pricing and matching with forward-looking buyers and...

MANUFACTURING & SERVICE OPERATIONS MANAGEMENTArticles in Advance, pp. 1–18

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

Pricing and Matching with Forward-Looking Buyers and SellersYiwei Chen,a Ming Hub

aCarl H. Lindner College of Business, University of Cincinnati, Cincinnati, Ohio 45220; bRotman School of Management, University ofToronto, Toronto, Ontario M5S 3E6, CanadaContact: [email protected] (YC); [email protected], http://orcid.org/0000-0003-0900-7631 (MH)

Received: March 24, 2017Revised: November 14, 2017; July 24, 2018;November 14, 2018Accepted: November 19, 2018Published Online in Articles in Advance:August 7, 2019

https://doi.org/10.1287/msom.2018.0769

Copyright: © 2019 INFORMS

Abstract. Problem definition: We study a dynamicmarket over a finite horizon for a singleproduct or service in which buyers with private valuations and sellers with private supplycosts arrive following Poisson processes. A single market-making intermediary decidesdynamically on the ask and bid prices that will be posted to buyers and sellers, respec-tively, and on the matching decisions after buyers and sellers agree to buy and sell. Buyersand sellers can wait strategically for better prices after they arrive. Academic/practicalrelevance: This problem ismotivated by the emerging sharing economy and directly speaksto the core of operations management that is about matching supply with demand. Meth-odology: The dynamic, stochastic, and game-theoretic nature makes the problem intractable.We employ the mechanism-design methodology to establish a tractable upper bound on theoptimal profit, which motivates a simple heuristic policy. Results: Our heuristic policy is: fixedask and bid prices plus price adjustments as compensation for waiting costs, in conjunctionwith the greedy matching policy on a first-come-first-served basis. These fixed base pricesbalance demand and supply in expectation and can be computed efficiently. The waiting-compensated price processes are time-dependent and tend to have opposite trends at the be-ginning and end of the horizon.Under this heuristic policy, forward-looking buyers and sellersbehave myopically. This policy is shown to be asymptotically optimal. Managerial impli-cations: Our results suggest that the intermediary might not lose much optimality bymaintaining stable prices unless the underlying market conditions have significantlychanged, not to mention that frequent surge pricing may antagonize riders and induceriders and drivers to behave strategically in ways that are hard to account for withtraditional pricing models.

Funding: Financial support from the National Natural Science Foundation of China [Grant 71825007]and theNatural Sciences and Engineering Research Council of Canada [Grant RGPIN-2015-06757] isgratefully acknowledged by M. Hu.

Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2018.0769.

Keywords: sharing economy • two-sided market • strategic customers • pricing • matching • mechanism design • asymptotic optimality

1. IntroductionMarket makers have long existed in many market-places. The rise of the sharing economy has allowedemerging marketplaces and market-making intermedi-aries to tap into our daily lives. They select and pub-licly announce their bid and ask prices. Buyers who arewilling to pay above the ask price would want tobuy from the intermediary, whereas sellers with anopportunity cost below the bid price would want tosell to the intermediary. Unlike the traditional mar-ket makers such as used-car dealers (see, e.g., Amihudand Hendelson 1980), the emerging market-makingintermediaries do not buy and sell assets with in-ventory holding. Instead, they focus on enabling thematching between buyers and sellers. For example,Uber and Lyft crowdsource freelance drivers andmatch them with riders by making pricing andmatching decisions. Postmates and Deliv pay free-lance couriers or even travelers to deliver parcelsto end consumers. In most of those applications, the

intermediary needs to decide not only on the inter-temporal ask and bid prices, but also on the detailedmatching decisions between buyers and sellers afterthey agree on the prices. For instance, Uber seemsto implement a greedy matching policy that is on afirst-come-first-served (FCFS) basis along the tem-poral dimension.1

Buyers and sellers can time their transactions basedon market prices and their expectation of the likeli-hood of being matched. On the demand side, Chenet al. (2015) observe that Uber’s customers oftenchoose to “wait out” the price surges. On the supplyside, freelance drivers can decidewhen towork. Then,it is desirable to design the pricing and matchingpolicies that take into account the forward-lookingbehavior of buyers and sellers.In some two-sidedmarkets, such as the ride-hailing

market, we observe that different ride-hailing plat-forms adopt different pricing strategies. On one hand,Uber implements surge pricing with likely varying

1

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price for the same trip at different times. On the otherhand, Gett, another ride-hailing platform that oper-ates in the United Kingdom, promises to passengersthat it sticks to no surge pricing even when demand ishigh.2 Therefore,we aremotivated to explorewhetherthe intermediary shall adopt a dynamic-pricing ver-sus fixed-pricing strategy in the two-sided marketwith forward-looking buyers and sellers. Althoughour motivation is ride-hailing, for simplicity of ex-posure, we abstract away the spatial dimension of thepractical context and focus on the temporal dimen-sion. In particular, wemodel a dynamicmarket over afinite horizon for a homogeneous product or service inwhich buyers with private valuations and sellers withprivate supply costs arrive following Poisson pro-cesses. A single market-making intermediary decidesdynamically on the ask and bid prices that will beposted to buyers and sellers, respectively, and on thematching decisions after buyers and sellers agree tobuy and sell. Buyers and sellers can wait strategicallyfor better prices after they arrive. Our model does notcapture the spatial dimension of ride-hailing such asdriving to pick up a rider and the possible reentry ofdrivers to the market at another or the same locationafter dropping off a rider.3

1.1. Results and ContributionsIndeed, the ride-hailing industry provides a rich con-text for studying operational problems. Very recently,we have seen a large number of research papers onthis topic from various angles by applying differenttools (see Hu 2019a). It is challenging to realisticallycapture all themain features of the practical context ofride-hailing. Our model focuses on the forward-looking behavior of riders and drivers along thetemporal dimension. The system, essentially a double-ended queue when our heuristic control is imple-mented, can be viewed as a transient (finite-horizon)version of Taylor (2018) and Bai et al. (2018), both castin the context of ride-hailing, in which the spatial di-mension is also abstracted away and the system hasreached a steady state. Moreover, the double-endedqueue can more accurately capture a matching marketwhere service providers do not come back after amatching, for example, shoppers at a grocery store helpmake a delivery for their neighbors and commuters pooltheir car with riders on the way to work or home.

As far as we know, this is the first paper that studiesan intermediary’s two-sided pricing and matchingproblem in which both buyers and sellers exhibitforward-looking behavior. Themain technical challenge,beyond the studies on one-sided revenue-managementproblemswith forward-lookingbuyers, isdue toademand–supply balancing constraint that the number of buyersand sellers who transact at any point of time must beequal along any sample path. This operational-level

constraint makes the two-sided pricing and matchingproblem different from the revenue-management andpricing problem with forward-looking buyers.We obtain the following set of results and consider

the first two results as the most significant.(i) We use the mechanism-design approach to es-

tablish a tractable upper bound on the intermediary’soptimal profit. Because of the system’s stochasticnature and buyers’ and sellers’ forward-lookingbehavior, it is very challenging to compute theintermediary’s optimal policy and profit. Therefore,we establish a tractable upper bound on the inter-mediary’s optimal profit, which serves as a bench-mark to evaluate the performance of any heuristicpolicy. This upper bound is obtained in the follow-ing way. First, we randomly generate a sample pathof buyers’ and sellers’ arrival processes and theirvaluations/costs. Second, for any such sample path,we solve a deterministic assignment problem, whichis a special type of 0-1 integer program. Third, wecompute the expected optimal value of these deter-ministic assignment problems. We show that thisprocedure indeed provides an upper bound by amechanism-design approach. We begin by provingthat the platform’s optimal profit is upper-boundedby the optimal value of an auxiliary two-sided dy-namicmechanism-design problem.We then show thatthe optimal value of this auxiliary problem is upper-bounded by the expected optimal value of thoseassignment problems mentioned above. Unlike thecounterpart in the revenue-management setting withforward-looking buyers, establishing such an upperbound for the two-sided market is technically chal-lenging, which is a main contribution of this paper.(ii) We propose a simple market-making pricing

and matching policy. On the pricing side, the marketmaker posts fixed ask and bid prices plus a (time-dependent) price adjustment as compensation forthe expected cost of waiting to be matched, and onthe matching side, the market maker implements thegreedy matching policy on a FCFS basis. Those fixedprices balance demand and supply and can becomputed efficiently. The commitment to the fixedbase prices with a price adjustment for the expectedwaiting cost induces the strategic buyers and sellersto behave myopically. They will submit a request formatching upon arrival without delay if the buyer’svaluation is no less than the fixed ask price or theseller’s cost is no more than the fixed bid price. Theintuition is as follows. First, the time-dependent priceadjustment on top of the fixed price for a buyer or aseller arriving at a specific time is the expected costof waiting to be matched after submitting her match-ing request. That is, the price-adjustment componentcompensates buyers’ and sellers’ waiting disutilities.As a result, the presence of the price adjustment only

Chen and Hu: Pricing and Matching with Strategic Buyers and Sellers2 Manufacturing & Service Operations Management, Articles in Advance, pp. 1–18, © 2019 INFORMS

requires every buyer to simply compare her valuationwith the demand-side fixed-price component whilemaking the purchasing decision and every seller tosimply compare her costwith the supply-sidefixed-pricecomponent while making the selling decision. Second,because the fixed-price component does not change overtime, buyers and sellers have no incentive to delay theirdecision making. The effective prices offered aftercompensating for waiting are in general time-varyingand tend to have opposite trends at the beginningand the end of the horizon. But we show that whenthe volumes of demand and supply are large, thecompensation for waiting becomes negligible (morespecifically, it isO(1/n1/3), where n is the scaling of thearrival rates) and the effective price trajectory tends tobe stationary; moreover, when customers are willingto wait for a certain length of time without beingcompensated,4 the heuristic price trajectory can be-come stationary. The waiting compensation is due tothe randomness in the buyers’ and sellers’ arrivalprocesses and the fact that, even if buyers and sellersare willing to transact, they may not be able to do sobecause of the shortage on the other side, an issue thatdoes not exist in the revenue-management setting.Characterizing the simple market-making pricing andmatching policy including the waiting compensationis another main contribution of this paper.

(iii) We show that our proposed simple market-making policy is near optimal. Put differently, weshow that the gap between the profit yielded underour simplemarket-making policy and our establishedprofit upper bound mentioned above diminishes tozero, as the arrival rates on both sides of the marketrise. The optimality gap is O(1/n1/2), where n is thescaling of the arrival rates. This implies that the benefitfrom contingent pricing—that is, making prices de-pendenton the realizeddemandandsupply—is second-order and diminishes as n grows. We conduct a seriesof numerical experiments to show that our proposedpolicy performs consistently well in a wide range ofmarket environments.

(iv) We conduct comparative statics analysis for thefixed base prices and other corresponding perfor-mance measures. We show that even in a market en-vironment with demand and supply in large quantities,the optimal fixed bid and ask prices should be updatedwhen the market environment significantly changes. Inparticular, when buyers’ [respectively (resp.) sellers’]arrival rate increases, both the optimal fixed bidand ask prices should increase (resp. decrease), and theresultingfluidmatching quantity and the correspondingintermediary’s profit would increase (resp. increase).These comparative statics are intuitive and desirable,but may become ambiguous in alternative queuingformulations (see, e.g., Bai et al. 2018 for comparativestatics of prices on the supply-side parameters).

1.2. Managerial ImplicationsOur asymptotic optimality result suggests that for amarket condition with large volumes, Uber maynot need to adopt rapidly contingent pricing (i.e., acontingent and rapid micromatching of supply withdemand), when the underlying market conditions,such as arrival rates and customer valuation and driveropportunity-cost distributions, have remained stable.5

Our fixed-pricing formulae can be used to computethe base fares of Uber, which balance supply anddemandat theaggregated level fora region.6 In contrast,a fully contingent policy would respond to any re-alization of the randomness in the arrivals of buyersand sellers aswell as that in their valuations and costs.There are two apparent downsides of a fully contin-gent pricing policy. First, contingent pricing results inprice fluctuations that antagonize riders, regulators,and sometimes evendrivers.7 Second, contingent surgeprices are usually adopted without taking riders’ anddrivers’ strategic behavior into account and may in-duce them to behave strategically in ways that arehard to account for. As one can see from our modelthat considers riders’ and drivers’ strategic behavior,it is unclear what the optimal contingent pricingpolicy is, even in such a simplified model.8

Our asymptotic optimality result for the simplepricing and matching policy implies that the un-certainty in the arrivals and their valuations andcosts may not alone be sufficient to justify contingentpricing, not to mention its potential downsides. It maybe safe to update prices less often for urban areas duringpeak hours, because rigid pricing does not lose muchoptimality for a thick market with large volumes ofdemand and supply, in addition to many other benefitsof pricing stability—for example, to provide con-sumers with clear price points. That advice is con-sistent with the practice of Gett and Via, emergingcompetitors of Uber in New York City, which chargeflat rates that do not surge in Manhattan and use thepractice as a competitive advantage in the competi-tion with Uber.9 On the other hand, if the matchingplatform is in its early stage or is operated in a thinmarket with low volumes of demand and supply—forexample, in rural areas or off-peak hours—contingentpricing may be of high value. Yet, as we show, theadditional benefits of contingent pricing are second-ary, whereas the simple heuristic could have capturedthe first-order benefit. In the long term, with machine-learning tools to help better predict demand and thenpreallocate drivers and with more autonomous vehi-cles on the road that can be dispatched, the incentive toadopt surge pricing shall be diminished over time.Moreover, the simple heuristic has the advantage of

deterring strategic behavior by buyers and sellers,whereas those contingent surge prices are usuallyadopted without taking forward-looking behavior

Chen and Hu: Pricing and Matching with Strategic Buyers and SellersManufacturing & Service Operations Management, Articles in Advance, pp. 1–18, © 2019 INFORMS 3

into account. This advantage is desirable in a matchingmarket in which it is hard to monitor the arrivals ofdemand and supply.10 Strategic behavior is also hardto detect11 and account for with pricing models. Therapidly changing demand and supply may very likelybe an outcome, not necessarily the cause, of the rapidchanges in prices. Our heuristic brings in all possibledemand and supply for a homogeneous good or ser-vice to a marketplace as they arrive, something that isappealing to the market maker, who can then makesupply meet demand more efficiently.

The rest of the paper is organized as follows.Section 2 provides a literature review. Section 3 in-troduces our model. Section 4 presents an upperbound on the intermediary’s optimal profit. Section 5proposes a simple heuristic policy and analyzes itsperformance. Section 6 conducts the comparativestatics analysis. Section 7 concludes the paper.

2. Literature ReviewThe following two papers are closely related to oursregarding managerial implications. Riquelme et al.(2015) study a two-sided platform’s pricing problemin the ride-hailing context. The authors assume thatdrivers can reenter the market after completing aride and make themselves available for another trip.However, they do not consider riders’ and drivers’forward-looking behavior. They show that a staticpricing policy is optimal in a so-called large-marketregime. However, their results also imply that with adifferent market environment, there exists a differ-ent optimal static price. Similarly, with a differentmodel, Cachon et al. (2018) also show that riders anddrivers are generally better off with different pricescontingent on varying demand and supply condi-tions. We further confirm these managerial implica-tions after taking into account the forward-lookingbehavior of buyers and sellers.

It is extremely desirable to provide a simple solu-tion to a complex system and yet show that the so-lution can still be close to optimal. Gallego and vanRyzin (1994) show that in the classic capacitateddynamic pricing setting, a fixed-pricing policy mayperform nearly as well as a fully dynamic pricingpolicy. Gallego and Hu (2014) extend this monopolysetting to account for competition. Our model can beconsidered as a two-sided extension of Gallego andvan Ryzin (1994) with the additional desirable featureof taking both sides’ forward-looking behavior intoaccount. On the other hand, more recently, Kim andRandhawa (2018) study the value of dynamic pricingto maximize revenues in queueing systems. Theyshow that static pricing leads to a revenue loss oforder n1/2, while the dynamic pricing can mitigate theloss to the n1/3-scale.

In the operations literature, there is a recent streamof studies on two-sided pricing decisions of an inter-mediary platform. Taylor (2018) studies an intermediary,on-demand service platform that simultaneously de-cides on a per-service price posted to customers anda wage posted to independent agents as service pro-viders. The author shows that the uncertainty in thedelay-sensitive customers’ valuations or the agents’opportunity costs can lead to counterintuitiveinsights—delay sensitivity can raise the optimal priceor lower the optimal wage. Complementing Taylor’swork, Bai et al. (2018) focus on the impact of thedemand rate, sensitivity to waiting time, service rate,and the size of available providers on the optimalprice, wage, and payout ratio (i.e., one minus thecommission rate; see also Hu and Zhou 2016b). Theauthors show that the optimal price, wage, and payoutratio increase in the potential customer demand rate.Bimpikis et al. (2019) consider the joint price andwageoptimization for a network of locations over a finitenumber of periods with deterministic patterns ofdemand, supply, and spatial transitions among lo-cations. Besbes et al. (2018b) consider a continuouslydispersed linear city where the drivers can repositionthemselves in a simultaneous-move game after theplatform sets location-specific prices andwages alongthe city. Chen et al. (2016) consider a similar settingof market making to ours. However, their paper isdistinguished from ours because it does not considerbuyers’ and sellers’ forward-looking behavior, and itallows the intermediary to hold inventory.On the other hand, because of the nature of strategic

interactions among the intermediary, sellers, andbuyers, our work contributes to a stream of studiesadopting the dynamic mechanism-design approachfor revenue-management problems with forward-looking customers. Vulcano et al. (2002) consider im-patient but strategic customers who arrive sequen-tially over a finite horizon; they propose a modifiedsecond-price auction for each period. Gallien (2006)considers a discounted, infinite-horizon revenue-management model with forward-looking customers.The author shows that the optimal dynamic mecha-nism can be applied as a sequence of increasing postedprices. More recently, Board and Skrzypacz (2016)consider a discrete-time revenue-management prob-lem of selling an initial inventory of one product over afinite horizon. The authors characterize the optimaldynamic mechanism, which allocates a unit of theproduct to the buyer with the highest valuation if hervaluation exceeds a cutoff. For the classic revenue-management problem with private, possibly corre-lated, customer time-discounting and price-monitoringcosts, Chen and Farias (2018) propose a class ofpricing policies, and, within this class, an efficientlycomputable policy is guaranteed to achieve 29%


optimality. Chen et al. (2019) further show that a staticpricing policy is asymptotically optimal, even in thepresence of forward-looking customers. One crucialdistinction between our paper and theirs is that thesupply capacity in ours stochastically evolves overtime, whereas the supply capacity in theirs is prefixed.Therefore, in our setting, a buyer (resp. seller) whoarrives at the system and is willing to buy (resp. sell)may not be instantly matched with a seller (resp.buyer) because of the supply (resp. demand) shortage.This feature is crucial and brings additional com-plexities in our analysis of the two-sided market. Thefirst complexity arises from the upper-bound optimi-zation problem. We have to take into account thedemand–supply mismatching cost. The presence ofthis cost significantly makes the computation of theupper-bound problem more difficult. The second com-plexity arises from our proposed heuristic policy. To in-duce each buyer to behave myopically as in Chen et al.(2019), we have to introduce a price-adjustment term ontop of a static base price. This term is time-dependentand depends on both the real-time number of out-standing buyers and sellers at present and stochasticarrivals of buyers and sellers in the future. Therefore,quantifying and computing this term is not trivial.

Beyond the papers mentioned above, a variety ofalgorithmic papers have studied a special class ofdynamic mechanisms for a monopolistic seller whofaces dynamically arriving forward-looking buyers—anonymously posted dynamic pricing mechanisms.See, for instance, Besbes and Lobel (2015), Borgs et al.(2014), Liu and Cooper (2015), and Lobel (2017). Inparticular, the paper by Myerson and Satterthwaite(1983) studies a two-sided mechanism-design prob-lem, but only in a static setting.

There is an emerging stream of research on thesharing economy in the operations literature (see Hu2019b for a collection of the latest research). First,focusing on the effect of the sharing economy onwelfare and consumption, Benjaafar et al. (2019)study an equilibrium model of product sharing inwhich individuals make ownership decisions andnonowners can rent the goods from owners on de-mand. Second, a couple of papers show that opera-tional efficiency in a matching market can bedetrimental to social welfare and emphasize themoderating role of the intermediary platform. Allonet al. (2012) study a moderating service platformthat can gain operational efficiency by pooling theagents virtually; they show that such efficiency maybe detrimental to the overall efficiency of the mar-ketplace because of intensified competition amongagents under pooling. Arnosti et al. (2015) show thatreductions in search costs in a matching market withasymmetric information may decrease social welfare.Third, like ours, a group of papers highlights the

importance of modeling individual agents on thesupply side as decision makers who also maximizetheir own utility. This is a unique feature in the ap-plications with crowdsourced freelancers. Gurvichet al. (2015) and Ibrahim (2018) study the capacitydecision of a service provider who crowdsourcesthe supply of self-scheduling agents. Yang et al.(2016) use the notion of mean-field equilibrium tocharacterize a rational agent’s strategic behavior in aresource-sharing setting. In contrast to these paperson agent behavior, we focus on the intertemporalbehavior of sellers who can time their sales. Fourth, inother sharing-economy settings, Shu et al. (2013),Kabra et al. (2016), and Henderson et al. (2016) studybicycle sharing; and He et al. (2017) study electricvehicle sharing systems. Fifth, several papers study thedetailedmatching decisions of an intermediary. Gurvichand Ward (2014) study the optimal control of matchingqueues and propose a heuristic that asymptotically ach-ieves an imbalance-based lower bound. Hu and Zhou(2016a) focus on the matching policy at the operationallevel for an intermediary who matches demand andsupply of different types, such as their geographic lo-cations. For a ride-hailing system, Ozkan and Ward(2016) establish the asymptotic optimality of a linear-programming-basedmatching policy in a large-marketregime in which the drivers are fully utilized.Finally, under our simple heuristic, the arrivals on

both sides behave myopically, and the system effec-tively operates as a double-ended queue with constantarrival rates on both sides. Double-ended queues areoften used to model matching markets. Kendall (1951)introduces the double-ended queuing model, withthe example of customers and taxis independentlyarriving at a taxi stand. Dobbie (1961) provides fur-ther solution schemes for the double-ended queuewith time-dependent arrival rates. Zenios (1999) andBoxma et al. (2011) apply double-ended queues withabandonment to model the organ-transplant waitinglist in the United States. Afeche et al. (2014) apply adouble-ended queue with batch arrivals and aban-donment tomodeling “dark pool” trading in financialmarkets (see also the literature review therein for asurvey). The matching scheme in those papers isFCFS, the same as our heuristic. However, our em-phasis is not on performance evaluation under thegiven arrival processes of two sides, as in those pa-pers. We focus on designing profit-maximizing pric-ing and matching schemes for the intermediary.

3. ModelConsider an intermediary who dynamically matchesdemand and supply of a single-type product (orservice). Buyers and sellers have heterogeneous val-uations and supply costs, respectively, and sequen-tially arrive in the market over a finite horizon [0,T].


The intermediary implements an anonymously pos-ted price mechanism on both demand and supplysides. That is, at each point of time t ∈ [0,T], the in-termediary posts demand-side priceπd

t (i.e., ask price)that she charges buyers for each unit of the productand supply-side price πs

t (i.e., bid price) that she paysto sellers for each unit of the product.

3.1. Buyers’ BehaviorOver the horizon [0,T], buyers arrive to the inter-mediary according to a Poisson process with rate λd.A buyer arriving at time t is endowed with a productvaluation v ∈ [v , v], as a realization from a willingness-to-pay distribution. We denote by

φ≜(tφ, vφ

),

the “type” of an arriving buyer, which specifies herarrival time tφ and valuation vφ. Every buyer pur-chases at most one unit of the product. Buyers areforward-looking and can strategically determinewhento request to buy the product. Specifically, for everybuyer φ, at each point of time after her arrival, shedecides to either continue to monitor price dynamicsor stop doing so by either sending a request to theintermediary for buying the product or permanentlyleaving the market without buying anything. Wedenote by τφ ∈ tφ,T

[ ]the time that buyer φ stops

monitoring price dynamics. We denote by aφ ∈ {0, 1}the indicator function of whether buyer φ requests tobuy the product at time τφ. A buyer who requests tobuy the product keeps on staying in the market untilher demand is matched with a supply. As an ex-ception, if a buyer’s demand cannot be matched bythe end of the horizon, she withdraws her demandrequest from the intermediary and leaves the market.We denote by sφ ∈ τφ,T

[ ]the time when buyer φ

leaves the market. We denote by mφ ∈ {0, 1} the in-dicator function of whether buyer φ’s demand re-quest is successfully matched with a seller at time sφ.Buyer φ pays pφ � πd

τφmφ to the intermediary—that

is, if buyer φ is successfully matched with a seller, shepays the demand-side price posted at the time that sherequests to buy the product, πd

τφ; otherwise, she makes

no payment to the intermediary.This business rule is consistent with the practice of

the ride-hailing industry. In that setting, riders sub-mit matching requests and then wait to bematched. Ifa rider is not paired with any driver for some reason,no payment will be made by the rider, and the riderwill typically not be compensated by the intermediaryfor the wait. This treatment is also consistent withalternative queueing formulations of the ride-hailingmarket such as Taylor (2018) and Bai et al. (2018) andmore generally with the queueing economics litera-ture in which the service provider does not com-pensate customers for the wait and the customers

need to take into account the expectedwait when theymake the decision of whether to make a service re-quest. It will be clear later that this setup is, in fact,more complicated than other alternative settings suchas the intermediary compensates both sides out ofpocket for any wait after a matching request is sub-mitted and before a matching is realized.We define the tuple

yφ ≜(τφ, aφ, sφ,mφ, pφ

).

Buyer φ garners utility

Ud(φ, yφ) � vφmφ − pφ − b(sφ − tφ

),

where b ∈ R+ ≜ {x : x ≥ 0} is buyer φ’s per unit of timedisutility from staying in the system over [tφ, sφ],hereafter referred to as the buyers’ waiting-cost pa-rameter, which is assumed to be common knowl-edge.12 In this paper, we follow many papers in theliterature—for example, Besbes and Maglaras (2009)andChen and Frank (2001)—to assume that all buyershave the same waiting-cost parameter b.As mentioned, we allow for the heterogeneity of

buyer arrival times and valuations. A buyer’s arrivaltime and valuation are pieces of private informationand are independent of each other. We denote thecumulative distribution function (c.d.f.) of the buyerproduct valuation by Fd(·) and the correspondingprobability density function (p.d.f.) by f d(·). We de-note Fd(·)≜ 1 − Fd(·). In addition to assuming Fd(·) hasan inverse function, denoted by Fd,−1(·) (see, e.g.,Gallego and van Ryzin 1994), we make a standardassumption on the valuation distribution:

Assumption 1 (Willingness-to-Pay). The buyer virtual

value function Vd(v)≜ v − Fd(v)f d(v) is increasing

13 in v ∈ [v , v].This assumption is widely adopted in the literature;

see, for example, Myerson (1981), Gallien (2006), andGallego and van Ryzin (1994). Many commonly useddistributions, suchasuniform, exponential, andGumbel,satisfy this assumption. All information above is com-mon knowledge, except for that specified as privateinformation.

3.2. Sellers’ BehaviorOver the horizon [0,T], sellers arrive to the inter-mediary according to a Poisson process with rate λs.A seller arriving at time t is endowed with c ∈ [c , c], aprocurement, production, and delivery cost for agood, or opportunity cost for providing a service. Weassume c ≤ v.14 We denote by

ψ≜ tψ, cψ( )

,

the “type” of an arriving seller. Every seller sells atmostone unit of the product. All sellers are forward-looking


and can strategically determine when to request tosell her product. For every seller ψ, at each pointof time after her arrival, she decides to either con-tinue to monitor price dynamics or stop doing soby either sending a request to the intermediary forselling the product or permanently leaving the mar-ket without selling anything.We denote by τψ ∈ tψ,T

[ ]the time that seller ψ stops monitoring price dynam-ics. We denote by aψ ∈ {0, 1} the indicator function ofwhether seller ψ requests to sell her product at timeτψ. A seller who requests to sell her product keeps onstaying in the market until she is paired with a buyer.Consistent with the demand side, as an exception, if aseller cannot be matched by the end of the horizon, thenshewithdraws her supply request from the intermediaryand leaves the market. We denote by sψ ∈ τψ,T

[ ]the

time when seller ψ leaves the market. We denote bymψ ∈ {0, 1} the indicator function of whether seller ψ’ssupply is successfully matched with a demand atsψ. The seller ψ is paid by the intermediary withpψ � πs

τψmψ; that is, if seller ψ’s supply is successfully

matched with a demand, then the intermediary paysher the supply-side price posted at the time that sherequests to sell the product, πs

τψ; otherwise, she receives

no payment from the intermediary.We define the tuple

yψ ≜ τψ, aψ, sψ,mψ, pψ( )

.

Seller ψ garners utility

Us ψ, yψ( ) � pψ − cψmψ − h sψ − tψ

( ),

where h ∈ R+ is seller ψ’s per unit of time disutilityfrom staying in the system over [tψ, sψ], hereafterreferred to as the sellers’ waiting-cost parameter, whichis also assumed to be common knowledge.15 In ad-dition, like the demand side, we assume there is noheterogeneity in sellers’ waiting-cost parameter h.

As mentioned, we allow for the heterogeneity ofsellers’ arrival times and supply costs. A seller’s ar-rival time and supply cost are pieces of private in-formation and are independent of each other. Wedenote the c.d.f. of the seller product producing anddelivering cost by Fs(·) and the corresponding p.d.f.by f s(·). In addition to assuming that Fs(·) has an in-verse function, denoted by Fs,−1(·), we make the fol-lowing assumption on the supply cost distribution:

Assumption 2 (Willingness-to-Sell). Vs(c)≜ c + Fs(c)f s(c) is

increasing in c ∈ [c , c].This assumption is also widely adopted; see, for

example, Myerson and Satterthwaite (1983) andNiazadeh et al. (2014). Many commonly used dis-tribution functions, such as uniform, exponential, andGumbel, satisfy this assumption. This function can beunderstood as the counterpart of the buyer virtual

value function on the supply side. We hereafter referto this function as the seller virtual-cost function. Allinformation above is common knowledge, exceptfor that specified as private information. In the rest ofthis paper, we may put the subscript ψ on seller ψ’sarrival time, tψ, and cost, cψ, to emphasize their het-erogeneity among sellers.As wewill see in the subsequent sections, similar to

the role of the buyer virtual-value function in com-puting the intermediary’s expected revenue collectedfrom buyers, this seller virtual-cost function plays akey role in computing the intermediary’s expectedcost that she pays to sellers. A near-optimal pricingpolicy requires the intermediary to strike a delicatebalance between both demand and supply sides,captured in terms of the buyer virtual-value functionand the seller virtual-cost function.

3.3. Game Dynamics and the EquilibriumBefore the start of the horizon, the intermediary de-termines a pricing policy on both demand and supplysides, π � (πd

t , πst) : t ∈ 0,T[ ]{ }

, and a demand andsupply matching policy, M � {(sφ,mφ), (sψ,mψ) : (τφ ∈[0,T], aφ � 1), (τψ ∈ [0,T], aψ � 1)}. The pricing policyπ and the matching policyM are, in general, dynamicpolicies depending on the realized uncertainty up to thedecision point. The intermediary commits to implementthis pricing policy π and matching policy M over theentire course of the horizon. The intermediary’s pric-ing and matching policy is common knowledge forall buyers and sellers. We denote by Ht ≜ {φ, ψ : tφ ≤t, tψ ≤ t} the set of buyers’ and sellers’ types that arriveup to time t. Define byφt ≜ {(τφ, aφ) : τφ ≤ t, aφ � 1} theset of demand-side information that the intermediarycollects up to time t. Define by φt

M ≜ {(sφ,mφ) : sφ ≤ t}the set of matching decisions the intermediary hasmade on the demand side up to time t. Similarly, definebyψt ≜ {(τψ, aψ) : τψ ≤ t, aψ � 1} the set of supply-sideinformation that the intermediary collects up to timet and by ψt

M ≜ {(sψ,mψ) : sψ ≤ t} the set of matchingdecisions the intermediary has made on the supplyside up to time t. Define by πd,t ≜ πd

t′ : t′ ∈ 0, t[ ]{ }

thehistoric demand-side prices posted up to time t. Defineby πs,t ≜ πs

t′ : t′ ∈{

0, t[ ]} the historic supply-side pricesposted up to time t. Define a filtration {^t : t ≥ 0}with^t � σ(φt−,φt−

M , ψt−, ψt−M , πd,t−, πs,t−). A feasible pricing

policy π requires πdt and πs

t to be ^t-progressive.16

Denote by Π the set of all feasible pricing policies.A feasible matching policyM requires {sφ ≤ t} and mφ,and {sψ ≤ t} andmψ to be^t-progressive and to satisfythe demand and supply balancing condition that∑

φ∈Ht

1 sφ � t,mφ � 1{ } � ∑

ψ∈Ht

1 sψ � t,mψ � 1{ }

,

∀ t ∈ [0,T].Denote by } the set of all feasible matching policies.


During the horizon, on the demand side, buyers areforward-looking and employ (symmetric) stoppingand purchasing rules contingent on their types thatconstitute a symmetric Markov perfect equilibrium.The following information structure is mainly moti-vated by the current practice of ride-hailing apps. Ourresults would still hold under alternative informa-tion structures with an update of the waiting timecompensation. For a given buyer φ, the informationavailable to her at time t ∈ [tφ,T] consists of demand-side price dynamics that she tracks during her stay inthe system, {πd

t′ : t′ ∈ [tφ, t]}, and the number of un-

matched supply during her stay in the system, {1st′− :

t′ ∈ [tφ, t]}, where 1st′− � ∑

ψ∈Ht′− 1{τψ < t′, aψ � 1, sψ ≥t′}.17 Therefore, at each point of time t ∈ tφ,T

[ ], the

event associated with the stopping decision {τφ ≤ t}and the purchasing decision aφ are adapted to σ(πd

t′ ,1s

t′− : t′ ∈ [tφ, t]). Under the intermediary’s pricingpolicy π and matching policy M, for a given buyertype φ, the optimal stopping rule τπ,Mφ and the optimalpurchasing rule aπ,Mφ are the optimal solution to thefollowing optimization problem:

supτφ∈[tφ,T],aφ∈{0,1}

E[Ud(φ, yφ)|πd

tφ ,1stφ−,φ

],

where the expectation assumes that other buyers usesymmetric stopping and purchasing rules.18

On the supply side, sellers are forward-looking andemploy (symmetric) stopping and selling rules con-tingent on their types that constitute a symmetricMarkov perfect equilibrium. For a given seller ψ, theinformation available to her at time t ∈ [tψ,T] consistsof supply-side price dynamics that she tracks duringher stay in the system, {πs

t′ : t′ ∈ [tψ, t]}.19 Therefore,

at each point of time t ∈ [tψ,T], the event associatedwith the stopping decision {τψ ≤ t} and the sellingdecision aψ are adapted to σ(πs

t′ : t′ ∈ [tψ, t]). Under the

intermediary’s pricing policy π and matching pol-icy M, for a given seller type ψ, the optimal stop-ping rule τπ,Mψ and the optimal selling rule aπ,Mψ are theoptimal solution to the following optimization problem:

supτψ∈[tψ,T],aψ∈{0,1}

E Us ψ, yψ( )∣∣πs

tψ , ψ[ ]

,

where the expectation assumes that other sellers usesymmetric stopping and selling rules.

Our goal in this paper is to construct a price processπ ∈ Π and a matching policy M ∈ } and characterizethe corresponding buyer stopping rule τπ,Mφ and pur-chasing rule aπ,Mφ and seller stopping rule τπ,Mψ and sellingrule aπ,Mψ tomaximize the intermediary’s expected profit

Jπ,M � E∑φ∈HT

pφ − ∑ψ∈HT

pψ

[ ].

4. BenchmarkIn this section, we establish a benchmark that serves as anupper bound of all dynamic pricing policies Π andmatching policies }. Because it is difficult, if not im-possible, to characterize the optimal pricing andmatching policy, the upper bound provides a bench-mark and enables us to analyze the relative perfor-mance of any dynamic pricing policy π ∈ Π andmatching policy M ∈ }.Consider the following problem (B) that assumes

that the intermediary is clairvoyant and thus sheknows buyers’ and sellers’ arrival processes,HT, over[0,T] at time 0:

max{xφψ :φ,ψ∈HT}

∑φ,ψ∈HT

Vd(vφ) − Vs(cψ)(

− b(tψ − tφ)+

− h(tφ − tψ)+)xφψ

s.t.∑ψ∈HT

xφψ ≤ 1, ∀ φ ∈ HT,

∑φ∈HT

xφψ ≤ 1, ∀ ψ ∈ HT,

xφψ ∈ 0, 1{ }, ∀ φ, ψ ∈ HT.

(B)

Because all information is known, problem (B) is sim-ply a deterministic assignment problem. The term in theparentheses of the objective function has the followinginterpretation. If buyerφ and sellerψ arematched, thenthe intermediary collects revenue from buyer φ withthe amount that is equal to her virtual value, Vd(vφ),subsidizes seller ψ with the amount that is equal toher virtual cost, Vs(cψ), and suffers from either buyerφ’s waiting for seller ψ or vice versa, depending onwhoever arrives earlier. We denote by J(HT) the optimalvalueofproblem (B) conditional on buyers’ and sellers’arrival processes HT. We have the following result:

Lemma 1. For any pricing policy π ∈ Π and matchingpolicy M ∈ }, we have

Jπ,M ≤ E J(HT)[ ].

It is worth discussing the salient properties of theupper-bound function E J(HT)[ ]

. First, because HT is astochastic process, J(HT) is a random variable. There-fore, E J(HT)[ ]

captures the randomness of buyers’ andsellers’ arrival processes. Second, E J(HT)[ ]

capturesbuyers’ and sellers’ waiting disutility. Finally, be-cause problem (B) is simply an assignment prob-lem, for each sample path HT, J(HT) can be efficientlycomputed.The proof of Lemma 1 consists of two steps. First,

we show that the intermediary’s optimal expectedprofit is upper-bounded by the optimal value ofan auxiliary two-sided dynamic mechanism-design


problem. Second, we show that the optimal value ofthis mechanism-design problem is upper-boundedby E J(HT)[ ]

.The intuition behind the proof is as follows. First,

following from a dynamic version of the revelation prin-ciple, we can show that the intermediary’s joint pricingand matching problem can be equivalently formulatedas a two-sided dynamic mechanism-design problem.Second, this mechanism-design problem has buyers’and sellers’ incentive compatibility constraints. Byexploiting the structural properties of these con-straints (e.g., applying the envelope theorem to theseconstraints; see Myerson 1981), we can show that theoptimal value of themechanism-design problem is upper-bounded by the optimal value of the intermediary’s pureoptimization problem in an idealized setting that sheknows every buyer’s valuation (resp. seller’s cost),except that she replaces every buyer’s valuation (resp.seller’s cost) with the virtual value function (resp.virtual cost function) that captures the intermediary’slack of knowledge of every buyer’s valuation (resp.seller’s cost) in our primary model. The detailedanalysis is available in the online appendix.

5. A Simple Dynamic Policy:Asymptotic Optimality

In this section, we begin by characterizing the in-termediary’s optimal policy in an auxiliary settingwherein all uncertainties are washed away, and allbuyers and sellers behave myopically. We then usethis policy as a basis to develop another policy for theprimary setting that takes into account buyers’ andsellers’ waiting disutilities.

5.1. Optimal Policy in an Auxiliary SettingIn this subsection, we consider an auxiliary version ofthe primary stochastic model introduced in Section 3.As we will see soon in the next subsection, theintermediary’s optimal policy in this auxiliary settingcan easily motivate us to propose another simplepolicy that can be proven to be near optimal in theprimary setting. In this auxiliary problem, the systemis entirely deterministic, and buyers and sellers areinfinitesimal and myopic. To be precise, in the aux-iliary problem, the intermediary solves the followingoptimization problem:

maxπ∈Π

∫ T

0λdπd

t Fd πd

t

( )dt −

∫ T

0λsπs

tFs πs

t( )

dt

s.t. λdFd πdt

( )� λsFs πs

t( )

, ∀ t ∈ [0,T],(D)

where π � (πdt , π

st) : t ∈ 0,T[ ]{ }

is an arbitrary mea-surable function from [0,T] to R2+.

In problem (D), the intermediary determines thedemand-side price trajectory πd

t : t ∈ 0,T[ ]{ }and the

supply-side price trajectory πst : t ∈ 0,T[ ]{ }

at time 0.The intermediary’s pricing policy π is feasible if itclears the market at each point of time t, that is,λdFd πd

t( ) � λsFs πs

t( )

. Under the pricing policy π, overthe entire season [0,T], the total revenue that theintermediary collects from buyers is

∫ T0 λdπd

t Fd πd

t( )

dt,and the total cost that the intermediary incurs fromcompensating sellers is

∫ T0 λsπs

tFs πs

t( )

dt. The inter-mediary aims at maximizing her net profit overthe entire horizon.Now, we characterize the intermediary’s optimal

policy and profit in this auxiliary setting.

Proposition 1 (Optimal Solution to the DeterministicProblem). The optimal solution to problem (D) is thatthe intermediary simply posts fixed prices p∗ and w∗ forbuyers and sellers, respectively, throughout the horizon,where prices p∗ and w∗ always exist and are determined bythe following conditions:

(i) (Demand-supply balancing condition)

λdTFd p∗( ) � λsTFs w∗( )≜μ∗; (1)

(ii) (Virtual value-cost balancing condition)

μ∗ � max μ ∈ 0,min λdT, λsT{ }[ ]

: V μ( ) ≥ 0

{ }, (2)

where V(μ)≜Vd(Fd,−1( μ

λdT

)) − Vs(Fs,−1( μλsT

)).

Moreover, p∗ ≥ w∗. The optimal value of program (D) is

J∗ � p∗ − w∗( )μ∗. (3)

We observe that the optimal price pair p∗,w∗( )is

determined by Equations (1) and (2). Equation (1)is the market-clearing condition. Under this condi-tion, the number of buyers who purchase the pro-duct is equal to the number of sellers who sell theproduct. Equation (2) entails that either the inter-mediary has matched the most number of buyers andsellers under the optimal price pair p∗,w∗( )

, and it isinfeasible to match an additional pair of buyer andseller, μ∗ �min λdT, λsT

{ }, or although it is feasible to

match more pairs of buyers and sellers, μ>μ∗, byadjusting the price pair p∗,w∗( )

, the marginal revenuethat the intermediary collects from enabling oneadditional buyer to get the product, Vd(Fd,−1( μ

λdT

)),

is less than the marginal cost that the intermediaryincurs from enabling one additional seller to sell theproduct, Vs(Fs,−1( μ

λsT

)); that is, the intermediary’s

marginal net profit from matching one additionalbuyer with one additional seller is negative, thatis, V μ

( )< 0.20


The intermediary’s optimal profit in this auxiliarysetting, J∗, has the following property.

Lemma 2. We have

E J HT( )[ ] ≤ J∗.

That is, the benchmark bound E J HT( )[ ]in Lemma 1

can be further upper-bounded by J∗. The upper boundJ∗ has the following notable features. First, J∗ is en-tirely determined by buyers’ and sellers’ arrival rates,the buyer virtual-value function, and the sellervirtual-cost function. The upper bound J∗ does notdepend on the buyer’s or the seller’s waiting cost,which can be avoided in the deterministic worldwith perfectly matched, myopic buyers and sellers,in a continuous quantity. Second, J∗ has the intuitiveeconomic interpretation that the intermediary earnsthe bid–ask spread (p∗ − w∗) from each matched buyer–seller pair.

5.2. Waiting-Adjusted Fixed-Pricing PolicyIn this subsection, we use the optimal policy in theauxiliary setting, characterized in the previous sub-section, to motivate a simple dynamic pricing andmatching policy and show that this policy is nearoptimal. We begin by presenting the matching part ofour policy, the greedy matching policy, denoted byMg. Under this policy, the intermediary matches eachdemand (resp. supply) request as soon as a supply(resp. demand) is available on a FCFS basis. There-fore, the intermediary minimizes demand and supplymismatch at each point of time.21 Define

It ≜∑ψ∈Ht

1 τψ ≤ t, aψ � 1{ } − ∑

φ∈Ht

1 τφ ≤ t, aφ � 1{ }

.

Therefore, at each point of time t, the number ofunmatched supply is It( )+, and the number of un-matched demand is It( )−. This matching policy isnatural, practical, and fair.

Along with the simple greedy matching part of ourpolicy, we next present the pricing part, the waiting-adjusted fixed-pricing (FP) policy, denoted by πWFP �πWFP,dt , πWFP,s

t : t ∈ [0,T]{ }

. This policy is constructed

in the following way. Recall from the previous sub-section that the intermediary’s optimal pricing pol-icy in the auxiliary deterministic myopic buyer andseller model is to simply post fixed prices p∗ andw∗ onthe demand and supply sides, respectively. However,in our original stochastic system, although this pol-icy is easy to implement, it does not lead to simplebuyers’ and sellers’ behavior. The presence of buyers’and sellers’ arrival uncertainties in the original modelcan cause them to wait to be matched with waitingdisutilities, even though buyers and sellers do not

strategize their entry to the matching pool given fixedprices. Therefore, if the intermediary posts merelyprices p∗ and w∗ on the demand side and the supplyside, respectively, then a buyer (resp. seller) cannotmake the purchasing (resp. selling) decision by merelycomparing her valuation (resp. cost) with p∗ (resp. w∗).She has to take into account the joint effects of theprice p∗ (resp. w∗) and the waiting disutility. How-ever, calculating the expected waiting disutility maynot be an easy job for buyers and sellers. Therefore, toalleviate buyers’ and sellers’ computational burdenand ease their decisions, we require the intermediaryto adjust the fixed prices p∗ and w∗ by taking intoaccount thewaiting disutilities that buyers and sellersincur, such that a buyer (resp. seller) canmake an easydecision by merely comparing her valuation (resp.cost) with p∗ (resp. w∗).Formally, under policy πWFP, the prices posted at

each point of time t ∈ [0,T] on demand and supplysides, respectively, are given by

πWFP,dt � p∗ − b

EIt− sφ − tφ|tφ � t, vφ ≥ p∗, It−( )+[ ]Pr mφ � 1|tφ � t, vφ ≥ p∗, It−( )+( ) ,

πWFP,st � w∗ + h

E sψ − tψ|tψ � t, cψ ≤ w∗[ ]Pr mψ � 1|tψ � t, cψ ≤ w∗( ) ,

where p∗ and w∗ are determined byEquations (1) and (2),respectively, and the expectations and the supply–demand mismatch quantity It are computed underthe assumption that all buyers (resp. sellers) behavemyopically; that is, every buyer φ (resp. seller ψ)makes her purchasing (resp. selling) decision at herarrival time, τφ � tφ (resp. τψ � tψ), and decides topurchase (resp. sell) if and only if her valuation (resp.cost) is no less (resp. more) than p∗ (resp. w∗).22The waiting compensation terms have the follow-

ing properties. First, the probability of being matchedfor a buyer (resp. seller) is in the denominator of thewaiting compensation, because the monetary fundsexchange hands only when a match is realized. Sec-ond, because of the different information structuresof buyers and sellers introduced in Section 3 (see alsoEndnote 19), on the demand side, each buyer φ’sexpected time of staying in the system and the prob-ability of being matched are conditional on the num-ber of unmatched supply, (Itφ−)+. Because (Itφ−)+ is arandom variable, the demand-side compensation termis random. As a result, the demand-side pricing policyπWFP,d is a contingent policy. In contrast, on the supplyside, because a seller does not have the informationof the number of unmatched supply or demand at anypoint of time, the seller’s expected time of staying inthe system and the probability of being matched arenot conditional on the number of unmatched supply


or demand. Therefore, the supply-side compensationfor each point of time is deterministic. As a result, thesupply-side pricing policy πWFP,s is a deterministicpolicy.23

Now, we establish an equilibrium stopping andrequesting rule for buyers and sellers when the in-termediary follows the waiting-adjusted FP and thegreedy matching policy.

Theorem 1 (Strategic Myopia). Assume that the interme-diary adopts the waiting-adjusted FP policy πWFP and thegreedy matching policy Mg. Then, there exists a perfectBayesian equilibrium (PBE), such that all buyers and sellersbehave myopically:24

(i) For each buyer φ, τπWFP,Mg

φ � tφ and aπWFP,Mg

φ �1 vφ ≥ p∗{ }

.(ii) For each seller ψ, τπ

WFP,Mg

ψ � tψ and aπWFP,Mg

ψ �1 cψ ≤ w∗{ }

.

In our model, the stochastic nature of buyers’ andsellers’ arrival processes makes them wait to bematched and hence to incur waiting disutilities. There-fore, the waiting-adjustment terms play a vital role incompensating their losses and then inducing them tobehave myopically.

So far, we have shown that with waiting compen-sation, the two-sided pricing policy πWFP is dynamic.The following proposition characterizes the vari-ability of the pricing policy in an asymptotic sense.

Proposition 2 (Waiting Compensation). For any t ∈ [0,T)and any k ∈ [0, 1), we have

EIt− p∗ − πWFP,dt

[ ]≤ b

min kT,T − t{ } + Tmin K, 1{ }1 −min K, 1{ } ,

πWFP,st − w∗ ≤ h

min kT,T − t{ } + Tmin K, 1{ }1 −min K, 1{ } ,

where K≜ 20μ∗ min{k2,(1− t

T)2}. In addition, we have the fol-

lowing results:(i) If buyers are fully patient—that is, b � 0—then

πWFP,dt � p∗. If sellers are fully patient—that is, h � 0—

then πWFP,st � w∗.

(ii) Consider a sequence of systems. In the n-th system,λd,(n) � nαdλd with αd > 0 and λs,(n) � nαsλs with αs > 0.Denote α ≜ min αd, αs{ }. For any t ∈ [0,T), we have

lim supn→∞

EIt− p∗,(n) − πWFP,d,(n)t

[ ]≤ O

1nα/3

( ),

lim supn→∞

πWFP,s,(n)t − w∗,(n) ≤ O

1nα/3

( ).

We make the following observations from thisproposition.

1. If buyers (resp. sellers) are fully patient—that is,b � 0 (resp. h � 0)—they do not incur any waitingdisutility, although they may spend time in waiting

to be matched. Therefore, πWFP does not need to beliterally adjusted from the base fixed prices p∗,w∗( )

.2. In the high-volume regime in which buyers’ and

sellers’ arrival rates grow large (scaled by n), re-gardless of whether they grow at the same or dif-ferent speeds (measured by αd and αs), the waiting-compensation terms on both demand and supplysides in the policy πWFP diminish to zero—that is,the policy πWFP tends to be the fixed-pricing policyp∗,w∗( )

. In addition, as n grows large, the variability ofπWFP decays to zero at a speed that is no slower than1/nα/3.Although it seems challenging to theoretically char-

acterize the monotonicity property of the waiting-adjusted FP policy πWFP with respect to time, it is stillworthwhile making some discussions here. Let usconsider the demand-side expected price trajectory{E[πWFP,d

t ] : t ∈ [0,T]}. First, consider the scenario thattime t is very far away from the end of the sellinghorizon T. On one hand, a buyer who requests to buythe product at time tneeds towait until all buyerswhorequest to buy at earlier times have been matched. Attime t, the remaining unmatched demand and supplyare [ND(μ∗t) −NS(μ∗t)]+ and [ND(μ∗t) −NS(μ∗t)]−, re-spectively, where ND(μ∗t) and NS(μ∗t) are two in-dependent homogeneous Poisson processes with thesame rate μ∗t, representing the arrival processes ofbuyers and sellers, respectively. As time goes by, theexpected numbers of unmatched buyers and sellerswho have arrived are increasing.25 On the other hand,because t is much smaller than T, a buyer who re-quests to buy the product at time t has sufficienttime and is very likely to be matched before the endof the horizon. Therefore, as t increases, the waiting-compensation term tends to increase, and, thus, thedemand-side expected price E[πWFP,d

t ] tends todecrease.Now, consider the other end of the spectrum,where

time t is very close to the end of the selling horizon T.As t increases, on one hand, for a buyer who requeststo buy the product at time t, her maximum time ofstaying in the system decreases as t increases. On theother hand, because t approaches T, a buyer whorequests to buy the product at time t can be eventu-ally matched only if the total number of supply re-quests up to time T exceeds the total number ofdemand requests up to time t. Under policy πWFP,because the arrival rates of buyers who request to buythe product and sellerswho request to sell the productare the same, as t is getting close to T, the probabilitythat a buyer who requests to buy the product at time tcan be eventually matched is approaching 50%.Therefore, as t increases, the waiting-compensationterm tends to decrease, and, thus, the demand-sideexpected price E[πWFP,d

t ] tends to increase.


Now, consider the supply-side price dynamicsπWFP,st : t ∈ [0,T]{ }

. First, consider one extreme thattime t is very far away from the end of the sellinghorizon T. Following the similar arguments as above,as t increases, thewaiting-compensation term tends toincrease, and, thus, the supply-side price πWFP,s

t tendsto increase. Consider the other extreme that time t isvery close to the end of the selling horizon T. Fol-lowing the similar arguments as above, as t increases,thewaiting-compensation term tends to decrease, and,thus, the supply-side price πWFP,s

t tends to decrease. Ingeneral, the effective ask-and-bid price processes

E[πWFP,dt ] :

{t ∈ [0,T]

}and

{πWFP,st : t ∈ [0,T]

}may not

always demonstrate a consistently upward ordownward trend, and they tend to have oppositetrends in the beginning and in the end of the horizon.

As an illustration, Figure 1 plots the price dynamics

E[πWFP,dt ], πWFP,s

t : t ∈ [0,T]{ }

for a set of market en-

vironments in which both buyers’ and sellers’ arrivalrates are equal toλ. The results are consistent with bothProposition 2 in that the variability of these two pricedynamics diminishes as the demand and supplyquantities grow up (λ increases) and our predictionabove on the temporal trends of these price dynamics.

5.3. Performance of the Waiting-Adjusted Fixed-Pricing Policy

In this subsection, we analyze the performance ofthe waiting-adjusted FP and the greedy matchingpolicy Mg. Under this simple heuristic, buyers and

sellers behave myopically, and, as a result, the in-termediary faces a stochastic system that is effectivelya double-ended queue, with the arrival rates of bothsides being μ∗ and that starts with being empty attime t � 0.

Theorem 2 (Performance Guarantee). Under the waiting-adjusted FP policy πWFP and the greedy matchingpolicy Mg,

JπWFP,Mg

E J HT( )[ ] ≥ JπWFP ,Mg

J∗≥ 1 − 1 + 2

3b + h( )Tp∗ − w∗

( )1��μ∗√ .

In addition, consider a sequence of systems. In the n-thsystem, λd,(n) � nαdλd with αd > 0 and λs,(n) � nαsλs withαs > 0. Denote α ≜ min αd, αs{ }. Therefore,

JπWFP,Mg,(n)

E J(n) HT( )[ ] ≥ JπWFP,Mg,(n)

J∗,(n)≥ 1 −O

1��nα

√( )

.

Theorem 2 has the following implications. First, asboth buyers’ and sellers’ arrival rates grow large(scaled by n), regardless of whether they grow up atthe same or different speeds (measured by αd andαs), the simple, waiting-adjusted FP policy πWFP andgreedy matching policy Mg are asymptotically opti-mal. Second, as n grows large, the relative profit lossof implementing the simple heuristic policy, com-paredwith the optimal mechanism, converges to zeroat a speed that is no slower than 1/

��nα

√, where

α � min αd, αs{ }. Put differently, the relative additionalbenefit of implementing any more sophisticated policy

Figure 1. (Color online) Price Dynamics

Note. Fd(·) ∼ U[0, 10], Fs(·) ∼ U[0, 4], b � h � 0.1 and T � 50.


than our simple heuristic is nomore than amagnitudeof 1/

��nα

√.

To better understand the impact of different market-environment parameters on the performance loss ofour proposed simple heuristic policy, we conduct aseries of numerical experiments. In these experiments,we use the following market environment as our base:λd � λs � λ � 2, Fd(·) ∼ U[0, 10], Fs(·) ∼ U[0, 4], b � h �0.1. For each market environment, we report theperformance of our heuristic against the performanceupper bound obtained by solving the clairvoyant’sproblem (B). We compute the sample means of thesetwo profit functions by randomly generating K =10,000 sample paths of buyers’ and sellers’ arrivalprocesses.

First, we study the impact of the identical buyers’and sellers’ arrival rate λ and the length of the horizonT on the performance of our heuristic. We choose λfrom the set 1, 2, 5, 10{ } andT from the set 1, 2, . . . , 50{ }.The results are reported in Figure 2. We observe that,for a very short time horizon, the end-of-horizoneffect immediately kicks in, leading to the under-performance of our heuristic. But the performance ofour heuristic stabilizes with a not-too-short horizon.Even for a small-scale system with λ � 1 or 2, ourheuristic has a performance guarantee of more than80% when T ≥ 15. Because we benchmark againstthe performance upper bound from problem (B), theactual performance should be even better. For asystem with λ � 10, our heuristic has a performanceguarantee of more than 90% when T ≥ 5.

Second, we study the impact of buyer (resp. seller)waiting disutility rate b (resp. h) on the performance

of our heuristic. In Figure 3(a), we fix h � 1 and chooseb from the set 0.02, 0.1, 0.2{ }. In Figure 3(b), we fix b � 1and choose h from the set 0.02, 0.1, 0.2{ }. We observethat when the length of the horizon T is not very short,the intermediary achieves a better performance whenbuyers (resp. sellers) incur less waiting disutility—that is, b (resp. h) is smaller. The intuition is as follows.Under our heuristic policy, to induce buyers and sellersto behavemyopically, the intermediary’s pricing policyneeds to include the waiting-compensation terms.When buyers (resp. sellers) incur less waiting dis-utility—that is, b (resp. h) is smaller—these waiting-compensation terms become smaller—that is, theintermediary can charge each buyer a higher pricethat is closer to p∗ (resp. pay each seller a lower wagethat is closer to w∗). As a result, the intermediary cancapture more profit.Third, we study the impact of the range of the

buyer’s valuation (resp. seller’s cost) on the perfor-mance of our heuristic. We assume that the buyer’svaluation (resp. seller’s cost) is uniformly distributedwith mean 5 (resp. 2). We choose the length of thebuyer’s valuation range (resp. seller’s cost range)from the set 2, 4, 6{ } (resp. 0.8, 1.6, 2.4{ }). The resultsare reported in Figures 4 and 5. We observe thatour heuristic performs consistently well regardless ofwhether the buyer’s valuation (resp. seller’s cost) isdispersed or concentrated. In addition, we also ob-serve that the performance of our heuristic seems notquite sensitive to the degree of the dispersion of thebuyer’s valuation (resp. seller’s cost).In all numerical experiments mentioned above,

we randomly generate K = 10,000 sample paths ofbuyers’ and sellers’ arrival processes. As mentioned,we compute the relative performance of our heuristicas the ratio of the sample mean of the profit under ourheuristic to the sample mean of the upper-boundprofit obtained by solving the clairvoyant’s prob-lem (B), hereafter referred to as the “empirical relativeperformance.” Therefore, wemay get different valuesof the empirical relative performance when we gen-erate K randomly drawn sample paths for multi-ple times. Hence, we can use the sample coefficientof variation26 of these different values to measurewhether the empirical relative performance com-puted from K sample paths gives us an accurate andreliable estimation of the true value of the relativeperformance. In the following experiment, we studythe impact of the number of sample paths, K, on thevolatility of the empirical relative performance. Wechoose K from the set {30; 100; 300; 1,000} and thelength of the horizon T from the set 1, . . . , 50{ }. ForeachK and T, we generateK sample paths for 30 timesand thus obtain 30 data points of the empiricalrelative performance. We then compute the sam-ple coefficient of variation of these data points.

Figure 2. (Color online) Impact of Buyers’ and Sellers’Arrival Rates λ and the Length of the Horizon T on thePerformance of Our Heuristic

Note. Fd(·) ∼ U[0, 10], Fs(·) ∼ U[0, 4], b � h � 0.1.


The results are reported in Figure 6. We observe thatwhen the number of sample paths increases, thesample coefficient of variation of the empirical rela-tive performance decreases. When K = 1,000 and T isnot too small, such a sample coefficient of variations isless than 0.01. This result guarantees that in all ex-periments above, the empirical relative performancesobtained with K = 10,000 are very robust. Also, weobserve that the sample coefficient of variation of theempirical relative performance tends to decreasewhen the length of the horizon T prolongs, which isconsistent with that the performance of our heuristicis, on expectation, more stabilized for a longer hori-zon (see Figure 2).

6. Effects of Market Conditions onPricing Heuristic

The pricing policy πWFP introduced in the previoussection depends onmarket conditions, such as themarketsize (characterized by buyer arrival rate λd and sellerarrival rate λs). In practice, the market conditions maychange over time. For instance, Uber’s market condi-tions during rush hour and nonrush hour are quitedifferent. During rush hour, more people need rides,and less so innonrushhour. Therefore, in this section,weare motivated to explore the effects of market conditionson the pricing policy πWFP, as well as the interme-diary’s matched quantity and the resulting profit. Wenotice that, in practice, an intermediary, such as Uber,

Figure 3. (Color online) Impact of Buyer (Resp. Seller) Waiting Disutility Rate b (Resp. h) on the Performance of Our Heuristic

Note. Fd(·) ∼U[0,10], Fs(·) ∼ U[0, 4].

Figure 4. (Color online) Impact of the Range of the Buyer’s Valuation on the Performance of Our Heuristic

Note. Fs(·) ∼ U[0, 4], b � h � 0.1.


is typically operating in a high-volume regime wherethe demand and supply sizes are sufficiently large.Therefore, following from the asymptotic conver-gence property established in Proposition 2 that πWFP

converges to fixed prices (p∗,w∗) and the asymptoticoptimality result established in Theorem 2, here, wefocus on the intermediary’s optimal pricing policyand her performance in the fluid regime—that is, theintermediary’s pricing policy is given by the fixed-price pair (p∗,w∗), her matching quantity is given byμ∗, and her profit is given by J∗.27

In this section, we focus on exploring the effectsof the market sizes (measured by λd and λs) on pricesp∗ and w∗, as well as on the intermediary’s matchingquantity μ∗ and her profit J∗.28

Theorem 3 (Comparative Statics on Market Sizes).(i) (Seller’s Market) p∗ and w∗ are increasing in λd, and

μ∗ and J∗ are increasing in λd.

(ii) (Buyer’s Market) p∗ and w∗ are decreasing in λs, andμ∗ and J∗ are increasing in λs.

As the demand rate λd increases and the supply rateλs does not change, the market becomes more tilted to-ward a seller’s market. The intermediary can addressthis demand and supply imbalance by adjustingprices on both demand and supply sides. On thedemand side, the intermediary raises up the price forbuyers (p∗ increases) to target a more selective fraction ofbuyers who are willing to pay more. On the supply side,the intermediary increases payments to sellers (w∗ in-creases) to encourage more sellers to deliver theproduct. As the demand rate λd grows, the interme-diary can match more pairs of buyers and sellers (μ∗increases) and gain a higher profit (J∗ increases).As the supply rate λs increases and the demand rate

λd does not change, the market becomes more tiltedtoward a buyer’s market due to the existence of moreexcess supply. The intermediary can also address thisdemand and supply imbalance by adjusting priceson both sides. On the demand side, the intermediarycuts down the price for buyers (p∗ decreases) to at-tract more buyers to request the product. On thesupply side, the intermediary reduces payments tosellers (w∗ decreases) to discourage more sellers todeliver the product. As the supply rate λs grows, theintermediary can again match more pairs of buyersand sellers (μ∗ increases) and gain a higher profit (J∗increases).In summary, the matching quantity and profit im-

plications of a more considerable demand or supplyvolume are unambiguously positive to the intermediary,whereas the price implications can go either way, de-pending on whether it is a seller’s or buyer’s market.

7. ConclusionIn this paper, we propose a simple heuristic pricingandmatching policy that is proven to performwell for

Figure 5. (Color online) Impact of the Range of the Seller’s Cost on the Performance of Our Heuristic

Note. Fd(·) ∼ U[0, 10], b � h � 0.1.

Figure 6. (Color online) Impact of the Number of SamplePaths K on the Performance Volatility

Note. Fd(·) ∼ U[0, 10], Fs(·) ∼ U[0, 4], b � h � 0.1.


an intermediary who operates in a two-sided marketwith forward-looking buyers and sellers. We showthat this simple policy is near optimal, as the arrivalrates on both sides of the market grow. Our re-sults help identify practical market conditions underwhich a market maker may not need much of a veryfrequent dynamic pricing policy. For a market withroughly constant rates of random arrivals of forward-looking buyers and sellers who tend to have the samedistribution of willingness to pay and to sell, re-spectively, fixed pricing with some waiting-cost ad-justment and greedy matching can be good enoughwhen the numbers of buyers and sellers are large.

Our results may be extended in the following set-tings in the presence of forward-looking buyersand sellers. First, one may consider the time-varyingarrival rates of buyers and sellers. Second, one canconsider the matching of buyers and sellers ofmultiple-type goods or services, where types can begeographic locations. For example, in the ride-hailingmarket, after a driver in one location accepts a bidprice, the matching policy can be more complicatedthan that for buying and selling a homogeneous (suchas single-location) service, because the driver can berouted to serve riders at different places. Third, onemay consider more than one intermediary platformcompeting for both demand and supply. Such situa-tions arise, for example, in the ride-hailing industry,where Uber and Lyft compete with each other. Themechanism-design problem for these situations in thepresence of strategic buyers and sellers is challenging.It is worthwhile exploring whether asymptotic opti-mality of some heuristic pricing and matching policy,similar to our results, may be still attainable.

AcknowledgmentsThe authors thank the department editor Hau Lee, theanonymous associate editor, the reviewers, Saif Benjaafar,Peter Frazier, Chen Peng, Garrett van Ryzin, and Zhixi Wanfor constructive and insightful suggestions, which signifi-cantly improved the paper.

Endnotes1Uber seems to apply the match-to-the-closest greedy policy in thespatial dimension. Hu and Zhou (2016a) show that, under somespecial conditions—for example, when all travelers are going in thesame direction—this greedymatching policy in the spatial dimensioncan be optimal.2 See http://time.com/4953495/uber-london-alternative.3Taylor (2018) and Bai et al. (2018) do consider the reentry of drivers,but do not consider drivers transitioning in a network of locations.Bimpikis et al. (2019) and Besbes et al. (2018b) focus on the drivers’spatial transitions and repositioning in a deterministic setting. Besbeset al. (2018a) take into account a driver’s travel time to pick up a riderin a queueing setting. Afeche et al. (2018) also consider a queueingformulation and focus on the performance impact of platform’s

repositioning control (or drivers’ self-repositioning) and demand-side admission control.4For example, Uber often maintains an estimated waiting time offewer than 5 minutes for riders (see Hall et al. 2015), and, accordingto Uber’s latest policy, a driver will be compensated for havingto wait more than 5 minutes after arriving at the pickup location.This suggests that a wait of fewer than 5 minutes on both sides ofdemand and supply seems to be tolerable without the need ofcompensation.5Chen et al. (2015) suggest that under normal circumstances, Uberappears to update prices every 5 minutes, which demonstrates somelevel of pricing stability.6The pricing formulae can also be used to compute the surge pricesfor a given region under a particular market condition.7Drivers who pursue surge prices may find that the surge is gone bythe time they arrive.8A recent paper, Hu et al. (2018), studies the optimal contingentpricing policy in a stylized two-period model with forward-lookingriders and drivers and shows that there could exist different typesof equilibrium pricing policies with qualitatively distinctive pricetrajectories.9Uber recently also offered a flat-fare price for a ride in the form of apackage of multiple rides for New York City.10 It may be relatively easy for Uber to monitor the arrival of ridersand drivers because they all need to open the Uber app to make atransaction. But that may not always be the case for other marketmakers.11For example, Carr (2015) reports that Uber drivers falsely triggersurge pricing by conspiring to shut down and restart their apps.12 In the ride-hailing industry, if rider φ starts to consider to have aride from time tφ and is eventually matched with a driver at time sφ,then her disutility that arises from her waiting for a car is captured byb(sφ − tφ). Over [tφ, τφ], rider φ not only incurs waiting disutility, butexerts effort to track how the ride-hailing platform dynamicallychanges prices. However, a rider can easily track price dynamics byfrequently refreshing the ride-hailing platform’s app. Therefore, arider’s disutility arising from the price-monitoring activity is negli-gible relative to the disutility incurred from waiting for a ride.13The monotonicity in this paper is in its weaker sense unless oth-erwise specified.14 If this condition is violated—that is, c> v—the intermediary cannotmake a positive profit given that buyers and sellers have the incentiveto participate in the market.15 Similar to the demand side, in the ride-hailing industry, the termh sψ − tψ( )

captures driver ψ’s waiting disutility that is from the timethat she is available to serve, tψ, to the time that she is matched with arider, sψ. We assume that over [tψ, τψ], driver ψ’s disutility arisingfrom her price monitoring activity is negligible relative to the dis-utility incurred from waiting to serve a rider.16When the intermediary makes the pricing decisions at time t, theinformation available to her consists of historic times at which buyersinform the intermediary about their willingness to buy the product,φt−, historic times at which sellers inform the intermediary abouttheir willingness to sell the product, ψt−, the intermediary’s historicmatching decisions, φt−

M and ψt−M , and the intermediary’s historic

pricing decisions, πd,t− and πs,t−.17When an Uber rider repeatedly refreshes her app, she can track thedynamics of the fares quoted to her. When an Uber rider keeps therider app open, she can also track the dynamics of available cars thatare around her.18We do not prove the existence of the equilibrium stopping time ingeneral. We only demonstrate the existence of such an equilibrium


http://time.com/4953495/uber-london-alternative/

stopping rule for a specific class of pricing and matchingpolicies—that is, static prices with expected waiting-cost adjustmentand the greedy matching policy that matches a demand or supplyrequest on a first-come-first-served basis.19When an Uber driver opens the driver app and sets the status to beonline, she can see a heat map. In the map, different colors in re-gions represent different surge rates. The colors (surge rates) in theheat map dynamically change over time. Although an Uber driveris notified of the payment only after she completes a ride, beforeaccepting a riding request, she can still use the heat map to estimatethe fare that she will receive from Uber by offering a ride. Unlike anUber rider, whose app shows the available cars around her, the Uberdriver app does not tell a driver any information about the otheravailable cars around or how many riders in the neighboring regionare requesting rides and where they are located. An Uber driver canonly passivelywait for a riding request sent fromUber. Therefore, ourmodel does not assume that sellers can observe the number of un-matched supply or demand.20Consider any price pair p,w

( )that clears the market—that is,

λdTFd p( ) � λsTFs w( )≜μ. Therefore, the intermediary’s marginal

revenue from enabling one additional buyer to get the product isd(pμ)/dμ�p + (dp/du)μ�p − Fd(p)/f d(p)�Vd(p)�Vd(Fd,−1(μ/(λdT))),and the intermediary’s marginal cost from enabling one addi-tional seller to sell the product is d wμ

( )/dμ�w+ (dw/du)μ� w+

Fs w( )/f s w( ) �Vs w( ) �Vs Fs,−1 μ/(λsT)( )( ). Therefore, the intermedi-

ary’s marginal net profit from matching one additional pair ofbuyer and seller is Vd Fd,−1 μ/(λdT)( )( )− Vs Fs,−1 μ/(λsT)( )( )�V(μ).21 If there is more than one unmatched supply (resp. demand) requestthat simultaneously arrives at the intermediary, then the interme-diary randomly chooses one among them to match with a newlyarriving buyer’s demand (resp. seller’s supply).22 In the presence of the waiting-compensation terms, we cannotexclude the possibility that sometimes the demand-side price issmaller than the supply-side price, πWFP,d

t <πWFP,st . However, as

we show in our numerical analysis (see Figure 1), for a wide rangeof parameters, the demand-side price is always higher thanthe supply-side price, which is also guaranteed to be true inexpectation.23 If alternative information structures are assumed, our resultswould not change after adjusting the waiting-compensation termsaccordingly.24The PBE that we present in this theorem is one equilibrium. We donot prove the equilibrium uniqueness. It is worth exploring whetherother equilibria exist.25Note that the difference between two independent Poisson randomvariables with the same mean μ follows the so-called Skellam dis-tribution, which has mean 0 and variance 2μ. Hence, the class ofrandom variables ND(μ∗t) −NS(μ∗t) parameterized by t is a class ofmean-preserving spreads. Thus,ND(μ∗t) −NS(μ∗t) is increasing in t inthe convex order. Because both x+ and x− are convex functions, by thedefinition of the convex order,E[ND(μ∗t) −NS(μ∗t)]+ andE[ND(μ∗t) −NS(μ∗t)]− are increasing in t.26The sample coefficient of variation is defined as the ratio of thesample standard deviation to the sample mean.27 It is tempting to derive comparative statics results for the matchingquantity and profit level in the original stochastic system. Because it ishard to rank Poisson random variables in usual stochastic order eventhough their means can be ordered, we analytically study the per-formance measures in the fluid regime, with which the performancemeasures in the original stochastic system are directly linked. Forexample, the matching quantity in the stochastic system under oursimple heuristic is min(ND(μ∗T),NS(μ∗T)). We numerically study theeffect of the market conditions on the compensation terms in thenonfluid regime with results reported in Figure 1.

28More studies on the effects of other market conditions are reportedin the online appendix.

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