aa r X i v : . [ ec on . T H ] N ov Platform-Mediated Competition
Quitz´e Valenzuela-StookeyNovember 5, 2020
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Abstract
Cross-group externalities and network effects in two-sided platform markets shapemarket structure and competition policy, and are the subject of extensive study. Lessunderstood are the within-group externalities that arise when the platform designs many-to-many matchings: the value to agent i of matching with agent j may depend on the setof agents with which j is matched. These effects are present in a wide range of settings inwhich firms compete for individuals’ custom or attention. I characterize platform-optimalmatchings in a general model of many-to-many matching with within-group externalities.I prove a set of comparative statics results for optimal matchings, and show how thesecan be used to analyze the welfare effects various changes, including vertical integrationby the platform, horizontal mergers between firms on one side of the market, and changesin the platform’s information structure. I then explore market structure and regulationin two in-depth applications. The first is monopolistic competition between firms on aretail platform such as Amazon. The second is a multi-channel video program distributor(MVPD) negotiating transfer fees with television channels and bundling these to sell toindividuals. Multi-sided platforms play a large and growing role in the economy. The core business ofsome of the world’s largest companies, including Amazon, Alibaba, Facebook, and Google,fall into this category. The defining feature of platforms is that they match different agents.In the case of multi-sided platforms, agents can be divided into distinct groups, and matchesoccur across groups. Increasing data availability and developments of matching technology,both facilitated in many cases by the internet, have fueled the rise in multi-sided platformbusinesses.An important feature of the environments in which many platforms operate is the ex-istence of cross-group externalities. A U.S. based retailer derives benefits from contracting1ith a Chinese manufacturer, and the manufacturer benefits from selling products to theretailer. In general, neither party appropriates the full surplus from their transaction. Thiswould not be a problem if it was easy for the retailer to find an appropriate manufacturer,or vice-versa. However when searching for a partner is costly, the presence of externalitiesmeans that agents will generally under-invest in search. This explains the existence of aplatform such as Alibaba. The platform facilitates matches, and internalizes the matchingexternalities through fees charged to agents on one or both sides of the market.In addition to cross-group externalities, platforms often take advantage of network effects.The benefit that one side of the market derives from the platform’s services depends on theset of agents on the other side with whom they may be matched. Software developers wouldlike to create programs for operating systems that have a large number of users, and usersprefer operating systems which support many programs. Modern platforms generally engagein more sophisticated matching than simply granting all or nothing access to a network.Search engines prioritize certain results, and curate results based on user preferences. Cableproviders allow customers to choose between many different packages consisting of differentbundles of channels. By doing so, platforms fine-tune the network effects within the platform.Cross-group externalities and network effects have long been central to the literature onmulti-sided platform design and regulation. However significantly less attention has been paidto with-group externalities in multi-sided settings. Consider again Alibaba’s role matchingretailers and manufacturers. Cross-group network effects are present; retailers would liketo search on a platform on which many manufacturers are available, and manufacturerswould like access to the largest set of potential customers. However manufacturers are alsocompetitors. Fixing the set of retailers using the platform, a manufacturer would preferto compete with as few other manufacturers as possible. As in this example, within-groupexternalities often work in the opposite direction as network effects.Within-group externalities have important implications for the regulation of platforms.As noted above, platforms add value by internalizing cross-group externalities. Indeed, whennetwork effects are large, it has been argued that the efficiency gains provided by largenetworks are sufficient to justify the existence of a monopolistic platform (Evans, 2003).However the platform also internalizes, to some degree, the effects of competition betweenfirms. That is, the platform internalizes within-group externalities. Thus the platform willhave an incentive to reduce competition between firms beyond the socially optimal level.Within-group externalities also affect the welfare implications of vertical integration by theplatform into the firm side of the market, as well has horizontal mergers between firms.This paper studies the implications of within-group externalities on the design and regu-2ation of platforms. I consider a monopolistic platform whose role is to match each agent onone side of the market with a set of agents on the other (so called many-to-many matching).For example, Google’s ad platform matches advertisers and websites. Each advertiser’s admay be shown on multiple websites, and websites display multiple ads. Matches are recip-rocal; i is matched with j if and only if j is matched with i . I will refer to one side of themarket as firms and the other side as individuals, although the analysis applies equally wellto a wide range of business-to-business activities. There are no within-side externalities forindividuals; an individual’s payoff depends on the set of firms with which they are matched.A firm’s payoffs, however, depend not only on the set of individuals with which it is matched,but also on the set of firms with which each of these individuals is matched. In the webadvertising example, the “individuals” are the websites, whose payoff depends only on whichads are being displayed on their site. The firms are the advertisers, who care not only aboutwhich sites display their ads, but also, potentially, about how many other adds are shown onthe same sites (due perhaps to viewers’ limited attention), and whether these ads are fromtheir competitors.The model accommodates both vertical and horizontal differentiation between agents. Anagent’s vertical type relates to their own marginal value for better matches, whereas theirhorizontal type captures their attractiveness to the other side of the market. I show thatwhen payoffs are suitably supermodular, optimal matches have a natural threshold structure,whereby agents are matched with those on the other side of the market who have high enoughvertical types. Using this characterization, I prove a set of general comparative statics resultson how the matching changes when payoffs shift in a way that makes some agents relativelymore important. I show how in a broad class of problems, including those in which one setof agents is privately informed about their type, these comparative statics results can beused to perform welfare analyses. Within-group externalities are an essential component ofcompetition analysis. For example, I show that when these effects are not present verticalmergers between the platform and firms will unambiguously benefit individuals in most cases.I explore two applications in depth. I modify the canonical Dixit-Stiglitz model of monop-olistic competition by giving a platform control over the set of firms that each individual hasaccess to. This model applies to many settings; for example, Amazon mediating interactionsbetween customers and vendors. I characterize the types of mergers between the platform andfirms, and the types of information acquisitions by the platform, that make individuals betteror worse off. I also study a multi-channel video program distributor (MVPD) negotiatingtransfer fees with television channels and bundling these to sell to individuals who are pri-vately informed about their value for programming. I show that horizontal mergers between3hannels which are included in the basic cable package will make all individuals worse off,even if the merger does nothing but create cost synergies and has no direct anti-competitiveeffects. On the other hand, all individuals will be made better off if the merger is betweenchannels only offered in the premium packages. Similar results obtain for vertical mergersbetween the MVPD and channels.The general intuition for a number of the results can be illustrated by focusing on theeffect of the platform acquiring a firm, denote by j . Assume that firms like matches withindividuals, but dislike competition; their payoff from a match with individual i is decreasingin the number of other firms that i is matched with. Suppose that before the merger theplatform was not able to capture the full surplus enjoyed by firm j . This will be the casewhen firms have bargaining power or private information. After the merger, on the otherhand, the platform will internalize j ’s surplus completely. This change has two effects onthe matching structure. First, the platform will want to increase the payoff to firm j bymatching more individuals with j . This effect is analogous to “eliminating markups”, asdiscussed in the classical vertical integration literature. Second, the platform wants toreduce the competition faced by j (analogous to “raising rivals’ costs”). To do this, theplatform matches the individuals who are matched with j with fewer additional firms. Whenthere are no within-group externalities only the first effect is present. In this case the mergerresults in larger matching sets for all individuals, which I show implies higher payoffs for allindividuals. The second effect however, which is driven by competition between firms, hasthe effect of shrinking individual matching sets. The overall welfare effect depends on whichof the two dominates. Using the characterization of optimal matches, I am able to identifycases in which the welfare effect is unambiguous. In general, an acquisition by the platformof a low-type firm will benefit individuals, and an acquisition of a high-type firm will harmindividuals.The literature on competition policy for multi-sided platforms is extensive, and will not besummarized in full here. For a recent review see Jullien and Sand-Zantman (2020). Much ofthis literature focuses on competition between platforms, and ignores within-group external-ities. Seminal theoretical contributions in this area were made by Rochet and Tirole (2003),Caillaud and Jullien (2003), and Armstrong (2006). My primary interest is on the implica-tions of platform mediation for competition between firms. I therefore focus on a monopolisticplatform, but introduce competition effects. Pouyet and Tr´egou¨et (2016) study vertical inte- See Riordan (2005). This conclusion holds even when individuals make monetary payments to the firm, in which case it is animplication of the envelope condition for the platform’s revenue maximizing mechanism. i ) a platform that controls the interactions betweenagents on different sides of the market via many-to-many matchings, and ii ) within-side com-petition effects on one side. While these features appear separately in the literature, they havenot been previously considered together. This paper builds on the matching design and pricediscrimination literature. The model of the platform is similar to that of Gomes and Pavan(2016), who also consider the design of many-to-many matchings by a platform. Howevertheir model does not allow for within-side competition. As in Gomes and Pavan (2016), theplatform in this paper may engage in price discrimination by offering a menu of matching setsand fees to each side of the market. The platform can flexibly design both the matching setsand fees. This is in contrast to the literature on two-sided markets, in which platforms sellaccess to a single network, or to different mutually exclusive networks (see Rysman (2009)for a survey and Weyl (2010), White and Weyl (2016) for more recent contributions).The remainder of this paper is organized as follows. Section 1 presents the basic model,characterizes optimal matchings, and discusses the main comparative statics results. Section2 discusses an extension of the model, which is useful in applications. Section 3 presents thetwo applications mentioned earlier. 5 The basic model
The model generalizes that of Gomes and Pavan (2016). There is a unit mass of individuals(side I ) and a set F of firms (side F). Depending on the setting, I will consider either finite F or F = [0 , λ will be used to denote either Lebesgue measure, in thecase of a continuum of firms, or the measure placing mass 1 on each firm when F is finite.Competition effects are present only on the firm side, and take a form that will be specifiedbelow. Matchings are reciprocal: individual i is matched with firm j iff j is matched with i . I present here the baseline model. Alternative payoff structures are explored in the extensionsin Section 2. Agent ℓ ∈ [0 ,
1] on side K ∈ { F, I } is characterized by a vertical type v ℓK and a horizontal type σ ℓK ≥
0. The platform’s objective function will be of the form Z F U F ( v jF , | s F ( j ) | S I ) dλ ( j ) + Z I U I ( v iI , | s I ( i ) | ) dλ ( i ) . (1)The components of (1) will be discussed below. What is important to note at present is thatthe platform’s payoff depends on the sum of some aggregate payoffs coming from the firmside and the individual side. If the platform’s objective is utilitarian welfare maximizationthen U F and U I will correspond to the utilities of firms and individuals respectively. Howeverthere are many other objectives of the form given in (1). One such setting, in which agentsare privately informed about their vertical type, will be discussed in Section 1.5. Furtherexamples will be illustrated in the applications of Section 3. For simplicity, in what followsI will refer to U F and U I as firm and individual payoffs respectively. Later, when discussingindividual welfare I will be careful to differentiate between the true utilities of agents and thepayoffs that are relevant for the firm’s objective in (1).The vertical type v ℓK determines the value that ℓ attaches to matches with agents onthe other side of the market, while the horizontal type σ ℓK , which I also refer to as salience ,represents how important ℓ is to agents on the other side. The be precise, for an individual i on side I the payoff of being matched with a set s I ( i ) ⊆ [0 ,
1] of firms is U I ( v iI , | s I ( i ) | )where | s I ( i ) | is the salience weighted size of the set s I ( i ), given by | s I ( i ) | = Z j ∈ s I ( i ) σ jF dλ ( j ) . It is also interesting to consider markets with competition effects on both sides. For now I will focus onthe one-sided case because most of the applications that I have in mind are of this form. j with matching set s F ( j ) when individuals have matchings S I = { ( s I ( i ) , i ) } i ∈ [0 , is given by U F ( v jF , | s F ( j ) | S I )where | s F ( j ) | S I = Z i ∈ s F ( j ) h ( | s I ( i ) | , σ iI , σ jF , ) dλ ( i ) , and h is non-negative. The function h captures competition effects. It can be thought of asdepending on the exogenous ( σ iI ) and endogenous ( | s I ( i ) | ) components of individual salience. In most applications h will be decreasing; individuals are less valuable to firms if they arematched with many other firms. However I will not assume that this is always the case.In the model described above, neither firms nor individuals care about the vertical typesof the agents they are matched with. The interpretation is the an agent’s vertical type isa private taste parameter that describes their value for matches. I will discuss settings inwhich this assumption is natural. On the other hand, in some situations we may derive thevertical type from characteristics of the agent, for example the cost function of a firm thatis also choosing prices, that may be relevant for agents on the other side. In Section 3.2 Iexplore payoffs of this form. The main qualitative conclusions of the model will be the samein both cases. For the time being, assume that there is no horizontal differentiation; σ iI = σ kI for all i, k and σ jF = σ lF for all j, l . If there is horizontal differentiation, the results of this section will holdconditional on horizontal types. Supermodularity.
Let v ′′ > v ′ , x ′′ > x ′ . Then U K ( v ′′ , x ′′ ) + U K ( v ′ , x ′ ) ≥ U K ( v ′′ , x ′ ) + U K ( v ′ , x ′′ ) for K ∈ { F, I } . Order-Supermodularity.
There exists a complete order on the agents on side K suchthat Supermodularity holds with respect to this order. That is, if type ˆ v is higher in thisorder than type ˜ v (not necessarily ˆ v > ˜ v ) and x ′′ ≥ x ′ , then U K (ˆ v, x ′′ ) + U K (˜ v, x ′ ) ≥ U K (ˆ v, x ′ ) + U K (˜ v, x ′′ ) for K ∈ { F, I } . h need not be a function of | s I | . The general results presented below apply as long as h is continuous (inan appropriate sense) in s I and independent of the vertical types v jF for j ∈ s I . For example, h could be afunction of λ ( s b ) rather than | s I | . Lemma 1.
Under Order-Supermodualrity, optimal matchings are monotone in the super-modular order: higher type individuals receive larger matching sets and higher type firmsreceive higher quality matching sets.
Proof.
Without loss, let the order be given by type. Consider first individual side mono-tonicity. Let v Ij ≥ v Ik and suppose | s I ( k ) | ≥ | s I ( j ) | . Then switch the matching sets. BySupermodularity, payoffs on the individual side increase. Moreover, payoffs on the firm sideare unchanged, since any firm that was matched with j is now matched with firm i that hasthe same matching set that j had. The same switching argument works on the firm side.For the remainder of this section assume that Supermodularity holds. All results extendimmediately to other supermodular orders.I now turn to establishing the aforementioned threshold structure of matching sets. Onemight be tempted to apply a similar proof as that of Lemma 1. To see why this does notwork, suppose g F is increasing and concave. Fix an individual, and suppose that they arematched with a firm with type v ′ but not with a type v ′′ firm, where v ′′ > v ′ . Simply droppingthe low type firm from and adding the high type firm to the individual’s set clearly does notchange the individual’s payoff, but will not necessarily improve payoffs on the firm side. Thisis because despite having a higher vertical type, the v ′′ firm will also have a higher qualitymatching set, by Lemma 1. Then by concavity this means precisely that the marginal changein match quality for the v ′′ firm is lower than for the v ′ firm. The proof of the followingproposition modifies the switching argument to accommodate this case. Proposition 1.
For an individual with type v I there is a threshold v ∗ ( v I ) such that theindividual is matched with a firm if and only if the firm’s type is above v ∗ F ( v I ). Proof.
First I claim that the optimal matching should be characterized by a threshold in themarginal firm utility U F ( v jF , | s F ( j ) | S I ) (or the discrete analog). If this was not the case thenswitching out low marginal utility firms for high marginal utility firms does not change thesize of the individual’s matching set, and thus has no effect on the individual’s payoffs ortheir endogenous salience. Moreover it increases firm side payoffs. I am using here the fact that monotonicity does not bid. If it did then I would have to guarantee thatthis switch does not violate monotonicity. This could probably be dealt with, but I don’t need to. v F , which is what I know show.Let { S F , S I } be the optimal matching. A necessary condition for { S F , S I } to be optimalis that each individual’s match be characterized by a threshold in the marginal firm utilitiesinduced by { S F , S I } , as discussed above. Suppose that U F ( v jF , | s F ( j ) | S I ) is not increasing infirm type, that is, there are firms i, j with v j > v k such that U F ( v kF , | s F ( k ) | S I ) > U F ( v jF , | s F ( j ) | S I ) . (2)For this to hold there must be a positive measure of individuals for whom the marginalutility threshold defining their matching set falls strictly above U F ( v jF , | s F ( j ) | S I ) and weaklybelow U F ( v kF , | s F ( k ) | S I ), (otherwise | s F ( v ′ ) | s I ) = | s F ( v ′′ ) | s I and so (2) does not hold). Then s F ( v ′′ ) ( s F ( v ′ ). Since h ≥
0, this implies | s F ( v ′ ) | s I > | s F ( v ′′ ) | s I , contradicting monotonicityfrom Lemma 1. Corollary 1.
The threshold v ∗ F ( v I ) is decreasing. Proof.
Immediate from Lemma 1 and Proposition 1
Corollary 2.
Firm matchings are characterized by a threshold v ∗ I ( v F ). Moreover v ∗ I ( v F ) isdecreasing. Proof.
Immediate from Corollary 1.
The previous section established that optimal matchings have a threshold structure whenthere is no horizontal differentiation. With horizontal differentiation, the same structurecontinues to hold conditional on horizontal types; each individual i ’s matching set will becharacterized by a threshold function v ∗ I ( σ F , i ) such that the individual matches with a firmof type v F and salience σ F if and only if v F ≥ v ∗ I ( σ F , i ). Similarly for firms. In this section Iwill be interested in the features of optimal threshold functions. In particular, I will identifywhen this functions are increasing/decreasing.The platform’s problem simplifies greatly when U F ( v, · ) is affine. In this case the firm’spayoffs can be written as U F ( v, x ) = a F ( v ) + b F ( v ) · x . If supermodularity holds then we can If g F is concave then we need not appeal to monotonicity, | s F ( v ′ ) | s I > | s F ( v ′′ ) | s I contradicts (2). Theargument can be modified to accommodate for negative values of h . b F ( v ) = v . Then the platform’sobjective function can be written as Z I U I ( v Ii , | s I ( i ) | ) + Z s I ( i ) v Fj · h ( | s I ( i ) | , σ Ii , σ Fj ) dλ ( j ) ! dλ ( i ) . This integral can be maximized point-wise for each individual. This is true even when matchqualities are constrained to be monotone in agent type, as will be the case when types areprivate infromation, since Lemma 1 tells us that these constraints will not bind. For simplicity,assume all primitive functions are differentiable. Consider the objective U I ( v Ii , | s I ( i ) | ) + Z s I ( i ) v Fj · h ( | s I ( i ) | , σ Ii , σ Fj ) dλ ( j ) . The marginal effect of adding a firm of type v F and salience σ F to the matching set ofindividual i is v F · h ( | s I ( i ) | , σ Ii , σ F ) + σ F · U I ( v Ii , | s I ( i ) | ) + Z s I ( i ) v Fj · h ( | s I ( i ) | , σ Ii , σ Fj ) dλ ( j ) ! . (3)The function v ∗ F ( σ F , i ) will be decreasing (increasing) if for σ ′′ F > σ ′ F the following singlecrossing property holds: if the marginal benefit in (3) is positive (negative) for σ ′ F then itis positive (negative) for σ ′′ F . Assume h is decreasing it its first and third arguments (If h is not decreasing in σ F then the threshold functions may be non-monotone). The term inparentheses in (3), U I ( v Ii , | s I ( i ) | ) + R s I ( i ) v Fj · h ( | s I ( i ) | , σ Ii , σ Fj ) dλ ( j ), is the key determinantof the slope of v ∗ F ( σ F , i ). The first part of this expression is the marginal benefit to theindividual of increasing the size of their matching set. The second part is the inframarginalcost to all firms matched with this individual of increasing the size of the matching set. Ifthe sum of these two is positive, it means that the benefit to the individual outweighs theexternality imposed on firms. But then this individual should be matched with all firmsthat have positive values v F ) j . Moreover, the individual should be matched with firms withnegative values only if σ Fj is large enough, so that the benefit the individual is sufficient tooutweigh both the cost to the new firm and the inframarginal cost to all other firms. If on theother hand U I ( v Ii , | s I ( i ) | ) + R s I ( i ) v Fj · h ( | s I ( i ) | , σ Ii , σ Fj ) dλ ( j ) < Lemma 2.
Assume h is decreasing in its first and third arguments (match size and firmsalience). Then the matching sets of individuals have the following structure: Just assuming that h is decreasing in match size, we can conclude that if U I ( v Ii , | s I ( i ) | ) + R s I ( i ) v Fj · h ( | s I ( i ) | , σ Ii , σ Fj ) dλ ( j ) ≥ There exists a threshold v ∗∗ such that an individual i ’s matching set contains firms withtypes v Fj < v Ii > v ∗∗ . • The individual’s threshold function v ∗ F ( σ F , i ) is downward sloping if v Ii > v ∗∗ andupward sloping otherwise.Lemma 2 has some interesting implications. If any firms have positive values then high-value, low-salience firms will be matched with the largest set of individuals (in the set inclusionsense). If no firms have positive values then the largest matching sets will instead go to high-value, high-salience firms.Suppose that firms vertical types are their private information, while horizontal typesare known to the platform. For example, the vertical type may reflect a firm’s marginalcost, while its horizontal type is the attractiveness of its product to consumers. Below, I willdiscuss more extensively the platform’s problem when types are private information. Supposethat the platform make monetary transfers with firms, and that firms payoffs are quasi-linearin money. In this context it is easy to see, from the usual Myerson (1981) argument, that theobjective of a revenue maximizing platform will look the same as 1, except that the firms’vertical types v Fj will be replaced with their “virtual values” ϕ ( v Fj , σ Fj ) = v Fj − − Q F ( v Fj | σ Fj ) q F ( v Fj | σ Fj ) .Virtual values may be negative even if all true values are positive. As a result, the secondbest matching, in which firm types must be elicited, and the first best matching in which theyare known may be qualitatively different: in the first best, conditional on values, low-saliencefirms will always be matched with a superset of the individuals matched with high-saliencefirms. In the second best the reverse may hold: for low enough values, high-salience firmsare matched with a superset of the individuals that low-salience firms with the same valueare matched with. Throughout this section, assume that there is no horizontal differentiation on either side. Iwill be interested in how the optimal matching changes when some of the firms become “moreprominent”, in a sense I will make precise. Intuitively, there are two effects of a platformplacing greater weight on the payoffs of firm j . For one, it should increase the size firm j ’s’ matching set. This effect is easy to see. On the other hand, it should change the otherfirms’ matching sets to increase the value of firm j ’s matching. If h is decreasing this meansreducing the size of the other firms’ matching sets. How these two effects interact is notobvious in general. However there will be cases in which the effect on the matching structurecan be identified. I explore such cases in this section.11y “more prominent” I mean that the marginal value of increasing the quality of a firm’smatching increases. Increasing differences change.
A change in the payoffs of firm j from U F to ˆ U F arean increasing differences change if for x ′′ > x ′ , ˆ U F ( v Fj , x ′′ ) − ˆ U F ( v Fj , x ′ ) ≥ U F ( v Fj , x ′′ ) − U F ( v Fj , x ′ ). Lemma 3.
The quality of firm j ’s matching set increases following an increasing differenceschange in its payoffs. Proof.
This follows from the resulting single-crossing property of the objective.To further understand the changes in the matching, is important to first clarify the nec-essary conditions for optimality of a matching. Assume that there are
N < ∞ firms anda continuum of individuals. The argument can be easily modified for the case of finitelymany individuals. For notational simplicity, assume that Supermodularity holds (all resultsgo through under Order-supermodularity). Label the firms in order of types, with firm 1being the highest type and firm N the lowest.If there is no horizontal differentiation h ( n ) is the endogenous salience of an individualmatched with n firms. Let v ∗ I ( j ) be the threshold type for firm j ’s matching set. If firm j marginally increases the size of its matching set by lowering the cut-off it benefits from alarger matching set, but affects the endogenous salience of the newly added individuals, whoare included in the matching sets of all higher types. Suppose that firm j adds a marginalindividual to its matching set. Assume that there is not another firm with the exact samematching set as j , and that U I is continuous in its first argument. Using the fact that theoptimal matching has the threshold structure described in Proposition 1, the FOC for sucha change is given by U F ( v Fj , | s F ( j ) | S I ) · h ( j ) − j − X k =1 U F ( v Fk , | s F ( k ) | S I ) · [ h ( j − − h ( j )]+ U I ( v ∗ I ( j ) , j ) − U I ( v ∗ I ( j ) , j − ≥ v ∗ I ( j ) is interior.The first term in (4) is the marginal benefit to firm j , the second is the inframarginalcost to all firms matched with the newly added individual, and the last is the direct benefitto the new individual. If another firm has the same matching set we just need to modify the term [ h ( j − − h ( j )] in equation(4). roposition 2. Suppose there is an increasing differences change in the payoffs of firm k ,and that the supermodular order remains unchanged. If U F ( v, · ) is concave for all v and h is decreasing then all firms j > k receive smaller matching sets and firm k receives a largermatching set. Proof.
Throughout this proof s and S will refer to the original matchings and ˜ s , ˜ S will referto the new matching.If k = N then the proposition follows immediately from Lemma 3, so assume k > Claim 1. There cannot exists an ℓ such that firms N, N − , . . . , ℓ receive larger matchingsets after the change. First, suppose that not all firms receive larger matching sets. Let j be the lowest value firm that receives a strictly smaller matching set ( Lemma 3 impliesthat j = k ). Since j receives a smaller matching set ˜ v ∗ I ( j ) must be larger than v ∗ I ( j ). ThenSupermodularity implies that U I (˜ v ∗ I ( j ) , j ) − U I (˜ v ∗ I ( j ) , j − > U I ( v ∗ I ( j ) , j ) − U I ( v ∗ I ( j ) , j − . Since all lower value firms receive larger matching sets and h is decreasing, firm j ’s matchingset is of lower quality. By concavity, U F ( v Fj , | ˜ s ( j ) | ˜ S I ) > U F ( v Fj , | s ( j ) | S I ). Then the FOC for j holds only if j − X t =1 U F ( v Ft , | ˜ s F ( t ) | ˜ S I ) · [ h ( j − − h ( j )]is larger as well. But then we have that j X t =1 U F ( v Ft , | ˜ s F ( t ) | ˜ S I ) · [ h ( j − − h ( j )] > j X t =1 U F ( v Ft , | s F ( t ) | S I ) · [ h ( j − − h ( j )] (5)is also larger. Since firm j + 1 received a larger matching set ˜ v ∗ I ( j + 1) is smaller. ThenSupermodularity implies that U I (˜ v ∗ I ( j + 1) , j + 1) − U I (˜ v ∗ I ( j + 1) , j ) < U I ( v ∗ I ( j + 1) , j + 1) − U I ( v ∗ I ( j ) , j ) . Then the FOC for j +1 and (5) imply that U F ( v Fj +1 , | ˜ s F ( j ) | ˜ S I ) is larger (or ˆ U F ( v Fj +1 , | ˜ s F ( j ) | ˜ S I )if j + 1 = k ). Proceeding in this way we conclude that U F ( v FN , | ˜ s F ( N ) | ˜ S I ) must be larger.But if firm N received a larger matching set then concavity implies that U F ( v FN , | ˜ s F ( N ) | ˜ S I ) isstrictly smaller after the merger (since k > U F ( v F , | ˜ s (1) | ˜ S I ) must be higher. Since firm 2receives a larger matching set ˜ v ∗ I (2) must be smaller. Then (4) implies that U F ( v F , | ˜ s (2) | ˜ S I )13ust be larger. Proceeding in this way we conclude that U F ( v FN , | ˜ s ( N ) | ˜ S I ) is larger after thechange, which we have already noted is a contradiction. Claim 2. No firm j > k can receive a larger matching set after the change.
Let j bethe lowest value firm that receives a larger matching set, and suppose j > k . By Claim 1, j < N . Since j receives a larger matching set ˜ v ∗ I ( j ) is smaller, so Supermodularity impliesthat U I (˜ v ∗ I ( j ) , j ) − U I (˜ v ∗ I ( j ) , j −
1) is smaller. Since all firms m > j receive smaller matchingsets, concavity implies that U F ( v Fj , | ˜ s ( j ) | ˜ S I ) is smaller. Then the FOC for j implies that j − X t =1 U F ( v Ft , | ˜ s F ( t ) | ˜ S I ) · [ h ( j − − h ( j )]is smaller. But then we have that j X t =1 U F ( v Ft , | ˜ s F ( t ) | ˜ S I ) · [ h ( j − − h ( j )] < j X t =1 U F ( v Ft , | s F ( t ) | S I ) · [ h ( j − − h ( j )] (6)Since j + 1 receives a smaller matching set ˜ v ∗ I ( j + 1) is larger. But then the FOC for j + 1implies that U F ( v Fj +1 , | ˜ s ( j + 1) | ˜ S I ) is smaller. Proceeding in this way we conclude that U F ( v FN , | ˜ s ( N ) | ˜ S I ) is smaller. But this contradicts the assumption that N receives a smaller(and thus worse) matching set, given concavity. Claim 3. Firm k receives a larger matching set. Suppose k receives a smaller set. Supposesome firm with a higher value than k receives a larger set, and let j be the lowest value suchfirm. Then using the same proof as in Claim 2 we can arrive at a contradiction, so no firmcan receive a larger set. If firm 1 receives a smaller matching set then the FOC for 1 impliesthat U F ( v F , | ˜ s (1) | ˜ S I ) (or U F ( v F , | ˜ s (1) | ˜ S I ) if k = 1) must be smaller. Then again we can usethe proof of Claim 2 to arrive at a contradiction.The assumption of concavity of U F ( v, · ) is used throughout the proof, but it is not neces-sary. This can be seen most easily by appealing to continuity of the objective function. Thestrict version of the comparative statics result of Proposition 2 holds when U F ( v, · ) is affine,and given appropriate continuity assumptions Berge’s theorem implies that it will continueto hold if U F ( v, · ) is perturbed slightly to be strictly convex. Thus it is not clear exactlywhat the role of concavity is in the result. To try to build some intuition, consider an in-creasing differences change to the payoffs of firm 1. At the new optimal matching, the sumof payoffs for all other firms must be smaller (otherwise it would have been worthwhile tochange their matchings before the change in firm 1’s payoffs). Roughly speaking, concavityof U F ( v, · ) implies that it is optimal to spread the reduction in payoffs across all these firms.This intuition is incomplete, and it would be valuable to know how far convexity of U F ( v, · )can be pushed. 14et v ∗ I ( k ) be the threshold individual type defining firm k ’s matching set. If all firmswith lower types than k receive smaller sets then all individuals with types higher then v ∗ I ( k )receive smaller matching sets. Corollary 3.
Under the conditions of Proposition 1, all individuals with types higher than v ∗ I ( k ) receive (weakly) smaller matching sets.If an individual’s utility is given entirely by U I , and is increasing in match size, thenCorollary 3 implies that all individuals with types above v ∗ I ( k ) are weakly worse off followingthe change in k ’s payoffs. Even if this is not the case, for example if there are transfersbetween individuals and the platform, it may be possible to determine the welfare effects ofsuch a change. I return to this question in the next section.It is easy to see from the proof of Proposition 2 that the result can be extended toincreasing changes in the payoffs of multiple firms. Lemma 4.
Under the conditions of Proposition 2, if there is an increasing differences changein the payoffs of a set C of firms firms the all firms j > max C receive smaller matching sets.Proposition 2 does not specify what happens to the matching sets for higher types thanthe one for which there is an increasing differences change. This will in general depend onthe function h . This is because it is not clear what the effect of the described changes is onthe value of the original matching sets for such types. The fact that all j > k receive smallermatching sets benefits types m < k . However these types are also hurt by the fact that k receives a larger matching set.If U F ( v, · ) is affine then comparative statics are much easier to identify. In this case, wecan write U F ( v, x ) = α F ( v ) + β F ( v ) · x In this case, we can look at the problem entirely fromthe point of view of an individual, and the problem separates across individuals. In otherwords, the objective function can be written as Z I U I ( v Ii , | s I ( i ) | ) + h ( | s I ( i ) | ) · Z s I ( i ) β F ( v Fj ) dλ ( j ) ! dλ ( i ) . An increasing differences change in payoffs here corresponds to an increase in β F ( v ). Thefollowing Lemma is immediate. Lemma 5. If U F ( v, · ) is affine then the size of firm j ’s matching set increases following anincreasing differences change in payoffs.In this case we can also identify what happens when there is an increasing differenceschange to the payoffs of the lowest type. The following result is immediate given the thresholdstructure of matching sets. 15 emma 6. Suppose there is an increasing differences change in the payoffs of firm N , andthat the supermodular order remains unchanged. If U F ( v, · ) is affine then either firm N isadded to an individual’s matching set or the individual’s matching set remains unchanged. Propositions 1 and 2 will be particularly interesting when there is a continuum of individu-als who have private information about their type which must be elicited by the platform.Assume that Supermodularity holds on side I (the arguments extend directly to Order-supermodularity). In this setting the usual Myerson (1981) argument implies that in anyincentive compatible mechanism the payoff of individual i can be written as V ( v Ii ) = V ( v I ) + v Ii Z v I U I ( v, | s I ( v ) | ) dv. (7)Suppose that the conditions of Proposition 2 are satisfied, and that there is a increasingdifferences change in the payoffs of firm 1. Corollary 3 says that all individuals with typesabove v ∗ I (1) receive smaller matching sets. Under Supermodularity, U F ( v, x ) is increasing in x . Then if V ( v ∗ I (1)) does not increase, the envelope condition in (7) implies that all individualswill be worse off. This will be the case if v ∗ I (1) = ¯ v I , which in turn will hold whenever both U F ( v F , · ) and U I ( v, · ) (or the variant of payoffs used in the platforms problem, if it is notthe same as (1)) are increasing. Lemma 7.
When individuals’ types are private information and v ∗ I (1) = ¯ v I , all individualsare worse off when all receive smaller matching sets, and better off when all receive largermatching sets.In order to apply Lemma 7 it need not be the case that the platform’s objective exactlycorresponds to that in (1). So long as the platform objective satisfies the conditions ofProposition 2.In many settings individual payoffs are quasi-linear and the platform wants to maximizetransfers from individuals. Normalize the outside option of the lowest type, U ( v, u I ( v, | s I ( v ) | ) − t ( v ) then the sum of transfers from allindividuals is given by ¯ v Z v (cid:20) u I ( v, | s I ( v ) | ) − − Q F ( v ) q F ( v ) u I ( v, | s ( v ) | ) (cid:21) q F ( v ) dv. (8)16nder Supermodularity, the matching mechanism is incentive compatible for individuals ifand only if individual match sizes are increasing in types and payments are constructed tosatisfy the envelope condition in (7). Proposition 1 will hold provided the integrand in (8) isSupermodular. Moreover, under this condition the monotonicity constraint will not bind. One extension of the model, which arises naturally in many applications, is that individualsand/or firms may care about the vertical types of their matches. In the MVPD exampleexplored above, we might think that channels prefer individuals with high viewing propensity,or individuals prefer channels with high quality programming (which in this case we interpretas high v F ). The main results will continue to hold when agents on one side prefer matcheswith higher type agents on the other side. This reinforces the optimality of giving betterquality match sets to high type individuals, which is the key driver of the threshold structure,and thus the results that follow. Another natural extension is if the endogenous salience of an individual depends on thevertical types of the firms they are matched with. Again, if the endogenous salience isincreasing in the types of an individual’s matches then there will be no complications to theprevious results. However in many cases this is not the direction we expect. An example,which I will explore in detail later on, is that monopolistically competitive firms may care notonly about how many other firms their customers have access to, but also about whether theseare high or low cost firms. Low cost (high type) are more damaging competitors because theyare able to charge a lower price. Nonetheless, the general welfare conclusions will continueto hold; broadly speaking, increasing differences payoff changes for high type firms will hurtindividuals, while such changes for low type firms will benefit individuals.In order to gain tractability when this holds, I assume that U F is affine in match quality.As we will see, this does not immediatly imply that the platform’s problem will separtedaccross individuals, as would be the case if veritcal types were not payoff relevant for agentson the other side of the market; when agents’ types are their private information the mono- Without this condition Proposition 1 will hold, i.e. the optimal matching will still have a thresholdstructure, but the proof of Proposition 2 does not go through. We could interpret the model in which agents care about the vertical types of those on the other sideas a special case of the model presented above, in which the horizontal types are perfectly correlated withthe vertical types. However under this interpretation the results of the previous section become trivial: athreshold structure of matchings conditional or horizontal type is meaningless when there is a single verticaltype for each horizontal type. Moreover, we will be intersted in settings in which vertical types are privateinformation. The case of privatly known horizontal types was not studied above. v I individual are given by U I ( v I , V I ( s I )) where V I ( s I ) = Z s I v j dλ ( j ) .U I is strictly increasing in its second argument.The payoffs of a type v F firm are given by β F ( v F ) · V F ( s F | S I ), where V F ( s F | S I ) = Z s F h ( v I ( i ) , V I ( s I ( i ))) dλ ( i )and h ≥ Z U I ( v I , V I ( s I ( v I ))) + h ( v I , V I ( s I ( v I ))) · Z s I ( v I ) β ( v F ) dQ F ( v F ) dQ I ( v I ) . (9)I will refer to maximizing the integrand in (9) separatly for each individual as maximizingpointwise. Denote the integrand by Π( s I | v I ). The solution to the platform’s problem maynot be given by pointwise maximization, in particular if they are subject to a monotonicityconstraint imposed by inceitnive compatability. We cannot appeal to Proposition 1 to estab-lish the threshold structure of optimal matchings in this setting, which would imply that themonotonicity constraint does not bind. This is because the proof of Lemma 1, which saysthat higher type firms receive larger matching sets, does not go through. The reason is thathigher type firms impose a larger negative externality on other firms by setting lower prices.However if the problem can be solved pointwise the solution is easy to characterize. Thischaracterization also reveals the relevant assumptions needed to guarantee that the matchingshave a threshold structure in vertical types. Lemma 8.
Pointwise maximization of (9) yeilds matchings for each individual type v I thatare characterized by a threshold in β ( v F ) /v F : a firm is included if and only if β ( v F ) /v F ishigh enough. Proof.
The integrand in (9) only depends on the matching through the terms V I ( s I ( v I )) and R s I ( v I ) β ( v F ) dQ F ( v F ). Consider the problem of maximizing R s I ( v I ) β ( v F ) dQ F ( v F ) subject to18 I ( s I ( v I )) = ¯ V . Since the objective and the constraint are linear, this amounts to includinga firm in the matching set if and only if β ( v F ) /v F is high enough.In immediate implication of Lemma 8 is that v F V F ( s F | S I ) is increasing in β ( v F ) /v F .However it says nothing about the comparison of matching sets for different individuals.Monotonicity properties of individual side matchings can be derived from the usual conditions(single crossing, interval order dominance, etc.) on the integrand. I will explore an applicationin detail, and delay further discussion of such results until then. Instead, I will present auseful comparative statics result on the matching set of a given individual that makes useof the threshold structure indentified in Lemma 8. As we will see, it will be intersting tocompare the optimal matchings under differnt firm payoffs, captured by different funtions β and ˜ β .Assume v β ( v ) /v and v ˜ β ( v ) /v are increasing. This is not necessary to obtain thetypes of results that follow, but it simplifies the notation, and will, in any case, be satisfied inmany applications. If this holds then each individual’s optimal matching set will be definedby a threshold in firm type. Let Π( s I | v I , β ) and be the integrand of 9 under β . Let v ∗ ( s I ) and˜ v ∗ ( s I ) be the threshold types that maximize Π( s I | v I , β ) and Π( s I | v I , ˜ β ) respectively; thesedefine the optimal matchings under pointwise maximization. The following comparativestatics result is straightforward, but will be useful in applications. Lemma 9.
Let h ( s I , · ) be strictly decreasing. Assume v β ( v ) /v and v ˜ β ( v ) /v areincreasing. If ˜ β ( v ) = β ( v ) for all v ≤ v ∗ ( s I ) then ˜ v ∗ ( s I ) ≥ v ∗ ( s I ). If ˜ β ( v ) = β ( v ) for all v ≤ v ∗ ( s I ) + ε for some ε > β ( v ) ≥ β ( v ) for all v > v ∗ ( s I ) + ε , with strict inequalityon a positive measure set, then ˜ v ∗ ( s I ) > v ∗ ( s I ). Proof.
Abusing notation, for a matching set s I defined by a threshold v ∗ , denote the integrandby Π( v ∗ | v I , β ). If ˜ β ( v ) = β ( v ) for all v ≤ v ∗ ( s I ), for any v ∗ < v ∗ ( s I ), we have Π( v ∗ | v I , ˜ β ) − Π( v ∗ ( s I ) | v I , ˜ β ) = Π( v ∗ | v I , β ) − Π( v ∗ ( s I ) | v I , β ). This implies that ˜ v ∗ ( s I ) ≥ v ∗ ( s I ).The second part of the lemma, the strict inequality, follows from the fact that h is strictlydecreasing and R ¯ vv ∗ β ( v ) dQ F ( v ) > R ¯ vv ∗ β ( v ) dQ F ( v ) under the stated assumptions.On the other hand, if the increase in β occurs for firms below v ∗ ( s I than the opositeconclusion will hold. The proof is symmetric to that of Lemma 9. Lemma 10.
Let h ( s I , · ) be strictly decreasing. Assume v β ( v ) /v and v ˜ β ( v ) /v areincreasing. If ˜ β ( v ) = β ( v ) for all v > v ∗ ( s I ) then ˜ v ∗ ( s I ) ≤ v ∗ ( s I ). If ˜ β ( v ) = β ( v ) for all If these functions are not increasing then we can attain similar comparative statics results with respect tothe order they induce. ≥ v ∗ ( s I ) − ε for some ε > β ( v ) ≥ β ( v ) for all v < v ∗ ( s I ) − ε , with strict inequalityon a positive measure set, then ˜ v ∗ ( s I ) < v ∗ ( s I ).Suppose that there is no direct individual-side component to platform payoffs, so U I = 0.For example, an online retail platform that generates revenue by charging fees to sellers,but which allows customers free access to the site. Under this assumption we can drawstronger comparative statics conclusions. Proposition 3 is relevant for considering the effectsof technological change on the matches. Proposition 3.
Assume h ( v I , · ) is decreasing and differentiable for all v I , U I = 0, β ( v ) /v is increasing, and R ¯ vv β ( v ) dv ≥ Suppose there exists an increasing and strictly positivefunction α such ˜ β ( v ) = α ( v ) β ( v ). Then under pointwise maximization all individuals and allfirms recieve smaller matching sets under ˜ β than under β. The proof of Proposition 3 makes use of the following result (see Quah and Strulovici(2009) for a proof).
Lemma 11.
Suppose [ x ′ , x ′′ ] is a compact interval or R and that α and k are real valuedfunctions on [ x ′ , x ′′ ], with k integrable and α increasing (and thus integrable as well). If R x ′′ x h ( t ) dt ≥ x ∈ [ x ′ , x ′′ ] then x ′′ Z x ′ α ( t ) h ( t ) dt ≥ α ( x ′ ) x ′′ Z x ′ h ( t ) dt. Proposition 3 follows from Proposition 2 and Theorem 1 of Quah and Strulovici (2009),and Lemma 11.
Proof. Proposition 3 . Consider an idividual with type v I . Lemma 8 means that the individ-ual’s matching set will be defined by a threshold firm type v ∗ . Then, abusing the definitionof V I , we can write Π( v ∗ | ˜ β ) = h ( v I , V I ( v ∗ )) · ¯ v F Z v ∗ ˜ β ( v ) q F ( v ) dv A similar result holds if β ( v ) /v is not increasing, we just have to define α to be increasing in the order ontypes induced by β ( v ) /v . V I ( v ∗ ) = R ¯ v F v ∗ vq F ( v ) dv . The derivative of the objective with respect to v ∗ isΠ ′ ( v ∗ | ˜ β ) = − h ( v I , V I ( v ∗ )) v ∗ q F ( v ∗ ) · ¯ v F Z v ∗ ˜ β ( v ) q F ( v ) dv − h ( v I , V I ( v ∗ )) · ˜ β ( v ∗ ) q F ( v ∗ ) ≥ − h ( v I , V I ( v ∗ )) v ∗ q F ( v ∗ ) α ( v ∗ ) · ¯ v F Z v ∗ ˜ β ( v ) q F ( v ) dv − h ( v I , V I ( v ∗ )) · α ( v ∗ ) ˜ β ( v ∗ ) q F ( v ∗ )= α ( v ∗ )Π ′ ( v ∗ | β ) . Proposition 2 and Theorem 1 of of Quah and Strulovici (2009) then implies that the optimalthreshold is higher for ˜ β than β , which means that all individuals recieve smaller matchingsets. Then all firms also recieve smaller matching sets. A monopolistic multi-channel video program distributor (MVPD), such as DirectTV or Com-cast, faces a population of viewers with unknown values for programming. The MVPD offersa menu of packages of different channels to viewers. At the same time, the MVPD negotiatescarriage fees with channels (referred to as video programmers in the IO literature). Videoprogrammers benefit from viewers through advertising revenue, but may differ in terms oftheir cost of producing programming or their attractiveness to advertisers. From a program-mers perspective, a viewer who has access to a large number of additional channels is lessvaluable, as they are likely to spend less time watching a given channel. The objective of theMVPD is to maximize revenue.The payoff of individual i with type v Ii who purchases package consisting of a set s orchannels at a price of p is given by v Ii · g I ( | s | ) − p , where g I is increasing and g I (0) = 0.Implicit in this formalization is the assumption that viewers like all channels equally, andonly care about the number of channels they have access to. I will discuss this assumptionlater on.As discussed above, the total revenue to the platform from viewer fees generated by adirect mechanism s : [ v I , ¯ v v ] F , where A is the set of channels, is given by ¯ v I Z v I (cid:18) v − − Q I ( v ) q I ( v ) (cid:19) g I ( | s ( v ) | ) dQ I ( v ) . (10)21ssume Q I is regular in the sense of Myerson (1981), so that the virtual values ϕ ( v ) = v − (1 − Q I ( v )) /q I ( v ) are increasing. Lemma 7 applies in this setting. After showing that theconditions of Propositions 1 and 2 are satisfied, we will identify changes to the environmentwhich make all individuals worse off.The payoffs of channel j matched with a set s F ( j ) of viewers is U F ( v j , | s F ( j ) | S I ). Iassume that the function h determining individuals’ endogenous salience is decreasing; themore channels an individual has access to the less valuable they are. Assume that U F issupermodular and concave in its second argument.The platform bargains with channels over a payment to be made for the right to carrytheir programming. In this industry, negotiations are generally over a monthly per subscriber“affiliate fee” that the MVPD pays the channel for every subscriber who has access thechannel, whether the subscriber watches it or not. Assume that the platform is able tocommit to a set of cable packages it will offer before negotiations over affiliate fees takeplace. The outcome of the multilateral bargaining game between the MVPD and channelsis described by the Nash-in-Nash solution concept. Assume that if negotiations betweena channel and the MVPD break down then the channel receives a payoff of zero. In thisscenario the MVPD simply drops this channel from the existing packages. This results ina new allocation for consumers. If the original allocation had a threshold structure thenthis new allocation remains implementable: monotonicity is preserved when when one of thechannels is dropped, holding fixed the remaining channels offered to each type. The totalnumber of consumers who have access to each channel does not change, and so the MVPD’srevenue from affiliate fees from other channels does not change. Thus the only effect of such abreakdown is the change it induces on the payments made by individuals for their packages.Thus we can think of negotiations as being simply over the total payment made from theMVPD to the channel, given the menu of bundles it plans to offer individuals. The pricenegotiated with channel j is given by p ∗ j = arg max p (cid:0) p + R ( s ) − R Oj ( s ) (cid:1) β ( U F ( v Fj , | s F ( j ) | S I ))= βU F ( v Fj , | s F ( j ) | S I ) + (1 − β )( R Oj ( s ) − R ( s ))where R ( s ) is the individual-side revenue given in (10) and R Oj ( s ) is the individual-siderevenue if channel j is dropped. The difference between the two is given by R O ( s F ( j )) − R ( s ) = Z s F ( j ) ϕ ( v ) · [ g I ( | s I ( v ) | − − g I ( | s I ( v ) | )] dQ I ( v ) . Using the expression for p ∗ j , we can write the MVPD firm-side revenue as a function of the22atching as X j p ∗ j = β X j U F ( v Fj , | s F ( j ) | S I ) + (1 − β ) X j Z s F ( j ) ϕ ( v ) · [ g I ( | s I ( v ) | − − g I ( | s I ( v ) | )] dQ I ( v )= β X j U F ( v Fj , | s F ( j ) | S I ) + (1 − β ) ¯ v I Z v ∗ I (1 | s ) ϕ ( v ) | s I ( v ) | · [ g I ( | s I ( v ) | − − g I ( | s I ( v ) | )] dQ I ( v )where v ∗ I (1 | s ) is the threshold type for firm 1’s matching set, which is also the lowest typeindividual that purchases a non-empty package. Then total MVPD revenue is given by β X j U F ( v Fj , | s F ( j ) | S I )+ Z ¯ v I v I ϕ ( v ) (cid:0) g I ( | s I ( v ) | ) + (1 − β ) | s I ( v ) | · [ g I ( | s I ( v ) | − − g I ( | s I ( v ) | )] (cid:1) dQ I ( v ) . (11)It is worth pointing out that if g I ( x ) = ax + b then the MVPD’s objective simplifies to β X j U F ( v Fj , | s F ( j ) | S I ) + ¯ v I Z v ∗ I (1 | s ) ϕ ( v ) (cid:0) βa · | s I ( v ) | + b (cid:1) dQ I ( v ) . In the special case of a = 1 and b = 1 maximizing this objective is the same as maximizingtotal (second-best) welfare, given by X j U F ( v Fj , | s F ( j ) | S I ) + ¯ v I Z v I ϕ ( v ) · g I ( | s I ( v ) | ) dQ I ( v ) . The general form of the objective in (11) will satisfy the conditions of Propositions 1 and2 if U F ( v, · ) is concave and the integrand in the second term of (11) is supermodular. Thissecond requirement will be satisfied iff g I ( x ) + (1 − β ) x · [ g I ( x − − g I ( x )] is increasing. Asufficient condition for this, given that g I is increasing, is that g I is concave. Lemma 12. If g I is concave then g I ( x ) + (1 − β ) x · [ g I ( x − − g I ( x )] is increasing for x ≥ Proof.
For x ′′ > x ′ we want to show g I ( x ′′ ) − g I ( x ′ ) + (1 − β ) x ′′ · (cid:2) g I ( x ′′ − − g I ( x ′′ ) (cid:3) − (1 − β ) x ′ · (cid:2) g I ( x ′ − − g I ( x ′ ) (cid:3) ≥ . By concavity, g I ( x ′ − − g I ( x ′ ) ≥ g I ( x ′′ − − g I ( x ′′ ) ≥
0. If x ′′ · [ g I ( x ′′ − − g I ( x ′′ )] ≥ x ′ · [ g I ( x ′ − − g I ( x ′ )] then we are done, so suppose this does not hold. Under this assumption,since (1 − β ) < g I ( x ′′ ) − g I ( x ′ ) + x ′′ · (cid:2) g I ( x ′′ − − g I ( x ′′ ) (cid:3) − x ′ · (cid:2) g I ( x ′ − − g I ( x ′ ) (cid:3) ≥ . x ′ [ g I ( x ′′ − − g I ( x ′ )], we can rewrite the left hand side as g I ( x ′′ ) − g I ( x ′ ) + ( x ′′ − x ′ ) · (cid:2) g I ( x ′′ − − g I ( x ′′ ) (cid:3) − x ′ · (cid:2) g I ( x ′ − − g I ( x ′ ) − [ g I ( x ′′ − − g I ( x ′′ )] (cid:3) If g I is concave, g I ( x ′′ ) − g I ( x ′′ − ≤ [ g I ( x ′′ − − g I ( x ′ − / ( x ′′ − x ′ ). Using this inequality,we obtain g I ( x ′′ ) − g I ( x ′ ) + ( x ′′ − x ′ ) · (cid:2) g I ( x ′′ − − g I ( x ′′ ) (cid:3) − x ′ · (cid:2) g I ( x ′ − − g I ( x ′ ) − [ g I ( x ′′ − − g I ( x ′′ )] (cid:3) ≥ g I ( x ′′ ) − g ( x ′′ − − [ g I ( x ′ ) − g ( x ′ − − x ′ · (cid:2) g I ( x ′ − − g I ( x ′ ) − [ g I ( x ′′ − − g I ( x ′′ )] (cid:3) = (1 − x ′ )[ g I ( x ′ − − g I ( x ′ )] − (1 − x ′ ) · [ g I ( x ′′ − − g I ( x ′′ )] ≥ g I ( x ′′ ) − g I ( x ′′ − ≤ [ g I ( x ′′ − − g I ( x ′ − / ( x ′′ − x ′ ),and the final inequality from x ′ ≥ g I .The discussion thus far of the MVPD problem is summarized in the following proposition Proposition 4.
Assume g I is concave and increasing and U F is supermodular and con-cave in its second argument. Then the MVPD’s objective in (11) satisfies the conditions ofPropositions 1 and 2.If we assume that U F ( v, · ) is increasing for all v then a sufficient condition for no individ-uals to be excluded is that ϕ ( v I ) >
0. If this holds then all individuals will be made worseoff by increasing differences shifts in the payoffs of the highest type channels.The assumption that viewers care only about the number of channels they have accessto may seem unnatural. High type channels as those which are best able to convert viewerswho have access to their channel into profits, which are generated by add revenue. A naturalinterpretation is that these channels attract a larger portion of the available viewers thenlow type channels, but this interpretation suggests non-trivial viewer preferences. Viewerswith the same ordinal preferences over channels can be accommodated, especially if theordering corresponds to the ordering on channel types. However if viewers have differentordinal preferences over channels then threshold matchings may not be optimal. None theless, conditional on the MVPD choosing to offer threshold matchings all viewers will preferlarger matching sets. In reality, MVPDs almost always offer a menu of nested bundles. Thismay be because there is uncertainty about which programs channels will offer, so that viewersdo not in fact have strong preferences over channels ex-ante, although they may develop suchpreferences after purchasing a bundle. It could also be because the distribution of preferencesconditional on vertical types v I is such that profitable screening along this dimension is notpossible. In this sense, the model seems to be a reasonable approximation of reality.24 will now discuss situations in which increasing differences changes in firm payoffs makeall individuals worse off. Changes in viewing patters.
The most obvious cause of an increasing differences change in firmpayoffs is technological change - direct shifts in firm payoffs. In this context, technologicalchange could mean better programming or changes in viewing patterns that make somechannels relatively more attractive to advertisers than others. A “high type” channel in thissetting is one that most effectively converts viewers who have access to the channel intoadd revenue. This could be because of the demographic composition of their viewers or thebroadness of their appeal. High type channels can be identified as those offered in the basiccable packages, such as CBS and FOX, whereas low type channels have more niche appeal,such as Animal Planet or HBO. Suppose viewers become less interested in watching thenightly news on NBC, and more interested in watching serial shows such as those offered byHBO. This can be interpreted as an increasing differences change in the payoffs of high typechannels relative to NBC. Assuming U F is affine in its second argument, by Lemmas 5 and7, we would expect such a change to lead to more packages that include HBO, with pricesadjusting in a way that makes consumers better off. Horizontal mergers.
Suppose there is a merger between two channels. Mergers are oftenexecuted because of “synergies” between the firms involved. Moreover, firms will often defendthe proposed merger against anti-trust challenges by arguing that synergies, in particularthose that reduce costs, will benefit consumers.As an example to illustrate mergers with cost synergies in the context of this model,suppose that given a matching set s , a channel chooses the quality q of its programming tomaximize ( r ( q ) − c ( v, q )) · | s | S I . Here | s | S I is interpreted as the channels viewership potential, r ( q ) is the realized add revenue per potential viewer, and c is the cost per-viewer of producingprogramming. Assume r is increasing and c is submodular (decreasing differences in v, q ).The firm’s optimal choice of program quality is independent of its viewership, and increasingin v , so U F ( v, x ) = max q ( r ( q ) − c ( v, q )) · | s | S I is supermodular and linear in its secondargument.Cost synergies entail a reduction in the marginal cost of production for the firms involved.In this context, suppose firms with values v ′′ and v ′ merge. Because of cost synergies, they willeach have new cost functions ˆ c that are point-wise lower than their original costs. Then theirmargins will increase, i.e. max q r ( q ) − ˆ c ( v, q ) will be larger. This is an increasing differences An alternative formulation for channel profits would be r ( q ) g ( ·| s | S I ) − c ( v, q ), i.e. zero marginal costs. Inthis case however the value function is supermodular, but it will only be concave if g is sufficiently concave. Vertical mergers.
Suppose the MVPD purchases a channel. This has two effects on itsmatching problem. One is that the MVPD will now appropriates the entire surplus generatedby the purchased channel. This will have the effect of an increasing differences shift in thischannel’s payoffs. However the acquisition also affects the negotiations with other firms. Ifa channel fails to reach an agreement with the MVPD and is dropped from the packages,some viewers who would have had access to both this channel and the MVPD-owned channelwill migrate to the latter. This benefits the MVPD, whereas before the merger the MVPDappropriated none of this additional benefit. This change in the outside option of the MVPDis termed the “bargaining leverage over rivals” (BLR) effect by Rogerson (2019). The effecton the structure of bundling will depend on the magnitude of the BLR effect.If the MVPD purchases firm 1 then the price negotiated with channel j is given by ˜ p ∗ j = arg max p (cid:16) p + R ( s ) − R Oj ( s ) + θ (cid:16) U F ( v F , | s F (1) | S I ) − U F ( v F , | s F (1) | ˜ S jI ) (cid:17)(cid:17) β ( U F ( v Fj , | s F ( j ) | S I )) (1 − β ) = βU F ( v Fj , | s F ( j ) | S I ) + (1 − β ) θ (cid:16) R Oj ( s ) − R ( s ) + U F ( v F , | s F (1) | ˜ S jI ) − U F ( v F , | s F (1) | S I ) (cid:17) where | s F ( j ) | ˜ S jI is quality of channel 1’s matching set when firm j is dropped from allmatchings. The parameter θ ∈ [0 ,
1] captures the degree to which the MVPD internalizesthe change in the viewership of the purchased channel should negotiations break down withchannel j . If θ = 0 then there is no MVPD effect. The MVPD’s total revenue is N X j =2 ˜ p ∗ j + U F ( v F , | s F (1) | S I )+ R ( s ) = N X j =2 p ∗ j +(1 − β ) θ N X j =2 (cid:16) U F ( v F , | s F (1) | ˜ S jI ) − U F ( v F , | s F (1) | S I ) (cid:17) + R ( s ) β N X j =2 U F ( v Fj , | s F ( j ) | S I ) + U F ( v F , | s F (1) | S jI )+ Z ¯ v I v I ϕ ( v ) (cid:0) g I ( | s I ( v ) | ) + (1 − β ) | s I ( v ) | · [ g I ( | s I ( v ) | − − g I ( | s I ( v ) | )] (cid:1) dQ I ( v ) − (1 − β ) Z s F (1) ϕ ( v ) · [ g I ( | s I ( v ) | − − g I ( | s I ( v ) | )] dQ I ( v )+ (1 − β ) θ N X j =2 (cid:16) U F ( v F , | s F (1) | ˜ S jI ) − U F ( v F , | s F (1) | S I ) (cid:17) . (12)The first two lines of (12) are the same as the original MVPD revenue in (11), except that theweight on channel 1’s revenue is now 1 instead of β . The third and fourth lines complicatethe analysis. The third line reflects the fact that the individual side payoffs in the even ofa breakdown in negotiations with firm 1 need no longer be considered. The fourth line isthe BLR effect. Here payoffs do not have the same form as discussed above, and we can notdirectly apply or previous results. Under some additional assumptions however we can drawwelfare conclusions. First, assume that there is no BLR effect, so θ = 0 and the fourth line of (12) disappears.Second, suppose g I affine with slope a . Then the second line of (12) reduces to(1 − β ) a Z s F (1) ϕ ( v ) dQ I ( v ) . The only effect of this on the MVPD’s problem is to increase the marginal benefit of addingviewers to the matching set of firm 1. Thus the result of Proposition 2 will continue to hold.
Lemma 13.
Assume that there is no BLR effect, g I is affine, and U F ( v, · ) is concave. Thenif there is no exclusion of viewers and the MVPD purchases the highest type channel, allindividuals will be worse off. If instead U F ( v, · ) is affine and the MVPD purchases the lowesttype channel then all individuals will be better off.The BLR effect reduces the price that the platform pays to all channels. However it is notimmediately clear how this affects the marginal benefit of increasing bundle size. Analyzingthe merger when the BLR effect is present is an interesting objective, but one which I leavefor future work. Just assuming that U F ( v, · ) is affine should be enough to do comparative statics, but this will requireadditional work. .2 Platform-mediated monopolistic competition Retailers, particularly on-line retailers such as Amazon, exert considerable control over the setof products to which consumers have access. This influence may take the form of placementof products in stores or in search results pages, advertising, or product recommendations,among others. Firms care about how many customers see their product and how many otherproducts these customers see. Additionally, a firm’s payoffs will depend on the prices offeredby the competing firms to which its customers also have access. The retailer may exertvarying levels of control over firm pricing decisions. A grocery store may have a high degreeof control, whereas Amazon may have little direct say in the prices set by firms.I will consider a version of the classic Dixit and Stiglitz (1977) model of monopolisticcompetition. There are a continuum of firms and individuals. For simplicity, assume thatindividuals are identical. I will discuss later how individual heterogeneity can be accommo-dated. The driving force here will be the desire of the platform to screen firms that havedifferent marginal costs, and thus value access to consumers differently. The platform cannotdirectly control the prices set by firms, although it may do so indirectly through the choiceof the matching sets. This assumption fits best with a model of an online platform such asAmazon. I will show that platform payoffs in this setting reduce to the form studied in 2.
I first describe the demand of a customer who is matched with a set s of firms, where firm j has set price p ( j ). The individual chooses a quantity q ( j ) to purchase from firm j and howmuch money m to hold. The individual’s choice solvesmax q ( · ) ,m = m + 1 θ (cid:18)Z s q ( j ) σ − σ dj (cid:19) θσσ − s.t. Z s p ( j ) q ( j ) dj + m ≤ w (13)where w is the individual’s wealth, σ > θ ∈ (0 ,
1) and σ (1 − θ ) > Assume that wealthis high enough that individuals hold a positive amount of money. Define the price index formatching set s as P ( s ) = (cid:18)Z s p ( j ) − σ dj (cid:19) − σ . Then demand for money can be written as m ( s ) = w − P ( s ) θθ − Preferences of the form in (13) have been used by Bagwell and Lee (2018), Helpman and Itskhoki (2010),and Helpman and Krugman (1989) among others. j is given by q ( j | s ) = p ( j ) − σ · P ( s ) σ (1 − θ ) − − θ . The indirect utility of an individual is given by1 − θθ P ( s ) θθ − + w. A firm matched with a set s F ( j ) of customers faces an aggregate demand given by p ( j ) − σ · Z s F ( j ) P ( s I ( i )) σ (1 − θ ) − − θ di. Define h ( s ) ≡ P ( s I ( i )) σ (1 − θ ) − − θ . Assume that marginal costs are constant for all firms, butdiffer across firms. Marginal costs are the firm’s private information, which the platform willlike to elicit. For simplicity, assume that firms have no fixed cost. The firm’s price settingproblem can be stated asmax p p − σ · Z s F ( j ) h ( i ) di − c j p − σ · Z s F ( j ) h ( i ) di. The solution to this problem is to set p ( j ) = σσ − c j . The fact that the price does not dependon the matching set or the prices of other firms is the main benefit of assuming CES utilityand constant marginal costs. Define v j ≡ c − σj (recall that σ > s F ( j ) are then given by v j γ · Z s F ( j ) h ( i ) di where γ = (cid:16) σσ − (cid:17) − σ − (cid:16) σσ − (cid:17) − σ > s I are givenby w + (cid:18) − θθ (cid:19) (cid:18) σσ − (cid:19) θθ − Z s I v j dj θ ( θ − − σ ) . (14)This can be written as w + g I ( s I ). Derivations can be found in Appendix A Adding fixed costs, even if they differ across firms, does not change anything as long as all firms want tocontinue operating. .2.3 The platform There are a number of platform objectives that can be entertained in this setting. I willassume that the goal of the platform is to maximize a weighted sum of customer surplus andrevenue collected from firms. The platform may care about customer surplus because it wantsto attract customers. Alternatively, if the platform was also screening on the individual sidewe have seen how the individual-side objective looks like welfare maximization, with virtualvalues replacing true values. I could also consider a platform that collects transaction feesfrom users, and thus wants to maximize a weighted sum of customer transactions and revenuesfrom firms. The results discussed here would not change.Payoffs for the firm and individual take the form studied in section 2: the endogenoussalience of individual i , given by h ( i ), depends on the types of the firms in s I ( i ). Using thefirm’s pricing decision, we can rewrite P ( s I ( i )) in terms of firm types: h ( s I ( i )) = P ( s I ( i )) σ (1 − θ ) − − θ = ψ · Z s I ( i ) v j dj κ where κ = σ (1 − θ ) − − θ )(1 − σ ) ∈ ( − ,
0) and ψ = (cid:16) σσ − (cid:17) κ (1 − σ ) >
0. The platform wants to screenfirms based on marginal costs. Firm payoffs are of the form v j · g F ( s F ( j )), where g F islinear. Assuming that the distribution of v j is regular, the platform will solve for the optimalmatching by replacing v j with ϕ F ( v j ) = v j − − Q F ( v j ) q F ( v j ) , subject to monotonicity of firm sidematch quality in type.Ignoring individual wealth, which is fixed, the platform payoff can be written as ¯ v I Z v I g I ( s I ( v )) + γh ( s I ( v )) · Z s I ( v ) ϕ F ( v j ) dj dQ I ( v ) . (15)where g I ( s I ) = (cid:0) − θθ (cid:1) P ( s I ) θθ − . The platform chooses s I ( · ) to maximize (15), subject tomonotonicity of firm match quality. If the platform cannot discriminate between customers,meaning that all customers must receive the same matching set, then monotonicity of the firmside matching implies that individual matching sets must be characterized by a threshold in v F . If the platform is allowed to offer different matching sets to different individuals however,the problem is not necessarily separable across customers due to the firm-side monotonicityconstraint. Given the separable structure of the objective, Lemma 8 implies that the thresholdstructure will hold, under a slight strengthening of regularity.30 emma 14. Maximizing the integrand in (15), i.e. solving the problem separately for eachindividual, yields matchings characterized by a threshold in ϕ F ( v ) /v ; a firm with type v ismatched with the individual if and only if ϕ F ( v ) /v is high enough.Under the assumption that ϕ F is increasing, ϕ F ( v ) /v will always be increasing over theset of v such that ϕ F ( v ) <
0. This means that if we know that all firms with positivevirtual values are matched with all individuals then under regularity we can conclude thatthe individuals’ matching sets are characterized by a threshold in v F . In general however wecan only conclude this when ϕ F ( v ) /v is increasing. Corollary 4. If ϕ F ( v ) /v is increasing then maximizing (15) pointwise yeilds matching setscharacterized by a threshold in v F .If ϕ ( v ) /v is increasing, or if the platform is not able to discriminate between individuals,then the problem is separable accross individuals, and can be solved by maximizing theintegrand in (15). Lemma 14 implies that the solution to this problem is unique, so allcustomers will receive the same matching set, and we can talk about the “representativecustomer”.Even when the problem is separable, comparative statics in this setting are complicatedby the fact that firm types enter into both the endogenous salience of individuals and theprofitability of firms. This is not an issue however if we consider changes that affect thevirtual values without changing the true firm types, in which case we can apply Lemma 9. Iwill consider two such changes; if the platform receives more precise information about firmtypes, and if the platform purchases a subset of high type firms. The following Lemma is theimmediate implication of 9 in this setting. Lemma 15.
Assume ϕ F ( v ) /v is increasing. Consider a set of firms, all of which are matchedwith the representative customer. If the virtual values of these firms increase then the repre-sentative customer will be worse off and receive a smaller matching set.Suppose the platform receives some information about the vertical types of firms. Forsome partition τ of the set [ v F , ¯ v F ] of potential firm types the platform learns to whichpartition cell each firm belongs. The platform therfore only needs to worry about firmsdeviating to types that are in the same cell; the mechanism design problem on the firm sideseparates completely accross cells. Thus the virtual values are computed cell-by-cell; for afirm v in cell [ v ∗∗ , v ∗∗ ] the virtual value is v − Q F ( v ∗∗ ) − Q F ( v ) q F ( v ) . Moreover the monotonicityconstraint is need only be satisfied within each cell. Say that a partition cell is included if allfirms in that cell are matched with the representative customer. The following are corollariesof Lemma 15. 31 orollary 5. Assume ϕ F ( v ) /v is increasing within each cell. If the platform purchases anincluded partition cell of firms then the customer recieves a smaller matching set and is worseoff. Proof.
If the platform purchases an included partition cell it replaces the virtual values ofeach firm in this cell with the true values, which are higher. There is no change in the virtualvalues of firms in other cells.Assume ϕ F ( v ) /v is increasing. Corollary 5 and Lemma 14 imply that if the inversehazard rate is decreasing then the customer receives a smaller matching set. Even withoutthis assumption, the merger makes the customer worse off, as the platform seeks to divertcustomers to the firms that it owns. Similarly, if the platform gets better information aboutthe values of firms that are already included, it will be able to extract more surplus fromthese firms, and would thus like to divert customers to them. Corollary 6.
Assume ϕ F ( v ) /v is increasing within each cell. If the platform’s partition overa set of included firms becomes finer then the customer recieves a smaller matching set andis worse off. Proof.
If a given partition cell [ v ∗∗ , v ∗∗ ] is subdivided then any firm in [ v ∗∗ , v ∗∗ ] will be in anew cell with an upper bound that is below v ∗∗ , and strictly so for some firms. Then thevirtual values of these firms will be higher.The natural counterpoints to 7 and 8 also obtain; if the merger or information aquisitionconcerns cells of excluded firms, firms with which the customer is not matched, than thecustomer recieves larger matching sets and is better off A natural dimension of customer heterogeneity is the degree to which customers value thegoods sold on the platform relative to money. With quasi-liner preferences wealth hetero-geneity does affect the purchase decisions. In some sense, the trade-off between goods andmoney is captured by the parameter θ , so we can consider heterogeneity in this dimension.However higher θ does not exactly capture an intuitive notion of valuing goods relativelymore. The quality of a customer’s matching set is given by R s I v j dj . Customer preferencesare supermodular in θ and match quality if and only if P ≥
1. In fact, if
P < θ . Thus, I will consider an slight modifi-cation of customer preferences which admits a more natural interpretation of the individualtype: let preferences for a type v I customer be represented by m + v − θI θ Z s q ( j ) σ − σ dj θσσ − where v I is the individuals type. It is easy to show, following the same steps as before, thatdemand for good j is given by q ( j | s, v I ) = v I · p ( j ) − σ · P ( s ) σ (1 − θ ) − − θ and customers’ indirect utility is given by v I (cid:18) − θθ (cid:19) P ( s ) θθ − . Define h ( s | v I ) ≡ v I · P ( s ) σ (1 − θ ) − − θ . The remainder of the derivations are as before.If the platform can observe individuals’ types, which may not be an unreasonable assump-tion for online platforms with access to detailed information about their customers, not muchchanges in the above analysis. Lemma 8 applies, so if ϕ F ( v ) /v is increasing then matchingsfor each individual have a threshold structure and the firm-side monotonicity constraint doesnot bind. Then the conclusions Corollaries 5 and 6 continue to hold for each individual:each individual is matched with a smaller set of firms if the platform purchases or gainsmore precise information about an included cell of firms. This makes individuals worse of ifthe platform is maximizing the weighted sum of customer welfare and firm-side revenue. Ofcourse, if the platform can use transfers to extract surples from individuals then individualsrecieve zero surpluss in either case.We can also consider the situation in which an individual’s type is their private infor-mation. If the platform can only control individual payoffs through the matching set, forexample if monetary transfers are not possible, then again all individuals will receive thesame matching set. Lemmas 14 and 15; and Corollaries 5 and 6 continue to hold for allindividuals.The primary case of interest is when individuals are privatly informed about their type,and transfers can be made between individuals and the platform. The necessary and sufficient There is no technical problem with considering heterogeneity in θ , although it does complicate the analysiswhen we consider the case in which the type is an individual’s private information and there are transfersbetween the platform and individuals. The issue is one of interpretation. g I be defined as above.The platform may still seek to maximize the sum of total individual-side welfare and netrevenue, meaning the sum of transfers from firms and customers. Aggregate individual-sidewelfare can be written as U ( v I ) + ¯ v I Z v I g I ( s I ( v ))(1 − Q I ( v )) dv. The platform does not benefit from transferring money to individuals since its payoffs arelinear in money, so it is without loss to set U ( v I ) = 0. The platform objective is to maximize ¯ v I Z v I g I ( s I ( v ))(1 − Q I ( v )) + vγh ( s I ( v )) · Z s I ( v ) ϕ F ( v j ) dj dQ I ( v ) (16)subject to monotonicity of firm and individual match qualities. Lemma 14 applies here, so if ϕ F ( v ) /v is increasing then maximizing the integrand in (16) yields threshold matching setsfor each individual.If we ignored the firm side, maximizing individual-side welfare would imply full ironingsince 1 − Q I ( v ) is decreasing. All individuals would receive the same quality matching set, andthus the same set given that threshold matchings are optimal. In fact, the firm side reinforcesthis effect. The platform faces a trade-off between higher customer welfare and greater firm-side revenue. Since firm side revenue scales with v I , while 1 − Q I ( v I ) is decreasing, theplatform will prioritize firm side revenue when dealing with higher type individuals. Lemma 16.
Assume ϕ F ( v ) /v is increasing. If the platform’s objective is to maximize (16),the sum of individual-side welfare and revenue, then there is full pooling: the optimal match-ing is the same for all individuals. Proof.
Suppose platform maximized its objective separately for each individual, ignoringthe monotonicity constraint. I want to show that higher type individuals would get worsematching sets. The objective for an individual with type v I can be written as v I g I ( s I ) (1 − Q I ( v I )) v I + γψ Z s I v j dj κ · Z s I ϕ F ( v j ) dj . This objective satisfies single-crossing in v I and − R s I v j dj , so higher types get worse matchingsets. Since point-wise maximization yields a decreasing allocation there is full ironing: theoptimal matching subject to monotonicity has the same quality for all individuals. Since ϕ F ( v ) /v is increasing this implies that all matching sets will be the same.34emma 16 is consistent with some observed patterns of platform structure. Early stageplatforms, which are focused on attracting new users, often offer the same service to allcustomers. Later, when the user base has been established, do platforms begin do discriminatebetween customers. As we will see below, such discrimination arises when the platform seeksto translate individual-side users into revenue.Given Lemma 16 we can again talk about the “representative customer” who in this casehas a type given by the average type in the population. Lemma 15 and Corollaries 5 and 6continue to apply.Finally, suppose the platform also wants to maximize firm-side and individual-side rev-enue. Individual side revenue can be written as U ( v I ) + ¯ v I Z v I ϕ I ( v ) · g I ( s I ( v )) dv where ϕ I ( v ) = v − − Q I ( v ) q I ( v ) . The platform can extract full surpluss from the lowest type, so U ( v I ) = 0. The platform objective is to maximize ¯ v I Z v I ϕ I ( v ) · g I ( s I ( v )) + vγh ( s I ( v )) · Z s I ( v ) ϕ F ( v j ) dj dQ I ( v ) (17)subject to individual and firm side monotonicity. We can rewrite the integrand as v ϕ I ( v ) v g I ( s I ( v )) + γh ( s I ( v )) · Z s I ( v ) ϕ F ( v j ) dj . This objective satisfies single-crossing in ϕ I ( v ) /v and R s I v j dj , which has the following impli-cation. Lemma 17.
Point-wise maximization of total revenue, given by (17), yields individual matchqualities that are increasing in ϕ I ( v ) /v .An immediate implication of Lemma 17 is that point-wise maximization will yield mono-tone allocations if ϕ I ( v ) /v is increasing. This claim has a partial converse: if ϕ I ( v ) /v is notincreasing then point-wise maximization will violate monotonicity, unless the violations ofincreasing ϕ I ( v ) /v happen to occur for individuals that are either excluded or fully matched.Assuming that both ϕ I ( v ) /v and ϕ F ( v ) /v are increasing. Then monotonoicity constraintsdo not bind for either individuals or firms. Thus we can apply Lemma 8. and Corollaries 5and 6 apply to each individual: each individual is matched with a smaller set of firms if theplatform purchases or gains more precise information about an included cell of firms. Usingthe envelope expression for individual payoffs we have the following welfare conclusions.35 orollary 7. Assume ϕ F ( v ) /v is increasing within each cell. If the platform purchases acell of firms that is included for all customers then all customers are worse off. Corollary 8.
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