Competition Between Two-Sided Platforms Under Demand and

MANUFACTURING & SERVICE OPERATIONS MANAGEMENTArticles in Advance, pp. 1–19

http://pubsonline.informs.org/journal/msom ISSN 1523-4614 (print), ISSN 1526-5498 (online)

Competition Between Two-Sided Platforms Under Demand andSupply Congestion EffectsFernando Bernstein,a Gregory A. DeCroix,b N. Bora Keskina

a Fuqua School of Business, Duke University, Durham, North Carolina 27708; bWisconsin School of Business, University ofWisconsin–Madison, Madison, Wisconsin 53706Contact: [email protected], https://orcid.org/0000-0002-1013-6898 (FB); [email protected],

https://orcid.org/0000-0003-2638-6522 (GAD); [email protected], https://orcid.org/0000-0003-1828-0533 (NBK)

Received: September 23, 2018Revised: March 30, 2019; September 16, 2019Accepted: October 22, 2019Published Online in Articles in Advance:October 1, 2020

https://doi.org/10.1287/msom.2020.0866

Copyright: © 2020 INFORMS

Abstract. Problem definition: This paper explores the impact of competition betweenplatforms in the sharing economy. Examples include the cases of Uber and Lyft in thecontext of ride-sharing platforms. In particular, we consider competition between twoplatforms that offer a common service (e.g., rides) through a set of independent serviceproviders (e.g., drivers) to a market of customers. Each platform sets a price that is chargedto customers for obtaining the service provided by a driver. A portion of that price is paidto the driver who delivers the service. Both customers’ and drivers’ utilities are sensitive tothe payment terms set by the platform and are also sensitive to congestion in the system(given by the relative number of customers and drivers in the market). We consider twopossible settings. The first one, termed “single-homing,” assumes that drivers workthrough a single platform. In the second setting, termed “multihoming” (or “multiapping,”as it is known in practice), drivers deliver their service through both platforms. Academic/practical relevance: This is one of the first papers to study competition andmultihoming inthe presence of congestion effects typically observed in the sharing economy. We leveragethe model to study some practical questions that have received significant press attention(and stirred some controversies) in the ride-sharing industry. The first involves the issue ofsurge pricing. The second involves the increasingly common practice of drivers choosing tooperate onmultiple platforms (multihoming).Methodology: We formulate our problem asa pricing game between two platforms and employ the concept of a Nash equilibrium toanalyze equilibrium outcomes in various settings. Results: In both the single-homing andmultihoming settings, we study the equilibrium prices that emerge from the competitiveinteraction between the platforms and explore the supply and demand outcomes that canarise at equilibrium. We build on these equilibrium results to study the impact of surgepricing in response to a surge in demand and to examine the incentives at play whendrivers engage in multihoming. Managerial implications: We find that raising prices inresponse to a surge in demand makes drivers and customers better off than if platformswere constrained to charge the same prices that would arise under normal demand levels.We also compare drivers’ and customers’ performance when all drivers either single-homeor multihome. We find that although individual drivers may have an incentive to mul-tihome, all players areworse off when all driversmultihome.We conclude by proposing anincentive mechanism to discourage multihoming.

Supplemental Material: The online supplement is available at https://doi.org/10.1287/msom.2020.0866.

Keywords: sharing economy • two-sided platforms • competition • congestion effects • ride sharing

1. Introduction1.1. MotivationOver the past decade, the growth of the sharingeconomy has had a substantial impact on severalestablished industries, on the ways that people canearn income, and on the utilization of otherwiseunderused assets. As well-known examples, the taxiservice and hotel industries have faced new compe-tition from ride-sharing services such as Uber andLyft and housing services such as Airbnb, respec-tively, and individuals with full-time jobs have begun

earning extra money on the side by providing rides orrenting out rooms. Though in many cases there isnothing new about the underlying services beingprovided, the structure of the system providing theservice has some distinctive characteristics. First,rather than consisting of afirmwith employeeswhosework schedules are directly controlled by the firm,the firm in a sharing-economy setting is typically aplatform that facilitates matches between indepen-dent service providers and customers. Second, inmany (though certainly not all) cases, the individuals

1

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https://orcid.org/0000-0002-1013-6898

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https://orcid.org/0000-0003-2638-6522

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providing the service are doing so as a side activityrather than as their primarymeans of earning income.As a result, service providers are able to decidewhether to participate in the market at any givenpoint in time (based on the current pricing structure,the intensity of demand, and other factors), andplatforms need to set high enough prices to incen-tivize a sufficient number of service providers to beactive in the marketplace. On the other hand, pricescannot be set too high, because higher prices will tendto reduce customer demand for the service. Furthercomplicating the problem faced by the platform is thefact that price is not the only factor affecting supplyand demand—there is a feedback loop between sup-ply and demand that also plays a role. At any givenprice, a service provider would be more likely toenter the market if demand is high, because that in-creases the likelihood of successfully completing aservice and earning income (or reduces the time spentwaiting to find a customer). Similarly, a customerwould be more likely to use the service if supply ishigh, because that reduces the time spent waiting toreceive service. This interaction complicates the plat-form’s pricing problem, for example, by causing sup-ply and demand to no longer be monotone in price.Choosing a good pricing policy is challenging enoughif a platform is the only one of its kind operating in agiven market, but it becomes even more challengingif multiple platforms are present, and they need to com-pete for customers and possibly share service providers.

This paper presents and analyzes a model cap-turingmany of the key elements of such a setting. Twocompeting platforms operate in a single market for aparticular service. Each platform sets a price forservice through that platform—each customer mustpay this full price to receive service, and each pro-vider who completes a service earns a portion of thisprice (after paying a fixed commission and a fractionof the service price to the platform). A potentialcustomer’s decision about whether to seek service,and which platform to choose, is influenced by herunderlying preference between the platforms, theprices offered by the two platforms, and the level ofcongestion in each platform—that is, the number ofcustomers seeking service relative to the numberof providers active in the market. A potential serviceprovider’s decision about whether to be active in themarket is influenced by the provider’s underlyingcost to provide the service, the price for service set bythe platform, and the level of congestion in theplatform. A key feature of our model is the inclusionof these congestion effects, where the relative levels ofsupply and demand influence each other through afeedback loop. Such congestion effects are typicallyobserved in ride-sharing services such as Uber andLyft, and because these applications serve as our

primary motivation (and are familiar to most readers),we will hereafter use the terms “service provider” and“driver” interchangeably. To reflect realities that areemerging in some ride-sharing markets, we analyzetwo separate scenarios—one in which drivers canonly operate on a single platform and one in whichthey operate on both platforms.We refer to the formerscenario as single-homing and the latter as multi-homing. (In the ride-sharing industry, the latter is alsoknown as multiapping.)Within this framework we examine a variety of

research questions. On a fundamental level, we seekto understand how supply and demand in the marketrespond to changes in platform prices. From thisunderstanding, we explore what types of supply anddemand outcomes can arise at equilibrium prices andhow various underlying cost and market parametersinfluence the relationships between supply and de-mand and how much of the customer market getsserved (at equilibrium). We further leverage theseinsights to study some practical questions that havereceived significant press attention (and stirred somecontroversies) in the ride-sharing industry. The firstinvolves the issue of surge pricing—the practice ofincreasing prices during times of high demand—which has been criticized by some as a form of pricegouging (Lowrey 2014, David 2016). We investigatehow equilibriumpriceswould change if the platformsare allowed to respond to a demand spike, and wealso explore whether the parties in the market arebetter or worse off than if surge pricing were pro-hibited. The second involves the practice of driverschoosing to operate on both platforms—for example,drivers choosing to drive for both Uber and Lyft. Onone hand, numerous ride-sharing platform-switchingapps have been recently released (e.g., Mystro, Up-shift, QuickSwitch), which allow drivers to operatefor multiple platforms without the inconvenience ofusing multiple phones or having to manually switchapps. At the same time, there are reports of theplatforms taking steps to prevent or minimize thispractice (Fink 2014). By leveraging our analysis ofboth scenarios (drivers operating on one or bothplatforms), we compare the two scenarios to deter-mine which leads to better outcomes for each of theparties in the market.

1.2. Main ContributionsFor each of the scenarios considered, we derive ex-pressions for how supply and demand on the plat-forms respond to any given pair of prices, and weestablish conditions under which a price equilibriumis guaranteed to exist. At a given equilibrium priceoutcome, platforms may or may not compete directlyfor customers indifferent between the two platforms,and all drivers wishing to be active in the market may

Bernstein, DeCroix, and Keskin: Competition Between Two-Sided Platforms2 Manufacturing & Service Operations Management, Articles in Advance, pp. 1–19, © 2020 INFORMS

or may not find a customer. We characterize regionsof problem parameters such that each of these out-comes occurs based on which region the parametersfall into. We find that platforms actively compete forcustomers when overall demand intensity is lower orcustomers are less sensitive to congestion. On theother hand, all drivers are able to find customerswhen overall demand intensity is high or customersare less sensitive to congestion. We also find thatplatform competition leads to lower prices and lowercongestion levels in equilibrium, compared with asetting with a monopolist platform. In addition, wefind that, rather than being a form of exploitation,adjusting (i.e., raising) prices in response to a surge indemand actually makes both drivers and customersin the market better off than if platforms were con-strained to hold prices fixed at the equilibrium thatwould arise at normal demand levels.Whereas higherprices in isolation would seem to harm customers,raising prices incentivizes additional drivers to enterthe market, thus reducing the scarcity of supply, withthe net effect being positive for customers. That in-crease in supply (relative to demand) is a negative fordrivers, but from their perspective, the higher pricesmore than compensate, making them better off aswell. By comparing the equilibrium outcomes whendrivers multihome with the scenario where theysingle home, we find that all parties are weakly betteroff when drivers single-home. At first, this may seemto run counter to anecdotal experience—for example,it is becoming more and more common to see carswith both Uber and Lyft stickers in the window.However, this result reflects a prisoners’ dilemmatype of outcome. All else held constant, any indi-vidual driver would be better off choosing to multi-home. However, if all (or a sufficient number of)drivers choose to do so, the platforms would respondwith a new price equilibrium, and at this new equi-librium, prices and demand are weakly lower, anddrivers and customers are weakly (and, in some pa-rameter regions, strictly) worse off than when driverssingle-home and prices are set accordingly. Finally,we propose an incentive mechanism that, if imple-mented by platforms, could helpmitigate the practiceof multihoming.

The rest of this paper is organized as follows. Wereview relevant streams of literature in Section 2and then present the basic problem formulation inSection 3. In Section 4, we analyze the case of single-homing and examine the impact of surge pricing. InSection 5, we analyze the case of multihoming, and inSection 6, we compare market outcomes and partic-ipant preferences between single-homing and mul-tihoming. Section 7 discusses some extensions and ro-bustness checks of our results, and provides concluding

remarks. The proofs of all results are deferred to anonline supplement.

2. Related LiteratureThis paper is related to two distinct streams of literature.The first consists of economics research studying pric-ing behavior of multiple platforms competing in two-sided markets, and the second is a stream of operationsmanagement papers that incorporate more detailedoperational considerations suchas congestioneffectsor thephysical distribution of drivers throughout the network.In the economics literature, Rysman (2009 p. 125)

defines a two-sided market as “one in which (1) twosets of agents interact through an intermediary orplatform, and (2) the decisions of each set of agentsaffects the outcomes of the other set of agents, typi-cally through an externality.” Examples of this bodyof work include Caillaud and Jullien (2003), Rochetand Tirole (2003, 2006), Armstrong (2006), Hagiu(2006, 2009), Armstrong and Wright (2007), Ambrusand Argenziano (2009), and Choi (2010). A commonfocus of much of this literature is the structure ofequilibrium prices, addressing issues such as asym-metry in pricing between the two sides of the market(e.g., one side being subsidized in order to attractmany customers, which in turn attracts participantsfrom the other side of the market), whether pricediscrimination is possible, and the impact of multi-homing on market behavior (as well as when it willarise in equilibrium). Rysman (2009) provides a broaddiscussion of some of this literature and how theanalysis of two-sided markets relates to variousquestions of economic and public policy interest. Onekey way that our model and analysis differ from thisresearch is that we include features that reflect op-erational realities that arise inmany sharing-economysettings. Specifically, we include a measure of marketcongestion—the relative balance between the numberof drivers and the number of customers—in the utilityfunctions of both drivers and customers. This intro-duces negative externalities within each side of themarket (e.g., all else equal, a customer/driver isworse off when more customers/drivers are par-ticipating in the market) that are absent in the bulk ofthe economics literature. (An exception to this is thepaper by Hagiu (2009), which assumes that compe-tition exists on the supply side, resulting in net profitper customer being a decreasing function of thenumber of suppliers on the platform. This is some-what different than the congestion we model, andHagiu’smodel does not include negative externalitieson the customer side.) Bai and Tang (2018) studywhether multiple platforms can simultaneously beprofitable in a two-sidedmarket. Theauthors characterizeconditions under which Bertrand-type equilibria may

Bernstein, DeCroix, and Keskin: Competition Between Two-Sided PlatformsManufacturing & Service Operations Management, Articles in Advance, pp. 1–19, © 2020 INFORMS 3

arise, making competing platforms earn no profit.Nikzad (2018) investigates how labor market size andcompetition between two-sided platforms affect driverand customer welfare, but unlike our work, this studydoesnot consider suppliers’ sensitivity to congestion—that is, suppliers are assumed to receive a fixed wage,and their net profit per customer does not depend onthe number of suppliers or customers on the platform.Siddiq and Taylor (2019) consider competition be-tween two ride-sharing platforms where one of theplatforms can generate some of its supply throughthe acquisition of autonomous vehicles. The authorsstudy how intensity of competition in the labor marketaffects prices and the size of the autonomous vehiclefleet and how the cost of autonomous vehicles affectsthe competing platform’s profits. Once again, thiswork differs from ours in that congestion effects arenot incorporated into the supply or demand functions.

The treatment of operational issues such as marketcongestion is naturally more prevalent in the opera-tions literature studying two-sided platforms. Severalof these papers construct formal queueing models tocapture these effects. Riquelme et al. (2015) develop aqueueing network model capturing real-time dy-namics of a market with Poisson customer and driverentries and exponential trip durations. Drivers decidewhether to offer their service based on expectedearnings (which are a function of the current price andexpected time spent waiting to obtain a customer),and customers accept the service if a driver is avail-able and the current price is acceptable. Using somelarge market limiting arguments, the authors deriveequilibrium behavior under dynamic threshold pric-ing policies (prices rise if the number of availabledrivers drops below a threshold). They show thatsuch dynamic pricing does not allow the platform toincrease throughput or revenues, but it is more robustthan fixed pricing to changes in problem parameters.Taylor (2018) uses an M/M/k queueing model tocapture customer waiting times and derives the equi-librium number of participating drivers and effectivecustomer arrival rate (the potential arrival rate timesthe probability that a customer chooses to seek service)givenfixedprices andwages, and for a given realizationof customer valuations. The paper focuses on howagentindependence (i.e., the platform cannot control theagents’ actions) and congestion disutility affect op-timal prices and wages. In the absence of uncertaintyin customer valuation and driver opportunity cost,the author shows that concerns about congestionshould be addressed by raising wages and loweringprices but that these insights can break down inspecific ways in the presence of such uncertainties.Note that one property of the underlying congestionmodel is that there are economies of scale on thedriverside—as the number of drivers (with a particular

opportunity cost) increases, the equilibrium demandallocated to each driver increases. This is in contrast toour model, where there are negative externalitieswithin each side of the platform (e.g., a given driver isharmed when there are more drivers participating).Bai et al. (2019) present a similarmodeling framework(an M/M/k queueing model for customer waitingtimes, with similar customer and driver utility func-tions) but with continuous customer valuation anddriver opportunity costs, as well as variable servicevolume demands by customers. Under a fixed price-to-wage ratio, the authors study how various parameters—demand rate, sensitivity to waiting, service rate, andsize of the driver pool—affect the optimal wage andprice decisions.Other papers capture supply-demand balance is-

sues in ways that do not involve detailed queueingmodels. Cachon et al. (2017) take a rationing ap-proach. The authors assume that if supply exceedsdemand, then demand is spread equally across alldrivers, and if demand exceeds supply, then capacityis rationed by randomly rejecting some customers.(For this paper and others using a rationing approach,note that there is no explicit customer disutility fromcongestion effects—customers are either served or notbased on available supply.) Drivers decide whether toparticipate based on their rational expectations ofearnings, and customer demand is captured by down-wardly sloping (in price) demand curves. The authorsconsider five different wage and price contract typesthat allow varying degrees of flexibility (in the face ofdifferent market conditions such as surges in demand).They find that contracts where the price/wage ratio isheld constant perform quite well relative to contractswith full flexibility, providing support for their use inpractice. They also explore surge pricing, finding thatcustomers may be better off if both wages and pricescan adjust to demand conditions but that customers arehurt by pricing flexibility if wages are fixed. Gurvichet al. (2019) also employ rationing to reject demandthat exceeds available capacity, but stochastic de-mand is exogenous and is not influenced by platformdecisions. The authors focus on choices the platformcan make to control supply—that is, limiting thenumber of potential drivers allowed in the systemandguaranteeing drivers a given wage rate per unit oftime (which can also be implemented via a piece-ratewage per service). The authors derive newsvendor-type results that provide insights into the “cost” ofhaving drivers free to choose whether to participateat a given point in time. They also find that drivers’wages are higher in high-demandperiods, but despitethis, a larger fraction of customers are rejected as aresult of supply shortages. Hu andZhou (2017)modelsupply and demand more abstractly using functionsthat are increasing in wage and decreasing in price,


respectively, but they allow for multiple differentversions reflecting different states of the world (e.g.,weather). The authors demonstrate that requiringhigher wages for drivers (which increases supply)may actually benefit customers, because this mightlead to lower prices to spur additional demand tomatch the new supply. Using similar arguments, theyalso note that surge pricing could benefit customers.The authors then analyze a fixed commission contract(and variations on it) and explore conditions underwhich that contract can be shown to be optimal orwithin certain bounds of optimality. Reflecting the “gigeconomy” aspect of ride-sharing platforms, Benjaafaret al. (2017) relax the distinction between drivers andcustomers, allowing for individuals to endogenouslychoose among several different transportation strat-egieswhere theymight take on either (or neither) role.The authors analyze the impact of the existence of aride-sharing platform on car ownership, transportationchoices, and congestion, and they explore how own-ership and usage costs affect these outcomes. In morerecent work, Benjaafar et al. (2018) study how thecongestion level in a two-sided platform affects laborwelfare. The authors identify two regimes, one inwhich expanding the labor pool improves welfareand another in which the opposite result holds.

A final group of operations papers focuses more ongeospatial issues and/or aspects of matching cus-tomers with drivers. Afeche et al. (2018) present amodel with two locations and four routes (withineach location and between locations in each direc-tion). Queueing models capture customer waiting ateach of the locations. The authors consider differentdegrees of platform control over customer admissionand driver repositioning from one location to another.They find that it is sometimes optimal to deny ad-mission to customers even in low-demand scenariosin order to incentivize drivers to reposition them-selves. Bimpikis et al. (2019) also model a network ofmultiple locations and explore how pricing can beused to motivate driver participation and reposi-tioning to meet demand that may not be evenlyspread across locations. In the face of such unbal-anced underlying demand, the authors find that it isoptimal for the platform to set prices and compen-sation in a way that increases the geospatial supply-demand balance. Besbes et al. (2018) consider a sim-ilar type of setting and derive structural results forsupply equilibria. The authors use their model toanalyze the optimal response to a demand shock andfind that it may be beneficial to adjust prices to reducedriver profitability around such a shock, thus push-ing additional supply toward the shock. Guda andSubramanian (2017) study platform use of forecastsharing and surge pricing to influence driver posi-tioning in a two-location, two-period model. The

authors show that drivers may not trust surge de-mand forecasts, so price changes can help increasecredibility. Counterintuitively, the authors show thatit may be most effective to increase price in the lo-cation with excess supply, thus choking off demandand giving drivers incentive to move away from thatlocation. Feng et al. (2017) present a stylized model ofa circular road and compare twomethods ofmatchingcustomers with drivers—one where the platformmakes the matches and one where drivers pick up thefirst customer they pass. The authors find that the firstmethod works well for high and low traffic intensity,whereas the second method works better for mediumtraffic. Ozkan and Ward (2020) also take supply anddemand to be exogenous, and they analyze the problemof matching drivers with customers with the goal ofmaximizing the number of customers served. The au-thors propose a policy that they show is optimal in alarge market asymptotic regime.Outside the context of platform research, Johari

et al. (2010) analyze oligopoly competition in mar-kets with congestion effects. Unlike our work and theaforementioned studies, the authors focus on a settingwhere market participants are assumed to set theirown prices and the market does not involve a two-sided platform.In addition to some of the modeling details and the

specific research questions addressed, a key distinc-tion between our paper and the preceding operationspapers is that the latter focus on a single platform,whereas we study two-sided markets with two com-peting platforms. Thus our work can be seen as in-corporating key elements of both streams of literature.We combine the multiplatform nature of the econom-ics literature with some of the operational aspects (i.e.,congestion effects) of the operations literature.As the preceding discussions indicate, the issue of

surge pricing has received a fair amount of attentionin recent years. Some existing analytical studies—forexample, Cachon et al. (2017) and Hu and Zhou(2017)—have identified scenarios where surge pric-ing can benefit customers. Our analysis adds evidencethat surge pricing is beneficial and extends theseresults to settings with customer congestion effectsand multiple competing platforms. Several empiricalstudies have also looked at this issue. UsingUber datagathered in New York and San Francisco, Chen et al.(2015) find that surge pricing seems to have a smalleffect on increasing supply and a larger effect ondecreasing demand. The data used in Diakopoulos(2015) (Uber data from five locations in Washington,DC) suggest that drivers may not start driving inresponse to surge pricing but that they seem to moveinto the surge area from nearby locations. In an ex-tensive study, Chen and Sheldon (2015) collected dataon more than 25 million Uber trips in five different


cities over a span of 10 months and found that surgepricing resulted in drivers working longer shifts andcompleting more rides. Hall et al. (2015) identifyspecific examples involving Uber in New York wheresurge prices helped keep waiting times for customersrelatively low (or the lack of surge pricing had theopposite effect), and they may have increased ag-gregate driver earnings. Note that these empiricalresults generally find that surge pricing helps betteralign supply and demand, by reducing the number ofcustomerswilling to pay for a ridewhile incentivizingdrivers to increase supply. Although this is consistentwith certain aspects of ourmodel and the dynamics ofour system, the empirical studies do not directlyaddress questions of individual customer and driverutility. Also, with the exception of Hall et al. (2015),increases in demand and surge pricing occur together,which makes it difficult to identify the potentialbenefits of adding surge pricing on top of a givendemand surge as we do.

3. Problem FormulationConsider two platforms, indexed by i ∈ {0, 1}, such thateach platform enables two types of agents—namely,customers and drivers—to interact with each other.

Before presenting the details of themodel, it may behelpful to clarify the time scale that it is intended tocapture. The decisions and actions in the model takeplace within a relatively short time frame—for ex-ample, on the order of minutes or hours. Events thatoccur on that time scale include (a) underlying mar-ket demand intensity arises (perhaps depending onweather, sporting events, or other factors), (b) plat-forms set the prices customers will pay for service,(c) customer demand is realized in light of those pricesand the level of supply in the market, and (d) driverparticipation in themarket arises in light of prices andthe level of demand in the market. Decisions that arelonger term in nature have been made prior to theshort-term decisions we analyze, and they are as-sumed to be fixed for the purposes of our basicanalysis. These include (a) setting the broader com-pensation structure between drivers and platforms(the commission rate and share of service price paidby drivers to the platforms) and (b) driver choice ofwhich platform to operate through (or whether tooperate through both).

3.1. Platform DemandThere is a population of customers with heteroge-neous preferences for the two platforms. We encodethe customers’ taste for platforms as a “location”parameter vd ∈ [0, 1]; higher values of vd representstronger affinity toward platform 1, whereas lowervalues of vd correspond to stronger affinity towardplatform 0. The customer population is composed of a

continuum of infinitesimal customers whose locationparameters vd are distributed uniformly over [0, 1]with intensity λ > 0. A customer’s utility from usingplatform i depends on (a) her location parameter, vd;(b) the price that the customer pays for using thatplatform, pi; and (c) the level of supply congestion inthat platform, measured by a convex increasing func-tion of the utilization rate in the system, ρi � di/si,where di is the mass of customers receiving service onplatform i and si is the mass of drivers active onplatform i. (These quantities will be explained inmoredetail below.) In particular, we focus on quadraticcongestion functions.The negative impact of congestion reflects the delay

experienced by the customer before the start of ser-vice. This consists of two elements—the time betweena service request and beingmatchedwith a driver andthe time between matching and the beginning ofservice. In services such as ride-sharing platforms,matching is nearly instantaneous inmost cases,whereasthe time for a driver to reach the customer is nontrivial,so our model focuses on the second element. To esti-mate the impact of utilization on this delay, we simu-lated a geographical region with 100 drivers and any-where from 1 to 100 customers (representing differentutilization levels). Customers and drivers were (inde-pendently) distributed according to a bivariate normaldistribution (mimicking a city center and outlyingareas). Customers and drivers were matched in a se-quential manner—customers were randomly orderedand assigned in this order to the driver nearest tothem. The average customer delay was computed ateach utilization level, and a quadraticmodelwas usedto capture delay as a function of utilization. Themodel fit the data very well, yielding an R2 value ofapproximately 0.98. This led to the choice of an in-creasing quadratic function to capture the negativeimpact of congestion on customer utility.A customer with location parameter vd derives the

following respective utilities from using platforms 0and 1:

ud0 � 1 − vd − p0 − γρ20, (1)

ud1 � vd − p1 − γρ21, (2)

where γ denotes the customers’ sensitivity to con-gestion in the system. On the other hand, if the cus-tomer uses neither platform, she derives a net utilityof 0 from using an outside option. Denote by

¯vd the

location parameter of the customer who is indifferentbetween using platform 0 and the outside option.Similarly, we let vd be the location of the customerindifferent between using platform 1 and the out-side option. Then,

¯vd � 1 − p0 − γρ2

0 and vd � p1 + γρ21.

(Note that we assume customers are able to use eitherplatform, which is consistent, for example, with the


ease of setting up an account with multiple ride-sharing platforms simply by downloading apps andentering some basic personal information. In that sense,customers are always assumed to multihome. How-ever, we assume that each customer will only choose touse one platform—the one that delivers higher utilityas expressed in (1) and (2).)

Depending on the values of¯vd and vd, there are two

possible cases regarding how the platforms share themarket. In the first case, when

¯vd ≤ vd, any given

customer derives positive utility from at most oneplatform, so that the customer segments served by theplatforms are separated from each other. We refer tothis case as a partially covered market, hereafter ab-breviated as case P. The respective demands forplatforms 0 and 1 in case P are

d0 � λ¯vd � λ 1 − p0 − γρ2

0

( ), (3)

d1 � λ 1 − vd( ) � λ 1 − p1 − γρ21

( ). (4)

Note that if the platforms’ prices exceed 1, then thepreceding demand expressions would become neg-ative. This observation implies that there is a naturalupper bound p ≤ 1 on the platforms’ prices such thatthe demand for the platforms are well defined if theirprices are below p.

Figure 1 illustrates the demand for both platformsin case P. As can be seen in this figure, the conditioncharacterizing case P—namely,

¯vd ≤ vd—holds if and

only if the demands as specified in (3) and (4) satisfyd0 + d1 ≤ λ.

The second case is the complement of the first case,characterized by the condition

¯vd > vd. In this case,

every customer derives positive utility from at leastone of the platforms, so between them, the platformsserve the entire customer population; that is, theplatforms actively compete for customers. We referto this case as a fully covered market, hereafter ab-breviated as case F. To express the customer de-mand in this case, let vd denote the location of the cus-tomer who is indifferent between platforms 0 and 1.Then, vd satisfies 1− vd − p0 − γρ2

0 � vd − p1 − γρ21; thus,

vd � 12 [1+ p1 − p0 + γ(ρ2

1 − ρ20)]. As a result, the respective

demands for platforms 0 and 1 in case F are

d0 � λvd � 12λ 1 − p0 + p1 − γ ρ2

0 − ρ21

( )[ ], (5)

d1 � λ 1 − vd( ) � 12λ 1 − p1 + p0 − γ ρ2

1 − ρ20

( )[ ]. (6)

Therefore, in case F, we have d0 + d1 � λ. Figure 2illustrates the demand for both platforms in case F.

3.2. Platform SupplyIn ride-sharing settings (and similarly, in other two-sided markets), the process for becoming a driver is abit more involved than it is for becoming a customer.As a result, the drivers’ platform choices occur on alonger time scale than the one modeled here, so wetake those choices as given in our analysis. In the basicmodel presented here, and in the analysis in Section 4,we focus on the case of single-homing, where eachdriver has selected a single platform in which tooperate. In Section 5, we extend our analysis to thecase ofmultihoming, and in Section 6, we compare theoutcomes between the two scenarios.The supply for platform i ∈ {0, 1} is generated by a

population of driverswith heterogeneous operationalcosts, denoted by ksi, which represent the cost ofexerting the effort to pursue a service transaction. Thedriver population for platform i consists of a con-tinuum of infinitesimal drivers whose operationalcosts ksi are distributed uniformly over [0, 1] with adensity normalized to 1. In addition to incurring theseeffort costs, to provide service to a customer usingplatform i, a driver must pay a commission c ≥ 0 andshare a fraction α ∈ [0, 1] of the revenue with theplatform. This contract scheme can be viewed as atwo-part tariff, which is commonly used in two-sidedplatforms in practice. For example, ride-sharing plat-forms Uber, Lyft, and Didi typically employ suchcontracts (Global Times 2016, Antczak 2017, Lyft 2018,RideGuru 2018).A driver’s utility from serving in platform i is

determined by (a) the driver’s operational cost, ksi;(b) the drivers’ share of the price that the customers

Figure 1. Demand in Case P (Partially Covered Market)

Notes. The dark shaded region labeled d0 on the left shows the de-mand for platform 0, whereas the light shaded region labeled d1 onthe right shows the demand for platform 1. The unshaded region inthe middle corresponds to the mass of customers who choose neitherplatform.

Figure 2. Demand in Case F (Fully Covered Market)

Note. The dark shaded region labeled d0 on the left shows the de-mand for platform 0, and the light shaded region labeled d1 on theright shows the demand for platform 1.


pay for using that platform, αpi; (c) the commissioncharged by the platform for a service transaction, c;and (d) the level of congestion in the platform ascaptured by the utilization ρi. Because the latterequals the ratio of customers to drivers in the plat-form, it can be interpreted as the probability that adriver who chooses to operate will successfully receivenet earnings of αpi − c. Therefore a driver with oper-ational cost ksi derives the following utility from beingactive in platform i:

usi � −ksi + αpi − c( )

ρi for i ∈ 0, 1{ }. (7)If the driver chooses not to be active, then she derivesa net utility of 0. Let ksi be the operational cost of thedriver who is indifferent between being active inplatform i and not. Then, ksi � (αpi − c)ρi, and there-fore, the supply for platform i is

si � ksi � αpi − c( )

ρi for i ∈ 0, 1{ }. (8)There is a natural lower bound p > 0 such that thesupply for a platform is nonnegative if its price isabove p. We restrict attention to parameter valuessuch that p < p, where p is the upper bound on pricesthat ensures that the demand for each platform isnonnegative (as explained in the paragraph followingEquations (3) and (4)). Figure 3 illustrates the supplyof platform i ∈ {0, 1}.3.3. Pricing Game FormulationThe relations in (3), (5), and (8) imply a supply-demandbalance for both platforms. For i ∈ {0, 1}, we letDi(pi, pj)and Si(pi, pj) denote the balanced demand and balancedsupply, respectively, for platform i as functions of pricespi and pj, where j ∈ {0, 1} \ {i}. In subsequent sections,we provide derivations of these quantities.

At different stages in their development, platformsmay focus on different objectives. Firms that are stillestablishing themselves may focus primarily on growth.This currently seems to be the case for ride-sharingplatforms such as Uber and Lyft (see, e.g., Knowledge@Wharton (2017), Sherman (2017), Wilhelm (2017), andCook and Price (2018)). Once firms are more estab-lished they may focus on maximizing profit instead.We assume the profit maximization objective; that is,

platform i ∈ {0, 1} selects price pi to maximizeΠi(pi,pj) :� [(1−α)pi+c]min{Di(pi,pj),Si(pi,pj)}. When α � 1, thisobjective is equivalent to maximizing the number ofcustomers served.Platforms choose prices in [p, p] such that balanced

demand and balanced supply are well defined andnonnegative. Aswill be shown later, there is a suitablechoice of [p, p] such that, for i ∈ {0, 1} and all pj, themapping pi �→ Πi(pi, pj) is concave, with its maximizerin the interior of [p, p]. The prices selected by theplatforms must also satisfy the following supplyconstraint:

Di pi, pj( ) ≤ Si pi, pj

( ). (9)

This constraint reflects the fact that, regardless ofwhich objective a platform focuses on, the firm wantsto avoid unsatisfied demand in order to establish andmaintain customer acceptance of and satisfactionwith the service. Note that under the supply con-straint (9), we have min{Di(pi,pj),Si(pi,pj)}�Di(pi,pj),and thus, Πi(pi,pj) � [(1−α)pi+ c]Di(pi,pj) for i ∈ {0, 1}.We also note that as long as the demand congestioneffect is not too small (i.e., γ ≥ max{[(λ − α)(α − c)]/(α2 + 3αλ), 0}), the supply constraint is redundant inequilibrium—that is, prices that constitute an equilib-riumwhen the supply constraint ispresent continue tobeequilibrium prices in the absence of this constraint.We say that a pair of prices (p∗0 , p∗1) ∈ [p, p] × [p, p] is

a Nash equilibrium if

Π0 p∗0 , p∗1( ) ≥ Π0 p0, p∗1

( )for all p0 ∈ [

p, p]satisfying (9) for i � 0,

Π1 p∗1 , p∗0( ) ≥ Π1 p1, p∗0

( )for all p1 ∈ [

p, p]satisfying (9) for i � 1.

As mentioned earlier, we focus on actions and eventsthat are short term in nature. As a result, this for-mulation focuses on price equilibria and does notincorporate the commission structure in the decisionvariables (i.e., α and c are exogenous problem pa-rameters), because that structure is a long-term de-cision that is typically not changed in response toshort-term shocks. (See, e.g., Uber (2018) and RideGuru(2018) for discussions of this issue.) By focusing themodel in this way, we can investigate how platformsreact to short-term changes in market conditions; forexample, we are interested in whether and how ride-sharing platforms should implement surge pricingwhen there is a positive demand shock. As will beexplained in the following section, we will concen-trate on symmetric equilibria to provide a crisp analy-sis of how platforms respond to shifts in problem pa-rameters. This is why we assume that the commissionstructure (i.e., α and c) and the form of the supplyfunctions (8) are the same for both platforms.

Figure 3. Supply of a Platform

Notes. The shaded region labeled si on the left shows the supply ofplatform i ∈ {0, 1}. The unshaded region on the right shows the massof drivers on platform i who choose not to be active.


4. Analysis for Single-Homing Drivers4.1. Characterizing EquilibriaThis section studies price equilibria in the setting ofsingle-homing drivers described in Section 3.

As alluded to earlier, the existence of price equi-libria depends on well-defined balanced supply anddemand functions. From Equations (3), (5), and (8), itis clear that supply and demand on a given platforminteract with each other through the supply and de-mand congestion effects. As a first step in analyzingthat interaction, we note that if drivers single-home,then the supply of a platform does not depend di-rectly on the other platform’s supply or demand. Tobe more precise, the supply of platform i ∈ {0, 1} canbe written as a function of the demand for the sameplatform: from (8), we have

si �αpi − c( )

di√

for i ∈ 0, 1{ }. (10)

An analogous statement can be made about demandin the case of a partially covered market, but not inthe case of a fully coveredmarket, and so the behaviorof balanced demand and balanced supply differsdepending on which of these two cases arises in re-sponse to the prices p0 and p1 charged by the plat-forms. We next derive the balanced demand andsupply functions for the single-homing setting.

Lemma1. The balanced demand for platform i, as a functionof prices pi and pj, is given by

Di pi, pj( )� dP

i pi( )

if dP0 p0( ) + dP

1 p1( ) ≤ λ Case P[ ],

dFi pi, pj( )

if dP0 p0( ) + dP

1 p1( )

> λ Case F[ ],

{(11)

where

dPi pi( ) � λ

αpi − c( )

1 − pi( )

αpi − c + λγ, (12)

dFi pi, pj( )� λ

αpi − c( )

λγ + αpj − c( )

1 − pi + pj( )[ ]

2 αpi − c( )

αpj − c( ) + λγ α pi + pj

( ) − 2c[ ] , (13)

for i, j ∈ {0, 1} with i � j. The corresponding balancedsupply for platform i is given by

Si pi, pj( )� sPi pi

( )if dP

0 p0( ) + dP

1 p1( ) ≤ λ Case P[ ],

sFi pi, pj( )

if dP0 p0( ) + dP

1 p1( )

> λ Case F[ ],

{(14)

where

sPi pi( ) � αpi − c

( ) λ 1 − pi( )

αpi − c + λγ

√, (15)

sFi pi, pj( ) � αpi − c

( )×

λ λγ + αpj − c

( )1 − pi + pj( )[ ]

2 αpi − c( )

αpj − c( ) + λγ α pi + pj

( ) − 2c[ ]√

,(16)

for i, j ∈ {0, 1} with i � j.In the expressions in (11)–(16), the superscript “P”

denotes partial market coverage (case P), whereas thesuperscript “F” denotes full market coverage (case F). Itfollows fromLemma 1 that the condition characterizingcase P (i.e.,

¯vd ≤ vd) holds if dP

0(p0)+dP1(p1) ≤λ, whereas

the condition characterizing case F (i.e.,¯vd > vd) holds

if dP0(p0) + dP

1(p1) > λ (see Figures 1 and 2 for illustra-tions of these two conditions). Note that in case P, theplatforms act similar to two local monopolies, settingthe same prices as they would if the other platform didnot exist. In case F, on the other hand, platforms ac-tively compete through prices for customers who havepositive utility for both platforms. We next show theexistence of a price equilibrium between the platforms.

Theorem 1. There exists a unique symmetric price Nashequilibrium for the platforms.

To understand the equilibrium behavior for each setof input parameters, for the remainder of this sectionwe assume that α � 1 for analytical tractability. Werevisit the case of general α in Section 7.The next result derives conditions under which the

equilibriumprices result in partial or fullmarket coverage(i.e., local monopolistic behavior or active competition).

Proposition 1. The following are the symmetric equilib-rium prices for each set of parameters λ, γ, and c:

a( ) p∗ � c + λ2γ2 + λγ 1 − c( )√ − λγ

for γ ≥ 12−c( )22λ and γ ≥ λ 1−c( )

1+2λ ,

b( ) p∗ � c + λ 1−γ−c( )1+λ

for γ ≥ 12 − λ

2 − c and γ < λ 1−c( )1+2λ ,

c( ) p∗ � c + λ2

for γ < 12 − λ

2 − c and γ < λ2 ,

d( ) p∗ � c +λγ2

√for γ <

12−c( )22λ and γ ≥ λ

2 .

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩In region (a), the platforms partially cover the market, andthe equilibrium demand is strictly lower than equilibriumsupply. In region (b), the platforms again partially coverthe market, but equilibrium demand equals equilibrium


supply for each platform. In regions (c) and (d), the plat-forms fully cover the market, with equilibrium demandequal to equilibrium supply for each platform in region (c)and equilibrium demand strictly lower than equilibriumsupply for each platform in region (d).

Figure 4 illustrates regions (a)–(d) in the (λ, γ) space,showing how some of the underlying parametersaffect the relative balance between the number ofcustomers who receive service and both the totalmarket size and the number of participating drivers.As the overall market size (represented by λ) getslarger, it is more likely that platforms will not directlycompete for marginal customers because some of thepotential market will go unserved (i.e., we movetoward regions (a) and (b) in Figure 4) and that allparticipating drivers will be occupied serving cus-tomers (i.e., we move toward regions (b) and (c) inFigure 4). As customers become more sensitive tocongestion (i.e., γ increases), it is also more likely thatsome of the potential market will go unserved, butthis is driven by reduced customer utility fromgettingservice, and thus reduced customer participation,whereas the impact of λ stems from the growth in thepotential market size outpacing the drivers’ will-ingness to participate. Note that γ has the oppositeeffect of λ on the supply-demand balance—as γ in-creases, the decreased willingness of customers toparticipate moves the system toward having moredrivers than customers.

The preceding visual representations of the equi-librium outcomes can also be useful for obtaininginsights into the impact of competition between plat-forms in this market. In regions (a) and (b) in Figure 4,the platforms effectively operate as local monopoliesbecause the customer segments they serve are distinctand separate. In these regions, neither platform isaffected by the presence of a competitor. In regions (c)

and (d), however, the platforms directly interact andcompete for customers who have positive utilities forboth platforms, yielding different outcomes than ifonly a single platform were operating in the market.The following result summarizes the impact on aplatform’s equilibrium outcomes of facing competi-tion from a second platform, compared with a sce-nario where the platform has a true monopoly in themarket. To compare utilities in this result, we refer to acustomer who uses either of the platforms as a pur-chasing customer and to a driver who offers service ineither of the platforms as an active driver.

Proposition 2. When a platform faces competition, theequilibrium values for platform price, supply, demand,utilization, and an active driver’s utility are all (weakly)lower, whereas the equilibrium value for a purchasingcustomer’s utility is (weakly) higher, relative to the case of amonopolist platform.

As one might expect, the presence of competitiondrives equilibrium prices lower. Because the twoplatforms split the market, each platform’s demand islower (making a platform worse off than if it were amonopoly), and so it does not need to attract as manydrivers. Proposition 2 also shows that competitionresults in lower congestion, benefiting customers.With lower prices and utilization, customers’ utilitiesare higher but drivers are worse off in the presence ofcompetition.

4.2. Analysis of Surge PricingWe next build on our price equilibrium results toexamine the impact of surge pricing on customers anddrivers. More specifically, we study the impact of asurge in potential demand—as represented by anincrease in the market size λ—on the equilibriumprices and other relevant quantities. Consistent with

Figure 4. Price Equilibrium Regions in the (λ, γ) Space When Drivers Are Single-Homing

Note. The cases (a), (b), (c), and (d) given in Theorem 1 are shown by regions (a), (b), (c), and (d), respectively, in the (λ, γ) space.


our focus on short time frames, the types of marketsize increase we consider are those of relatively shortduration. Such market shifts can be the result ofunusual weather events (e.g., rain, snow) or increasedtraffic related to public events (e.g., sport games,shows). In regions that are supply constrained (re-gions (b) and (c) in Figure 4), it is clearly necessary forthe platforms to adjust prices to ensure sufficientsupply to meet the higher demand levels resultingfrom an increased market size. For regions that aresupply unconstrained, it is less clear whether allowingsuch an adjustment would benefit various participantsin themarket, and sowe focus on these regions. It isfirsteasy to verify that the equilibrium prices are increasingin λ, indicating that a surge in the market size is ac-companied by surge pricing in equilibrium. Ourfollowing result establishes the impact of an increasein the market size on customer and driver utilities. Tostate this result, we let p∗(λ) be the equilibrium pricecharacterized in Theorem 1, with its dependence onthe market size λ expressed explicitly. As before, letus refer to a customerwho uses either of the platformsas a purchasing customer and to a driver who offersservice in either of the platforms as an active driver.With a slight abuse of notation, we also let udi(p, λ)and usi(p, λ) denote the utilities of a purchasing cus-tomer and an active driver, respectively, for platformi ∈ {0, 1}, when both platforms charge the price p andthe market size is λ.

Proposition 3. Consider two levels of the market size,λ < λ, such that the pairs (λ, γ) and (λ, γ) are in regions ofunconstrained supply (i.e., regions (a) and (d) in Figure 4).i. If the platforms adjust their equilibrium prices in

response to a market size surge from λ to λ, then a pur-chasing customer’s utility decreases and an active driver’sutility increases. That is, for i ∈ {0, 1},

udi p∗ λ( ), λ( )> udi p∗ λ

( ), λ

( )and

usi p∗ λ( ), λ( )< usi p∗ λ

( ), λ

( ).

ii. Given λ, there exists Λ(λ, γ) ∈ (λ,∞] such that forλ ∈ (λ,Λ(λ, γ)], purchasing customers and active driversare better off if the platforms adjust their equilibrium pricesin response to a surge in the market size than if the plat-forms keep their original equilibrium prices. That is, fori ∈ {0, 1},

udi p∗ λ( ), λ( )< udi p∗ λ

( ), λ

( )and

usi p∗ λ( ), λ( )< usi p∗ λ

( ), λ

( ).

Both parts of Proposition 3 are illustrated in Figure 5.This figure displays two instances of the supply anddemand curves for a particular platform (as a func-tion of that platform’s price)—the lower instance ofeach curve corresponds to the original market size λ,whereas the higher one corresponds to the largermarket size λ.

Figure 5. Surge Pricing When Drivers Are Single-Homing

Notes. The lower and upper bold curves display the balanced demand functions when the market size is λ � 0.6 and λ � 0.8, respectively.Similarly, the lower and upper dashed curves display the balanced supply functions when the market size is λ � 0.6 and λ � 0.8, respectively.Vertical coordinates of points �, �, and � are Di(p∗(λ), λ), Di(p∗(λ), λ), and Di(p∗(λ), λ), respectively, (and their horizontal coordinates arep∗(λ), p∗(λ), and p∗(λ), respectively). The other problem parameters are c � 0.2 and γ � 0.4.


Part (i) of Proposition 3 reflects a comparison be-tween point�, which shows the original equilibriumprices at the original market size, and point�, whichshows the new (higher) equilibrium prices at thelarger market size. The proposition states that anincrease in the market size accompanied by surgepricing results in a higher utility for drivers and alower utility for customers. Note that there is a larger“supply cushion” of drivers at point � (the distancebetween that point and the lower supply curve) thanat point � (the distance between that point and thehigher supply curve), and this corresponds to a higherutilization ρi at point � than at point �. This makesany original customers worse off at point �, as doesthe higher price, and so their utility drops. Driversexperience the opposite—the higher utilization com-bined with the higher price causes their utilities toincrease. Following the common belief on surge pric-ing (Lowrey 2014, David 2016), higher prices come atthe expense of lower customer utility. However, thiscomparison is somewhat misleading. Given that anexogenous event caused demand to surge from itsoriginal level λ to a higher level λ, the question iswhether customers are better off when this demandsurge is accompanied by a surge in prices, rather thanby platforms keeping their original equilibriumpricesthat were associated with the original market size λ.Part (ii) of Proposition 3 explores that question.

A comparison of point � and point � in Figure 5helps illustrate part (ii). The move from point � topoint� reflects just each platform’s reoptimization ofprice given the new larger market size λ. Both cus-tomers and drivers experience a trade-off. As onemight expect, and as opponents of surge pricingmight argue, on its own, the increased price hurtscustomers but helps drivers. On the other hand, thesupply cushion (the gap between the upper demandcurve and the upper supply curve) increases as wemove from point� to point�—that is, the utilizationdecreases, which is good for customers but bad fordrivers. The net impact of these contrary effects is notobvious a priori, but Proposition 3 shows that bothcustomers and drivers are better off at point �—thatis, they both benefit from allowing surge pricing. Thisis perhaps least easy to anticipate from the customer’sperspective, and this further illustrates the impor-tance of capturing the congestion effects on customerand driver utilities. Finally, whereas part (ii) of theproposition explicitly states that utilities increase forpurchasing customers and active drivers, we notethat this also implies that some additional customersand/or driversmay become active (because they nowhave positive utility), so all customers and drivers areat least weakly better off.

5. Analysis for Multihoming DriversThis section extends our formulation and analysis to asetting inwhich driversmultihome (i.e., simultaneouslyparticipate in both platforms). Recall that we focus ouranalysis on events that happenwithin short time frames.Because the choice of which platform(s) to participate inis a longer-term decision by the drivers, we take thischoice (including the choice of whether to single- ormultihome) as exogenous. In this setting, the driverpopulations of the two platforms jointly serve the entirecustomer population that generates the demand forplatforms. Accordingly, the utilizations of the platformsare based on aggregate supply and demand; that is,

ρ0 � ρ1 � d0 + d1s0 + s1

. (17)

In the single-homing setting studied in precedingsections, a driver’s utility from serving in platform i ∈{0, 1} (expressed as usi in (7)) depends only on the pricefor using that platform, pi. In themultihoming setting,drivers face a mixed population of customers acrossboth platforms, implying that usi depends on bothprices. Supposing that the likelihood of serving agiven platform’s customer is proportional to the de-mand for that platform, we replace the price term piin (7) with di

d0+d1 pi +dj

d0+d1 pj to obtain the followingexpression for usi:

usi � −ksi + αdi

d0 + d1pi + dj

d0 + d1pj

( )− c

[ ]ρi

for i, j ∈ 0, 1{ } with i � j.(18)

Based on this, the supply of platform i is given by

si � αdi

d0 + d1pi + dj

d0 + d1pj

( )− c

[ ]ρi

for i, j ∈ 0, 1{ } with i � j.(19)

By (17) and (19), we can also express the supply ofplatforms as follows:

s0 � s1 �αp0 − c( )

d0 + αp1 − c( )

d12

√, (20)

which is analogous to (10) in the single-hominganalysis. As in the preceding section, the analysisof price equilibria depends on well-defined balanceddemand and supply functions. We next derive thebalanced demand and supply functions for the multi-homing setting.

Lemma 2. There exists a compact set 3 ⊆ [p, p] × [p, p]with 3 ⊃ {(p0, p1) ∈ [p, p] × [p, p] : p0 � p1} such that for


all price pairs (p0, p1) ∈ 3, the balanced demand for platformi is well defined and given by

Di pi, pj( )

�dPi pi, pj( )

if dP0 p0, p1( ) + dP

1 p1, p0( ) ≤ λ

Case P[ ],dFi pi, pj( )

if dP0 p0, p1( ) + dP

1 p1, p0( )

> λ

Case F[ ],

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩(21)

where

dPi pi, pj( ) � λ

2ξ+ p0, p1

( ) − pi + pj[ ]

, (22)

dFi pi, pj( ) � λ

21 − pi + pj( )

, (23)

for i, j ∈ {0, 1} with i � j, and ξ+(p0, p1) �−B(p0,p1) +

B(p0,p1)2−4A(p0,p1)C(p0,p1)

√2A(p0,p1) with A(p0, p1) � 2[λγ+

α(p0+p1)2 − c], B(p0, p1) � 4[c − (α+c)(p0+p1)

2 + αp0p1], andC(p0, p1) � α(2 − p0 − p1)(p0 − p1)2. Moreover, for all pricepairs (p0, p1) ∈ 3, ξ+(p0, p1) is concave in p0 and p1. Thecorresponding balanced supply for platform i is

Si pi, pj( )

�sPi pi, pj( )

if dP0 p0, p1( ) + dP

1 p1, p0( ) ≤ λ

Case P[ ],sFi pi, pj( )

if dP0 p0, p1( ) + dP

1 p1, p0( )

> λ

Case F[ ],

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩(24)

where

sPi pi, pj( )� 12

λξ+ p0, p1

( )α p0 + p1( ) − 2c

[ ] − λα p0 − p1( )2√

,

(25)sFi pi, pj( ) � 1

2

λ α p0 + p1

( ) − 2c[ ] − λα p0 − p1

( )2√, (26)

for i, j ∈ {0, 1} with i � j.

Because of the interdependence of supply anddemand quantities, the existence of a well-definedsupply-demand balance entails additional conditionsin themultihoming setting.We note that in the single-homing setting, both platforms havewell-defined andconcave payoff functions. To have a well-defined pricinggame with concave payoffs in the multihoming setting,one needs to restrict the action space of the two platforms.For thispurpose,we showinLemma 2 that there exists arestricted subset of the price space—namely, 3—inwhich both platforms have well-defined and concavedemand functions. As will be shown later, this alsoyields well-defined and concave payoff functions for

both platforms. Therefore, the multihoming settingcan accommodate a well-defined pricing game with arestricted action space that contains all candidates forsymmetric equilibria (i.e., the price pairs on the linep0 � p1).It follows from Lemma 2 that the condition for case

P (i.e.,¯vd ≤ vd) holds if dP

0(p0,p1) +dP1(p1,p0) ≤λ, and the

condition for case F (i.e.,¯vd > vd) holds if dP

0(p0, p1) +dP1(p1, p0) > λ (see Figures 1 and 2 for illustrations of

these two conditions). Given (21), platforms chooseprices p0 and p1 from3 in equilibrium,with platform imaximizing its profit Πi(pi, pj) while satisfying thefollowing joint supply constraint:

D0 p0, p1( ) +D1 p1, p0

( ) ≤ S0 p0, p1( ) + S1 p1, p0

( ). (27)

As in single-homing, this constraint is redundant inequilibrium if γ ≥ max{[(λ − α)(α − c)]/(α2 + 3αλ), 0}.In the context of multihoming drivers, we say that a

pair of prices (p∗0 , p∗1) ∈ 3 is a Nash equilibrium if

Π0 p∗0 ,p∗1( ) ≥Π0 p0,p∗1

( )for all p0 satisfying p0,p∗1

( ) ∈3 and (27),Π1 p∗1 ,p∗0

( ) ≥Π1 p1,p∗0( )

for all p1 satisfying p∗0 , p1( ) ∈3 and (27).

In the next result, we show the existence of a priceequilibrium between the platforms.

Theorem 2. There is a set Θ of problem parameters suchthat if (λ, γ, α, c) ∈ Θ, then there exists a unique symmetricprice Nash equilibrium for the platforms.

As in the preceding section, with the goal of un-derstanding the equilibrium behavior for each set ofinput parameters, we assume that α � 1 going for-ward (andwe revisit the case of generalα in Section 7).The following result characterizes the set Θ in closedform and the equilibrium in the different cases ofmarket coverage.

Proposition 4. Let Θ � Θ1 ∩Θ2, where Θ1 � {(λ, γ, c) ∈R3+ : 2(1 − c)2 ≥ λγ(1 − c) + λ2γ2} and Θ2 � {(λ, γ, c) ∈R3+ : γ ≤ 1

2 − λ2 − c or γ ≥ 3

3−2c+c2√ +c−5

2λ }. If (λ,γ,c)∈Θ thenthe following are the symmetric equilibrium prices for eachset of parameters λ, γ, and c:

a( ) p∗ � c +λ2γ2+λγ 1−c( )

2

√− λγ

for γ≥ 33−2c+c2√ +c−5

2λ and γ ≥ 2λ 1−c( )1+2λ−λ2 ,

(b) p∗ � c + λ 1−γ−c( )1+λ

for γ ≥ 12 − λ

2 − c and γ < 2λ 1−c( )1+2λ−λ2 ,

c( ) p∗ � c + λ2

for γ < 12 − λ

2 − c.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩


In region (a), the platforms partially cover the market, andthe equilibrium demand is strictly lower than equilibriumsupply. In region (b), the platforms again partially coverthe market, but equilibrium demand equals equilibriumsupply. In region (c), the platforms fully cover the market,with equilibrium demand equal to equilibrium supply.

Figure 6 depicts regions (a)–(c) in the (λ, γ) space.From this figure we see that the impacts the param-eters have on driver utilization and whether themarket is fully covered are qualitatively the same asfor the case of single-homing. Specifically, as themarket size λ increases, platform competition forcustomers is less intense, and all drivers are generallymore likely to be occupied serving customers; ascustomer sensitivity to congestion γ increases, it isalso more likely that the market will be partiallycovered but that there will be more drivers thancustomers in the market.

As stated in Theorem 2 and Proposition 5, there is aprice equilibriumbetween the platforms as long as theproblem parameters reside in the setΘ. If the problemparameters are outside Θ, then there is no priceequilibrium because the balanced demand expressionwould either become negative or exceed the totalmarket size, meaning that the balanced demand wouldbe ill-defined in equilibrium. In particular, in the contextof Proposition 5, if (λ, γ, c) /∈ Θ1, then the balanceddemand becomes negative, whereas if (λ, γ, c) /∈ Θ2,the balanced demand exceeds the total market size.

6. Comparison of Single-Homingand Multihoming

In this section, we compare the equilibrium outcomesin the single-homing and multihoming settings. Inparticular, we study how price, demand, and utilities

of customers and drivers change in equilibriumwhendrivers switch from single-homing (SH) to multi-homing (MH). Consistent with our earlier focus onsymmetric platforms and symmetric equilibria, in thissection we generally suppress the platform index fornotational simplicity.Given problem parameters for which price equi-

libria exist as in Propositions 1 and 4,we let p∗SH and p∗MH

be the equilibrium prices in the settings of single-homing and multihoming, respectively. To comparethe performance of platforms in these two settings,wealso let D∗

SH and D∗MH be the equilibrium demand

quantities under single-homing and multihoming,respectively. For a purchasing customer (i.e., a customerwho uses either of the platforms), let u∗d,SH and u∗d,MH

bethe equilibrium utilities of said customer under single-homing and multihoming, respectively. Then,

u∗d,SH − u∗d,MH � p∗MH + γ ρ∗MH

( )2 − p∗SH − γ ρ∗SH( )2,

where ρ∗SH and ρ∗MH are the equilibrium utilizationsunder single-homing and multihoming, respectively.Similarly, for an active driver (i.e., a driver who offersservice in either of the platforms), we let u∗s,SH and u∗s,MH

be the equilibrium utility of this driver under single-homing and multihoming, respectively. That is,

u∗s,SH − u∗s,MH � p∗SH − c( )

ρ∗SH − p∗MH − c( )

ρ∗MH.

We note that the parameter regions characterized inPropositions 1 and 4 are distinct. Therefore, to com-pare the equilibrium outcomes in each possible case,we need to consider how the regions in Propositions 1and 4 overlap in the (λ, γ) space. To that end, wedefine region (x-y) as the intersection of region (x) inthe single-homing setting and region (y) in the mul-tihoming setting, for x ∈ {a, b, c, d} and y ∈ {a, b, c} (see

Figure 6. Price Equilibrium Regions in the (λ, γ) Space When Drivers Are Multihoming

Notes. The cases (a), (b), and (c) given in Theorem 2 are shown by regions (a), (b), and (c), respectively, in the (λ, γ) space. In the dark shaded areaat the top left, there is no equilibrium in the multihoming setting.


Propositions 1 and 4 and Figures 4 and 6 for thedefinitions of these regions). Thismeans that if (λ, γ) isin region (x − y), then (λ, γ) is in Region (x) whendrivers single-home and in region (y) when driversmultihome.

Based on this construction, the following resultcompares the equilibrium outcomes under single-homing and multihoming.

Proposition 5. Let (λ, γ, c) ∈ Θ � Θ1 ∩Θ2, where Θ1 �{(λ, γ, c) ∈ R3+ : 2(1 − c)2 ≥ λγ(1 − c) + λ2γ2} and Θ2 �{(λ,γ,c) ∈R3+ :γ≤ 1

2− λ2− c or γ≥ 3

3−2c+c2√ +c−5

2λ }. We havep∗SH ≥ p∗MH, D∗

SH ≥ D∗MH, u∗d,SH ≥ u∗d,MH

, and u∗s,SH ≥ u∗s,MH.Moreover, if (λ, γ) is outside regions (b-b) and (c-c), thenp∗SH > p∗MH, u

∗d,SH > u∗d,MH

, and u∗s,SH > u∗s,MH,whereas if (λ,γ) isoutside regions (b-b), (c-c), and (d-c), then D∗

SH > D∗MH.

A simple verbal paraphrase of Proposition 5 is thatunder multihoming, equilibrium prices, demands,and the utilities of purchasing customers and activedrivers are all lower than they are in the single-homing setting. Moreover, they are strictly lowerunder the conditions that are provided in the prop-osition. This also implies that the platforms are betteroff under single-homing. The regions referenced inthis proposition are depicted in Figure 7.

Although one might intuitively anticipate the factthat the platforms prefer single-homing (i.e., pre-venting drivers from working for the competingplatform), it may not be obvious that customers anddrivers share that preference. This is because multi-homing introduces externalities into the pricing de-cisions by the platforms. First, price increases by aplatform are less effective at motivating driver par-ticipation under multihoming. In single-homing, thefull impact of a platform’s price influences driverson that platform (as seen in (8)). In multihoming(see (19)), a platform’s price has only a partial impactonmotivating drivers from its own platform, but nowit has an equal impact on motivating drivers from theother platform (who can also serve on the platform inquestion as a result of multihoming). However, ahigher price causes that platform’s demand to drop,altering the weights on the prices in (19) such that theoverall motivating impact of a price increase is lowerin multihoming. Second, if a platform uses higherprices to induce a multihoming driver to enter themarket, that platform receives only part of the benefitof having that driver active—because a fraction of thedriver’s services will be through the competing plat-form. As a result, each platform gets less benefit fromincreasing prices under multihoming, and thereforeequilibrium prices are lower than under single-homing.These lower prices cause drivers to earn less. Customersbenefit from the lower prices but suffer from reduceddriver participation—and it turns out that the latter

dominates, making customers worse off. This in turnreduces demand in the market.Given Proposition 5, one might wonder why mul-

tihoming is often observed in practice. As statedearlier, it is, for example, quite common to see a carwith both Uber and Lyft stickers on it. The key ob-servation here is that the equilibrium prices, supply,and demand under single-homing represent an equi-librium under the assumption that driversmust single-home (and similar for the multihoming case). If weconsider a longer time frame in which drivers are ableto choose which platform(s) to join, it turns out thatthey have an incentive to multihome.To see this, consider a scenario where platform

prices are fixed and each driver’s single/multihomingstatus is fixed (though any configuration of single/multihoming across drivers of both platforms isallowed). To determine which customers get matchedwithwhich drivers, one can imagine a sample pathwitha random ordering of customers and a random order-ing (ranking) of drivers. On this sample path, a cus-tomer is assigned to the first driver that can accom-modate her (i.e., the first driver that either single-homeson that customer’s platform or multihomes). For anysample path (random sequence of customer arrivalsand random ranking of drivers), a failure of the fo-cal driver to obtain a customer under multihomingmust be the result of all customers being served by a

Figure 7. Changes in Price Equilibria When Drivers Switchfrom Single-Homing to Multihoming

Notes. The dashed and bold curves show the boundaries of the re-gions in Figures 4 and 6, respectively. In the dark shaded area at thetop left, there is no equilibrium in the multihoming setting. In themoderately shaded area at the top right, upon transitioning fromsingle-homing to multihoming, price, demand, and utilities of cus-tomers and drivers all decrease in equilibrium. In the light shadedarea in the bottom left, similar effects occur, except that equilibriumdemand remains the same. In the unshaded area at the bottom right,equilibrium outcomes do not change.


higher-ranked driver (because the focal driver couldserve either type of customer). As a result, the fo-cal driver would also fail to obtain a customer if thatdriver chose single-homing. Therefore the focaldriver’s probability of obtaining a customer (which isjust the expected value of the number of customersobtained across all possible sample paths) must beat least weakly larger under multihoming than un-der single-homing. One can construct at least onepositive-probability sample path such that the focaldriver obtains a customer under multihoming but notunder single-homing. As a result, the probability thatthe focal driver is matched with a customer is strictlyhigher under multihoming. Note that utilization ρican be interpreted as the probability that a driverobtains a customer. Thus, given the drivers’ utilityexpression (7), a higher probability of getting a cus-tomer in multihoming corresponds to higher driverutility, so the driver prefers to multihome. Because thepreceding argument holds for any configuration ofdrivers, one can iteratively apply it to each driver inthe market to conclude that all drivers ultimatelyhave an incentive to choose multihoming and that ifa driver makes that choice she will maintain thatpreference regardless of subsequent choices by otherdrivers. The end result is a situation where all driversprefer to multihome.

To summarize, one can imagine the evolution of amarket with competing platforms as follows. At first,when the platforms are relatively new, all driversmayaffiliate with a single platform (single-home), and theplatforms would select pricing policies under thatassumption. Over time, the incentives described ear-lier may lead more and more drivers to choose tomultihome, eventually leading to full multihomingacross both platforms. At that point, the platformswould recognize this fact, resulting in a new multi-homing price equilibrium with the properties de-scribed in Proposition 5. (Note that the platformscould, in fact, adjust their prices along the way torespond to partial multihoming—this would not alterthe driver incentives described previously.)

Because equilibriumprices and demands are lower,and customers and drivers are worse off under multi-homing, platforms might wish to identify mechanismsto deter this behavior. For example, a platformmay seekto structure driver compensation in a way that differ-entiates between single- and multihoming drivers to asufficient extent that drivers would never choose tomultihomewhenmaking their up-front platformchoice.This might be achieved in practice through the use ofsome kind of loyalty or incentive program. For example,if a driver completes a certain number of rides τwithin afixed number of driving opportunities T, the com-mission could be reduced on all rides given by thatdriver during that time window—say, from c + δ to c.

The threshold τ could be set high enough that itwouldbe difficult to reach if a driver is engaging in signif-icant multihoming but relatively easy to reach if thedriver primarily or exclusively single homes. Forexample, for a given utilization ρi, the expectednumber of successful rides given in T attempts wouldbe ρiT, whereas that number would be about 1

2ρiT ifthe driver is splitting efforts between the two plat-forms. Setting τ such that 12ρiT < τ < ρiTwould allowsingle-homing drivers to pay the lower commission c,whereas multihoming drivers would pay c + δ.We seek to formalize this idea and establish how

large δwould need to be for a single-homing driver tohave no incentive to multihome. To that end, supposethe system starts with all drivers single-homing andprices at the resulting unique symmetric equilibrium(with all secondary quantities di, ρi, etc., at corre-sponding values). Also, suppose that the threshold τand the opportunity window T are chosen in such away that all single-homing drivers pay commission cand any driver choosing to multihome will pay c + δ.

Proposition 6. If drivers on platform i pay commission cif they single-home and c + δ if they multihome, then forany δ > (1 − ρi)(pi − c), every platform i driver prefers tosingle-home.

The lower bound on δ given in the proposition hasan intuitive interpretation. Under single-homing, adriver earns an average of ρi(pi − c) per service at-tempt. At best, multihoming could increase the proba-bility of earning the margin pi − c from ρi to 1, leadingto a potential gain of (1 − ρi)(pi − c). As long as theadditional commission incurred by pursuing thatoption is high enough to outweigh that gain, thedriver will stick with single-homing. That bound alsoprovides some insight into the types of settings whereplatforms might need to worry the most about mul-tihoming and how big of a commission differentialmight be needed to deter it. In settings with highutilization ρi under single-homing, the bound on δ isquite small, becoming 0 when utilization is equal to 1.In such settings, multihoming is not as much ofa concern and can be countered with a relativelymodest incentive plan. (Note that this observation isconsistent with Proposition 5, which states that whena platform is supply constrained—i.e., ρi � 1—theequilibrium outcomes are the same for both single-and multihoming.)

7. ConclusionIn this paper, we present a model of a two-platformsharing-economy system where each platform canonly indirectly influence supply and demand throughprice. A key element of this model is the inclusion oftwo-sided congestion effects. On the demand side, alarge number of customers seeking service relative to


the number of drivers offering service can result inlonger customer delays before service begins, so thatpotential customers are less willing to participate inthe market. On the supply side, a large number ofdrivers offering service relative to the number ofcustomers seeking service reduces the likelihood thata driver will successfully be able to find a customerand earn income, which makes drivers less willing tobe active. This creates an interaction effect betweensupply and demand that captures relatively subtlesystem behaviors and allows us to obtain deeperinsights into system performance and questions ofpractical interest. For both single-homing and mul-tihoming settings, we establish conditions under whicha price equilibrium exists, and we identify regions inthe parameter space under which different types ofoutcomes can arise. Examination of these regions pro-vides insights regarding the impact that factors such asmarket size or customer sensitivity to delays have onoutcomes such as the intensity of competition betweenthe platforms or the level of congestion in the market. Italso reveals insights regarding the impacts on prices,congestion, and participant utilities of having a secondplatform competing in the market.

We build on these results to make some compari-sons that address managerial and policy questionsthat are currently being debated in sharing-economycircles. First, we examine the effect of surge pricing byplatforms. We find that, despite resulting in higherprices for customers during periods of high demand,the net effect of that price shift is to make both cus-tomers and drivers better off. The additional supplyinduced by the higher price more than compensatesfor the negative price impact on customers, but it isnot so large that it dominates the benefit drivers ex-perience from the higher price. Next, we explore thedifferences—in terms of outcomes for customers anddrivers—between single-homing and multihomingsettings. We find that demand and prices are lower,and both customers and drivers are weakly (andsometimes strictly) better off under single-homing.On the surface, the conclusion regarding driversseems to run counter to anecdotal observation, be-cause it is common to see drivers working under bothUber and Lyft platforms. However, we identify aprisoners’ dilemma type of phenomenon that canexplain this apparent inconsistency. Starting withsingle-homing, any single driver is better off multi-homing, and this preference continues for each driverin turn until all drivers shift to multihoming. Onceplatforms adjust prices to reflect this new reality,however, demand and prices are lower, and bothcustomers and drivers are weakly worse off. Wepropose an incentivemechanism that, if implementedby platforms, could help deter the practice of multi-homing, thus mitigating these negative effects. By

analyzing one form of such amechanism, we find thatless incentive is needed to deter multihoming insettings where single-homing results in high systemutilization.To test the robustness of our managerial insights

(i.e., those discussed in Propositions 2, 3, and 5), weconducted an extensive numerical study where wedropped the supply constraint (9) or (27) and al-lowed for general α. We used the following parametervalues: c ∈ {0.1,0.2}, α∈ {0.8,0.9,1}, γ∈ {0.1,0.2, . . . ,1},and λ ∈ {0.1, 0.2, . . . , 3}, for a total of 1,800 cases. Theseinclude parameter combinations under which thesupply constraint is not guaranteed to be redundant(i.e., γ < max{[(λ − α)(α − c)]/(α2 + 3αλ), 0}). Almostall of the results continued to hold inmore than 98%ofthe cases considered (andmany of them in 100% of thecases). The only exceptions to this are the results thatdemand and customer utility are higher in single-homing than in multihoming; these results contin-ued to hold in 94% of the cases. On the basis of thisnumerical study, our managerial insights do notappear to rely on the supply constraint in general,and they appear to be fairly robust to the value of α.The analysis in this paper reflects two key features

of our model: platform competition and market con-gestion. If competition between platforms were ig-nored, then it would not be possible to conduct acomparison of the single-homing and multihomingsettings. Furthermore, modeling platform competi-tion allows us to analyze (a) how platforms adjustprices and, in turn, congestion levels in response tocompetition and (b) how platforms respond to posi-tive demand shocks in a competitive fashion. Therefore,explicitly modeling competition makes our work per-tinent to some of the current challenges in the ride-sharing industry.Our model captures the impact of congestion on

customers and drivers via disutility terms that de-pend on the level of utilization in a platform (theadditive term −γρ2

i in (3.1) and the multiplicativeterm ρi in (7)). If we suppress the operational featuresof our model by removing the congestion effects(i.e., replacing the additive term −γρ2

i in (3.1) by 0and the multiplicative term ρi in (7) by 1), then theresulting supply and demand curves would be linear,and the platforms’ pricing problems would reduce tointersecting these supply and demand curves to de-termine the equilibrium price. Ignoring congestioneffects would result in platforms no longer main-taining supply cushions (as illustrated in Figure 5),and it would not be possible to obtain our findings onthe impact of surge pricing. Moreover, the equilib-rium outcomes in the single-homing and multihomingsettings would become the same, implying that allparties would be indifferent between these scenarios.Incorporating congestion effects allows us to explore a


richer set of questions that result from themore nuancedfeedback between supply and demand.

In order to make more extensive analysis tractableand focus attention on the phenomena being studied,our model assumes that platforms are symmetric. Tocheck the robustness of our equilibrium results, how-ever, we have analyzed two types of platform asym-metry. The first modifies the customer location distri-bution shown in Figures 1 and 2 to have a nonzeromass at one end of the distribution, reflecting a groupof customers with strong loyalty to one platform.Under that extension, for the case of single-homing,we are still able to (a) derive expressions for balanceddemand and supply, (b) establish the existence ofequilibrium prices, (c) (implicitly) characterize fourregions with different outcomes similar to Figure 4,and (d) (implicitly) characterize the equilibrium pricesin those regions. (For three of the regions, we are able toestablish uniqueness of the equilibrium analytically,whereas in the fourth region, numerical trials alwaysresulted in a unique equilibrium.) In addition, theplatform with the loyal customer base charges higherprices but still enjoys higher demand in equilibrium.The second version of asymmetry allows the commis-sion c to be different for each platform. Under thatgeneralization, for single-homing, we can show that aunique equilibrium exists, and we can characterizeanalogues of the four regions in Figure 4, as well asequilibrium prices in each of these regions. One newoutcome in this setting is that a new subregion mayarise in which one platform’s drivers are fully utilizedwhile the other has some drivers idle. The analysisunder either of these generalizations quickly becomesintractable, however, and further exploration of thesecases is beyond the scope of this paper.

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Competition Between Two-Sided Platforms Under Demand and