Aggregate Modeling and Equilibrium Analysis of the Crowdsourcing Market for Autonomous Vehicles
AAggregate Modeling and Equilibrium Analysis of theCrowdsourcing Market for Autonomous Vehicles
Xiaoyan Wang a Xi Lin a Meng Li a,b,* a Department of Civil Engineering, Tsinghua University, Beijing 100084, P.R. China b Center for Intelligent Connected Vehicles and Transportation, Tsinghua University, Beijing 100084, P.R. China
Abstract
Autonomous vehicles (AVs) have the potential of reshaping the human mobility in a wide va-riety of aspects. This paper focuses on a new possibility that the AV owners have the option of”renting” their AVs to a company, which can use these collected AVs to provide on-demandride services without any drivers. We call such a mobility market with AV renting optionsthe ”AV crowdsourcing market”. This paper establishes an aggregate equilibrium model withmultiple transport modes to analyze the AV crowdsourcing market. The modeling frameworkcan capture the customers’ mode choices and AV owners’ rental decisions with the presence oftraffic congestion. Then, we explore different scenarios that either maximize the crowdsourc-ing platform’s profit or maximize social welfare. Gradient-based optimization algorithms aredesigned for solving the problems. The results obtained by numerical examples reveal thewelfare enhancement and the strong profitability of the AV crowdsourcing service. However,when the crowdsourcing scale is small, the crowdsourcing platform might not be profitable.A second-best pricing scheme is able to avoid such undesirable cases. The insights generatedfrom the analyses provide guidance for regulators, service providers and citizens to makefuture decisions regarding the utilization of the AV crowdsourcing markets for serving thegood of the society.
Keywords : Crowdsourcing; Autonomous vehicles; Mobility market; Equilibrium; Pricing
With the rapid development of artificial intelligence and communication technology in recentyears, autonomous vehicles (AVs) are undergoing rapid development. There has been an intenseeffort by researchers and manufactures to enable AVs to handle a wide range of driving condi-tions that can be encountered during the road travel. In 2018, public road tests for AVs to provideon-demand ride service were performed, and in 2020, some major transportation network com-panies (TNCs) such as Lyft, Waymo and Baidu have launched commercialized AV on-demandride service programs in certain areas around the globe. It is predicted that approximately 40%of vehicle travel could be autonomous in the 2040s and traveling by AV may become the predom-inant transport mode in the future. It is widely believed that due to the increased mobility and * Corresponding author. E-mail address: [email protected]. a r X i v : . [ ec on . GN ] F e b afety, reduced congestion, emissions and travel costs, AVs have the potential to reshape futurehuman transport.High-tech AVs can perform tasks that cannot be performed by manually driven vehicles(MVs). One of the most distinct differences between AVs and MVs is that AVs owned by aperson can be used for other tasks when not used by the owner such as traveling to other placeswithout drivers to complete certain tasks. One potential use is to serve other riders (Figure 1 forillustration). This would provide opportunities for people who do not own AVs to utilize theservice. Figure 1: One day’s task chain of an autonomous vehicleWhen a large number of AV owners are able to share their AVs to provide ride servicesto others, a platform is required to aggregate the information of those AVs with that of thecustomer demands, and act as a bridge connecting the AV owners with the customers. Similarto the current TNCs such as Uber and Lyft, the platform identifies the dynamically-updatedsupply from AV owners, and then matches the available AVs with the customers generatedwithin the region in a real-time fashion. The platform needs to pay these AV owners a certainamount of money to encourage them to share their AVs, and meanwhile the platform chargesthe customers a certain fee for their trips; the difference between the platform gain from travelersand the platform payment to AV owners is the revenue that should be equal to a considerableamount so that people have the incentive to establish and operate such a service.The aforementioned new type of mobility market with AV renting options is called ”the AVcrowdsourcing market” in this study. The concept of crowdsourcing, first introduced by Brab-ham in 2008 and later perfected by other researchers using case studies, refers to the act of acompany handing over a part of its tasks, particularly needs-based problems, such as hatchingideas, designing algorithms and public supervision to the public crowd who have the poten-tial to finish the job better than employees. Developed communication technologies enable suchbusiness model to be realized, through which a wide range of the public including but not lim-ited to social experts are attracted and involved to produce a better outcome for the companies.For the AV crowdsourcing market, our primary interest is to answer two key questions: 1) canAV crowdsourcing benefit society? and 2) can the operating agency gain from operating thecrowdsourcing platform? The first question relates to the utilitarian objective of such businessmodel that tells social managers whether they should allow or regulate the market, and the sec-ond question is related to the spontaneity problem whether people have the incentive to runsuch a business. To answer these questions, we establish an aggregate equilibrium modelingframework in this study. The model considers three market subjects: travelers, AV owners andthe crowdsourcing platform, where AV owners and travelers may have some coincidence withina time period. The platform determines the payment to the AV owners and the charge to thetravelers, and the travelers and AV owners determine their mode choices and rental choices. Un-2er a user’s utility-maximization framework, we can capture the equilibrated state of the wholesystem. Then, based on the equilibrium state, the platform can adjust the payment and pricingstrategies to achieve different goals, including profit maximization, social welfare maximization(first-best), and constrained social welfare maximization (second-best). Numerical experimentsand sensitivity analyses are conducted to draw a variety of insights on the AV crowdsourcingmarket.The rest of this paper is organized as follows. Section 2 reviews related literature on AVs,ride-providing services and crowdsourcing. In Section 3, we develop the static equilibrium modelfor the AV crowdsourcing market, and investigate some properties of the model. Section 4formulates the platform’s decision framework across different market scenarios with varioustransport modes, and then propose a gradient-based algorithm to solve the problem. In Section5, we use numerical examples to illustrate key points and insights of our model, and then presentsome discussions. Finally, Section 6 concludes the paper. The body of literature on AVs has been continuously growing in recent years. Control of fullyAVs may be the best investigated topic (Wu et al., 2020), including route planning (Wang et al.,2019; Bang and Ahn, 2018), longitudinal control, i.e. Cooperative Adaptive Cruise Control (Wanget al., 2014; Gong and Du, 2018), and lateral control of autonomous vehicles (Luo et al., 2016; Yanget al., 2018). As a sophisticated scenario in transportation field, control of an isolated intersectionhas been studied with ”signal-free” schemes (Lee and Park, 2012; Xu et al., 2018) and ”signalized”schemes (Li et al., 2014; Yu et al., 2018) where the aim is to increase intersection capacity andreduce delay on the basis of driving safety. Recently, Chen et al. (2020) proposed a rhythmiccontrol scheme where CAVs pass through an intersection with a preset rhythm, and Lin et al.(2021), further expended the rhythmic control to a grid network. Meanwhile, studies found thatAVs with communication technologies that can smooth oscillations and reduce crash probability,will greatly improve the capacity of a highway (Michael et al., 1998; Tientrakool et al., 2011; Bianet al., 2019) and reduce road congestion (Fagnant and Kockelman, 2015; Sun et al., 2020). AVsare highly advantageous for providing high-level transport experience so that they will changeor even reshape future travel modes. One change will revolutionize the taxi industry. Literatureon AV on-demand ride service mainly focuses on its operation. For example, Vosooghi et al.(2019), considering multi-modal dynamic demand, investigated SAV operations including thefleet size, vehicle capacities, ridesharing and rebalancing strategies through simulations. Tanget al. (2020) proposed an advisor-student reinforcement learning framework to organize theautonomous electric taxi fleet in an online manner.Although there is scarcely any literature on AV on-demand ride service, a wide range ofstudies on the traditional taxi market and on-demand ride service market have been performedthat provide great inspiration for our study. Yang and Wong (1998) used a network model toestablish the equilibrium of the cruising taxi market and offered some policy-relevant results in-cluding average taxi utilization and average customer waiting time for decision making. Daganzoand Ouyang (2019) presented a general analytic framework to model transit systems that providedoor-to-door service, including non-shared taxi and shared taxi service. Studies pointed out thatboth the traditional taxi market and on-demand ride market has similar economic rules. Arnott(1996) performed an economic analysis of the dispatch taxi market and found that the first-besttaxi pricing entails operation at a loss under economies of density. Zha et al. (2016) using an3ggregate model of the on-demand ride market, pointed out that under economies of scale, thecompany will suffer a deficit in the first-best scenario. Vignon and Yin (2020) considered thedual modes of ride-sourcing and ride-sharing and also pointed out that when congestion is low,the service provider must be subsidized to achieve the first-best scenario. Vignon and Yin (2020)developed several concrete approaches to achieve the second-best scenario, including regulationof prices, vehicles’ vacant duration and the number of drivers. In this paper, we will show that insome circumstances, the AV crowdsourcing platform can be profitable in the first-best scenario.Other influential works on the regulation of the MV on-demand ride market include Yu et al.(2017) that analyzed the regulation of a market with both traditional taxis and on-demand rideservice, and further proved that on-demand ride service is competitive with the traditional taxiservice. He et al. (2018) modeled the effects of the customers’ reservation cancellation behaviorson the network and designed the pricing and penalty strategy. Wang and Yang (2019) have madea comprehensive review of the literature on on-demand ride services. However, none of theabove works considered the future transport modes in which AVs can drive autonomously andcan serve customers on their own (Zmud et al., 2018; Stocker and Shaheen, 2018; Narayanan etal., 2020). The potential AV crowdsourcing market enabled by the driverless feature of AVs hasbeen seldom investigated in the existing literature.Crowdsourcing has boomed in various industries in recent years (Brabham, 2008). It canbe applied to collect ideas to improve products or services (Bayus, 2010; Poetz and Schreier,2012; Bayus, 2013), city inspection (Kang et al., 2013; Glaeser et al., 2016) and other applications.Platforms such as Threadless and IStockPhoto collect and share designer’s talents around theworld; Kaggle and TunedIT gather studies and solutions to difficult problems; other companiessuch as PeoplePerHour, Crowdspirit, Editzen and MusikPitch are examples of crowdsourcing(Zhao and Zhu, 2014). A business model in the hotel industry similar to AV crowdsourcingis shared lodging where one of the typical companies is Airbnb (Zervas et al., 2017). The AVcrowdsourcing that we propose in this paper has been referred to in Stocker and Shaheen (2018)and described as hybrid AV ownership with the same entity operating. Stocker and Shaheen(2018) preliminary listed potential shared autonomous vehicle business models but did not makea detailed analysis of them; to bridge this gap, such models are investigated in the present paper.
This section introduces the equilibrium model of the AV crowdsourcing market. We consider atime period with duration h (e.g., 2 hours) for which the market conditions are time invariant,and therefore we can treat it as a stationary state. In this hypothetical market, there exists acrowdsourcing platform; the AV owners have the option to rent their AVs to this crowdsourcingplatform, and the crowdsourcing platform collects these AVs to provide on-demand ride servicesfor other riders. In addition to the collected AVs, the crowdsourcing platform can also buy someAVs from external markets to provide the services. On the other hand, travelers can choose touse the on-demand ride services, to drive private (manual or autonomous) vehicles or to takepublic transit to fulfill their travel needs (the average trip distances of taking different modes areidentical); meanwhile, AV owners can choose to rent their cars to this crowdsourcing platform.Traffic congestion induced by these trips is considered in the framework. A schematic illustrationis provided in Figure 2.In the following, we will introduce the details of the equilibrium model establishment. Wefirst describe the equilibrium state of the on-demand ride service; and then, we model the travel-4igure 2: Conceptual design of an AV crowdsourcing marketers’ mode choices as well as the AV owners’ rental choices based on the utilities associated withthese selections; and finally, we describe the equilibration of supply and demand in a steady-statemobility market with AV renting options. We separate one-day time into several periods and use piece-wise fixed level of demand to ap-proximate the time-variant demand. In this section, we focus on a single period where theplatform set prices to equilibrate the supply and demand of on-demand rides. On-demand rideservices are composed of matching, pick-up and on-travel processes. The matching is assumedto be a bilateral process where both the available AVs and the customers actively search for eachother. Since this process does not display distinctive differences from the existing on-demandride services, we can adopt Zha’s formula to describe the relation between the customers’ aver-age searching time w c and the AVs’ average searching time w t . Derived from the Cobb-Douglasmatching function, the customers’ average searching time function at the equilibrium can beformulated as follows: w c = q ( − α − α ) / α o A − α ( w t ) − α / α (1)Upon being matched, the AV drives itself to the customized location to pick up the cus-tomer. We assume that the matching algorithm assigns to every request one of the closest ( w t q o ) available vehicles. Given the size of the city R , the average pick-up time t p ≈ k ( w t q o ) − √ R / v ,where k is a parameter depending on the network topology and k ≈ v is the cruising speed. The average trip time from picking up acustomer to arriving at the destination is t r = κ √ Rv , where κ is a constant reflecting the averagetravel distance. Then, we obtain the following equation between the average pick-up time t p andthe average trip time from picking up a customer to the arrival at the destination t r : t p = θ (cid:0) w t q o (cid:1) − t r (2)where θ = k / κ is a city-specific constant. 5ext, we discuss the modeling of traffic congestion. Since we have constructed the relationbetween the average pick-up time t p and the average trip time t r , in the following we focus onthe effect of traffic congestion on the average trip time t r . The functional form is given by thefollowing equation: t r = T s (cid:104) q m + α ( q a − q o ) + α q o ( + θ ( w t q o ) − ) ; R , v (cid:105) (3)In the above equation, T s ( · ) is a continuously differentiable, monotonically increasing andconvex function; q m , q a , q o are unit-time travel demands for manual driving, autonomous driv-ing (including both private vehicles and crowdsourcing vehicles) and on-demand ride service,respectively; α ∈ (
0, 1 ) is a parameter representing the ”relative occupation” of AVs comparedto MVs (because the AV technology can reduce the vehicle headway in traffic flows). The equa-tion states that the average trip time t r is a function related to city size R and free-flow cruis-ing speed v , and it is a monotonically increasing and convex function with regards to the term q m + α ( q a − q o ) + α q o ( + θ ( w t q o ) − ) . In this term, q m represents the MVs’ contribution to conges-tion, α ( q a − q o ) represents privately-used AVs’ contribution to congestion, and α q o ( + θ ( w t q o ) − ) represents the crowdsourcing AVs’ contribution to traffic congestion; the latter has a factor of ( + θ ( w t q o ) − ) because it includes both the pick-up trips and delivery trips, where θ ( w t q o ) − isassociated with Eq. (2). The functional form of T s ( · ) can be obtained from realistic data.Despite their analytical formulation, there are some mathematical difficulties associated withEqs. (1)-(3). The main difficulties lie in the terms ( w t ) − α / α in Eq. (1) and ( w t q o ) − in Eqs. (2)and (3), where there are negative powers associated with these variables; this gives rise to dis-continuity and possibly results in imaginary numbers in the process of solving the equilibrium.To address this issue, we introduce some continuous counterparts to replace the above discon-tinuous functions. The idea is as follows (taking ( w t ) − α / α as example). We first choose a verysmall positive number (cid:101) . Then, when w t ≥ (cid:101) , the counterpart takes exactly the same form as ( w t ) − α / α ; and when w t < (cid:101) , the counterpart is a linear function such that the whole function iscontinuous and smooth at (cid:101) . The same approximation is used for ( w t q o ) − . This approximationis reasonable because w t and w t q o only have physical meaning when they are positive, and a suf-ficiently small (cid:101) can ensure that the approximations are almost the same as the original values onthe positive horizon. In the following, we denote the two approximations as ξ ( w t ) and ξ ( w t q o ) .The graphical illustration of the approximation is shown in Figure 3.With the above-mentioned function approximation scheme, we can express the approxi-mated counterparts of Eqs. (1)-(3) as described below. These expressions will be used in themodel analysis and algorithmic development. w c = W c ( q o , w t ; A , α , α ) (4) t p = T p ( w t , q o , t r ; θ ) (5) t r = T r ( q m , q a , q o , w t ; R , v , θ ) (6) AV owners have the incentive to rent their cars to the crowdsourcing platform only if thisaction can earn profit. Suppose that the rental depends on the serving rates, i.e., for a period6igure 3: Illustration of function approximationwith duration h , the rental revenue for the AV owners is given by ( pn ) , where p is the paymentper ride and n is the expected number of on-demand rides served by an AV in this period.The rental revenue can be viewed as the ”commission fee” for the vehicle providers. Then, theutility of a citizen renting his/her private AV to the crowdsourcing platform for each period withduration h is given by ( pn − m ) , where m is a constant indicating the expected additional costof sharing the private vehicle for public use, including energy consumption, vehicle depreciationand psychological costs. The AV owners who choose not to rent their cars will receive zero rentalutility.Next, we discuss the mode choices for citizens with travel needs in this period. The travel(dis)utility includes time costs and monetary costs. We respectively specify the perception ofon-board time for AVs, public transit and MVs as β A , β P and β M , and the relation among themis β A < β P < β M ; this is because driving a car costs the most human effort, and taking publictransit should be generally less comfortable than taking an AV. The perception of waiting fora crowdsourcing AV is specified as γ (The waiting time is composed of matching and pick-uptime). On the other hand, it is assumed that the average travel time t n and fare F n of takingpublic transit are constant, i.e., unaffected by traffic congestion (e.g., BRT vehicles or subway).Given the average fare F o for each trip on a crowdsourcing AV, the utility of taking on-demand7ide service can be stated as ( − F o − β A t r − γ ( w c + t p )) . The utilities of other transport modescan be similarly expressed. The utilities for all choices are shown in Table 2, and the (dis)utility V ( i , j ) x can be calculated by adding up the rental utility and travel utility in each choice ( i , j ) forthe citizens of class x .In the AV crowdsourcing market, we assume that sensitivity to travel utility and rental utilityfor one class of people are identical; and the random variables are identically distributed with aGumbel density function. Then, the probability for the choice ( i , j ) by people of class x can becaptured by the following logit model: π ( i , j ) x = e µ x V ( i , j ) x ∑ ( i , j ) ∈C x e µ x V ( i , j ) x ∀ ( i , j ) ∈ C x , x ∈ {
1, 2, . . . , 6 } (7)where µ x is a nonnegative parameter representing the degree of uncertainty for the choices from8he perspective of people of class x . A larger parameter µ x means that people have more knowl-edge about the AV crowdsourcing market and thus can be more certain about their utilities, i.e.,an increasing number of class x citizens would give up the choice ( i , j ) when V ( i , j ) x decreases bya unit. In modeling customer choices, however, it is currently unknown that whether the sensi-tivity coefficients associated with travel and rental choices are similar. If they are not identical,we need a nested logit model to capture the choice probabilities. The detailed discussions of thenested logit models are presented in Appendix B. This section establishes the equilibration between supply and demand. For crowdsourcing AVs,the supply equation can be written as: N = N s + N r (8)where N is the total number of usable AVs for the crowdsourcing platform in this period; N s isthe number of pre-purchased AVs by the crowdsourcing platform; and N r is the number of AVscollected from crowdsourcing in this period. The number of AVs crowdsourced from the societycan be expressed by the following equation: N r = d a (cid:16) π P , Ra + π O , Ra (cid:17) + d a (cid:48) π − , Ra (cid:48) + d b (cid:16) π P , Rb + π M , Rb + π O , Rb (cid:17) + d b (cid:48) π − , Rb (cid:48) (9)where the first to last terms of the RHS represent the number of AVs from citizens belonging toclasses 3-6, respectively.Similarly, on the demand side, the equilibration can be expressed as: q o h = d n π O , − n + d r π O , − r + d a (cid:16) π O , Na + π O , Ra (cid:17) + d b (cid:16) π O , Nb + π O , Rb (cid:17) (10) q a h = q o h + d a π A , Na + d b π A , Nb (11) q m h = d r π M , − r + d b (cid:16) π M , Nb + π M , Rb (cid:17) (12) q p h = d n π P , − n + d r π P , − r + d a (cid:16) π P , Na + π P , Ra (cid:17) + d b (cid:16) π P , Nb + π P , Rb (cid:17) (13)In Eq. (10), the LHS is the total number of customers using the on-demand ride services inthis period, and the RHS is the sum of the travelers choosing the associated mode. Eqs. (11)-(13)express similar equilibria for the AV riders (including privately used ones and crowdsourcingones), for manually driving, and for taking public transit, respectively.According to the stability requirement, the number of trips of the on-demand rides shouldbe equal to the total number of crowdsourcing AV riders during a period with duration h : Nn = q o h (14)9here N is the total number of crowdsourcing AVs and n is the expected number of the on-demand rides served by an AV in this period.Lastly, the sum of searching time, pick-up time and on-board time for all AVs put into useshould be equal to their total service time. Therefore, the service time constraint in view of unittime is modeled as follows: N = q o · (cid:0) w t + t p + t r (cid:1) (15)where q o is the number of crowdsourcing trips and (cid:0) w t + t p + t r (cid:1) is the service time for a singletrip by a crowdsourcing AV.Aggregating Eqs. (4)-(15) yields the equilibrium model of the AV crowdsourcing market.The following subsection will specify some basic properties of the model. For the platform, on the demand side, the trip fare of on-demand ride service F o is treated asthe decision variable; on the supply side, the commission fee per ride p is treated as the decisionvariable. When fixing these two decision variables, the utility equations along with Eqs. (4)-(15)together constitute a system of nonlinear equations that can be used to obtain the equilibriumstate of the AV crowdsourcing market, i.e., the intermediate variables including choice splits ofdifferent travel modes, on-demand ride service rate and total travel costs.We examine the existence of the equilibrium solution in this subsection. The statement ispresented as follows. Proposition 1.
The system of nonlinear equations (4)-(15) has at least one solution when fixing F o and p.Proof. We apply the fixed point theorem to prove the existence. The proof is based on Schauder’sfixed-point theorem: let C be a closed convex subset of the Banach space and suppose f : C (cid:55)→ C and f are compact (i.e., bounded sets in C are mapped into relatively compact sets), and then f has a fixed point in C .Let the constant parameters d mv and d av respectively denote the total number of MVs andAVs owned by the citizens. We focus on four variables ( t r , n , t p , w c ) , and the system (4)-(15) canbe treated as a mapping Φ such that ( ˆ t r , ˆ n , ˆ t p , ˆ w c ) = Φ ( t r , n , t p , w c ) , and the equilibrium requires ( ˆ t r , ˆ n , ˆ t p , ˆ w c ) = ( t r , n , t p , w c ) . The mapping works as follows: it first computes all choice splitswith the logit model Eq. (7), and obtain ˆ n by Eq. (14), obtain w t by Eq. (15); and then with w t we obtain ˆ t r , ˆ t p and ˆ w c by Eqs. (4)-(6). Clearly, the mapping Φ is continuous. To proceed, we firstprove the following result. Claim : when max ( t r , t p ) → + ∞ , ˆ t r , ˆ t p are upper bounded by constants.To prove this claim, we note a simple fact that when t r or t p approaches infinity, q o willapproach zero with an exponential rate due to the presence of constant-utility public transit, sothat q o ( t r + t p ) will also approach zero. Therefore, by Eq. (15), q o w t ≈ N . Meanwhile, N islower bounded by a positive constant due to the presence of citizens with types 4 and 6 and thefact that n ≥
0. Thus, q o w t is lower bounded by a constant. Meanwhile, q m and q a are upperbounded by d mv and d av , respectively. Combining the above facts, based on Eqs. (5) and (6), weknow that ˆ t r , ˆ t p are upper bounded by constants. The claim is proved.10n the other hand, ˆ n is always upper bounded by a constant because N is lower boundedby a positive constant, and with Eq. (14), we can easily identify the bounded nature of ˆ n .Now, we define the ranges of t r , t p and n to be within [ M ] , where M is a large positivenumber. We show that ˆ w c is upper bounded by a constant; with Eq. (4), this is equivalent toshowing that w t is lower bounded by a constant. By Eq. (15), we obtain: w t = Nq o − t p − t r ≥ − t p − t r ≥ − M and with which the upper boundedness of ˆ w c is confirmed.Finally, we define the feasible ranges of t r , t p , w c and n to be all within [ M ] , and thisregion is denoted as D . When M is sufficiently large, by the above reasoning we know that ˆ w c and ˆ n are upper bounded by constants smaller than M , and based on the claim stated aboveas well as the continuity of Φ , we know that ˆ t r , ˆ t p are also upper bounded by constants smallerthan M . Therefore, Φ ( t r , n , t p , w c ) ∈ D , and by Schauder’s fixed-point theorem the existence isguaranteed. The proof is completed. Using the equilibrium model proposed in the previous section, we can then proceed to analyzeseveral scenarios with different objectives for decisions regarding the platform payment p and theplatform charge F o . The scenarios of interest include a monopoly scenario, a first-best scenarioand a second-best scenario. The first intends to maximize the crowdsourcing platform’s profit;the second intends to maximize social welfare; and the third intends to maximize social welfareunder the constraints that the crowdsourcing platform’s profit is guaranteed to be no less thana preset threshold. For practical consideration, in this section we assume that the time-of-daycan be divided into several heterogeneous periods according to the levels of demands. In thefollowing, we use P to denote the set of all homogeneous periods, and H k represents the durationof the k th period, ∀ k ∈ P . Below, we first present the optimization models in the above-mentionedthree scenarios, and then develop the algorithms for solving these models.In this section, we use a tuple τ to collectively denote all of the equilibrium variables definedin the last section; these are treated as intermediate variables in the established optimizationmodels. In the monopoly scenario, the crowdsourcing platform decides some key variables to maximizeits daily profit. The decision variables include the period-specific average fare for the on-demandride service F o , k and the period-specific payment to AV owners per customer p k . The total numberof purchased AVs, i.e., N s , is a parameter that is not considered as a decision variable. Let q o , k denote the period-specific unit-time customer demand for the on-demand ride service; N r , k denote the period-specific number of rented AVs; and n k denote the period-specific number ofcustomers served by a crowdsourcing AV per period of duration h . The optimization model forthe profit maximization problem is then stated as:11ax F o , p , τ (cid:40) ∑ k ∈P H k h ( F o , k q o , k h − N r , k p k n k ) (cid:41) − N s ( g + z ) − C f (16)s.t. ( ) − ( ) ∀ k ∈ P F o , k , p k ≥ ∀ k ∈ P (17)where g and z are the unit-day amortized purchase cost and maintenance cost of an AV respec-tively; and C f represents the basic operational cost of the crowdsourcing platform per day.In the objective function (16), H k F o , k q o , k represents the revenue of the platform in the k th period, and H k h N r , k p k n k is the total payment to the AV owners in the k th period. Eq. (17) statesthat the fares and payments should be nonnegative. It should be noted that the constraints (4)-(15) are period-specific; i.e., for each period k , there is a corresponding constraint set stating theequilibration of the AV crowdsourcing market. Now we consider the case that a government agency is operating the crowdsourcing platform,and its goal is to maximize social welfare. This scenario is called the first-best scenario. Bydeciding the period-specific fare F o , k and payment p k , the optimization model for the operatingagency can be written as:max F o , p , τ ∑ k ∈P H k h S k (18)s.t. ( ) − ( ) , ( ) ∀ k ∈ P where S k represents the unit-time social welfare in the k th period, and the length of such timeduration is h . The constraints are the same as in the monopoly scenario. The social welfare iscomposed of the total time cost by various transport modes and an additional cost of sharingprivate AVs with other people; its detailed form is given by: S k = ∑ x ∈{ } µ x d x , k ln ∑ ( i , j ) ∈C x e µ x V ( i , j ) x , k + H k h ( F o , k q o , k h − N r , k p k n k ) k ∈ P (19)In Eq. (19), µ x d x , k ( ln ∑ ( i , j ) ∈C x e µ x V ( i , j ) x , k ) is the total utility of class x citizens, where µ x is the corre-sponding scale parameter; d x , k is the period-specific population of class x citizens; and V ( i , j ) x , k is theperiod-specific (dis)utility of choice ( i , j ) by class x citizens. The social welfare should excludemonetary utility, and therefore we compensate by adding H k h ( F o , k q o , k h − N r , k p k n k ) in calculatingthe welfare terms. From the platform’s perspective, the on-demand ride service can produce a deficit under thefirst-best case, and a similar phenomenon is identified in the traditional taxi and ride-sourcing12arkets. Specifically, in the early stage of AV adoption, it is likely that only a few citizens ownAVs, and there will be a shortage of AVs that can be rented. In this case, if the crowdsourc-ing platform lacks start-up fund to purchase AVs, the on-demand ride service provided by thecrowdsourcing AVs will be of little attraction since it requires excessively long pick-up distances,cutting down the platform’s profit. To guarantee a certain level of profitability for the serviceprovider, in this subsection we propose a second-best operational strategy that maximizes socialwelfare under the constraint of a lower bound on the platform profit. The model formulation isillustrated as follows:max F o , p , τ ∑ k ∈P H k h S k (20)s.t. ( ) − ( ) , ( ) ∀ k ∈ P ∑ k ∈P H k h ( F o , k q o , k h − N r , k p k n k ) ≥ ρ (cid:2) N s ( g + z ) + C f (cid:3) ∀ k ∈ P (21)where ρ ≥ The optimization models presented above are all nonlinear programs with complicated nonlinearconstraints. To efficiently solve these models, we treat them as bi-level forms containing anequilibrium problem in the lower level, and then incorporate them into a gradient projectionalgorithmic framework. In detail, we first obtain the equilibrium given the fare F o , k and payment p k ; and then we calculate the gradient of the objective functions to F o , k and p k in the currentequilibrium state, according to which we update the decision variables and project them to thefeasible domain when they are to cross the boundary. Thus, the process is carried out iterativelyuntil the termination criterion is met (Figure 4).We use the implicit function theorem to compute the gradient of equilibrium state ∇ f withregard to F o , k and payment p k . Applying the chain rule, we have: J ( ∂ τ ∂ F o , ∂ τ ∂ p ) = − J T ( ∂ f ∂ F o , ∂ f ∂ p ) · J − ( ∂ f ∂ τ ) (22)Here, f u are the choice utility functions; f π are the choice probability functions for whichthe original forms are given by Eq. (7); f n are the supply/demand functions for which theoriginal forms are given by Eqs. (8)-(15); and f t are the traffic-related functions for which theoriginal forms are given by Eqs. (4)-(6). Then, the equilibrium can be compactly expressed as f = f u f π f n f t = . 13igure 4: Solution algorithmsThe Jacobian matrix J ( ∂ f ∂ F o , ∂ f ∂ p ) can be obtained as follows. With the exception of f u that islinearly related with F o and p , all other functions in f have zero partial derivatives with regard to F o and p . To obtain the Jacobian matrix J ( ∂ f ∂ τ ) , we observe that f u and f n are linearly related with τ . For probability functions, f π , we have ∂ f ( m , n ) π , x ∂ V ( m , n ) x = µ x e µ x V ( m , n ) x ∑ ( i , j ) ∈H x e µ x V ( i , j ) x − e µ x V ( m , n ) x ∑ ( i , j ) ∈H x e µ x V ( i , j ) x (23) ∂ f ( m , n ) π , x ∂ V ( w , z ) x = − µ x e µ x V ( m , n ) x e µ x V ( w , z ) x ∑ ( i , j ) ∈H x e µ x V ( i , j ) x (24)where ( w , z ) ∈ C x , ( m , n ) ∈ C x and ( w , z ) (cid:54) = ( m , n ) . The traffic-related functions f t are piece-wisefunctions consisting of a linear function and a power function. ∂ f t ∂ τ can be obtained using thederivative rule for complex functions.In the second-best scenario, there exists a nonlinear constraint Eq. (21) that can make thegradient projection method proposed above ineffective. To handle this issue, we introduce aLagrange multiplier λ to transfer the constraint into a term in the objective function. Specifically,the altered objective function can be written as:max F o , p , τ ∑ k ∈P H k h S k + λ (cid:40)(cid:34) ∑ k ∈P H k h ( F o , k q o , k h − N r , k p k n k ) (cid:35) − ρ (cid:2) N s ( g + z ) + C f (cid:3)(cid:41) (25)14ince the Lagrange multiplier λ is unknown prior to solving the problem, in the solutionprocess we must adjust its value accordingly in order to obtain a maximum objective functionvalue while ensuring the feasibility of the constraint. In most cases, the optimal solution isidentified at the points where ∑ k ∈P H k h ( F o , k q o , k h − N r , k p k n k ) = ρ (cid:2) N s ( g + z ) + C f (cid:3) . We report the equilibrium results under different scenarios in this section.
We consider a city with the area of approximately 400 km . The population density of this cityis 5, 000 persons per square kilometer, and 30% of the residents of the city own MVs or AVs. Weset the AV ownership as approximately 3%; i.e., the number of AVs owned by citizens is equalto 3% of the population. The time of a day can be divided into two periods according to traveldemand: peak and off-peak hours that last 4 h and 20 h, respectively. In the peak-hours, the tripgeneration per hour equals 15% of total population; in the off-peak hours, the trip generationper hour equals 5% of total population. The model input data for the population of six types ofcitizens are shown in Table 3.AV technological maturity index α is set to 0.7. The matching function of the AV crowdsourc-ing is captured by A = α = α = N s =
0; i.e.,all supplied AVs are crowdsourced from the AV owners. The cost incurred by the crowdsourcingplatform is set to $6 × per day. The additional cost of AV owners is set as m = $20. Publictransit is captured by the average travel time t n = F n = $6. For simplic-ity, we consider µ x = µ for all classes of citizens. Traveler’s VOTs are β A = $20/h, β P = $30/h, β M = $40/h, γ = $30/h.Setting κ =
1, the average trip time with free-flow speed by private vehicle is 30 min. Withoutloss of generality, the trip time function considering congestion T s ( · ) is set as a quadratic function,i.e., T s ( · ) = a + b [ q m + α ( q a − q o ) + α q o ( + θ ( w t q o ) − )] , and the approximation takes similarform. The basic trip time term is a = b = × − h, indicating that traffic congestion can lead to a delay of at most 1 h. The first-best scenario reduces people’s travel cost and increases the AV owners’ revenue sig-nificantly (Table 4) illustrating that the first-best scenario encourages AV owners to share the15rivate AVs with others and encourages citizens to travel by on-demand ride service. Paymentin the first-best scenario is high: considering an AV owner renting out their private AV in off-peak hours every day, it requires approximately 1.12 years for the owner to recoup the purchasecost of the vehicle. This suggests that AV crowdsourcing may be a worthwhile investment forsome citizens; in turn, this will promote AV acceptance by the public. Different from the cur-rent manual driving on-demand ride providers, we find the AV crowdsourcing platform can stillgreatly benefit from the first-best scenarios (Table 4). However, the results show that the crowd-sourcing platform must cut approximately 62% of its profits to achieve the first-best scenario.To balance the crowdsourcing platform’s profits and the overall social welfare, we explore thesecond-best scenario and find a trade-off pricing strategy in which the social welfare is close tothat of the first-best scenario while the crowdsourcing platform only loses approximately 35%of its monopoly profits. By applying this second-best pricing strategy, it is easier to obtain theagreement of the crowdsourcing platform to achieve a relatively high social welfare. We note that µ = In this section, we examine the influence of a series of model parameters on the equilibriumresults, including people’s recognition of utility uncertainties (i.e., the scale parameter in thelogit model), population density, AV market penetration rate, AV technology maturity and theadditional cost for sharing one’s AVs with the society (i.e., m ). Without loss of generality, we assume that all classes of citizens are characterized by thesame degree of uncertainty with regard to choices, i.e., µ x = µ ( ∀ x ∈ {
1, 2, ..., 6 } ) . As shownin Figures 6(a) and 6(b), when people are more sensitive to utilities, it will be harder for thecrowdsourcing platform to attract citizens, and thus the trip fare decreases, and the paymentincreases. It is observed from Figures 6(c) and 6(d) that the crowdsourcing platform’s profit andthe social welfare decline overall. Trips by public transit decrease, implying that the shortcomingof its high time cost is amplified with increasing µ , and trips by other three transport modesincrease (Figure 7). One noticeable deviation is that the payment during the peak hours forthe first-best scenarios decreases (Figure 6(b)), leading to slightly fewer crowdsourcing trips andslightly greater platform profit. This indicates that more crowdsourcing trips during the peakhours are discouraged with increasing µ , most likely due to the amplification of the congestionexternality. Another deviation is the minor increase of the platform profits for the monopolyscenario for µ ≥ µ has a great effecton price setting and the equilibrium, and thus it is important for the crowdsourcing platform to17 a) (b)(c) (d) Figure 6: Influence of people’s sensitivity to utilities.investigate this parameter prior to setting the optimal prices in order to maximize its profits; µ also influences the urban planner’s decision whether to regulate the AV crowdsourcing market,because the first-best scenarios present a more limited improvement with a higher µ , i.e., 11.4%for µ = µ = µ = Figures 8(a) and 8(b) show that the population density generally has little impact on priceswith exceptions for a few cases, namely the payment during peak-hours declines for the first-bestscenarios, and the fare during the off-peak hours increases quickly when the population densitygrows beyond 5000 persons per km . This discourages people to rent AVs or take crowdsourcingservice and thus alleviate the congestion externality that grows with the population density. This18 a) (b)(c) (d) Figure 7: Influence of people’s sensitivity to utilities on mode split.implies that main impact of the population density on the equilibrium is the congestion exter-nality. Figure 8(c) shows that the crowdsourcing platform’s profits increase, indicating that thecrowdsourcing platform must expect to be launched in more densely populated areas, such as ametropolis. Figure 8(d) shows that individual welfare first increases slightly and then decreases,most likely because a low population density induces insufficient number of crowdsourcing AVs,increasing the individual travel cost. The highest individual welfare is obtained when the popu-lation density is approximately 2500 persons per km . We denote the autonomous vehicle market penetration rate as the ratio of the population whoown AVs to the population who own AVs or MVs. A greater market penetration rate meansa larger pool for AV crowdsourcing. As shown in Figures 9(a) and 9(b), fare and paymentdecline due to a more sufficient supply and the rate of decline decreases with growing AV market19 a) (b)(c) (d)
Figure 8: Influence of population density.penetration rate. As observed from Figure 9(c), the crowdsourcing platform’s profits increasewith AV popularization, demonstrating the economy of scale. Figure 9(d) demonstrates thatsocial welfare increase significantly, suggesting that the urban planner should have a supportiveattitude toward AV popularization. Moreover, the wider gap in the social welfare between thefirst-best and monopoly scenarios suggests that regulation will become increasingly beneficialwith increasing AV market penetration.
The maturity of AV technology ( − α ) represents the degree of improvement on road capacity,where α represents the ”relative occupation” of AVs compared to MVs. As Figures 10(a) and10(b) show, AV technology maturity’s impact on prices is ignorable except that the platform raisespayment in peak hours for the first-best scenarios to encourage more crowdsourcing trips, whichresults from higher road capacity. Figure 10(c) shows the crowdsourcing platform’s profits are20 a) (b)(c) (d) Figure 9: Influence of the autonomous vehicle market penetration rate.improved in the monopoly scenario, but are deteriorated under regulation. Figure 10(d) pointsout that the advances of AV technology help improve the social welfare. However, the maturityof AV technology has limited impact on price setting and the equilibrium state.
As shown by Figures 11(a) and 11(b), the optimal fare and payment remain almost unchanged.The results presented in Figure 11(c) indicate that the crowdsourcing platform should purchaseas many AVs as possible to obtain a higher daily profit. An examination of Figure 11(d) showsthat social welfare improves with greater number of purchased AVs. Nevertheless, the impactof the number of vehicles pre-purchased is much smaller than the impact of the AV marketpenetration rate on the equilibrium state. 21 a) (b)(c) (d)
Figure 10: Influence of autonomous vehicle technology maturity.
Figures 12(a) and 12(b) present the rise of fare and payment with increasing additional cost. Thecrowdsourcing platform is more difficult to collect AVs and thus must provide greater compen-sation to the AV owners. The decline of supply leads to a decreasing number of crowdsourcingtrips. One noticeable difference is that the payment during the peak hours for the first-best sce-nario first decreases and then increases. In our model, travel time and additional cost are twocomponents of social welfare. When additional cost is low, the payment in peak hours decreasesto discourage crowdsourcing trips for less congestion; however, when additional cost continues togrow, additional cost becomes more dominant for social welfare and thus, the payment increasesto compensate the AV owners. An examination of the results presented in Figure 12(c) showsthat although the monopoly profits decrease monotonically with additional cost m , profits in thefirst-best scenario first increase and then decrease after m exceeds $30. The first half of the curvesuggests that under the first-best case, a higher m means a higher profit for the crowdsourcing22 a) (b)(c) (d) Figure 11: Influence of the number of autonomous vehicle purchased.platform. However, when m is quite high, the first-best scenario and the monopoly show littledifference in social welfare and platform’s profits. These tests suggest that it is important for thecrowdsourcing platform to investigate the AV owners’ additional cost prior to setting prices; itmay also be important for the urban planner to determine m when deciding whether to imposeregulations because a small m means that regulation has a more beneficial effect on the society,as shown in Figure 12(d). This paper investigates the AV crowdsourcing market that may emerge in the future. We firstproposed an equilibrium model with multiple transport modes: AV on-demand ride service,private AV, private MV and public transit, and proved the existence of the equilibrium solution.Then, several scenarios, namely, the monopoly scenario, the first-best case and the trade-off23 a) (b)(c) (d)(e)
Figure 12: Influence of the additional cost to rent out private autonomous vehicles.24econd-best case were explored.We use numerical examples to illustrate some insightful findings concerning the AV crowd-sourcing market, including the comparison of optimal pricing for trip fare and payment duringthe peak and off-peak hours, the crowdsourcing platform’s profits, the social welfare and trans-port modes in each scenario. Sensitivity analysis of the optimal pricing and the equilibrium statewith regard to different parameters in the model are presented in a thorough manner. The mainfindings from the numerical examples are summarized below.1. AV crowdsourcing service is beneficial for the society overall. This provides guidance forurban planners to have a supportive attitude toward AV development and popularization.2. Renting out private AVs to the crowdsourcing platform provides high return, encouragingpeople to rent their cars instead of driving them themselves. This verifies a new opportunity foradventurous investors such as the wealthy to invest into AVs.3. Regulation may be easier for the crowdsourcing platform to accept in some second-best sce-nario where the social welfare is close to that of the first-best scenario. Regulation is particularlynecessary when people are not sensitive to utilities, or the AV owners are more inclined to rentout their AVs to gain profits.4. It is important for the crowdsourcing platform to know about the people’s sensitivity toutilities and AV owners’ additional cost prior to setting prices, because these two factors signifi-cantly affect the optimal pricing. Additionally, the crowdsourcing platform should expect to belaunched in areas with high population density and should purchase as many AVs in advanceas possible to obtain higher profits. Nevertheless, even without pre-purchased vehicles, the plat-form is highly likely to gain profits under the premise that the market penetration rate of AVs issufficiently high (above 10%).This paper could be further advanced by considering competition between MV and AV on-demand ride service in the mobility market, as is closer to the reality before the society is fullyaccustomed to AV technologies. Additionally, as shared autonomous vehicles will probably beoperated together with public transit, the joint planning and operation of two modes to forman integrated mobility-as-a-service (MaaS) system should be explored in the future. In terms ofwider implications, our results suggest that with the presence of AV crowdsourcing business,even when most AVs are owned privately, these AVs will be mostly used for serving othercustomers in the society, which motivates the discussion of AV ownership in the future, includingwhere it will converge and how it will evolve. On the other hand, for these AV owners, AVs willnot be purchased merely for personal usage, but also for investment (buying vehicles to earnmoney from day to day), and the capital value of AVs merits careful investigation.
Acknowledgements
The research is supported in part by Tsinghua-Daimler Joint Research Center for SustainableTransportation. 25 eferences
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InChipot, M., editor, Handbook of Differential Equations, volume 6 of Handbook of DifferentialEquations: Stationary Partial Differential Equations, pages 503 – 583. North-Holland.28 ppendix A. Nomenclature ppendix B. Discussion about logit model in Section 3.2 Let µ rx denote the scale parameter of logit model with regard to rental choices; µ tx denote thescale parameter of logit model with regard to transport mode choices. If µ rx > µ tx , citizens firstchoose travel modes and then choose whether to rent their AVs to the crowdsourcing platform.Let T x denote the choice set of travel modes by class x citizens and T ix denote the rental choiceset after they have chosen travel mode i . Therefore, the probability of class x citizens choosing ( i , j ) ∈ C x can be obtained using the nested logit model: π ( i , j ) x = e µ rx V ( i , j ) x ∑ j ∈T ix e µ rx V ( i , j ) x ( ∑ j ∈T ix e µ rx V ( i , j ) x ) µ tx / µ rx ∑ i ∈T x ( ∑ j ∈T ix e µ rx V ( i , j ) x ) µ tx / µ rx ∀ ( i , j ) ∈ C x , x ∈ {
1, 2, . . . , 6 } (B-1)If µ rx < µ tx , citizens first choose whether to rent their AVs to the crowdsourcing platform andthen choose travel modes. Let R x denote the rental choice set by class x citizens and R jx denotethe choice set of travel modes after they have made rental choice j . Therefore, the probability ofclass x citizens choosing ( i , j ) ∈ C x can be obtained using the nested logit model: π ( i , j ) x = e µ tx V ( i , j ) x ∑ i ∈R jx e µ tx V ( i , j ) x ( ∑ i ∈R jx e µ tx V ( i , j ) x ) µ rx / µ tx ∑ j ∈R x ( ∑ i ∈R jx e µ tx V ( i , j ) x ) µ rx / µ tx ∀ ( i , j ) ∈ C x , x ∈ {
1, 2, . . . , 6 } (B-2)If µ rx = µ tx = µ x , people decide simultaneously whether to rent their AVs and the type oftravel mode they take. Therefore, the probability of class x citizens choosing ( i , j ) ∈ C x is reducedto the logit model: π ( i , j ) x = e µ x V ( i , j ) x ∑ ( i , j ) ∈C x e µ x V ( i , j ) x ∀ ( i , j ) ∈ C x , x ∈ {
1, 2, . . . , 6 }}