Supply and Demand Management in Ride-Sourcing Markets

By Jintao Ke, Hai Yang, Hai Wang and Yafeng Yin

Supply and Demand Management in Ride-Sourcing Markets offers a fundamental modeling framework for characterizing ride-sourcing markets by spelling out the complex relationships among key endogenous and exogenous variables in the markets. This book establishes several economic models that can approximate matching frictions between drivers and passengers, describes the equilibrium state of ride-sourcing markets, and more. Based on these models, the book develops an optimum strategy (in terms of trip fare, wage and/or matching) that maximizes platform profit. While the best social optimum solution (for maximizing the social welfare) is generally unsustainable, this book provides options governments can use to encourage second-best solutions.

In addition, the book's authors establish models to analyze ride-pooling services, with traffic congestion externalities incorporated into models to see how both new platforms and government designs can optimize operating strategies in response to the level of traffic congestion.

• Serves as a foundation for subsequent research studies that investigate ride-sourcing services through mathematical modeling
• Offers valuable managerial insights for ride-sourcing platforms and helps them develop more efficient and effective operating strategies
• Assists the governments or social planners in designing appropriate regulatory schemes to achieve more sustainable and societally beneficial market outcomes

• Language: English
• Publisher: Elsevier Science
• Release date: Apr 30, 2023
• ISBN: 9780443189388

Jintao Ke

Dr. Jintao Ke received his B.S. degree (2016) in civil engineering from Zhejiang University, and his Ph.D. degree (2020) in Civil and Environment Engineering from Hong Kong University of Science and Technology. He is now an Assistant Professor at the University of Hong Kong. His research interests include smart transportation, smart city, urban computing, shared mobility, machine learning in transportation, operational management for transportation studies, etc. He has published over 20 SCI/SSCI indexed research papers in in top-tier transportation journals, such as Transportation Research Part A/B/C/E and IEEE Transactions on Intelligence Transportation System. He serves as an Advisory Board Member of Transportation Research Part C: Emerging Technologies, and referees for a few top transportation journals.

Book preview

Supply and Demand Management in Ride-Sourcing Markets

Jintao Ke

Assistant Professor, Department of Civil Engineering, University of Hong Kong, Hong Kong

Hai Yang

Chair Professor, Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong

Hai Wang

Associate Professor, School of Computing and Information Systems, Singapore Management University, Singapore

Yafeng Yin

Professor at Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor

Table of Contents

Cover image

Title page

Copyright

Contributors

About the authors

Preface

Chapter 1. Introduction of ride-sourcing markets

1.1. Background

1.2. Theoretical developments

1.3. Outline of this book

Chapter 2. Fundamentals of ride-sourcing market equilibrium analyses

2.1. Introduction

2.2. Matching frictions (inductive approaches)

2.3. Matching frictions (deductive approaches)

2.4. Market measures

2.5. Discussion

Glossary of notation

Chapter 3. Calibration and validation of matching functions for ride-sourcing markets

3.1. Introduction

3.2. Matching functions and market metrics

3.3. Experimental settings

3.4. Analysis of experimental results

3.5. Summary

3.6. Discussion and conclusion

Appendix 3.A

Chapter 4. Government regulations for ride-sourcing services

4.1. Properties of the pareto-efficient solutions

4.2. An alternative method to obtain and analyse pareto-efficient solutions

4.3. Regulations

4.4. Discussion and conclusion

Chapter 5. Equilibrium analysis for ride-pooling services

5.1. Introduction

5.2. Pool-matching schemes

5.3. Equilibrium analyses

5.4. Market measures

5.5. Numerical illustrations

5.6. Conclusion

Chapter 6. Ride-pooling services and traffic congestion

6.1. Introduction

6.2. Equilibrium analyses

6.3. Market measures

6.4. Conclusion

Chapter 7. Equilibrium analysis for ride-pooling services in the presence of traffic congestion

7.1. Introduction

7.2. Equilibrium analyses

7.3. Market measures

7.4. Numerical studies

7.5. Conclusion and remarks

Chapter 8. Revisiting government regulations for ride-sourcing services under traffic congestion

8.1. Introduction

8.2. Theoretical analyses

8.3. Numerical studies

8.4. Conclusion

Chapter 9. Third-party platform integration in ride-sourcing markets

9.1. Background

9.2. Market equilibrium and optimal strategies

9.3. Evaluation of the performance of platform integration

9.4. Numerical studies

9.5. Conclusion

Appendix 9.A. Proof of Lemma 9-1

Appendix 9.B. Proof of theorem 9-1

Appendix 9.C. Proof of Lemma 9-2

Appendix 9.D. Proof of theorem 9-2

Appendix 9.E. Proof of Lemma 9-3

Appendix 9.F. Proof of Lemma 9-4

Appendix 9.G. Proof of theorem 9-3

Appendix 9.H. Proof of Lemma 9-5

Appendix 9.I. Proof of theorem 9-4

Appendix 9.J. General matching function

Chapter 10. Ride-sourcing services and public transit

10.1. Background

10.2. Model description

10.3. Optimal strategy design

10.4. Numerical case study

10.5. Conclusion

Chapter 11. Optimization of matching-time interval and matching radius in ride-sourcing markets

11.1. Research problem

11.2. Modelling and optimising the matching process

11.3. Model properties in imbalanced scenarios

11.4. Numerical studies

11.5. Conclusion

Chapter 12. Labour supply analysis of ride-sourcing services

12.1. Background

12.2. Related literature

12.3. Labour supply model

12.4. Modelling endogeneity of income rates and self-selected participation in the labour force

12.5. Research design

12.6. Results and discussion

12.7. Conclusion

Chapter 13. Some empirical laws of ride-pooling services

13.1. Introduction

13.2. Literature review

13.3. Optimisation framework and data descriptions

13.4. Empirical laws

13.5. Conclusions

Appendices

Glossary of notation

Chapter 14. Summary

Glossary of abbreviations

Index

Copyright

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Contributors

Siyuan Feng,     Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China

Sun Hao,     Faculty of Business, The University of Hong Kong, Pokfulam, Hong Kong, China

Jintao Ke,     Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China

Xinwei Li,     School of Economics and Management, Beihang University, Beijing, China

Zhixi Wan,     Faculty of Business, The University of Hong Kong, Pokfulam, Hong Kong, China

Hai Wang,     School of Computing and Information Systems, Singapore Management University, Bras Basah, Singapore

Shuqing Wei,     Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China

Hai Yang,     Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China

Yafeng Yin,     Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, United States

Zhengfei Zheng,     Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China

Zhu Zheng,     Department of Civil Engineering, Zhejiang University, Hangzhou, China

Yaqian Zhou,     School of Automation, Chongqing University, Chongqing, China

About the authors

Dr Jintao Ke is an Assistant Professor in the Department of Civil Engineering, the University of Hong Kong. He received his bachelor's degree from Zhejiang University, China, and his PhD from the Hong Kong University of Science and Technology. His research interests include smart transportation, smart city, urban computing, shared mobility, machine learning in transportation, operational management for transportation studies, etc. He has published over 30 SCI/SSCI-indexed papers in top transportation journals, including Transportation Research Part A–E, IEEE Transactions on Intelligence Transportation System. He serves as an Advisory Board Member of Transportation Research Part C and Transportation Research Part E.

Professor Hai Yang is a Chair Professor in the Department of Civil and Environmental Engineering, the Hong Kong University of Science and Technology. He received his bachelor's degree from Wuhan University, China, and his PhD from Kyoto University, Japan. Professor Yang is internationally known as an active scholar in the field of transportation, with more than 300 papers published in SCI/SSCI-indexed journals. He has received a number of national and international awards, including the JSCE Outstanding Paper Award (1991); the Distinguished Overseas Young Investigator Award from the National Natural Science Foundation of China (2004); the National Natural Science Award bestowed by the State Council of China (2011); HKUST School of Engineering Research Excellence Awards (2012); 2020 Frank M. Masters Transportation Engineering Award, American Society of Civil Engineers (ASCE) and 2021 ASCE Francis C. Turner Award. Professor Yang served as the Editor-in-Chief of Transportation Research (TR) Part B: Methodological from 2013 to 2018, a prestigious journal in the field of transportation. Currently, Professor Yang serves on the Distinguished Editorial Board of TR Part B, Scientific Council of TR Part C: Emerging Technologies, and serves as an Advisory Editor of Transportation Science.

Dr Hai Wang is an Associate Professor in the School of Computing and Information Systems at Singapore Management University and a visiting faculty at the Heinz College of Information Systems and Public Policy at Carnegie Mellon University. He received his bachelor's degree from Tsinghua University and his PhD from MIT. His research focuses on the methodologies of analytics and optimisation, data-driven modelling, computational algorithms and machine learning and relevant applications in smart cities, including innovative transportation, advanced logistics, modern e-commerce and intelligent healthcare. He serves as the Department Editor for Transportation Research Part E, Associate Editor for Transportation Science and Service Science, Special Issue Editor for Transportation Research Part B, Transportation Research Part C and Service Science and Editorial Board Member for Transportation Research Part C. Dr Wang was selected as Chan Wu and Yunying Rising Star Fellow in Transportation, received Lee Kong Chian Research Excellence Awards, was nominated for MIT's Top Graduate Teaching Award and won the Excellent Teaching Award at SMU. During his PhD at MIT, he also served as the Co-President of the MIT Chinese Students and Scholars Association and Chair of MIT-China Innovation and Entrepreneurship Forum.

Dr Yafeng Yin is a Professor of Civil and Environmental Engineering and Professor of Industrial and Operations Engineering at the University of Michigan, Ann Arbor. His research aims to analyse and enhance multimodal transportation systems towards efficiency, resilience and environmental sustainability. Currently, he focuses on developing innovative mobility solutions and services by leveraging vehicle connectivity and automation. Dr Yin has published nearly 150 refereed papers in leading academic journals. He was the Editor-in-Chief of Transportation Research Part C: Emerging Technologies between 2014 and 2020 and currently serves as Area Editor of Transportation Science and Associate Editor of Transportation Research Part B: Methodological, another two flagship journals in the transportation domain. Professor Yin received his PhD from the University of Tokyo, Japan, in 2002, and his master's and bachelor's degrees from Tsinghua University, Beijing, China, in 1996 and 1994, respectively.

Preface

Current disruptive trends in transportation, such as driving automation, increased connectivity, vehicle electrification and shared mobility, are altering traditional thinking on transportation and changing our daily travel modes. For example, the prevalence of shared mobility services, which typify emerging on-demand ride-sourcing services, has dramatically increased over the past decade. This has led to research in many interesting areas, including the modelling of passengers' mode choices and drivers' mode participation; investigations of ride-sourcing platforms' optimal decisions on pricing, wages and matching; the design of effective government regulations and analyses of the social effects of ride-sourcing services. Ride-sourcing markets are challenging to research as they are two-sided markets in which demand and supply interact in a complex manner. Therefore, there is a pressing need for mathematical models that can precisely characterise passengers' and drivers' ride-sourcing behaviours and perform efficient on-demand matching of passengers and drivers. Such models will assist ride-sourcing platforms to develop operational strategies to maximise their profits and assist governments to design effective regulatory schemes to enhance social welfare.

This book addresses this need by detailing the methodological development of a series of advanced mathematical models that delineate the complex and intriguing relationship between a system's endogenous variables (such as effective passenger demand and driver supply) and a platform's decision variables (such as price, wage and matching rules) in the stationary equilibrium state of ride-sourcing markets. These models are intended to enable effective research on several current topics, including Pareto-efficient frontiers and the design of Pareto-efficient government regulation schemes; pricing and matching operations for ride-pooling services; the effects of traffic-congestion externalities on ride-sourcing markets; the design of operational strategies for ride-pooling services in the presence of traffic congestion; the reanalysis of government regulations by considering traffic congestion and the heterogeneity of drivers' reservation rates and the optimisation of on-demand matching of passengers with drivers.

This book also describes the first systematic modelling framework for ride-sourcing markets, which aims to address key operational and planning aspects from the viewpoint of ride-sourcing platforms or social planners. This framework is based on state-of-the-art research that has primarily been conducted by us and our colleagues. Accordingly, this book will be a useful reference for all students, scholars, scientists, and professionals studying and/or working in the fields of shared mobility, especially those focused on ride-sourcing services.

We are sincerely grateful to several researchers for their collaborations with us in this endeavour. In particular, we thank Dr Xinwei Li from Beihang University, Dr Zheng Zhu from Zhejiang University, Dr Hao Sun and Prof. Zhixi Wan from the University of Hong Kong and Dr Xiaoran Qin, Dr Yaqian Zhou, Mr Zhengfei Zheng, Ms Shuqing Wei and Mr Siyuan Feng from the Hong Kong University of Science and Technology for their valuable contributions to the theoretical analyses and numerical studies presented in this book. Finally, we acknowledge funding support from the Research Grants Council of the Hong Kong Special Administrative Region (HKSAR).

Jintao Ke

Hai Yang

Hai Wang

Yafeng Yin

January, 2023

Chapter 1: Introduction of ride-sourcing markets

Jintao Ke ¹ , Hai Yang ² , Hai Wang ³ , and Yafeng Yin ⁴       ¹ Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China      ² Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China      ³ School of Computing and Information Systems, Singapore Management University, Bras Basah, Singapore      ⁴ Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, United States

Abstract

The popularity of smartphones and the mobile Internet has catalysed the rapid growth of on-demand ride-sourcing services provided via online platforms by transportation network companies, such as Uber, Lyft, Didi, Grab and Ola. These emerging mobility services have had profound effects on the way we travel and on multimodal urban mobility systems. This book provides an analytical framework for the investigation of on-demand ride-sourcing services and offers some managerial insights that can aid platforms to enhance their operations and help the government reform their regulation of the market.

This chapter offers a brief introduction to on-demand ride-sourcing markets by outlining the intriguing relationships between platform decision variables (such as the trip fare, the wage, and the matching strategies) and system endogenous variables (such as the passenger demand, the driver supply, and the waiting time). In addition, this chapter reviews related studies on ride-sourcing services that examine a variety of topics, including pricing, equilibrium analysis, revenue management, government regulation analysis, matching and pooling, traffic congestion externality, platform competition and integration, and coordination with other transportation modes.

Keywords

On-demand ride service; Ride-hailing; Ride-sourcing; Shared mobility; Transportation network companies

1.1. Background

Urban mobility has undergone dramatic changes around the world in recent years with the introduction of ride-sourcing services (or on-demand ride-hailing services) provided by transportation network companies (TNCs). These companies, such as DiDi, Uber, Grab, Lyft, Careem and Ola, efficiently connect passengers and dedicated drivers via online platforms. Ride-sourcing services are playing an increasingly important role in meeting on-demand mobility needs and reshaping the conventional taxi industry. Moreover, the accelerating development of mobile Internet-based technologies has led to a rapid expansion in ride-sourcing services. For example, since its official launch in 2011, Uber has expanded its business to more than 700 metropolitan areas in 65 countries and served over five billion on-demand trips (DMR, 2019a). It offers riders a menu of services, including UberX (the basic service provided by four-seater sedans), Uber Black (an executive luxury service), UberPool (a ride-pooling service that enables one driver to serve two or more passengers with different requests in each ride), SUV (a six-seater vehicle luxury service) and Taxi (an e-hailing taxi service requested via the platform). Similarly, DiDi is the largest ride-sourcing company in China, and since its launch in 2012 has served more than 550 million users in 400 cities throughout China (DMR, 2019b). Its services include Express (a basic service), Premier (an upgraded service), Luxe (an executive luxury service), ExpressPool (a ride-pooling service), Hitch (a ride-sharing service offered by non-dedicated drivers with their own trip plans) and Minibus (an on-demand minibus for shared ride services). In addition, these companies are now deploying electrified vehicles to reduce fuel costs and developing autonomous vehicles to reduce human labour costs.

On a typical ride-sourcing platform, a passenger makes a request that contains his/her detailed trip information, i.e., origin and destination location, departure time and service option. At the same time, drivers affiliated with the platform are cruising around the city waiting for requests or parked in a specific waiting area (e.g., an airport or railway station). Thus, the platform matches waiting passengers and idle drivers and aims to maximise the matching rate (the number of driver–passenger pairs matched per unit time) and minimise the average pick-up time (the time taken for drivers to pick up their assigned passenger(s)). The platform also collects trip fares from passengers and pays wages to drivers, and the difference between the fares collected and wages paid is the commission, which represents the platform's profit.

Wang and Yang (2019) developed a general framework to describe the intrinsic and complex interactions between various endogenous and decision variables of ride-sourcing market stakeholders and agents. As shown in Fig. 1.1, a ride-sourcing market is a typical two-sided market that comprises a supply side (drivers) and a demand side (passengers) that interact with each other. On the demand side, potential passengers compare ride-sourcing services with other transportation services, such as conventional street-hailing taxis and public transit, by evaluating the trip fare and quality of these services (e.g., waiting time until pick-up). On the supply side, drivers decide whether to work and when and for how long to work, according to their expected income level or hourly wage. A ride-sourcing platform therefore aims to maximise profit or social welfare by leveraging various decision variables, such as the trip fare charged to passengers and the hourly wage paid to drivers, or by directly controlling the vehicle fleet size, while taking into account the effect of its decisions on passengers and drivers.

Figure 1.1  General framework of ride-sourcing markets. Adapted from Wang, H., Yang, H., 2019. Ridesourcing systems: a review and framework. Transportation Research Part B: Methodological 129, 122–155.

An essential characteristic of a ride-sourcing system is the matching friction between passengers and drivers. Although ride-sourcing platforms perform efficient on-demand matching that often reduces matching friction more than in conventional street-hailing taxi services, matching friction cannot be completely eliminated. In particular, the equilibrium quantity of services supplied by drivers is generally greater than that consumed by passengers, which results in a certain surplus supply. This surplus is measured in terms of idle vehicle hours and is an important measure of service quality, as it governs passengers' waiting time until pick-up. Thus, the supply surplus indirectly influences passengers' generalised cost and passenger demand, which means that passenger demand, service quality (measured as passengers' waiting time until pick-up), and drivers' idle time (or number of idle vehicle-hours) interact with each other in a complex manner. These are key endogenous variables that influence platforms' operational efficiency, and therefore serve as crucial references that guide platform operations and decisions on aspects such as pricing (trip fares and drivers' wages), vehicle fleet-size management, empty vehicle repositioning, ride-pooling assignments and fare splitting, and dispatching and matching.

The design of operational strategies for maximising ride-sourcing platform profit or achieving optimal social welfare therefore requires a precise understanding of the intricate relationships between a platform's decision variables and a system's endogenous variables. Thus, there is a pressing need for the development of efficient mathematical models to describe ride-sourcing markets, which can be used to determine optimal operating strategies and regulatory policies for ride-sourcing platforms.

1.2. Theoretical developments

Various aggregate models (e.g., Zha et al., 2016) and disaggregate models (for example, He et al., 2018; Xu et al., 2021) have been proposed to describe supply–demand conditions and properties at stationary equilibria (He et al., 2018; Xu et al., 2017; Zha et al., 2016). Due to the similarities between ride-sourcing markets and conventional taxi markets, studies have been rooted in research on street-hailing taxi services (Yang and Yang, 2011; Yang et al., 2010) and Internet-based-hailing (e-hailing) taxi services (He and Shen, 2015; He et al., 2018; Wang et al., 2017). Aspects of ride-sourcing markets that have been examined include the coordination of supply and demand using prices and wages (Bai et al., 2019; Taylor, 2018); pricing and surge-pricing strategies (Cachon et al., 2017; Castillo et al., 2017; Yang et al., 2020b; Zha et al., 2016); government regulations and policies (Yu et al., 2020); the effects of ride-sourcing markets on conventional taxi markets (Nie, 2017; Wallsten, 2015); geometrical matching and order dispatching (Lyu et al., 2019; Xu et al., 2017; Xu et al., 2018; Yang et al., 2020a; Zha et al., 2018; Zhang et al., 2017); driver labour supply (Sun et al., 2019a,b; Zha et al., 2017); supply and demand predictions (Ke et al., 2017, 2019b; Tong et al., 2017); repositioning and subsidies for empty vehicles (Wang and Wang, 2020; Zhu et al., 2021); ride-pooling services (Ke et al., 2020b); and electrified ride-sourcing vehicles (Ke et al., 2019a). We do not exhaustively review this large and growing body of literature here; instead, we outline several important topics and some relevant analytical studies. Readers are invited to refer to Wang and Yang (2019) for a general framework and comprehensive review of research problems in ride-sourcing markets.

1.2.1. Stationary equilibrium state

Most previous studies have focused on a stationary equilibrium state in which the rate of arrival of passengers, the service quality (i.e., passengers' waiting time) and the combined number of idle/in-trip vehicles are invariant over time. This equilibrium state is affected by decision variables, such as trip fares and vehicle fleet sizes, and exogenous variables, such as potential demand, trip distances, and city sizes and topologies. As a result, the equilibrium states at different times of the day or on different days of the week can be compared by putting different exogenous variables into a model.

On the supply side, at any given instant of the equilibrium state (which can be obtained by taking a snapshot of the market), a vehicle is in one of three phases: an idle phase (i.e., waiting for passengers, and thus parked at a specific region or being driven around a city), an in-trip phase (delivering a passenger to his/her destination) or a pick-up phase (en route to pick up a passenger assigned by online matching). On the demand side, service quality is measured in terms of passengers' waiting time, which consists of two parts: the time passengers spend waiting online, after submitting a request for transport, to be matched with drivers, and the time passengers spend waiting to be picked up by drivers with whom they have been matched. We denote the first part of waiting time the matching time and the second part of waiting time the pick-up time.

The distributions of vehicles in each phase on the supply side and the service quality on the demand side are endogenously and interactively dependent. First, the average matching and pick-up times depend on both the number of idle vehicles and the number of waiting passengers. The more idle vehicles and waiting passengers there are the shorter the average pick-up time is, as under these circumstances it is easier for a platform to match vehicles and passengers. Second, the average matching and pick-up times are crucial service-quality measures that influence passenger demand, which in turn affects the distribution of vehicles in each phase, e.g., the higher the passenger demand, the higher the number of in-trip vehicles (i.e., vehicles transporting passengers), and thus the lower the number of idle or pick-up vehicles. Moreover, the interdependencies of these parameters, such as the relationship between the average pick-up time and the number of idle vehicles and waiting passengers, are also influenced by the matching technologies and algorithms implemented by a platform.

Over the past few decades, various mathematical models have been developed to analyse aspects of the market equilibria of ride-sourcing services (or taxi services). Arnott (1996) studied the marketplace offered by a taxi call centre that operated its service based on a first-come-first-served (FCFS) mechanism, which immediately matches a passenger who submits a request with the closest idle taxi driver. Arnott (1996) further assumed that idle drivers' entry into the marketplace followed a spatial Poisson process, and thus developed an analytical approximation to the average waiting time that is inversely proportional to the square root of the number of idle vehicles. However, this model implicitly neglects the effect of the time vehicles spend in pick-up on their utilisation; that is, it assumes that vehicles are either in an in-trip or idle phase, and never in a pick-up phase. To deal with this problem, Castillo et al. (2017) considered a modified FCFS matching mechanism that assumes a vehicle is idle, en route (to pick up passengers) or in-trip. They found that this created a wild goose chase regime, as matching failure occurred when idle drivers were matched with distant passengers because drivers wasted substantial time in the pick-up phase. Castillo et al. (2017) showed that such a matching failure caused the trip supply curve to bend backwards, and that the failure could be alleviated or prevented by the use of well-designed surge pricing.

In actual operations, instead of using an FCFS scheme, ride-sourcing platforms such as DiDi and Uber use a batch-matching mechanism that accumulates a certain number of waiting passengers and idle vehicles in a matching pool before performing online matching. In addition, some platforms divide a marketplace into numerous subregions and then perform matching between idle drivers and waiting passengers within each subregion. These matching strategies can effectively prevent distant matching and reduce pick-up time. However, a ride-sourcing market that uses these matching strategies cannot be well characterised by the models that have been proposed by Arnott (1996) and Castillo et al. (2017), as pick-up time depends on the number of waiting passengers in addition to the number of idle vehicles. Thus, the matching efficiency measured in terms of passengers' waiting time is governed by the size of these two groups of agents.

Yang and Yang (2011) and Zha et al. (2016) have both used a Cobb–Douglas-type meeting function to characterise the searching and meeting frictions between drivers and passengers in such a two-sided matching mechanism. In their models, the rate of matching between drivers and passengers is an increasing function of the number of waiting passengers and the number of idle drivers. The matching functions can exhibit increasing, constant or decreasing returns to scale. Under certain conditions, such matching function-based models can be reduced to the model of Arnott (1996). However, this two-sided matching model is unable to consider the effects of pick-up time on the distribution of vehicle phases, which means that although it is a good approximation for markets in which matchings are made between drivers and passengers within small blocks (where the pick-up time is relatively short, and thus can be neglected), it is not a good approximation for markets in which matchings are made between drivers and passengers who are distant from each other (where the pick-up time is long, and thus cannot be neglected).

Alternatively, a queuing model can be used to approximate the waiting time. For example, Bai et al. (2019) proposed a queueing model that analytically approximates the average waiting time of passengers by assuming that drivers are servers and passengers are arriving customers. In another example, Banerjee et al. (2015) combined a theoretical queueing model with underlying stochastic dynamics to determine the stationary equilibrium solutions that capture the choices of drivers and passengers, and the maximum profit for a platform. Xu et al. (2019) constructed a double-ended queueing model to analyse the supply curve of an e-hailing system with a constrained matching radius, which revealed that a smaller matching radius decreased the backward bending of the supply curve. Such queueing theoretical models are a flexible and trackable framework with which to describe the matching process and market equilibria and can generate interesting analytical results, but they rely on strict assumptions regarding the birth and death processes of a queue, and the spatiotemporal distributions of the arrivals of passengers and drivers.

In sum, the complexity of the marketplace means that there is always a trade-off in the use of mathematical models for describing ride-sourcing market equilibria, as these models must balance interpretative ability with mathematical tractability.

1.2.2. Monopoly optimum, social optimum, and Pareto-efficient solutions

Analyses of ride-sourcing markets must determine optimal operating strategies by tuning platform decision variables, such as trip fares, wages, vehicle fleet sizes, and matching strategies. However, ride-sourcing platforms (private firms) and governments (the public sector) may have different interests and objectives: the former are typically only interested in maximising their profits, whereas the latter also wish to maximise the total social welfare of a ride-sourcing system, which requires maximising the benefits of various stakeholders and agents (such as passengers' surplus and drivers' welfare). The set of platform decision variables that generate the maximum profit for a platform in a monopoly market is regarded as the monopoly optimum solution, whereas the set of platform decision variables that generate the maximum social welfare is regarded as the social optimum solution. It is also important to analyse the Pareto-efficient frontier, along which neither stakeholder (platform or government) can increase its own benefit (profit or social welfare) without decreasing the other stakeholder's benefit. As a result, Pareto-efficient solutions are a set of operating strategies that achieve the best results when considering both the platform and the government, as deviating from these strategies cannot simultaneously improve the results for both stakeholders. The monopoly optimum and social optimum are thus the two polar points of the Pareto-efficient frontier.

Yang and Yang (2011) sought a set of Pareto-efficient solutions by simultaneously considering two objectives–the maximisation of platform profit and the maximisation of social welfare–which naturally gave rise to a bi-criteria or bi-objective maximisation problem. They showed that the utilisation rate of vehicles and the service quality (measured in terms of the waiting/searching time of passengers) were constant along the Pareto-efficient frontier and equal to that at the social optimum. Analyses of the properties of the Pareto-efficient frontier would assist governments to design suitable regulations, such as a price cap, a minimum wage level, a maximum-allowed fleet size or a minimum vehicle utilisation rate, to achieve a desirable level of social welfare without deflecting the optimal strategies from the Pareto-efficient frontier, thereby preserving market efficiency.

1.2.3. Regulations

The emergence of ride-sourcing services brings convenience to travellers but also creates many questions and challenges. A major question is whether and how a government should regulate a ride-sourcing market. Regulations have already been established in some locations, particularly in metropolitan cities. For example, New York City requires ride-sourcing platforms to guarantee that the hourly wage of drivers is higher than the minimum wage (US$15/h). This was extended to a ‘minimum per-trip formula’ stipulating that the wage per trip should not be less than US$23 for a 30 min/7.5 mile ride. In June 2019, New York City imposed a more stringent regulation on Uber, Lyft and their competitors, which requires drivers to carry a passenger at least 69% of the time they are operating in Manhattan below 96th Street, or the companies will be subject to penalties. Similarly, in January 2020, California Assembly Bill (AB5) was passed, which classifies hundreds of thousands of independent contractors, including ride-sourcing drivers, as full-time employees. Uber and Lyft were denied exemption from this legislation, but nevertheless refused to reclassify their drivers as employees and declared that they planned to continue ‘business as usual’, which exposes them to litigation from state agencies. In addition, since 2016, the authorities of Beijing and Shanghai have required DiDi to only employ drivers who are registered residents in their cities. This regulation is similar to the fleet-size control rules imposed in taxi markets, which allow only drivers with local taxi licences to provide ride services.

The effects of existing and proposed regulations have been extensively studied and debated. Li et al. (2019) argued that although imposing a minimum wage can motivate ride-sourcing companies to hire more drivers and serve more passengers, it will cause companies' profits to decrease. They also found that fleet size control (i.e., maximum fleet size) regulation reduces driver income, as it motivates a platform to hire cheaper labour by reducing drivers' average pay. Parrott and Reich (2018) examined the likely effects of the regulations imposed in New York City by carrying out simulation studies based on TNC administrative data. This revealed that a regulation guaranteeing minimum wage will increase drivers' income by 22.5%, but will also increase passengers' trip fares and waiting times by 5% and 12–15 s, respectively. Yu et al. (2020) argued that traditional taxi services will die out if no government regulation is applied, and therefore agreed that the Chinese government's new regulations effectively balance multiple objectives, namely, business and job creation, the viability of taxi services and consumer welfare.

These relatively mixed empirical findings on the effects of regulation on ride-sourcing markets are partially due to the fact that passenger demand, driver supply and other characteristics of markets vary from city to city. Therefore, it is critical that mathematical models are established to investigate the effects of regulatory policies currently applied by various cities to their respective ride-sourcing markets, as this will facilitate the development of new regulatory policies that better coordinate the interests of all stakeholders in these markets.

1.2.4. Ride-pooling services

Recently, several TNCs have launched on-demand ride-pooling services (Chen et al., 2017), which enable a driver to serve two or more passengers per ride. Typical examples include UberPool, DiDi Express Pool, Lyft Line and GrabShare, with Lyft aiming for 50% of its rides to be shared by 2022 (Schaller, 2018). Ride-pooling services are expected to improve vehicle utilisation and alleviate traffic congestion, and help solve the first-/last-mile problem in public transit (Wang et al., 2019; Wang and Odoni, 2016). However, this new mobility service brings new challenges, such as determining a discounted trip fare that will attract ride-pooling passengers.

When passengers launch a TNC application (app), they can choose to submit an order for a ride-pooling service or a normal (non-pooling) service. The fare for a ride-pooling service will typically be discounted to a predetermined cost that is less than the trip fare of a non-pooling service for the same distance. A key concern for ride-sourcing platforms is the pool-matching probability, i.e., the proportion of passengers who are pool-matched and thus share their rides with other ride-pooling passengers, as ride-pooling can have adverse effects on passengers' service experience or platforms' profits. For example, pool-matched passengers may experience a longer trip time than they would by using a non-pooling service, due to detours being made to service the transportation needs of their fellow pool-matched passengers. Alternatively, platforms may suffer a loss of revenue because up-front discounts decrease predetermined fares.

The relationships between the system endogenous variables and decision variables of ride-pooling services are more complicated than those between the variables of regular ride-sourcing services. First, the pool-matching probability depends on the passenger demand for ride-pooling and the pool-matching strategies, which generally impose matching radii to pool-match passengers with similar origins and destinations. A platform will also typically perform pool-matching after a certain period of waiting (by passengers) for a certain length of matching window, as this accumulates passengers to enable better pool-matching. Thus, it is crucial to determine the length of the matching window and the matching radius, as these influence the pool-matching probability, passengers' detour time and vehicle utilisation. Second, the predetermined fare discount rate directly affects a platform's profits and passenger demand, which in turn affects pool-matching probability and thereby indirectly affects a platform's profits. A precise