Revenue from Matching Platforms * Philip Marx † James Schummer ‡ February 28, 2019 Abstract We consider the pricing problem of a match-making platform where heterogeneous agents create exclusive, stable matchings and pay match- contingent fees. Absent prices, agents on the short side of such markets capture relatively greater surplus than those on the long side (Ashlagi et al. (2017)). Nevertheless we show that this need not induce the platform to bias its price allocation on the basis of market imbalance: with independently drawn preferences, optimal price allocation deci- sions are independent of market size or imbalance. Larger markets induce higher price levels, leading to price increases for both sides of the market; consequently the “cost of stability” imposed on the plat- form becomes negligible. In contrast, preference correlation leads to price allocation bias that depends on an interaction between market imbalance and preference correlation; changes in market imbalance lead to opposing price changes for the two sides. This interaction is a novel effect that arises from the exclusivity of interactions in a two-sided market. * We are grateful to multiple referees for comments that have improved the writing of this paper. This project was partially funded by NSF grant #1534138. † Education Innovation Laboratory, Harvard University, 1280 Massachusetts Ave, Cam- bridge, MA 02138. Email: [email protected]. ‡ MEDS Department, Kellogg School of Management, Northwestern University. Email: [email protected]. 1
Transcript
Revenue from Matching Platforms∗
Philip Marx† James Schummer‡
February 28, 2019
Abstract
We consider the pricing problem of a match-making platform whereheterogeneous agents create exclusive, stable matchings and pay match-contingent fees. Absent prices, agents on the short side of such marketscapture relatively greater surplus than those on the long side (Ashlagiet al. (2017)). Nevertheless we show that this need not induce theplatform to bias its price allocation on the basis of market imbalance:with independently drawn preferences, optimal price allocation deci-sions are independent of market size or imbalance. Larger marketsinduce higher price levels, leading to price increases for both sides ofthe market; consequently the “cost of stability” imposed on the plat-form becomes negligible. In contrast, preference correlation leads toprice allocation bias that depends on an interaction between marketimbalance and preference correlation; changes in market imbalancelead to opposing price changes for the two sides. This interactionis a novel effect that arises from the exclusivity of interactions in atwo-sided market.
∗We are grateful to multiple referees for comments that have improved the writing ofthis paper. This project was partially funded by NSF grant #1534138.†Education Innovation Laboratory, Harvard University, 1280 Massachusetts Ave, Cam-
bridge, MA 02138. Email: [email protected].‡MEDS Department, Kellogg School of Management, Northwestern University. Email:
The proliferation of online matching platforms has led to intensified interestin the study of platform pricing. The topic becomes increasingly importantas dominant platforms emerge in “winner-take-all” environments.1 While anestablished literature on two-sided markets explains much about pricing oncertain kinds of platforms (see Subsection 1.3), it has mostly set aside twomarket characteristics described below that apply to many real world plat-form environments that we consider here. Specifically, we are interested insettings where exclusive (one-to-one) partnerships are created by horizontallydifferentiated agents.
(I) Exclusivity. Canonical models of two-sided market platforms repre-sent environments in which each participating agent interacts with all (or aconstant fraction) of the agents on the other side of the market. These mod-els accurately portray oft-cited examples of platforms such as credit cards(connecting consumers and merchants), video game consoles (game playersand developers), and newspapers (readers and advertisers). Yet many promi-nent platforms exist specifically to create one-to-one matchings, such as asAirbnb, Uber, TaskRabbit, online dating services, and others.2
(II) Heterogeneity. A second characteristic common to many two-sidedmarket models is the assumption that an agent obtains the same transactionvalue from any partner with whom they transact. This homogeneity assump-tion, like non-exclusivity, is realistic in certain environments (e.g. an agent’svalue for a credit card platform is roughly proportional to its adoption rateby agents on the other side of the market). In other environments—especiallythose exhibiting exclusivity—agents have heterogeneous preferences over po-tential matches on the other side. Obvious examples include online dating,
1Considering what regulatory rules ought to apply in such settings, The Economist(“Online Platforms: Nostrums for Rostrums,” May 28, 2016) convincingly argues that“established rules of regulation often do not apply.”
2Even when matchings are not perfectly exclusive, such as with UberPool or casualdating sites, capacity constrained agents may be better approximated by one-to-one mod-els than by “all-to-all” models in which payoffs scale proportionally with the number ofparticipants on the other side.
2
or Airbnb guests whose tastes differ over housing types and locations.3 Inthese kinds of settings, heterogeneous preferences on one side of the markethorizontally differentiate the agents on the other side.
Our interest in characteristics (I) and (II) leads to a third distinctionbetween existing work and our own involving the matching process itself.Under exclusive matchings and heterogeneous preferences, agents care notonly about the price they pay to the platform but about the identity ofthe partner with whom they are matched.4 While the platform controls thesetting of prices, the matching outcome is often determined by the behaviorof the agents, especially on decentralized platforms. This leaves the platformwith the question of which matching outcomes actually occur.
This question is addressed by both theoretical and empirical work suggest-ing that, at least when search frictions are low, agents on such decentralizedplatforms create stable matchings in the sense of Gale and Shapley (1962).Adachi (2003) considers a decentralized matching model in which, in everyround, each agent randomly encounters an agent from the other side of themarket. A pair either becomes matched (if both agents mutually agree toit) or continue to another random encounter. Adachi’s elegant conclusionis that, as search frictions (via discount rates) become negligible, the set ofequilibrium payoff profiles in this game converges to the set of payoff profilesobtained over the set of all stable matchings.
Empirically bolstering this result, Hitsch et al. (2010) examine the char-acteristics and behavior of users on an online dating site. Their centralfinding is that the matching outcomes observed on the site are similar tothose that would be obtained by running Gale and Shapley’s (1962) De-ferred Acceptance algorithm on the estimated preferences of the agents. Inother words, the decentralized, low-friction platform they study creates thekinds of matching outcomes predicted by Adachi (2003).5 Motivated by the
3Even Airbnb hosts have varying preferences over types of guests, e.g. with pets, chil-dren, or other particular needs. Numerous subjective articles on the internet advise Airbnbhosts on strategies for selecting among guest applications, strengthening our opinion thatpreferences are heterogeneous on both sides of this market.
4In contrast, the matching process is somewhat trivialized when homogeneous agentsface no capacity constraint: a pair of agents becomes matched if and only if it is mutuallybeneficial with respect to prices.
5Importantly, Hitsch et al. (2010) also deduce that “sorting is largely driven by prefer-ence heterogeneity.” This heterogeneity not only strengthens our motivation for charac-teristic (II) above, but shows that stability on their platform is not simply a consequence
3
these findings, this paper specifically considers matching platforms that yieldstable matchings.
Our focus on stability has been motivated by the above-cited work ondecentralized platforms that predicts stable outcomes on decentralized plat-forms. This permits us to “black box” the physical matching process itself,and bypass any modeling of the agents’ behavior on the platform. At thesame time, of course, our results would also apply to settings where a cen-tralized platform can implement stable matching outcomes through the useof some centralized mechanism. Whether such centralized implementationis possible depends on informational and strategic assumptions about theparticipating agents; e.g. see Roth (1984a), Kara and Sonmez (1996), orAlcalde (1996).
1.2 Overview
Though characteristics (I)—(II) have not been the primary focus of the two-sided markets literature pioneered by Rochet and Tirole (2003) (see Sub-section 1.3), a separate literature dating back to Gale and Shapley (1962)considers two-sided matching models in which these three characteristics playa central role. Despite the natural connection between these matching modelsand real world platform environments, to date there has been no work apply-ing those models to the platform’s pricing problem. Our work fits betweenthese two literatures by considering how a monopolistic platform would setseparate prices to two sides of a matching market based on market parame-ters, such as the distributions of agents’ values, correlation in values, marketsize, and the degree of market imbalance (relative sizes of the two sides).
It is important to note that in classic (many-to-many) two-sided marketsmodels, only the first of these parameters—distributions of agents’ values—is relevant in determining the platform’s expected revenue. In contrast, ourfindings collectively show that all four of these parameters are relevant in oursetting. Generally speaking, market size drives the platform’s price level,while an interaction between preference correlation and market imbalancedrives the platform’s price allocation, i.e. its tendency to charge one side ofthe market a relatively higher price than the other.
To demonstrate how, we begin with the price allocation question in asetting with independently drawn preferences (Section 3). Our results in
of assortative matching among vertically differentiated agents.
4
this baseline case appear counterintuitive with respect to a recent result inthe literature on stable matching. Namely, in a classic stable marriage modelwith independently drawn preferences (and no pricing), Ashlagi et al. (2017)show the following. In large markets with essentially any degree of marketimbalance, average (normalized) payoffs are notably higher for agents on theshort side of the market than for agents on the long side.
Following their result, one might infer that a monopolist platform, actingas gatekeeper to the stable matching process, should capture these imbal-anced payoffs by charging a relatively higher price to the short side of themarket. This turns out not to be the case when platforms charge agents inthe form of match-contingent fees. Instead, we prove a “symmetric-pricing”result stating that a relative size imbalance between the two sides is notper se a justification for such third-degree price discrimination, despite theasymmetric-payoff result of Ashlagi et al. (2017). In fact, holding fixed anyprice level, the platform’s price allocation decision is independent of the de-gree of market imbalance. In the special case that agents’ values on bothsides are independently drawn from the same distribution, a standard haz-ard rate condition leads the platform to charge the same price to both sidesof the market, regardless of their relative sizes. Consequently, optimallypriced agents on the short side of such markets can capture higher averagenet surplus than those on the long side, even conditional on being matched.
These no-price-bias results might appear to resemble a similar phenomenonin many-to-many, two-sided market models (Subsection 1.3). In those mod-els, conventional wisdom is to subsidize the price-sensitive side of the market.This implies, for example, that if all agents’ values are identically distributed,then the platform equates prices across the two sides of the market. Thoughwe obtain the same conclusion (with independently drawn preferences), thecomparison is misleading. In many-to-many models without capacity con-straints, the price sensitivity argument applies regardless of any correlationin agent’s values. Intuitively, since the platform is essentially pricing eachpotential transaction separately in such models, correlation plays no role inevaluating expected revenue. On the other hand, when values are correlatedin capacity constrained models such as ours, market imbalance does affect theplatform’s price allocation decision, e.g. leading to different pricing across thetwo sides even when match values are identically distributed. In Section 5 weexplore this issue further, showing that both the direction and scale of priceallocation bias depend on an interaction between market imbalance and theform of preference correlation.
5
Turning to the platform’s choice of price level (Section 4), we first providean expression that yields a lower bound for the stable platform’s expectedrevenue at any given prices. The expression corresponds to the expectednumber of matches created under a “constrained serial dictatorship” whereagents on one side sequentially choose their favorite mutually compatiblepartner from the other side.6 We use this relationship to show that as marketswith independently drawn preferences grow arbitrarily large, the platform’s“cost of stability” vanishes, in the sense that, fixing any prices, the absolutenumber of matched pairs that the platform fails to create due to the stabilityconstraint is bounded by a constant across all market sizes. That is, evenif the platform could rematch agents in any arbitrary, individually rationalway, its revenue would improve by only a vanishingly small percentage.
1.3 Related literature on two-sided markets
The two-sided markets literature pioneered by Rochet and Tirole (2003) con-siders profit-maximizing platforms that match two sets of agents who derivevalue from interacting with each other. In contrast to our characteristics (I)–(II) described earlier, the models in this literature typically exhibit primitiveswith the following two features:
All-to-all: each agent receives constant marginal benefit from each addi-tional participant on the other side of the market;
No differentiation: each agent perceives the other side’s participants asindistinguishable.
These features are implied by the modeling assumption that an agent’s payoffis some affine function, say a + bn, of the number of agents n on the otherside of the market. The agents’ fixed (a) and per-transaction (b) benefitsmay or may not be assumed to vary across agents.7
This affine payoff structure easily captures so-called cross-network effects,where agents benefit from additional participation on the other side. These
6A ranking between stability and a form of serial dictatorship is established in a relatedmodel by Arnosti (2016).
7This is the essential payoff structure in models such as Rochet and Ti-role (2003),(2006), Armstrong (2006), and Weyl (2010). More broadly, the exclusive-matching model of Caillaud and Jullien (2003) also yields this structure of expected pay-offs.
6
effects yield one of the fundamental lessons taken from the two-sided marketsliterature: A profit-maximizing platform should not set prices to the two sidesof the market independently, as if it were pricing two unrelated products.Instead, an increase in the per-transaction price charged to one side of themarket should be viewed as a decrease in the marginal cost of providingtransactions to the other side of the market.
Two-sided market models with the features described above elegantlycapture these kinds of effects. On the other hand, the all-to-all and nodifferentiation features tend to neutralize both market size effects and pref-erence correlation effects on the platform’s pricing decision.8 The intuitionfor this is that, since agents cannot be crowded out (all-to-all) and viewall potential partners identically (no differentiation), the platform can es-sentially view each potential pair of agents as a separate, individual pricingproblem, independent of the presence of other agents: Prices that maximizerevenue across the entire platform are identical to those that would maxi-mize expected revenue from any individual pair. Therefore these prices areindependent of absolute and relative market sizes, and of any correlation inagents’ preferences.
In contrast, market size does impact pricing in the absence of the all-to-alland no differentiation features of the classic models. In fact our one-to-onemodel exhibits both absolute and relative market size effects. The absolutesize effect is intuitively straightforward: as markets grow thicker (i.e. largerin absolute size), optimal prices increase due to the increased likelihood thata large fraction of the agents can simultaneously be matched with unique,high-quality partners. The relative size effect, as discussed in Subsection 1.2,only comes about when preferences are correlated, and depends on the formof correlation (Section 5).
Various other topics beyond this paper’s scope are addressed in the two-sided markets literature, such as platform competition and information struc-ture. Work on the former topic emphasizes the “divide and conquer” theme,formalizing the idea that platforms may subsidize a “critical” side of themarket and recover profits from the other. As Armstrong (2006) points out,if agents on only one side of the market must “single-home” (commit to a sin-gle platform), then platforms will compete for them (through low/subsidized
8An interesting exception unrelated to our work is Ambrus and Argenziano (2009) whoconsider competing platforms that endogenously create a kind of market size effect. Intheir equilibrium, one platform ends up being cheaper and larger on one side of the marketwhile the other ends up cheaper and larger on the other side.
7
pricing) while charging the “multi-homing” side monopoly prices.Damiano and Li (2008) consider competing platforms where agents are
randomly matched one-to-one and each payoff is the product of the pair’stypes (see also Damiano and Li (2007)). One of their main points is thatprices can lead to an assortative segmentation of agents across platforms.Besides the issue of competition, there are other critical differences betweentheir model and ours. Damiano and Li’s agents have the same (ordinal) pref-erences over the “vertically differentiated” agents on the other side (analogousto one form of correlation we consider in Section 5). Second, their randommatching assumption bypasses market size effects in the same way that all-to-all models do; this is natural since their motivation is the question ofmarket composition or quality. Third, any agent in their model is unaffectedby the addition of a new agent to the same side of the market, which is notthe case in our model (Gale and Sotomayor (1985a)).
Finally, a series of papers considers the impact of information structureon a (many-to-many) platform’s mechanism design problem. Fershtman andPavan (2016) consider settings where private information is persistent overtime. Where matchings can change over time, they show that optimal mecha-nisms are dynamic auctions that determine matchings by applying a scoringrule to reported preferences. Gomes and Pavan (2016) consider the inter-action between pricing and matching rules in large, many-to-many marketswhere agents have private information about their own values and those ofagents on the other side of the market. Jullien and Pavan (2016) study thedesign of information management policies when agents are uncertain aboutthe participation decisions of other agents; this uncertainty does not applyin our setting since our platform guarantees ex post individual rationality.
2 Model
2.1 Primitives
There are two finite sets of agents, referred to as men M = {1, 2, . . . ,M}and women W = {1, 2, . . . ,W}. A (one-to-one) matching is a functionµ : M ×W → M ×W satisfying the following, usual conditions for all m ∈M,w ∈ W : (i) µ(m) ∈ W ∪ {m}, (ii) µ(w) ∈ M ∪ {w}, and (iii) µ(m) = wif and only if µ(w) = m. Agent i ∈ M ∪W is unmatched (or single) at µwhen µ(i) = i.
8
If man m ∈ M is matched to woman w ∈ W , m obtains value um(w) ∈[0, 1] and w obtains uw(m) ∈ [0, 1]. The value of being unmatched is zero(denoted ui(i) ≡ 0 when necessary). These normalizations are not critical toour results. Each value um(w) is randomly drawn according to a marginaldistribution FM, and each uw(m) is drawn according to FW , where the cor-responding densities are continuously differentiable with positive support on[0, 1]. A random economy is one in which each value ui(j) is drawn inde-pendently of all other values. We consider random economies in Section 3and Section 4 and introduce correlated values in Section 5.
Agents make transfers to the platform as described in Subsection 2.2.Under a matching µ, an agent i who makes a payment x to the platformreceives a payoff of ui(µ(i))− x.9
2.2 The matching platform
Our interest is in platforms that charge prices in the form of “match-contingentfees.” Fixing M and W (i.e. the size of the market), we assume that theplatform can charge an agent only as a function of (i) the side of the mar-ket to which the agent belongs, and (ii) whether the agent ends up beingmatched. That is, prices are a pair p = (pM, pW) ∈ R2, where matchedmen and women are respectively charged pM and pW , while the payments ofunmatched agents are normalized to zero.10
Motivated by the theoretical and empirical work discussed in Subsec-tion 1.1 (e.g. Adachi (2003), Hitsch et al. (2010)), our focus is on decentralizedplatforms that yield stable matchings in the sense of Gale and Shapley (1962).Since we bypass the process by which these matchings are actually created,we have no need to make informational assumptions on the part of the agents.
The definition of stability is partially endogenous when a platform has theability to set prices. An agent who is comparing his/her current match statusto a departure from the platform (individual rationality) or to an alternativepartner (pairwise blocking) does so in consideration of the platform’s prices.11
9The quasilinear form of payoffs is without loss of generality as long as agents’ prefer-ences are assumed to be monotonic in value and transfers.
10In our model it is without loss of generality to restrict the ranges of prices to be thesupports of FM, FW .
11Clearly the pricing problem we consider is relevant when the agents require the plat-form’s technology in order to be matched; the platform has no pricing power otherwise.
9
Definition 1. Fix prices p = (pM, pW) and values u = ((um)M , (uw)W ). Amatching µ is p-stable (with respect to u) if it is both
1. individually rational at p: for all (m,w) ∈ M × W , µ(m) = wimplies um(w) ≥ pM and uw(m) ≥ pW ; and
2. not p-blocked: there is no (m,w) ∈M ×W such that
um(w)− pM > um(µ(m))− pM ∗ 1µ(m)∈W , and
uw(m)− pW > uw(µ(w))− pW ∗ 1µ(w)∈M
where 1 is the indicator function.
We say that a platform itself is p-stable when, for any realization of valuesu, it yields some p-stable match with respect to u. When such a platform isdecentralized, it may or may not have control over which stable matching isultimately created. Regardless, this ambiguity is inconsequential in terms ofthe platform’s revenue due to the Rural Hospital Theorem (Roth (1984b)):since all p-stable matchings yield the same number of marriages, they allyield the same revenue.
Theorem (Rural Hospital Theorem). Fix prices p = (pM, pW). For any u,all p-stable matchings contain the same number of marriages.
2.3 Algorithms
It is well known that the Deferred Acceptance algorithm yields a p-stablematching. However some of our results are obtained through a compar-ison to the number of marriages created by another algorithm, namely aprice-constrained form of serial dictatorship. The remainder of this sectionintroduces these algorithms.12
A p-stable matching can be found by running the Deferred Acceptance al-gorithm (Gale and Shapley (1962)) after the agents’ preferences are truncatedwith respect to prices p. To state the definition, for any prices p = (pM, pW)we say that (m,w) ∈ M × W are p-compatible if um(w) > pM anduw(m) > pW .
12Interestingly, at least in terms of expected revenue, both of these algorithms can beviewed as special cases of a broader class of “Meet-and-Propose” algorithms that we useto prove additional technical results. Due to the technicality, we relegate this material tothe Appendix.
10
Definition 2 (DAp algorithm). The algorithm takes a profile of values, u,as input and initializes all men to be single. In each round t = 1, 2, . . ., thefollowing two steps are executed.
Step t.1: Each manm who is single “meets” his favorite13 woman, w, amongthose to whom he has not already proposed. (If no such women exist, heremains single.) He proposes to w if and only if they are p-compatible.
Step t.2: Each woman becomes matched to her favorite man among thosewho have proposed to her. (If none exist, she remains single.) All othermen become (or remain) single. If each man is either matched or has“met” every woman, the algorithm ends; otherwise begin round t+ 1.
Definition 2 is standard, but with a slight redundancy in the statementthat men “meet” women before possibly proposing. This specification isintentional, allowing us to relate DAp to a the broader class of algorithmsdiscussed in the Appendix.
The following Serial Dictatorship algorithm also is standard, with thestipulation that only p-compatible pairs can be matched.
Definition 3 (Serial Dictatorship algorithm (SDp)). The algorithm takesa profile of values, u, as input and initializes all men to be single. In eachround t = 1, 2, . . . ,W , the following step is executed:
Step t: Woman w = t is matched with her most preferred p-compatible,single man, m (if any exist; otherwise she remains unmatched). Manm is removed from the set of single men.
Though we introduce SDp mainly as a tool to bound (and approximate)the p-stable platform’s expected revenue, the study of pricing decisions un-der SDp could be interesting in its own right. For example, the algorithmdescribes matching outcomes in environments with myopic (or satisficing)agents on one side of the market who accept the first (p-compatible) proposalthey receive. It is worth noting that all of the results of Section 3—statedfor p-stable platforms—are also proven for SDp, and more generally for theclass of Meet-and-Propose algorithms described in the Appendix.
13In the case of independent preferences, we ignore the issue of ties which happen withzero probability. Ties can occur under one kind of preference correlation considered inSection 5, but they can be broken arbitrarily without affecting the results.
11
3 Price allocation
At prices p = (pM, pW), the platform earns a total price of pT ≡ pM + pWfor every realized p-stable marriage. The total number of p-stable mar-riages created in a random economy is a random variable, denoted KDA
p ,that depends on the realization of values, u. The platform’s expected rev-enue is simply the product of these “margin” and “volume” components,pT · E(KDA
p ) ≡ (pM + pW) · E(KDAp ).
Observe that even if we fix a price level pT = (pM + pW), the allocationof pT between pM and pW affects the distribution of KDA
p and hence affectsthe platform’s expected revenue.14 Thus the platform’s pricing decision ofchoosing p = (pM, pW) can be viewed as a sequential choice of a total pricelevel and an allocation of that price level amongst the two sides:
maxp=(pM,pW )
(pM + pW)E(KDAp ) = max
pT
(pT · max
pM+pW=pTE(KDA
p )︸ ︷︷ ︸Allocation Problem
).
Throughout this section we focus on the platform’s price allocation problem.We begin with a result showing that the price allocation decision affects
the distribution of KDAp only to the extent that it affects the probability
that any one man-woman pair is p-compatible. To formalize this, let theincompatibility parameter q(p) denote the probability that an arbitrarypair (m,w) ∈M ×W is incompatible at prices p = (pM, pW).
The following result states that, in a random economy of M men andW women, the distribution of p-stable marriages is a function only of q(p).Consequently, two price lists p and p′ yielding the same incompatibility pa-rameter must yield the same expected number of marriages.15
Lemma 1 (The distribution of p-stable marriages is a function of q(p)).Fix M and W , and let KDA
p be the random variable denoting the number ofp-stable marriages generated by a random economy. For any two price lists
14 In fact Rochet and Tirole (2006) define two-sided markets to be those in which revenueis affected by changes in such total price allocation; our model fits this definition.
15All of the results in this section hold not only for p-stable platforms, but for plat-forms that yield matchings created by SDp or by any of the Meet-and-Propose algorithmsdescribed in the Appendix.
12
p = (pM, pW) and p′ = (p′M, p′W), q(p) = q(p′) implies that KDA
p and KDAp′
have the same distribution.
For a straightforward intuition behind the result, first imagine each pairbeing randomly determined to be compatible or not, with respect to givenprices. Second, once each pair’s compatibility is determined, prices playno additional role in determining the platform’s revenue: the number of p-stable marriages that will be created now depends only on the realization ofthe agents’ ordinal preferences over their (previously determined) compatiblepartners. Since prices affect only the probability of each pair’s compatibility,the result follows.
One general implication of Lemma 1 is that the platform does not biasits price allocation against any one side of the market based on the degree of“market size imbalance” (i.e. the size of M relative to W ). We formalize thisobservation in two ways, through Theorem 1 and Theorem 2. First, considerthe special case in which there are no ex ante differences between the twosides of the market other than size, i.e. where FM = FW . It immediatelyfollows from the symmetry of q() (Equation 1) that the expected number ofp-stable marriages (and hence expected revenue) is a symmetric function ofpM and pW , regardless of the sizes of M and W .
Theorem 1 (Revenue symmetry). Fix arbitrary M and W , and supposeFM = FW . Let p = (pM, pW) and p′ = (p′M, p
′W) be such that pM = p′W and
pW = p′M. Then p and p′ yield the same expected revenue to the platform.
It follows that any revenue that the platform can earn by charging arelatively higher price to the short side of the market could be achieved byinstead reversing its price list. This observation is interesting in light of theasymmetric welfare results of Ashlagi et al. (2017). Their results imply thatif M < W , FM = FW , and pM = pW = 0, then the average payoff to menexceeds the average payoff to women under stability. Despite this asymmetry,the platform does not charge a higher price to the advantaged (short) sideof the market, as naive intuition might suggest.
A broader observation in the general case (FM 6= FW) is that the plat-form’s optimal price allocation decision—whether symmetric or not—is inde-pendent of market size. First observe that, as is intuitive, a higher incompat-ibility parameter q lowers the expected number of p-stable marriages. Thisfollows from a classic result by Gale and Sotomayor (1985b) showing that forany realized profile of values u, a price increase (i.e. preference truncations)cannot increase the number of stable marriages.
13
Lemma 2 (Expected p-stable marriages decreasing in q; Gale and Sotomayor(1985b)). The expected number of p-stable marriages, E[KDA
p ], is a strictlydecreasing function of q(p).
Combining this result with Lemma 1, it is easy to see that for any givenprice level, the platform’s price allocation problem is solved by minimizingincompatibility q(). Since this minimization exercise is independent of thesizes of M and W , optimal price allocation is independent of the degree ofmarket imbalance.
Theorem 2 (Price allocation is market-size independent.). For any totalprice level pT and prices p∗M + p∗W = pT , the following two statements areequivalent.
1. (p∗M, p∗W) minimizes q(pM, pW) subject to the constraint pM+pW = pT .
2. For any M and W , (p∗M, p∗W) maximizes expected revenue subject to the
constraint pM + pW = pT .
Thus the optimal price allocation of pT is independent of M and W .
The result tells us that optimal prices minimize incompatibility, but notwhich prices do so. One might expect that, at least for symmetric distribu-tions FM = FW , this can be achieved through symmetric pricing. This isindeed the case under a standard hazard rate condition. For i ∈ {M,W}, we
say that Fi has a strictly increasing hazard rate if hi(x) ≡ fi(x)1−Fi(x)
is strictly
increasing in x ∈ [0, 1].
Proposition 1 (Symmetry with monotone hazard rate). Fix M , W , and anytotal price level pT that can yield positive revenue (i.e. 0 < pT < 2). Supposethat FM = FW = F , where F has a strictly increasing hazard rate. Subject tothe constraint pM+ pW = pT , the unique expected-revenue maximizing pricesare p∗M = p∗W = pT/2.
Consequently, the unconstrained maximizers of expected revenue satisfyp∗M = p∗W when FM = FW = F satisfies the hazard rate condition. Onthe other hand, when the hazard rate condition fails the platform mightstrictly benefit from unequal pricing even if FM = FW ; see Example 1 in theAppendix.
The results above—particularly Theorem 2—establish the principle thatprice allocation decisions are not directly impacted by changes in market
14
imbalance, i.e. the relative sizes of M and W . The optimal allocation of agiven price level pT is market-size independent. On the other hand, as wediscuss in Section 4, changes in M and W can typically change the platform’schoice of price level pT which, in turn, can lead to a relative change in priceallocation.
Our final result in this section implies that, under the monotone hazardrate condition, any such changes in market sizes would lead the platform tochange both of its prices in the same direction. Proposition 2 states that ifthe platform is optimally allocating total price, then any increase in pricelevel pT (whether optimal or not) leads to an increase in both (optimallyallocated) prices.
Proposition 2 (Comparative statics). Fix any total price level pT that canyield positive revenue (i.e. 0 < pT < 2), and suppose that FM and FW havestrictly increasing hazard rates.
1. Subject to the constraint pT = pM + pW , there are unique expected-revenue maximizing prices p∗M(pT ) and p∗W(pT ).
2. Both p∗M(pT ) and p∗W(pT ) are nondecreasing in pT .
Combined with Theorem 2, this implies that if a change in market sizesleads to some change in optimal price level, then both sides’ optimal priceswould move in the same direction.
4 Price level
In order to consider the p-stable platform’s choice of price level (pT ), wewould like a tractable expression for its expected revenue as a function ofmarket size (M,W ), prices (pM, pW), and value distributions (FM, FW). Dueto the combinatorial nature of this problem an exact expression remains outof reach. Nevertheless we demonstrate that a p-stable platform’s expectedrevenue is well-approximated (and bounded) by the expected revenue of aplatform that produces matchings via the (price-constrained) serial dictator-ship algorithm, SDp, described in Subsection 2.3. We provide an expressionfor the latter, using it to draw conclusions about pricing on stable platforms.
In particular, we use it to show that as the market grows large, theplatform can capture an increasingly large fraction of the maximum feasiblesurplus. Even at arbitrarily high prices (relative to the support of FM, FW),
15
the platform creates close to the maximum feasible number of marriagesin sufficiently large markets. The price-volume tradeoff present in smallermarkets starts to disappear. In this sense, the relative “cost of stability” fromthe platform’s perspective decreases to zero as markets become arbitrarilylarge.16
The relationship we see between p-stability and SDp is consistent with arelated asymptotic result of Arnosti (2016), who considers matching marketsin which one side has “short” preference lists. Arnosti shows that as marketsgrow large (while preference lists remain “short”) the expected number ofstable marriages exceeds the expected number of marriages under a randomdictatorship mechanism.17 Simulations demonstrate a similar bound with re-spect to SDp (which satisfies the IR constraint with respect to prices p). Ournext step is then to use a closed form expression for our bound (Equation 2)to derive the pricing conclusions described above. To provide that expressionwe introduce the following notation.
Definition 4. For any real number q ∈ [0, 1], the q-analog of integerj ∈ Z and the q-factorial of j are, respectively, defined as follows.
[j]q ≡ 1 + q + · · ·+ qj−1 =1− qj
1− q[j]q! ≡ [j]q[j − 1]q · · · [1]q
The q-binomial coefficient for integers k,n ∈ Z+ (k ≤ n) is[n
The distribution of marriages under SDp can be stated as follows.18
16We do not wish to interpret the convergence to “zero” too literally since, in reality,many seemingly large markets are in fact a patchwork of smaller, disconnected marketsto which asymptotic results need not apply. Instead our results (e.g. Figure 3) should beinterpreted more as a bound on the platform’s cost of providing stable outcomes.
17Arnosti’s dictatorship mechanism occurs via what he calls common preferences underDeferred Acceptance.
18Equation 2, known as the absorption distribution, was first described byBlomqvist (1952). Kemp (1998) finds its moments and shows that it is log-concave. Itis used by Ebrahimy and Shimer (2010) to describe the employment rate in a stock-flowmatching model with heterogeneous workers and jobs.
16
Theorem 3 (Distribution of marriages under SDp). Fix M,W , and pricesp = (pM, pW) with incompatibility parameter q = q(p). Let KSD
p be a randomvariable representing the number of marriages in a random economy createdunder SDp. The probability distribution of KSD
The first two terms in Equation 2 have a straightforward interpretation:(1− q)k is the probability that k given pairs of agents are compatible, whileq(M−k)(W−k) is the probability that all possible pairs of the remaining agentsare incompatible. The remaining q-analog terms in Equation 2 are a proba-bilistic analog to the number of ways to form k man-woman pairs from themarket (M,W ), i.e.
(Mk
)(Wk
)k!.
Figure 1 graphs Equation 3, fixing the number of men at M = 50 whilevarying the levels of incompatibility q(p) and number of women W . Thegraph illustrates that the more interesting pricing decisions occur for fairlybalanced markets (W close to 50). When the market is very unbalanced,the platform creates close to the maximum feasible number of marriages(min{W, 50}) even at relatively high prices (q(p) close to 1). In more balancedmarkets, on the other hand, the platform faces a richer tradeoff between priceand volume.19 Since the same observation is true for p-stable platforms, wefocus on the more interesting case of balanced markets below.
We now compare the expected number of marriages created under SDp
(using Equation 3) to that under DAp (via simulations), and conclude thatthe former is both an approximation and a bound for the latter. As arguedabove, the approximation would be unsurprising in sufficiently imbalancedmarkets: both algorithms would create close to the maximum feasible number
19To illustrate this in a qualitative example, when M = W = 50 and FM, FW are bothU [0, 1], revenue maximizing prices under SDp are approximately p∗M = p∗W = 0.718. Theseprices (q(p∗) ≈ 0.92) yield approximately 41.9 expected marriages, i.e. 16% of the marketis left unserved. While these numbers obviously should not be taken too seriously, theyillustrate a nontrivial tradeoff between price and volume.
17
0
10
20
30
0 5 10 15
0
10
20
30
40
50
0 10 20 30 40 50 60 70 80 90 100
E(SD
mar
riag
es)
W
q = 0.8
q = 0.9
q = 0.95
q = 0.97
Figure 1. The expected number of marriages under serial dictatorship,E(KSD
p ), fixing M = 50 and varying q(p),W .
of marriages (min{M,W}), thus approximating each other. Therefore wefocus on the “worst case” of balanced markets (M = W ) where, for a broaderrange of prices, the expected number of marriages under either algorithm isnot close to min{M,W}. Our conclusions are then easily extended to theremaining cases.
For various incompatibility parameters q(p), Figure 2 graphs the percent-age by which E(KSD
p ) underestimates E(KDAp ) as a function of (balanced)
market size n = M = W (formally, [E(KSDp ) − E(KDA
p )]/E(KDAp )). The
figure demonstrates the approximation mentioned above; e.g. for the param-eter values considered, Equation 3 is within 2.5% of the expected numberof p-stable marriages estimated through simulations. More importantly, thevalues in the graph are nonnegative, demonstrating that Equation 3 servesas a lower bound for E(KDA
p ); this fact is utilized below. As discussed ear-lier, this observation is consistent with a result of Arnosti (2016) in a relatedmodel, showing an analogous bound in asymptotically large markets.
The figure also suggests that, fixing any pair of prices (i.e. any q(p)), thepercentage difference between E(KSD
p ) and E(KDAp ) converges to zero. In
fact the following theorem states a slightly stronger conclusion: under SDp,
18
Figure 2. The percentage difference between expected p-stable marriages(simulated) and SDp marriages (Equation 3) in balanced markets.
the expected absolute number of unmatched agents converges to a constantas n = M = W grows large. Figure 3 graphs this expectation as a functionof q(p).20
Theorem 4 (Expected number of unmatched agents under SDp). Fix prices
p = (pM, pW) with incompatibility parameter q = q(p) < 1. Let SSDpn be
a random variable representing the number of unmatched men (hence un-matched women) under SDp, in a random economy of size n = M = W . Itsasymptotic expectation is
limn→∞
E(SSDpn ) ≡ SSDp
∞ =
[∞∏i=1
(1− qi)
][∞∑s=0
s · qs2
((1− q) · · · (1− qs))2
]. (4)
The result is significant since it allows us to draw the same conclusionabout DAp in economies of arbitrary size and imbalance. Specifically, firstconsider a fixed, balanced economy size of n = M = W . We clearly haveE(S
SDpn ) < S
SDp∞ , i.e. there are fewer total expected single men/women in
finite economies than in the limit. Next is the observation that E(SSDpn ) is
20Fixing any q(p), the distribution of SSDpn has thin tails (see Equation 12 in the Ap-
pendix). Therefore Figure 3 approximates SSDp∞ by summing the first (sufficiently many)
terms of Equation 4.
19
0
5
10
15
20
25
30
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
E(si
ngl
es)
q
Figure 3. The expected number of unmatched men (women) under SDp inlarge, balanced markets as a function of q = q(p) (Equation 4).
an upper bound on the expected number of unmatched men (resp. women)under DAp, for n = M = W (as demonstrated in Figure 2), hence the latter
is also bounded by SSDp∞ . Finally, consider an arbitrary economy of size
M = n and W > n. The expected number of unmatched men under DAp
(i.e. the platform’s “shortfall” relative to the maximum feasible number ofmatches n = min{M,W}) is clearly no greater than it would be if we hadM = W = n. Hence even in this case, the expected shortfall of the p-stableplatform is bounded above by S
SDp∞ .
In summary, Equation 4 thus provides an upper bound on the expectednumber of couples that a p-stable platform fails to produce by chargingprices p = (pM, pW) > (0, 0), as a function of the incompatibility param-eter q(p) > 0. At any fixed prices, the fraction of potential couples that theplatform fails to create—but that could have been created at lower prices—becomes vanishingly small as the market grows. Not only does a largermarket benefit the platform in the obvious way of increasing the number ofpotential matches, but also does so by increasing the quality of potentialmatches, yielding higher surplus extraction per agent. This establishes amarket-size effect under which the relative “cost of stability” suffered by theplatform in smaller markets vanishes as the market grows larger.
Of course the extremity of this conclusion relies on two attributes of ourstylized model that depart from some real world settings. The first is that wehave eliminated any search frictions that might prevent agents from beingable to find or match with any potential partner on the other side of themarket. While search frictions can be an important consideration in various
20
applications, online settings allow for increasingly frictionless interactions.21
A second relevant assumption we have made so far is that agents’ preferencesare independently randomized. In the next section we relax this assumption,showing that the platform’s decision to bias its price allocation towards oneside of the market depends on the interaction between market imbalance andthe form of correlation in the agents’ preferences.
5 Correlated Preferences
We consider how correlation among agents’ preferences affects the interactionbetween the platform’s prices and the expected number of marriages it cre-ates. Correlation alters our earlier price-symmetry results with independentpreferences (Theorem 1 and Theorem 2), leading a platform to bias its priceallocation towards one of the two sides of the market. However the direc-tion of price discrimination—whether towards the short or long side of themarket—depends on the form of preference correlation. Furthermore an in-crease in market imbalance amplifies the magnitude of price discrimination.Consequently, the presence of preference correlation does not validate thefalse intuition (discussed in the Introduction) that a stable platform shouldcharge relatively higher prices to the short side of a matching market. Infact, under one form of preference correlation considered below, the platformdoes the opposite.
Our observation that correlation affects pricing is important for a secondreason. It contrasts with the fact that, in the classic two-sided market mod-els (Subsection 1.3) where agents form multiple partnerships, the platform’soptimal pricing decision is independent of whether agents’ preferences arecorrelated. To elaborate on this, imagine a many-to-many platform beingallowed to price each potential transaction separately, i.e., in our language,imagine the platform setting a personalized pair of match-contingent pricesto each potential man-woman pair (m,w). With no capacity constraints, thewillingness of m and w to match with each other is unaffected by their otherpotential partnerships; these two agents will match with each other if andonly if they are compatible at their personalized prices. Thus the platformmaximizes this pair’s expected revenue by treating (m,w) as an independentpricing problem. Any correlation across the men’s preferences (or across the
21Hitsch et al. (2010) conclude from their analysis that “the online dating site that westudy provides an efficient, frictionless market environment.”
21
women’s preferences) has no impact on this pairwise pricing problem. Fur-thermore, when all men’s values are drawn from the same FM (and women’sfrom FW), the optimal personalized prices charged to each pair (m,w) areidentical. That is, the constraint to charge all men (and all women) thesame, non-personalized price does not bind for the risk-neutral platform:non-personalized, revenue-maximizing prices can be calculated without con-sidering preference correlation within any one side of the market. In factthis argument illustrates why neither preference correlation nor market im-balance plays a role in the pricing problem for such models, in contrast toours.
5.1 Two forms of correlation
While agents’ values could be correlated in various ways, we can illustrateour main points by focusing on two specific forms which are in a sense dual toeach other. In the first form, which we call same-side correlation, all agentson a given side of the market (e.g. W ) tend to agree on the desirability ofany given agent (e.g. m) on the other side; to put it roughly, the ui(j)’sare correlated across i’s. For example, Airbnb hosts may tend to agree thatclean, well-behaved guests are the most desirable ones. Under this formof correlation, agents have a common value component in assessing each(heterogeneous) agent on the other side.
In the second form—cross-side correlation—any given agent has corre-lated values for all agents on the other side of the market, i.e. the ui(j)’s arecorrelated across j’s. Under this form of correlation each agent has privatevalues for partners, but tends to view those potential partners as homo-geneous. Put differently, agents with heterogeneous outside options differprimarily in their willingness to participate on the platform. For example,Uber passengers (or drivers) may be roughly indifferent between encounters,but place different values on obtaining service (or jobs).
To demonstrate our qualitative results it is sufficient for us to consider thefollowing definitions in which these two forms of correlation hold perfectly.22
Previously we assumed that the values (ui(j)’s) were drawn independentlyaccording to the marginal distributions FM, FW . Throughout this section
22Of course most real world applications might involve both forms of correlation. Sinceour intention is to consider the different effects these two forms have on pricing, we considereach one separately.
22
we assume that the entire profile of values, u, is drawn from some jointdistribution, but under one of the following two sets of assumptions.
Definition 5. Preferences exhibit (perfect) same-side correlation when uis drawn in such a way that
• for each w ∈ W , the men have a common value UM(w) drawn fromFM (so ∀m ∈M, um(w) ≡ UM(w));
• for each m ∈M , the women have a common value UW (m) drawn fromFW (so ∀w ∈ W, uw(m) ≡ UW (m));
• these (W +M) different values are drawn independently.
Preferences exhibit (perfect) cross-side correlation when u is drawn insuch a way that
• each m ∈ M has a participation value UM(m) drawn from FM (so∀w ∈M, um(w) ≡ UM(m));
• each w ∈ W has a participation value UW (w) drawn from FW (so∀m ∈M, uw(m) ≡ UW (w));
• these (M +W ) different values are drawn independently.
When preferences are drawn independently, Figure 2 shows that the ex-pected number of marriages under SDp approximates the expected numberof p-stable marriages. Under either of the above forms of correlation, thesetwo expected values coincide.
Proposition 3. Suppose preferences exhibit either same-side correlation orcross-side correlation. Then for any prices p = (pM, pW), the expected num-ber of marriages under SDp is equal to the expected number of p-stable mar-riages.
The following explanation of the proof of Proposition 3 also provides aconvenient way to set up our discussion of the pricing problem that mo-tivates Subsection 5.2. Consider same-side correlation, where the womencommonly find any given man “acceptable” with probability 1 − FW(pW).The total number of acceptable men is a binomial random variable, kM ∼B(1−FW(pW),M). Similarly the number of acceptable women is a binomialrandom variable, kW ∼ B(1−FM(pM),W ). It is easy to see that under DAp,
23
k = min{kM, kW} pairs of agents will be matched (assortatively). UnderSDp, the kM acceptable men sequentially propose to (only) the kW acceptablewomen; regardless of the proposal order, this also yields k = min{kM, kW}marriages.
An analogous argument holds under cross-side correlation, except thatthe random number of men and women who are “willing to participate” onthe platform are k′M ∼ B(1 − FM(pM),M) and k′W ∼ B(1 − FW(pW),W ),respectively. It is straightforward to see that k′ = min{k′M, k′W} marriagesare created under both p-stability and SDp.
Now consider the platform’s expected revenue under same-side correla-tion.
(pM + pW) ∗ E(min{kM, kW}) (5)
The distribution of kM is a function of parameters (pW ,M) while kW dependson (pM,W ). An analogous revenue expression holds for cross-side correlation,except that k′M and k′W depend on (pM,M) and (pW ,W ), respectively. Thatis, the form of correlation determines which price (pM vs. pW) interacts withwhich market size (M vs. W ), which in turn determines the direction of pricediscrimination when the market is imbalanced.
Since the maximization of Equation 5 is generally an intractable problem,we formalize this interaction between correlation and market imbalance usinga version of our model with a continuum of agents.23 This allows a more directdescription of the interaction effect, while a similar exercise in the discretemodel would add little if any insight.
5.2 Price allocation and correlation
Consider our original model but with a continuum of agents: a mass M ofmen and a mass W of women. Since our definitions extend to this setting ina straightforward way, we omit their reformalization for brevity.
Under same-side correlation, prices pM, pW yield a (now deterministic)mass of acceptable men, κM = (1− FW(pW)) ∗ M , and a mass of acceptablewomen, κW = (1 − FM(pM)) ∗ W . This results in a mass of (assortative)p-stable marriages, κ = min{κM, κW}. Clearly the platform wants to set
23The intractability comes from the lack of a simple expression for the expected mini-mum of two binomials. The continuum model eliminates the uncertainty in kM, kW .
24
men
κM
women
κW
Figure 4. Under same-side correlation, M > W leads the platform to setFW(pW) > FM(pM) (so that κM = κW), i.e. to charge a relatively higherprice to the short side of the market.
prices so that κM = κW (see Figure 4),24 i.e. so that
1− FW(pW)
1− FM(pM)=W
M(6)
This immediately implies that the short side of the market is charged arelatively higher price than the long side in the following sense: if W < Mfor example, then we will have FW(pW) > FM(pM). Thus in the special casethat FM = FW , the short side of the market is charged a higher price inabsolute terms.
Under cross-side correlation, the platform equates the two masses ofagents “willing to participate” by setting (1 − FW(pW)) ∗ W equal to (1 −FM(pM)) ∗ M . This inverts the relationship in Equation 6, reversing thedirection in which the platform biases its price allocation. These argumentsare summarized as follows.
Proposition 4 (Price allocation: market imbalance and preference corre-lation). Consider an imbalanced market where (without loss of generality)M > W .
• Under same-side correlation, the revenue maximizing platform chargesa relatively higher price to the short side: FW (pW ) > FM(pM).
• Under cross-side correlation, the revenue maximizing platform chargesa relatively higher price to the long side: FM(pM) > FW (pW ).
24Otherwise the platform could increase one of its prices without reducing the numberof marriages.
25
For balanced markets (M = W ), we have FM(pM) = FW (pW ).
This result can be contrasted with our earlier results on both price alloca-tion and price level in the case where preferences are independently drawn. Inthat case, the platform does not use market imbalance as a basis for biasingits price allocation toward either side of the market (Theorem 1, Theorem 2).In terms of price level, the combined results of Section 4 imply that, as themarket grows in size (thickness), the revenue maximizing platform wouldpost increasingly high prices to both sides of the market, in the sense thatFM(pM), FW(pW)→ 1. This does not occur in large markets with correlatedpreferences (see Figure 4), where the platform continues to face a non-trivialtradeoff between price and volume.
Next we extend this idea to a comparative statics result with respect tochanges in the degree of market imbalance. To relate it to a practical exam-ple, consider the setting of a ride-sharing platform, where preferences typi-cally exhibit cross-side correlation: agent heterogeneity comes about mostlythrough differences in their willingness to participate in the platform ratherthan their individual values as match partners. In such a setting, imagine anincrease in the relative number of passengers. Depending on the parametersof the model, an effect of this change might be to intensify the conclusions ofProposition 4, leading the platform both to raise its price to the passengerside, and to lower its price to the driver side (through higher wages). Indeedthis is precisely the idea of “surge pricing.” When FM and FW satisfy themonotone hazard rate condition, Proposition 5 shows that this surge priceeffect necessarily holds.
Analogously, same-side correlation leads to the opposite conclusion. In-tuitively, an increase in the relative size of the men’s side of the market giveseach woman a more valuable partner. The platform might extract some ofthis value by raising pW . Since this would make marriages more profitable,the platform might generate more of them by lowering pM. Again, under themonotone hazard rate condition, this necessarily occurs. Interestingly, whilethe latter part of this intuition sounds similar to the intuition underlyingthe “see-saw effect” of Rochet and Tirole (2006),25 it comes about in oursetting from an interaction between preference correlation and market sizeimbalance—two parameters that play no role in their model.
25The see-saw effect is the idea that a price-increase on one side of the market incentivizesthe platform to lower its price to the other side.
26
Proposition 5 (Comparative statics). Suppose FM and FW have strictlyincreasing hazard rates. An increase in the relative proportion of men, M/W ,causes revenue-maximizing prices (p∗M, p
∗W) to change as follows.
• Under same-side correlation, p∗M weakly decreases and p∗W weakly in-creases.
• Under cross-side correlation, p∗M weakly increases and p∗W weakly de-creases.
This result also can be contrasted with the corresponding one for indepen-dently drawn preferences: by Proposition 2, any change in market imbalancethat would affect revenue-maximizing prices must move p∗M and p∗W in thesame direction. Intuitively, the difference in the two results follows from thedifference in price allocation objectives across the two settings. In the caseof independently drawn preferences, price allocation is done to minimize theincompatibility rate for any single pair of agents (Theorem 2). Under thetwo forms of correlation considered above, price allocation is done to achievea kind of market clearing, balancing the number of compatible agents on thetwo sides of the market; e.g. in the case of large markets this occurs preciselythrough Equation 6.
6 Conclusion
We have studied pricing decisions within a stylized model of monopolistic,match-making platforms. In contrast to a canonical literature on two-sidedmarkets, our setting involves heterogeneous agents who use the platformto create exclusive pairings in a way that results in stable matchings a laAdachi (2003) or Hitsch et al. (2010). Our model thus sits between a two-sided markets literature (that prices non-exclusive interactions) and a stablematching literature (that allows for exclusivity while setting aside pricingquestions). Our results lead to insights that cannot be drawn from either ofthese two literatures individually, in ways that we now summarize.
We first establish the idea that price allocation decisions are independentof market size or imbalance when agents have independently drawn prefer-ences. Theorem 2 states this directly: fixing any total price per marriage pT ,the optimal allocation of pT between the two sides of the market is indepen-dent of market size (M,W ). Of course, market size can indirectly influence
27
the platform’s price allocation decision by changing its revenue-maximizingchoice of pT . Nevertheless, Proposition 2 states that any such change in pTwould change the two sides’ prices in the same direction under a standardmonotone hazard rate condition.
As discussed in Subsection 1.2 these results compare interestingly withthose of Ashlagi et al. (2017), who show that agents on the short side of sta-ble marriage models obtain relatively better matching outcomes than thoseon the long side. Theorem 1 and Proposition 1 show that this asymmetryin relative gross payoffs is not extracted by the platform via asymmetrictransaction fees. For instance, when FM = FW satisfies a standard hazardrate condition, revenue-maximizing prices typically give higher net payoffsto matched agents on the short side of the market than to matched agents onthe long side. Likewise, consider a small change in market size parametersthat reverses the direction of market imbalance. This reversal would generatesignificant welfare gains (losses) for agents on the now-short (now-long) sideof the market. Nevertheless, a revenue-maximizing platform would respondto this reversal by changing both sides’ prices in the same direction, if at all(Proposition 2).
Turning to the issue of price level we formalize the idea that larger marketsinduce higher prices. The underlying cause is that, for any fixed prices, theexpected number of unmatched pairs that are “priced out of the market” isbounded by a constant (described in Theorem 4) that is independent of mar-ket size. Thus, while there is typically a substantial margin-volume tradeoffin smaller markets, the tradeoff is less substantial in larger markets: the plat-form creates close to the maximal number of matches even at relatively highprices. Consequently, the implicit “cost of stability” imposed on the revenue-maximizing platform becomes negligible as the market grows. It is interestingto contrast this market-size effect with the fact that, in non-exclusive match-ing environments (e.g. Rochet and Tirole (2003)), transaction-based pricelevels are independent of market size; see Subsection 1.3.
Finally we establish that preference correlation does cause price allocationto be biased in a way that depends on market imbalance. However the direc-tion of bias depends on the form of correlation, specifically on whether val-uations ui(j) are correlated across the i’s (“same-side correlation”) or acrossthe j’s (“cross-side correlation”). When agents have similar preferences overthe other side’s agents (same-side), the platform charges relatively higherprices to the short side of the market, extracting the relatively higher valuethat side obtains from an assortative matching (Proposition 4). On the other
28
hand, when agents have heterogeneous outside options (cross-side), the biasis reversed: a higher relative price on the long side of the market profitablybalances the number of participants across the two sides. Unlike our earlierfinding for independently drawn preferences, a change in market imbalancenow causes revenue-maximizing prices to move in opposite directions acrossthe two sides of the market (Proposition 5), with the direction of the pricemovements depending on the form of correlation. In particular, cross-sidecorrelation leads to a “surge pricing” result: an increase in the number ofpassengers increases both the passengers’ price and the drivers’ wage.
The results on correlation are perhaps our most significant demonstra-tion of the importance of our exclusivity assumption. In models with non-exclusive partnerships, expected revenue maximizing prices are independentnot only of market size but of preference correlation (again see Subsec-tion 1.3). When agents match exclusively, in contrast, we show that aninteraction between market imbalance and preference correlation is crucialin determining the optimal allocation and level of prices. Since it is no longersufficient for the platform to know (or estimate) the marginal distributionsof agents’ match values, our work reveals additional empirical requirementsfor such platforms that are unnecessary in environments with non-exclusivematching.
7 Appendix: Proofs and Technical Results
7.1 Meet and Propose
We describe a class of “Meet and Propose” (MAP) algorithms that, in termsof expected marriages, captures both DAp and SDp as special cases. Thisclass of algorithms is parameterized by meeting orders that are determinedexogenously, rather than by the endogenous order in which men meet womenin Step t.1 of DAp. This decouples the men’s preferences from the orderingin which they make (p-compatible) proposals to women. At the same time,this class of algorithms maintain the flavor of SDp, which operates as if menare required to make (p-compatible) proposals to women in a common order.
Define a meeting order for man m ∈ M , denoted Bm, to be a linearorder on W . We denote a profile of meeting orders by B = (Bm)m∈M .
Definition 6 (MAPBp algorithm). The algorithm is parameterized by a pro-
file of meeting orders B. It takes values u as input and initializes all men to
29
be single. In each round t = 1, 2, . . ., the following two steps are executed.
Step t.1: Each man m who is single “meets” the woman w ranked highestunder Bm among those to whom he has not already proposed. (If nosuch women exist, he remains single.) He proposes to her if and onlyif they are p-compatible.
Step t.2: Each woman becomes matched to her favorite man among thosewho have proposed to her. (If none exist, she remains single.) All othermen become (or remain) single. If each man is either matched or has“met” every woman, the algorithm ends; otherwise begin round t+ 1.
Each MAPBp differs from DAp to the extent that each Bm differs from
m’s relative preference order over women according to um(). If (by chance)a realization of u is such that each man ranks women identically under umand Bm, then the two algorithms produce the same matching. Typically, ofcourse, this does not happen and the algorithms produce different outcomesex post. Nevertheless, in random economies there is a close ex ante relation-ship between p-stability and a randomized version of MAP: running DAp ona random economy generates the same distribution of marriages as uniformlyrandomly generating a profile of meeting orders, B, and then running MAPB
p
on a random economy.
Lemma 3 (randomized-MAPBp marriages
d= p-stable marriages). Fix M , W
and prices p = (pM, pW). Let KDAp be a random variable representing the
number of p-stable marriages in a random economy. Let KrMAPp be a random
variable representing the number of marriages created under MAPBp for a
random economy when each meeting order Bm is independently drawn froma uniform distribution over all orders. Then KDA
p and KrMAPp have the same
probability distribution.
Proof of Lemma 3. To randomly generate a realization of KDAp , we ran-
domly generate preferences, and then run DAp. This is probabilisticallyequivalent to the following:(i) randomly generate whether each w is acceptable to each m (given pM),(ii) randomly order each m’s acceptable women to determine ordinal pref-
erences, and then(iii) run standard Deferred Acceptance with respect to these ordinal prefer-
ences.
30
To randomly generate a realization of KrMAP, we would randomly gen-erate preferences and Bm’s, and then run MAPB
p . This is probabilisticallyequivalent to the following:(i) randomly generate whether each w is acceptable to each m (given pM),
(ii) randomly order each m’s acceptable women to determine m’s meetingorder over just those women,
(iii) run standard Deferred Acceptance, interpreting these meeting orders asmen’s preferences.
It is easy to see that these two processes generate the same distribution overmatchings.
The intuition behind the result is straightforward.26 It follows from thefact that the outcomes of both algorithms depend on how the realizationof values u determines (i) which pairs of agents are p-compatible, and (ii)the realized order of men’s proposals. In the case of MAPB
p , (i) and (ii)are independent, since the latter is exogenously randomized. In the caseof DAp, however, the relative order of a man’s potential proposals betweenany two women is conditionally independent, given that the two women arep-compatible with him.
On the other hand, as mentioned earlier, SDp is just a special “corner”case of MAPB
p in which all men have the same meeting order Bm.
Observation 1. SDp is equivalent to MAPBp when Bm = Bm′ for all m,m′.
7.2 Proofs
The following lemma is used to prove Lemma 1.
Lemma 4 (The distribution of MAP marriages is a polynomial function ofq). Fix M , W , and profile of meeting orders B. For any prices p = (pM, pW),let KB
p be a random variable denoting the number of marriages created underMAPB
p for a random economy. For any two price lists p = (pM, pW) and p′ =(p′M, p
′W), q(p) = q(p′) implies that KB
p and KBp′ have the same distribution.
This distribution is a polynomial function of q(p).
Proof: It is well known that the outcome of DAp would be unaffected if,in each Step t.1, men instead propose one at a time. For the same reason,it is easy to see that the outcome of MAPB
p also would be unaffected by
26All proofs appear in the Appendix.
31
such a change in each Step t.1. Throughout this proof, we suppose that ineach Step t.1, only the lowest-indexed man (among single men who have notexhausted their list of women) performs the meet/propose step toward thehighest ranked remaining woman in his ordering Bm.
Fixing B and p, if we observe an execution of MAPBp , then after any given
Step t of the algorithm we would observe the following variables:• nw: for each woman w, the number of (p-compatible) proposals she has
received so far;• Sm: for each man m, the set of women m has already met; and• µ: the current matching of women to their favorite proposers so far.
As a function of only these variables, let πk(n, S, µ) denote the probabilitythat MAPB
p terminates with exactly k marriages for a random economy, giventhat the values of n = (n1, . . . , nW ), S = (S1, . . . , SM), and µ are observedat some arbitrary step of the MAPB
p algorithm. We show recursively that πkis a polynomial function of q.
First, MAPBp terminates once every unmatched man has proposed to every
woman, at which point πk is either zero or one. That is, whenever µ and Sare such that, for all m, µm = m implies Sm = W , we have
πk(n, S, µ) =
{1 if |µ| = k
0 else
regardless of n.In the remaining cases of (n, S, µ) where MAPB
p has not terminated, thealgorithm proceeds to the next step, where some currently unmatched manm meets the next woman w whom he has not already met (determined bySm and Bm). In that step, one of three things could happen.• m and w are incompatible. This has probability q.• m and w are compatible but w prefers a previous proposer. This occurs
with probability (1− q)(1− 1/(nw + 1)).• m and w are compatible and w prefers m to all previous proposers.
This occurs with probability (1− q)(1/(nw + 1)).Therefore, for m and w determined as above, we have the following recursiverelationship.
πk(n, S, µ) = qπk(n, S′, µ) + (1− q)
(1− 1
n′w
)πk(n
′, S ′, µ)
+ (1− q) 1
n′wπk(n
′, S ′, µ′)
32
where n′ = n + 1w augments the wth value of n by 1, S ′ augments S byadding w to Sm, and µ′ is obtained from µ by matching w to m instead ofto µ(w).
Therefore since each “terminal” πk() is degenerately zero or one, all othervalues of πk() are polynomial functions of q. In particular, the initial valueπk((0w)W , (∅m)M , µ∅) (where µ∅ is the null matching) is polynomial in q.
Proof of Lemma 1. By Lemma 3, the distribution of KDAp is the average
of the distributions of KBp ’s (over all meeting orders B). The theorem then
follows from Lemma 4.
The following result implies Lemma 2.
Lemma 2*. Under either DAp or any arbitrary MAPBp algorithm, the ex-
pected number of marriages is a strictly decreasing function of q(p).
Proof: Fix q, q′ ∈ [0, 1] with q < q′. Define price pairs p, p′ to satisfypM = p′M = 0, pW = F−1W (q), and p′W = F−1W (q′). Observe that q(p) = q andq(p′) = q′. By Lemma 1 the choice of prices is without loss of generality.
Given any realization of u, each woman’s preferences over men at price p′Wis a truncation of her preferences at pW . It follows from Theorem 2 in Galeand Sotomayor (1985a) that any agent who would be unmatched under DAp
would be unmatched under DAp′ . Therefore the number of marriages at udecreases weakly from p to p′. Furthermore, because FM and FW have strictlypositive densities on [0, 1], there exists a positive mass of realizations of u forwhich the number of marriages strictly decreases from p to p′. Therefore inexpectation over all possible realizations of u we have E[KDA
p ] > E[KDAp′ ].
For the case of MAP, fix a profile of meeting orders B and compare MAPBp
to MAPBp′ . Observe that, since pM = p′M = 0, each man finds every woman
acceptable. Thus, each man m meets women according to the ordering Bm
and proposes to a woman if and only if she finds m acceptable with respectto pW . This makes the MAPB
p algorithm equivalent to operating the DAp
algorithm when we instead use B as the profile of men’s ordinal preferences.(More precisely, if we run DAp using B to represent the men’s ordinal pref-erences, Step t.1 of DAp becomes equivalent to Step t.1 of MAPB
p . This isbecause, when pM = 0, p-compatibility is determined solely by the women’spreferences. Step t.2 is already the same in both algorithms.) Thus, fixingany u and B, the above result of Gale and Sotomayor (1985a) applies alsoto MAP. Taking expectations over possible u again yields K(q) > K(q′) forthe case of MAP.
33
Proof of Theorem 1. The result follows from the fact that q() is symmet-ric in its two arguments. For the same reason, the result also holds for anyMAPB
p algorithm with arbitrary B, including SDp.
Proof of Theorem 2. The result follows from Lemma 1: maximizing ex-pected revenue subject to pM + pW = pT is equivalent to minimizing q()subject to the same constraint, due to Lemma 2.
Proposition 1 and Proposition 2 mainly follow from the following charac-terization of revenue-maximizing prices under the hazard rate condition.
Lemma 5. Suppose that FM and FW have strictly increasing hazard rates.For any 0 < pT < 2, there is a unique pair of prices, (p∗M, p
∗W), that maximize
expected revenue subject to the constraint pM + pW = pT . These pricesminimize the absolute difference in the two sides’ hazard rates:
(p∗M, p∗W) = arg min
pM,pW : pM+pW=pT
∣∣hM(pM)− hW(pW)∣∣ (7)
Proof: By Theorem 2, it suffices to show that there is a unique pair ofprices that minimize q(pM, pW) subject to the constraint pM+pW = pT , andthat these prices satisfy (7). From our assumptions on FM, FW , prices thatminimize q() must satisfy (pM, pW) ∈ [0, 1]2; hence we can restrict attentionto pM, pW ∈ [0, 1].
Defining q(pM) ≡ q(pM, pT − pM), the minimization problem can bewritten as
minpM
q(pM) s.t. max{0, pT − 1} ≤ pM ≤ min{1, pT} (8)
where the constraints on pM are derived from the fact pM, pW ∈ [0, 1].Differentiating q on this range, we have
where division by zero is avoided for pM < min{1, pT}. Furthermore thisinequality also implies
(1− FW(pT − pM))(1− FM(pM)) > 0
meaning that, for interior values of pM, dq(pM)/dpM has the same sign asthe difference
(hM(pM)− hW(pT − pM)). (10)
By the monotone hazard rate assumption, (10) is strictly increasing in pM.Hence on the range max{0, pT−1} ≤ pM ≤ min{1, pT}, the sign of dq(·)/dpMis either (i) always negative, (ii) always positive, or (iii) crosses zero frombelow at exactly one price p∗M. In these three respective cases, q() is min-imized at (i) p∗M = min{1, pT}, (ii) p∗M = max{0, pT − 1}, or (iii) wherehM(p∗M) = hW(pT − p∗M). In each case, this is the price that minimizes (7),proving the lemma.
Proof of Proposition 1. The result follows immediately from Lemma 5and observing that the solution to (7) is given by p∗M(pT ) = pT/2 whenFM = FW .
Without the hazard rate condition, revenue maximizing prices may beunequal even when FM = FW , as demonstrated in the following example.The idea does not rely on the discreteness of the example.
Example 1 (Optimal, unequal prices). Consider one man and one woman(M = W = 1). The value that each agent assigns to the potential mate is(independently) either 0.1 (probability π) or 0.9 (probability 1−π). One canrestrict attention to prices pM, pW ∈ {0.1, 0.9} and check by inspection thatthe following price pairs maximize expected revenue.
In the case 4/9 ≤ π ≤ 4/5, it is strictly optimal to charge unequal prices.Even in this case, however, the set of optimal price lists is symmetric inaccordance with Theorem 1.
Proof of Proposition 2. For any 0 < pT < 2, Lemma 5 states that thereare unique revenue-maximizing prices, (p∗M(pT ), p∗W(pT )). We show thatp∗M(pT ) is nondecreasing in pT . An identical argument applies to p∗W(pT ).
The strict inequalities follow immediately from the hazard rate assumption.To derive the weak inequality, observe that if hM(p′M) > hW(p′W) then forsmall ε > 0, p′M − ε > 0 and p′W + ε < 1 would strictly reduce the absolutedifference in hazard rates, i.e.∣∣hM(p′M − ε)− hW(p′W + ε)
∣∣ < ∣∣hM(p′M)− hW(p′W)∣∣
in contradiction to (7).By definition, (p′′M, p
′′W) minimizes (7) with respect to p′′T . However,
hM(p′′M) < hW(p′′W) implies that for small ε > 0, p′′M + ε < 1 and p′′W − ε > 0strictly reduces the absolute difference in hazard rates, i.e.∣∣hM(p′′M + ε)− hW(p′′W − ε)
∣∣ < ∣∣hM(p′′M)− hW(p′′W)∣∣
which contradicts (7).
Proof of Theorem 3. Fix prices p = (pM, pW) with incompatibility pa-rameter q = q(p). We prove Equation 2 for all M,W , by an inductionargument on M , holding fixed an arbitrary W . In addition, rather thanrunning SDp, the proof considers executing a MAPB
p algorithm in a case inwhich all men have identical meeting orders, Bm ≡ Bm′ (see Observation 1),where one man meets/proposes at a time as in the proof of Lemma 4.
Fixing W , it is clear that Equation 2 holds for M = 1: the lone man in theeconomy is either incompatible with each woman (P (0; q, 1,W ) = qW ), or not(P (1; q, 1,W ) = 1− qW ). Inductively, for some M suppose that Equation 2accurately describes P (·; q,M − 1,W ). Consider running MAPB
p only until
36
man mM−1 is matched (or is rejected by all women); call this the end of stageM − 1. The probability that k of the first M − 1 men are married at thispoint in the algorithm is precisely P (k; q,M − 1,W ), since a complete run ofthe algorithm for a randomized economy of size (M − 1,W ) is equivalent toa run of the algorithm to the end of stage M − 1 for a randomized economyof size (M,W ).
Furthermore for the economy (M,W ) to end up with k marriages it mustbe that, at the end of stage M − 1, there were either k or k − 1 temporarymarriages. We separately consider these two cases.
Case 1: at the end of stage M − 1, k men are temporarily matched.There are thus W − k women currently unmatched. The algorithm nowintroduces man mM , who begins to sequentially meet women. If w1 is cur-rently unmatched, there is probability (1 − q) that she accepts a proposalfrom mM (ending the algorithm), and probability q that he must continue bymeeting w2 (if she exists). But if w1 was temporarily matched, then with cer-tainty some man—either mM or her temporary partner—will be permanentlymatched to her, and the other man continues by meeting w2 (if she exists).In this latter case, it is probabilistically irrelevant which man continues onto meet w2 (by the i.i.d. assumption on values).
This process continues for each woman in turn until the algorithm ends.Each temporarily married woman wj keeps some offer and sends the otherman on to meet wj+1. Each currently single woman (if met) ends the algo-rithm with an accepted proposal with probability (1 − q). Therefore withprobability qW−k, “stage M” does not add an additional marriage to thealready existing k marriages.
Case 2: at the end of stage M − 1, k − 1 men are temporarily matched.There are thus W − k+ 1 women currently unmatched. As above, the intro-duction of man mM in stage M fails to yield an additional match preciselywhen each of the W − k + 1 is incompatible with the unique man who pro-poses to her. Therefore with probability 1 − qW−k+1, “stage M” adds anadditional marriage to the already existing k − 1 marriages.
Combining Case 2 and Case 1 respectively, P (k; q,M,W ) equals
P (k − 1; q,M − 1,W ) · (1− qW−k+1) + P (k; q,M − 1,W ) · qW−k
Using Equation 2 to substitute for P ( · ; q,M − 1,W ) this becomes
(1− q)k−1q(M−k)(W−k+1)
[M − 1
k − 1
]q
[W
k − 1
]q
[k − 1]q!(1− qW−k+1)
37
+ (1− q)kq(M−k−1)(W−k)[M − 1
k
]q
[W
k
]q
[k]q!(qW−k)
=
(qM−k
[k]q (1− q)
)(1− q)kq(M−k)(W−k)
[M − 1
k − 1
]q
[W
k − 1
]q
[k]q!(1− qW−k+1)
+ (1− q)kq(M−k)(W−k)[M − 1
k
]q
[W
k
]q
[k]q!
=
(qM−k
[k]q (1− q)
)(1− q)kq(M−k)(W−k)
[M
k
]q
[k]q[M ]q
[W
k
]q
[k]q[W − k + 1]q
[k]q!(1− qW−k+1)
+ (1− q)kq(M−k)(W−k)[M
k
]q
[M − k]q[M ]q
[W
k
]q
[k]q!
= qM−k(1− q)kq(M−k)(W−k)[M
k
]q
[k]q[M ]q
[W
k
]q
[k]q!
+ (1− q)kq(M−k)(W−k)[M
k
]q
[M − k]q[M ]q
[W
k
]q
[k]q!
= (1− q)kq(M−k)(W−k)[M
k
]q
[W
k
]q
[k]q!
(qM−k [k]q + [M − k]q
[M ]q
)
= (1− q)kq(M−k)(W−k)[M
k
]q
[W
k
]q
[k]q!
([M ]q[M ]q
)
Proof of Theorem 4. Fix any prices p with q = q(p). In a balanced mar-ket of size n = M = W , the probability that SDp yields a perfect matching,i.e. of having zero unmatched agents, is given by Equation 2.
P (KSDp = n; q, n, n) = (1− q)nq(n−n)(n−n)
[n
n
]q
[n
n
]q
[n]q!
= (1− q)n [n]q!
= (1− qn) · · · (1− q1)
As n goes to infinity, P (n; q, n, n) converges to the following expression.
φ(q) ≡∞∏i=1
(1− qi) (11)
38
Observe that for 0 ≤ q < 1 we have φ(q) > 0 (Lemma 6 below). ThereforeP (n; q, n, n) is bounded away from zero by the constant φ(q) across all marketsizes n.
More generally, even in imbalanced markets of arbitrary sizes M,W , theprobability that there are exactly k marriages, and hence g = M − k singlemen and h = W −k single women, can be expressed by rewriting Equation 2in terms of g and h as follows.
P (KSDp = k; q,M,W ) = P (KSD
p = k; q, k + g, k + h)
= (1− q)kqgh[k + g
k
]q
[k + h
k
]q
[k]q!
Letting the market grow large (i.e., k → ∞, so M = k + g → ∞ andW = k + h → ∞), the probability that there are g single men and h singlewomen converges to
limk→∞
P (KSDp = k; q, k + g, k + h)
= limk→∞
(1− q)kqgh[k + g
k
]q
[k + h
k
]q
[k]q!
= limk→∞
(1− q)kqgh(
(1− qk+1) · · · (1− qk+g)(1− q) · · · (1− qg)
)((1− qk+1) · · · (1− qk+h)
(1− q) · · · (1− qh)
)[k]q!
= limk→∞
(1− qk) · · · (1− q)qgh(
(1− qk+1) · · · (1− qk+g)(1− q) · · · (1− qg)
)((1− qk+1) · · · (1− qk+h)
(1− q) · · · (1− qh)
)= φ(q)qgh
1
(1− q) · · · (1− qg) · (1− q) · · · (1− qh)(12)
In the case of balanced markets, we must have g = h. Applying Equa-tion 12 (or Equation 11 for the case g = h = 0), the expected number ofsingle men (women) converges to
=∞∑s=0
sφ(q)qss1
(1− q) · · · (1− qs) · (1− q) · · · (1− qs)
which yields Equation 4.
The function φ() in Equation 11 is the topic of Euler’s Pentagonal NumberTheorem. The following fact appears to follow from arguments made inEuler’s own writings; nevertheless we provide a proof for accessibility.
39
Lemma 6. For all q ∈ [0, 1), φ(q) > 0.
Proof: We wish to show
limn→∞
n∏i=1
(1− qi) > 0
so it suffices to show∞∑i=1
ln(1− qi) > −∞.
Rewrite the fundamental logarithmic inequality ln(x) ≥ 1 − 1/x to stateln(1− qi) ≥ −qi/(1− qi). Then we have
∞∑i=1
ln(1− qi) ≥ −∞∑i=1
qi
1− qi≥ −
∞∑i=1
qi
1− q=
−q(1− q)2
> −∞
completing the proof.
Proof of Proposition 3. First consider same-side correlation. Fix anyprices pM, pW and a realization of values u. Given pW , man m is acceptable toeach woman if and only if his common value to them exceeds pW ; there is somenumber kM of such “acceptable” men. Similarly there is some number kW ofacceptable women. Hence any IR matching has at most k ≡ min{kM, kW}marriages.
On the other hand it is simple to see that the unique p-stable matching isthe “assortative” one, matching the k “best” men and women, i.e. it containsk marriages. In the case of SDp, the kW acceptable women will sequentiallychoose their favorite among the (remaining) kM men, until one of the setsbecomes empty. The particular matching outcome depends on the order inwhich women choose, but the number of marriages creates is k, regardless.
A similar argument can be made for cross-side correlation. An inconse-quential difference in this case is that preferences have indifference, but themethod of tie-breaking does not impact the expected number of marriagesunder either algorithm.
Proof of Proposition 4. Follows from Equation 6.
Proof of Proposition 5. Fixing FM, FW , we prove that under cross-sidecorrelation, the optimal (revenue maximizing) price charged to the men is
40
weakly increasing in η ≡ M/W , the relative proportion of men. By relabelingthe sides of the market, the same proof thus shows that the optimal pricecharged to the women weakly increases in 1/η, i.e. decreases in η.
For each side of the market i and any x ∈ [0, 1], define pi(x) ≡ F−1i (1−x)to be the price at which the proportion of agents on side i who are “willingto match” is x. Our assumptions on FM and FW imply that pM() and pW ()are strictly decreasing and continuously differentiable.
Recall that the platform sets prices in a way that equates the two massesof agents “willing to match,” which we write as κM = κW = κ, a la Figure 4.Thus, revenue maximization can be written as the following optimal choiceof κ.
maxκ≤min{M,W}
κ ·[pM
(κ/M
)+ pW
(κ/W
)]Equivalently, the objective can be written in terms of choosing the proportionof men willing to match, which we denote κ ≡ κ/M .
maxκ≤min{1,1/η}
Mκ · [pM (κ) + pW (ηκ)]
The first-order condition necessary for an interior optimum 0 < κ <min{1, 1/η} is
By definition x = 1−Fi(pi(x)) and by the inverse function theorem, p′i(x) =−1/fi(pi(x)). Substituting these terms and denoting the hazard rates hi(x) =fi(x)
1−Fi(x), the first-order condition (13) is:
G(κ, η) ≡ pM(κ) + pW(ηκ)− 1
hM(pM(κ))− 1
hW(pW(ηκ))= 0
Each pi() is strictly decreasing κ; the negative inverse hazard rate−1/hi(pi(·))is also strictly decreasing in κ by the increasing hazard rate assumption.Therefore G() is strictly decreasing in κ and thus the second-order condi-tions are also satisfied.
Observe that G(κ, η) is continuous in κ and strictly positive when eval-uated at κ = 0. Therefore there are two possibilities: either there ex-ists a unique interior optimizer 0 < κ < min{1, 1/η} satisfying (13), orG(κ, η) > 0 for all interior κ, the constraint binds, and the unique optimizeris κ = min{1, 1/η}.
41
Regardless of the case, let κ∗(η) denote the unique optimizer as a functionof η. Consider an increase in the ratio of men to women from some η′ to someη′′ > η′. As η increases, G(κ, η) decreases and the constraint κ ≤ min{1, 1/η}becomes weakly tighter. Therefore this must lead to a weakly lower optimalproportion of matched men: κ∗(η′′) ≤ κ∗(η′), whether the constraint bindsor not. This implies a weak increase in the optimal price charged to the men,
pM(κ∗(η′′)) ≥ pM(κ∗(η′)).
By relabeling the sides of the market, the same conclusion can be drawn forthe optimal price charged to the women’s side.
The proof for the case of same-side correlation is analogous to this one,but instead relates the proportion of “acceptable men” κ to the women’s pricerather than the men’s. Hence the opposite conclusion is reached, yielding theproposition.
References
Adachi, Hiroyuki. “A Search Model of Two-Sided Matching Under Non-transferable Utility.” Journal of Economic Theory 113.2 (2003): 182-198.
Alcalde, Jose. “Implementation of Stable Solutions to Marriage Problems.”Journal of Economic Theory 69.1 (1996): 240-254.
Ambrus, Attila, and Rossella Argenziano. “Asymmetric Networks in Two-sided Markets.” American Economic Journal: Microeconomics 1.1(2009): 17-52.
Arnosti, Nick. “Centralized Clearinghouse Design: AQuantity-Quality Tradeoff” Working Paper (2016).http://dx.doi.org/10.2139/ssrn.2571527
Ashlagi, I., Kanoria, Y., and Leshno Jacob D. “Unbalanced Random Match-ing Markets: The Stark Effect of Competition.” Journal of PoliticalEconomy 125.1 (2017): 69-98.
Blomqvist, N. “On an Exhaustion Process.” Scandinavian Actuarial Journal35 (1952): 201-210.
42
Caillaud, Bernard, and Bruno Jullien. “Chicken & Egg: Competition amongIntermediation Service Providers.” RAND Journal of Economics 34.2(2003): 309-328.
Damiano, Ettore, and Hao Li. “Price Discrimination and Efficient Match-ing.” Economic Theory 30.2 (2007): 243-263.
Damiano, Ettore, and Li Hao. “Competing Matchmaking.” Journal of theEuropean Economic Association 6.4 (2008): 789-818.
Ebrahimy, Ehsan, and Robert Shimer. “Stock-Flow Matching.” Journal ofEconomic Theory 145.4 (2010): 1325-1353.
Fershtman, Daniel, and Alessandro Pavan. “Dynamic Matching Auctions.”Working Paper (2017).
Gale, David, and Lloyd S. Shapley. “College Admissions and the Stabilityof Marriage.” American Mathematical Monthly 69.1 (1962): 9-15.
Gale, David, and Marilda Sotomayor. “Some Remarks on the Stable Match-ing Problem.” Discrete Applied Mathematics 11.3 (1985): 223-232.
Gale, David, and Marilda Sotomayor. “Ms. Machiavelli and the StableMatching Problem.” American Mathematical Monthly 92.4 (1985): 261-268.
Gomes, Renato, and Alessandro Pavan. “Many-to-many Matching and PriceDiscrimination.” Theoretical Economics 11.3 (2016): 1005-1052.
Hitsch, Gunter J, Hortacsu, Ali and Ariely, Dan. “Matching and Sorting inOnline Dating.” American Economic Review 100.1 (2010): 130-163.
Jullien, Bruno, and Alessandro Pavan. “Information Management and Pric-ing in Platform Markets.” Working Paper (2016).
Kara, Tarik, and Tayfun Sonmez. “Nash Implementation of MatchingRules.” Journal of Economic Theory 68.2 (1996): 425-439.
Kemp, Adrienne W. “Absorption Sampling and the Absorption Distribu-tion.” Journal of Applied Probability 35.2 (1998): 489-494.
McVitie, D. G., and Leslie B. Wilson. “Stable Marriage Assignment forUnequal Sets.” BIT Numerical Mathematics 10.3 (1970): 295-309.
Rochet, Jean-Charles, and Jean Tirole. “Platform Competition in Two-sided Markets.” Journal of the European Economic Association 1.4(2003): 990-1029.
43
Rochet, Jean-Charles, and Jean Tirole. “Two-sided Markets: A ProgressReport.” RAND Journal of Economics 37.3 (2006): 645-667.
Roth, Alvin E. “Misrepresentation and Stability in the Marriage Problem.”Journal of Economic Theory 34.2 (1984): 383-387.
Roth, Alvin E.. “The Evolution of the Labor Market for Medical Internsand Residents: A Case Study in Game Theory.” Journal of PoliticalEconomy 92.6 (1984): 991-1016.
Roth, Alvin E. “On the Allocation of Residents to Rural Hospitals: aGeneral Property of Two-sided Matching Markets.” Econometrica 54.2(1986): 425-427.
Roth, Alvin E. “The Economist as Engineer: Game Theory, Experimenta-tion, and Computation as Tools for Design Economics.” Econometrica70.4 (2002): 1341-1378.
Sonmez, Tayfun. “Strategy-Proofness and Essentially Single-Valued Cores.”Econometrica 67.3 (1999) 677-689.
Weyl, E. Glen. “A Price Theory of Multi-sided Platforms.” American Eco-nomic Review 100.4 (2010): 1642-1672.
