not impose that the agents’ and the platforms’beliefs be generated from a common prior. All the results are consistent with the possibility of multiple priors, which is perhaps the most relevant case empirically. To see how the special case of a common prior …ts into the framework introduced above, it is convenient to think of the following set-up. The aggregate state (that is, the cross-sectional joint distribution of stand-alone valuations on the two sides of the market) is parametrized by
2
. All players believe that
is
drawn from a distribution F : In each state ; each agent’s di¤erential in stand-alone valuations, vil , is drawn from a distribution
i,
with density
i
(v) ; independently across agents, i = 1; 2; l 2 [0; 1].
The mass of agents from side i with valuation no greater than v in state
is then equal to
i
(v) :
In this case, the platforms’and the agents’beliefs are given by k i (v)
=E [
i
(vi )] and Mji (vj jvi ) =
E [
j
(vj )
E [
i
i
(vi )]
(vi )]
where both expectations are computed under the common prior F :15 Timing. At stage 1, platforms simultaneously set prices on each side. At stage 2, after observing the platforms’ prices, and after observing their own stand-alone valuations, all agents simultaneously choose which platform to join. At stage 3, payo¤s are realized. As mentioned above, we focus on equilibria in which agents single-home and where no agent opts out of the market. In the baseline model we thus assume that all agents must join one of the 15
Note that, as is standard in the analysis of games of incomplete information, irrespective of whether or not beliefs
are consistent with a common prior, the mappings Mji ( jvi ) from types to beliefs are common knowledge. A common prior simply imposes restrictions on these mappings as well as on the platforms’ beliefs functions
i
k i:
Finally note that the
and Mji do not provide a complete description of the platforms’and of the agents’beliefs. However, as
we will show below, they summarize all the information that is relevant for the determination of the equilibrium prices and the agents’participation decisions.
8
two platforms and cannot join both. In the Supplementary Material, however, we identify conditions under which the equilibrium allocations in the baseline model are also equilibrium allocations in a more general game in which all agents can opt out of the market or multihome. Finally, hereafter, we say that a strategy pro…le for the agents constitutes a continuation equilibrium in the game that starts after the platforms announce their prices if each agent’s strategy is a best response to all other agents’strategies.
3
Benchmarks
As a warm-up (but also as a useful step toward the analysis in the subsequent sections), we start by considering two benchmarks. The …rst one is a market without network e¤ects. The second is a market with complete information. These benchmarks also serve the purpose of illustrating that the novel e¤ects we document below originate precisely in the combination of network e¤ects with dispersed information. Benchmark 1 (absence of network e¤ects). Suppose that
1
=
2
= 0: In this case, the
demand that each platform expects on each side is independent of the price it sets on the opposite side of the market. B Given the prices pA i and pi set by the two platforms on side i, each agent l from side i whose
di¤erential in stand-alone valuations, vil , exceeds the price di¤erential, pB i whereas each agent with di¤erential vil < pB i
pA i joins platform A. The (direct residual) demand
curve that platform A thus expects on each side i is given by QA i = expected by platform B is QB i =1
B (pB i i
pA i ; joins platform B,
A (pB i i
pA i ); while the one
pA i ): To be able to go from the direct demands to the
inverse demands, we introduce the functions ViA (Qi )
A i
1
(Qi ) and ViB (Qi )
B i
1
(1
Qi ):
Note that the function Vik (Qi ) identi…es the critical agent on side i such that, when all agents from side i join platform A when vil < Vik (Qi ) and platform B when vil > Vik (Qi ), the side-i participation expected by platform k = A; B is equal to Qi . Given this notation, the price that platform k needs to set to induce an expected demand Qi on side i is then the one that makes an agent with valuation B V A (Q ) for platform Vik (Qi ) indi¤erent between the two platforms. Such a price is equal to pA i i = pi i A B A and pB i = pi + Vi (Qi ) for platform B: The inverse semi-elasticity of the residual demand curve
expected by each of the two platforms is then equal to16 Qi
@pA i @Qi
= pB i =const
Qi and A (V A (Q)) i i
Qi
@pB i @Qi
= pA i =const
Qi : B (V B (Q)) i i
(1)
The expected participation rates under the pro…t-maximizing prices must then satisfy the familiar 16
This semi-elasticity is referred to as the market power in Weyl (2010).
9
optimality conditions pki + Qki
@pki @Qki
= 0: pi k =const
Equivalently, the equilibrium prices, along with the participation thresholds they induce, must satisfy the following conditions pki = where QA i (vi )
A (v ) i i
Qki (^ vi ) ; and v^i = pB i k (^ v ) i i
and QB i (vi )
B (v ) i i
1
pA i , i = 1; 2; k = A; B,
(2)
are the demands expected by the two platforms.
Note that these conditions are the familiar conditions for a duopoly equilibrium with horizontally di¤erentiated products. Clearly, in this benchmark, the dispersion of information is irrelevant. The prices set by the two platforms re‡ect only the platforms’own beliefs. Benchmark 2 (complete information). Next, reintroduce the network e¤ects, but assume information is complete, meaning that the cross-sectional distribution of stand-alone valuations is commonly known by the platforms and by all agents. The platforms’beliefs then coincide with the beliefs of any other agent. Formally, there exist functions k j (vj )
Mji (vj jvi ) =
=
j (vj )
j(
) such that
for all (vi ; vj ) 2 R2 ; i; j = 1; 2; j 6= i; k = A; B:
A Now suppose that platform A sets prices pA 1 and p2 on the two sides. Given that the sideB j participation to platform A cannot exceed 1, and given that mA j + mj = 1; any agent from side
i 6= j whose di¤erential in stand-alone valuations vi exceeds vi0 = pB i
pA i +
i
joins platform B,
irrespective of his beliefs about the other agents’ participation decisions. In turn, this means that any agent from side j should expect the participation of side i to platform A not to exceed Using again the fact that B for all vj
vj1 =
pB j
B mA i + mi = 1; we pA j + j (2 i (vi0 )
i (vi0 ).
then have that any such agent should join platform 1): But then each agent from side i should expect
the participation of side j to platform A to not exceed
j (vj1 )
< 1: Iterated deletion of strictly
dominated strategies then leads to a pair of thresholds v i and vi for each side i = 1; 2 such that it is iteratively dominant for each agent l from each side i to join platform B for vil > vi and to join platform A for vil < v i . These observations in turn suggest existence of a continuation equilibrium in threshold strategies whereby each agent l from each side i joins platform B if vil > v^i and platform A if vil < v^i . The thresholds v^i ; i = 1; 2; must satisfy v^i = pB i Note that the term 2
vj ) j (^
pA i +
i [2
vj ) j (^
1] ; i; j = 1; 2; j 6= i:
(3)
1 in the left-hand side of (3) is simply the di¤erential between the
side-j participation to platform A and the side-j participation to platform B. In the Appendix (proof of Lemma 1) we show that, as long as network e¤ects are not too large, for any vector of A B B prices p = (pA 1 ; p2 ; p1 ; p2 ) set by the two platforms, there exists one and only one solution to the
system of equations given by (3). Furthermore, when this is the case, the continuation equilibrium 10
is unique and hence so are the demand functions. Because of the one-to-one mapping from prices to participation thresholds, we write the demand functions as follows QA vi ) = i (^
vi ) i (^
and QB vi ) = 1 i (^
vi ); i (^
i = 1; 2;
(4)
where (^ v1 ; v^2 ) is the unique solution to the system of equations given by (3).17 Equivalently, for any B vector of participation rates QB = (QB 1 ; Q2 ); the prices that platform B must set on the two sides B to implement the participation rates QB 1 and Q2 are given by A B B B pB i = pi + Vi (Qi ) + 2 i Qj
where ViB (QB i )
(
i)
1
(1
i;
i; j = 1; 2; j 6= i:
(5)
QB i ): Platform B’s pro…ts (expressed as a function of the participation
B rates QB = (QB 1 ; Q2 )) are thus equal to B
=
X
B B B pA i + Vi (Qi ) + 2 i Qj
i
QB i :
(6)
i;j=1;2;j6=i
Combining (5) and (6), we have that the participation rates that maximize platform B’s pro…ts must solve the following optimality conditions pB i +
dViB (QB i ) B Qi + 2 j QB j = 0: dQi
(7)
Similar conditions apply to platform A: These conditions require that pro…ts not vary when a platform changes its price on side i and, at the same time, adjusts its price on side j so as to maintain the demand on side j constant. Equivalently, the platforms’ equilibrium prices under complete information must satisfy pki =
Qki k i (Vi
Qki )
2 j Qkj ;
(8)
where the demands Qki = Qki (^ vic ) are given by (4) and where the participation thresholds (^ v1c ; v^2c ) = V1k Qk1 ; V2k Qk2 , k = A; B; are the unique solution to (3): Naturally, under complete information, the participation rate expected by each platform on each side i coincides with the one expected by each agent from side j: The pro…t-maximizing prices then solve the familiar optimality conditions (8) according to which each platform’s price on each side equals the platform one-sided inverse semi-elasticity, Qki (^ vic )= i (^ vic ); adjusted by the e¤ect of a variation in the side-i’s participation on the side-j’s revenues (the term 2 j Qkj in (8)— see, for example, Weyl (2010)).18 17
The unique continuation equilibrium is also the unique rationalizable strategy pro…le. It thus does not require a
high ability to coordinate with other agents. This is appealing, especially in large markets, which are the focus of the paper. 18 The reason why, in a duopoly model, this last e¤ect is scaled by a factor of two is that, under single-homing and full market coverage, the bene…t of attracting more agents on board on side i combines the direct bene…t of providing the side-j agents with a larger user base on side i with the indirect e¤ect of reducing the attractiveness of the other platform.
11
4
Dispersed Information
We now turn to the interesting case where network e¤ects interact with dispersed information.
4.1
Preliminaries: Monopoly Analysis
As a …rst step toward the characterization of the duopoly equilibrium, but also as a convenient tool to introduce the key ideas in the simplest way, suppose for a moment that only platform B is present in the market. Each agent l 2 [0; 1] from each side i = 1; 2 then joins the platform if si +
vil + 2
B i E[mj
j vil ]
pB i
(9)
and abstains from joining if the above inequality is reversed (recall that si + vil =2 is the agent’s stand-alone valuation for platform B). Provided the network e¤ects are not too large, we then have, B for any vector of prices (pB 1 ; p2 ); the demand expected by the platform on each side i = 1; 2 is given
by QB vi ) = 1 i (^
B (^ i vi ),
where (^ v1 ; v^2 ) is the unique solution to the system of equations given by19
si +
v^i + 2
i [1
Mji (^ vj j v^i )] = pB i
i; j = 1; 2; j 6= i:
(10)
B Now suppose that the platform aims at getting on board QB 1 agents from side one and Q2 agents
from side two. Which prices should it set? Obviously, because the platform does not know the exact state of the world (i.e., the exact distribution of stand-alone valuations in the cross-section of the B population), the quantities QB 1 and Q2 must be interpreted as expected participation rates, where
the expectation is taken over all possible states of the world using the platform’s own beliefs. Given B the platform’s beliefs, the prices (pB ^1 1 ; p2 ) should be set so as to induce participation thresholds v B and v^2 satisfying v^i = ViB QB i ; where recall that Vi (Q)
B i
1
(1
Q); i = 1; 2: Using (10),
we then have that the prices should be set equal to pB i = si +
ViB (QB i ) + 2
i
B B Mji VjB (QB j ) jVi (Qi )
1
:
Di¤erences in beliefs between the platform and the agents then manifest themselves in the B B B B di¤erence between QB j and 1 Mji Vj (Qj ) jVi (Qi ) ; where the former is the side-j participation
rate expected by the platform and the latter is the side-j participation rate expected by the side-i marginal agent (the one who is just indi¤erent between joining and not joining the platform). What is interesting here is how incomplete information a¤ects the slope of the inverse demand function on each of the two sides of the market, which is given by @pB 1 dViB (QB i ) i = B 2 dQi @Qi 19
B B @Mji VjB (QB j ) j Vi (Qi ) dViB (QB i ) : i @vi dQi
(11)
The precise bound on the network e¤ects is teh same as the one identi…ed by Conditions (M) and (Q) below —
see the Supplementary Material (Lemma S1) for details.
12
Note that the …rst term in (11) coincides with the slope of the inverse demand curve that would obtain if all agents were to share the same beliefs as the platform’s, which is necessarily the case under complete information. The novel e¤ect is the one captured by the second term in (11). To understand this term, note that, as the side-i demand expected by the platform changes, the side-i’s marginal agent and hence the latter’s expectation of the side-j’s participation to the platform also changes according to @E[mB vi ] j j^ @^ vi
=
@[1
v^i =ViB (QB i )
Mji (^ vj j^ vi )] @^ vi
:
(12)
v^i =ViB (QB vj =VjB (QB i );^ j )
When network e¤ects are positive, as assumed here, and valuations are positively a¢ liated between the two sides, this novel e¤ect contributes to steeper inverse demand curves, whereas the opposite is true under negative a¢ liation.20 This is intuitive. When valuations are positively a¢ liated, the new marginal agent that the platform brings on board when it lowers its price on side i is less optimistic about the side-j’s participation than those agents who are already on board (the infra-marginal agents). This means that, to get the new marginal agent on board, the reduction in the side-i price must be larger than the one necessary to induce the same increase in the side-i participation under complete information. Interestingly, note that this novel e¤ect is present even if the platform adjusts its price on side j so as to maintain the side-j participation threshold v^j = VjB (QB j ) …xed (which amounts to maintaining the side-j’s demand QB j =1
B (^ j vj )
expected by the platform constant).
A noteworthy implication of this e¤ect of dispersed information on the slope of the inverse demand curves is that the side-i’s inverse demand curve naturally depends on the intensity
i
of
side-i’s own network e¤ects. In contrast, under complete information, the slope of the side-i inverse demand is independent of
i.
These novel e¤ects, of course, play an important role for the equilibrium
prices, as we show next. B and the participation thresholds Given the bijective relationship between the prices pB 1 ; p2
(^ v1 ; v^2 ) given by pB i = si +
v^i + 2
i [1
Mji (^ vj j v^i )];
(13)
B maximize the platform’s pro…ts if and only if the participation we have that the prices pB 1 ; p2
thresholds (^ v1 ; v^2 ) they induce solve the following problem: o X n vi max ^ B (v1 ; v2 ) si + + i [1 Mji (vj j vi )] 1 2 (v1 ;v2 )
B i (vi )
:
i;j=1;2;j6=i
We then have the platform’s pro…t-maximizing prices, along with the participation thresholds they induce, must satisfy the following optimality conditions for i; j = 1; 2; j 6= i, pB i = 20
QB vi ) i (^ B 2 i (^ vi )
@Mji (^ vj j v^i ) QB vi ) i (^ i B @vi vi ) i (^
Recall that dViB (QB i )=dQi is negative.
13
vj ) @Mij (^ vi j v^j ) QB j (^ j B @vi vi ) i (^
(14)
where QB vi ) = 1 i (^
B
(^ vi ) ; i = 1; 2; and where the participation thresholds v^1 and v^2 are given by
(10).21 Note that the price formulas in (14) are the incomplete-information analogs of the familiar complete-information optimality conditions @Pj (Q1 ; Q2 ) Qj : @Qi
@Pi (Q1 ; Q2 ) Qi @Qi
Pi =
(15)
They require that pro…ts do not change when the platform increases the expected side-i participation and then adjusts the side-j price to maintain the expected side-j participation constant. In particular, the …rst term in (14) is the familiar inverse semi-elasticity of the demand curve in the absence of network e¤ects; this term is completely standard and entirely driven by the platform’s beliefs over the distribution of the stand-alone valuations. The third-term in (14) captures the extra bene…t of cutting the price on side i that comes from the possibility to raise the price on side j. When side j bene…ts from the presence of side i; that is, when
j
> 0, as assumed here, this term is known to contribute negatively to the price charged by
the platform on side i (see, e.g., Rochet and Tirole, 2006 and Armstrong, 2006). The novelty, relative to complete information, comes again from the fact that the variation in the side-i demand that the platform expects to trigger by varying its side-i price now di¤ers from the variation expected by the side-j marginal agent. This novel e¤ect is captured by the term @E[mB ^j ] i j v B @Qi
= v^j =const
@E[mB ^j ] i j v @^ vi
v^j =const
@Mij (^ vi j v^j ) dViB (QB i ) = dQi @vi
1 B v ) i i (^
in (14), which measures the sensitivity of the beliefs of the side-j’s marginal agent to changes in the demand expected by the platform on side i.22 Note that this term is always positive, thus contributing to a lower price on side i: Interestingly, also note that, under a common prior, this term is the same across the two sides, @M12 (^ v1 j v^2 ) @v1
v1 ) 1 (^
=
@M21 (^ v2 j v^1 ) @v2
v2 ) 2 (^
=
E E [
v1 ) 1 (^ v1 )]E 1 (^
v2 ) 2 (^ [
v2 )] 2 (^
:
The second term in (14) is absent under complete information. As anticipated above, this term originates in the fact that a variation in the side-i demand now implies a variation in the side-i marginal agent and hence in the latter’s expectation about the side-j’s participation, as captured by the term @E[mB vi ] j j^ @QB i
21
= v^j =const
@E[mB vi ] dViB (QB j j^ i ) @^ vi dQi
= v^j =const
@Mji (^ vj j^ vi ) @^ vi
1 B v ) i i (^
:
These optimality conditions are simply the FOCs of the problem consisting in the maximization of the platform’s
pro…ts ^ B with respect to the participation thresholds. 22 Note that, under complete information, instead, @Mij (^ vi j v^j ) =@vi = in (14) reduces to
B vj ): j Qj (^
14
vi ) i (^
=
B vi ) i (^
in which case the third term
This term contributes to higher equilibrium prices under positive a¢ liation, because, as discussed above, the inverse demand curves steepen, and hence the platform’s incentives to lower the side-i price to accommodate additional participation are reduced. The opposite is true under negative a¢ liation. Once again, observe that this term is present despite the fact that the side-j demand expected by the platform (equivalently, the side-j participation threshold v^j = VjB (QB j )) does not change, given the adjustment in the side-j price.
4.2
Duopoly Analysis
We now reintroduce platform A and consider the duopoly game between the two platforms. As usual, we solve the game backwards by considering …rst the continuation game that starts after the A B B platforms announce the prices p = (pA 1 ; p2 ; p1 ; p2 ). Each agent l from each side i = 1; 2 then chooses
platform B when vil +
B i E[mj
B mA j j vil ] > pi
pA i
(16)
and platform A when the above inequality is reversed. Note that the left-hand side in (16) combines the di¤erential in stand-alone valuations with the di¤erential in the participation rates expected on B the other side of the market. Under full participation and singlehoming, mA i + mi = 1; i = 1; 2: B i E[2mj
We can then rewrite Condition (16) as vil +
1 j vil ] > pB i
pA i : Arguments similar to
those in the monopolist case then imply that, when network e¤ects are not too large (formally when Conditions (M) and (Q) below hold), iterated deletion of strictly dominated strategies identi…es unique thresholds v^i , i = 1; 2, such that, in the unique continuation equilibrium, each agent l from each side i = 1; 2 joins platform B if vil > v^i and platform A if vil < v^i . The thresholds (^ v1 ; v^2 ) solve the following indi¤erence conditions v^i
2 i Mji (^ vj j v^i ) +
i
= pB i
pA i
i; j = 1; 2; j 6= i:
(17)
Notice that the left-hand side of (17) is the gross payo¤ di¤erential of joining platform B relative to joining platform A for the marginal agent v^i on side i, when agents on both sides follow threshold strategies with respective cuto¤s v^1 and v^2 . As we show in the Appendix (Lemma 2), a solution to A B B the indi¤erence conditions in (17) exists and is unique, for any vector of prices p = (pA 1 ; p2 ; p1 ; p2 );
when the following two conditions jointly hold: Condition (M): For any i; j = 1; 2; j 6= i; any v1 ; v2 2 R; 1
2
i
@Mji (vj jvi ) > 0: @vi
Condition (Q). For any i; j = 1; 2; j 6= i; any v1 ; v2 2 R; 1 2
0, as assumed here. Contrary to complete information, though,
the impact of this e¤ect now depends on the ability of the side-j’s agents to forecast variations in the side-i demand. We summarize the implications of the above results in the following corollary:25 Corollary 1 Assume competition is symmetric. As in the complete-information case, equilibrium duopoly prices (i) increase with the inverse-semi-elasticity of the component of the demand that 24 25
Recall (8). It should be obvious that the conclusions in the corollary below cease to hold if beliefs about the distribution of
stand-alone valuations depend on the intensity of the network e¤ects. When this is the case, there need not be any speci…c relation between the intensity of the network e¤ects and the equilibrium prices.
18
comes from the stand-alone valuations and (ii) decrease with the intensity of the network e¤ ect from the opposite side. However, contrary to complete information, equilibrium prices under dispersed information increase with the intensity of the own-side network e¤ ects when stand-alone valuations are positively a¢ liated between the two sides, and decrease when they are negatively a¢ liated. As mentioned already, the above results pertain to a simpli…ed game in which agents singlehome (that is, they join only one platform) and cannot opt out of the market. In the Supplementary Material, we show that, under reasonable parameter con…gurations, the equilibrium allocations identi…ed above remain equilibrium allocations in richer environments in which agents can multihome and opt out of the market, provided that platforms cannot set negative prices.26
5
Advertising and platform design
Having characterized how the dispersion of information a¤ects the elasticities of the demands and the equilibrium prices, we now investigate the e¤ects on equilibrium pro…ts of variations in (i) the quality of the agents’information and (ii) the distribution from which stand-alone valuations are drawn. For simplicity, we focus on environments in which beliefs are consistent with a common prior, competition is symmetric, and information is Gaussian, as de…ned below. However, we generalize the analysis in the previous section by (a) allowing the agents to possess imperfect information about their own stand-alone valuations, and (b) by considering markets in which network e¤ects can be negative on one of the two sides (we continue to assume that
1
+
2
> 0, which appears the relevant case for
most markets of interest, but also discuss how the results change when
1+ 2
< 0). These extensions
permits us to uncover interesting comparative statics which we then use to shed light on the e¤ects of advertising campaigns and platform design.27
5.1
Gaussian information
Suppose that, on each side i = 1; 2, each agent l’s di¤erential in stand-alone valuation is given by vil =
i
+ "il with
( 1;
2)
drawn from a bivariate Normal distribution with mean (0; 0) and
variance-covariance matrix = with the parameter
"
( p
1)
1
1 2
p
(
1 2
2)
1
#
indexing the coe¢ cient of linear correlation between
1
and
2:
To capture the
possibility that agents may possess imperfect information about their own valuations, assume that each agent l from each side i = 1; 2 privately observes a signal xil = 26
i
+
il
that is only imperfectly
See Armstrong and Wright (2006), Bergemann and Bonatti (2011), Athey, Calvano and Gans (2012), and Ambrus,
Calvano and Reisingerx (2013) for models that allow for multihoming under complete information, and Amelio and Jullien (2010) for how the impossibility to set negative prices may lead to tying. 27 The reader interested in how similar campaigns a¤ect pro…ts in monopolistic markets is referred to the corresponding section in the Supplementary Material.
19
correlated with his stand-alone valuations: More precisely, assume that each ("il ;
il )
is drawn from
a bivariate Normal distribution with mean (0; 0) and variance-covariance matrix 3 2 ( i" ) 1 p "i i i 5 i =4 p "i ( i) 1 i
with the parameter ("il ;
il )l2[0;1];i2f0;1g
i
0 denoting the coe¢ cient of linear correlation between "i and
i
i:
The pairs
are drawn independently across agents and independently from :
The above structure is commonly known and the platforms do not possess any additional information. Now, given the signal xil , agent l from side i = 1; 2 believes that the di¤erential vil in his stand-alone valuations is Normally distributed with mean Vil
E [vil j xil ] =
Hereafter, we refer to Vil
i xil
with
cov[vil ; xil ] = var[xil ]
i
i
+
i i i
p
+
i
=
" i
:
(23)
i
E [vil j xil ] as to the estimated stand-alone di¤ erential. Notice that Vil
summarizes both the agent’s preferences over the platforms’products and the agent’s information. This means that the analysis in the previous section extends to the environment under examination here by replacing the agents’true di¤erentials in stand-alone valuations vil by their estimated values Vil . As we discuss below, allowing the estimated stand-alone di¤erentials to di¤er from the true values, while it does not change the equilibrium analysis, has important implications for the e¤ects of variations in the quality of the agents’information. Now let
denote the c.d.f. of the Standard Normal distribution (with density ). With an abuse
on notation, then denote by
k i
(V ) the measure of agents from side i that platform k = A; B expects
to have an estimated stand-alone di¤erential no greater than V . Likewise, denote by Mji (Vj j Vi )
the measure of agents from side j 6= i with di¤erential in estimated stand-alone valuations no greater than Vj , as expected by each agent from side i with estimated stand-alone di¤erential equal to Vi .
Using the properties of the Normal distribution, we then have that28 q q p k VV 2 and M (V j V ) = 1 + (V ) = ji j i i i
where
(
V i i 28
i
+
+ i i
i) p
i i i=
" i
2,
cov (V1l ; V2l0 )
p = var (V1l ) var (V2l0 )
V
s
V j Vj
q
V i Vi
(24)
1 2
(
1
+
1) ( 2
+
2)
(25)
To compute the formulas in (24) observe that, from an ex-ante standpoint (and hence also from the perspective
of each platform), each Vil , i = 1; 2; l 2 [0; 1]; is drawn from a Normal distribution with mean zero and variance var(Vil ) = 1=
V i
; whereas, from the perspective of any agent l 2 [0; 1] from side i = 1; 2 with estimated stand-alone
di¤erential equal to Vil , each Vjl0 , l0 2 [0; 1]; j 6= i; is drawn from a Normal distribution with …rst and second moments equal to
E [Vjl0
cov[Vjl0 ; Vil ] j Vil ] = Vil = var[Vil ]
V
s
V i V j
Vil and var [Vjl0 j Vil ] = var[Vjl0 ](1
respectively.
20
2 V
)=
2 V
1 V j
and
V i
Note that 1=
q 1
V 2 V
= var[Vil ] is the unconditional variance of the estimated stand-alone di¤erential of
each agent l 2 [0; 1] from side i,
V
is the coe¢ cient of linear correlation between the estimated
stand-alone di¤erentials of any pair of agents from opposite sides, and j j measures the ability of
each side to forecast the distribution of estimated stand-alone di¤erentials on the opposite side of the market. Hereafter, we refer to the term
as to the coe¢ cient of mutual forecastability.
It is then easy to see that, by varying the coe¢ cient terms ("il ;
il )
i
of correlation between the two idiosyncratic
while keeping all other parameters …xed, one can capture variations in the agents’
ability to estimate their own stand-alone valuations, holding …xed the agents’ ability to estimate the participation decisions on the other side of the market. Likewise, by varying by varying
V
(for example
) holding …xed all other parameters, one can capture variations in the agents’ability
to estimate the participation decisions on the other side of the market, holding …xed their ability to estimate their own stand-alone valuations. The following result then translates to the Gaussian environment under examination the general results derived in the preceding section: Proposition 3 Suppose that (a) competition is symmetric, (b) information is Gaussian, and (c) agents are only imperfectly informed about their own stand-alone valuations, as described above. The equilibrium prices that both platforms charge on each side are given by pi = where
V i
(0) =
p
var[Vil ] 2 (0)
V i
(0) +
i
j
p
1+
2
i; j = 1; 2; j 6= i;
(26)
is the inverse semi-elasticity of the component of the demand curve that
comes from the estimated stand-alone valuations and where
is the coe¢ cient of mutual forecasta-
bility between the two sides. Under a Gaussian information structure, the equilibrium price on each side thus depends on the details of the information structure only through the coe¢ cient of mutual forecastability sign of of
is the same as the sign of
: The
= corr(Vil ; Vjl0 ): It should thus not surprise that the sign
V
re‡ects whether estimated stand-alone di¤erentials are positively or negatively a¢ liated across
the two sides (equivalently, of whether each agent becomes more or less optimistic about the other side’s participation as his own appreciation for a platform’s product increases). As a result, the sign of
is what determines whether the equilibrium price pi on each side increases or decreases with
the intensity
i
of that side’s own network e¤ects (recall the discussion in the general model in the
previous section). In contrast, when it comes to the impact on equilibrium prices of the intensity of the network e¤ects
j
on the opposite side, what matters is only the square of
. To interpret this
result, observe that 1+
2
=
var[Vil0 ] : var[Vil0 E[Vil0 jVjl ]] 21
(27)
The right-hand side in (27) measures the ability of each side to forecast variations in participation decisions on the opposite side. The larger this ratio (i.e. the smaller the variance of the forecast error), the better each side is in forecasting variations in the participation of the opposite side: It is then natural that the sensitivity of the equilibrium price on side i to the intensity e¤ects on the opposite side depends on
only through
of the network
j
2.
The above properties also suggest that equilibrium prices need not be too sensitive to the speci…c way the information is distributed across the two sides. In particular, in the Gaussian case, any two information structures with the same coe¢ cient of mutual forecastability yield the same equilibrium prices.
We now discuss how the equilibrium prices depend on the various structural parameters of the model. First, observe that the inverse semi-elasticities of the components of the demand curves that come from the estimated stand-alone di¤erentials are proportional to the dispersion of the estimated stand alone di¤erentials. Then use (25) to observe that
Naturally,
V (0) i
p p " var[V ] + 1 i i il V i i= i = p : i (0) = 2 (0) ( i + i ) i i 2 (0) increases with the correlation
between the noise
i
il
in the agents’ information
and the idiosyncratic taste shock "il in the stand-alone di¤erentials: It is also easy to see that V (0) i
decreases with
" i;
this is because a higher
" i
implies a lower dispersion of idiosyncratic taste
shocks which increases competition between the platforms. On the other hand, non-monotone in
i
and in
precision of the prior about
i
. The non-monotonicity with respect to
i)
re‡ects the fact that a higher
i
i
V (0) i
is typically
(which parametrizes the
implies a lower dispersion of stand-
alone di¤erentials but also a higher precision of the agents’ information. Because the latter e¤ect makes the agents respond more to their information, it contributes to a higher dispersion of estimated di¤erentials. The non-monotonicity with respect to the precision
i
turn re‡ects the fact that, holding constant the correlation coe¢ cient
of the agents’ information in i;
a higher
i
implies a lower
covariance between the noise in the signals and the idiosyncratic taste shocks in the di¤erentials. Because a lower covariance between the noise in the signals and the taste shock in turn contributes to a lower sensitivity of estimated di¤erentials to the agents’signals, the net e¤ect of a higher
i
on
var[Vil ] is typically non-monotone. Next, consider the coe¢ cient formation of the coe¢ cient
V
of mutual forecastability. Recall that
is an increasing trans-
of linear correlation between the estimated stand-alone di¤erentials
on any pair of agents from opposite sides, which in turn determines the two sides’ability to forecast each other. The ability of side i to forecast the distribution of the estimated stand alone valuations on side j clearly increases as the noise in the side-j’s signals decreases (that is, as also increases with side-i’s ability to forecast the correlated taste shock 22
j
j
increases). It
in the side-j’s valuations,
which can be measured by (the inverse of) var[
E[ j jVil ]] =
j
Not surprisingly, the ability of side i to forecast
1
2 i
j
1
i
+
i
:
(28)
j
increases with j j and
i
; and decreases with
i.
Building on the above observations, we now investigate the platforms’incentives to take actions that a¤ect either (i) the agents’ ability to estimate their own stand-alone valuations as well as those of the agents from the opposite side (e.g., through informative advertising and/or marketing campaigns, as well as through personalized disclosures aimed at facilitating consumers’learning of their own personal match with the product’s characteristics), or (ii) the distributions from which the agents’ stand-alone valuations are drawn (e.g., through platform design). We examine each of the two channels separately.
5.2
Informative advertising campaigns
Think of a smart-phone company entering the market with a new operating system. The …rm must decide how much information to disclose to the public about the various features of its product. We think of these disclosures as a¤ecting both the developers’and the end-users’ability to estimate their own stand-alone valuations (both in absolute value and relative to the product provided by a rival incumbent …rm), as well as their ability to forecast the distribution of valuations on the other side of the market. A similar role is played by the release of a freeware version of a new software or of a new application. Formally, we think of these disclosure and advertising campaigns as a¤ecting the information available to the two sides of the market, for …xed distribution of the true stand-alone valuations. That is, …x the parameters (
1;
2;
;
" 1;
" 2)
de…ning the prior join distribution from which individual
stand-alone valuations are drawn and consider the e¤ects on pro…ts of variations in the agents’ability to estimate (i) their own stand-alone valuations (as measured by the inverse of the volatility of the forecast error var[vil
Vil ]), and (ii) the distribution of stand-alone valuations on the other side of
the market (as measured by the inverse of var[
E[ j jVil ]]). Hereafter, we isolate the e¤ects of the
j
variations in (i) by looking at changes in the coe¢ cient
i
of correlation between the noise
il
in the
agents’signals and the idiosyncratic taste shock "il . We then isolate the e¤ects of the variations in (ii) by looking at joint changes in (
i
; i )i=1;2 that leave var[vil
Vil ], i = 1; 2; …xed.
Proposition 4 Informative advertising and marketing campaigns that increase the agents’ability to estimate their own stand-alone valuations without a¤ ecting their ability to forecast the distribution of (true or estimated) stand-alone valuations on the other side of the market always increase pro…ts. Conversely, campaigns that increase the agents’ ability to forecast the distribution of (true or estimated) stand-alone valuations on the other side of the market without a¤ ecting their ability to estimate their own stand-alone valuations increase pro…ts if inequality is reversed. 23
> 0 and reduce pro…ts if the above
The …rst part of the result is intuitive. Campaigns that help agents understand their own needs and preferences, without a¤ecting their ability to forecast other agents’needs and preferences, make agents more responsive to their own idiosyncrasies. As such, these campaigns increase the ex-ante dispersion of estimated stand-alone valuations, thus increasing the inverse semi-price elasticity of the component of the demand on each side that comes from the stand-alone valuations. These campaigns are thus similar to those that increase the degree of di¤erentiation between platforms under complete information (e.g., Johnson and Myatt, 2006, Anderson and Renault, 2009). By reducing the intensity of the competition between the platforms, such campaigns unambiguously contribute to higher prices and hence to higher equilibrium pro…ts. Next, consider campaigns whose primary e¤ect is to make agents more informed about what is likely to be "hip" on the other side of the market (formally, that help agents better predict the distribution of true and estimated stand-alone valuations on the other side), for example through articles in the specialized press. As we show in the Appendix, these campaigns impact the coe¢ cient of mutual forecastability
; without a¤ecting the ex-ante dispersion of the estimated stand-alone
valuations, var[Vi ] (and hence the terms
V (0) i
in the equilibrium price formulas (26)). From (26),
one can then see that, depending on the intensity of the network e¤ects, such campaigns may either increase or decrease the equilibrium prices. To see the e¤ect of such campaigns on equilibrium pro…ts, then note that, in a symmetric equilibrium, the latter are given by 1n 1 = (p1 + p2 ) = 2 2 and are increasing in
V 1
(0) +
V 2
(0) + (
1+
p 1+
2)
2
o
;
(29)
: The e¤ect of such campaigns on equilibrium pro…ts is then determined by
whether increasing the agents’ ability to forecast the distribution of stand-alone valuations on the other side of the market (which, by (25), corresponds to an increase in the precisions
i
, i = 1; 2;
of the agents’information) increases or decreases the coe¢ cient of mutual forecastability
. As the
latter is increasing in the quality of the agents’information
1
and
2
positively a¢ liated between the two sides (that is, if and only if
if and only if preferences are > 0), we then have that the
e¤ect of such campaigns on equilibrium pro…ts is positive if and only if the a¢ liation in the agents’ preferences is positive.29 To better understand the result, consider the case in which preferences are positively a¢ liated between the two sides and network e¤ects are positive on each side. Recall that the term
i
in
the price formulas captures the e¤ect of the dispersion of information on side-i’s own-price elasticity. More precise information (that is, an increase in either
1
or
2)
makes the marginal agent on
each side more responsive to his private information (equivalently, to his estimated stand-alone di¤erential). Such e¤ect in turn contributes to a lower elasticity of the side i’s demand and hence to a higher side-i equilibrium price. 29
Notice that, if the sum of the network e¤ects were negative (that is, if
1
+
2
< 0) then the conclusion would be
reversed: The e¤ect of such compaigns on equilibrium pro…ts is positive if and only if a¢ liation is negative.:
24
At the same time, however, more precise information also contributes to a higher sensitivity of p the side-j demand to variations in the side-i price. These e¤ects are captured by the terms j 1 + 2 in the price equations. While the net e¤ect of more precise information on the side-i equilibrium price then depends on the relative strengths of the network e¤ects
1
and
2,
the net e¤ect on total pro…ts
is unambiguously positive. This is because any loss of pro…ts on one side is more than compensated by an increase in pro…ts on the opposite side, as one can see from (29). What is interesting about the results in the proposition is that they identify two fairly general channels through which information a¤ects pro…ts, without specifying the particular mechanics by which advertising and marketing campaigns operate. In reality, most campaigns operate through both channels. That is, they a¤ect both the agents’ ability to understand their own preferences and their ability to understand what agents from the opposite side are likely to …nd attractive. For instance, as mentioned in the Introduction, discussion forums are used in online marketing to exploit social interactions. By sharing personal experiences, consumers help each other learning about own preferences as well as expected interactions on the platform. Opening such forums to both sides (say customers and designers for a fashion website) may also help users predict the other side’s preferences. The results in the proposition then indicate that such campaigns unambiguously increase pro…ts in markets in which (i) preferences are positively a¢ liated between the two sides and (ii) the sum of the network e¤ects is positive. Finally note that the above results refer to informative campaigns. They do not apply to campaigns that distort the average perception the agents have about the quality di¤erential between the two platforms. These campaigns could be modelled in our framework by allowing the platforms to manipulate the mean of the distributions from which the signals are drawn. However, because in our environment platforms do not possess any private information and the agents are fully rational, the e¤ect of such campaigns on pro…ts is unambiguously negative. This is because each agent can always “undo” the manipulation by adjusting the interpretation of the information he receives. As discussed in the "signal-jamming" literature (e.g., Fudenberg and Tirole (1986)), platforms may then be trapped into a situation in which they invest resources in such campaigns, despite the fact that, in equilibrium, such campaigns have no e¤ect on the agents’decisions.
5.3
Platform design
We now turn to the e¤ects on equilibrium pro…ts of changes in the distribution from which the true stand-alone valuations are drawn. As anticipated in the Introduction, these e¤ects should be interpreted as the result of platform design. For example, an increase in
" 1
and/or
" 2
should be
interpreted as the choice (by one, or both platforms) to o¤er a product that is more similar to the one o¤ered by the rival (in which case the dispersion of stand-alone di¤erentials is smaller). An increase in , instead, should be interpreted as the choice to favor platform characteristics that are expected to appeal to both sides. An interesting new development along these lines is the design of a collaborative 25
architecture (see, e.g., Pisani and Verganti, 2008), e.g. the organization of the process allowing the users’ community to contribute to the design of the platform through interactive communication. We then have the following result: Proposition 5 Platform design that reduces the ex-ante dispersion of the stand-alone valuations (as captured by an increase in (
" 1;
" 2 ))
always decreases pro…ts. Conversely, an increase in the alignment
of the stand-alone valuations between the two sides (as captured by an increase in
) always increases
pro…ts. When platforms o¤er products that are less likely to be sensitive to idiosyncratic appreciations, they reduce the ex-ante dispersion of the stand-alone valuations. Not surprisingly, such forms of platform design contribute negatively to pro…ts, given that they lead to …ercer competition. The result pertaining to the decision to favor dimensions that appeal to both sides is less obvious. Observe from the price equation (26) that an increase in the alignment of preferences (which amounts to an increase in the coe¢ cient
of mutual forecastability) may increase prices on one side while
decreasing prices on the opposite side. To see this, consider a market in which network e¤ects are positive on both sides. A design that increases
reduces the own-price elasticity of the demand
on each side by making the marginal agent’s beliefs about participation decisions on the opposite side more sensitive to his private information. As discussed above, this e¤ect contributes to higher equilibrium prices on both sides. At the same time, a higher
also implies a higher sensitivity
of the beliefs of the marginal agent on the opposite side to variations in the side-i’s price, which contributes negatively to the equilibrium prices. While the net e¤ect on the equilibrium prices on each side depends on the importance that the two sides attach to interacting with one another, the proposition shows that the net e¤ect on total pro…ts is always unambiguously positive. The conclusion extends to markets in which network e¤ects are positive on one side but negative on the other, provided that
1+ 2
> 0: When, instead,
1+ 2
< 0, the conclusion is reversed: more
alignment in stand-alone valuations across the two sides contributes unambiguously to lower pro…ts.
6
Extensions
We consider three extensions. The …rst one is a stylized dynamic market in which agents revise their beliefs about the distribution of stand-alone valuations over time, for example by observing participation decisions in earlier periods. The second extension is a market in which cross-side network e¤ects interact with within-side network e¤ects (e.g., congestion). The last extension is a market in which platforms compete in two-part tari¤s (for example, insulating tari¤s).
6.1
Dynamics
Consider the following stylized dynamic market whose role is to shed light on how the results in the baseline model must be adapted to accommodate for the possibility that agents may revise their 26
decisions over time in response to the arrival of new information.30 To maintain the analysis as simple as possible, we assume that there are two periods, and allow agents to switch platforms only in the second period. We assume that switching costs are negligible and completely disregard them. The cross-sectional distribution of stand-alone valuations is constant over the two periods. Information is dispersed in the …rst period and all agents perfectly learn the aggregate state at the beginning of the second period (for example, but not necessarily, by observing the aggregate participation decisions in the …rst period31 ). Preferences are time-separable (i.e., the sum of the ‡ow payo¤s, with the latter de…ned as in the benchmark model) with no discounting.32 Finally, to simplify the exposition, we assume that beliefs are consistent with a common prior, so that
k (v ) i i
=
i (vi )
=E
i (vi )
;
k = A; B; i = 1; 2: Our prime interest is in demand adjustments when platforms expect agents to revise their beliefs about the distribution of stand-alone valuations at a higher frequency than the one at which the platforms adjust their prices.33 We capture such a possibility in the starkest possible way by assuming that, while agents learn the state perfectly in the second period, platforms maintain their prices constant over the two periods (the alternative case in which platforms adjust prices at the same frequency at which agents revise their beliefs is examined in the Supplementary Material). A B B Because switching costs are negligible, given any collection of prices p = (pA 1 ; p2 ; p1 ; p2 ); the
period-1 demands expected by the two platforms continue to be given by QA i = 1
vi ) i (^
vi ) i (^
and QB i =
with the thresholds (^ v1 ; v^2 ) given by (17). The period-2 demands in each state , instead,
are given by qiA ( ) =
i (vi )
vi
and qiB ( ) = 1 2
i
j (vj )
+
i
i (vi ),
= pB i
with the thresholds (v1 ; v2 ) solving34
pA i ;
i; j = 1; 2; j 6= i:
(30)
B Now consider platform A’s best response to platform B’s prices (pB 1 ; p2 ): As in the baseline
model, the latter can be described in terms of the period-1 participation thresholds (^ v1 ; v^2 ) that A the choice of (pA 1 ; p2 ) induces on each side of the market, and then noting that, in each state ;
the period-2 participation thresholds (v1 ; v2 ) are a deterministic function of the period-1 thresholds B (^ v1 ; v^2 ). More precisely, …xing (pB v1 ; v^2 ); any ; the period1 ; p2 ); for any pair of period-1 thresholds (^
2 participation thresholds vi (^ v1 ; v^2 ); i = 1; 2; are implicitly given by the solution to the system of 30
See also Mitchell and Skrzypacz (2006), and Cabral (2011) for dynamic models of network competition focusing,
however, on di¤erent issues. 31 The results extend to markets in which the resolution of uncertainty is more gradual. 32 Note that those agents who do not change platform must still renew their membership in the second period by paying to the platform the same participation fee they paid in the …rst period. 33 As in the benchamark model, we assume that agents face no uncertainty about their own stand-alone valuations. 34 We assume a unique solution to the system of demands given by (30). From the result in Lemma 1 in the Appendix, a su¢ cient condition for this to be the case is that, for any ; 1 2
0: j
where the inequality follows from positive a¢ liation. We conclude that d pi (0)=d
< 0, which
establishes the result. Proof of Proposition 8. Assume the following conditions hold, which are the analogs of Conditions (M) and (Q) in the baseline model: Condition (M-C). For any i; j = 1; 2; j 6= i; any v1 ; v2 ; vil 2 R; 1
2
i
@Mji (vj jvil ) @vil
2
i
@Mi (vi jvil ) > 0: @vil
Condition (Q-C). For any i; j = 1; 2; j 6= i; any vj 2 R; vi
2 i Mji (vj jvi )
2 i Mi (vi )
is strictly increasing in vi : Furthermore, for any v1 ; v2 2 R; h ih 1 2
v^j and platform A when vj < v^j and (b) all agents from his own side i to join platform B when vi
v^i and platform A when vi < v^i is strictly increasing in v^i : From the
A B B intermediate and implicit function theorems, this means that, given the prices p = (pA 1 ; p2 ; p1 ; p2 ),
for any v^j 2 R, there exists a unique solution v^i = %i (^ vj ) to the equation v^i +
i
+
2 i Mji (^ vj j^ vi )
i
2 i Mi (^ vi ) = pB i
pA i ;
with %i (^ vj ) satisfying %0i (^ vj )
@Mji (^ vj j%i (^ vj )) @vj : @Mji (^ vj j%i (^ vj )) dM (% (^ v )) 2 i i dvi j i @vi
2
= 1
2
i
(44)
Note that the denominator in (44) is strictly positive under Condition (Q-C). Next observe that, under Condition (M-C), any agent from side i with stand-alone di¤erential vi < %i (^ vj ) strictly prefers joining platform A to joining platform B if he expects (a) all agents from side j 6= i to follow
a threshold strategy with cut-o¤ equal to v^j and (b) all agents from his own side i to follow a threshold strategy with cut-o¤ equal to %i (^ vj ): Likewise, any agent from side i with the same expectations as above but with stand-alone di¤erential vi > %i (^ vj ) prefers joining platform B to joining platform A: Next, let Lj (^ vj ) = v^j +
j
+
j
2 j Mij (%i (^ vj )j^ vj )
2 j Mj (^ vj )
denote the gross payo¤ di¤erential that an agent from side j = 1; 2 derives from joining platform B relative to joining platform A when he expects (a) all agents from side i 6= j to join platform B when vi
%i (^ vj ) and platform A when vi < %i (^ vj ) and (b) all agents from his own side j to join platform
B when vj
v^j and platform A when vj < v^j : The function Lj (^ vj ) is di¤erentiable with derivative
equal to L0j (^ vj ) = 1
2
4 =1 1
@Mij (%i (vj )jvj ) 0 @Mij (%i (^ vj )j^ vj ) %i (^ vj ) + @vi @vj
j
@Mji (^ vj j%i (^ vj )) @Mij (%i (vj )jvj ) @vj @vi @Mji (^ vj j%i (^ vj )) dM (% (^ v )) 2 i i dvi j i @vi
i j
2
44
2
j
2
j
dMj (^ vj ) dv
@Mij (%i (^ vj )j^ vj ) @vj
2
j
dMj (^ vj ) : dv
Together, Conditions (M-C) and (Q-C) imply that the function Lj (^ vj ) is strictly increasing. Because limvj !
Lj (^ vj )
vj ) = 1 1 Lj (^ = pB pA j j exists
and limvj !+1 Lj (^ vj ) = +1; we then have a solution to the equation
and is unique. This in turn implies that there exists one and only one
solution to the system of equations given by (33). Step 2. This step parallels the proof for Proposition 1 in the baseline model. First one expresses each platform’s pro…ts as a function of the participation thresholds. Fixing the prices o¤ered by the rival …rm and di¤erentiating pro…ts with respect to the participation thresholds yields the price equations in (36). Step 3. The …nal step consists in observing that, when competition is symmetric, the participation thresholds are given by v^i = 0; i = 1; 2: Furthermore, because the two platforms expect the k (0) i
same participation rates,
= 1=2, k = A; B. From (36), we then have that the equilibrium
prices must satisfy pA i = pB i =
1 2 1 2
1 A i (0)
1 B i (0)
@Mji (0 j 0) @vi @Mji (0 j 0) i @vi
@Mij (0 j 0) @vi @Mij (0 j 0) j @vi
i
@Mi (0) @v @Mi (0) i @v
j
i
Clearly, the terms in curly brackets cannot be equal to zero, for otherwise the platforms’ prices would be equal to zero and platforms would have a pro…table deviations. For the platforms’prices to coincide, it must then be that
B i (0)
=
B i (0)
=
i (0);
i = 1; 2: The above results imply that,
when competition is symmetric, the equilibrium prices must satisfy the formulas in the proposition.
B Proof of Proposition 9. Suppose that platform B o¤ers an insulating tari¤ pB i = ti +
B i mj
on each of the two sides of the market and suppose that platform A responds by o¤ering any pair of A A A two-part tari¤s pA i = ti + wi mj i; j = 1; 2; j 6= i; satisfying the following two conditions:
C1: For any i; j = 1; 2; j 6= i; any v1 ; v2 ; 2 R; 1 + wiA
i
@Mji (vj jvi ) > 0; @vi
C2: For any i; j = 1; 2; j 6= i; any v1 ; v2 ; 2 R; h ih A @M21 (v2 jv1 ) 1 w 1 1 1 @v1 A A w w < 1 2 1 2 @M (v jv ) @M 12
1
@v1
2
2 21 (v2 jv1 ) @v2
w2A
@M12 (v1 jv2 ) @v2
i
:
Then in the continuation game among the agents, the continuation equilibrium is unique and is in threshold strategies. Also note that, because the environment satis…es Conditions (M) and (Q), there exist bi 2 R++ , i = 1; 2; such that Conditions C1 and C2 are satis…ed for any pair of two-part tari¤s by platform A with wiA 2 [0;
i
+ bi ]; i = 1; 2:
Hence, for any pair of two-part tari¤s by platform A satisfying the above two conditions, the de-
mands expected by platform A are given by QA i = 45
vi ) ; i (^
i = 1; 2; with the participation thresholds
v^1 and v^2 given by the unique solutions to the equations39 v^i = tB i
tA i +
wiA Mji (^ vj j v^i ); i; j = 1; 2; j 6= i:
i
(45)
Next observe that, for any pair of two-part tari¤s satisfying the above two conditions, platform A’s pro…ts, expressed as a function of the participation thresholds they induce and the sensitivity of its prices to the participation rates, wA A
2
(^ v1 ; v^2 ; wA ) = E 4
X
(wiA )i=1;2 , are given by tB i
v^i +
wiA Mji (^ vj j v^i ) + wiA
i
i;j=1;2; j6=i:
X
=
tB i
i;j=1;2; j6=i:
X
+
wiA E
i;j=1;2; j6=i:
v^i + h
i
vj i Mji (^
(^ vi )
j v^i ) E [
vj ) j (^
i
j
(^ vj )
i
(^ vi )]
Mji (^ vj j v^i )
i
(^ vi )
3
(^ vi )5
i
A A A Notice that there are in…nitely many pairs (tA 1 ; w1 ; t2 ; w2 ) that, together with platform B’s insulating
tari¤s, permit platform A to induce the desired participation thresholds (^ v1 ; v^2 ) while satisfying Conditions C1 and C2 above (this can be seen from (45)). Also note that, …xing the participation A (^ v1 ; v^2 ; wA )
thresholds (^ v1 ; v^2 ); platform A’s expected pro…ts
are linear in wiA ; i = 1; 2: This means
that, for platform A to …nd it optimal to respond to platform B’s insulating tari¤s by o¤ering itself a pair of insulating tari¤s, it must be that E
h
i
(^ vi )
i h (^ v ) = M (^ v j v ^ )E j ji j i j
i
i (^ vi ) ; i; j = 1; 2; j 6= i:
(46)
Replicating the same arguments for platform B; we have that, for platform B to …nd it optimal to respond to platform A’s insulating tari¤s by o¤ering itself a pair of insulating tari¤s, it must be that E
h (1
i
i (^ v )) = (1 j j
(^ vi ))(1
or, equivalently, E
h
i
(^ vi )
j
i h (^ vj ) = Mji (^ vj j v^i )E
i
Mji (^ vj j v^i ))E
i h (^ vi ) + E
j
(^ vj )
i
h
1
i
i (^ vi ) ; i = 1; 2:
Mji (^ vj j v^i ) i; j = 1; 2; j 6= i
(47)
Combining (46) with (47) we thus conclude that an equilibrium in insulating tari¤s exists only if E 39
h
j
i (^ vj ) = Mji (^ vj j v^i ) i; j = 1; 2; j 6= i
(48)
Note that an agent from side i with stand-alone di¤erential vi who expects all agents from side j 6= i to follow a
cut-o¤ strategy with cut-o¤ v^j is indi¤erent between joining one platform or the other if and only if vi +
i [1
2Mji (^ vj j vi )] = tB i
tA i +
46
i
(1
Mji (^ vj j vi ))
wiA Mji (^ vj j vi )
and E
h
v1 ) 1 (^
i h (^ v ) = E 2 2
i h (^ v ) 1 1 E
i (^ v ) 2 2 :
(49)
Condition (48) means that the side-j demands expected by the side-i marginal agent coincide with the side-j demand expected by the platforms. Condition (49) in turn is equivalent to Cov where qiA ( ) = 1
qiB ( ) =
i
q1A ( ) ; q2A ( ) = 0
(^ vi ) is the side-i participation to platform A in state : We conclude
that an equilibrium in insulating tari¤s exists only if, on each side i = 1; 2, (i) the marginal agent has the same beliefs as the platforms’about the expected participation of the opposite side, and (ii) the demands on the two sides are statistically independent.
47
Table 1: Symbols symbol
description
k = A; B
platforms
i; j = 1; 2
sides
l 2 [0; 1]
agents on one side
k vil
stand alone valuation for platform k of agent l of side i intensity of network e¤ects on side i
i
mki pki
mass of agents of side i that join platform k price charged by platform k to side i vilB
vil k i (v) k i (v)
A vil ; di¤erential in stand alone valuations; type
measure of agents from side i that platform k expects to have a type less than or equal to v d
k i (v) dv
Mji (vj j vi )
measure of agents from side j with type
Qki
(direct residual) demand on side i expected by platform k
Vik (Qki )
cuto¤ type that platform k needs to induce to obtain Qki (assuming agents follow cuto¤ strategy)
v^i
valuation of critical agent of side i
vj as expected by agents from side i with type = vi
Table 2: Symbols when assuming common prior symbol
description aggregate state; cross-sectional joint distribution of stand-alone valuations of the two sides
F
CDF of i
CDF of types for any agent of side i given state
i
PDF of types for any agent of side i given state
Table 3: Symbols when assuming common prior and Gaussian structure symbol vil =
i
description +
il
B vil
A vil ; di¤erential in stand alone valuations; type
precision of
i
i
correlation between xil =
i+
il
precision of
il
i
precision of
il
i
correlation between
V
and
2
private signal of agent l of side i
i
Vil
1
il
and
il
E [vil j xil ]; estimated stand-alone di¤erential of agent l of side i; estimated type
corr(Vil ; Vjl0 ); correlation between estimated types of any pair of agents from opposite sides p V 2 ; coe¢ cient of mutual forecastability 1
V
48
