Dynamic Matching for Real-Time Ridesharing

In a ridesharing system, arriving customers must be matched with available drivers. These decisions affect the overall number of customers matched, because they impact whether or not future available drivers will be close to the locations of arriving customers. A common policy used in practice is the closest driver (CD) policy that offers an arriving customer the closest driver. This is an attractive policy because it is simple and easy to implement. However, we expect that parameter-based policies can achieve better performance.<br><br>We propose matching policies based on a continuous linear program (CLP) that accounts for (i) the differing arrival rates of customers and drivers in different areas of the city, (ii) how long customers are willing to wait for driver pick-up, (iii) how long drivers are willing to wait for a customer, and (vi) the time-varying nature of all the aforementioned parameters. We prove asymptotic optimality of a forward-looking CLP-based policy in a large market regime and of a myopic LP-based matching policy when drivers are fully utilized. When pricing affects customer and driver arrival rates, and parameters are time homogeneous, we show that asymptotically optimal joint pricing and matching decisions lead to fully utilized drivers under mild conditions.


Introduction
We consider the control of a two-sided matching system with multiple item types in both sides arriving randomly to the system with potentially time-varying rates; see Figure 1. The items arriving to the circles must be matched at the time of their arrival, and there is a timevarying probability that the match is acceptable. The items arriving to the queues will wait to be matched, but are impatient, and may leave their queue if left waiting for too long. The goal of the system controller is to maximize the cumulative number of (weighted) matchings in a finite time horizon.
Our main motivation for studying the system in Figure 1 is the customerdriver matching in ridesharing platforms. The N queue locations in Figure   Keywords and phrases: ridesharing, dynamic matching, fluid model, asymptotic optimality plied locations. However, surges in price can lead to negative publicity (cf. White (2016), The Economist (2016), and Michallon (2016)). This leads to the question of whether better matching decisions could reduce the need for price surges. An ideal is to solve a joint pricing and matching problem that accounts for the impact of differing customer and driver location. We do this in a static environment, but that joint problem is very difficult in a timevarying environment. As a first step to making progress in a time-varying environment, we assume customer and driver arrival rates are given, and focus on optimizing the matching decisions. The aforementioned time-varying arrival rates could be thought of as the result of a pricing policy as well as other underlying incentives that may be given to drivers or customers, such as demand information sharing or discount coupons -but we do not explicitly model that.
The CD policy is a greedy policy, and, therefore, a forward-looking policy is likely to perform better than the CD policy does. The following example illustrates this point. Figure 2 represents a region of a city, partitioned into ten disjoint hexagonal areas (cf. Figure 3 in Chen and Sheldon (2015) for motivation that ridesharing companies find such a representation useful). Suppose a customer arrives at area 1 and requests a ride. There are three drivers idle in area 4 and there is a single idle driver in area 3. Moreover, a concert recently ended in area 8, which implies a high potential customer arrival rate in that area. If the destination of the current area 1 customer is far away, the driver assigned to that customer will not return to area 1 for a long time. Then, the system controller has two options: He can either offer one of the drivers in area 4 or the driver in area 3 to the customer in area 1. Under the CD policy, the system controller offers an area 4 driver. However, offering the driver in area 3 saves all drivers in area 4 to match with the potential customers departing the concert in area 8. This prevents the potential future need to offer an area 8 customer a far-away area 3 driver, whom the customer will likely refuse, ending in no match being made. We conclude there is a nontrivial tradeoff between offering the closest driver in accordance with the CD policy and offering a farther driver in order to maximize the future number of customers matched.
1.1. Contributions of the Paper We analyze the two-sided matching system depicted in Figure 1 that is motivated by ridesharing systems. The objective is to maximize the cumulative number of matchings in a finite time horizon. Our main contributions are as follows.
Proposing asymptotically optimal matching policies based on a continuous linear program (CLP) and a linear program (LP): We consider a large matching market, in which the arrival rates of the customers and drivers Electronic copy available at: https://ssrn.com/abstract=2844451 grow without bound, so as to approximate the case of a large city. For any matching policy that does not know the future with certainty, we establish that the solution to a CLP is an asymptotic upper bound on the cumulative number of matchings done in a finite time horizon under fluid scaling (cf. Theorem 1). That upper bound is very strong in the sense that it is valid on almost every sample path. Then, we propose a matching policy based on an optimal solution of the CLP which is asymptotically optimal (cf. Corollary 1). The CLP leads to an LP when drivers are fully utilized or the CLP parameters are time homogeneous, which motivates an asymptotically optimal LP-based matching policy in each case (cf. Theorems 3 and 4, respectively). When pricing affects customer and driver arrival rates and parameters are time homogeneous, we provide an asymptotically optimal pricing and matching policy (cf. Corollary 3), and show drivers are fully utilized under that policy under very mild conditions (cf. Example 1).
Providing simulation experiments illustrating the superior performance of the proposed policies against the CD policy: Our simulation experiments in Section 5 show that both the CLP-based and the LP-based proposed matching policies can significantly outperform the CD policy. Consistent with intuition, we see that demand spikes coupled with low nearby driver availability, such as in Figure 2, lead to the poor performance of the CD policy. This is exactly when we recommend spending the extra effort of estimating parameters, such as customer and driver arrival rates, in order to be able to implement a CLP-or an LP-based matching policy. (In comparison, the CD policy requires no network information to implement.) The remainder of this paper is organized as follows. We conclude this section with a literature review (cf. Section 1.2) and a summary of our mathematical notation (cf. Section 1.3). Section 2 presents our model. We formalize our large matching market regime, an asymptotic upper bound on the cumulative number of matchings, and prove that a CLP-based matching policy achieves that upper bound in Section 3. We provide conditions under which an LP-based matching policy achieves the asymptotic upper bound and consider a joint pricing and matching problem in Section 4. Section 5 presents some simulation experiments. Finally, we make concluding remarks in Section 6. The proofs of all results can be found in the appendix.
1.2. Literature Review Dynamic matching control has been studied in the literature in the context of kidney exchanges (cf.Ünver (2010)), housing markets (cf. Leshno (2016)), online matching platforms such as Upwork or Airbnb (cf. Arnosti, Johari and Kanoria (2016)), assemble-to-order manufacturing systems (cf. Plambeck and Ward (2006) and Reiman and Wang (2015)), and more abstract queueing models (cf. Gurvich and Ward (2014)). However, the ridesharing model is different enough that it is not clear any of the results of these studies carry over.
The spirit of our methodology is drawn from the queueing control literature, where (i) an asymptotic regime is defined, (ii) a control policy is derived from the solution of an optimization problem, and (iii) asymptotic optimality of the control policy is proven (cf. Harrison (2000)). We consider a large market asymptotic regime in which number of drivers and customers grow without a bound. A similar large market regime is considered in Plambeck and Ward (2006), Gurvich and Ward (2014), Arnosti, Johari and Kanoria (2016), and Leshno (2016).
In recent years, academic research related to ridesharing platforms has grown rapidly, alongside the use of these platforms. An overview of research problems on ridesharing platforms can be seen in Azevedo and Weyl (2016). Although the effects of pricing have been well-studied by Chen, Mislove and Wilson (2015), Chen and Sheldon (2015), Riquelme, Banerjee and Johari (2015), Bimpikis, Candogan and Saban (2016), Banerjee, Freund and Lykouris (2016), Castillo, Knoepfle and Weyl (2016), Hall, Kendrick and Nosko (2016), Cachon, Daniels and Lobel (2017), Guda and Subramanian (2019), and Besbes, Castro and Lobel (2018), none of those papers optimize the matching decisions. Ozkan (2018) studies joint optimization of the pricing and the matching decisions in a setting with time-homogeneous parameters. He shows that optimizing the pricing decisions while fixing the matching decisions to same area matchings (i.e., customers can be matched only with the drivers from the same area) can be suboptimal. Therefore, matching decisions have firstorder importance for the ridesharing firms.Özkan (2018) considers a steadystate fluid model, whereas we rigorously prove the convergence of a pre-limit model to a fluid model in a large market regime.
There are only a few papers that study the effect of matching decisions to the ridesharing firms (cf. Hu and Zhou (2015), Banerjee, Kanoria and Qian (2018)). Hu and Zhou (2015) consider dynamic matching control of a twosided, discrete-time matching system where the objective is to maximize the expected total discounted profit. They derive conditions under which the CD policy is optimal. Banerjee, Kanoria and Qian (2018) consider dynamic matching control of a two-sided, continuous-time matching system where the objective is to maximize the number of matchings. They propose a state-dependent matching policy which achieves the asymptotically optimal system performance with the fastest possible rate as the market size increases. A main modeling difference is that in both Hu and Zhou (2015) and Banerjee, Kanoria and Qian (2018), matchings between different types occur with either probability zero or one, whereas we assume probabilistic matching. Another difference is that Banerjee, Kanoria and Qian (2018) consider a closed network in the sense that number of drivers in the network is constant at all times, whereas Hu and Zhou (2015) and our study consider an open network such that drivers can enter and leave the network.
There are also ridesharing studies focusing on fairness of the carpooling decisions to the customers (cf. Gopalakrishnan, Mukherjee and Tulabandhula (2016)), competition between ridesharing firms (cf. Nikzad (2017)), optimal driver decisions (cf. Chaudhari, Byers and Terzi (2018)), and empty car routing (cf. Braverman et al. (2016)). Braverman et al. (2016) consider a closed network where the drivers are centrally controlled (e.g., driverless vehicles) and propose asymptotically optimal routing policies for empty vehicles in a large market regime. However, customers can be matched only with the vehicles from the same area in their setting, i.e., there is no matching optimization.
There are many ridesharing papers which consider a model with timehomogeneous parameters in steady state (cf. Riquelme, Banerjee and Johari (2015), Banerjee, Freund and Lykouris (2016), Braverman et al. (2016), Bimpikis, Candogan and Saban (2016),Özkan (2018)). However, customer demand and driver supply can change dramatically within minutes in practice (see Figures 1-3 in Hall, Kendrick and Nosko (2016)), which implies that the system may never reach steady state. Consequently, the time-varying nature of the problem is important. Among all of the aforementioned ridesharing studies, the only ones that allow time-varying parameters are Hu and Zhou (2015) and our study.

Notation and Terminology
The set of nonnegative integers and strictly positive integers are denoted by N and N + , respectively. The k dimensional (k ∈ N + ) Euclidean space is denoted by R k and R + denotes [0, +∞). For x, y ∈ R, x ∨ y := max{x, y}, x ∧ y := min{x, y}, and (x) + := x ∨ 0. We let B(R k ) denote the Borel σ-algebra on R k for all k ∈ N + and L(R) denote the collection of Lebesgue measurable subsets of R, which is a σ-algebra on R. For all T ∈ R + , B([0, T ]) and L([0, T ]) denote the Borel and Lebesgue σ-algebra on the interval [0, T ], respectively. A function f : X → R defined in measure space (X, X ) is called X -measurable (denoted by f ∈ X ), if it is (X , B(R))-measurable. If the measure space (X, X ) is (R, B(R)) ((R, L(R))), we say that f is Borel (Lebesgue) measurable.
For each k ∈ N + , D k denotes the the space of all ω : R + → R k which are right continuous with left limits. Let 0, e ∈ D be such that 0(t) = 0 and e(t) = t for all t ∈ R + . We abbreviate the phrase "independent and identically distributed" by "i.i.d.", "almost surely" by "a.s.", and "uniformly on compact intervals" by "u.o.c.". The notation a.s. − − → denotes almost sure convergence. For f ∈ D and t ∈ R + , we let f t := sup 0≤s≤t |f (s)|. Let {X n , n ∈ N} be a sequence in D and X ∈ D. Then X n → X u.o.c. as n → ∞, if X n − X t → 0 as n → ∞ for all t ∈ R + . We assume that all of the random variables and stochastic processes are defined in the same complete probability space (Ω, F, P), E denotes the expectation under P, and P(A, B) := P(A ∩ B). We let σ{X} denote the σ-field generated by the random variable X, I denote the indicator function, and ⊥ denotes independence.

The Ridesharing Model
We partition the considered region of the city into N ∈ N + disjoint areas, as illustrated in Figure 2, and assume that in the aggregate the individual driver decisions (regarding when to begin and end working, where to relocate if left waiting too long to be matched with a customer, and where to go after dropping off a customer) result in Poisson process arrivals to each area, independent of the matching decisions made by the system controller (see Remark 2). Specifically, type i drivers arrive at area i ∈ N := {1, 2, . . . , N } according to a non-homogeneous Poisson process having instantaneous rate λ i (t) at time t ∈ R + , and cumulative rate function Λ i (t) := t 0 λ i (s)ds. We further assume that the amount of time a type i driver will wait in an area to be matched with an arriving customer is exponentially distributed with time dependent rate θ i (t) for all i ∈ N and t ∈ R + .
Customers arrive in the system and request to be matched. The J ∈ N + different customer types are categorized by factors such as their origin and destination area, and their priority status. Type j ∈ J := {1, 2, . . . , J} customers arrive in accordance with a non-homogeneous Poisson process having instantaneous rate µ j (t) at time t ∈ R + , and cumulative rate function Γ j (t) := t 0 µ j (s)ds. A matching policy π = (π 1 , . . . , π J ) determines which driver type to offer an arriving customer. Each component π j tracks the sequence of driver types offered to type j customers, i.e., when π j (k) = i, then the system controller attempts to match the kth arriving type j customer with a type i driver, for j ∈ J , k ∈ N + , i ∈ N ∪ {0}. The notation π j (k) = 0 implies no driver is offered to the customer, in which case the customer is lost. The customer accepts the offered driver if the waiting time required for pick-up is not too large (in a sense specified precisely below), and otherwise the customer is lost and the offered driver stays in his current area. The implication is that customers are classified as matched or unmatched (i.e., lost) at the time of their arrival, even though a matched customer must still wait to be picked up by a driver. The process D π ij (t) tracks the cumulative number of type i drivers matched to type j customers under policy π in [0, t].
The number of drivers in area i at time t ∈ R + depends on the matching policy π. Then, for A i and R i unit rate Poisson processes, the number of unmatched type i drivers at time t is: where {Q i (0), i ∈ N } are random variables independent of all other stochastic primitives. The second term in the right-hand side of (1) represents the cumulative number of driver arrivals to area i in [0, t], while the third and fourth terms represent the cumulative number of driver departures from area i in [0, t] by unmatched and matched drivers, respectively. We would like to match as many customers with drivers as possible. This is straightforward if there are many matching policies under which Q π i (t) > 0 for all i ∈ N and t ∈ R + , because then there is always a driver near to an arriving customer. The difficulty is that in general not every area will have an available driver -and which areas have available drivers depends on earlier matching decisions. In this case, the arriving customer is matched only if the time required for the driver to pick-up the customer is less than the amount of time that customer is willing to wait for a driver.
The time a customer is willing to wait for a driver can depend on the time of day. For example, during working hours, a customer may be more timeconstrained than during non-working hours. We represent this using a step (piecewise constant) function that depends on the customer arrival time. First, we partition the time horizon into countably many disjoint intervals by defining the deterministic sequence {τ m , m ∈ N} such that τ m ∈ R + and τ m < τ m+1 for all m ∈ N, and τ m → ∞ as m → ∞. Second, to allow for potentially changing traffic conditions, we define the pick-up time of a type i driver for a type j customer who arrived in the system at time t ∈ [τ m , τ m+1 ) by t ij (m) ∈ R + for all i ∈ N , j ∈ J , and m ∈ N. Third, we denote the time the kth type j customer arrival is willing to wait for pick-up given the arrival occurred at time t ∈ [τ m , τ m+1 ) by the random variable a k j (m) for all k ∈ N + , j ∈ J , m ∈ N. The sequence {a k j (m), k ∈ N + } is i.i.d., and independent of all other stochastic primitives for all m ∈ N and j ∈ J . Then, the probability that the kth type j customer accepts a type i driver, given the arrival time is t, is for all i ∈ N , j ∈ J , k ∈ N + , and t ∈ R + . The closest driver (CD) policy, denoted by π CD , offers a type j customer that arrives at time t ∈ R + a driver type from the set If the set in (3) is not a singleton, the offered driver is chosen randomly from the closest drivers. The question is: Does the CD policy match as many customers as possible?
The total cumulative number of matchings under any policy π depends on the number of customers willing to wait for their offered driver. Specifically, if E j is a unit rate Poisson process and ν j (k) := inf{t ∈ R + : E j (Γ j (t)) = k} is the arrival time of the kth type j ∈ J customer, k ∈ N + , then (4) for all i ∈ N , j ∈ J , t ∈ R + , and under any policy π. Our objective is to find a policy π that maximizes the total cumulative number of matchings made in a finite time horizon over a specified class of admissible matching policies Π; that is, to solve (5) max π∈Π i∈N ,j∈J for all ω ∈ Ω. In studying this problem, we obtain some results on the more general objective that also allows the system manager to provide weights {w ij , i ∈ N , j ∈ J }, on the possible matchings. The objectives (5) and (6) are very strong objectives because solving either requires specifying a policy that maximizes the number of matchings on every sample path.
Remark 1. In our model formulation, the driver types are formed only based on the arrival locations of the drivers. However, we can extend our results to an arbitrary (but finite) number of driver types I ∈ N + by updating a k j (m) to a k ij (m) for all k ∈ N + , j ∈ J , m ∈ N, and i ∈ I := {1, 2, . . . , I}. Then, a k ij (m) denotes the patience time of the kth type j customer for a type i driver given that she arrived in the system on the time interval [τ m , τ m+1 ), for all k ∈ N + , j ∈ J , m ∈ N, and i ∈ I. All of our results hold under this extension.
Remark 2. In reality, customer and driver arrival rates and driver departure rates in an area can depend on the decisions of the system controller, i.e., the customer and driver behaviors are endogenous. For simplicity, we assume exogenous customer and driver behaviors. There is some support for such an assumption in Zhong, Wan and Shen (2019), who use data from the ridesharing company Didi to show that a Markovian queueing model with exogenous customer and driver behaviors provides good estimates for system performance measures during rush hours in China.
2.1. Admissible Policies Roughly speaking, an admissible matching policy cannot anticipate the future. This is formalized mathematically by defining the filtration F := {F(t), t ∈ R + } such that The information in (7) includes past observations on the amount of time customers have been willing to wait for pick-up (the a k j (m)'s), which is Electronic copy available at: https://ssrn.com/abstract=2844451 generally not available in practice. In that case, any proposed policy should not rely on those observation (and our proposed policies in the next Section do not). The reason we include such information in the filtration is to make our asymptotic optimality proof stronger, because the upper bound result we prove later (in Theorem 1) is with respect to a larger policy class. However, the information in (7) cannot include how long a customer arriving in the system at time t will wait for driver pick-up, because that would allow the ability to cherry-pick certain customers to offer faraway drivers without consequence. The technical implication is that the filtration F is not right continuous.
The information available to an admissible policy at the arrival epoch of the kth type j customer is for all k ∈ N + and j ∈ J . Since each of the stochastic processes that generate the σ-field in (7) is either right or left continuous and ν j (k) is a stopping time with respect to the filtration F for all j ∈ J and k ∈ N + , the σ-field in (8) is well defined. Since ν j (k) ≤ ν j (k + 1) for all k ∈ N + , F j := F j (k), k ∈ N + is a filtration for all j ∈ J .
Definition 1. (Admissible Policies) For all j ∈ J , let Π j denote the set of discrete-time stochastic processes with domain N + ×Ω and range N ∪{0}, such that for all π j ∈ Π j , π j is F j -adapted (i.e., π j (k) ∈ F j (k) for all k ∈ N + ) and if Q i (ν j (k)−) = 0 for some i ∈ N , then π j (k) = i. Let Π be the set of J-dimensional discrete-time stochastic processes such that for all π ∈ Π, we have π = (π 1 , π 2 , . . . , π J ) where π j ∈ Π j for all j ∈ J . Then, Π is the set of admissible policies.
Lemma 1. The CD policy is admissible; that is, π CD ∈ Π.
The proof of Lemma 1 is presented in Appendix E.1.
Electronic copy available at: https://ssrn.com/abstract=2844451 Remark 3. Although the system controller cannot anticipate the arrival times of the customers and drivers, how long customers will wait for driver pick-up, or how long drivers will remain in their current area, he can accurately forecast the arrival rates of the customer and driver types and the driver acceptance probabilities associated with the customer types. In other words, he knows the functions Λ i , Γ j , andF ij for all i ∈ N and j ∈ J .
Remark 4. The filtration F can be augmented to include additional stochastic processes Υ m : R + × Ω → R with right or left continuous sample paths for all m ∈ N, provided that the sequence {Υ m , m ∈ N} does not contain any future information related to the processes that generate F(t) for all t ∈ R + . For example, if the system controller randomly chooses a driver to offer an arriving customer, then {Υ m , m ∈ N} includes a sequence of i.i.d. random variables that reflect the outcome of an N -sided die roll.
Remark 5. Since the arrival process of each customer type is a nonhomogeneous Poisson process, the probability that more than one customer arrives in the system simultaneously at some time epoch is zero. Therefore, the range of all π j ∈ Π j is chosen as N ∪{0} instead of the m-fold Cartesian product of N ∪ {0} for some m ≥ 2 for all j ∈ J .
Remark 6. Let F(ν j (k)) denote the sigma algebra defined by the stopping time ν j (k) as in Definition 1.2.12 of Karatzas and Shreve (1988). Another way to define Π j is such that for all π j ∈ Π j , π j (k) ∈ F(ν j (k)) for all k ∈ N + . We do not choose this option because proving a k j (m) ⊥ F(ν j (k)) for all j ∈ J , m ∈ N, and k ∈ N + is difficult 1 , but a k j (m) ⊥ F j (k) is by construction (cf. (8)). This result is exactly what prevents the system controller knowing how long an arriving customer is willing to wait for driver pick-up, and so is crucial to our model and analysis.
3. An Asymptotically Optimal CLP-Based Matching Policy It is very difficult to solve the optimization problem (6) exactly. Even if we can accomplish this very challenging task, the optimal matching policy will most likely be sample path dependent and will be very complicated. Hence, we consider a large market where the arrival rates of the customers and drivers grow without bound, and solve (6) under fluid scaling in that limiting regime. Section 3.1 specifies the large market limiting regime and the fluid scaling.
Section 3.2 establishes that the solution to a CLP serves as an asymptotic upper bound on the objective (6) under fluid scaling. Section 3.3 provides a simple policy that can asymptotically mimic the performance of any feasible matching process for the CLP, and, therefore, can be used to specify an asymptotically optimal policy when the CLP is solvable.

A Large Matching Market
We consider a sequence of matching systems indexed by n, n ∈ N + . Each matching system has the same primitives with the one introduced in Section 2 except that the arrival rates of the drivers and customers, and departure rates of unmatched drivers from their current areas depend on n. In the nth matching system, for all i ∈ N , j ∈ J , and t ∈ R + , let where λ n i : R + → R + and µ n j : R + → R + are Lebesgue measurable rate functions for all i ∈ N , j ∈ J , and n ∈ N + . Moreover, θ n i ∈ D and θ n i ≥ 0 for all i ∈ N and n ∈ N + . A policy π = {π n , n ∈ N + } is a sequence that specifies a policy for each n, and π is admissible if π n is admissible for all n ∈ N + . For a policy such as CD that does not change with n, in a slight abuse of notation, we specify π (i.e., π = π CD ) and assume π n = π for all n ∈ N + . Our notational convention is to denote a process (or random variable) X in the nth system under the admissible policy π by X π,n .
Increasing the arrival rates without a bound and keeping the departure rates of unmatched drivers from their areas bounded results in a crowded matching system where we can use law of large numbers type of results to obtain tractable approximations for the processes of interest.
Assumption 1. (Large Market) We assume that Λ n i /n → Λ i , µ n j /n → µ j , and θ n i → θ i u.o.c. as n → ∞ for all i ∈ N and j ∈ J such that Λ i = t 0 λ i (t)dt for all t ∈ R + , the functions λ i : R + → R + and µ j : R + → R + are Lebesgue measurable, and the function θ i is defined such that θ i ∈ D and θ i ≥ 0 for all i ∈ N and j ∈ J . We also assume that sup t∈R + λ i (t) < ∞, sup t∈R + θ i (t) < ∞, and sup t∈R + µ j (t) < ∞ for all i ∈ N and j ∈ J . The unit rate Poisson processes A i , R i , and E j are mutually independent for all i ∈ N and j ∈ J .
The reader will have noticed that Assumption 1 re-uses the notation Λ i , λ i , µ j , and θ i for all i ∈ N and j ∈ J . In Section 2, the notation refers to parameters associated with a particular pre-limit matching system having index n. In this section and the next, the parameters without the superscript n refer to the limiting quantities in Assumption 1.
Assumption 1 does not necessarily imply that the arrival rates of all customer and driver types grow to infinity for all t ∈ R + as n → ∞. In particular, arrival rates can be zero in some areas during some time periods (e.g., λ n i (t) = µ n j (t) = λ i (t) = µ j (t) = 0 for some i ∈ N , j ∈ J , and t ∈ R + and for all n ∈ N + ), as may be true in parts of the city during some time intervals.
We focus on the first-order imbalances between the driver supply and customer demand by considering fluid scaling. For all i ∈ N , j ∈ J , t ∈ R + , n ∈ N + , and admissible policy π, definē Λ n i := Λ n i /n,Γ n j := Γ n j /n,Q π,n i := Q π,n i /n,D π,n ij := D π,n ij /n.
By (1), (10a), and (10b), for all i ∈ N , t ∈ R + , n ∈ N + , and admissible policy π, We make the following assumption about the initial number of drivers.
Assumptions 1 and 2 are in force throughout the paper.

An Asymptotic CLP Upper Bound
In this section, we derive an asymptotic upper bound on the fluid scaled objective (6) by solving a CLP. The decision variables are {q, x} := {q i , x ij , i ∈ N , j ∈ J } such that q i , x ij : [0, T ] → R + , q i (t) denotes the number of type i drivers at time t, and x ij (t) denotes the fraction of type j customers who are offered type i drivers at time t, so that µ j (t)F ij (t)x ij (t) approximates the instantaneous matching rate between type i drivers and type j customers for all i ∈ N and j ∈ J . The relevant CLP is: Electronic copy available at: https://ssrn.com/abstract=2844451 q i and x ij are Lebesgue measurable for all i ∈ N and j ∈ J . (12e) Constraint (12b) is the queue length equation (cf. (11)). Constraint (12c) implies that a customer cannot be offered more than one driver. Any q i feasible for CLP (12) is Lipschitz continuous by (12b) and Assumption 1.
The proof of Lemma 2 is presented in Appendix E.3. We denote an optimal solution of the CLP (12) by the process {q,x} : The following theorem establishes that the optimal objective function value of the CLP (12) is an asymptotic upper bound on the fluid scaled objective (6) under any admissible policy for almost all sample paths. The proof of Theorem 1 is presented in Appendix A. The key challenge is to show that there exists a feasible matching process {x ij (t), i ∈ N , j ∈ J , t ∈ [0, T ]} (not necessarily optimal) such that the derivative of the limit of the fluid scaled cumulative matching process {D π,n ij (t), i ∈ N , j ∈ J , t ∈ [0, T ]} corresponds to the process {µ j (t)F ij (t)x ij (t), i ∈ N , j ∈ J , t ∈ [0, T ]}. If the aforementioned feasible matching process isx, then the upper bound in Theorem 1 is attained, meaning the policy is asymptotically optimal.

A CLP-Based Randomized Policy
We would like to find an admissible matching policy that can reproduce any feasible queue length process and matching process {q, x} for the CLP (12) (including an optimal one, {q,x}). This is important because finding a feasible queue length and matching process pair that improves on any myopic matching policy (such as CD) may be possible even when finding an optimal CLP solution is not. The question is: How do we translate between a feasible process pair {q, x} and the decision of which driver to offer an arriving customer?
Definition 2. (Randomized Policy) If a type j customer arrives in the system at time t, the system controller makes a random selection from the set N ∪ {0} such that the probability that the outcome is i is x ij (t) for all i ∈ N and the probability that the outcome is 0 is 1 − i∈N x ij (t). If the outcome is i for some i ∈ N and there is a type i driver in the system, then the system controller offers a type i driver to the customer. If the outcome is i for some i ∈ N but there is no type i driver in the system or if the outcome is 0, then no driver is offered to the customer (and so the customer is lost).
Because the randomized policy does not use the queue length process q and by Lemma 2 Part 1, we denote it by π R and write π R (x) when we want to emphasize its dependence on the particular feasible matching process x. Under a technical condition on the associated feasible matching process for the CLP (12), π R is admissible.
Lemma 3. Suppose that the feasible matching process for the CLP (12), x, is such that x ij is a Borel measurable simple function for all i ∈ N and j ∈ J . Then, π R (x) is admissible.
The proof of Lemma 3 is presented in Appendix E.2. The randomized policy is a simple policy, because no state information is needed for its implementation. Still, its performance replicates any feasible process pair for the CLP (12).
Theorem 2. Let {q, x} be a feasible process pair for the CLP (12) such that x satisfies the condition in Lemma 3. For all i ∈ N , as n → ∞, The proof of Theorem 2 is presented in Appendix D. Theorem 2 shows that when an optimal matching processx of the CLP (12) is a Borel measurable simple function, then π R (x) attains the upper bound in Theorem 1. However,x may not satisfy the aforementioned condition. Then, we do not know the associated randomized policy is admissible, Electronic copy available at: https://ssrn.com/abstract=2844451 and so we cannot use Theorem 2 to ensure the upper bound in Theorem 1 is achieved. However, we can approximate an optimal matching process of CLP (12) with a sequence of Borel measurable simple functions to ensure the upper bound is nearly achieved. Specifically, by Theorem 2.10 and Proposition 2.12 of Folland (1999), there exists a sequence {x r , r ∈ N + } such that x r := {x r ij (t), i ∈ N , j ∈ J , t ∈ [0, T ]}, for all i ∈ N , j ∈ J , and r ∈ N + ,x r ij is a Borel measurable simple function, 0 ≤x r ij (t) ≤x r+1 ij (t) ≤x ij (t) for all t ∈ [0, T ] except on a set of zero measure, andx r ij (t) →x ij (t) as r → ∞ for all t ∈ [0, T ] except on a set of zero measure. By Lemma 2 Part 1 and Lemma 12 Part 3, for eachx r , there exists a uniqueq r such that {q r ,x r } is a feasible process pair for the CLP (12). Then, by Theorem 2 and the bounded convergence theorem used on the sequence {µ j (t)F ij (t)x r ij (t), t ∈ [0, T ], r ∈ N + }, we have the following corollary.
Corollary 1. For any > 0, there exists an r 0 ( ) ∈ N + such that if r ≥ r 0 ( ), Solving the CLP (12) or accurately approximating an optimal solution of it is a very challenging task (cf. Perold (1981)). If µ j (t)F ij (t) was a constant function of t and θ i = 0 for all i ∈ N and j ∈ J , then the CLP (12) would become a Separated Continuous Linear Program (SCLP, cf. Anderson, Nash and Perold (1983)). Although an SCLP is solvable (cf. Anderson, Nash and Perold (1983) and Weiss (2008)), it is an NP-hard problem (cf. Bertsimas, Nasrabadi and Paschalidis (2015)). Hence, we will derive conditions under which the CLP (12) can be simplified into an LP in the following section.
4. An Asymptotically Optimal LP-Based Randomized Policy LP-based matching policies arise when the system controller does not need to "save" drivers for future customers, i.e., when the system controller can be myopic. Section 4.1 shows a myopic LP-based randomized policy is asymptotically optimal when the drivers are always busy, or fully utilized. A myopic LP-based randomized policy is also asymptotically optimal when parameters are time homogeneous, regardless of whether or not drivers are fully utilized, and that LP can be modified to include pricing. We show in Section 4.2 that when pricing affects driver supply and customer demand, jointly optimizing over pricing and matching decisions results in fully utilized drivers, which provides a partial justification of the aforementioned "fully utilized driver" condition.

An LP-Based Randomized Matching Policy
The following LP is relevant at each fixed t ∈ [0, T ]: where the decision variables are {x ij (t), i ∈ N , j ∈ J }. The main difference between the LP (13) and the CLP (12) is that (13a) is the derivative of (12a) and (13b) is a capacity constraint instead of a queue length equation (meaning we do not need q(t) as a decision variable). By Assumption 1 and the fact that x ij (t) = 0 for all i ∈ N and j ∈ J is feasible for the LP (13) for all t ∈ [0, T ], there exists an optimal solution of the LP (13) for all t ∈ [0, T ]. We denote an optimal solution of the LP (13) If each of λ i and µ j is a step function for all i ∈ N and j ∈ J , then there exists a function which satisfies Assumption 3, and that function can be chosen as a step function. Assumption 3 is valid in Section 4.1.
The CLP (12) can be solved using the LP (13) when the initial driver mass is zero and drivers are fully utilized.
for some matching process x that is feasible for the CLP (12) 2 . If w ij = 1 for all i ∈ N and j ∈ J , then there exists a process q * such that the pair {q * , x * } is an optimal solution of the CLP (12).
The proof of Lemma 4 is presented in Appendix E.4. The following asymptotic LP-based upper bound on the fluid scaled objective (5) follows from Lemma 4 combined with Theorem 1.
Corollary 2. If the conditions (14) and (15) hold, under any admissible policy π, we have lim sup In order to obtain the upper bound in Corollary 2, the LP (13) must be solved infinitely many times (for all t ∈ [0, T ]). However, in practice, the LP (13) can be solved at finitely many time epochs, and the remaining x * ij (t) values can be approximated by, for example, linear interpolation or assuming that x * ij is a step function. Moreover, the time-varying parameters of the LP (13) can be estimated in real-time, because the matching decisions do not depend on future estimated arrival rates of the customers and drivers.
If the initial driver mass is zero and drivers are fully utilized, there is no need to be forward looking. Consequently, the myopic LP-based randomized policy π R (x * ) is asymptotically optimal.
Theorem 3. Suppose x * satisfies the condition in Lemma 3, so π R (x * ) is admissible. If conditions (14) and (15) hold and w ij = 1 for all i ∈ N and j ∈ J , then π R (x * ) achieves the asymptotic upper bound given in Corollary 2, i.e., The proof of Theorem 3 is presented in Appendix D.
4.2. Jointly Optimizing Pricing and Matching when Parameters are Time-Homogeneous One natural intuition is that the required balance in (15) arises naturally when the system controller can use pricing to influence customer and driver behavior. This is because a "smart" system controller will not raise prices beyond what is needed to match driver supply with customer demand. This leads to a joint pricing and matching problem. However, any time-varying problem formulation is very difficult to solve. Therefore, we focus on a static formulation to show how (15) can be viewed as a consequence on good pricing decisions.
We begin with the observation that we do not need the condition (15) to show that the randomized policy based on the solution to the LP (13) is asymptotically optimal when parameters are time homogeneous.
Theorem 4. If λ i , µ j , andF ij are constant functions of t for all i ∈ N and j ∈ J , we can choose x * as a constant function of t. Suppose that condition (14) holds. Then, there exists a process q * such that the pair {q * , x * } is an optimal solution of the CLP (12) and π R (x * ) is admissible. Moreover, if w ij = 1 for all i ∈ N and j ∈ J , then π R (x * ) achieves the asymptotic upper bound given in Theorem 1, i.e., The proof of Theorem 4 is presented in Appendix D.
The question we address is: When the LP (13) formulation does not vary with time, and is expanded to include pricing, does an optimal solution satisfy (15)? To do this, suppose that, for all i ∈ N , j ∈ J and n ∈ N + , λ n i and µ n j are constant functions that depend on the prices determined by the system controller at time 0,F ij is a time-homogeneous function independent of the prices, and θ n i can depend on both the prices and the time. Specifically, first, the system controller sets prices p = {p ij , i ∈ N , j ∈ J } at time 0. The prices can depend on both the customer's type j ∈ J (and so can be based on the customer's origin and destination area) and the area i ∈ N where the assigned driver is currently located. Second, the time-homogeneous arrival rates of the customers and drivers (λ n i and µ n j ) and time-varying departure rates of unmatched drivers from their current areas (θ n i ) are realized, and matchings are performed continuously over the time horizon [0, T ]. We extend Assumption 1 in the following way: , ∀i ∈ N , j ∈ J } be the set of possible price vectors wherep ∈ R + is the maximum chargeable price. For all i ∈ N , j ∈ J , n ∈ N + , p ∈ P, we have λ i , µ j , λ n i , µ n j : P → R + , the functions θ i , θ n i : The system controller sets prices at time 0 by solving the following optimization problem: Electronic copy available at: https://ssrn.com/abstract=2844451 where the decision variables are x ij and p ij for all i ∈ N and j ∈ J . The optimization problem (16) is the LP (13) which is time homogeneous but modified to include pricing decisions. Since x ij = 0 and p ij ∈ [0,p] for all i ∈ N and j ∈ J is feasible for (16), the feasible region is nonempty.
We cannot know whether or not there exists an optimal solution without additional assumptions on the functions λ i and µ j , i ∈ N , j ∈ J . For example, if λ i and µ j are continuous functions of p for all i ∈ N and j ∈ J , then the feasible region is compact and so an optimal solution of (16) exists. If the price(s) in an area is equal to 0, then one can expect the total customer arrival rate to be very large, but the driver arrival rate to be 0 and increasing in price. Thus, an optimal solution should have nonzero prices.
We associate the set of admissible policies given in Definition 1 with a price vector p ∈ P, which determines the rates of the processes generating the filtration F = {F(t), t ∈ R + } representing the available information as time moves forward (cf. (7)). Specifically, for a given p ∈ P, a policy π(p) = {π n (p), n ∈ N + } is a sequence that specifies a policy for each n, and π(p) is admissible if π n (p) is admissible for all n ∈ N + . Let {x, p} be feasible matching fractions and prices for (16) and π R,p (x) denote the randomized policy associated with the matching fractions x (cf. Definition 2) under the price vector p, i.e., the system controller sets the price vector p at time 0 and then makes the matching decisions using the randomized policy with matching fractions x. Then, we have the following corollary to Theorems 1 and 4.
Corollary 3. Suppose that condition (14) holds and {x * , p * } is an optimal solution of (16). Then, under any admissible policy π(p), p ∈ P, Moreover, if w ij = 1 for all i ∈ N and j ∈ J , then π R,p * (x * ) is asymptotically optimal, i.e., If the constraint (16b) is binding for all driver types under given feasible matching fractions and prices, then condition (15) holds. Hence, we would like to know whether the constraint (16b) binds under "good" pricing decisions, e.g., under an optimal solution of (16) (if exists). The intuition that a "good" pricing policy is one under which the constraint (16b) binds is natural when lowering prices results in less driver arrivals but more customer arrivals. Hence, there is no need to have idle drivers in the system at any time. That intuition is consistent with a result proved in Bimpikis, Candogan and Saban (2016) (see Proposition 1 and the second and third paragraphs on page 20 therein) and results proved inÖzkan (2018) (see Propositions 1 and 3 therein) showing that drivers never idle. We provide sufficient conditions in Example 1 below under which the constraint (16b) binds under an optimal solution of (16).
Example 1. There exists a customer specific baseline price for riding, denoted by c j > 0, and an area specific surge multipliers s i ∈ [0,s] for i ∈ N , wheres ∈ R + is the maximum possible surge multiplier. Suppose that c j is constant but the system controller determines the surge multipliers at time 0. Let α : J → N be a function which specifies the arrival location of each customer type. Then, the price that a type j customer needs to pay for a ride provided by a type i driver is p ij = c j s α(j) for all i ∈ N and j ∈ J . Suppose that there exists C 2 ≥ C 1 > 0 such that Condition (17a) states that as the surge multiplier in an area decreases, the total customer arrival rate in that area increases and the driver arrival rate in that area decreases. Condition (17b) implies that the total customer arrival rate in an area is not affected by surge multipliers in other areas. Condition (17c) roughly requires that the change in the surge multiplier in an area affects that area more than it affects the other areas. Condition (17d) enforces that drivers do not work for free.
Lemma 5. Under the conditions (17a)-(17d), there exists an optimal solution of (16) in which the constraint (16b) is binding for all driver types.
The proof of Lemma 5 is presented in Appendix E.5.

Performance Evaluation
We begin by observing that the performance of the randomized policy can likely be improved by incorporating state information. This is because the randomized policy can match a customer with an area in which there are no drivers, leading to that customer being lost. To correct this, in Section 5.1, we introduce state-dependent LPand CLP-based policies, that require knowledge of driver locations. Then, in Section 5.2, we compare the performance of those policies and the LPand CLP-based randomized policy against the benchmark CD policy. We do this first when parameters are time homogeneous and second when they vary with time. In the first case, the LP-and CLP-based policies coincide (cf. Theorem 4), whereas in the second, when the importance of considering future customer and driver arrival rates becomes important, they do not.

Additional Proposed Matching Policies
We drop the superscript n in this section to be clear that any matching policy we propose must be welldefined and admissible for the model given in Section 2. We associate the proposed matching policies with x, which is a feasible matching process for the CLP (12) for an appropriately defined q (cf. Lemma 2 Part 1). We begin by modifying the randomized policy to incorporate information regarding which areas have no drivers.
Randomized Weighted Queue (RWQ) Policy: If a type j customer arrives in the system at time t, the system controller offers a type i driver with probability , if the denominator in (18) is strictly positive. Otherwise, the system controller does not offer any driver to the customer. Under the the randomized and the RWQ policies, if x ij (t) = 0 for some i ∈ N and j ∈ J , then the system controller will never offer a type i driver to a type j customer, even if the only drivers in the system are type i. In comparison, the benchmark CD policy in (3) is match conserving in the sense that a customer arriving in a system with at least one driver available is always offered a driver, regardless of the driver types present. This motivates us to introduce two additional policies that are match conserving, one deterministic and one not.
Deterministic Policy: If a type j ∈ J customer arrives in the system at time t, the system controller offers a driver type from the set If the set in (19) is not a singleton, then the system controller can use any tie breaking rule which does not use any future information. If there is no driver in the system, then no driver is offered.
Hybrid Policy: Suppose that a type j customer arrives in the system, and there is a driver in the system. Then the system controller makes a random selection in the same way explained under the randomized policy (cf. Definition 2). If the outcome is i ∈ N and there is a type i driver in the system, the system controller offers a type i driver to the customer. If the outcome is i ∈ N but there is no type i driver in the system or if the outcome is 0, the customer is offered the driver type specified in (3) by the CD policy. If there is no driver in the system, then the customer is lost.
Remark 7. If x ij = 0 for all i ∈ N and j ∈ J such that i = α(j), then the Hybrid policy is the CD policy defined in (3).
Lemma 6. The RWQ, deterministic, and the hybrid policies are admissible under the condition on x in Lemma 3.
The proof of Lemma 6 is presented in Appendix E.2. The RWQ, deterministic, and the hybrid policies can be implemented regardless of whether or not they are admissible. However, in order to prove asymptotic optimality results parallel to those for the randomized policy (specifically, Theorems 2, 3, and 4), admissibility is required. We conjecture that the statements of Theorems 2, 3, and 4 hold under the RWQ, deterministic, and hybrid policies. However, the proofs are more difficult due to the state dependence of the aforementioned policies.

Simulation Experiments
In this section, we present simulation experiments where we test the performances of the CD policy and the LPand CLP-based matching policies. We associate the LP-and CLP-based policies based on an optimal solution of the LP (13) and a solution of the CLP (12) with at most 1.02% optimality gap (that we find by solving CLP (12) when θ i = 0 for all i ∈ N ), respectively. We present an experiment with time-homogeneous parameters in Section 5.2.1 and an experiment with time-varying parameters in Section 5.2.2.
Electronic copy available at: https://ssrn.com/abstract=2844451 The term "offered driver" refers to the driver offered to an arriving customer, that is determined by the matching policy. The customer may or may not accept being matched with the driver offered at the arrival time t ∈ [0, T ], depending on the acceptance probabilityF ij (t). Any variation of the word "match" means that the relevant customer has both been offered a driver, and has accepted that driver, meaning that driver picks up that customer.
We assume that the larger theF ij (t) at time t ∈ [0, T ], the smaller the pick-up time of a type i driver for a type j customer. Then, under the CD policy, the offered driver set for a type j customer arriving at time t in (3) is exactly argmax {i∈N : Q i (t−)>0}Fij (t) for all j ∈ J and t ∈ R + (and the match occurs if also the resulting pick-up time is lower than the time that customer is willing to wait for pick-up).
We choose N = J = 3 in both simulation experiments (so the customer type represents the customer's arrival location). This is the smallest network size possible that allows for us to illustrate the more general insight behind when and why LP-based policies outperform CD. The reason a two area network will not work is that as long asF ii (t) ≥F ij (t) for all i, j ∈ {1, 2, 3} and t ∈ [0, T ], the LP-based policies give more priority to within-area matchings than they give to between-area matchings, which coincides with the CD policy. In contrast, the reason CLP-based policies outperform CD (as well as any other myopic policy such as an LP-based one) has to do with their potential to be forward-looking (which can be seen in a two-area network).
Implementation Details. We used Omnet++ discrete-event simulation freeware. In each experiment, we started with an empty system, i.e., Q i (0) = 0 for all i ∈ N . At each simulation instance associated with each matching policy, we have done Rep number of replications. The maximum of the margin of errors associated with the 95% confidence interval for the percentage of all customers matched (= t 0.025,Rep−1 × Sample Stdv./ √ Rep) among all policies in all instances is less than 0.42. Fig 3: The parameters of the first simulation experiment are: Q i (0) = 0, 1/θ i = 10 time units,F ii = 1, w ij = 1 for all i, j ∈ {1, 2, 3}, and n ∈ {1, 10, 100}.

Time-Homogeneous Parameters
We present a simulation experiment which shows that our LP-based policies can potentially match more customers and drivers in a finite time horizon than the CD policy does. Figure 3 provides all input parameters. Since all parameters are time homogeneous, the LP-and CLP-based policies coincide (cf. Theorem 4). We omit "(t)" from the notation. Moreover, n ∈ {1, 10, 100} measures the market size determined by the number of arrivals per time unit, as in Assumption 1.
The intuition for why we do not expect the CD policy to perform well in this example is as follows. Under the CD policy, when a type 1 or 2 customer arrives in the system, the offered driver is type 2 (if there is one in the system) because λ 1 = 0,F 21 >F 31 , andF 22 >F 32 . Since µ 1 + µ 2 = 2n > λ 2 = n, some of the type 1 or 2 customers cannot be offered type 2 drivers and must be offered type 3 drivers. The type 2 customers who are not offered type 2 drivers are offered type 3 drivers and so 98% of those customers are matched. However, the type 1 customers offered type 3 drivers are lost becauseF 31 = 0. In comparison, the optimal solution of the LP (13) has x * 21 = 1, x * 22 = 0.01, and x * 32 = 0.99. In words, the LP-based policies match type 1 customers with type 2 drivers and type 2 customers with type 3 drivers, to prevent losing a non-trivial percentage of the type 1 customers.    (5), since w ij = 1 for all i, j ∈ {1, 2, 3}) under the CD policy and the LP-based policies. "LP Ub." shows one hundred times the optimal objective function value of the LP (13) divided by the total customer arrival rate to the system, which is an approximate upper bound on the percentage of all customers matched under any admissible policy by Theorems 1 and 4. The key observations are that (i) the LP-based policies outperform the CD policy in all traffic intensi-Electronic copy available at: https://ssrn.com/abstract=2844451 ties; (ii) the LP-based policies are very close to the approximate upper bound based on Theorem 1 as the arrival rates become large, which verifies Theorem 4 numerically.
, -"+ # , -+" # *  Figure 2 in Section 1, our next simulation experiment is designed to show that CLPbased policies can significantly outperform LP-based policies and the CDpolicy by taking into account the future customer and driver arrival rates. We assume that the customer arrival rate in one area will spike; in particular, we assume µ 1 (t) = 0 for all t ∈ [0, T /2] and µ 1 (t) = 2n for all t ∈ [T /2, T ], where n ∈ {0.1, 1, 10, 100} is used to determine the customer and driver arrival rates to the system. Figure 5 provides the other input parameters, which are time homogeneous, and so "(t)" is omitted. The average driver patience time is 1/θ ∈ {1.667, 16.67, 166.7, 1667} min, and so noting that T = 30 min, 1/θ = 1.667 min represents impatient drivers and 1/θ = 1667 min represents very patient drivers.

Time-Varying Parameters Motivated by the example in
Since the total customer and driver arrival rate in the first half of the simulation (t ∈ [0, T /2]) are n and 2n, respectively, at least half of the arriving drivers will be idle in the first half of the simulation. This implies that the condition (15) requiring drivers to be fully utilized does not hold in this experiment.
In the first half of the simulation, i.e., in the time interval [0, T /2], the CD policy offers type 1 drivers to type 2 customers. In [T /2, T ], the CD policy offers type 1 drivers to both type 1 and type 2 customers. However, in [T /2, T ], arrival rate of the type 1 drivers is n but the arrival rate of the customers who will accept a match with type 1 drivers if offered is 2n + 0.99n = 2.99n. Therefore, only 1/2.99 of the type 1 customers are offered and matched with type 1 drivers and the rest will be lost. Table 1 The percentage of all customers matched in the simulation experiment corresponding to Figure 5, for T = 30 min. The optimal solution of the LP (13) is such that x * 12 (t) = 1 for all t ∈ [0, T /2] and x * 11 (t) = 0.5 and x * 32 (t) = 1 for all t ∈ [T /2, T ], and all other decision variables at all other times are 0. Hence, in [0, T /2], type 2 customers are offered type 1 drivers; and in [T /2, T ], type 2 customers are offered type 3 drivers so that type 1 customers can be offered type 1 drivers. Therefore, unlike the CD policy, the LP-based policy considers the fact that type 1 customers accept only type 1 drivers. However, in [T /2, T ], the arrival rate of type 1 customers is 2n but the arrival rate of the type 1 drivers is n, which implies that half of the type 1 customers will be lost.
When θ i = 0 for all i ∈ N , an optimal matching process for the CLP (12) has: x 32 (t) = 1 for all t ∈ [0, T ], x 11 (t) = 0 for all t ∈ [0, T /2], x 11 (t) = 1 for all t ∈ [T /2, T ], all other decision variables are 0. (To see that process is optimal, notice that as many drivers as possible are matched.) We consider the CLP-based matching policy which always offers type 3 drivers to type 2 customers and offers type 1 drivers to type 1 customers as much as possible. This solution is forward-looking because the type 1 drivers arriving before time T /2 are "saved" to be offered to the type 1 customers arriving after time T /2. Furthermore, for any given θ ∈ R + , the optimality gap under this policy is at most 1.02%. To see this, under this policy, type 1 customers are matched as much as possible, the number of matched type 2 customers is 0.98nT and bounded above by 0.99nT . Therefore, the loss compared to an optimal CLP solution is at most (0.99 − 0.98)/0.98 = 1.02%. Table 1 shows the percentage of all customers that are matched with drivers in the simulation experiment (i.e., the objective (5), since w ij = 1 for all i, j ∈ {1, 2, 3}) under the CD policy, the LP-based policies, and the CLP-based policies. Table 1 confirms that the CLP-based policies outperform the LP-based policies, and the LP-based policies outperform the CD policy, although condition (15) does not hold. Furthermore, the performance of the CLP-based policies improves as θ becomes small, whereas the performance of the CD and LP-based policies is independent of θ. This is because the CLP-based policies are forward-looking but the CD and LP-based policies are not.

Concluding Remarks
The decisions on which driver to offer to each arriving customer in a ridesharing system impact the overall number of customers matched. This is because those decisions determine whether or not future available drivers will be close to the locations of arriving customers. We have formulated an optimization problem whose solution serves as an asymptotic upper bound on the cumulative number of matchings as the market becomes large. That optimization problem accounts for (i) the differing arrival rates of customers and drivers in different areas of the city, (ii) how long customers are willing to wait for driver pick-up, (iii) how long drivers are willing to wait for a customer, and (vi) the time-varying nature of all the aforementioned parameters.
The aforementioned optimization problem is in general a CLP, which can be difficult to solve. We establish that a simple randomized matching policy can asymptotically mimic the performance of any feasible matching process for the CLP, so that there is potential to develop "good" CLPbased matching policies, even when an optimal CLP solution is unknown. When an optimal CLP solution is known, then a CLP-based randomized policy asymptotically achieves the aforementioned optimization problem upper bound.
Under the assumption that drivers are fully utilized or when the CLP parameters are time homogeneous, the CLP solution can be specified through LP solutions. Then, a LP-based randomized policy asymptotically achieves the aforementioned optimization problem upper bound. In the time-homogeneous setting, when customer and driver arrival rates depend on price, we establish an asymptotic upper bound on the cumulative number of matchings by solving an optimization problem that jointly optimizes over prices and matchings, and provide a joint pricing and matching policy that achieves that upper bound. Excellent questions for future research include better understanding the joint pricing and matching problem in time-varying settings and how to incorporate customer and driver behaviors.
We do not track drivers after they drop off customers, or after they choose to relocate from one area to another while waiting to be matched. Tracking drivers is an interesting future research topic. This is clearly doable when the parameters are time homogeneous (cf. Banerjee, Freund and Lykouris (2016), Braverman et al. (2016), Banerjee, Kanoria and Qian (2018)), but is challenging when the parameters are time-varying. First, let us consider time-homogeneous parameters. Then, the CLP (12) simplifies into the LP (13) without requiring the condition (15) (cf. Theorem 4). Let d : J → N be a function denoting the destination area of the customer types. In order to track the drivers after being matched, we can modify the constraint (13b) as follows.
where λ i denotes the external idle driver arrival rate in area i ∈ N (i.e., λ i denotes the rate of drivers becoming online at area i) and the second term in the right-hand side of (20) denotes the rate of drivers who drop customers and become available in area i ∈ N . Notice that (20) is a linear constraint, thus the resulting optimization problem is still an LP. We conjecture that the randomized policy associated with an optimal solution of the modified LP is asymptotically optimal under the objective of maximizing the steady-state total matching rate.
Next, let us consider time-varying parameters. Let f ij denote the deterministic travel time of a driver from area i to j for all i, j ∈ N and recall that α(j) denotes the arrival area of type j customers. Then, the constraint (12b) is modified as follows.
where Λ i (t) denotes the expected cumulative number of external idle driver arrivals in area i ∈ N up to time t ∈ R + and the term in (22) denotes the cumulative number of drivers who drop customers and become available in area i ∈ N up to time t ∈ R + . Notice that solving the CLP under the constraint (21) is very challenging independent of the system being in transient or steady state.
Acknowledgements We would like to thank Siddhartha Banerjee and Kimon Drakopoulos for their valuable comments that helped us to write Section 4.2. We would also like to thank the associate editor and two anonymous referees whose comments significantly improved the paper.

APPENDIX
Section A provides the proof of Theorem 1, which provides an asymptotic upper bound on the fluid-scaled objective (6). That proof relies on the fluid equations, and the details of their derivation are given in Section B. Section C provides a new regulator mapping result which is used in the proofs for asymptotic optimality results in Section D. The proofs of Lemmas 1, 2, 3, 4, 5, and 6 are found in Section E. Finally, we present a relative compactness result in space D, which is used in the derivation of fluid equations, in Section F.

APPENDIX A: PROOF OF THEOREM 1
The proof of Theorem 1 relies on understanding the behavior of fluid limits. In particular, the proof uses the fact that all fluid limits satisfy a set of equations that can be connected to the constraints of the CLP (12). The implication is that any fluid limit gives a feasible matching process for the CLP (12), which implies the optimal objective function value of the CLP (12) is an asymptotic upper bound for the objective (6) under fluid scale.
For all i ∈ N , j ∈ J , t ∈ R + , n ∈ N + , and admissible policy π = {π n , n ∈ N + }, let us define which is the cumulative number of times the system controller offers a type i driver to a type j customer up to time t and letḠ π,n ij := G π,n ij /n be the associated fluid scaled process. Clearly, D π,n ij ≤ G π,n ij for all i ∈ N , j ∈ J , n ∈ N + , and admissible policy π by (4) and (23). For all n ∈ N + and admissible policy π, let where "•" denotes the composition map. Then, X π,n (·, ω) ∈ D 2N J+3N +J for all ω ∈ Ω. LetX π,n := X π,n /n be the fluid scaled version of X π,n . We define fluid limit(s) of {X π,n , n ∈ N + } similar to the definition on page 299 of Dai and Tezcan (2011) in the following way.
Proposition 1. (Fluid Equations) For any admissible policy π = {π n , n ∈ N + }, there exists a full set A π ⊂ Ω (i.e., P(A π ) = 1) such that for any ω ∈ A π , {X π,n (·, ω), n ∈ N + } is relatively compact (i.e., every subsequence has a convergent subsequence) in the Skorokhod space D 2N J+3N +J endowed with the u.o.c. topology. Thus, fluid limits exist in almost all sample paths and any fluid limit and satisfies the following equations for all t ∈ R + : D π ij andḠ π ij are nondecreasing for all i ∈ N and j ∈ J , (25c)D π ij ,Ḡ π ij , andQ π i are Lipschitz continuous for all i ∈ N and j ∈ J , ∀j ∈ J and t 1 , t 2 ∈ R + such that t 2 ≥ t 1 , × I (t ∈ [τ m , τ m+1 )) = 0 for all m ∈ N, i ∈ N , and j ∈ J .
The proof of Proposition 1 is presented in Section B. Let us fix an arbitrary admissible policy π and an arbitrary ω ∈ A π . Then there exists a subsequence {n k , k ∈ N + } such that lim sup n→∞ i∈N ,j∈J w ijD π,n ij (T, ω) = lim k→∞ i∈N ,j∈J w ijD π,n k ij (T, ω).
Proposition 2. Let π = {π n , n ∈ N + } denote an arbitrary admissible policy. Under any fluid limit of {X π,n , n ∈ N + }, we have Next, we will prove Proposition 2. Let us consider the following CLP which has the decision variables {q, z} : q i and z ij are Lebesgue measurable for all i ∈ N and j ∈ J .
By an argument similar to Lemma 2, there exists an optimal solution of the CLP (26) which is denoted by {q * i (t), z * ij (t), i ∈ N , j ∈ J , t ∈ [0, T ]}. We consider the CLP (26) instead of the CLP (12) because of three main reasons. First, the feasible region of the CLP (26) is smaller than the one of the CLP (12) in the sense that there exist a surjective but not bijective mapping from the feasible region of the CLP (12) to the one of the CLP (26). Second, it is easier to connect a fluid limit of {D π,n ij , i ∈ N , j ∈ J , n ∈ N + } with the decision variables of the CLP (26) than the decision variables of the CLP (12). Third, we have the following result.
Lemma 7. The optimal objective function values of the CLP (12) and the CLP (26) are equal.
Proof. Recall that {q i (t),x ij (t), i ∈ N , j ∈ J , t ∈ [0, T ]} is an optimal solution of the CLP (12). Consider the process pair It is easy to see that this process is feasible for the CLP (26). Thus, Second, for all i ∈ N , j ∈ J , and t ∈ [0, T ], let It is easy to see that the process {q * i (t), y ij (t), i ∈ N , j ∈ J , t ∈ [0, T ]} is feasible for the CLP (12), thus i∈N ,j∈J where the equality in (29) is by (26d) and (26e). Hence, the optimal objective function values of the CLP (12) and the CLP (26) are equal to each other by (27) and (29). Let us consider an arbitrary fluid limit of an arbitrary admissible policy π. By (25d) and Theorem 3.35 of Folland (1999), bothD π ij andḠ π ij are differentiable almost everywhere on [0, T ] for all i ∈ N and j ∈ J . By (25c), let d π ij and g π ij be the nonnegative derivatives ofD π ij andḠ π ij , respectively, Electronic copy available at: https://ssrn.com/abstract=2844451 on the points where they are differentiable and without loss of generality be equal to 0 on the points where they are not differentiable for all i ∈ N and j ∈ J on the interval [0, T ]. By (25b), (25d), and the fundamental theorem of calculus for Lebesgue integrals (cf. Theorem 3.35 of Folland (1999)), for all i ∈ N , j ∈ J , and t ∈ [0, T ], Then, {Q π i (t), d π ij (t), i ∈ N , j ∈ J , t ∈ [0, T ]} satisfies (26b) by (25a) and (30); (26f) and (26g) by (25a), (25c), (25d), Theorem 3.35 of Folland (1999), and construction.

APPENDIX B: PROOF OF PROPOSITION 1 (PROPERTIES OF FLUID LIMITS)
Let us fix an arbitrary T 1 ∈ R + . For all k ∈ N + , let D k [0, T 1 ] denote the the space of functions with domain [0, T 1 ] and range R k which are right continuous with left limits. We will first prove Proposition 1 except the property (25f) with respect to processes defined in D 2N J+3N +J [0, T 1 ], then we will extend these results to the processes in D 2N J+3N +J . Lastly, we will prove (25f). By the functional strong law of large numbers, random time-change theorem (cf. Theorems 5.10 and 5.3 of Chen and Yao (2001), respectively), and Assumption 1, we have as n → ∞ for all i ∈ N and j ∈ J . Let o.c., as n → ∞ for all i ∈ N and j ∈ J .
Then, A 1 is a full set (i.e., P (A 1 ) = 1) by (35) and Assumption 2. Let us fix an arbitrary ω ∈ A 1 and an arbitrary admissible policy π = {π n , n ∈ N + }. We omit π and ω from the notation up to Lemma 8 below for notational convenience.
Clearly, for all i ∈ N and j ∈ J , {Ā n i •Λ n i , n ∈ N + } and {Ē n j •Γ n j , n ∈ N + } are relatively compact in the u.o.c. topology by (36). Moreover, Λ i and Γ j are Lipschitz continuous by Assumption 1; and for all i ∈ N , j ∈ J , and t 1 , t 2 ∈ R + such that t 2 ≥ t 1 , we have where (37) is by the fact that the system controller can offer a driver to a customer only at the customer arrival epochs, and (38) is by (4) and (23). Then by Lemma 19 (cf. Section F), {Ḡ n ij , n ∈ N + } and {D n ij , n ∈ N + } are relatively compact in D[0, T 1 ] endowed with the u.o.c. topology for all i ∈ N and j ∈ J such that all of their subsequential limits are Lipschitz continuous. Moreover, sinceḠ n ij andD n ij are nondecreasing andḠ n ij (0) =D n ij (0) = 0 for all i ∈ N , j ∈ J , n ∈ N + , any fluid limits of these processes satisfy (25b), (25c), and (25d). Furthermore, (25e) follows by (36) and (37).
By Assumption 1, letλ i := sup t∈R + λ i (t) < ∞ andμ j := sup t∈R + µ j (t) < ∞ for all i ∈ N and j ∈ J . Moreover, there exists a constantθ ∈ R + and n 0 ∈ N + such that if n ≥ n 0 , θ n i (t) ≤θ for all t ∈ R + and i ∈ N by Assumption 1. Let us fix an arbitrary i ∈ N and consider the process · 0 θ n i (s)Q n i (s)ds. By (36) Hence, there exists an n 1 ∈ N + and a constant C < ∞ such that if n ≥ n 1 ,Q n i (0) +Ā n i •Λ n i (T 1 ) ≤ C. Then for all n ≥ n 0 ∨ n 1 and t 1 , t 2 ∈ [0, T 1 ] such that t 2 ≥ t 1 , and the second inequality in (39) is by (11). Then, · 0 θ n i (s)Q n i (s)ds, n ∈ N + is relatively compact in the u.o.c. topology with Lipschitz continuous subsequential limits by Lemma 19 (cf. Section F). By the fact thatR n i → e u.o.c. (cf. (36)) and Lemma 11 of Ata and Kumar (2005), R n i · 0 θ n i (s)Q n i (s)ds , n ∈ N + is relatively compact in the u.o.c. topology and all of its limits are Lipschitz continuous. Hence, {Q n i , n ∈ N + } is also relatively compact in the u.o.c. topology with Lipschitz continuous and nonnegative limits by (11) and (36). Hence, fluid limits of {Q n i , n ∈ N + } satisfy (25d). Let us fix an arbitrary subsequence {n l , l ∈ N + } such that · 0 θ n l i (s)Q n l i (s)ds, l ∈ N + converges to a Lipschitz continuous limit under the uniform norm. Then, there exists a subsequence of {n l , l ∈ N + }, denoted by {n k , k ∈ N + }, such that Q n k i −Q i T 1 → 0 as k → ∞ for all i ∈ N , whereQ i ∈ D[0, T 1 ] is nonnegative and Lipschitz continuous for all i ∈ N . Then, if n k ≥ n 0 , by Assumption 1. Therefore, each convergent subsequence of the sequence · 0 θ n i (s)Q n i (s)ds, n ∈ N + converges to · 0 θ i (s)Q i (s)ds u.o.c. whereQ i is a fluid limit of {Q n i , n ∈ N + } for all i ∈ N . Then, by Lemma 11 of Ata and Kumar (2005), each convergent subsequence of R n i · 0 θ n i (s)Q n i (s)ds , n ∈ N + converges to · 0 θ i (s)Q i (s)ds u.o.c. whereQ i is a fluid limit of {Q n i , n ∈ N + } for all i ∈ N . This proves (25a).
By Theorems 16.2 and 16.4 of Billingsley (1999) and the fact that T 1 ∈ R + is arbitrarily chosen, we can extend the results above to the processes in D 2N J+3N +J , which proves Proposition 1 except (25f). Lastly, in order to prove (25f), it is enough to prove the following result. From this point forward, we keep π and ω in the notation.
We complete the proof of Proposition 1 by defining the full set A π := A 1 ∩ A π 2 . B.1. Proof of (43) Since the indices in the sum in (43) τm)+k ] = 0 for all n ∈ N + and k ∈ N + , so that we can use a Martingale strong law of large numbers result for triangular arrays in order to prove (43).
For notational convenience, we omit the superscript π from the notation. Remember that we fix arbitrary i ∈ N , j ∈ J , m ∈ N in Lemma 8. Let LetF n j (k) := F n j (E j •Γ n j (τ m )+k) for all k ∈ N + (cf. (8)) andF n j (k) := {∅, Ω} for all k ∈ Z\N + , where Z denotes the set of integers. Since ν n j (E j •Γ n j (τ m )+ k) is a stopping time with respect to the filtration F n for all k ∈ N + and is increasing in k,F n j (k) is well defined and F n j (k), k ∈ Z is a filtration. Let S n k : R + × Ω → R N be defined such that Electronic copy available at: https://ssrn.com/abstract=2844451 for all k, n ∈ N + , t ∈ R + , and ω ∈ Ω. By (8), Let (R N ) N := R N × R N × . . .. Then we have the following result.
Lemma 9. Fix arbitrary k, n ∈ N + and j ∈ J . Let g be an arbitrary F n j (k)-measurable function. Then, there exist a B((R N ) N )-measurable function f and a sequence of nonnegative real numbers {t l , l ∈ N} such that g(ω) = f (S n k (t l , ω), l ∈ N) for all ω ∈ Ω.
Proof. The proof follows by Exercise 1.5.6 of Stroock and Varadhan (2006) and the fact that R, R N , and (R N ) N are all Polish spaces (cf. Corollary 3.39 of Aliprantis and Border (2006)).
Lemma 10. We have π n j E j •Γ n j (τ m )+k ∈F n j (k) and a E j •Γ n j (τm)+k j (m) ∈ F n j (k + l) for all j ∈ J , k, l, n ∈ N + , and m ∈ N.
Since f r ∈ B((R N ) N ) by definition, the mapping in (55) isF n j (k)-mesurable.
Then, we have the following result.
Lemma 11. We have a E j •Γ n j (τm)+k j (m) ⊥F n j (k) for all j ∈ J , k, n ∈ N + , and m ∈ N.
Proof. For notational convenience, again letẼ := E j • Γ n j (τ m ). Let us fix arbitrary j ∈ J , k, n ∈ N + , m ∈ N, and c ∈ R and an arbitrary set B ∈F n j (k). Then Electronic copy available at: https://ssrn.com/abstract=2844451 where (56) is by Lemma 9. (57) is by (53) and the fact that a r+k j (m) ⊥ F n j (r+k) (cf. (8)), I Ẽ = r ∈ F n j (r+k) for all r ∈ N, and f S n r+k (t l ), l ∈ N ∈ F n j (r+k) (see proof of Lemma 10 for a similar argument explained in detail). (58) is by the i.i.d. property of the sequence {a r j (m), r ∈ N}, and the equality in (59) is by the fact that a r+k j (m) ⊥ F n j (r + k) and I Ẽ = r ∈ F n j (r + k) for all r ∈ N. By definition of independence (cf. page 41 of Durrett (2010)), the equality in (60) proves the desired result.
Remark 8. If the filtration defined in (7) is generated by the sequence of processes {Υ m , m ∈ N} defined in Remark 4, we can still prove Lemmas 10 and 11 by updating the definition in (52) such that the infinite dimensional process S n k includes the sequence {Υ m , m ∈ N}.
Now we are ready to prove (43) (or equivalently (51)) by the technique introduced by de Jong (1996). For any x ∈ R, let x denote the greatest integer which is smaller than or equal to x. Then, 1 n n k=1X n k = 1 n n k=1 X n k − E X n k F n j (k + n 0.25 − 1) We will consider each of A n 1 , A n 2 , and A n 3 separately. Let us choose an arbitrary > 0. First, let us consider Notice thatZ n k isF n j (k)-measurable by Lemma 10 andỸ k ⊥F n j (k) by Lemma 11. Moreover, where (63) is by the fact that a r+k j (m) ⊥ F n j (r + k) (cf. (8)) and I E j • Γ n j (τ m ) = r ∈ F n j (r + k) for all k ∈ N + and r ∈ N, and (64) is by definition ofF ij (τ m ) (cf. (2)). Hence, on the one hand, for all k, l, n ∈ N + , by (65). On the other hand,X n k isF n j (k + l)-measurable for all k ∈ N + and l ∈ N + by Lemma 10. Hence, for all k, n ∈ N + , Since X n k ≤ 1 for all k ∈ N + and n ∈ N + , the sum in the the right-hand side in (62) is less than or equal to 15 < ∞ by (67).
Second, by (66), we have Electronic copy available at: https://ssrn.com/abstract=2844451 Third, First, E M n l (r) ≤ 2r < ∞ and M n l (r) ∈F n j (l + r) (even when l + r ≤ 0 in which M n l (r) = 0 becauseF n j (k) = {∅, Ω} for k ∈ Z\N + ) for all r ∈ N + . Second, Therefore, for all fixed n ∈ N + and l ∈ Z, {M n l (r),F n j (l + r), r ∈ N + } is a martingale sequence such that E[M n l (r)] = 0. Moreover, M n l (r) − M n l (r − 1) ≤ 2 for all r ∈ N + , i.e., the martingale differences are bounded by 2. Then, the sum in (69)  where the inequality is by Azuma's inequality (cf. Theorem 6.3.3 of Ross (1996)).
Finally, (51) follows by the fact that > 0 is arbitrary and where the second equality is by the fact that the sum in (62) (2010)).

APPENDIX C: A REGULATOR MAPPING RESULT
We need regulator mapping results in order to prove Theorems 2, 3, and 4 in Section D. Since the generalized one-sided regulator mappings defined in the literature (cf. Ward (2004, 2008) and Ward and Kumar (2008)) are not applicable to our case, we introduce a new one-sided and nonlinear regulator mapping in this section. Let φ, ψ : D → D be such that for all x ∈ D and t ∈ R + , Then, φ is the conventional one-sided and one-dimensional regulator mapping (cf. Chapter 13.5 of Whitt (2002)). We define the following regulator mapping.
Definition 4. (A time-dependent, one-sided, and nonlinear regulator mapping): Let x, y ∈ D be such that x(0) ≥ 0 and sup t∈R + |y(t)| < ∞. The time-dependent, one-sided, and nonlinear regulator mapping φ M , ψ M : If y = 0 in Definition 4, then the time-dependent, one-sided, and nonlinear regulator mapping becomes the conventional one-sided and one-dimensional regulator mapping defined in (71) (cf. Theorem 6.1 of Chen and Yao (2001)); if y ∈ D is a constant function, then it becomes the linearly generalized onesided mapping described in Ward and Kumar (2008).
In order to write the time-dependent, one-sided, and nonlinear regulator mapping in terms of the conventional one-sided and one-dimensional regulator mapping defined in (71), we make the following definition.
Definition 5. (Integral equation): Let x, y ∈ D be such that sup t∈R + |y(t)| < ∞. Let M : D 2 → D be a mapping such that u := M(x, y) solves the integral equation A mapping g : D → D is Lipschitz continuous with respect to the uniform norm, if for all T 1 ∈ R + , there exists a constant κ ∈ R + which may depend on T 1 such that g(x) − g(y) T 1 ≤ κ x − y T 1 for all x, y ∈ D. Then, the first main result of this section is the following. The following lemma will be useful in the proof of Proposition 3.
1. There exists a unique u ∈ D which solves the integral equation (72). Moreover, the mapping M(·, y) : D → D is Lipschitz continuous with respect to the uniform norm. 2. If x is nondecreasing and absolutely continuous and x(0) ≥ 0, then M(x, y) ≥ 0.
3. Let x 1 and x 2 be two absolutely continuous functions such that x 1 ≥ x 2 and x 1 ≥ x 2 , where denotes the derivative. If M(x 2 , y) ≥ 0, then M(x 1 , y) ≥ M(x 2 , y).
Proof. Part 1: Fix arbitrary x, y ∈ D such thatȳ := sup t∈R + |y(t)| < ∞. Let η y : D → D be such that for all u ∈ D and t ∈ R + , η y (u)(t) := y(t)φ(u)(t). Then for all u 1 , u 2 ∈ D and t ∈ R + , where the last inequality is by the fact that the mapping φ is Lipschitz continuous with respect to the uniform norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)). Thus, η y is Lipschitz continuous with respect to the uniform norm. Since (72) is equivalent to there exists a unique u ∈ D which solves (72) and the mapping M(·, y) : D → D is Lipschitz continuous with respect to the uniform norm by Lemma 1 of Reed and Ward (2004).
Next, let us consider arbitrary x 1 , x 2 ∈ D such that x 1 (0) ≥ 0 and x 2 (0) ≥ 0. Let u 1 := M(x 1 , y) and u 2 := M(x 2 , y). By Lemma 12 Part 1, the mapping M(·, y) : D → D is Lipschitz continuous with respect to the uniform norm. Let κ y (t) be the corresponding Lipschitz constant for t ∈ R + . Then, for all t ∈ R + , where the first inequality is by the fact that the mapping φ is Lipschitz continuous with respect to the uniform norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)). Hence, φ M (·, y) is Lipschitz continuous with respect to the uniform norm. Lastly, for all t ∈ R + , where the first inequality is by the fact that the mapping ψ is Lipschitz continuous with respect to the uniform norm with Lipschitz constant 1 (cf. Lemma 13.4.1 of Whitt (2002)). Hence, ψ M (·, y) is also Lipschitz continuous with respect to the uniform norm.
Next, we present the following preliminary result.
Lemma 13. Let T 1 ∈ R + be an arbitrary constant, x, y ∈ D be such that y ≥ 0 and sup t∈R + y(t) ≤ K for some K ∈ R + . Then, there exists a constant C = C(K, T 1 ) ∈ R + such that M(x, y) T 1 ≤ C x T 1 .
Proof. Let u := M(x, y) be the unique solution of the integral equation (72). Notice that φ(u) ≥ 0 by (71). Then by (72) and the fact that y ≥ 0 and φ(u) ≥ 0, Let a denote the smallest integer which is greater than or equal to a for all a ∈ R. Let us partition the interval [0, T 1 ] into subintervals with length 1/(4K) except the last interval which is [( 4KT 1 −1)/(4K), T 1 ]. Then there are 4KT 1 subintervals. Let, Electronic copy available at: https://ssrn.com/abstract=2844451 By (72), By (79) and the fact that the mapping φ is Lipschitz continuous with respect to the uniform norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)), where the second inequality above is by (77) and (78). By (79), we have for where the last inequality follows by (77) and (78). By induction, we can show that By (80) and (81), one can show that We complete the proof by defining C(K, T 1 ) := ( 4KT 1 + 4)!.
The second main result of this section is the following.
Electronic copy available at: https://ssrn.com/abstract=2844451 Lemma 14. Let {x n , n ∈ N} and {y n , n ∈ N} be sequences in D such that x n (0) ≥ 0 and y n ≥ 0 for all n ∈ N, there exists a constant K ∈ R + such that y n (t) ≤ K for all n ∈ N and t ∈ R + , and there exist x ∈ D and y ∈ D such that x n → x u.o.c. and y n → y u.o.c. as n → ∞. Then, Proof. For each n ∈ N, let u n ∈ D be the unique solution of the equation which exists by Lemma 12 Part 1. For each n ∈ N, let us define Notice that f n is continuous for all n ∈ N and u n = x n − f n for all n ∈ N by (82) and (83). Let T 1 be an arbitrary constant in R + . We will show that the sequence {f n , n ∈ N} restricted to the compact domain [0, T 1 ] is relatively compact by the Arzelà-Ascoli Theorem (cf. Theorem 7.2 of Billingsley (1999)). First, sup n∈N |f n (0)| = 0 < ∞. Second, where the first inequality in (84) is by the fact that the mapping φ is Lipschitz continuous with respect to the uniform norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)). The inequality in (85) is by Lemma 13. Notice that x n → x u.o.c where x is a bounded function in [0, T 1 ], thus there exists a sufficiently large n 0 ∈ N such that sup n≥n 0 x n T 1 < ∞, and without loss of generality we assume sup n∈N x n T 1 < ∞. Hence, we obtain the convergence result in (85). Therefore, {f n , n ∈ N} restricted to the compact domain [0, T 1 ] is relatively compact.
Since both {x n , n ∈ N} and {f n , n ∈ N} are relatively compact in D[0, T 1 ] endowed with the u.o.c. topology, so does {u n , n ∈ N} by (82) and (83). Let Electronic copy available at: https://ssrn.com/abstract=2844451 us consider an arbitrary subsequence of {u n , n ∈ N}, denoted by {u n k , k ∈ N}, such that u n k → u u.o.c. as k → ∞ where u ∈ D[0, T 1 ]. Then, As k → ∞, all of the terms in (87) converges to 0, so (86) is equal to 0. Thus, u = M(x, y) and u is the unique solution of (72) by Lemma 12 Part 1. Therefore, each subsequence of {u n , n ∈ N} has convergent subsequence which converges to the same limit, which implies that u n → u u.o.c. as n → ∞ where u = M(x, y). Notice that the process φ M , ψ M (x n , y n ) ∈ D 2 is well defined for all n ∈ N by Proposition 3. Since the mappings φ and ψ are Lipschitz continuous with respect to the uniform norm, φ M , ψ M (x n , y n ) = (φ, ψ) (M(x n , y n )) = (φ, ψ) (u n ) → (φ, ψ) (u) = (φ, ψ) (M(x, y)) = φ M , ψ M (x, y), u.o.c. as n → ∞, which completes the proof.

APPENDIX D: PROOFS OF THEOREMS 2, 3, AND 4
We prove Theorems 2, 3, and 4 by representing the queue length process of each driver type with the time-dependent, one-sided, and nonlinear regulator mapping defined in Definition 4. We first present some preliminary results that will be used in the proofs of all of the three theorems in Section D.1. Then, we prove Theorems 2, 3, and 4 in Sections D.2, D.3, and D.4, respectively.

D.1. Preliminary Results
In this section, we present some preliminary results that will be used in the proofs of Theorems 2, 3, and 4. We consider π R (x) where {q, x} = {q i (t), x ij (t), i ∈ N , j ∈ J , t ∈ [0, T ]} is a feasible process pair for the CLP (12) such that x ij is a Borel measurable simple function (cf. page 46 of Folland (1999)) for all i ∈ N and j ∈ J in this section. We start with the formal definition of the randomized policy given in Definition 1.
Let {x ij (m), i ∈ N , j ∈ J , m ∈ U} be a sequence of real numbers in R + such that U is a finite subset of N. Let {B m , m ∈ U} be a disjoint partition of the interval [0, T ] such that B m is a Borel measurable set for all m ∈ U. Then, F ij has finite range on the interval [0, T ] for all i ∈ N and j ∈ J by (2) and the fact that τ m → ∞ as m → ∞. Thus, without loss of generality, we choose the partition {B m , m ∈ U} such that for each m ∈ U, there exists l ∈ N such that B m ⊆ [τ l , τ l+1 ). By (12c), i∈Nx ij (m) ≤ 1 for all j ∈ J and m ∈ U. Let us define the sequence of independent random variables {p k j (m), k ∈ N + , j ∈ J , m ∈ U}, which is independent of all other stochastic primitives and F(0)-measurable such that, P(p k j (m) = i) =x ij (m) and P(p k j (m) = 0) = 1 − i∈Nx ij (m) for all i ∈ N , j ∈ J , k ∈ N + , and m ∈ U. Notice that the sequence {p k j (m), k ∈ N + , j ∈ J , m ∈ U} corresponds to {Υ m , m ∈ N} defined in Remark 4 associated with the randomized policy. With a slight abuse of notation, let π R (x) = (π n 1 , π n 2 , . . . , π n J ) in the nth system. Then, (89) π n j (k) = i∈N m∈U for all j ∈ J and k, n ∈ N + . Then, clearly, π n j (k) is F n j (k)-measurable for all j ∈ J and k, n ∈ N + , thus π R (x) is admissible by Definition 1.
Let β : N → N be a function such that β(m) Electronic copy available at: https://ssrn.com/abstract=2844451 Let us define the following stochastic process such that for all i ∈ N , j ∈ J , n ∈ N + , and t ∈ [0, T ], where the second equality is the Lebesgue-Stieltjes integral (cf. page 107 in Folland (1999)), and the third equality is by (2) and (88). LetH π R (x),n ij := (1/n)H π R (x),n ij . Then we have the following result whose proof is presented in Section D.1.1.
Lemma 15. For all i ∈ N and j ∈ J , we have We omit the superscript π R (x) from the notation for convenience in presentation in the rest of Section D.1. Let us define the following stochastic process such that for all i ∈ N , j ∈ J , n ∈ N + , and t ∈ [0, T ], LetĪ n ij := (1/n)I n ij for all i ∈ N , j ∈ J , and n ∈ N + . Then, by (11) and some algebra, for all i ∈ N , n ∈ N + , and t ∈ [0, T ], wherē for all i ∈ N , n ∈ N + , and t ≥ T for mathematical completeness, so (91) is well defined for all t ∈ R + . We approximateX n i by a deterministic process in the following result.
Lemma 16. For all i ∈ N and j ∈ J , as n → ∞, we have for all t ∈ [0, T ] andX i (t) :=X i (T ) for all t ≥ T . Moreover,X i is nonnegative and Lipschitz continuous for all i ∈ N .
The proof of Lemma 16 is presented in Section D.1.2. Notice that, for any t ∈ [0, T ], I n ij (t) can increase only if there is a type j customer arrival at time t and Q n i (t−) = 0 by (90). Since probability that there is also a type i driver arrival at time t is 0, then Q n i (t) = 0 with probability 1. Hence, for all i ∈ N and n ∈ N + , Electronic copy available at: https://ssrn.com/abstract=2844451 for all i ∈ N and n ∈ N + . There exists a constantθ ∈ R + and n 0 ∈ N + such that θ n i (t) ≤θ for all t ∈ R + , i ∈ N , and n ≥ n 0 by Assumption 1 and (92). Then, by Assumption 1, Lemma 14, and Lemma 16, as n → ∞ for all i ∈ N . Lastly, we have the following result.
Proof. By (94) and the fact that ψ M X i , θ i = 0, Then, where the inequality is by triangular inequality and the convergence result is by Lemma 15,(95), and Lemma 16.

D.1.1. Proof of Lemma 15
The proof of Lemma 15 is very similar to the one of Lemma 8. For notational convenience, we omit the superscript π R (x) from the notation in this section. Let us fix arbitrary i ∈ N and j ∈ J . Then, by definition, we have For all k, n ∈ N + and m ∈ U, let Then, the right-hand side of (96) is equal to Notice that, if we can prove that (98) 1 n n k=1W n k a.s.
− − → 0, as n → ∞, then we can prove that the term in (97) converges to 0 as n → ∞ by the same way we prove Lemma 8. Hence, in the remaining part of this section, we will prove (98), whose proof is very similar to the one of (43). For all k ∈ Z and n ∈ N + , if k ∈ Z\N + , we let G n k := {∅, Ω}; and if k ∈ N + , we let Then, {G n k , k ∈ Z} is a filtration for all n ∈ N + . Moreover, Y k (m) ⊥ G n k and Z n k (m) ∈ G n k for all k, n ∈ N + and m ∈ U; and W n k (m) ∈ G n k+l for all k, n, l ∈ N + by construction. First, for all k, n ∈ N + , Second, for all k, n, l ∈ N + , We will consider each term in the right-hand side (RHS) of (102) separately. The RHS of (102a) converges to 0 a.s. by Assumption 2, (35), and Lemma 15, respectively.
Next, let us consider the term in (102b). Let us fix an arbitrary ω ∈ A 1 (cf. (36)). From the proof of Proposition 1, we know that each subsequence of the term in (102b) has a subsequence which converges to 0 on the sample path ω, which implies that the term in (102b) itself converges to 0 as n → ∞ on the sample path ω. Since P(A 1 ) = 1, the term in (102b) converges to 0 a.s as n → ∞.
Next let us consider the term in (102c) and fix an arbitrary i ∈ N and j ∈ J . By (88), and the construction of the sequence of Borel measurable sets {B m , m ∈ U}, Let denote the Lebesgue measure on R. Then, for all > 0, there exists a sequence of sets {B  Folland (1999). Hence, for all > 0, there exists a simple function ξ such that ξ(t) := m∈U (2) z m I(t ∈ B (2) m ) for all t ∈ [0, T ], U (2) is a finite subset of N, {z m , m ∈ U (2) } is a nonnegative sequence of real numbers bounded above by 1, B (2) m is finite union of open intervals for all m ∈ U (2) , and (t ∈ [0, T ] : ξ(t) =F ij (t)x ij (t)) < . Moreover, for all > 0, there exists a continuous function g : R → [0, 1] such that g(t) = 0 for all t / ∈ [0, T ], g has bounded variation, and (t ∈ [0, T ] : g(t) = ξ(t)) < (see the proof of Theorem 2.26 of Folland (1999) for a specific construction and the reason why g has bounded variation is that ξ has finite range and B (2) m is finite union of open intervals for all m ∈ U (2) ). Let U (3) := {t ∈ [0, T ] : g(t) =F ij (t)x ij (t)}. Then, (U (3) ) < 2 . LetL n j :=Ē n j •Γ n j − Γ j . Then, First, let us consider the first term in (103), which is less than or equal to Electronic copy available at: https://ssrn.com/abstract=2844451 is uniformly distributed on [0, T ] is in the Borel measurable set U (3) is (U (3) )/T . Therefore, (105) and (106) imply that (104) is bounded above by √ +4μ j with probability 1 − 4C √ . Since > 0 is arbitrary, the first term in (103) is equal to 0 with probability 1 for all n ≥ n 0 .
Next, let us consider the second term in (103) and and fix an arbitrary ω ∈ A 1 (cf. (36)). Then,L n j (·, ω) → 0 u.o.c. as n → ∞. By Theorem 3.36 of Folland (1999) (integration by parts), the second term in (103) is equal to Notice that the Lebesgue-Stieltjes measure induced by g can be a signed measure. By Theorem 3.3 of Folland (1999) (the Hahn Decomposition Theorem), there exists a positive set P and negative set N for the Lebesgue-Stieltjes measure induced by g such that P ∪ N = [0, T ] and P ∩ N = ∅. Let V(g) < ∞ denote the total variation of g. Then, the sum of the terms in (107)  L n j (t, ω) (1 + 2V(g)) → 0 as n → ∞.
Therefore, the sum in (103) converges to 0 as n → ∞ a.s. so do the the term in (102c) and Lastly,X i is nonnegative by (12b) and (12d) and is Lipschitz continuous by Assumption 1.
Electronic copy available at: https://ssrn.com/abstract=2844451 D.2. Proof of Theorem 2 By (12d), (71), and the fact that q i is Lipschitz continuos, we have φ(q i ) = q i for all i ∈ N . Hence, by (12b), Therefore, q i = M(X i , θ i ) by Lemma 12 Part 1. Then, we have ψ M X i , θ i = ψ M X i , θ i = ψ (q i ) = 0 for all i ∈ N by Proposition 3, (12d), and (71). Then, the first part of the proof follows by Lemma 17. Furthermore, by Proposition 3, (12d), and (71), for all i ∈ N , we have where the convergence is by (94).

D.3. Proof of Theorem 3
The process x * is a feasible matching process for the CLP (12) by Lemma 4 and it is a Borel measurable simple function for all i ∈ N and j ∈ J by assumption. Then, the results of Section D.1 apply to this process. Moreover, we can extend the results of Lemma 16 such thatX i is also nondecreasing for all i ∈ N by (13b). Then, by Lemma 12 Part 2, M X i , θ i ≥ 0 for all i ∈ N . By (71) and the fact that D.4. Proof of Theorem 4 Since the parameters of LP (13) are time homogeneous, x * can be chosen as a constant function of time and so we let x * ij (t) = x * ij for all i ∈ N and j ∈ J . Then, for all i ∈ N , Then,X i is nonnegative and nondecreasing by (13b) and it is Lipschitz continuous. By Lemma (12) Part 2, q * i ≥ 0 and so φ(q * i ) = q * i for all i ∈ N by (71). Thus, {q * , x * } satisfies (12b), satisfies (12e) because x * is a constant, and satisfies (12c) and (12d) by satisfying (13c) and (13d), respectively. Therefore, {q * , x * } is a feasible process pair for the CLP (12). Thus, Letx ij := (1/T ) T 0x ij (s)ds for all i ∈ N and j ∈ J . Then it is easy to see that {x ij , i ∈ N , j ∈ J } is feasible for the LP (13) when the parameters are time homogeneous, thus i∈N ,j∈J By (108) and (109), we see that {q * , x * } is an optimal solution of the CLP (12). Since x * is constant, π R (x * ) is admissible by Lemma 3.
The rest of the proof of Theorem 4 is very similar to the one of Theorem 3. Since the process x * is a feasible matching process for the CLP (12) and x * ij is a constant function of time and so Borel measurable simple function for all i ∈ N and j ∈ J , the results of Section D.1 apply to this process. (71). Then, by Lemma 17, (108), and (109), for all i ∈ N , which gives us the desired result.

APPENDIX E: LEMMA PROOFS
We first prove Lemma 1, second prove Lemmas 3 and 6 (because they are related to Lemma 1), third prove Lemma 2, then prove Lemma 4, and lastly prove Lemma 5.
Notice that Q i (ν j (k)−) = lim t→∞ Q i ((t ∧ ν j (k))−) ∈ F j (k) by Proposition 2.7 of Folland (1999) and definition of F j (k) (cf. (8)) for all i ∈ N , j ∈ J , and k ∈ N + . By (8), (110), and the fact that ν j (k) ∈ F j (k), the random variables I (|S j (k)| = 1), I (|S j (k)| > 1), I (i ∈ S j (k)), and I (S j (k) = N 1 ) are F j (k)-measurable for all i ∈ N , j ∈ J , k ∈ N + , and N 1 ⊂ N . Lastly, Therefore, π j ∈ F j for all j ∈ J under the CD policy, so it is an admissible policy. Notice that, the sequence {q k j (l), k ∈ N + , j ∈ J , l ∈ N N } corresponds to {Υ m , m ∈ N} defined in Remark 4 associated with the CD policy.
E.2. Proofs of Lemmas 3 and 6 First, the randomized policy given in Definition 2 is admissible by the fact that (89) is F n j (k)-measurable for all j ∈ J and k, n ∈ N + . Second, the admissibility proof of the hybrid policy follows by the fact that it is a hybrid of the randomized policy given in Definition 2 and the CD policy defined in (3), and both of the latter two policies are admissible. Third, the deterministic policy is admissible because both Q n i (ν n j (k)−) and D n ij (ν n j (k)−) are F n j (k)-measurable for all i ∈ N , j ∈ J , and k, n ∈ N + (cf. (8)) and the tie breaking rule does not use any future information.
Lastly, we will prove that the RWQ policy is admissible. We let π = (π 1 , π 2 , . . . , π J ) denote the RWQ policy and we omit the superscript π from the notation for notational convenience in this proof. Recall from Section D.1 that {x ij (m), i ∈ N , j ∈ J , m ∈ U} is a sequence of real numbers in R + , U is a finite subset of N, and {B m , m ∈ U} is a disjoint partition of the interval [0, T ] such that B m is a Borel measurable set for all m ∈ U. Let us define the sequence of independent random variables {q k j,m (l), k ∈ N + , j ∈ J , m ∈ U, l ∈ N N }, which is independent of all other stochastic primitives and F(0)-measurable, such that q k j,m (l 1 , l 2 , . . . , l N ) = for all k ∈ N + , j ∈ J , m ∈ U, and l ∈ N N such that l = (l 1 , l 2 , . . . , l N ). Under the RWQ policy, for all j ∈ J and k ∈ N + , let (8), (18), and (88) for all j ∈ J and k ∈ N + . By (8) and the fact that B m is Borel measurable for all m ∈ U (by definition), π j (k) ∈ F j (k) for all j ∈ J and k ∈ N + . Therefore, π j ∈ F j for all j ∈ J under the RWQ policy, so it is an admissible policy. Notice that, the sequence {q k j,m (l), k ∈ N + , j ∈ J , m ∈ U, l ∈ N N } corresponds to {Υ m , m ∈ N} defined in Remark 4 associated with the RWQ policy.

E.3. Proof of Lemma 2
Proof of Part 1 For all i ∈ N and t ∈ [0, T ], let By (12d) and (71) Proof of Part 2 We use the proof technique introduced by Levinson (1966). We first present a preliminary result from Levinson (1966), then prove Lemma 2. Let L 2 ([0, T ]) denote the space of Lebesgue measurable functions with domain [0, T ] and range R such that if f ∈ L 2 ([0, T ]), then T 0 f (t) 2 dt < ∞. We let w − → denote weak convergence in L 2 ([0, T ]) as defined in Proposition 6 in Section 8.2 of Royden and Fitzpatrick (2010).
Lemma 18. (Lemma 2.1 of Levinson (1966)) Let {f r , r ∈ N} be a uniformly bounded sequence of functions in T ] except on a set of zero measure.
By (12b), (12d), and Assumption 1, q i T ≤Q i (0) +λT for all i ∈ N . Second, x ij is nonnegative and bounded for all i ∈ N and j ∈ J by (12c) and (12d). Third, both µ j and Λ i are bounded processes on [0, T ] by Assumption 1, and w ij andF ij are bounded by definition for all i ∈ N and j ∈ J .
Electronic copy available at: https://ssrn.com/abstract=2844451 Let x ij = 0 for all i ∈ N and j ∈ J and consider the equation By Lemma (12) Parts 1 and 2, there exists a unique solution of (111) such that q i = M(Q i (0) + Λ i , θ i ) ≥ 0. Then, q i = φ(q i ) for all i ∈ N and so q i = M(Q i (0) + Λ i , θ i ) and x ij = 0 for all i ∈ N and j ∈ J is a feasible pair for the CLP (12). Hence, the feasible region of the CLP (12), denoted by H := {(q, x) : (q, x) satisfies (12b)-(12e)}, is nonempty. Let Then, there exists a sequence of feasible matching processes for the CLP (12), denoted by {(q r , x r ), r ∈ N}, such that (q r , By Theorem 14 in Section 8.3 of Royden and Fitzpatrick (2010), there exists a subsequence of {(q r , x r ), r ∈ N} which is again denoted by {(q r , x r ), r ∈ N} for notational convenience such that q r Hence, it is enough to prove that a modification of (q,x) is in H. First, let us consider (12b). Let us fix an arbitrary i ∈ N and t ∈ [0, T ]. By definition of weak convergence, we have Electronic copy available at: https://ssrn.com/abstract=2844451 By (12b) and (113), q r converges point-wise to some nonnegative function q as r → ∞ andq =q almost everywhere (a.e.) on [0, T ] by Lemma 18.
Next, let us consider (12c) and (12d). Notice that x r ij w − →x ij for all i ∈ N and j ∈ J implies i∈N x r ij w − → i∈Nx ij for all j ∈ J . Then, by Lemma 18, for all i ∈ N , j ∈ J , and t ∈ [0, T ] except on a set of zero measure, Hence,x satisfies (12c) and (12d) for all t ∈ [0, T ] except on a set of zero measure, and we denote this set of zero measure by H and its complement by H c , i.e., H c : for all t ∈ H c , i ∈ N , and j ∈ J , andx ij (t) := 0 for all t ∈ H, i ∈ N , and j ∈ J . Notice that (q,x) is a feasible pair for the CLP (12). To see this (q,x) satisfies (12b), (12c) and (12d) by construction, and satisfies (12e) by Proposition 2.11 of Folland (1999). Furthermore, (q,x) is an optimal solution of the CLP (12) by (112) and the fact that it is equal to (q,x) a.e. on [0, T ].
E.4. Proof of Lemma 4 Suppose that w ij = 1 for all i ∈ N and j ∈ J . By (12b), (12d), and (14), we have Then, by summing (115) in i ∈ N , we can see that an upper bound on the objective function value of the CLP (12) is i∈N Λ i (T ). Any feasible matching process that satisfies condition (15) has the objective function value i∈N Λ i (T ), thus it is an optimal CLP solution. Moreover, by (115), condition (15) implies that (115) binds for all i ∈ N and t ∈ [0, T ] under at least one optimal CLP (12) solution.
Let {q,x} be an optimal solution of the CLP (12) under which (115) is binding for all driver types at all times and soq i = 0 for all i ∈ N by (12b) and (12d). Notice thatx satisfies (13c) and (13d) because these constraints are exactly the same of (12c) and (12d), respectively. Since (115) is binding for all driver types at all times underx, then by taking the derivative of both right-and left-hand sides of (115) with respect to t, we can see that {x ij (t), i ∈ N , j ∈ J } (or a modification of it) satisfies (13b) for all t ∈ [0, T ], so it is feasible for the LP (13) for all t ∈ [0, T ]. Then, i∈N ,j∈J Electronic copy available at: https://ssrn.com/abstract=2844451 which implies (116) i∈N ,j∈J T 0 µ j (s)F ij (s)x ij (s)ds ≤ i∈N ,j∈J T 0 µ j (s)F ij (s)x * ij (s)ds.
E.5. Proof of Lemma 5 Since λ i and µ j are continuous functions of the surge multipliers and s i has a compact domain (so does p ij ) for all i ∈ N and j ∈ J , then an optimal solution of (16) exists. Let {p ij , x ij , i ∈ N , j ∈ J } be feasible prices and matching fractions for the optimization problem (16) such that the constraint (16b) is not binding for some driver type(s). Let {s i , i ∈ N } be the surge multipliers corresponding to the {p ij , i ∈ N , j ∈ J }. Let z i := j∈J µ jFij x ij , z := i∈N z i , G i := λ i − z i , and G := i∈N G i for all i ∈ N . Then, z i denotes the matching rate of type i drivers, z denotes the total matching rate, G i ≥ 0 for all i ∈ N , G denotes the gap between the total driver arrival rate and the total matching rate, and G > 0 because the constraint (16b) is not binding for some driver type(s). We will show that the system controller can increase z and decrease G to 0 by changing the surge multipliers. This implies that there exists an optimal solution of (16) in which the constraint (16b) is binding for all driver types.
Consider an area i ∈ N such that G i > 0 (since G > 0, such an area exists). If we decrease s i , then j∈J :α(j)=i µ j will increase and λ i may decrease by condition (17a). Let us decrease s i and match the additional customers arriving at area i with the excess type i drivers so that z i increases. We should decrease s i until all type i drivers are matched with customers because decreasing s i more results in excess customers which lowers z i , i.e., we should decrease s i until G i becomes equal to 0. Since we match the additional customers arriving at area i with the excess type i drivers, for j ∈ J such that α(j) = i, we should increase x ij and decrease x kj for all k ∈ N \{i} such that the constraints (16c) and (16d) hold.
Next, let us consider the change in z and G. If there exists k ∈ N \{i} such that ∂λ k /∂s i > 0, then λ k decreases in area k which can decrease z k . However, total decrease in matching rates in all other areas is less than the increase in z i by condition (17c). Hence, z strictly increases. When λ k decreases, G k does not increase and it is possible that some of the customers matched with type k drivers may not find an available type k driver anymore and so leave the system without being matched. In such a case, we should decrease x kj for some j ∈ J such that constraints (16c) and (16d) hold.
If there exists k ∈ N \{i} such that ∂λ k /∂s i ≤ 0, then λ k is nondecreasing. If λ k is strictly increasing and there are unmatched customers in the system, we prefer to not to match those customers with the additional type k drivers arriving in the system for mathematical simplicity, thus z k stays constant but G k increases. However, the increase in k∈N \{i} G k is less than the decrease in G i by condition (17c). Hence, G strictly decreases.
In summary, we can increase z and decrease G by decreasing s i until G i becomes equal to 0. By condition (17d), such an s i exists. We propose the following algorithm which decreases G to 0 and increases z: Let G i = max k∈N G k > 0. Decrease s i until G i becomes equal to 0. Then, update the surge multipliers, customer and driver arrival rates, and feasible matching fractions. If G decreases to 0, stop. Otherwise, repeat the procedure.
Under this algorithm, at each step, z strictly increases, G strictly decreases. Let ∆s i and ∆G denotes the total change in s i and G, respectively, in a step where G i = max k∈N G k . Then, G i /(2C 2 ) ≤ |∆s i | ≤ G i /C 1 by condition (17a). Moreover, ∂G/∂s i < 0 and ∂G ∂s i ≥ ∂λ i ∂s i + j∈J :α(j)=i ∂µ j ∂s i − k∈N \{i} ∂λ k ∂s i ≥ C 1 by (17c). Since G i ≥ G/N , Therefore, G decreases C 1 /(2N C 2 ) × 100% > 0% at each step which implies that G can be made arbitrarily close to 0 in finite steps and converges to 0 as the number of steps increases to infinity.

APPENDIX F: RELATIVE COMPACTNESS IN SPACE D
In this section, we present a relative compactness result in space D that we use in the proof of Proposition 1 (cf. Section B). Although this result is known in folklore, we could not find a specific theorem to refer to, thus we provide one.
Lemma 19. Let {Y n , n ∈ N + } be a relatively compact sequence in D[0, T 1 ] endowed with the u.o.c. topology such that all of its subsequential limits are uniformly continuous. Let {X n , n ∈ N + } be a sequence in D[0, T 1 ] such that for all t 1 , t 2 ∈ [0, T 1 ] and n ∈ N + , where K ∈ R + is a constant. Then, {X n , n ∈ N + } is relatively compact in D[0, T 1 ] endowed with the u.o.c. topology and all of its subsequential limits are uniformly continuous.
Moreover, if all of the subsequential limits of {Y n , n ∈ N + } are absolutely (Lipschitz) continuous, then all of the subsequential limits of {X n , n ∈ N + } are also absolutely (Lipschitz) continuous.