Dynamic Pricing for On-Demand Ride-Sharing: A ...

Dynamic Pricing for On-Demand Ride-Sharing: A Continuous Approach Qi Luo Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, [email protected]

Romesh Saigal Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, [email protected]

We investigate the dynamic pricing problem in on-demand ride-sharing using a continuous-time continuousspace approach. A monopolistic ride-sharing platform controlls two sides of the market, supply (vacant vehicles) and demand (customers’ trip requests) via dynamic pricing with the objective of maximizing its expected revenue in infinite horizon. The dynamic model of supply is described by the multi-population traffic flow with intergroup transfer (a system of hyperbolic stochastic partial differential equations); the demand subjects to independent stochastic processes. This continuous setting allows to use dynamic pricing without treating combinatorial explosions in controls. In both deterministic and stochastic cases, we solve the revenue maximization by dynamic programming and find the optimal prices and commissions. This work provides a macroscopic perspective in handling the complicated spatiotemporal pricing problem in ride-sharing and similar matching markets. Key words : Dynamic pricing, continuous approach, optimal control, ride-sharing, two-sided market. History : This paper was first submitted on October 20, 2017.

1. Introduction On-demand ride-sharing have transforms the mobility services in populated urban areas by the “sharing economy”. This is a typical two-sided market: on one side of the platform are drivers who use their own private cars as taxi cabs and the other side are passengers. A passenger can submit a trip request through an mobile app instead of hailing a taxi on the roadside, and the platform dispatches the nearest vacant car to pick him or her up. After finishing the trip, it will charge passenger’s account electronically with a fixed percentage of the fare (20% -30%) shared by the platform itself. Since Uber was founded in 2009, this new business model has achieved huge success 1

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Luo and Saigal: Dynamic Pricing for On-Demand Ride-Sharing : A Continuous Approach

. This start-up company has completed about 2 billion trips in the U.S. (Forbes (2016)). In New York City where the yellow cabs have been a fixture of the streets, there are more trips finished by on-demand private car drivers than cab drivers in 2017 (Schneider (2016)). Ride-sharing transportation network companies are leading the race in taxi service markets partly because they are able to match the supply (vacant taxis) and demand (passengers) dynamically based on streaming data. These data include but not limited to taxi trips and ride requests from mobile apps, as well as real-time traffic conditions. In particular, the platform can not only match a passenger with nearest drivers at the first place, but also be able to balance the excess demand and supply by mechanisms like dynamic pricing (e.g. surge pricing in Uber). Most pricing algorithms are based on network capacity control with certain service level constraints. The forms of pricing have been extended from time-of-use rates (charge depends on time during the day) to real-time pricing in response to any scenario that causes supply-and-demand imbalance: weather, peak hours or special events (Hall et al. (2015)). However, network capacity control in ride-sharing market faces two challenges that have not been fully addressed in previous studies: (a) Modeling the congestion resulted from control, because the realization of control will affect traffic conditions iteratively; (b) Modeling the interactions between supply and demand because of the cross-side network effects in pricing. In this paper, we argue that a continuous spatiotemporal model has substantial advantages over discrete network models with regard to stochastic controls. The curse of dimensionality is eliminated, and variation of on-line version is feasible. The resulting pricing is smooth in space span, which prevents drivers or passengers from taking advantage of ’jumping prices’ on the boundaries as discrete models do. In Section 2, we briefly review recent literature that attempt to improve the dynamic matching or pricing in a network setting. In Section 3, we incorporate these concerns into a continuous model that consists of conservation laws for multi-population traffic, transfer rate functions and fundamental diagram of traffic flow. We prove that such a system of equations is hyperbolic so that the numerical method to solve it is stable. In Section 5, we develop a optimal control framework to


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provide incentive for drivers to move towards regions with high demand, as well as encourage more inactive drivers to participate with the objective of maximizing the platform’s discounted revenue in infinite horizon.

2. Literature Review Several papers have proposed different pricing policies to improve the simple nearest neighbor matching in ride-sharing market. Pricing discrimination in equilibrium is the main focus in the literature of the platform’s long-term decision. Bimpikis, Candogan and Daniela explore spatial price in the context that a platform serves a network of locations. Their results show that the platform’s profits and consumer surplus are monotonic with the balancedness of the demand pattern, and maximized when the pattern is “balanced” across the locations in the presence of heterogeneity in different locations (Bimpikis et al. (2016)). The short-term decision intends to smooth the instantaneous imbalance in demand and supply by pricing, for example, the “surge pricing” or “boost” used by Uber. The platform optimizes profit-based or fulfillment-rate-based objectives with various capacity, time window and level of service constraints. Zha, Yin and Du study the impacts of surge pricing in labor supply by using a bi-level programing framework: the lower-level problem captures the equilibrium models of labor supply in a ride-sourcing market, and the upper-level problem represents revenue-maximizing surge pricing. They conclude that the platform and drivers share higher revenue and customers are worse off by using such a dynamic pricing scheme comparing to static pricing (Zha et al. (2017)). Ozkan and Ward investigate the driver allocation policies without using surge pricing. They propose an asymptotic continuous linear program upper bound for the matching in a queuing network by assuming the driver is a scarce resource and fully utilized (Ozkan and Ward (2016)). Similar routing and matching models are mainly based on queuing with congestion and network optimization techniques (Braverman et al. (2016), Banerjee et al. (2015), Alonso-Mora et al. (2017)). Even though the effectiveness of dynamic pricing in the drivers allocations is well-recognized, debates has arouse over its consequence in social welfare. By analyzing the extensive individual-level


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taxi trips data and costs for traveling, behavior economics studies are able to quantify customers surplus growth by introducing dynamic pricing in ORP (Azevedo and Weyl (2016), Chen and Sheldon (2015), Cohen et al. (2016)). Different metrics they applied made contradictory conclusions on whether ORP is displacing taxi services or exploiting an under-served market. Cachon, Daniels and Lobel (Cachon et al. (2017)) compare static contracts and dynamic contracts for service providers and customers. They find that the contract that chooses both prices and wages contingent on demand substantially increases the profit relative to contracts that have fixed prices or fixed wages. Our paper contribute to the literature on on-demand ride-sharing management by: (a) Introducing a continuous-time-continuous-space framework to prevent the curse of dimensionality in dynamic programming; (b) Including the reactions to pricing in a macroscopic fashion; (c) Solving the platform’s revenue maximization by stochastic control.

3. State Dynamic Model The couple of state variables demand (number of trip requests) and supply (number of vacant taxis) in space x at time t are denoted as D(t, x) and S (t, x) respectively. Supposing that at each time t ∈ [0, ∞) the ride-sharing platform announces the prices p(t, x) only to passengers at location x, and commissions c(t, x) as a public information to all drivers. (p(t, x), c(t, x)) becomes a decoupled pair of control variables for state variables (D(t, x), S (t, x)). We assume that both p and c are piecewise continuous in time. In this section, we describe the dynamics of state variables under fixed controls on an infinitesimal domain I (t, x) = [t, t + dt) × [x, x + dx]. We assume that the demand D(t, x) follows a Ornstein - Uhlenbeck process: D(t, x) = (µ(t, x) − D(t, x))d(x · t) + δ(t, x)dB(t, x)

(1)

W (t, x) is a Brownian sheet adapted to filtrations Ft N x . The reason for using the Ornstein Uhlenbeck process is that it is a continuous analogue of the autoregressive process in discrete time so we can use linear approximation in deterministic case1 . 1

Our framework does not limit to particular stochastic model for demand; theoretically any controllable demand

model will fit.


5

The difficulty of using network model for control is due to the explosion of state space by keeping track of taxis 2 . Notice that this deficiency vanishes in continuous-time-continuous-space setting since the state spaces are infinite by default. We label a taxi that is not with passengers as “0” (“vacant”), and that is with passengers as “1” (“occupied”)3 . To model the dynamic of supply, we need to have observations on the superset of vacant taxis. T

Let a vector of multi-group traffic density be ρ = (ρT , ρ0 , ρ1 ) , which is in unit of vehicle per unit space. The physics behind this is to simulate the complex interactions of a large number of vehicles by cluster formation and shock wave propagation4 . • ρ0 : density of vacant taxis on [x, x + dx] such that supply S = ρ0 · dx. • ρ1 : density of occupied taxis whose transfers to ρ0 simulate the pick-ups and drop-offs of

passengers. • ρT : density of total traffic flow (including taxis and other vehicles) to include the impact of

traffic conditions on the supply dynamics. Two additional intermediate variables in the system of equations (2) are v(t, x) and λ(t, x): • v: space average velocity of all cars in the traffic (group “T ”) in unit of space per unit time5 . • λ is the net transfer flux (vehicles per unit time) from the group “1” to group “0” . This

variable represents that taxis’ pick-up and drop-off behaviors from a macroscopic perspective and capture the interactions between D and S explicitly. We describe the dynamics of ρ by a fundamental law in physics, the mass conservation in differential form with source term h : ρ t + divx (Q) = h(ρρ)

(2)

2

Combinatorial explosion in describing state variables is obvious. For instance, the total number of configurations of N taxis in V vertices is NV+V−1−1 , and the decision space in next time period grows exponentially in horizon. 3

For simplicity of notation, the arguments (t, x) of variables are sometimes omitted in the following content.

4

Similar setting can be referred in multi-population macroscopic traffic flow model literature (Zhang et al. (2006),

Logghe and Immers (2008), Herty and Moutari (2009), Chu et al. (2011)) . 5

We assume that the average velocity of multi-population is identical, which is called a Keyfitz-Franzer system (Zhang

et al. (2006)). This coincides with the fact that subgroups of traffic can’t exceed the speed of other cars on average.


6

T

where the flux (number of cars per unit time) is a vector Q = (vρ0 + λ, vρ1 − λ, vρT ) . Subscripts in Equations (3) are abbreviated for partial differential derivatives such as ρTt := ∂ρT /∂t. divx is the total derivative with respect to space vector x. The homogeneous form of mass conservation of Equations (2) (i.e. h (ρρ, t, x) = 0) states that the number of vehicles in each group is conserved. They can only transport from one region to its neighbors, or transfer from one group to another group:

     ρ0t + divx (vρ0 + λ) = 0      ρ1t + divx (vρ1 − λ) = 0         ρT + divx (vρT ) = 0.

(3)

t

System of equations (3) is undetermined as it consists of three equations and five random variables (ρT , ρ0 , ρ1 , v, λ). We require two additional equations that describe the relations between these variables to have determined solutions as follows:

s.t.

v = f (ρT )

(4)

λ = λ1→0 (ρ1 ) − λ0→1 (ρ0 , µ)     λ1→0 = αρ1

(5)

   λ0→1 = min{βρ0 , µ · dx}. Equation (4) is called the fundamental diagram of traffic flow f : R+ → R+ that maps density ρT to velocity v 6 . We require that f (ρT ) is a non-increasing function and differentiable almost everywhere. It indicates their average velocity will be slower with higher traffic density. The intergroup transfer rate λ in Equation (5) is assumed to be linearly separable as the difference between the drop-off rate λ1→0 and the pick-up rate λ0→1 . For simplicity of analysis, the drop-off rate is assumed to be linear and the pick-up rate λ0→1 is bounded by demand and vacant taxis and α, β are independent of v. 6

There are many models describing this relationship since Frank Knight first produced the analysis of traffic equi-

librium in 1924, as well as recent developments as (Jiah et al. (2016)).


7

There are two sources of stochasticity in the dynamics of supply. Since drivers are self-scheduling labor, they may leave or enter the system randomly and thus break the conservation of mass in Equation (3). Another unobservable action is deviations from arterial network to local roads, for example, picking up customers at doors. We capture these randomness by adding a vector of T

stochastic source terms h(ρρ, t, x) = (h1 (ρT , t, x), h2 (ρ0 , t, x), h3 (ρ1 , t, x)) on the right-hand sides of Equations (3) respectively:      h1 (ρ0 , t, x) = [a1 (t, x) + b1 (t, x)ρ0 ] d(x · t) + σ1 (t, x)dB1 (t, x)      h2 (ρ1 , t, x) = [a2 (t, x) + b2 (t, x)ρ1 ] d(x · t) + σ2 (t, x)dB2 (t, x)         h3 (ρT , t, x) = [a3 (t, x) + b3 (t, x)ρT ] d(x · t) + σ3 (t, x)dB3 (t, x),

(6)

where the “mean-reverting” term [a(t, x) + b(t, x)ρ] dt dx indicates that the process tends to drift towards its long-term mean over time. The means are assumed to be a linear form of the density (refer to Yang (2012)). The diffusion terms are volatility σ(t, x) times the infinitesimal increment of Brownian Sheet dB(t, x) (2-parameter Gaussian random field). We justify such a supply model reflects the main features in traffic by a series of statistical tests in EC.2. Two separate traffic datasets that contains the vehicle trajectories and traffic flow information are used to verify two major assumptions in this continuous supply model: (a) The average velocity of subgroup vehicles is identical to that to that of total traffic flow; (b) The form of the source term h (t, x,ρρ) is valid. The conservation laws is valid by nature. One prominent advantages of using such a systems of equations to describe the taxis states in fleet is that, ideally, we can predict the state of S at any time t > 0 given initial conditions ρ (t = 0, x) by numerical methods. This predictability can be guaranteed by answering following questions: (a) There always exist theoretical solutions with any well-defined initial conditions; (b) The numerical method to compute the solutions is stable, which is not an easy task even for one-dimensional system of conservation laws (LeVeque (1992)). We first prove that the existence and uniqueness of solutions for the system of equations in Theorem 1. The proof and theoretical solutions are attached in EC.1:


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Theorem 1 (hyperbolicity). The system of stochastic partial differential equations ρ t + Dx (Q) = h(ρρ, t, x) is strictly hyperbolic if: 1. Generalized gradient of f is nonpositive. 2.

∂λ ∂λ 6= . ∂ρ0 ∂ρ1

Therefore ρ is uniquely determined by a system conservation-laws, a transfer rate equation and a fundamental diagram of traffic flow. To guarantee the reachability of states in dynamic programming formulated in the next section, we also describe the numerical method to calculate the systems of SPDE in EC.3. In the appendix, we argue the stability of using Godunov method to simulate the evolution of multi-group supply model given well-defined initial conditions. The important result is Proposition 1 that probes into the exact solutions for the Riemann’s problem, which is the basis for numerical methods. Proposition 1 (Fields of solutions to supply model). There are three of nonlinear waves appearing in the system depending on the initial condition: two fields are linearly dependent and one field is genuinely nonlinear. Extensibility is another major advantage in using continuous approach for dynamic matching. First, it is easy to extend the dimension of ρ to model multi-level service systems and the theoretical results and numerical methods remain viable . Second, comparing to network flow model where the computational efforts will grow exponentially when adding new vertices, our model can be linearly scaled up by adjusting the size of finite elements in numerical solutions.

4. Revenue Maximization The admissible control strategy is g is a family of functions such that u(t) = (p(t, x), c(t, x))x∈[x,¯x] = ¯

gt (ρρ(s, x), u(s, x); 0 ≤ s ≤ t)). This admissible strategy space is enormous with regard to decision horizon, especially because we have two state variable and two control variables, and they are all variables of space and time. In this section, we first describe how the controls affect the demand and supply process, and then formulate the revenue maximization as an continuous stochastic control.


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4.1. Control Strategy on Demand and Supply As described in Section 3, each passenger at location x only observe the current price for ridesharing service p(t, x)7 . We assume that the elasticity of demand D(t, x) = µ(t, x) dt dx is linear to price p such that the intensity function follows: p(t, x) · µ0 (t, x) µ(t, x) = 1 − pmax

(7)

The drivers are assumed to be more strategic as they are directly affected by commissions c(t, x), and constrained by matching results. We assume that each driver have the full observations on the commissions at all locations and they make non-myopic decision individually. Their action spaces include: (a) Being matched with passengers at (t, x). The specific matching strategies is not of interest in this paper, and we assume perfect matching for all regions at all time period, so the realized trips are min{D, S}. (b) Some drivers who are not matched with passengers at (t, x) may decide to relocate to x0 which adding the supply of S (t + 4t, x0 ) with time lag 4t8 . From a microscopic perspective, we can simplify the decentralized decision by assuming that drivers have uniformly distributed preference to relocation such that (1 − θ(c(t, x))) percentage of drivers will relocate if S (t, x) > D(t, x). In case that S (t, x) < D(t, x), we assume all drivers are matched so that θ = 09 ; 7

That being said, passengers make myopic decision immediately based on the current price, and they are not allowed

to reject the service once being assigned a taxi. 8

Potentially, their decentralized decision will introduce a dynamic game and may result in spatial pricing discrimina-

tion (as in Bimpikis et al. (2016)). However, optimizing the platform’s problem using macroscopic model only focus on the output from their decisions at each (t, x). Plus short-term decision should not focus on the equilibrium but the arbitrage in pricing. 9

The exact local matching policy does not affect the dynamics if the policy is not allowed to hold the trip requests

for next time period. We argue that the platform is worse off when it holds trip requests instead of matches with supply immediately in case that discount factor 0 < r < 1.


10

Approximately we can treat these relocated taxis as occupied taxis such that the density after introducing commission becomes: 



ρ(t, x)       0 ρ (t+ , x) =   ρ (t, x)θ(c(t, x))     ρ1 (t, x, c(t, x)) + (1 − θ(c(t, x)))ρ0 (t, x)

(8)

(c) Since ρ1 contains relocated drivers, the drop-off rate λ1→0 is also affected by commissions satisfying dα/dc > 0. We use a linear form such that α = α0 +

c(t, x) α1 in Equation (5). α0 is cmax

the rate of drivers with passengers which is not affected by dynamic commissions; α1 is the rate of relocating drivers that are more willing to start cruising for passengers by observing increasing commissions at location x. Consequently the evolution of traffic flow during (t, ∞) can capture the diffusion process of relocating drivers at t . (d) Activating or deactivating as service providers immediately10 . We capture this disturbance to conservation laws by setting the intercept of drift term as:   c(t, x)   a1 (t, x) = a ˜1 cmax   c(t, x)  a2 (t, x) = 1 − a ˜2 cmax

(9)

4.2. Optimal Control for Deterministic Case For the simplicity of computations, we first consider the deterministic case by ignoring the stochasticity on state variables without loss of generality in control frameworks. We denote the source terms ¯ the demand is approximated by a linear model. Since in supply model with only drift terms as h we decouple the price and commission in control policy g, the platform’s r-discounted expected revenue in infinite horizon is: Vg=

Z 0

10

∞Z x ¯

e−rt [min{D(t, x), S (t, x)} (p(t, x) − c(t, x))] dx dt

(10)

x ¯

For example, increasing local commissions c(t, x) provides incentive for inactive drivers to enter the system at

(t, x). and vice versa.


11

To simplify the notations, we separate the integral of x from the objective function as: Z φ(D, S , c, p) =

x ¯

min{D(t, x), S (t, x)}(p(t, x) − c(t, x))dx,

(11)

x ¯

The initial conditions are set to be D(0) = D0 , S (0) = S0 . The dynamic of state variables are expressed as follows11 :   dD    = χ(p) = µ(p) dx dt   ∂S  ¯ 1 (c, ρ0 ) − Dx (vρ0 + λ(c, p)) + θt (c)ρ0 dx.  = ψ(D, S , c, p) = θ(c) h ∂t

(12)

Notice that the integral on space x is inside the function φ, therefore we only need to consider the calculus on t in the optimal control problem. Both state variables D and S are corresponding continuous and piecewise differentiable12 . In addition we can observe that φ, χ and ψ are their partial derivatives with regard to D and S are continuous. Therefore the state-control pair (D, S , c, p) are admissible for all t. The platform solves a optimal control problem in infinite horizon with dynamic model described in Equation (12).such that ∞

Z

e−rt φ(D, S , c, p)dt

F (D0 , S0 ) = max g

(13)

0

We define the optimal current value function as W : Z W (t) = max g

∞

e−r(s−t) φ(D, S , c, p)ds

(14)

t

It is trivial that F (D0 , S0 ) = W (D0 , S0 ) at t = 0. Comparing to optimizing F , using W (t) simplifies the infinite horizon discounted problem because the integral part of W is independent of t. We have the Theorem 2 to compute the optimal values of F (t) for all t ∈ R+ . Theorem 2 (Bellman equation for current value function). Since the platform has full observations of D(t, x), S (t, x), the Bellman equation for F is: rF (D, S ) = max {φ(D, S , c, p) + χ(p) c(t),p(t)

∂F (D, S ) ∂F (D, S ) + ψ(D, S , c, p) } ∂D ∂S

(15)

11

Demand function in deterministic case uses a linear function to approximate AR process.

12

Demand function is continuous everywhere, and supply model derived from PDE is partially differentiable almost

everywhere depending on the exact form of the fundamental diagram of traffic flow.


12

The proof for the condition of optimality is in EC.4. Since the state at t in deterministic case is only dependent on initial values D0 and S0 , we can calculate the maximum revenue F by solving the partial differential equation in Equation (15). The commonly-used method is “guess-and-verify”. We first guess the exact form of F , solve a maximization problem to obtain (c∗ , p∗ ), then plug in the optimizers and solve a partial differential equation to get F˜ : rF (D, S ) = φ(D, S , c∗ , p∗ ) + χ(p∗ )

∂F (D, S ) ∂F (D, S ) + ψ(D, S , c∗ , p∗ ) ∂D ∂S

(16)

By verification theorem, we find the optimal control if F˜ = F . This guess-verify method can be tedious without strategically choosing the correct form in guessing step. There are also attempts that finding the optimal control using numerical method (for example, linear programming) without solving the Bellman equation directly (Chow (1993)). When implementing the continuous method in practice, the platform needs to solve the dynamic programming numerically in a backward scheme from t = T to t = 0 to obtain the optimal strategies g ∗ (t) = (c∗ (t), p∗ (t)) for each iteration.The computational effort is discussed in Section 5. A trivial variation of on-line (adaptive) control can be achieved by adding an estimation phase in parallel that updates the state variables (D(t, x), S (t)) in each iteration. According to classic separated control rules, this estimation phase is independent of the controls so the conclusion we made are still valid.

4.3. Optimal Control for Stochastic Case The stochasticity in supply model is not trivial when the platform needs to estimate the source terms in conservation laws. In addition, it is possible that the demand model has uncertainty in a more general setting. For the completeness of proof, we also provide the dynamic programming for the stochastic case. The platform’s problem is as follows: g

g

"Z

∞Z x ¯

V =E

# −rt

e 0

[min{D(t, x), S (t, x)} (p(t, x) − c(t, x))] dx dt ,

x ¯

¯ subject to the dynamic of state variables with h instead of h.

(17)


13

Difficulty in formulating the HJB equation for the stochastic case is because of the SPDE formulation of S . Numerically, we can use Monte Carlo method: first sampling paths for h, and solving PDE for each path. This works well is we also solve the PDE by numerical methods. However, an alternative way to stick with the continuous control framework is to use the stochastic representation of weak solutions of conservation laws. This will give a more concise expression for the dynamic of supply in form of backward stochastic differential equation (BSDE). The idea is to solve the Cauchy problem for systems of parabolic perturbations of conservation laws: ρt +

2 + div(Q) = h(ρρ) 2

(18)

In special cases13 the solutions of Equation (18) converges to the solution of hyperbolic conservation laws as → 0 (Rozkosz (2013)). What coincides the celebrated results of the Feynman-Kactype of formulas for viscous conservation laws is that, for each unique bounded weak solution ρ, there exists an unique corresponding solution %14 to a system of BSDE as follows : Z ∞ Z ∞ ∂ρ(t, x) ∂ρ(t, x) ∂Q(t, x) ·· ds − dB(s, x) · %(t, x) = −1 ∂ρ ∂x ∂x t t

(19)

To simplify the notation, we approximately denote the dynamic of supply S with regard to the BSDE form as → 0 as follows15 : dS (t, x) = η(S (t, x))dt + γ(S (t, x))dB(t, x)

(20)

With both dynamics of demand and supply written as a ODE or SDE, we have the following Theorem 3 describing the HJB equation for the stochastic case. Theorem 3 (Bellman equation for stochastic case). The Hamilton-Jacob-Bellman equation for F in stochastic case is: n ∂F ∂F 1 2 ∂2F 1 ∂2F o + η(D, S , c, p) + δ (D ) 2 + γ 2 (S ) 2 φ(D, S ) + χ(p) D,S,p,c ∂D ∂S 2 ∂D 2 ∂S

rF (D, S ) = max

(21)

13

One requirement is that h is a Lipschitz-continuous function of ρ .

14

Uniqueness of solutions of Equation (19) is not guaranteed for systems of conservation laws but a slightly weather

result is shown in literature. 15

Equation (20) is not exactly rigorously proved to uniquely define the weak solution of hyperbolic PDE and we

require more conditions on the quadratic variation of (Q, ρ) to guarantee the convergence to weak solutions.


14

5. Discussion and Conclusion 5.1. Computational Effort The computational effort by using the continuous-time-continuous-space approach is relatively enormous but easy to be parallelized. The effort of guess-and-verify is largely dependent of the choice of initial guess form. What interests us more is the case that the platform instead implements the optimal control by solving the finite horizon version dynamic programming with T is very large16 . Suppose the interval of time and space in discretization are 4t and 4x. In each iteration of the dynamic programming on interval t to t + 4t as shown Section 4, we need to finds the one-step optimizer as a pair of (c∗ , p∗ ) for the state space that consists of all combinations of (D, S ). Each state variable is a size of (¯ x − x)/4x vector. In this local optimization, since the supply model S (t) ¯ is determined by a system of SPDE, we solve a subproblem to compute the evolution from S (t) to S (t + 4t). Numerical method for PDE (Godunov method) solves a Riemann’s problem on the boundaries of neighboring grids in 3-D space. However, there is hope in handling this large optimization. First, for each time interval (t, t + 4t), we can compute optimal control for each state in parallel. Second, we can use look-up tables and specific data structurer to accelerate the computation of PDE on large-size grids of supply. Thirdly, using Feynman-Kac type conversion to SDE can reduce the costs of solving PDE directly. Finally, there are many advanced approximate dynamic programming methods to hasten the processes of value and policy evaluations.

5.2. Future Work To the best of our knowledge, we propose the very first continuous model for ridesharing management. The strength of using continuous approach for dynamic pricing is obvious: the curse of dimensionality disappear naturally in the supply state model and optimal control, and the prices are set to be smooth in space and time. Consequently, heavy computation efforts are required to 16

For instance, the platform develops an adaptive control algorithm using the continuous approach.


15

solve a system of hyperbolic PDE. In addition, we are not sure if this is possible to obtain intuitive heuristics or insights for revenue maximization because of the complicatedness on supply side. Approximate dynamic programming and BSDE are two promising area to make this formulation more tractable. In addition, one can introduce more specific drivers’ decentralized decision model into the supply model. Notice that we do not allow loss of passengers in the matching process, which is not the case in practice. In sum, we develop a continuous-time-continuous-space framework that solves the ridesharing platform’s revenue maximization problem. The platform use decoupled prices and commissions to control the supply and demand dynamically in both deterministic and stochastic cases. The existence and uniqueness of solutions to the supply state dynamics as a system of conservation laws is proved. We solve a dynamic programming to optimize the r-discounted expected revenue in infinite horizon.

Acknowledgments The authors gratefully acknowledge our colleagues, Robert Hampshire, Yafeng Yin, Jiah Song, Abdullah Alshelahi and Smadar Karni from the University of Michigan, who provided insight and expertise that greatly assisted the research.

ec1

e-companion to Luo and Saigal: Dynamic Pricing for On-Demand Ride-Sharing : A Continuous Approach

Appendix Proofs of statements and data validations are described in this E-companion.

Appendix EC.1: Proof of Theorem 1. Theorem 1. The system of stochastic partial differential equations ρ t + Dx (Q) = h(ρρ, t, x) is strictly hyperbolic if: 1. Generalized gradient of f is nonpositive. ∂λ ∂λ 6= . ∂ρ0 ∂ρ1

2.

Thus ρ is uniquely determined by a system conservation-laws, a transfer rate equation and a fundamental diagram of traffic flow. Proof of Theorem 1.

We first show that differential form of conservation laws holds for the case

with inter-group transfer rate λ. In one-dimensional case, the integral form of conservation law for any group of density ρ(t, x) is as follows: Z

x2

Z

x2

ρ(t2 , x)dx = x1

Z

t2

x1

Z

t2

ρ(t, x1 )v(t, x1 )dt −

ρ(t1 , x)dx + t1

Z

t2

ρ(t, x2 )v(t, x2 )dt − t1

¯ λdt, (EC.1)

t1

where we can rewrite the integrals in terms of differential forms in Equations (EC.2),   R t ∂ρ(t, x)    dt ρ(t2 , x) − ρ(t1 , x) = t12   ∂t    R x ∂ (ρ(t, x)v(t, x)) ρ(t, x2 )v(t, x2 ) − ρ(t, x1 )v(t, x1 ) = x12 dx   ∂x     R   ¯ = x2 ∂λ(t, x) dx. λ(t) x1 ∂x

(EC.2)

The left-hand-side of Equation (EC.1) is the number of vehicles within region [x1 , x2 ] at time t2 . It is equal to the number of vehicles at time t1 at the same location, plus the number of vehicle entering the region during time (t1 , t2 ], minus that of leaving the region, and plus the transfer from other groups during time (t1 , t2 ]. Plugging Equations (EC.2) into Equation (EC.1) gives: Z

t2 t1

Z

x2

x1

∂ρ(x, t) ∂(ρ(x, t)v(x, t)) ∂λ(x, t) + − dxdt = 0. ∂t ∂x ∂x

(EC.3)

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Since this must hold for any section [x1 , x2 ] and over any time interval (t1 , t2 ], we conclude that the integrand must be identical zero, so that we get the differential form of the conservation laws, which are first three equations in Equation (3). We define the transfer rate λ(x, t) is the net transfer rate from the occupied taxis group to the vacant one. During time (t, t + 4t], the transfer rate from the group 1 to group 0 is λ1→0 (i.e. dropping off customers), and the reverse process has rate of λ0→1 (i.e. picking up customers). After justifying the differential form of conservation laws in our setting, we need to analyze the characteristics of the system. Assuming that f 0 (ρT ) exists almost everywhere in the range where the solution is known to lie, the Jacobean of Q(ρρ) is as follows:   T 0 T f (ρ ) + ρf (ρ ) 0 0       0 0 T T Dρ (Q(ρρ)) =   ρ f (ρ ) f (ρ ) + λ0 λ1     ρ1 f 0 (ρT ) −λ0 f (ρT ) − λ1

(EC.4)

∂λ(ρ0 , ρ1 ) dλ0→1 (ρ0 ) ∂λ(ρ0 , ρ1 ) dλ1→0 (ρ1 ) = , λ = = . 1 ∂ρ0 dρ0 ∂ρ1 dρ1 The eigen-decomposition of Dρ (Q(ρρ)) are in Equation (EC.5) (v is eigenvalue and r is the

Where λ0 =

corresponding eigenvector):              ν1 = f (ρT ) + λρ0 − λρ1 ,                          ν2 = f (ρT ),                           ν3 = f (ρT ) + ρf 0 (ρT ),           





0        r1 =  1       −1    0         r2 =  λ  1        −λ0    1       ρ0   r3 =   T ρ   1 ρ  ρT

(EC.5)

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Equation ρ t + Dx (Q(ρρ)) = 0 can be written in the quasilinear form: ρ t + Dρ (Q(ρρ))ρρx = 0

(EC.6)

The system is hyperbolic if Dρ (Q(ρρ)) is diagonalizable with real eigenvalues for all values of ρ . It is strictly hyperbolic if the eigenvalues (v1 , v2 , v3 ) are distinct for all ρ . The first assumption required is that the fundamental diagram of speed-density is the (generalized) gradient of the speed is nonpositive, which indicates that the speed of roadway decreases when density increases. The last eigenvector also implies that ρ0 λ0 + ρ1 λ1 = 0, so we can assume that λ0 6= λ1 to guarantee the strict hyperbolicity of the system. In case that λ0 = λ1 , we can add a disturbance term into the Jacobean of Q(u) as follow:   0 0 0 f (ρ) + ρf (ρ)      2 DQ(u) =  ρ0 f 0 (ρ)  f (ρ) + λ0 + λ1     ρ1 f 0 (ρ) −λ0 f (ρ) − λ1 The third field eigenvalue and eigenvector is to solve:      2 λ1 f (ρ) + λ0 +  x0  x0      = v3 ·   −λ0 f (ρ) − λ1 x1 x1 Proposition EC.1 (Fields of Solutions). There are three of nonlinear waves appearing in the system depending on the initial condition. • Field I corresponding to (v1 ,rr 1 ) is linearly dependent, a contact discontinuity. • Field II corresponding to (v2 ,rr 2 ) is linearly dependent, a contact discontinuity. • Field III corresponding to(v3 ,rr 3 ) is genuinely nonlinear, who creates a shock or rarefaction

wave.

Appendix EC.2: Validation of Supply Dynamic Model Two critical assumptions in the state dynamic model are: (a) The average velocity of total traffic flow and subset traffic flow are equal; (b) The source term of the conservation law h is valid. We make hypothesis and test them by real-world traffic data in the following sections.

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EC.2.1. Hypothesis Test I: Equal Speed Assumption of Set-Subset The first test we made is fitting speed in order to test the assumption that subset of traffic (“0” and “1”) has equal speed in traffic flow “T ”. We use the Mobile Century data to verify this assumption Herrera et al. (2010). The source of speed measurements for set-subset are from two types of sensors in this dataset : The average speed of traffic flow T is measured at each loop detectors installed on the fixed locations on highway; The speed of subset are measured from GPS-enabled mobile devices carried on each vehicle. This guarantees that there is no correlation resulting from the measurements. Each point in the histogram is the difference between a pair of matched velocity measurements from two sources, which should be equal within tolerance of errors (as shown in Figure EC.2.1). We can see that the subset of traffic is long-tailed on the left, probably because of individuals who brake harder than the rest. In general, the speed of set-subset is equal.

(a) Figure EC.1

(b)

Regression analysis on the speed of traffic and a random subgroup. (a) histogram of the residuals. (b) qq-plot of the residuals.

EC.2.2. Hypothesis Test II: Source Term The second assumption we need to check is if the formulation of source term of both conservation equations are accurate. Because the source consist of mean reverting term and Brownian sheet term, our strategy is to fit the mean reverting term using least square, and then test if the residual fit normal distribution. We use the NGSIM dataset because it can provide detailed trajectories

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of all vehicles within the interested region (Punzo et al. (2011)). Another convenient feature of NGSIM dataset is that we can calculate density and flow directly from space headway hs and time headway ht . The definition of space headway is the distance from front vehicle’s bumper to the following vehicle’s bumper, and time headway is the same measure in terms of time. Density of the traffic flow is therefore a function of average space head: ρ=

n n 1 = Pn i = ¯ s2 − s1 hs i=1 hs

Similarly the flow is a density of time headway. q = vρ =

(EC.7)

1 ¯t . h

The fitting of source term is a two-step process. We fit the homogeneous conservation equation first, and then calculate the residual after subtracting the mean reverting term. Discretization is required for calculating the partial derivatives and parameters in the mean reverting term17 . The color map of average speed profile is shown in Figure EC.2.

Figure EC.2

NGSIM I-80 data speed profile (ft/sec) in 15-minute, x-axis is time in second, y-axis is distance to staring point. We aggregate all trajectories on lane 2 to 4 to exclude high-occupancy toll lane.

For each grid, compute ρt and ρsub by t ∂ρ ρt+∆t − ρt ≈ ∂t ∆t

(EC.8)

In case that x is one-dimensional, we have (vρ)s = 17

∂vρ vρ(i+1,j) − vρ(i−1,j) ≈ ∂s 2∆s

(EC.9)

This is achieved by dividing time-space region into grids, which are 10 seconds × 100 feet. Choosing this size is

because the average length of car is 16 feet so that 100 feet can at least catch one data in 10 second interval.

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Compute D(vρ) and D(vρsub ) in two-dimension by D(vρ) =

∂(vρ) (∂vρ) vρ(i+1,j) − vρ(i−1,j) vρ(i,j+1) − vρ(i,j−1) + ≈ + ∂x ∂y 2∆x 2∆y

(EC.10)

The estimators of the mean reverting term and the standard deviation of the Brownian sheet are shown in Figure EC.3 with their legends on the right. We can observe that b is non-positive in most regions, and the residual σ is small.

Figure EC.3

Estimators of Conservation Laws based on I-80 data.

For a given grid, if the flow data or (density and speed data) is missing in the dataset, we can use the average value from nearest neighbor grids. After cleaning the data, the residuals fit our normality assumption with R2 close to one for both total traffic and subset traffic cases, and the data points in QQplot lie along to the cross diagonal line. The results are shown in Figure EC.4.

Appendix EC.3: Numerical Method for Supply Dynamic Model EC.3.1. Godunov’s Method for PDE The numerical method starts from solving the system without source terms. The plane of time t and space x is divided to grids with size (4t, 4x) and Godunov method solves Riemann problems

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Figure EC.4

Normality test for residuals. The first row is the residual histogram and QQplot of the total traffic; the second row is the residual histogram and QQplot of the subset of traffic.

forward in time at the boundary of two neighboring grids. In order to guarantee that the waves do not interact with each other, we require that the average velocity of traffic within the grid is less than the ratio 4s/4t. Equivalently, 4t, 4s need to satisfy the Courant-Friedrichs-Lewy (CFL) condition18 where rk is the eigenvalues in Equation (EC.5) : |rk |4t < 4s, ∀rk

(EC.11)

The next step is to add the stochastic source term on right-hand side. The mean reversion term is estimated from the regression analysis as EC.2.2. The solver generate a random path of diffusion term (a Brownian sheet since it is a variable of time and location) in each iteration, and take the average of all sample paths as the estimator for h . The simulation of Brownian sheet dW (s, t) is to generation normally distributed variable dW ([s, s + ds] × [t, t + dt]) ∼ N (0, V ar(dW (s, t))). 18

It means that the solver needs to adjust the time step in each iteration to satisfy this condition.

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From Ito’s integral, dW (s, t) = W (s + ds, t + dt) − W (s, t + dt) − W (s + ds, t) + W (s, t), and its mean and variance are:     E[dW (s, t)2 ] = dx · dt

(EC.12)

   V ar(dW (s, t)) = 2 · dx2 · dt2 EC.3.2. Exact Solutions for Riemann’s Problem In this section, we discuss the basic for numerical method solving the system of PDE in Equation (2). From the proof of Theorem 1, it can be shown using the implicit function theorem that these solutions curves (called “Hugoniot Locus”) exist locally in a neighborhood of current point and these curves are smooth if the flow function is smooth. The numerical method solves Riemann’s problem on each boundary of two neighboring grids. The initial condition of the Riemann Problem is a simple jump of density ρ with values u at a given point x ˆ. We denotes u − as the left-side density value of the jump point and u + as the right-side value. From the Hugoniot locus, there are four cases of the jump depending on the initial condition as described in the figures below. u∗ is the intermediate state we need to determine according to Lax Entropy Condition (LeVeque (1992)). For the 3-D system of conservation laws, we know that the Hugoniot locus are 3-D curves in (ρ, ρ0 , ρ1 ) space. From Theorem 1 the system is strictly hyperbolic. According to the Proposition EC.1, the genuinely nonlinear field (field 3) are straight lines from origin with slope of

ρ0 ρ

on ρ-ρ0

projection, and contact discontinuity are hyperplanes vertical to ρ axis. The routine of contact discontinuity on ρ0 − ρ1 plane is the degenerate of eigenvectors on the first two fields. The Hugoniot locus is determined by states before and after the jump and hierarchy of curves. The eigenvalues v1 , v2 , v3 have three possible orders (Case I, Case II and Case III), and the inequality of ul and pr within each group may also vary, in total there are 24 Hugoniot Locus as follows. Case I: v1 > v2 > v3 ⇔ λ0 − λ1 > 0. Case II: v2 > v1 > v3 ⇔ ρf 0 (ρ) < λ0 − λ1 < 0. Case III: v2 > v3 > v1 ⇔ λ0 − λ1 < ρf 0 (ρ).


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For each case, ρl < (>)ρr , ρ0l < (>)ρ0r and ρ1l < (>)ρ1r with their combinations consist 8 initial conditions. We only plot the ρl < ρr , ρ0l < ρ0r , ρ1l < (>)ρ1r condition in Figure EC.5, Figure EC.6 and Figure EC.7. Notice that the order of eigenvalues implies whether intergroup transfer or flow-in and flow-out has greater impacts on the state of system.

(a) Figure EC.5

(b)

(c) 0

Hugoniot Locus for Case I: (a) 3-D Hugnoit Locus; (b) Hugnoit Locus on ρ -ρ1 plane of ρ = 20; (c)Solution of Case I on x-t plane with initial jump on x0 at t = 0; v3 curve is shock wave, v2 and v1 are contact discontinuity.

(a) Figure EC.6

(b)

(c) 0

Hugoniot Locus for Case II: (a) 3-D Hugnoit Locus; (b)Hugnoit Locus on ρ -ρ1 plane of ρ = 20; (c)Solution of Case II on x-t plane with initial jump on x0 at t = 0; v3 curve is shock wave, v2 and v1 are contact discontinuity.

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(a) Figure EC.7

(b)

(c) 0

Hugoniot Locus for Case III: (a) 3-D Hugnoit Locus; (b) Hugnoit Locus on ρ -ρ1 plane of ρ = 20; (c)Solution of Case III on x-t plane with initial jump on x0 at t = 0; v3 curve is shock wave, v2 and v1 are contact discontinuity.

Here we show a simulation of this system of SPDE on 2-D space. The initial condition is a smooth function ρ = C · y exp(−(x2 + y 2 )). Constant C = 500 for ρ0 and 2500 for ρT . We use splitting method to decompose the problem in space with first-order accuracy.

(a)

(b)

(c) Figure EC.8

Evolution of subgroup density ρ

(d) sub

on the x-y plane.


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Appendix EC.4: Proof of Theorem 2. Theorem 2. Since the platform has full observations of D(t, x), S (t, x), the Hamilton-JacobBellman equation for the F is: rF (D, S ) = max {φ(D, S , c, p) + χ(p) c(t),p(t)

Proof of Theorem 2.

∂F (D, S ) ∂F (D, S ) + ψ(D, S , c, p) } ∂D ∂S

The platform’s problem is: F (D(t, x), S (t, x)) = max V g g   dD    = χ(p) dt   ∂S   = ψ(D, S , c, p) ∂t

(EC.13)

The Hamiltonian is defined as: ˆ, H(t, D, S , c, p, λ) = e−rt (φ(D, S , c, p) + λ1 χ(p) + λ2 ψ(D, S , c, p)) = e−rt H

(EC.14)

ˆ . The maximum where λ1 and λ2 are co-state multipliers for the current-value Hamiltonian H principle gives necessary conditions for optimality as follows: ˆ ˆ ∂H ∂H 1. = 0 and = 0 for all t ∈ R+ (or weaker first-order condition satisfying that (c∗ , p∗ ) is ∂c ∂p ˆ ). the optimizers for H ˆ ˆ dλ1 ∂H dλ2 ∂H = rλ1 − and = rλ2 − for all t ∈ R+ . ∂D dt ∂S dt ˆ = 0. 3. Transversality conditions: limt→∞ e−rt H ˆ ˆ ∂H ∂H = χ(t), = ψ(t). In addition we have ∂λ1 ∂λ2 2. Wherever g ∗ is continuous,

The sufficient conditions are provided by Mangasarian sufficient conditions for discounted infinite-horizon problems (Mangasarian (1966)) and Arrow sufficient theorem with regard to the concavity of the Hamiltonian. We derive the Bellman equation for the revenue maximization by define W (t) as19 : Z W (t) = max g

19

∞

e−r(s−t) φ(D, S , c, p)ds

t

Introducing W is a common technique in solving infinite horizon autonomous problem in economics.

(EC.15)

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Since F is differentiable in t and almost differentiable everywhere for D and S , we have the Bellman’s Equation of F as: n ∂F ∂F o −Ft = max φ(D, S ) + χ(p) + ψ(D, S , c, p) D,S,p,c ∂D ∂S

(EC.16)

Notice that:      Ft = −re−rt W (D, S )      ∂W FD = e−rt   ∂D     ∂W   FS = e−rt ∂S

(EC.17)

Substituting it into the Bellman’s Equation of F (t) and we have: n ∂W (D, S ) ∂W (D, S ) o φ(D, S , c, p) + χ(p) + ψ(D, S , c, p) c(t),p(t) ∂D ∂S

rW (D, S ) = max

(EC.18)

It provides a natural Dynamic Programming scheme to calculate W (t) for all t ∈ R+ and the optimal value of revenue F (D, S ) = W (t). Alternatively we can rewrite the Equation (EC.18) as Theorem 2 with regard to F .

Appendix EC.5: Proof of Theorem 3 Theorem 3. The Hamilton-Jacob-Bellman equation for F in stochastic case is: n ∂F ∂F 1 2 ∂2F 1 ∂2F o φ(D, S ) + χ(p) + η(D, S , c, p) + δ (D ) 2 + σ 2 (S ) 2 D,S,p,c ∂D ∂S 2 ∂D 2 ∂S

rF (D, S ) = max Proof of Theorem 3

The platform’s problem in stochastic case is: g

s.t.

Z

∞ −r(s−t)

φ(D, S , c, p)ds

F (D0 , S0 ) = max E e D,S,p,c 0     dD(t, x) = (µ − D)d(x · t) + δdB(t, x)

(EC.19)

   dS (t, x) = η(S )dt + γ(S )dB(t) The current value function is: g

∞

Z

W (t) = max E g

e t

−r(s−t)

φ(D, S , c, p)ds

(EC.20)

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The optimal value function corresponds to filtration Ft is: g

∞

Z

e

V (t) = max E g

−rs

t

Z

e−rs φ(s)ds + W (t)

φ(s)ds|Ft =

0

0 ∗

V (t) is a martingale only if g is the optimal strategy such that Eg [V (t)|Fs ] = V (s) for all s < t. By taking a small interval of time and drive it to 0, we have the HJB equation as: n o 1 φ(D, S ) + E[dF (D, S )] D,S,p,c dt

rF (D, S ) = max

(EC.21)

We can substitute F by W and notice that at t = 0 they are equal. We argue that, similar to the proof in EC.4, it is equivalent to plugging in F . Applying Ito’s lemma with regard to F : ∂F ∂F 1 2 ∂2F 1 ∂2F 1 E[dF (D, S )] = χ(p) + η(D, S , c, p) + δ (D ) 2 + γ 2 (S ) 2 dt ∂D ∂S 2 ∂D 2 ∂S

(EC.22)

We have the final HJB without stochasticity as follows, and complete the proof. n ∂F ∂F 1 2 ∂2F 1 ∂2F o φ(D, S ) + χ(p) + η(D, S , c, p) + δ (D) 2 + γ 2 (S ) 2 D,S,p,c ∂D ∂S 2 ∂D 2 ∂S

rF (D, S ) = max

(EC.23)

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