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- Permanent Link:
- https://ufdc.ufl.edu/UFE0050878/00001
## Material Information- Title:
- Modeling and Analysis of On-Demand Ride-Sourcing Markets
- Creator:
- Zha, Liteng
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2017
- Language:
- english
- Physical Description:
- 1 online resource (182 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Civil Engineering
Civil and Coastal Engineering - Committee Chair:
- YIN,YAFENG
- Committee Co-Chair:
- SRINIVASAN,SIVARAMAKRISHNAN
- Committee Members:
- WASHBURN,SCOTT STUART
SLUTSKY,STEVEN M - Graduation Date:
- 4/30/2017
## Subjects- Subjects / Keywords:
- competition -- labor-supply -- matching -- on-demand -- ride-sourcing -- surge-pricing
Civil and Coastal Engineering -- Dissertations, Academic -- UF Customers ( jstor ) Pricing ( jstor ) Market prices ( jstor ) - Genre:
- Electronic Thesis or Dissertation
born-digital ( sobekcm ) Civil Engineering thesis, Ph.D.
## Notes- Abstract:
- Due to the innovative business mode and advanced dispatch technology, ride-sourcing companies such as Uber and Lyft have attracted many riders, eroding the traditional taxi market. Meanwhile, substantial controversies arise along their way of expansion. Among those often highlighted are dynamic pricing, service competition and regulation. In this dissertation, we propose both analytical and simulation frameworks to address these key issues. The proposed methodologies and tools could serve as important references for regulatory agencies in evaluating and managing the ride-sourcing markets. Our first investigation is on the regulation and competition of the ride-sourcing market. An aggregate economic model is developed where the matching between customers and drivers are captured by an exogenous matching function. It is found that a monopoly ride-sourcing platform will maximize the joint profit with its drivers without any regulatory intervention. We establish the conditions for regulating only the amount of commission charged by the platform to achieve the second best. Then the model is extended into a duopoly setting and unveils that competition does not necessarily lower the price or improve the social welfare. In the latter case, regulators may rather encourage the merger of the platforms and regulate them directly as a monopolist. The exploration of dynamic pricing is divided into two main thrusts. As drivers may adjust work schedules to cover more profitable periods, we first capture drivers' work hour choices under dynamic pricing utilizing a bi-level programming framework. The patterns of the market dynamics and the trade-offs among market players are then numerically discussed. Next, we extend the model into a spatially differentiated market. The spatial variations of market frictions and the welfare of involved agents under dynamic pricing are explicitly explored. Based on an empirical data set, we demonstrate the equilibrium outcomes under dynamic pricing and evaluate the effectiveness of a commission cap regulation. In parallel with the analytical models, an agent-based simulation is developed to better understand the ride-sourcing markets. We find that platform's matching technology exhibits increasing returns to scale property; and higher matching ranges lead to more transactions but lower magnitude of the returns to scale. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2017.
- Local:
- Adviser: YIN,YAFENG.
- Local:
- Co-adviser: SRINIVASAN,SIVARAMAKRISHNAN.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-11-30
- Statement of Responsibility:
- by Liteng Zha.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Applicable rights reserved.
- Embargo Date:
- 11/30/2017
- Classification:
- LD1780 2017 ( lcc )
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PAGE 1 MODELING AND A NALYSIS OF ON DEMAND RIDE SOURCING M ARKETS By LITENG ZHA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PH ILOSOPHY UNIVERSITY OF FLORIDA 2017 PAGE 2 2017 Liteng Zha PAGE 3 T o my family PAGE 4 4 ACKNOWLEDGMENTS Ph.D. is a bitter sweet journey After academic training, what I have gained is not just the knowledge and skills, but the confidence and persistence when confronted with challenges As the final deliverable, this dissertation could not be done by me alone. First and foremost, I appreciate the support from my beloved parents Yuqiang Zha and Yuchen Chen as well as my dear girlfriend Yinan Zheng. T hank you for accompanying me through th e ups and downs. You are the ones always t here that I can turn to. Next, my thank s go to my advisor Dr. Yafeng Yin, the most diligent (y es, not one of) professor I am fortunat e to work with. I v alue the discussions and tips from Dr. Yin both in research and daily life I s guidance and genero sity in my career design There are fantastic faculty members at UF from who I build up my knowledge set I tha nk Dr. Steven Slutsky, Dr. Siva Srinivasan and Dr. S cott Washburn for being on my committee and offering constructive comments on this dissertation ; Dr. Lily for the supervision on the FDOT project; Dr. Yongpei Guan, Dr. William Hager and Dr. J P Richard for the awesome courses on Operations Research stimulates my frenzy on classical economic theory. Last but not the least, I wou ld like to pass my thanks to my fr iends at UF, without who m my Ph.D. life would become les s colorful I cherish the happy time with Zhibin Chen, Zhengtian Xu, Xiaotong Sun, Zibin Bian Shanyue Guan, Do n Watson and Stephen Spana I also thank Ms. Ines, a lovely lady, for her consistent effort s in coordinating the transportation engineering grou p. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 4 LIST OF TABLES ................................ ................................ ................................ ........................... 9 LIST OF FIGURES ................................ ................................ ................................ ....................... 10 ABSTRACT ................................ ................................ ................................ ................................ ... 12 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .................. 14 1.1 Background ................................ ................................ ................................ ....................... 14 1.2 Dissertation Objectives ................................ ................................ ................................ ..... 18 1.2.1 Platform Regulation and Competition ................................ ................................ .... 18 1.2.2 Dynamic Pricing ................................ ................................ ................................ ..... 19 1.2.3 Si mulation Modeling ................................ ................................ .............................. 19 1.3 Dissertation Outline ................................ ................................ ................................ .......... 20 2 LITERATURE REVIEW ................................ ................................ ................................ ....... 21 2.1 Regulation of Taxi and Ride Sourcing Markets ................................ ............................... 21 2.2 Modeling and Analysis of Taxi Markets ................................ ................................ .......... 23 2.3 Dynamic Pricing ................................ ................................ ................................ ............... 25 3 ECONOMIC ANALYSIS OF ON DEMAND RIDE SOURCING MARKETS .................. 27 3.1 Basic Model ................................ ................................ ................................ ...................... 27 3.1.1 Matching Function ................................ ................................ ................................ .. 28 3.1.2 Customer Demand ................................ ................................ ................................ .. 31 3.1.3 Comparative Statics ................................ ................................ ................................ 32 3.2 Market Scenarios of Single Ride Sourcing Platform ................................ ....................... 33 3.2.1 Monopoly Scenario ................................ ................................ ................................ 33 3.2.2 First Best Scenario ................................ ................................ ................................ 35 3.2.3 Second Best Scenario ................................ ................................ ............................. 36 3.2.4 Discussions on the Single Platform ................................ ................................ ........ 38 3.2.4.1 Effects of homogeneous value of time ................................ ......................... 38 3.2.4.2 Contract solution set and properties ................................ ............................. 39 3.2.5 Regulation Policies ................................ ................................ ................................ 40 3.2.5.1 Regulating F ................................ ................................ ................................ 41 3.2.5.2 Regulating N ................................ ................................ ................................ 41 3.3 Competing P latforms ................................ ................................ ................................ ........ 42 3.3.1 Competition between Ride Sourcing Platforms ................................ ..................... 43 3.3.2 Second Best for Dual Ride Sourcing Platforms ................................ ..................... 45 PAGE 6 6 3.3.3 Welfare Changes between Single and Dual Platforms ................................ ........... 48 3.3.3.1 Analytical comparison ................................ ................................ .................. 49 3.3.3.2 Numerical analysis ................................ ................................ ....................... 50 3.4 Summary ................................ ................................ ................................ ........................... 52 4 DYNAMIC PRICING AND LABOR SUPPLY IN ON DEMAND RIDE SOURCING MARKETS ................................ ................................ ................................ ............................. 60 4.1 Literature on Labor Supply ................................ ................................ ............................... 61 4.2 Basic Modeling Considerations ................................ ................................ ........................ 62 4.2.1 Time Expanded Network ................................ ................................ ........................ 62 4.2.2 Demand and Revenue ................................ ................................ ............................. 63 4.2.3 Cost for Drivers ................................ ................................ ................................ ...... 66 4.3 Equilibrium Models with Endogenous Labor Supply ................................ ...................... 68 4.3.1 Neoclassical Equilibrium Model ................................ ................................ ............ 68 4.3.2 Incom e Targeting Equilibrium Model ................................ ................................ ... 71 4.3.3 Enhanced Equilibrium Models with Long Trip Duration ................................ ...... 74 4.4 Numerical Examples ................................ ................................ ................................ ......... 75 4.4.1 Set up ................................ ................................ ................................ ...................... 75 4.4.2 Equilibrium Solutions ................................ ................................ ............................. 76 4.4.3 Labor Supply Elasticity ................................ ................................ .......................... 77 4.5 Dynamic Pricing and Its Regulation ................................ ................................ ................. 78 4.5.1 Modeling Framework ................................ ................................ ............................. 78 4.5.2 Commission Cap Regulation ................................ ................................ .................. 79 4.5.3 Numerical Experiments ................................ ................................ .......................... 80 4.6 Summary ................................ ................................ ................................ ........................... 83 5 MODELLING SPATIAL EFFECTS OF DYNAMIC PRICING IN RIDE SOURCING MARKETS ................................ ................................ ................................ ............................. 94 5.1 Modeling Components ................................ ................................ ................................ ...... 94 5.1.1 Ge ometrical Matching ................................ ................................ ............................ 95 5.1.1.1 Technology ................................ ................................ ................................ ... 95 5.1.1.2 Returns to scale ................................ ................................ ............................ 98 5.1.2 Customer Demand ................................ ................................ ................................ .. 99 5.1.3 Vehicle Supply and Spatial Distribution ................................ ................................ 99 5.1.4 Market Equilibrium ................................ ................................ .............................. 101 5.2 Model Estimation ................................ ................................ ................................ ............ 103 5.2.1 Dataset ................................ ................................ ................................ .................. 103 5.2.2 Estimation of System Response Time and A verage Waiting Time ..................... 104 5.2.3 Demand Estimation ................................ ................................ .............................. 106 5.2.4 Labor Supply ................................ ................................ ................................ ........ 107 5.3 Equilibrium Results ................................ ................................ ................................ ........ 108 5.4 Dynamic Pricing ................................ ................................ ................................ ............. 109 5.4.1 Formulation ................................ ................................ ................................ .......... 110 5.4.2 Numerical Experiments ................................ ................................ ........................ 110 5.5 Commission Cap Regulation ................................ ................................ .......................... 113 PAGE 7 7 5.5.1 Formulation ................................ ................................ ................................ .......... 114 5.5.2 Numerical Experiments ................................ ................................ ........................ 114 5.6 Summary ................................ ................................ ................................ ......................... 115 6 AN AGENT BASED SIMULATION FOR ON DEMAND RIDE SOURCING MARKETS ................................ ................................ ................................ ........................... 130 6.1 Simulation Framework ................................ ................................ ................................ ... 130 6.1.1 Simulation Test Bed ................................ ................................ ............................. 131 6.1.2 Basic Agents Module ................................ ................................ ........................... 131 6.1.3 Transaction Module ................................ ................................ .............................. 132 6.1.4 Zonal Choice Module ................................ ................................ ........................... 133 6.1.5 Market Entry and Equilibrium ................................ ................................ .............. 134 6.2 Simulation Evaluation Framework ................................ ................................ ................. 135 6.3 Results and Discussions ................................ ................................ ................................ .. 1 36 6.3.1 Analysis of the Output Data ................................ ................................ ................. 136 6.3.1.1 Equilibrium property ................................ ................................ .................. 136 6.3.1.2 Matching frictions ................................ ................................ ...................... 137 6.3.1.3 Success rate of zonal transition ................................ ................................ .. 139 6.3.2 Regression Analysis ................................ ................................ ............................. 139 6.4 Summary ................................ ................................ ................................ ......................... 141 7 CONCLUSIONS AND FUTURE RESEARCH ................................ ................................ .. 151 7 .1 Conclusions ................................ ................................ ................................ ..................... 151 7 2 Future Research ................................ ................................ ................................ .............. 153 APPENDIX A RELAXATION OF P2 ................................ ................................ ................................ ......... 154 B PROOF OF PROPOSITION 1 ................................ ................................ ............................. 155 C .................... 156 D SENSITIVITY ANALYSIS ................................ ................................ ................................ 157 E EQUILIBRIUM MODEL WITH LONG TRIP DURATION ................................ .............. 160 E.1 Model Development ................................ ................................ ................................ ....... 160 E.2 Equilibrium definition and formulation ................................ ................................ ......... 162 E.3 Numerical Results ................................ ................................ ................................ .......... 162 F NOTATIONS ................................ ................................ ................................ ....................... 168 G EXISTENCE OF EQUILIBRIUM ................................ ................................ ....................... 169 H OVERVIEW OF THE SIMULATION PACKAGE ................................ ............................ 173 PAGE 8 8 LIST OF REFERENCES ................................ ................................ ................................ ............. 176 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ....... 182 PAGE 9 9 LIST OF TABLES Table page 4 1 Equilibrium Flow Distributions under ME N. ................................ ................................ ... 92 4 2 Equilibrium Flow Distributions under ME I. ................................ ................................ .... 93 5 1 Demand Estimation for Different Parameterizations. ................................ ...................... 128 5 2 Summary of the Parameters for t he Supply Side. ................................ ............................ 129 6 1 ................................ ................................ 146 6 2 Distributions of Trip Distances (mile). ................................ ................................ ............ 147 6 3 Suggested Value of the Parameters. ................................ ................................ ................. 148 6 4 Success Rate for Zonal Choice. ................................ ................................ ....................... 149 6 5 Summary of Regression Results. ................................ ................................ ..................... 150 F 1 Summary of Parameters. ................................ ................................ ................................ .. 168 PAGE 10 10 LIST OF FIGURES Figur e page 3 1 Cost Minimization of the Matching Technology ................................ .............................. 54 3 2 Contract Solution Set for Single Ride Sourcing Platform ................................ ................ 55 3 3 Contract Solution Set for Dual Ride Sourcing Platforms. ................................ ................. 56 3 4 Change in Social Welfare between Duopoly and Monopoly Solu tions ............................ 57 3 5 Average Searching Time ( and ). ................................ ............... 58 3 6 Best S olutions (at Zero Profit Level) ................................ ................................ ......................... 59 4 1 Time Expanded Network for Work Hour Choices. ................................ ........................... 85 4 2 The Relationship between Utility and Working Hours with Varying Wage Rates. .......... 86 4 3 Distribution of Daily Base Demand. ................................ ................................ .................. 87 4 4 Equilibrium Work Hours and Hourly Revenue with Varying Surge Multipliers .............. 88 4 5 Market Outcomes under Dynamic Pricing ................................ ................................ ........ 89 4 6 under Dynamic Pricing. .................. 90 4 7 Market Outcomes under Different Commission Caps ................................ ....................... 91 5 1 Estimation of The System Response Time, Cust erage Waiting and Matching Time ................................ ................................ ................................ ................. 117 5 2 Display of the Fraction of The On Line Drivers by Hour of Day. ................................ .. 118 5 3 Market Fri ctions with Different Z one and Fleet Size Combinations ............................... 120 5 4 Rate under Varying Fleet Sizes .................... 122 5 5 Display of Surge Mu ltipliers under Dynamic Pricing ................................ ..................... 123 5 6 Spatial Variations of Aver age Searching and Waiting Times ................................ ......... 124 5 7 Temporal Display of Welfare Changes under Dynamic Pricing ................................ ..... 125 5 8 Spatial Display of Welfare Changes under Dynamic Pricing. ................................ ......... 126 5 9 Equilibrium Outcomes unde r the Commission Cap Regulation ................................ ...... 127 PAGE 11 11 6 1 Simulation Configuration ................................ ................................ ................................ 142 6 2 Demand and Supply Equi librium in the Ride Sourcing Market ................................ ..... 143 6 3 Display of the Ratio of ( w v / w c ) to ( N v / N c ) for Each Simulation Run ............................ 144 6 4 Custome Radii ................................ ................................ ................................ ................................ 145 E 1 Display of A Sample Work Schedule with Corresponding Modeling Components. ...... 164 E 2 Distribution of Average Speed. ................................ ................................ ........................ 165 E 3 Average Searching Time and Service Duration with Time Varying Trip Durations. ..... 166 E 4 Average Waiting Time with Time Varying Trip Durations. ................................ ........... 167 G 1 Average Searching Time and Its Matching Time Portion. ................................ .............. 171 G 2 Matching Frictions. ................................ ................................ ................................ .......... 172 H 1 Display of the Simulation Interface. ................................ ................................ ................ 175 PAGE 12 12 Abstract of Dissertation Presented to the Graduate School of the U niversity of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MODELING AND ANALYSI S OF ON DEMAND RIDE SOURCING MARKETS By Liteng Zha May 2017 Chair: Yafeng Yin Major: Civil Engineering Due to the innovative business mode and advanced dispatch technology, ride sou rcing companies such as Uber and Lyft have attracted many riders, eroding the traditional taxi market. Meanwhile substantial controversies a rise along their way of expansion. Among those often highlighted are dynamic pricing service competition and regulation In this dissertation we propose b oth a nalytical and sim ulation frameworks to address these key issues. The proposed methodologies and tools could serve as important ref erences for regulatory agencies in evaluating and managing the ride sourcing market s Our first investigation is on the regulation and competition of the ride sourcing market. A n aggregate economic model is developed where the matching between customers and drivers ar e captured by an exogenous matching function. It is found that a monopoly ride sourcing platform will maximize the joint profit with its drivers without any regulatory intervention We establish the conditions for regulating only the amount of commission c harged by the platform to achieve the second best. The n the model is extended into a duopoly setting and unveils that competition does not necessarily lower the price or improve the social welfare. In the latter case, regulators may rather encourage the me rger of the platforms and regulate them directly as a monopolist. PAGE 13 13 The exploration of dynamic pricing is divided into two main thrusts. As drivers may adjust work schedules to cover more profitable periods, we first capture choices under dynamic pricing utilizing a bi level programming framework T he patterns of the market dynamics and the trade offs among market players are then numerically discussed N ext we extend the model in to a spatial ly differentiated market The spatial variations of market frictions and the welfare of involved agents under dynamic pricing are explicitly explored Based on an empirical dataset, we demonstrate the equilibrium outcomes under dynamic pr icing and evaluate t he effectiveness of a commission cap regulation In par allel with the analytical models, a n agent based simulation is developed to better understand the ride sourcing market s We find that matching technology exhibits increasing returns to scale property; and higher matching ranges lead to more tran sactions but lower magnitude of the returns to scale. PAGE 14 14 CHAPTER 1 INTRODUCTION 1.1 Background The number of smart mobile devices in the U.S. has been rising steadily and a study suggests that nearly two thirds of Americans now own at least one such device ( Pew Research Center (2015) ). T locations, enable ubiquitous communications and allow instant peer to peer interaction, giving rise to a new class of firms, on demand companies, which aim to effectiv ely bring together consumers and suppliers of resources (e.g., houses and parking spaces) and services (e.g., home cleaning and computer programming) with very low transaction costs. These companies are shaking up their industries and reshaping our daily l ives. On demand r ide sourcing companies such as Uber, Lyf t and Didi Chuxing a typical example of on demand economy are transforming the way we travel in cities. The companies provide ride hailing apps, which are real time and internet based platforms tha t intelligently source participating drivers to riders. A rider can monitor in real time the location of the coming vehicle and receive notification when it arrives. These apps are free to use but usually charge a com mission for each transaction (20 25 % of the fare paid to drivers). Thanks to their convenience a nd lower prices, on demand ride sourcing services have successfully attracted many riders, eroding the traditional taxi market. For example, in 2013, the revenue of Uber, Lyft and Sidecar (now bankru pted) is $140M in San Francisco, half what the established cab companies made. Uber now operates in more than 200 cities in approximately 50 countries ( The Economist, 2015 ) Several terms exist for descri bing services provided by Uber like companies, such as ridesharing, for profit rid esharing, on demand ridesharing and dynamic ridesharing, which may PAGE 15 15 ( Anderson, 2014 ; Rayle et al., 2014 ) However, the services provided by Uber like companies have distinct features and the be confusing Unlike taxis or jitneys, ridesharing is conventionally not f or profit For Uber like services, however, it is largely the financial motivation that brings in private car owners to participate and some even become a ful l like companies. Considering these companies are essentially sourcing a ride from a driver pool, Rayle et al. (2014) oning that major ride sourcing companies also allow ride splitting on their platforms to encourage multiple people to share a ride, e.g., UberPOOL and Lyft Line which may further blur the line between ridesharing and ride sourcing. However, we choose to u se ride sourcing (which is a reduced form from on demand ride sourcing for t he convenience of presentation) to highlight its distinctive features, i.e., private car owners drive their own vehicles to p rovide on demand taxi like services for profit Since their advent in 2009, ride sourcing companies have enjoyed huge success, but also created many controversies. The major opposition comes from the traditional or regular taxi industry, as ride sourcing platforms become a growing threat for their profit. The regular taxi market is usually regulated in terms of price, entry and service quality while comparatively fewer regulatory requirements have been imposed on or proposed for ride sourcing companies. Unfair competition is argued particularly by professional cab drivers and their employers, who have organized strikes and filed lawsuits around the world. Ride sourcing companies have also brought headache to government officials and legislators, because they do not know how to deal with them. While many are sti ll wondering what to do, some ( e.g., cities in Netherlands and Japan) have decided to ban them or treat their services to be illegal; others (e.g., Stockholm, PAGE 16 16 Sweden; Salt Lake City, Utah) embrace them as a new type of transit provider or have passed ride sourcing legislations and regulations. Although these legislations and regulations have some differences, they all essentially codify the insurance coverage, background check of drivers and inspection protocols that ride sourcing companies already have in place. There is no intervention on the fare of their services or the affiliated fleet size (or the number of vehicles in service) The success of ride sourcing platforms casts doubts on the regulation of the taxi industry that is often criticized for limit ed supply ( Badger, 2014 ) It also challenges some of the fundamentals for the tradit ional taxi regulation by significantly reducing information imperfection. The automatic matching algorithms offered by the platforms lead to significant mechani sm (the reputation system), the interaction between customers and drivers is enhanced. Moreover, all trip related fares are processed by built in pricing and payment functions, which eliminate the chaos caused by soliciting and bargaining that often emerge after a taxi market is deregulated ( ECMT, 2007 ) Last but not least, some argue that the competition among multiple ride souring platforms may lower the prices and reduce the market power of a particular predominate platform. Should we simply deregulate the taxi industry and leave rid e sourcing companies alone, and then let the market decide the winners? What are the implications and welfare impacts of such a deregulation? Should we conduct a systematic reform of regulation of the ride for hire market? Presumably the reform needs to be tailored to specific cities considering their demographics, mobility options, patterns, and culture. Is there a unified theoretical framework to guide such a reform? Many such questions remain unanswered Besides, t he improvement in the technology and the lack of regulatory inter vention have stimulated new business modes and distinctive features in ride sourcing market s One of the PAGE 17 17 issues centered in the discussion is surge pricing which is essentially a dynamic pricing scheme that adjusts trip fare in re al time. Based on the market condition of a geographic area, the base information of the surge multipl ier (SM) is provided to both customer s and driver s before a transaction happens. With the dual effect of suffocating dem and and increasing supply, dynamic pricing is advocated to guarantee a reasonable amount of waiting time for customers. In most cases, the surge multiplier falls below 1.5 times. However, it ca n soar 7 times or higher without a cap limit ( Curley, 2014 ; The Economists, 2014 ) Platforms generally benefit from the surges since commission is charged by a fixed percentage of the final trip fare. Therefore, platforms may surge unnecessarily high or more fr equently to exploit customers, given that no regulation is imposed and the pricing algorithms are proprietary Despite the heated discussion over the social media the temporal and spatial impact s of dynamic pricing have not been quantitatively modeled T he problem is manifested when we consider the flexibility ride sourcing drivers enjoy in choosing their work hours and locations. On one hand, t hey may adjust work schedules to cover more profitable periods; and the temporal variation i n the wage rate indu ced by dynamic labor supply decision s, e.g ., how long they would like to work ( Chen and Sheldon, 2016 ) On the other hand drivers are attracted to highly surged zone s for better trip opportunities. Little is known for the hic areas Unfortunately, related research on the ride sourcing market is limited. To the best of our knowledge, 1) there is no analytic al framework in studying the regulation and service competition i n the ride sourcing market; 2) a lthough several studie s empirically investigate the PAGE 18 18 labor supply under dynamic pricing ( Hall and Krueger, 2015 ; Chen and Sheldon, 2016 ) there is an increasing need to explore the temporal and spatial impact s of dynamic pricing and provide guidance for management strategies. 1.2 Dissertation Objectives This dissertation does not attempt to solve all the puzzles and controversies associated with the ride sourcing companies. Specifically, we propose to establish methodologies and tools to analyze the structure, service comp etition and dynamic pricing of the ride sourcing market s and then der ive insights on the management policies The se topics focus on different analytical domains and the proposed frameworks are thus characterized by different modeling resolution and complexity. In a parallel effort, w e develop an agen t based simulation to valid ate part of the assumptions and results 1.2.1 Plat form Reg ulation and Competition We start our first model by treating the r ide sourcing platform as a two sided market which matches users from both demand and supply side s ( Rochet and Tirole, 2003 ; Armstrong, 2006 ; Rochet and Tirole, 2006 ) The m atching process is featured by the nega tive same side externality and the positive cross side externality. That is, increasing one customer (driver) raises the average waiting (searching) time for the other customers (drivers) on the same side but reduces the average searching (waiting) time for the drivers (customers) cross the side. To capture such a market structure, w e revisited an aggregate economic model with an exogenous matching function ( Yang and Yang, 2011 ) The temporal and spatial variations of market dynami cs are assumed away and we essentially focus on the static market equilibrium in the long run Based on this, w e first examine the economic outcomes under different market scenarios and investigate the optimal regulation strategy that requires the minimum regulatory burden. We next extend our analysis via incorporating platform competition and explore the changes in trip fares PAGE 19 19 and social welfare to shed light on whether competition is beneficial to customers and the society at large 1 .2.2 Dynamic Pricing Dynamic pricing may adjust the distributions of demand and supply both temporally (e.g., encourage drivers to work during more profitable periods) and spatially (e.g., induce drivers to transition to highly surged zones). Our investigation of dynamic prici ng is divided into two main thrust s T o anal yze the temporal effect of dynamic pricing we develop a time expa nded network that outline s s and scheduling Then we propose a bi level programming framework wit h the upper level being specified with a revenue maximizing surge schem e while the lower level capturing hour choices In the numerical experiments we compare the patterns of the market dynamics under dynamic pricing and discuss the trade of f s among market players Subsequently i n a separate chapter we extend the framework to the spatially differentiated market and explore the spatial heterogeneity of market frictions and welfare of market players under dynamic pricing. The matching proc ess between the customers and drivers is further derived from a spatial point process If market power is a concern, we present a regulation scheme to enhance market efficiency U tilizing a public dataset from Did i Chuxing, we numerically explore the equil ibrium outcomes u nder dynamic pricing and evaluate the performance of the proposed regulation 1.2.3 S imulation Modeling We d evelop an agent based simulation in Netlogo to provide a better understanding of the working mechanism of ride sourcing platforms and validate some of the previous results and key assumptions The simulator attempts to represent various behaviors of the simulated agents, such as ride ing and PAGE 20 20 and sensitivity to waiting time as well as the pricing strategies and matching techniques of ride sourcing platforms. Based on a simulation test bed, we explore the market dynamics under a variety of market scenarios that characterize different demand levels and matching ran ge s between customers and drivers We also calibrate and discuss the employed matching functions using the simulated data 1.3 Dissertation Outline The remainder of this dissertation is organized as follows. Chapter 2 provides an overview of the literature on the analytical modeling of traditional taxi market s some recent development on the regulation and dynamic pricing of the ride sourcing market s Chapter 3 presents an aggregate economic model and discusses platform regulation and competition Chapter 4 models the work schedu ling of the ride sourcing drivers and investigates the temporal effect s of dynamic pricing. Chapter 5 investigate s the spatial heterogeneity under dynamic pricing and discusses the poten tial regulation policies Chapter 6 introduces an agent based simul ation and evaluates the Finally, the overall conclusions and recommendations for futur e work are provided in Chapter 7 PAGE 21 21 CHAPTER 2 LITERATURE REVIEW The literature on ride sourcing markets is rather thi n as these services were launched into the market quite recently. Therefor e, our review of literature interrelates some from the traditional taxi market whic h shares the similarity in the considerations of regulation and some modeling set ups. More specifi cally, we summarize our review of literature on three aspects: 1) regulation of taxi and ride sourcing markets ; 2) modeling and analysis of taxi marke ts ; and 3 ) dynamic pricing in the ride sourcing markets 2.1 Regulation of Taxi and Ride Sourcing Markets A regular taxi industry usually consists of dispatch, taxi stand and cruising segments The re gulat ion of the industry is to ensure the temporal and spatial stability and availability of taxi services and guarantee the public safety. It often takes the fo rm of price, entry and service quality ( Frankena and Pautler, 1986 ) The rationale for regulating a cruising market is mainly to correct the information imp erfection. It i s unlikely for a rider to have a prior experience with the service quality of the cab during a random hailing ( ECMT, 2007 ) Also, neither the cab driver nor the customer knows p. Consequently, a driver can potentially charge slig htly higher than a customer s ask, considering the cost for her to wait for the next available cab if s he turns do wn. For dispatch market, the reliable service quality (in terms of acceptable average waiting time and the 2 4 hour accessibility) needs to be guaranteed particularly in less popular areas ( Frankena and Pautler, 1986 ; ECMT, 2007 ) On the other hand regulation has been criticized for the lack of information and resources for regulators to take appropriate actions that maximize social benefit ( Schaller, 2007 ) The most controversial aspect comes from the medallion system, a common tool for entry control. The value of a medallion has once topped 1 PAGE 22 22 million in New York ( Badger, 2014 ) Th e medallion system has long been blamed to create a cartel that operates f or its own benefit rather than a simple tool to control the number of taxis ( Schaller, 2007 ; Badger, 2014 ) B esides, possible occurrence of regulation capture and rent seeking is often cited by the opponent s to taxi regulation ( Rauch and Schleicher, 2015 ) The majority of American cities still set limits on both the entry and price of the taxi market s The advent of ride sourcing companies has casted doubts on these regulations and caused sliding values of taxi medallions ( Badger, 2014 ) Features such as real time driver rider matching and mutual rating possessed by ride sourcing platforms can enhance the interaction among platform users and may reduce information imperfection. A reduction in search ing and waiting time of drivers and riders has been reported in some preliminary studies on ride sourcing companies ( Rayle et al., 2014 ) The built in pricing and payment functions have also eliminated soliciting and bargaining behaviors. Some have suggested that the ride sourcing market is self regulatory because the mutual rati ng system can lead to better service quality, and the competition among multiple platform s will lower the prices and reduce the market power of a predominant platform ( Koopman et al., 2015 ) These statements need to be examined wit h caution. Competition can be socially inefficient given the dominance of the cross side effect in a two sided market ( Wright, 2004 ; Hagiu, 2006 ) The altitude towards regulation of ride sourcing companies and the level of regulation vary among governments around the world. A detailed review on the process of regulation in major states in the U.S. can be found in Rayle et al. (2014) The core of the proposed or passed regulations is remarkably similar, codifying the insurance coverage, background check poli cy, and vehicle inspection protocol, in addition to other requirements such as voice enabled apps and more detailed reporting of activities. PAGE 23 23 2.2 Modeling and Analysis of Taxi Markets A number of analytical models have been developed ( Vany, 1975 ; Frankena and Pautler, 1986 ; Cairns and Liston Heyes, 1996 ; Salanova et al., 2011 ) after the seminal work of Douglas (1972) These models are all agg regate in nature, analyzing the properties of taxi market equilibrium. One of the most distinguishable features these models c apture is the searching and waiting frictions, since the taxi market is not merely cleared by price. On the demand side, a custome r will consider both the trip fare and waiting time when deciding whether to take a taxi. On the supply side, the profitability of a taxi will depend on its utilization rate (the fraction of time it is occupied) and operating cost In a series of work, Yan g and his collaborators developed the network equilibrium models to capture the spatial structure of a taxi market ( Yang and Wong, 1998 ; Yang et al., 2002 ) They further expanded the models to incorporate congestion externality ( Yang et al., 2005b ) temporal variation of the demand and supply ( Yang et al., 2005a ) user heterogeneity and modal competition ( Wong et al., 2008 ) and nonlinear pricing ( Yang et al., 2010a ) The form of the average waiting time function reflects the mechanism of how a customer and a driver are applying the stochastic geometry techniques and the double queuing model, Douglas (1972) Arnott (1996) and Matsushima and Kobayashi (2006) derived the average waiting time function for the stand, cruising and dispatching taxi market, respectively. Despite the difference in the analytical forms, they all confirm that the matching in the taxi market exhibits increasing returns to scale a phenomenon quite common in the queuing of gener al transportation systems ( Mohring, 1972 ) reduced if the number of waiting customers and the number of vacant vehicles are doubled. The first best solution is at deficit when the ma tching is increasing returns to scale, since the trip fare only covers the cost of PAGE 24 24 occupied taxi hours ( Vany, 1975 ; Arnott, 1996 ; Cairns and Liston Heyes, 1996 ) Accordingly, two remedies have been suggested. One is to subsidize the t axi service to obtain the first best; t he other is to regulate both the fare level a nd fleet size to achieve the second best, which shares the same objective of maximizing the social welfare but subject to a break even constraint of the industry profit ( Cairns and Liston Heyes, 1996 ; Yang et al., 2002 ) Despite the insightful implication of an increasing return s to scale matching function, previous derivations are somewhat flawed. The application o f the stochastic geometry ignores the competition among customers over the same vehicle so that the average waiting time is only a function of the density of vacant vehicles ( Yang et al., 2010b ) ; direct employment of the queuing methods forces the first in first out (FIFO) rule (for either the queue of waiting customer s or the vacant vehicles) which may be too strict for the spatial queuing process in the general taxi market (with the exception of the taxi stand market ) Therefore Yang et al. (2010 b) revisited the Cobb Douglas function first introduced by Schroeter (1983) to delineate the bilateral search and meeting process in the taxi market In fact, such an exogenous matching approach has long been considered for labor mark ets (e.g., see a recent short review by Moscarini and Wright (2010) ). A static market equilibrium can be found where dr and social surplus are shown to depend critically on the returns to scale of the search and meeting technology ( Yang and Yang, 2011 ) Intuitively, the search friction still exists in the ride sourcing or e hailing platforms (e.g. Curb and Hailo, which serves exclusively professional cab drivers, Shaheen (2 014) ). However, its properties may be different since the matching technology is significantly improved. A few recent studies still rely on the Cobb Douglas matching function (but with varying parameter values to differ entiate the matching technology) to investigate the competition between PAGE 25 25 traditional street hailing and e hailing services ( He and Shen, 2015 ) as well as the optimal price perturbation for the platform ( Wang et al., 2016 ) W e also assume a Cobb Douglas matching function in the discussion of platform competition and regulation in Chapter 3 due to its simple form; but will formall y derive the form using a spatial point process in Chapter 5 The calibration of the Cobb Douglas matching function and the discussion over the relationship between matching ranges and the returns to scale property are provided in Chapter 6. 2 .3 Dynamic P ricing D ynamic (surge) pricing is a common ly seen in the field of revenue management Companies dynamically adjust prices corresponding to the variations in demand for profit maximization ( Talluri and Van Ryzin, 2006 ) Two recent publications focus on the perfor mances of dynamic pricing implemented by the ride sourcing platforms. Cachon et al. (2015) fou nd dynamic pricing generally work ed of ch arging a fixed commission and yield ed very close results to the optimal contra ct where the platform dynamically determine d both the price (for the customers) and the wage (for the drivers) Co mpared to the static case, dynamic pricing increases the pl atform s profit; t he results on the and depends on the entry cost of the drivers ( Cachon et al., 2015 ) Dynamic pricing increases the hen ( supply is limited ) but hurts it when the entry cost is low ( supply is amber ) ( Cachon et al., 2015 ) Using a n M/M (k)/1 queuing model, Banerjee et al. (2015) explored t he effect of dynamic pricing and assumed t he platform was either revenue or transact ion maximizi ng. They modeled dynamic pricing via a threshold based dynamic pricing approach where price was adjust ed discretely and was exclusively based on the number of available vehicles in the queue. Interestingly, they found dynamic pricing under performed the op timal static pricing under both surge schemes. T he most PAGE 26 26 significa nt benefit from dynamic pricing i s however its robustnes s, i.e., dynamic pricing performs better when the platform has limited information ( Banerjee et al., 2015 ) The studies mentioned above are both based on a demand, which is only assumed to be a function of price. In Cachon et al. (2015) a (random) rationing on either the supply or demand side would occur to guarantee the market equilibrium. Comparatively in Banerjee et al. (2015) customers were assumed to abandon if not served immediately In our frameworks, customers are sensitive to both the pri ce and the average waiting time The form of the average waiting time is induced from t he spatial matching process that is either captured in an aggregate matching function as in Chapter 3 or derived based on the spati al point process as in Chapter 5 In t erms of dynamic pricing, w e focus on its impact in adjusting s in Chapte r 4 ; i n chapter 5, the discussion is extended to the spatial heterogeneity of market frictions and agents welfare W e also explor e the possible regulation polici e s of dynamic pricing if market power is a concern PAGE 27 27 CHAPTER 3 ECONOMIC ANALYSIS OF ON DEMAND RIDE SOURCING MARKETS This chapter makes a preliminary attempt to provide a quantitative analysis on the market structure of ride scouring services and explores ef fective re gu lation policies. W e consider hypothetical situations when ride sourcing companies become self sustainable and dominate the ride for hire market. We are subsequently interested in knowing whether and how to regulate the ride sourcing market. Fol lowing Yang and Yang (2011) we develop an aggregate model with a Cobb Douglas matching function to examine different market scenarios, properties and economic outcomes of a hypothe tical ride sourcing market with a single platform. In view of the potential market distortion, we investigate effective regulation policies that require minimal regulatory variables. The analysis is further extended to consider a duopoly market to examine the effects of platform competition. Analyzing the tradeoff in the pricing formula under the Nash equilibrium, we observe that competition does not necessarily lower the price level. Via a numerical analysis, we explore the conditions where competition is socially inefficient and a regulated monopoly market can be more efficient than a regulated duopoly one. The chapter is organized as follows. The basic aggregate model for a hypothetical ride sourcing market along with some comparative statics is presente d in Section 3.1. Section 3.2 explores the pricing structure, solution properties of the single platform across different market scenarios as well as the regulation policies. In Section 3.3, the discussion is extended to a duopoly market with a particular focus on the changes in price and social welfare. The summary of research findings and policy implications are provided in Section 3.4. 3.1 Basic Model This section introduces an aggregate model that captures a hypothetical ride sourcing market with a focu s on the matchings between customers and drivers. The model is established PAGE 28 28 by extending the work of Yang and Yang (2011) We firstly assume a hypothetical ride sourcing market with a single platform, a group of customers and a group of affiliated drivers. The ride sourcing platform serves as an intermediary that matches customers with potential drivers. The platform decides the fare paid by a customer, i.e., F and charges a commissi on, i.e., P from the payment by the customer at each transaction; the driver receives the remaining payment, i.e., It is assumed that the hypothetical ride sourcing market is mature such that the platform will gain profit from providing the services. A few other things are worth noting here: 1) unlike ride sourcing platforms, e hailing platforms for traditional regulated taxi market do not have price setting power ( He and Shen, 2015 ) ; 2) although in practice ride sourcing platforms charge commission as a percentage of the fare, such a distinction is immater ial in the context of this study ; 3) congestion externality (caused by both ride sourcing and regular vehicles) i s not considered ( Yang et al., 2005b ) 3.1.1 Matching Function The matchings between customer s and drivers are completed via a matching algorithm implemented at the ride sourcing platform. Taking Uber as an example, it dispatches one of the vehicles within a coverage radius of a requesting customer. The dispatching is made to minimize the estimate d waiting time for the customer ( Ranney, 2015 ) At the aggregate level, we assume that a matching function (a production function) can be used to characterize such a process. Note that aggregate matching functions have been calibrated for traditional street hailing (e.g., Yang et al. (2014) or radio dispatch taxi market (e.g., Schroeter (1983) Although being much more efficient with a larger matching area and more complete information of drivers and customers, the ma tching technology offered by a ride sourcing platform is actually similar to the one adopted by radio dispatch taxi companies. We thus assume that that aggregate matching functions may still be valid for repre senting the matching technology PAGE 29 29 More specifica lly, we consider a stationary state where variables such as the numbers of waiting customers ( ) and vacant ride sourcing vehicles ( ) are time invariant. The matching function then relates the rate of matchings (more precisely, meetings) per hour ( ) to and at any instant ( Yang and Yang, 2011 ) Note where is the average is the arrival rate of vacant vehicles per hour. where is the average customer waiting time and is the customer demand per hour. The matching function can be formally written as: ( 3 1 ) where are assumed so that the increase of e ither vacant vehicles or waiting customers will increase the meeting rate. We define the elasticities of the matching function with respect to and as and respectively. The elasticities reflect the matching technology of the ride sourcing platform. We hereinafter assume them to be constant and within the range of [0, 1]. This assumption leads to a Cobb Douglas matching function: ( 3 2 ) where is a scaling parameter, which depends on the unit of the meeting rate and encapsulates other factors in the matching technology that are not fully captured by and Moreover, the parameter can be interp reted to be related to the area of the ride sourcing market and the cruising speed of vacant vehicles (we assume waiting customers remain stationary until being picked up). is obtained by considering at the stationary state together with Eqs. ( 3 1 ) ( 3 2 ) : PAGE 30 30 ( 3 3 ) Setting yields: ( 3 4 ) depends on the number of vacant vehicles. This is true particularly when the supply of vacant vehicles is more than sufficient to serve the customers and there is no competition among customers over a particular vehicle. Further assuming and Eq. ( 3 4) will be reduced to the waiting time functions derived by Douglas (1972) and Arnott (1996) for cruising and radio dispatching taxi market respectively where represents the area of the market divided by the running speed of vacant vehicles. The matching function is increasing, constant or decreasing returns to scale when the sum of and is larger than, equal to or smaller than one ( Yang and Yang, 2011 ) By analyzing the radio dispatch taxi market in Minneapolis, U.S. and the overall taxi market in Hong Kong, China, respectively, both Schroeter (1983) and Yang et al. (2014) reported increasing returns to scale matching functions. The degree varies from 1.13 to 1.16. T he increasing returns to scale property is commonly seen in a queueing process that exists in many transportation systems ( Mohring, 1972 ; Schroeter, 1983 ; Arnott, 1996 ; Yang and Yang, 2011 ) In a ride sourcing both customers and drivers may increase the matching probability (Although this benefit may be marginal given the advanced m atching technology ride sourcing companies currently possess) After being matched, the matched driver will have to travel to pick up the customer, and the average travelling distance decreases with the increases in the numbers of customers and drivers. PAGE 31 31 Du e to the economy of density, the average waiting times of customers and drivers is expected to decrease more than linearly if both the numbers of customers and drivers increase. Consequently, the meeting rate increases more than linearly with the numbers o f customers and drivers. This observation has also been confirmed in an agent based sim ulation study conducted in Chapter 6 Therefore, the analyses hereinafter primarily focus on cases with an increasing returns to scale matching function. 3.1.2 Customer Demand Consider a stationary state where the hourly demand of customers (passengers) is Each customer consumes exactly one trip and faces two transportation modes: the ride sourcing service and alternative modes such as transit. The customer derives a d eterministic utility from completing the trip while incurring a generalized cost for using the ride sourcing service and for the other options. Note is determined endogenously while is exogenously given as a constant. With error terms capturing unmeasurable attributes, the utility from utilizing each mode can thus be specified as follows: ( 3 5 ) ( 3 6 ) where consisting of the trip fare ( ), waiting time cost ( ) and in vehicle trav el time cost ( ). This specification implicitly assumes that customers are homogen e ous in their values of waiting time ( ) and in vehicle travel time ( ). Moreover, is determined endogenously in Eq. ( 3 3 ) while represents the average trip time and is assumed constant. Each customer is assumed to choose the option that maximizes her utility. Consequently, the demand PAGE 32 32 for the ride sourcing service ( ) will dep end on the distribution of the error terms but can be specified as a decreasing function of the generalized cost: ( 3 7 ) where over the domain 3.1.3 Comparative Statics At any instant of the stationary state, total vehicle equals the sum of the numbers of vacant vehicles and occupied vehicles i.e., The matching function essentially dictate s the form of the waiting time function as defined in Eq. ( 3 3 ) So far, we have identified the following three equations: ( 3 8 ) ( 3 9 ) ( 3 10 ) The unknowns are Q, F, w v w c and N Treating F and N as decision variables, we present some derivatives with respect to F and N as follows: ( 3 11 ) ( 3 12 ) where It can be shown that ( 3 13 ) Similarly we have: PAGE 33 33 ( 3 14 ) ( 3 15 ) ( 3 16 ) ( 3 17 ) where are defined in Eqs. ( 3 11 ) and ( 3 12 ) 3.2 Market Scenarios of Single Ride Sourcing Platform In this section, three market scenarios with a single ride sourcing platform are examined. maximizing behavior without any regu latory intervention. The first best solution maximizes social welfare, but the platform and its drivers may be in deficit. We thus examine the second best scenario by adding additional constraints to guarantee the reservation profits of the platform and dr ivers. 3.2.1 Monopoly Scenario In this scenario, the ride sourcing platform determines the trip fare and commission to attract both drivers and customers to the platform to facilitate their matchings in order to maximize the profit of the platform. Due to the static nature of our model, advanced pricing features such as surge pricing are not considered. The platform essentially provides a two sided market and its decision making can be described as a leader followers game where the monopoly platform serves as the leader who determines the trip fare F and the commission to maximize its profit while customers and drivers are the followers. The former decides whether to use the ride sourcing service while the latter decide whether to provide the service. With a free entry and can thus be written as follows : PAGE 34 34 (P1) ( 3 18 ) s.t. ( 3 19 ) where is the c ost function of the ride sourcing platform and c captures the average hourly operation cost of a vehicle and the opportunity cost of the driver. For simplicity, we hereinafter do not consider non negative constraints of decision variables and only focus on interior solutions. Define the Lagrangian function of P1 as follows: ( 3 20 ) The first order nec essary conditions (FONCs) of P1 yield: ( 3 21 ) ( 3 22 ) ( 3 23 ) Eq. ( 3 21 ) indicates the profit of the drivers and that of the platform are substitutes while Eq. ( 3 23 ) shows is proportional to at optimality. Eq. ( 3 22 ) is the monopoly pricing formula Define the price elasticity of demand as It can be revised as: ( 3 24 ) which follows the fo rm of the Lerner formula ( Lerner, 1934 ) The right hand side of the pricing formula in Eq. ( 3 22 ) consists of f our terms: the marginal cost of the platform ( ), the average cost for a driver to serve a new customer ( ) and the monopoly mark up ( PAGE 35 35 to the returns to scale of the matching fu nction. As will be shown shortly, the realized waiting (or searching) frictions are lower if the matching function is increasing returns to scale. The incoming customer will thus be charged lower compared to the cases when the matching function is constant or decreasing returns to scale. Such an externality also exists in the optimal pricing formulae in other investigated scenarios. When substi tuting in the objective function using Eq. ( 3 19 ) we have the following equivalent formulation to P1: (P2) ( 3 25 ) s.t. ( 3 26 ) which indicates the platform maximizes its joint profit with its drivers. In fa ct, Eq. ( 3 26 ) can be safely dropped if we assume the resulting profit is nonnegative at optimality. (See Appendix A for more information). This implies that albeit not owning any vehicle, the platform essentially behaves like a traditional cab company that has a monopoly on the ride for hire market and thus determines its price and fleet size to maximize its profit. 3.2.2 First Best Scenario The first best s cenario represents an ideal case where a social planner or the platform maximizes total social welfare instead of its own profit by deciding the trip fare and fleet size. The commission does not impact social welfare but the revenue sharing between drivers and the platform. Therefore, the welfare formulation is similar to that in the traditional taxi literature except the additional cost incurred by the platform ( Yang and Yang, 2011 ) The correspo nding maximization problem can be written as: (P3) ( 3 27 ) PAGE 36 36 The FONC of the above problem leads t o: ( 3 28 ) ( 3 29 ) In a regular taxi market where the taxi trip production is increasing returns to scale, the first rofit is negative ( Douglas, 1972 ; Arnott, 1996 ; Cairns and Li ston Heyes, 1996 ) Utilizing Eq. ( 3 28 ) the joint hourly profit obtained by the ride sourcing platform and its drivers is given as: ( 3 30 ) Define the elasticity of the cost function as If and then That is, if the ma tching function exhibits increasing returns to scale and the cost function of the platform shows economies of scale, the profits for the platform and drivers will be negative, making the first best solution unsustainable. 3.2.3 Second Best Scenario Since t he profits for the platform and its drivers may be in deficit in the first best scenario, we consider the following second best pricing (P4) ( 3 31 ) s.t. ( 3 32 ) ( 3 33 ) PAGE 37 37 where represents the reservation profit of the platform and is nonnegative. Define , the Lagrangian function assoc iated with P4 can be written as: ( 3 34 ) The FONC will then give: ( 3 35 ) ( 3 36 ) ( 3 37 ) As shown in Eq. ( 3 35 ) the Lagrangian multipliers associated with the reservation profit constraints are the same. We can thus sum up Eqs. ( 3 32 ) and ( 3 33 ) to bound the joint profit of the platform and its drivers. Eq. ( 3 37 ) follows the Ramsey pricing ( Oum and Tretheway, 1988 ; Yang et al., 2005b ) It can be seen as a convex combination of the pricing formulas in the first best and monopoly solutions Further, the reservation profit constraints are binding given that the matching function exhibits increasing returns to scale. We summarize the results in the following proposition. Proposition 1. If and then The proof of Proposition 1 is given in Appendix B. In our modeling framework, the commission serves as an instrument to split the revenue obtained by the platfo rm and its drivers. Given a solution of F and N (and thus Q ), P is appropriately defined by the reservation profit constraints. PAGE 38 38 When the matching function shows increasing returns to scale, P4 is equivalent to the ubject to the same constraints ( Douglas, 1972 ; Vany, 1975 ; Frankena, 1983 ; Frankena and Pautler, 1986 ; Yang et al., 2002 ) 3.2.4 Discussions on the Single Platform 3.2. 4 .1 Effects of homogeneous value of time We have investigated the optimality conditions of three different scenarios. Despite the difference in the pricing formula, the optimality conditions all unveil the fact: ( 3 38 ) which means the average customer waiting time is proportional to the average driver searching time at optimality. Figure 3 1 illustrates an economic intuition, following Yang and Yang (2011) The meetings between customers and drivers can be viewed as the production of the ride sourcing platform with inputs and The monopolist and the social planne r essentially differ in the chosen meeting rates (realized demand levels). At each demand level, the optimal matching is characterized by cost minimizing production of the ride sourcing platform. Therefore, the proportional relationship in Eq. ( 3 38 ) results from the tangency of the matching function and the total external cost curve. The assumption on the homogeneous value of time of the customers guarantees t he external cost curve is a line, presenting a constant slope for all the scenarios examined 1 When the heterogeneity of value of time is modelled, however, such a tangency condition generally does not hold (See Appendix C for a detailed discussion). 1 The effect of homogeneous value of time is most evident when the matching function exhibits constant returns to scale. With the tangency condition, it is straightforwa rd to show the average waiting and searching times are constant ( Yang and Yang, 2011 ) measurement. Consequently, the monopoli st and customers are homogeneous in their value of time ( Spence, 1975 ; Yang and Yang, 2011 ) PAGE 39 39 3.2.4 .2 Contract solution set and properties The solution set for all the investigated cases are conceptually displayed in Figure 3 2 The monopoly and first best solutions are marked as the end points M and S respectively along a contract curve. The remaining points on the curve correspond to the second best solution of varying joint profit levels between the platform and its drivers. Each point on the contract curve r and social welfare contour ( Spence, 1975 ) That is, for a given reservation profit, the associated point maximizes the total welfare and vice versa. Define the average joint profit as It is intriguing to investigate the state where the increase of will be mutually beneficial to all the participants, i.e., and ( Yang and Yang, 2011 ) Consider the process of gradually increasing from the monopoly to the first best solution, it is generally expected the total joint profit keeps decreasing (and so does the average joint profit, i.e. ) while the social welfare is strictly increasing 2 In fact, the contract curve is characterized by Eqs. ( 3 8 ) ( 3 10 ) together with Eq. ( 3 38 ) Total differentiating Eqs. ( 3 8 ) and ( 3 38 ) yields : ( 3 39 ) ( 3 40 ) 2 Parallel to Proposition 1 in Yang and Yang (2011) we can treat as decision variables and evaluate and It can be shown the mutually beneficial situation occurs only when the matching function shows increasing returns to scale and the waiting time elasticity of demand is sufficiently large. However, su ch a solution is not on the contract curve and thus not of particular interest here. PAGE 40 40 Eliminating and re arranging the terms we have: ( 3 41 ) which is negative given the matching function shows increasing returns to scale. 3.2.5 Regulation Policies As previously shown, the monopoly ride sourcing market is suboptimal in terms of social welfare and thus re gulation may be necessary. This section seeks for regulation strategies by assuming the regulator has complete information. We examine possible combinations of regulatory variables to identify the most efficient regulation strategy that induces the second best and requires the minimum number of regulatory variables. The potential regulation variables include Let solve P4. Regulating all three variables definitely works. Since the break even constraint for drivers is binding at the second best solution a s given in Proposition 1, together with Eqs. ( 3 8 ) ( 3 10 ) t he nonlinear equation system has only two degrees of freedom (four equations and six unknowns). Therefore, regulating any two of these three variables at the second best level should work. Below we show that properly regulating the commission charged by th e platform can also yield the second best. To see this, recall that the second best solution can also be achieved by maximizing the customer demand subject to the reservation profit constraint. If the commission is regulated, the proprietary ride sourcing platform will maximize the customer demand, given that the cost function of the platform is assumed to exhibit economies of scale. Therefore, a proper choice of commission by the regulator would trigger the platform to maximize its profit to the reservat ion level, leading the final solution to coincide with the second best one Mathematically, when only is regulated at the second best level, the FONC of P1 is given by PAGE 41 41 and Correspondingly we have and from the FONC of P4. Both systems of equation s have two unknowns and share the same equations. Therefore, they will have the same solution. However, regulating F or N only will not work. 3.2.5.1 Regulating F When only restricting the FONC of P1 is given by and Correspondingly from the FONC o f P4, we have and Both systems of equations have two unknowns: but they differ in one equation and thus generally admit different solutions. 3.2.5.2 Regulating N When only restricting the FONC of P1 is given by and Correspondingly from the FONC of P4, we have and Following the same argument as above, we conclude that regulating N only may not achieve the second best. In summary, we have the following proposition. Proposition 2. Assuming customers are homogeneous in their value of time a nd the regulator has complete information, regulating the commission alone will achieve the second PAGE 42 42 best if the matching function exhibits increasing returns to scale and the cost function of the ride sourcing platform shows economies of scale. Admittedly, the discussion so far has been based on the assumption of homogeneous value of time for the customers. If a continuous distribution is adopted to capture the heterogeneity in the value of time, merely regulating the commission may not be enough. The main r eason is that the tangency condition given in Eq. ( 3 38 ) may no longer hold. The regulator may then have to regulate two variables to achieve the second best. 3.3 Competing Platforms The above analyses on a hypothetical single platform shed some light on the properties of the ride sourcing market. However, the real world situation is more complicated since several ride sourcing platforms are often competing for the market (e.g., Uber and Lyft in the U.S.). Some proponents of ride sourcing companies have stated that competition may lower down the price level and improve social welfare. In this section, we investigate such a statement in a duopoly ride sourcing ma rket. We consider that a driver will only work for a particular platform and a customer only uses one platform for a particular trip. It is not uncommon that a customer or driver installs more than one ride sourcing apps. Such a multi h oming issue is not c onsidered here ( Rochet and Tirole, 2003 ; Armstrong, 2006 ) With one more ride sourcing platform, the utility function for a customer to choose each option is considered as: ( 3 42 ) ( 3 43 ) All the other specifications are the same as the single platform case. The demand functions of the two platforms depend on the distribution of the error terms but can generally be PAGE 43 43 represented as and To facilitate the analysis, we make the following assumptions: Assumption 1. The own price effect is negative: The cross price effect is positive and symmetric: Further, and The above assumption generally hold s for the demand functions of two competing firms. Similarly, we assume a Cobb Douglas matching function for each ride sourcing platform. To simplify our analysis, we further assume the same parameter set as that in the single platform for each platform. Given the interpretation of the parameters in Section 3.1 we essentially assume that both platforms have the same geographic coverage of users, running speed of vehicles, and matching technology as the single platform. To summarize, we have the following assumption: Assumption 2. The matching function for each ride sourcing platform follows the Cobb Douglas type with the same parameters as those being used for the single platform. Namely, and 3.3.1 Competition between Ride Sourcing Platforms As seen p reviously, the commission can be substituted into the objective function using the reservation profit constraint. It is positive with the assumption on the non negativity of the joint profit between the platform and its drivers in a mature ride sourcing PAGE 44 44 market. We further assume both platforms choose simultaneously for profit (P5) ( 3 44 ) By setting and we have: ( 3 45 ) ( 3 46 ) Note that and where Next, dividing Eq. ( 3 45 ) by Eq. ( 3 46 ) and substituting and we obtain: ( 3 47 ) Similar to the single platform scenario, the tangency condition still holds, i.e., the cost minimizing behavior of each c Given Eq. ( 3 47 ) one can verify Define the price elas ticity of as Eq. ( 3 45 ) can be spelled out as: ( 3 48 ) PAGE 45 45 It is expected that and for the platform(s) to charge positive trip fares. The pricing formula in Eq. ( 3 48 ) is of similar form to that in Eq. ( 3 24 ) apart from being multiplied by a scaling factor It is strictly larger than one given and The ratio of price levels under duopoly and monopoly is given as follows: ( 3 49 ) where the platform production cost is assumed linear, i.e. For a symmetric NE, it is expected Similar to Eqs. ( 3 39 ) ( 3 41 ) the average waiting and searching times increase as the decrease of the platform specific demand given the matching function has increasing returns to scale. It follows that ( ), indicating more matchi ng frictions. Accordingly, the first component in the RHS of Eqn. ( 3 49 ) is no less than one while the second component is related to the price elasticities o f demand. Its value is unclear without fully specifying the demand function. If the price level under NE is strictly larger than that under the monopoly. Otherwise, the ratio of over remains indeterminate. Generally, one needs to explore the change of the price elasticity of demand and that of the matching friction. The conventional wisdom that competition lowers the price level does not stand if the increase of the former is dominated by the latter. 3.3.2 Second Best for Dual Ride Sourcing Platforms This section directly investigates the second best outcome with the dual platforms. The first best solution can be obtained by setting the Lagrangian multiplier associated with the reservation profit constraint to zero in the optimality conditions of the second best problem. PAGE 46 46 Define regularity conditions ( Sheffi, 1985 ) the following properties hold by construction: ( 3 50 ) To simplify the derivation, we treat as the decision variables. Mathematically, The maximization program is formally written as: (P6) ( 3 51 ) s.t. ( 3 52 ) Define its Lagrangian function as follows: ( 3 53 ) By setting it is immediate to show: ( 3 54 ) which again follows the tangency condition. Next, setting yields: ( 3 55 ) ( 3 56 ) where Substituting in Eq. ( 3 55 ) we then focus on the pricing formula for platform 1: ( 3 57 ) PAGE 47 47 where For more insight, further assume a symmetric solution where and Then is reduced to which is str ictly larger than one given Assumption 1. Define the price elasticity of as Eq. ( 3 57 ) can be rewritten as: ( 3 58 ) The pricing formula for platform 2 can be obtained by symmetry. The formula structure is the same as that in the single platform case, except being adjusted by the factor which has its roots in platform collusion (a case where the two platforms cooperatively maximize the total profit). The fact that it is greater than one is by the nature of the pricing for substitute goods 3 The firs t best pricing formula for the dual platforms can be obtained by setting Similarly, the first best solution is not sustainable if the matching function is assumed to exhibit increasing returns to scale and the cost function of the platform has economies of scale. The second best pricing formula can be thought of the convex combination of the first best and the collusion formulas. The solution sets obtained in the above scenarios can also be displayed over a contract curve that connects the social optimal (Point S) and NE solution (Point N) in Figure 3 3 (Point C 3 Assume a market without friction or externality. Define to be demand functions for two competing firms. Further assume ; the total cost for production is linear: The collusion pricing formula under the symmetric solution is: PAGE 48 48 represents the solution that two platforms collude to maximize the joint profit. It corresponds to the monopoly solution). The remaining points on the contract curve represent a continuum of the s econd best solutions characterized by different regulated profit levels. Define the average joint profit for platform i one can verify for the solutions on the contract curve. is strictly negative when the matching function shows increasing returns to scale. 3.3.3 Welfare Changes between Single and Dual Platforms In this section, we attempt to investigate whether or not competition yields a more efficient market outcome and whether a regulated duopoly market is more efficient than a regulated monopol y market. To address the former, we compare the social welfare under the monopoly solution and duopoly solution. For the latter, we focus on the welfare change under the regulated second best solutions. To proceed, we specify the demand function by assumin g that the error terms of the utility functions are identically and independently Gumbel distributed for both the single (Eqs. ( 3 5 ) ( 3 6 ) ) and dual platforms 4 (Eqs. ( 3 42 ) ( 3 43 ) ). Therefore, the demand functions for the single platform and the other transportation modes are: ( 3 59 ) where For consistency, we assume the total base demand remains the same for both scenarios and the demand for the dual platforms are given as: 4 The error terms are likely dependent in the duopoly market. This assumption is made for t he ease of welfare comparison. PAGE 49 49 ( 3 60 ) where In addition to specifying an increasing return to scale matching function as highlighted in Assumption 2, we limit oursel ves to the following aspects to further facilitate the comparison: The solutions for the dual ride sourcing platforms are symmetric. The related variables are marked by the subscript d e.g. Total cost incurred by the platform is linear: 3.3.3 .1 An alytical comparison Utilizing the assumption on symmetric solution and linear platform cost, the welfare change between the single and dual platforms for a given market scenario is as follows: ( 3 61 ) ( Train, 2009 ) while the between the platform and its drivers. For the duopoly and monopoly solutions, it is anticipated the decrease from the joint profit. Consider the extreme case that approaches to from the right. Then the change of pproximately zero. Eq. ( 3 61 ) is reduced to: ( 3 62 ) PAGE 50 50 One can verify when That is, competition is welfare improving if the increased searching friction is bounded above by a constant. The same condition holds when evaluating the welfare change at the second best solutions (e.g. at zero profit level), the explicit relationship between and (or equally and ) appears hard to obtain and neither has a close d form. We therefore resort to a numerical experiment to quantify the welfare changes. 3.3.3.2 Numerical analysis Define to be the experimental parameter set. are directly included in the matching function while and influence the base demand and re alized demand splits. Since our focus is on the matching technology, the other parameters (e.g., ) are held constant. In fact, it can be shown for all the solutions on the contract curve (The derivation for the sign of is given in Appendix D for illus tration). Marginal increase of each of these parameters reduces the matching friction and thus indicates more efficient matching technology. It is assumed ($/trip), ($/hr), ($/hr), (1/$), ($/hr), (hr), ($/trip). For demonstration, we investigate two base demand levels: and (trip/hr) and the outside transportation costs: and 15 ($/trip). For each combination of and a continuum of solutions are examined over a range of ( ). PAGE 51 51 We explored three levels of returns to scale with respectively. For symmetry, are set equal. The welfare changes between duopoly and monopoly solutions are given in Figure 3 4 For a meaningful comparison, we assume the joint platform demand will account for 10% 90% of the total passenger demand in the equilibrium states This in turn indicates that the difference between the generalized cost of using the ride sourcing service and the outside options cannot be very large. Accordingly, two dot curves are introduced to bound the solution region. Each solid red curve represe nts the computed welfare change over a feasible range of A for a given elasticity pair ( ). The little bar at the left end of the curve means the location where no solution is available for either problem P1 or P5 if we further reduce A The welfare diffe rence curve slopes upwards, indicating increases as the increase of For a given the increase of elasticities ( ) generally yields larger In Figure 3 4 (a), the changes of the total social welfare are positive but for the left end in the case of When the base demand increases from 50,000 to 100,000 (trip/hr), the total pattern moves upwards as seen in Figure 3 4 (b). All the stable solutions in the region are characterized by indicating competition between the platforms is welfare improving. However, when increases from 14 to 19 ($/trip) as shown in Figure 3 4 (c), some left portion of all curves fall in the solution region where It should be pointed out that the welfare difference curve is plotted over the interval (0.12, 0.3) of whic h is different from Figure 3 4 (a) and (b). In the cases shown in Figure 3 4 (c), there are more matching frictions. Figure 3 5 shows the average searching times under duopoly and monopoly solutions when and (trip/hr). It can b e observed that the average searching times for drivers are PAGE 52 52 larger in duopoly than those in monopoly due to the decrease of platform specific demands. For the cases presented in Figure 3 4 (c) where competition is not efficient, the average search time is much higher, approximately 6 12 minutes. The welfare comparison between the regulated dual and single platforms is given in Figure 3 6 The general tendency is similar to that in Figure 3 4 In many cases, competition increases social welfare. However in F igure 3 6 (c), all the meaningful solutions are characterized by In those situations, a regulated monopoly platform will be superior to the regulated duopoly platforms in term of efficiency. Although the numerical example is made from the arbitrary spec ification of function forms and parameter values, its interpretation is straightforward. Competition may not improve the social welfare when the matching technology is less efficient. Given the assumption of the increasing returns to scale of the matching function, its efficiency is not all about the advanced algorithms used by the ride sourcing platform but also the size of the market, i.e., the number of users. To derive effective policies, the regulatory agency may need to obtain a good estimate on custo the regulatory agency may rather encourage the merger of the ride sourcing companies and then regulate them as a monopolist. 3.4 Summary We analyze the ride sourcing service using an aggregate model with the matchings between customers and drivers captured by an exogenous Cobb Dougla s matching function. We examine different market scenarios, solution properties and general economic outcomes of a hypothetical monopoly ride sourcing market It is found that without regulatory intervention the monopoly ride sourcing platform would maximize the joint profits with its drivers. The first best solution is not sustainable when the matching function is increasing returns to sca le and the cost PAGE 53 53 incurred by the platform exhibits economies of scale Therefore, we further analyze the second best scenario with varying reservation profit levels for the ride sourcing platform. In terms of market frictions, all the examined scenarios are characterized by a proportional relationship between the average waiting and searching times that implies the cost minimization of the matching production. In view of the ma rket distortion, we demonstrate the feasibility of regulating two variables to ach ieve the second best. With the assumption of homogeneous value of time of customers, we further show that regulating only the commission should guarantee the second best. To address the effects from p latform competition, we extend our analysis by consideri ng a duopoly setting. I t is found that competition may not necessarily lower down the price levels when the increase of the matching friction overrides that of price elasticity of demand after the introduction of the competing p latform. We further investig ate the effects of competition on the welfare change when the matching function is increasing returns to scale. Based on the s ensitivity analysis, we observe that competition can reduce social welfare when the matching technology is less efficient, and the increased matching friction for each platform dominates the surplus generated by having one additional option. In this case, the regulator may rather encourage the merger of the platforms and regulate them directly as a monopolist PAGE 54 54 Figure 3 1 Cost M inimization of the Matching Technology PAGE 55 55 F igure 3 2 Contract Solution Set for Single Ride Sourcing Platform. PAGE 56 56 Figure 3 3 Contract Solution Set for Dual Ride Sourcing Platforms. PAGE 57 57 A B C Figure 3 4 Change in Social Welfare between Duopo ly and Monopoly Solutions. A) (unit: $/ trip, trip/hr) B) (unit: $/ trip, trip/hr), C) (unit: $/ trip, trip/hr) PAGE 58 58 Figure 3 5. Average Searching Time ( and ). PAGE 59 59 A B C Figure 3 6 latforms at Second Best Solutions (at Zero Profit Level). A) (unit: $/trip, trip/hr) B) (unit: $/trip, trip/hr), C) (unit: $/ trip, trip/hr) PAGE 60 60 CHAPTER 4 DYNAMIC PRICING AND LABOR SUPPLY IN ON DEMAND RIDE SOURCING MARKET S I n this chapter we relax from the static set ting in Chapter 3 and aim to analy ze the temporal effects of dynamic pricing. Note that ride sourcing companies provide flexibility for drivers to choose their work schedules. Some may work full time like professional taxi drivers while others only provide service for limited hours (e.g., on their way home from work). Such surge. On one hand, drive r s may adjust their work schedules to cover m ore profitable periods; on the other hand, temporal variation i n the wage rate induced by dynamic pricing may further ( Chen and Sheldon, 2016 ) Therefore, it is intriguing to investigate the effect of dynamic work hour choices in a dynamic context. In the lit erature of labor economics, competing theories or hypotheses exist in understandin g how a driver determines her shift length. The neoclassical theory expects drivers to work longer when their wage rate is higher, while the income targeting theory speculate s that drivers have target levels after which they are more likely to stop ( Camerer et al., 1997 ; Farber, 2015 ) supply decisions, even though empirical analyses have been carried out. Furthermore, the analyses mainly focus on inputs such as wage rate and work hours while struc tural information such as shift starting and ending times is largely overlooked. In Section 4.1 we conduct a brief review of the literature in the labor supply. Subsequently in Section 4.2 4.3, we construct a time expanded network scheduling under both labor supply assumptions PAGE 61 61 are all endogenously determined Section 4.4 illustrates the equ ilibrium outcomes through numerical examples. In Section 4.5, we investigate the impact of dynamic pricing using a bi level programing framework. maximization while the lower level captures the equilibrium work hour choices of the drivers. A simple regulation scheme is presented when market power is a co ncern. Further insights on dynamic pricing and regulation outcomes are discussed via numerical experiments. Section 4.6 summarizes the chapter. 4 .1 Literature on Labor Supply response to the variations of hourly wage rate. The income targeting theory was initially proposed by Camerer et al. (1997) to explain the phenomenon of negative labor supply elasticity found in a New York taxi dataset (i.e., an increased wage rate leads to shorter work hours). It can be seen as a special case of reference dependent preferences ( ) where a ginal utility of income at the reference income level. Drivers are loss averse in the sense that they suffer more when the target level is not achieved while the motivation of continuing working beyond the target vanishes. A dual reference approach was pro posed by Crawford and M eng (2011) where drivers were assumed to have targets for both daily income and working hours. Their model confirms the role of reference depen dence in labor supply and is flexible in accounting for the case when income effect is not significant. Compara tively, the neoclassical theory suggests drivers will work longer given a higher wag e rate, indicating a positive hourly wage rate elasticity of working hours ( Farb er, 2005 ; Farber, 2015 ) As often argued by the proponents of the neoclassical theory, the existence of a gainst hourly wage rate or the improper use of instrument for wage rate ( Farber, 2005 ; Farber, PAGE 62 62 2015 ) Most rec ently, Farber (2015) and Chen and Sheldon (2016) both reported positive labor supply elasticity using a large scale New York taxi dataset and a high resolution Uber dataset, respectively. Considering drivers may adjust their target income gradu ally based on their historical experience, Farber (2015) further argued that the income targeting hypothesis may only make sense for temporary unanticipated wage variations. ( Hall and Krueger, 2015 ; Chen and Sheldon, 2016 ) the research o f labor supply in the ride sourcing market is limited. As there is no clear evidence yet which theory would outperform, we will propose formulations based on both theories. 4 .2 Basic Modeling Considerations We report the basic modeling considerations in th is section. A time expanded network is first constructed in a similar spirit as that of Yang et al. (2005a) to delineate the work schedules of ride sourcing drivers. Then we specify the demand form for customers, as well as the revenue and cost structures for drivers. 4.2 .1 Time Expanded Network A time expanded network is presented in Figure 4 1 where and represent the set of nodes and links, respectively. The nodes represent time points for an aggregate ride sourcing market and no spatial structure of the market is explicitly specified. All drivers will start from t he same hypothetic origin and end at the hypothetic destination For each driver, a path between the O D pair defines a work schedule, which provides the complete in traditional taxi studies (e.g., Yang et al., 2005), the work schedule here also contains the break periods between subsequent working sessions. PAGE 63 63 More specifically, we divide a whole day into n time periods of an equal length of T and then denote the s ubset of work nodes T 2 T nT and the subset of the associated auxiliary nodes. Without loss of generality, we set T to 1 hour and n to 24 as shown in Figure 4 1. So, the set of nodes in corresponds to clock hours. A driver at a work node can either traverse the subsequent work link in search of customers or temporarily take a rest by heading to the corresponding auxiliary node. When at an auxiliary node, the driver can travel to the adjacent work, auxiliary or destination node, which respectively cor responds to going back to work, continuing in the resting mode and ending the work completely. The set of links can be divided into five mutually exclusive and collectively inclusive subsets. denotes the set of departure, work, transition, rest and e nd links. Note that passenger demand only arises on work links that belong to at which drivers can be matched i.e., when a link As an example, the path O 1 2 2 3 3 4 5 6 7 7 D defines a work schedule in which a driver starts working at 0:00 a.m., takes a break for one hour between 2:00 and 3:00 a.m., goes back to work until 7:00 a.m. and then stops. 4.2 .2 Demand and Revenue The ride sourcing market is not merely cleared by price, and each customer has to wait for some time until being served. Analytically, we assume that the passenger demand is a decreasing function of both a verage trip fare, and average waiting time, : ( 4 1 ) PAGE 64 64 where is the flag drop fee; is the time based charge equal to a unit time charge multiplied by the average trip ti me ; and is the surge multiplier that only applies to the time based portion. We have and by assumption. ride sourcing platform, i.e., the algorithm that matches cu stomers with drivers. Several recen t studies adopted a Cobb Dougla s function to delineate the matching process from which the waiting time can be derived ( He and Shen, 2 015 ; Wang et al., 2016 ) For simplicity, we assume that the ri de sourcing platform matches a waiting customer with the closest vacant vehicle. Under this assumption, the average waiting time can be approx imated as follows ( Arnott, 1996 ) : ( 4 2 ) where is a scaling pa rameter that adjusts the Euclidean distance to the Manhattan distance; is the average speed of the vehicles; is the vacant vehicle hours and is the area of the ride sourcing market. The derivation of Eq. ( 4 2 ) does not account for the case where a vacant vehicle turns out to be the closest one for multiple customers and thus the formula Considering that a ride sourcing vehicle is either occupied or vacant when traversing a work link, the following relationship holds for an analysis period of one hour : ( 4 3 ) PAGE 65 65 where is the occupied vehicle hours and is the total vehicle hours. denote as the average searching (w aiting) time until finding a customer. When the average service duration ( ) is far less than T we have: ( 4 4 ) The nonlinear equation system of Eqs. ( 4 1 ) ( 4 4 ) behaves properly. More specifically, for the given surge multipliers, the base trip fare as well as the total fleet size, the existence and uniqueness of the market equilibrium can be proven by means of the fixed point theorem as shown in Yang et al. (2005a) The ride sour cing industry features a revenue sharing structure. For each completed trip, the platform charges a fixed proportion (e.g., 20% 25%) from the total fare as the commission while the driver keeps the remaining The average revenue for a driver in a worki ng hour is then written as: ( 4 5 ) It is interesting to note: ( 4 6 ) ( 4 7 ) where is the elasticity of demand with respect to the vehicle hours and is a function of In addition, and PAGE 66 66 Considering Eqs. ( 4 6 ) ( 4 7 ) we have when It is possible when In the latter case, more vehicle hours would increase the average hourly revenue for the drivers and simultaneously reduce average waiting time of customers. However, such a Pareto improving scenario may only occur with a n unrealistically small vehicle size where the marginal reduction on the average waiting time stimulates significantly more demand ( Yang et al., 2005a ; Yang and Yang, 2011 ) For the analysis hereinafter, we limit our discussion to the following: Assumption 1 Given the surge multipliers, the base trip fare and the total fleet size, the average hourly revenue of ride s ourcing drivers is strictly decreasing in vehicle hours, i.e., Without further referencing Eqs. ( 4 1 ) ( 4 5 ) we denote the average revenue as a function of the total vehicle hours, i.e., ( 4 8 ) 4.2 .3 Cost for Drivers A substantial level of heterogeneity exists in the work scheduling of ride sourcing drivers. We def ine the set of drivers with elements denoted by m We assume drivers differ in their preferred start period and work duration. As previously mentioned, a path in the proposed network defines a work schedule. Following a path, a d river incurs two types of variable costs. The first cost is path dependent, denoted as measuring the disutility from cumulative work hours, which is assumed to increase more than linearly with the cumulative PAGE 67 67 hours. The other is the link specific cost that captures the cost a driver experiences when traversing a specific link. It represents: driving (e.g., from home ) to a normal business area ( ); the operating cost during one work period ( ) or driving back home ( ); a sufficiently small modeling artifact to avoid unrealistic loops between wo rk and rest links ( ); ). We define as the set of all possible simple paths betw een the O D pair. The total cost incurred by a dri ver of class on path p is written as follows: ( 4 9 ) where is the link path incidence; is the cost associated with cumulative working hours ; and captures the level of aversion to long work dura tions and is assumed to be larger than 1. For a simple quantification of available vehicles when the average service duration is rather small (much less than a modelling period T ), we assume: Assumption 2 All ride sourcing vehicles that provide service du ring a t ime period will be available in the following period. The relaxation of Assumpti on 2 is discussed in Section 4. 3 .3. We denote the number of drivers of type m choosing path p Then the relationship between the link and pat h vehicle flow follows: ( 4 10 ) PAGE 68 68 4.3 Equilibrium Models with Endogenous L abor Supply We provide formulations and algorithms for the two labor supply assumptions. An enhanced treatment for long trip duration is also discussed. 4.3 .1 Neoclassical Equilibrium M odel Given the surge multipliers and average trip duration, we consider a short run market equilibrium with a fixed total fleet size. It is reflected by a multiclass network flow equilibrium on the time expanded network, formally defined as follows: Definition 1 At equilibrium, for each driver class all paths that carry positive vehicle flows yield equal profit (total revenue minus all cost components) which is no less than that of any unused path. The equilibrium defined above can be viewed as a consequence of the neoclassical labor supply theo ry ( Farber, 2015 ) However, we incorporate the interactions among different drivers in competing for trip opportunities. Note this definition, together with our s pecification of hourly demand, implies that drivers and customers make decisions at different time scales, a modeling feature also highlighted by Banerjee et al. (2015) Drivers typically consider profits at a daily level while customers are sensitive to the price and waiting time only when they request rides. We are now ready to present the following mathematica l program whose solution describes the equilibrium of the ride sourcing market: (ME N) ( 4 11 ) s.t. ( 4 12 ) ( 4 13 ) PAGE 69 69 where is defined in Eq. ( 4 10 ) Given Assumption 1, it is straightforward to verify that ME N is convex in with linear constraints. However, multiple solutions may exist as strict convexity does not hold. The optimality conditions of ME N yield the following system of complementarity equations: ( 4 14 ) ( 4 15 ) where rep resents the average profit of the chosen path. Let be the vector of the Lagrangian multipliers associated with constraints ( 4 12 ) It is nonnegative and can be interpreted as the equilibrium profit. The complementary slackne ss condition in constraint ( 4 14 ) corresponds to our definition of market equilibrium. We make the following assumption to limit our discussion to a reasonable profit domain. Assumption 3 The equilibrium profit is strictl y positive. Assumption 3 avoids the unrealistic situation where the maximum possible profit that a driver could earn is zero. It implies that constraints ( 4 12 ) are always binding. Besides, t he gradients of constraints ( 4 12 ) and t hose binding in constraints ( 4 13 ) are linearly independent. This regularity condition guarantees the uniqueness of ( Bertsekas, 1999 ) The formulation of ME N bears some similarity with the one in Yang et al. (2005). The key difference is that our formulation considers that i ndividual drivers drive their own cars to provide for of ME N has two distinctive features. Namely, is not explicit in and path costs are not link additive. The former makes it cumbersome to evaluate the objective function while the latter precludes the possibility of using efficient link based solution algorithms for network PAGE 70 70 equilibrium flow problems ( Patriksson, 2015 ) Instead, various path based solution algorithms can be applied to solv e ME N, e.g., the one based on a gap function proposed by Lo and Chen (2000) which direct ly aims to solve the nonlinear complementarity problem (NCP) characterized by Eqs. ( 4 14 ) ( 4 15 ) The gap function is defined as: ( 4 16 ) where Note that and =0 if and only if Eqs ( 4 14 ) ( 4 15 ) hold ( Lo and Chen, 2000 ) The solution to ME N can be obtained by solving an unconstrained minimization program GAP N whose optimal value is zero, i.e., and ( Lo and Chen, 2000 ) Recognizing eventually only a small fraction of paths carry positive vehicle flow, we apply a column gene ration scheme to gradually expand a pre defined small path set. To ensure the validity of Assumption 1, the initial path set should be chosen properly to cover every work link The column generation approach requires iteratively solving a shortest path fin ding problem with a non additive path cost conditional on the current optimal link/path flows for each driver class: (SP) ( 4 17 ) s.t. ( 4 18 ) ( 4 19 ) PAGE 71 71 where is the node link incidence and is a column vector with element 1 and 1 corresponding to the origin and destination nodes and 0 for the other intermediate nodes. At each iteration, at most one path (with profit level larger than the current equilibrium profit) is added to the path set. We denote this growing path set as The detailed steps of the solution procedure are summarized as follows: Step 0: Select several paths that cover the work links and construct the path set Step 1: Given solve GAP N and obtain the optimal solution and Step 2: For each solve SP and obtain the optimal path with the minimum cost If add into If for all stop; otherwise, go to Step 1. 4.3 .2 Income Targeting Equilibrium M odel The income targeting theory assumes that drivers work until reaching a daily target, beyond which the gain from working is more likely to be offset by the disutility of working long hours. Specifically, drivers of class are assumed to choose their work schedules to maximize their utility, which is defined as follows ( Farber, 2015 ) : ( 4 20 ) where is the target income level; represents the total revenue of a chosen work schedule; controls the degree of loss aversion and is assumed to vary between The above specification can be viewed as th e consumption part of the utility augmented with the gain loss component ( Farber, 2015 ) For the work hour c hoice problem on the proposed network, a constant is added to guarantee Drivers gain more from PAGE 72 72 making money when their target level is not achieved and gain relatively less if they continue working aft revenue ( Farber, 2015 ) Note that that Eq. ( 4 21 ) can be further reduced to: ( 4 21 ) where To shed light on the income target assumption, we deviate to parameterize which makes explici t; and assume that is a differentiable function of Note ( Farber, 2015 ) The plot in Figure 4 2 assumes is concave in and treats it as a continuous variable for demonstration purposes. For sufficiently low or high average hourly wages (e.g., and respectively), the motivation for utility maximization yields the neoclassical fashion of labor supply. In the former, drivers stop working earlier because of bad t rip opportunities, while in the latter they work longer because the high wage rate overrides the disutility from cumulative work hours. However, for the intermediate wage levels (e.g., and ), the optimal decision is to stop right after hitting the target income level. This is because the wage rate is high enough to motivate drivers to work longer when but too low to offset the disutility from the increas ed work hours when For example, an increase of wage rate from to reduces the work hours from to indicating a negative elasticity of labor supply. Although in our model is implicit with the resolution equal to one modeling period, we expect the general intuition still holds. W e as sume is exogenously given and remains fixed. However, the target income PAGE 73 73 level can be endogenized as the hour choices is considered ( Crawford and Meng, 2011 ; Farber, 2015 ) See Xu et al. (2011) for a similar treatment to develop a prospect based user equilibrium model with endogenous reference points. We define the set of feasible path flow as The equilibrium condition can be captured by the following NCP: (ME I) ( 4 22 ) ( 4 23 ) where and When the equilibrium condition degrades to that of the neoclassical case. It is well known that the solution to the above NCP, i.e., in the vector form, can be chara cterized by the following variational inequality ( Na gurney, 2013 ) : ( 4 24 ) where Note that is continuous in and is further continuous in The set is convex and compact. Therefore, a solution always exists. However, path flows may not be uniq ue, as is not strictly monotone in The procedure for solving the income targeting model is similar to the one for solving the neoclassical model, with some modifications to the sub problems of flow equ ilibration and shortest path finding. The absolute operator in Eq. ( 4 21 ) poses a challenge for classical derivati ve based algorithms in minimizing the gap function. We thus provide a reformulation PAGE 74 74 based on the fact that is equivalent to the sum of two non negative auxiliary variables, i.e., where , The complementary feature of this reformulation fits well in the adopted gap function. If we denote the vector of and as and then the program for the income targeting model can be reformulated: (GAP I) ( 4 25 ) where is by construction. An optimal equilibrium solution is characterized by For the shortest path finding, we have: (S P I) ( 4 26 ) s.t. Eqs. ( 4 18 ) ( 4 19 ) ( 4 27 ) 4.3 .3 Enhanced Equilibrium Models with Long Trip Duration The previous two formulations rely on Assumption 2 and are valid if Th e assumption can be violated in hours of heavy congestion. When is close to or more than T a vehicle will not be available in the beginning of the next time period after picking up a customer. The key challenge of modelling pote ntial long trip duration is to properly trace the availability of a vehicle, which is jointly determined by how long the vehicle is PAGE 75 75 occupied in the previous period(s) and its status in the current period (e.g., to continue working or not). The reason for t he former is obvious since a vehicle previously occupied with long trip movement of the vehicle consistent with its path: a driver can never take a tr ip longer than one period if she is about to exit the market. The detailed formulations and dis cussions are given in Appendix E for interested readers. 4.4 Numerical Examples We first specify the functional forms and parameter values necessary for the discussion. Then, w e numerically investigate the equilibrium solutions of the proposed formulations and demonstrate their implication on the labor supply elasticity. 4.4 .1 Set up We set T equal to one hour and assume an exponential (hourly) demand function for each work link : ( 4 28 ) where is the base demand; is the demand sensitivity parameter ; and are the value of waiting time and in vehicle traveling time, respectively. The specification above assumes customers are homogen e ous in the value of time for a given time period, and the temporal variations are captured by Daily distribution of the base demand is depicted in Figure 4 3, with two peak periods at 7:00 9:00 and 17:00 19:00. Accordingly, we set and at $33/hr and $16/hr for peak hours while $20/hr and $10/hr for the remaining hours. We let be 0.03 and 0.05 for peak and off peak hours, respectively. These specifications reflect the assumption that customers value time more during peak hours and become less sensitive to price ( Small, 2012 ) PAGE 76 76 We consider four driver classes, each of which consists of 2,000 drivers. Specifically, we denote drivers who prefer to start late and work long/short hours as class and those preferring t o start early and work long/short hours as class Let denote the index of the starting node of link For a driver of that starts on link we set when and 10 otherwise. For a driver of on link we set when and 10 ot herwise. For the work, transition, rest and end links of all driver types, we have ; and respectively. To for and for All costs are in the unit of $/hour. The income target levels are set at $300, $200, $300 and $200 for m = 1 to 4. V alues for all param eters are summarized in Table F 1 of Appendix F for the convenience of the readers. 4.4 .2 Equilibrium Solutions We present the equilibrium solutions to ME N and ME I in Table 4 1 and Table 4 2, respectively. For ME N, the equilibrium profits for all chosen paths (work schedules) are $107.4, $65.3, $107.4, $74.3 for class 1 to 4, respectively, with average work hours equal to 8.52, 4.83, 7.58 and 4.95 respectively. It can be observed that although drivers of m = 1 tend to start later than those of m = 3, the distinction among the starting hour is not significant for those preferring shorter work hours ( m = 2, 4). Another set of sufficiently differentiated link costs may be used to better demonstrate such heterogeneity. Large variations exist for the rest (bre ak) hours. For example, some drivers of class 2 do not even take a rest while others may have a break time as long as 7 hours. Our modeling framework clearly shows the capability of capturing the flexibility of ride PAGE 77 77 For M E I, all paths that finally carry positive flow are characterized by the same utility levels of $281.3, $350.9, $281.3 and $359.9 for classes 1 4. Note that the presented values are all added by a constant The interpretation of work/rest hours as well as starting time choice is similar to that for ME N. Due to the existence of the income target, one can easily verify that for each driver class the profit of used paths may not necessarily be equal. Moreover, several chosen paths f or m = 1 and 3 have revenue around $293, which is very close to the specified target. Given the complex structure of the proposed network, however, cases where drivers stop exactly at their target levels are rarely seen. 4.4 .3 Labor Supply Elasticity To in vestigate the sign of the labor supply elasticity under both assumptions, we present the relationship between average work hours and hourly revenue in Figure 4 4. For demonstration purposes, we simply increase the surge multipliers for all work hours from 1 to 2 every 0.25 unit. Although a verage hourly rev enue increases with SMs different patterns in average work hours are uncovered. Higher hourly revenue corresponds to longer work hours in the neoclassical assumption while in the income targeting theory, this relationship only holds when SMs are larger than 1.25. In this case, the hourly revenue is high enough to offset the disutility from continuously working even after the income target has been achieved. The income target fashion of labor supply occurs when SMs increase from 1 to 1.25: an increase in hourly revenue comes with a reduction in average work hours. Further examination of the output when SMs are equal to 1.25 reveals that the average revenue is $314.6, $191.9, $305.1 and $193.1 for m = 1 to 4, respectively, which is close to the corresponding target, i.e., $300, $200, $300 and $200. PAGE 78 78 4.5 Dynamic Pricing and Its Regulation introduce a bi level prog ramming framework to study dynamic pricing. A simple regulation scheme to enhance market efficiency is also presented. To facilitate the presentation, we adopt the neoclassical assumption to characterize the labor supply in this section. 4.5 .1 Modeling Framework We now mod el the effect of dynamic pricing using bi level programming where the upper level problem represents the behavior of the platform while the lower level problem captures the response of the ride sourcing system to the decision made by the p latform. In the s hort run, dynamic pricing has a limited impact on attracting additional drivers to register to the platform. We thus assume that the total fleet size remains fixed. We further assume that the base fare structure is given and surge multipliers (SMs) become the only control variable for the platform, with the objective to maximize its daily revenue, i.e., The operation cost of the platform is not considered for simplicity. In general, the platform solves the following program: (BI) ( 4 29 ) s.t. ( 4 30 ) where is the control objective and is the gap function previously di scussed, with being the vector of the equilibrium profit. PAGE 79 79 For a given path set, BI is difficult to solve with Eq. ( 4 30 ) being a co nstraint. A relaxation of it to an allowable error, i.e., will make the problem more solvable by commercial nonlinear solvers. A good solution can be obtained by iteratively reducing If paths cannot be enumerated, it remains feasible to apply a column generation scheme to determine the effective path set where the shortest path finding problems are solved based on the currently optimized SMs. It is worth mentioning that this type of bi level programming problem has been extensively studied. Various locally convergent algorithms proposed in the literature may also apply here ( Marcotte and Zhu, 1996 ; Meng et al., 2001 ) 4.5 .2 Commission Cap Regulation If empirical evidence confirms significant mark urther intervention from the regulatory agency is needed. Below we present a simple but insightful regulation scheme that suggested in Chapter 3 : capping the amount of commission charged by the platform. More specifically, under such a scheme, the monopoly ride sourcing platform charges at most the capped amount of commission from each transaction. For revenue maximization, it will maximize the total transaction (i.e., realized demand) which is positively related to surplus Whenever surge pricin g is in place, the surged part completely goes to the drivers. Different choices of the commission cap essentially provide varying profit margins for the ride sourcing company. In a static setting with strict homogeneity assumptions, we showed that capping commission alone could achieve the second best. Given the heterogeneity among trip distance and duration in practice, the proposed regulation may be implemented in the form of distanced based charge, or time based charge, or the combination of both. PAGE 80 80 The market outcomes under the proposed regulation can be captured by the following program: (BI R) ( 4 31 ) s.t. ( 4 32 ) ( 4 33 ) ( 4 34 ) where and are respectively the vectors commission charged from each trip and the fleet size is the commission cap set by the regulatory agency. The notations for the other variables are the same as previously given. BI R is similar to BI ex cept that 1) the direct controls under the platform are surge multipliers and commission; 2) entry is necessary since otherwise the platform may set an extre mely low trip fare (to attract more customers) while leaving drivers in deficit. entry decision is captured. Note that any, to exploit either side wh en faced with the proposed regulation. 4.5 .3 Numerical Experiments The tolerance rate is initia lly set at 1 for constraint ( 4 30 ) a nd can be reduced to 0.01. Values for the other parameters r emain the same as in Section 4.4 We f irst explore the effect of dynamic pricing. The corresponding plots on SMs, average waiting times and searching times are given in Figure 4 5. In Figure 4 5 ( a), a revenue maximizing platform surges in peak periods when customers value time more and become less sensitive to price. Figure 4 5 (b) illustrates PAGE 81 81 that changes in average waiting and searching times are gene rally opposite each other. Dynamic pricing su vacant vehicles, customers on average enjoy lower waiting times. In return, it becomes more difficult for a driver to be matched with a customer, and so their average searchi ng time increases. To answer t he question whether or not dynamic pricing is beneficial, we construct a static pricing counterpart where a uniform SM is applied across all periods for revenue maximizing. The market outcomes under such (optimal) static scena rio (with the SM equal to 2.02) are used for comparison. The difference is computed as the metric of interest under dynamic pricing of the passengers, which is c alculated under a hypothetical market demand curve where ( Cairns and Liston Heyes, 1996 ; Yang et al., 2002 ) 5 The total social surplus is the sum of In Figure 4 between the platform and the drivers are plotted 6 With dynamic pricing, the platform tends to charge less during off peak hours b ut more during peak hours. Figure 4 5 shows that the joint are better off during the off peak hours, while during the peak hours, they have to pay significantly mo re and are thus worse off. 5 When the income targeting model is adopted at the lower level, it remains valid to use the total revenue for the 6 The calculation o PAGE 82 82 It is interesting to note though dynamic pricing has the potential to simultaneously optimal static pricing level as in off peaks). How ever, the net change of the total social surplus aggregated in a day can be positive or negative, depending on the demand pattern and other factors. Finally, the results are similar when we employ the income targeting assumption in the lower level problem Note the income targeting assumption relies on exogenous parameters such as the value of which affects the equilibrium flow distribution and eventually the market dynamics. Therefore, a head to head comparison of these two ass umptions on market dynamics is not meaningful. We now investigate the performance of the proposed regulation scheme when the market power of the ride sourcing platform is a concern. We assume the reservation profit levels are $120, $80, $120 and $80 for dr iver types 1 4. The operation cost of the platform, and the sunk costs of the platform and the drivers are not considered in quantifying the corresponding surplus. The left vertical axis of Figure 4 the commission cap increases. The right vertical axis of Figure 4 7 (a) shows the ratio of the social surplus under different regulated commissions to that of a (quasi) second best case where the keeps decreasing as the commission cap increases. The commission cap corresponding to the maximum social surplus is very close to costs are not specified in our analysis. Market efficiency is enhanced when compared to the PAGE 83 83 unregulated monopoly (as the commission cap goes to infinity), which only accounts for approximately 70% of the second best social surplus. Figure 4 rough estimate of market power. With the increase of the commission cap, the platform obtains a larger share from the social surplu s. In summary, the proper choice of commission cap essentially provides a healthy profit margin for the development of the ride sourcing company while limiting its market power. 4.6 Summary study has proposed formulations under different behavioral assumptions of labor supply to investigate the effects of dynamic pricing in the ride sourcing industry. A time expanded network is first constructed to represent the work scheduling of ride sourci ng drivers. Based on such a network representation, formulations and algorithms are presented to describe the equilibrium of the ride sourcing market for both the neoclassical and income targeting assumptions of labor supply. Through numerical experiments, we demonstrate that our models generate outcomes consistent with the definitions of market equilibrium, and are able to uncover the work schedule for multiple driver classes with different break durations, start and end times. We further show that a highe r average revenue rate corresponds to longer work hours in the neoclassical assumption but may lead to a reduction in work hours in the income targeting one. We next investigate the impact of dynamic pricing based on a bi level programing framework. With t he lower level capturing the equilibrium work schedule choices, the upper level is tailored to represent the scenario where a platform aims to maximize daily revenue. Patterns of average waiting and searching times are generally opposite each other. Our nu merical results indicate that the platform and drivers enjoy higher revenue while customers may be at a PAGE 84 84 loss during highly surged periods. We investigate a commission cap regulation where the platform can charge at most a fixed amount from each transaction The proposed regulation demonstrates the potential to increase market efficiency and limits the market p ower of the monopoly platform. In Chapter 5 w e will evaluate the effect of dynamic pricing and the proposed commission cap regulation using an empiri cal dataset. PAGE 85 85 Figure 4 1. Time Expanded Network for Work Hour Choices. PAGE 86 86 Figure 4 2. The Relationship between Utility and Working Hours with Varying Wage Rates. PAGE 87 87 Figure 4 3. Distribution of Daily Base Demand. PAGE 88 88 A B Figure 4 4. Equilibriu m Work Hours and Hourly Revenue with Varying Surge Multipliers. A) Neoclassical, B) Income Targeting PAGE 89 89 A B Figure 4 5. Market Outcomes under Dynamic Pricing. A) Surge Multipliers, B) Market Frictions PAGE 90 90 Figure 4 6. us an d Joint Revenue under Dynamic Pricing. PAGE 91 91 A B Figure 4 7. Market Outcomes under Different Commission Caps. Revenue to Social Surplus PAGE 92 92 Table 4 1. Equ ilibrium Flow Distributions under ME N. Start time End time Work hours (hr) Rest hours (hr) Revenue ($) Cost ($) Profit ($) Path flow(veh) Class 1 15 1 9 1 317.7 210.3 107.4 1182 1 9 8 0 286.5 179.1 107.4 818 Class 2 1 7 6 0 214.4 149.1 65.3 49 7 12 5 0 176.4 111.1 65.3 546 10 22 5 7 183.8 118.5 65.3 380 10 20 5 5 181.8 116.5 65.3 337 6 12 5 1 177.6 112.3 65.3 125 10 22 5 7 183.8 118.5 65.3 66 7 15 5 3 179.6 114.3 65.3 497 Class 3 7 17 9 1 317.7 210.3 107.4 268 9 19 9 1 317.7 210.3 107.4 248 1 9 8 0 286.5 179.1 107.4 266 8 18 9 1 317.7 210.3 107.4 289 7 20 6 7 218.7 111.3 107.4 930 Class 4 7 12 5 0 176.4 102.1 74.3 345 7 14 5 2 178.6 104.3 74.3 755 6 11 5 0 177.0 102.7 74.3 110 9 22 5 8 184.8 110.5 74.3 302 10 22 5 7 183.6 109. 3 74.3 488 PAGE 93 93 Table 4 2. Equilibrium Flow Distributions under ME I. Start time End time Work hours (hr) Rest hours (hr) Revenue ($) Cost ($) Utility ($) Path flow(veh) Class 1 15 1 9 1 293.0 210.3 281.3 1294 1 9 8 0 267.0 179.1 281.3 706 Clas s 2 8 20 5 7 174.3 118.3 350.9 1482 7 12 5 0 168.3 111.1 350.9 518 Class 3 7 17 9 1 293.0 210.3 281.3 840 1 9 8 0 267.0 179.1 281.3 547 8 17 9 1 293.0 210.3 281.3 613 Class 4 8 20 5 7 174.3 109.3 359.9 210 7 22 5 10 177.0 112. 5 359.9 381 6 22 5 11 178.1 113.9 359.9 41 7 15 5 3 171.1 105.5 359.9 488 7 22 5 10 177.0 112.5 359.9 631 7 22 5 10 177.0 112.5 359.9 249 PAGE 94 94 CHAPTER 5 M ODELLING SPATIAL EFFECTS OF DYNAMIC PRICING IN RIDE SOURCING MARKETS In Chapter 4, we capture dri addition, d rivers are free to choose where to search for customers upon completing a trip Given the ever changing market conditions, it is often the case that ride sourcing dri vers may cruise to/ne ar geographic areas which they deem the most profitable. In this sense, d ynamic pricing is also likely to stimulate the redistribution of vehicles among locations In this chapter, we focus on the induced spatial variations in market frictions, and further different geographic areas under dynamic pricing. This chapter is organized as follows. In Section 5.1, we first propose a novel spatial model that features the equilibration of demand and supply, while explicitly capturi ng the matching technology from a geometric probability perspective The description of an empirical data set and the algorithms for parameter estimation are given in Section 5.2. Section 5.3 demonstrates the equilibrium outcomes of the proposed model. Sub sequently, in sections 5 .4 and 5.5 we introduce and numerically explore the framework for studying dynamic pricing and its regulation. Summary of this chapter is given in Section 5.6 5.1 Spatial Equilibrium Model This section presents our model of the ride sourcing market. For simplicity, we only consider a single ride sourcing platform, and each vehicle can only pick up one customer during vehicle pairs from a geometrical probability persp ective, based on which the waiting and searching frictions are endogenously across a PAGE 95 95 spatial market The mathematical formulation of market equilibrium and the solution algorithms are discussed subsequently. 5.1.1 Geometrical Matching Below we describe a matching scheme in the spatial market. The framework is general enough to depict the matching technologies of both ride sourcing and street hailing services and to diffe rentiate the features in their returns to scale property. We consider a urban region divided into different geographic zones indexed by with the area of The market conditions in each zone are assumed h omogeneous. We discretize the total study horizon into periods of equal length, each indexed by Without loss of generality, we assume the length of one study period is one hour. 5.1.1.1 Technology The matchings between customers and drivers are completed automatically by the ride sourcing platform. The algorithm can be simplified as matching a requesting customer to her closest vehicle within a coverage radius ( Ranney, 2015 ) To delineate such a process, we discretize one analysis period into smaller time steps of equal length For each time step, we consider the following procedure: At the beginning of each time step, unmatched and newly ar riv ed customers send their requests to the platform and remain stationary at the ir current locations. T he platform randomly loops through all requesting customers. For each customer, it checks whether a vacant vehicle is available within distance r from t he customer. If so the platform assign s the closest vehicle to her; otherwise, it do es nothing. The matched pairs disappear from the market while those who remain unmatched wait until the next step. We assume the area of each zone is sufficiently large co mpared to the matching radius (one can always merge areas to satisfy this requirement if necessary), and thus both the customer and driver in a successful matching come from the same zone. We assume the arrivals of PAGE 96 96 customers and vehicles follow the spatial Poisson Point Process (SPPP) ( Chiu et al., 2013 ) and the market is stable. Therefore, the dynamics in each matching step have the same mean statistics through one study period. In contrast to Xu et al. (2017) our approach is customer oriented and relies on an exogenous matching radius that limits the matching within a study zone. me as the expected time from the moment a customer request s a ride until she is picked up. It can be decomposed into two parts: ( 5 1 ) where is the matching time, more specifically, the expected time for the requesting customer to be matched; represents the meeting time, i. e ., the e xpected travel time for the driver to pick up t searching time until picking up a customer is: ( 5 2 ) where is the expected time for a vacant vehicle to be matched. A vehicle is considered to be either occupied or vacant. A vacant vehicle is either waiting to be matc hed or en route to pick up a customer. Therefore, consists of the contribution of the vehicles that have searched in zone i during period t regardless whether they are successfully matched. The specific forms of the waiting and searching frictions are induced by the proposed matching procedure. Let and denote the intensities of unmatched vehicles and customers. Given the assumptions above, the probability of the existence of n unmatched vehicles in a circle of radius r centered at a requesting customer is: ( 5 3 ) PAGE 97 97 We denote by the distance of the closest unmatched vehicle with a cumulative distribution function and density function : ( 5 4 ) ( 5 5 ) a waiting customer The average number of potential drivers is approximately given the sufficiently large matching radius. The probability of a customer not being matched in a time Her expected matching time can be roughly estimated as times the number of steps she is going to wait: ( 5 6 ) Note that by construction and is a modeling parameter. We interpret it as the system response t ime to be estimated in Section 5.2.2 Conditional on a successful ma tching, is a function of denoted as : ( 5 7 ) where is a scaling parameter that adjusts the Euclidean distance to the Manhattan distance ( Arnott, 1996 ) ; is the a verage speed of the vehicle and is the Gauss error function. When is sufficiently large degrades to the PAGE 98 98 waiting time formula used by Daganzo (1978) and Arnott (1996) for the demand responsive transit and radio dispatched taxi market, respecti vely. The average matching time for vacant vehicles is still missing. By we have the following relationship at the steady state and F ollowing Eq. ( 5 6 ) we have: ( 5 8 ) The above scheme also applies to traditional street hailing, although the matching radius and the duration of the matching step may be technology specific. It is reasonable to argue that is much smaller than that of the ride sourcing technology Consequently, the average number of potential vehicles identified in the labeling process is approximately Given that the meeting time is often negligi ble, the average waiting time contains only the matching time: ( 5 9 ) wh ich is of the same form as provided by Douglas (1972) 5.1.1.2 Returns to scale The waiting time formulae above de monstrate the trade offs between component s for both technologies. In what follows, we discuss the returns to scale properties assuming a constant operating speed of vehicles. The argument still holds as long as traffic congestion is not severe. In the matching stage describ ed by Eqs. ( 5 6 ) and ( 5 9 ) the ride sourcing technology exh ibits constant returns to scale while street hailing technology exhibits increasing returns to scale. Specifically, d oubling and simultaneously leaves the matching time unch anged for the former but reduc ing the latter by half Such a tremendous reduction in matching time for PAGE 99 99 street hailing implies the information imperfection confined by the limited matching radius. As indicated by Eq. ( 5 7 ) however, the ride sourcing technology enjoys a significant reduction in meeting time due to the economies of density. Despite the differences in specific components, both technologies sho w increasing returns to scale at t he aggregate leve l, which is commonly seen in a spatial queuing system ( Yang and Yang, 2011 ) One can further observe that street hailing technology generally has higher returns to scale than rid e sourcing, an observation also verified by our extensive simulation in Chapter 6 5.1.2 Customer Demand We aggregate passenge rs arrival from each origin and focus on the hourly specific demand. It is well recognized that the ride sourcing market is not o nly cleared by price, since each customer has to wait for some time before getting served ( Douglas, 1972 ; Arnott 1996 ; Taylor, 2016 ) We assume that travel demand is a function of the average trip fare average waiting time and other control variables X in the following exponential form: ( 5 10 ) where is the flag drop fee, and is the surge multiplier adjusted by the platform. The surge multiplier is the same for all customers in zone i at period t is the vector of parame ters to be estimated. It is anticipated that and .The temporal or spatial correlations among the parameters are not considered. 5.1.3 Vehicle Supply and Spatial Distribution W hen completing a trip a driver obtains of the trip fare while the remaining serves as the commission to the ride sourcing platform. We consider a fixed fleet size PAGE 100 100 in the short run and each driver works for an average hour h We assume h is given exogenously, although serval research incentives in supplying work hours with respect to wage variations. See Farber (2015) and Chen and Sheldon (2016) for more details. To account for the temporal variation s in vehicle supply, we impose the constraint that the percentage of on line vehicles at period t is fixed : ( 5 11 ) where is the number of vehicles during period t Given that the length of one period is one hour, is also the to tal vehicl e hours during the period. The justification for the assumption likely subject to latent budget constraints that tend to be stable, at least in t he short run. T his results in similar day to day patterns (e.g., in weekdays) often observed in empirical data. Notice that and can be easily estimated as the average of the observations. For the conser vation of vehicle hours: ( 5 12 ) where is the number of vehicles in zone i at period t Given that a vehicle is either occupied or vacant, the following conservation equation holds at the steady state: ( 5 13 ) Note that when an occupied vehicle traverses several zones, it only contributes to its originated zone at each matching process along the journey We assume that r ide sourcing drivers search for customers in zones where they perceive to offer the highest payoff. The drivers are further assumed to have perfect information on the PAGE 101 101 profitability of each zone. Consequently, t he distribution of the vacant vehicles among a ll zones at equilibrium is to make each zone equally profitable ( Lagos, 2000 ) : ( 5 14 ) where and can be respectively treated as the average reward for completing a trip and the service duration for completing a trip; can be interpreted as the equilibrium hourly revenue. The quantification above assumes drivers are well positioned so that each zone is equally cho ice and is likely to overestimate the total vehicle supply. In contrast, Yang et al. (2010b) and Buchholz (2015) explicitly specify the transition cost in a logit type zonal choice model; and they assume drivers are not able to provide rides when traversing to their desired destination zones Such an assumption may be reasonab le for traditional street hailing technology. Ride sourcing drivers, however, can be matched to customers as long as their apps are on. A direct application of their approach will underestimate the total vehicle supply. Given that the current dataset does seems to be the simpler and feasible one. 5.1.4 Market Equilibrium For a ride souring market where trip fares and fleet size are given, market equilibrium is defined by the following set of nonli near equations : ( 5 15 ) ( 5 16 ) ( 5 17 ) PAGE 102 102 ( 5 18 ) ( 5 19 ) ( 5 20 ) ( 5 21 ) ( 5 22 ) The nonlinear system has a total unknowns and equations. Eq. ( 5 15 ) and Eq. ( 5 22 ) may pose challenges to the existence of a solution. The former requires to be bounded below by while the latter ma y force to be extremely small (even less than ) in zones with short trip durations. The following proposition guarantee s the existence of equilibrium under sufficient labor supply. The detailed proof is provided in the A ppendix G Proposition 1 There exists a constant k > 0 such that when the above nonlinear system admits at least one solution. We adopt a partially augmented Lagrangian method to solve for the equilibrium solu tion ( Bertsekas, 1999 ) We denote the vector of variables, the vector of multipliers, the functional mapping which corresponds to Eq. ( 5 15 ) and which rep resents the remaining constraints. The method solves a sequence of problems of the following form: (RP) ( 5 23 ) s.t. ( 5 24 ) PAGE 103 10 3 where is the penalty parameter updated via ; is updated via The choice of requires some trial and error ( Bertsekas, 1999 ) Under some regularity assumptions ( Bertsekas, 1999 ) will converge respectively to the solution of the original nonlinear system and the Lagrangian multipliers associated with Constraint ( 5 15 ) The augmented Lagrangian approach is quite stable and usually converges to at least a local minimum ( Bertsekas, 1999 ) We terminate the iterative process when is less than a tolerance value 5.2 Model Estimatio n We first introduce a dataset from Didi Chuxing, based on which we describe the Finally, we estimate the proposed log linear demand function as well as the mean statistics in labor supply 5.2.1 Dataset The original dataset contains 8 540 614 order requests from Jan. 1 st 21 st 2016 in a Chinese c ity and is publicly available by Didi Chuxing (2016a) Didi anonymously divides t he study region int o 66 zones and w e follow the default indexing of zones. Each request records the st occurs as well as the IDs of the origin and destination. In addition, there are supplementary datasets t hat contain other information such as weather, PM2.5 indices and the number of community facilities in a zone. After removing the observations with missing values and duplicate IDs, the master dataset retains 7,816,328 request records. Overall, approximate ly 83% of the time a customer succee ds in being matched to a driver in one request PAGE 104 104 Despite the high resolution order data, some key information is missing which includes the adjacency of the study zones, trip duration (distance) and custo ting time. Intentional ly, the proposed framework does not rely on zonal adjacency. The raw trip duration is calcu lated using the fare structure of 3 86 Yuan/mile+0.35 Yuan/min ( Didi Chuxing, 2016b ) Considering potential price surging, we approximate the trip duration between each OD pair based on the m edian of the corresponding raw trip durations. Zone 39 is identified as the airport area with an average trip duration of 1.03 hours. We filter all trips originating from Zone 39 The ma ster dataset is used to estimate system response time and a verage waiting time in Section 5 .2. 2. Two subsets are constructed from the master dataset. One summarizes observations of hourly requests in the daytime (6:00 a.m. 17:00 p.m.) for all workdays. It is used for demand estimation in Section 5.2. 3. The other summarizes the work hour information of drivers under the same restrictions, from which we obt ain the mean statistics presented in Section 5.2.4 5.2.2 Estimation of System Response Time and Average Waiting Time Note that the estimation of is non trivial. It is often observed that customers whose ride request s failed may wait for several minutes before requesting again. We account for these lagged times in determining We use the hat notation to denote the estimate of a parameter. In the estimation process, is first numerated from an ordered candidate set Conditional on the estimated key variables that include are then estimated mainly based on the probability of successful matching We settle with the that yields the minimum error. To define a counter n so that refers to the n th element from and n is initialized to be 1. The process is described as follows: PAGE 105 105 Step 1 Choose Compute the benchmark average matc hing time as follows: when a customer is successfully matched to a driver in her first trial, the customer incurs a matching time of ; otherwise the matching time is obtained as the elapsed time from her first trial until the latest trial of being matched Step 2 Compute the proportion of requests that is successfully matched, where is the rate of requests from zone i during period t Then, we have ( 5 25 ) Step 3 Estimate the intensity of the unmatched vehicles given the fact that : ( 5 26 ) where should be significantly larger than 1 to represent sufficient vehicle supply. Step 4 Calculate according to Eqs. ( 5 1 ) ( 5 6 ) ( 5 7 ) when (with some abu se of notations); otherwise set them to NAs. Step 5 Compute the augmented percentage error : ( 5 27 ) and set n = n +1; go to Step 1 In Step 5, the calculation of the estimation error excludes from each successful matching and thus focuses exclusively on the scenario not filled, PAGE 106 106 We let seconds since the system response time is typically short based and a matching radius mile a priori, which may be subject to certain biases. A rigorous estimation of these two parameters is left for future study, particularly when the geographic information of study zones and the traces of drivers are available. The estimated outcomes are given in Figure 5 1. Figure 5 1 (a) shows the curve for when is increased from 10 seconds to 20 seconds. The optimal of 13 seconds corresponds to a minimum error rate of 0.28 The rapid change of distributions of average waiting time and matching time are given in Figure 5 1 (b) and (c). The average waiting time ranges from 0.33 min to 7.52 min with a mean of 3. 61 min while the average matching time varies between 0.21 min to 2.93 min with a mean of 0.38 min. 5.2.3 Demand Estimation Consider the log transformed regression formula from Eq. ( 5 10 ) with the Gaussian error : ( 5 28 ) where we choose X to be the number of community facilities in the zone to capture a certain level of spatial heterogeneity ( Didi Chuxing, 2016a ) The estimation of demand may be subject to endogeneity problems particularly with the exi stence of dynamic pricing. Specifically, b oth trip fare and average waiting time are likely correlated to the error term due to the factors not the most chal lenging tasks in demand estimation. To partially alleviate the endogeneity problem, we implement a two stage least squares (2SLS) approach ( Hansen, 20 17 ) We instrument by the fare in the closest zone to i in PAGE 107 107 the first stage. Note that is positively related to but is less likely to be related t o In the second stage, we replace with the exogenous components identified in the first stage for demand estimation. Due to the limited data, we do not model the probable variation of with trip durations which may require an o rigin d estination demand specification or a segmentation of trip durations ( Buchholz, 2015 ) The 2SLS estimation is listed in Model III of Table 5 1 The estimated results for the intercept terms and variable X are omitted given the limite d space. We also report the ordinary least square (OLS) results via gradually modifying the regressors: without instrumenting trip fares (Model II); further removing the average waiting time (Model I). The estimated parameters are all negative and signific antly different from zero at the 0.05 significance level. The differences in the magnitude of parameters associated with trip fares ( ) between Model I and II imply a certain level of correlation between and The difference in between Model II and III is possibly due to the endogeneity bias. It is also interesting to note that the price elasticity of demand varies by time of day and is relatively smalle r in early morning (e.g., period from 6:00 7:00). We utilize the results from the 2SLS estimation for the remaining analysis 5.2.4 Labor Supply We h total fleet size N as well as the percentage of on line drivers i n each time period using their mean values. may bias the estim ation of working hours, the first step in estimating h is to define a working session ( Chen and Sheldon, 2016 ) We formally define a working session for a driver as a collection of trips with the inter trip time no longer than j during period t Working PAGE 108 108 hour is then calculated as the length of the working session 7 Other subroutines are also developed to handle extreme cases (e.g., when a driver only take s one ride in one day). Total fleet size N represents the number of drivers who provide services during the study period and is simply computed as the average of the chosen weekdays. The display of across the days is given in Fig ure 5 2 The percentage of on line drivers at each hour is qu ite stable, which supports the assumptions in Eq. ( 5 11 ) of Section 5 1. 3. The corresponding mean values are summarized in Table 5 2 5.3 Equilibrium Results We demonstrate the equilibrium outcome s when the surge multipliers are set to one (so that total trip fares are given) To facilitate the presentation, we choose Zone 2 and Zone 23 as two represe ntatives with an average trip duration of 4.1 min and 12.2 min respectively We also evaluate the changes of equilibrium outcome s when reducing the fleet size from 31 097 to 25 097 Other parameters remain the same as estimated from Section 5.2 The fare structure of 3 86 Yuan /mile +0.35 Yuan/min suggests an hourly rate of 81 Yuan assuming a constant speed of 15.6 mile/hr ( Didi Chuxing, 2016c ) At each iteration, RP is solved in GAMS with the nonlinear solver CONOPT ( Drud, 1994 ) For the augmented Lagrangian scheme, we update the penalty with and set the tolerance rate to 0.01 The method often converges within 10 iterations. We outline the components in searching and waiting frictions in Figure 5 3 with different combinations of zone ID and fleet size. Combination s I and II correspond respectively to the case with the fleet size of 31,097 and 25,097 in Zone 2; combination s III and IV correspond respectively to the case with the fleet si ze of 31,097 and 25,097 in Zone 23. The average waiting 7 PAGE 109 109 time and its matching time portion are presented in Figure 5 3 (a) (d). Customers in Zone 2 incur a higher average waiting time than those in Zone 23, as more vehicles are attracted to the latter (for longer trip duration once occupied). For the same zone, the reduction of fleet size raises the average waiting time. The average searching time and its matching time portion are given in Figure 5 3 (e) (h), which shifts in an opposite direction to their w aiting time counterparts. As an example, it is easier for vehicles in Zone 2 to find a customer; and the reduction in fleet size further decreases the average searching time. Under sufficient supply (e.g., N =31,097, h =2.7), the scenario that customers are not able to be matched in their first trial does not occur (Figure 5 3 (a) and (c)). But as we decrease the fleet size, it becomes harder for customers in zones with shorter trip durations to hail ride sourcing vehicles (Figure 5 3(b)). On the other hand, the matching time portion accounts for a average trip duration (e.g., Zone 23). ted in Figure 5 4 (a) (b). Apart from the early morning period 6:00 7:00 a.m., the equilibrium hourly revenue seems stable, ranging from 21.9 25.4 Yuan/hr and 27.1 31.2 Yuan/hr respectively for the fleet size of 31,097 and 25 097 A reduction in fleet size corresponds to a higher wage rate as drivers on average enjoy more trip opportunities. 5.4 Dynamic Pricing We present a framework for studying dynamic pricing in this section and then evaluate its impacts on the market dynamics and the welfare of involved market players. PAGE 110 110 5.4.1 Formulation We consider the market outcomes under dynamic pricing in the short run where the total fleet size is fixed. It is assumed that the platform adjusts period specific surge multipliers for revenue maximizing, subject to the equilibration of demand and supply : (DP) ( 5 29 ) s.t. ( 5 30 ) where is the vector of var iables and represents the nonlinear equation system characterized by Eqs. ( 5 15 ) ( 5 22 ) operation cost. Note that DP may be best seen as an approximation of the pricing practice for recurrent market patterns as it assumes that the platfor m has perfect anticipation. Other possible pricing heuristics that are passively trigged by random market events are beyond the scope of DP. 5.4.2 Numerical Experiments To evaluate the performance of dynamic pricing, we numerically compare the equilibrium outcomes under dynamic pricing with those of optimal static pricing. The formulation of the latter is a trivial adaptation from DP The proposed formulations can be reformulated using the augmented Lagrangian scheme and solved in a similar fashion as in Se ction 5.1. 4. A fixed fleet size of 31,097 is assumed. The surge multipliers are uniformly set at 1.21 in the optimal static pricing. Figure 5 5 (a) (b) display the period specific surge multipliers in two selected zones under dynamic pricing. The patterns of the se two share great similarity as the same sensitivity parameters (i.e., ) are used in the demand function. However, the platform has to increase the surge multipliers PAGE 111 111 more aggressively in Zone 2 to attract enough drivers. As long as the waiting cost is not severe, the objective function of DP suggests that the platform tries to set the total trip fare negative to the inverse of the price sensitivity parameter (i.e., ). Zones with shorter average trip durations achieve this optimality condition at higher surge multipliers. Although is fixed among trip s from zone i at period t s ( Buchholz, 2015 ) As will be seen, this observation has fundamental effects on the variations of market frictions and changes in the welfare of the market players. We measure the spatial variations in waiting and searching frictions using the coefficient of variation (CV), which is compute d respectively as and and are the average waiting and searching times across all study zones at period t ; and are the corresponding standard deviations. The temporal displays of the CVs are given in Figure 5 6 (a) (b). Compared to static pricing, dynamic pricing induces much smaller CV in average searching time but slightly larger CV in average wa iting time. Eq. ( 5 14 ) positively related to the average trip duration when there is no price surge. Under dynamic pricing, surge multipliers change to the opposite direction of the average trip duration as is seen in Figure 5 5 This in return reduces the spatial variation of the average searching time. The spatial variations in average waiting time are and trip durations. A few notations are introduced for better explanation. We denote the possible combinations of pricing types and trip durations as S and D are short for static and dynamic (p ricing) while s and l are short for short and long (trip durations). We retain the PAGE 112 112 basic notation for average waiting time with elements from as subscripts for distinction. For example, represents the average waiting time for a zone characterized by time is demonstrated in Figure 5 6 (c). Dynamic pricing still gives the average w aiting time with a smaller spread at a given demand level. However, our data simply has fewer zones characterized by the combinations of high demand with long trip duration or low demand with short trip duration (highlighted in the shaded areas). Therefore we tend to see and at the extremes which explains the slightly larger variation under dynamic pricing. For the analysis of the trade surplus to represent the welfare of passengers from taking ride sourcing services (see Cairns and Liston Heyes (1996) on the comput ation of consumers surplus when the average waiting time enters demand function) ; and the (quasi revenue between the surplus. We define the welfare change as the metric of interest under dynamic pricing minus that under static pricing. Figure 5 7 and Figure 5 8 present the welfare changes both temporally (at two selected zones) and spatially. With some abuse of unit, the average adjusted su rge multipliers are deployed in Figure 5 8 Three observations are worth highlighting. First, there exists temporal and s are in general better off while customers are worse off in highly surged periods. For example, the 1 6 :00 in Zon e 23 and throughout the whole study period in Zone 2. Customers in Zone 2 suffer more from dynamic pricing than those from Zone 23, a source of inefficiency due to spatial heterogeneity. Finally, dynamic pricing PAGE 113 113 may have the potential to simultaneously inc fraction of zones demonstrate this win win situation when price is dynamically adjusted below its static counterpart Note that these zones are all characterized by relatively longer trip durations, which corresponds again to the observation in Figure 5 5 In reality, trip durations vary even from the same zone. The current practice of setting a zonal specific surge multiplier may hurt those customers with longer trip durations and subsequently the revenue as well. As we pointed out previously in Chapter 4 the net outcome of the welfare changes depends on demand patterns and other factors. In the current setting, static pricing yields almost the same social surplus as the dynamic one: a rough estimate of Yuan/weekday and Yuan/weekday, respectively. In general, the comparisons on dynamic pricing and static pricing are mixed, particularly for c ustomer s. In addition equilibrium outcomes under both strategie s are likely to deviate from the socially optimal levels. 5.5 Commission Cap Regulation We evaluate the performance of the proposed commission cap regulation based on the empirical data Under the proposed regulation, a revenue maximizing platform has the incentive The additional revenue from price surging completely goes to the drivers. In Chapter 3, we showed that the commission cap regulation could achieve the second best under ideal homogeneity assumptions. In the current spatial market, a strict second best is barely achievable. However, the equilibrium market outcomes may still be superior to those from an unregulated monopoly, a statement yet to be verified PAGE 114 114 5.5.1 Formulation Given that no congestion is considered and trip duration varies, we implement the proposed regulation as the time based charge. As the platform may lower trip fare to raise the performance of the regulation. Mathematically, market outcomes are given by solving the following problem: ( DP R ) ( 5 31 ) s.t. ( 5 32 ) ( 5 33 ) ( 5 34 ) where N is the fleet size that is endogenously determined; is the average revenue obtained in a working section; is the vector of commission rate per unit time; and are the cap of commission rate per unit time working section, respecti vely; is the average trip duration from zone i during period t Note that Constraint ( 5 34 ) implicitly assumes that vehicle supply is perfectly elastic in the long run. 5.5.2 Numerical Experiments The equilibrium outcomes under different commission caps are presented in Figure 5 9 We assume the annual reservation income is 52 935 Yuan ( Didi Chuxing, 2016c ) which makes the hourly reser vation income around 25 Yuan. Note that when the commission cap goes to infinity, the platform becomes an unregulated monopoly with a commission rate of 97 Yuan/hour ; the commission rate will be bounded at this value for any larger commission cap. For a PAGE 115 115 c ommission rate of less than 97 Yuan/hour the increase in commission cap corresponds to an Figure 5 9 (a). Figure 5 9 (b) gives the ratio of the total social surplus to that under a (quasi ) second best case (i.e., the platform maximizes the social welfare subject to the same constraints as in D P R). The socially optimal commission cap is very close to zero given that the operation cost for the platform is not specified. Notice that the social surplu s by the unregulated monopoly is only approximately 62% of that under the second best ; and the proposed regulation improves market efficiency for a wide range of commission caps The platform is able to exploit the driver s with high commission cap s which, as shows in Figure 5 9 (c), results in a reduction of the fleet size in the long term. In general, the equilibrium outcomes under the proposed regulation are consistent with However, the proper choice of the commission c ap for practical implementation should fully consider the cost components of the platform which i s beyond the scope of this dissertation 5.6 Summary This chapter presents a spatial equilibrium model for studying dynamic pricing in the ride souring market. We first present a deduc tive approach for depicting the technology adopted by a ride sourcing platform for matching customers and drivers and discuss its difference s from traditional street hailing tech nology. We then specify both the demand and supply s ides; and give the conditions for the existence of the equilibrium when price is fixed. Utilizing a public dataset from Didi Chuxing, we estimate the key modeling parameters and discuss the patterns for market frictions and equilibrium revenue levels. We e valuate the effect of dynamic pricing assuming a revenue maximizing platform. It is found that the platform will set surge multipliers opposite to the average trip durations of a study zone. For this reason, drivers under dynamic pricing will see less vari ation in their average PAGE 116 116 searching time when compared to the optimal static pricing. In the welfare analysis, both the platform and the drivers are better off under dynamic pricing. However, the changes of during highly surged periods. Interestingly, c ertain zones adjusted below its static counterpart In general, the comparisons on dynamic and static pricing are mixed. We then evaluate the performance of the suggested commission cap regulation It reaps the responsive feature of dynamic pricing while incentivizing the platform to maximize the total transactions for revenue maximization. O ur experiments demonstrate that such a regulation policy generates higher social surplus for a wide range of commission caps compared to an unregulated monopoly. A good choice of commission cap balances the welfare between the platform and its users. PAGE 117 117 A B C Figure 5 1. Estimation of The System Average Waiting and Matching Time. A ) Augmented Absolute Error Rate versus System Response Time B ) Distribution Average Waiting Time, C) Distribution of Average Matching Time PAGE 118 118 Figure 5 2. Display of t he Fraction of The On Line Drivers by Hour of Day. PAGE 119 119 A B C D PAGE 120 120 E F G H Figure 5 3. Market Frictions with Different Zone and Fleet Size Combinations. A) Waiting Time and The Matching Time Portion (I), B) Waiting Time and The Matching Time Portion (II), C) Waiting Time and The Matching Time Portion (III), D) Waiting Time and The Matching Time Portion (IV), E) Searching Time and The Matching Time Portion (I), F) Searching Time and The Matching Time Portion (II), G) Searching PAGE 121 121 Time and The Matching Time Portion (III), H) Searching Time and The Matching Time Portion (IV) PAGE 122 122 A B Figure 5 4. Display A ) Equilibrium Revenue with Fleet = 31097, B ) Equilibrium Revenue with Fleet = 25097 PAGE 123 123 A B Figure 5 5. Display of Surge Multipliers under Dynamic Pricing. A) Surge Multipliers in Zone 2, B) Surge Multipliers in Zone 23 PAGE 124 124 A B C Figure 5 6. Spatial Variations of Average Searching a nd Waiting Times A) Temporal Displa y of CV for Average Searching Time B) Temporal Display of CV for Average Waiting Time C) Spreads of Average Waiting Times under Dynamic and Static Pricing PAGE 125 125 A B C D Figure 5 7 Temporal Display of Welfare Changes under Dynamic Pricing. A) Cha nge of CS in Zone 2, B) Change of CS in Zone 23, C) Change of Joint Revenue in Zone 2, D) Change of Joint Revenue in Zone 23 PAGE 126 126 Figure 5 8 Spatial Display of Welfare Changes under Dynamic Pricing. PAGE 127 127 A B C Figure 5 9 Equilibrium Outcomes under the Commission Cap R egulation. A) Share of evenue, B) The Ratio of The Social Welfare under The Proposed Regulation to That of The (Quasi ) Second Best C) Fleet Size versus Commission Cap PAGE 128 128 Table 5 1. Demand Estimation for Different Parameteri zations Model I Model II Model III Variables Parameter S.E. Parameter S.E. Parameter S.E. Fare6 0.033 0.004 0.008 0.003 0.033 0.008 Fare7 0.087 0.006 0.026 0.004 0.052 0.009 Fare8 0.088 0.006 0.034 0.004 0.081 0.012 Fare9 0.049 0.005 0.0 21 0.003 0.071 0.015 Fare10 0.042 0.004 0.009 0.002 0.068 0.015 Fare11 0.048 0.005 0.011 0.003 0.045 0.01 Fare12 0.049 0.005 0.014 0.003 0.06 0.015 Fare13 0.049 0.005 0.011 0.003 0.039 0.009 Fare14 0.048 0.005 0.009 0.003 0.043 0.02 F are15 0.042 0.005 0.011 0.003 0.029 0.016 Fare16 0.050 0.005 0.013 0.003 0.042 0.01 Fare17 0.048 0.005 0.015 0.003 0.065 0.018 Waiting time6 NA NA 1.618 0.074 1.894 0.093 Waiting time7 NA NA 3.556 0.128 3.649 0.348 Waiting time8 NA NA 2. 899 0.127 2.743 0.355 Waiting time9 NA NA 2.476 0.089 2.677 0.183 Waiting time10 NA NA 1.97 0.064 2.199 0.13 Waiting time11 NA NA 2.023 0.066 2.351 0.133 Waiting time12 NA NA 2.341 0.074 2.616 0.161 Waiting time13 NA NA 2.189 0.072 2.494 0. 14 Waiting time14 NA NA 2.067 0.075 2.322 0.196 Waiting time15 NA NA 2.291 0.08 2.649 0.167 Waiting time16 NA NA 2.95 0.105 3.171 0.277 Waiting time17 NA NA 2.677 0.103 2.738 0.315 Intercept Yes Yes Yes X Yes Yes Yes *robust standard error s are reported. PAGE 129 129 Table 5 2. Summary of the Parameters for t he Supply Side. Parameter Value Parameter Value H (hr) 2.7 n 11 0.070 N (veh) 31,097 n 12 0.074 n 6 0.025 n 13 0.079 n 7 0.092 n 14 0.078 n 8 0.114 n 15 0.083 n 9 0.097 n 16 0.098 n 10 0.071 n 17 0. 119 PAGE 130 130 CHAPTER 6 AN AGENT BASED SIMULATION FOR ON DEMAND RIDE SOURCING MARKET S Different analytical models have been proposed in previous three chapters, which simplify the behavioral interactions among market players and rely on the hypothetical equili b rium state s to generate insight s Realistically, each market participant should have specific attribute and preference T he inter actions among the involved agents and the evolution of market dynamics can be well captured by the agent based simulation ( Wilensky, 2015 ) The goal of this chapter is two fold. We first develop a simulation tool and demonstrate its capability of analyzing the dynamics of the ride sourcing ma rket. Using the simulation output we next calibrate and discuss the matching func tion employed in Chapter 3 In Section 6.1 we build a simulation test bed in NetLogo while specifying customers g Interface (API). Section 6.2 presents a simulation evaluation framework where d ifferent scenarios are created via modifying the zonal demand levels and the matching ranges. In S ection 6.3, w e demonstrate the market equilibrium, calibrate the Cobb Dougla s matching function for a range of matching radii and compare their efficiencies and returns to scale properties. Conclusion s are summarized in S ection 6.4 A detailed introduction on the simulation interface and its fu nctionality is attached in A ppendix H for interested readers. Note that some of the notations in this c hapter may be different from previous chapters 6.1 Simulation Framework This section mainly introduces the configuration of the simulation test bed, the specifications of the behavioral heu ristics of drivers and customers as well as the concept of market equilibrium. PAGE 131 131 6.1.1 Simulation Test Bed Figure 6 1 shows our simulation test bed that is based on the Manhattan network. The total network is divided into 13 z ones. We index 12 of them as fe asible zones where customers and ride sourcing vehicles are interacted (The center zone is the Center Park area and is treated as the infeasible zone in our simulation). As no local streets or roads is modeled, our simulation is to the resolution of the zo nal level. 6.1.2 Basic Agents Module Three types of agents are considered, namely, a ride sourcing platform, potential customers and the affiliated drivers. The role of the ride sourcing platform is to determine the prices (e.g., trip fare and the commissi on) and the matching algorithms. We assume all these components are exogenously given. Therefore, our focus is on the interactions between the Figure 6 1 Both agents are assumed to be homogeneous in their attributes to be mentioned. We assume a predetermined origin destination (OD) trip demand table for customers that use the ride sourcing services. Customers are constantly generated according to the OD table and randomly distributed i n the feasible zones. Location s of the customers are fixed once generated; they keep sending requests until being matched with the ride sourcing vehicles nearby. All customers are assumed to wait up to a maximum time above which they give up the service demand is thus elastic. We assume a total number of N ride sourcing vehicles throughout the simulation. The value of N shall be determine d endogenously at the market equilibrium, given that the ride used when the simulation is initialized. When being vacant, ride sourcing vehicles search in each PAGE 132 132 zone with a constant speed To specify the local searching behavior, we define an allowable that represents the potential rotation from their previous direction. When they get close to the b oundary of an unsearchable area (i.e., Center Park zone and areas outside of the simulation network), the ride sourcing vehicles are forced to face the center of the zones they are currently in and rotates an angle for their new direction. Drivers will consider to transition to other zones periodically for better trip opportunities. Their decision is triggered either right after successfully finishing a transaction or remaining vacant after certain time spent in the local search. The specification of the decision rules will be explained in the Zonal Choice Module. 6.1.3 Transaction Module At each time step, the platform loops through the queue of the waiting customers; and for each customer, the platform matches a vacant vehicle closest to her within a matching radius r If no vehicle is available, the customer will wait for the next ti me step until being matched or the maximum waiting time is reached whichever comes first. When a customer is successfully duration), the customer is assumed to be served and is removed from the simulation. The vehicle and de termines which zone to go for customers. We assume the travel distance table for inter zone trips is given which induces the inter zo ne travel times. Trip length for intra zone trips is randomly drawn with a minimum value We as sume that vehicles are always running either occupied or not. Drivers incur an average cost c per veh icle per unit time, which represents the operating cost and the opportunity as the set of origination and des tination zones. The price cha rged to a customer traveling from zone to zone is given by: ( 6 1 ) PAGE 133 133 where is the flag drop fee, represents the incremental fee per unit time, is the corresponding trip time. We do not consider the distance based charge as it duplicate s the time bas ed charge given that traffic congestion is not considered here. is fi xed and there is no surge in price. As the rule of operation, a certain percentage is changed by the ride sourcing company from the f Given the purposes of this study, there is no need to capture the dynamics in the temporal dimension. The presentation focus es on the spatial domain. 6.1.4 Z onal Choice Module Drivers will consider to transition to other zones for better trip opportunities periodically. They have the perceived information about the market dynamics across all zones. A driver in zone will choose zone to maximize her utility. The utility is assumed positively related to the average profit ; and negatively related to the sum of transition time and the average searching time ( Yang et al., 2010b ) The perception er ror is captured by the random variable Together, we have : ( 6 2 ) where and is the average trip fare and trip time from zone adjusted by the demand rate from zone ; is the average i If we assume the error term is independently and identically Gumbel distributed, the probability of a driver at zone j choosing to travel to zone i ca n be represented in the following multinomial logit form: PAGE 134 134 ( 6 3 ) where calibrated using the real world data ( Yang et al., 2010b ) Note that if the driver will stay in the current zone and continue the local search for another period of Driv ers always follow this probabilistic approach for zonal choice. It is still possible, however, for them to be matched with any customers on the way when transitioning. This occurs as long as a driver happens to be the closest to a waiting customer within t he matching radius. The increasing probability of being matched essentially reflects the improvement of the matching technology over the traditional street hailing. Given no street system has been specified, we predefine a set of seudo paths and project th e zonal crossing movement of the ride sourcing vehicles onto these seudo paths. In this manner, the location of vehicles during transitioning can be explicated traced at each time step. Note that such a restriction is only initiated in the Zonal Choice Mod ule while drivers still follow the random search behavior locally as is specified in the Basic Agent Module 6.1.5 Market Entry and Equilibrium Our simulation is capable of delineating the evolution of market dynamics. For the purpose of this study, we only capture a stable state that approximates the static equilibrium. To explain the concept of the equilibrium, we temporally omit the spatial dimension and introduce a fluid queuing process in Figure 6 2 As is shown in the left reservoir of Figure 6 2 the incoming rate of customers Q equals to the rate of the served customers (otherwise the queue of the waiting customers grows). A circular flow is used in the right reservoir to represent the conservation between the rate of the oc cupied and vacant ride sourcing vehicles, represented as and respectively (otherwise the queue of the vacant vehicles grows). The red dashed line PAGE 135 135 indicates which means the equi librium between demand and supply. Therefore, must hold at the equilibrium ( Yang and Yang, 2011 ) L evels of the fluid are proportional to the number of waiting custome rs ( ) and searching vehicles ( ) in the system. Both and remain constant at the equilibrium; the corresponding average waiting and searching times ( and respectively) can be obtained ( 6 4 ) ( 6 5 ) Multiple equilibria can be achieved with different fleet size N We assume an exogenous reservation profit level for the drivers, which reflects their preferences and the competition level in the labor market. Drivers are implicitly assumed to be p erfectly elastic, which is likely true in the long run: i f the average profit is larger than more drivers will be attracted to enter; otherwise a random portion of the drivers will exit. The fleet size is adjusted dynamically in this matter until the equilibrium profit matches the reservation one. Therefore, we are essentially considering a long run equilibrium with free entry. 6.2 Simulation Evaluation Framework This section explains the simulation evaluation framework. We first clarify the simulation tasks based on which we develop the simulation scenarios. The parameters used in the simulation are also specified. To examine the effectiveness of the agent based simulation in characterizing the market equilibrium, we are interest ed in verifying the queuing status highlighted in Eqs. ( 6 4 ) ( 6 5 ) To better understand the impact of the improved technology on the market dynamics, we analyze the patterns of matching frictions and the success rate of the zonal choice for different combinations PAGE 136 136 of demand levels and matching radii. Based on the da ta collected at the equilibrium state, we estimate the matching function for each matching radius and discuss its returns to scale property. We consider 4 scenarios by increasing r from 1 to 15 unit distance, with the lowest and highest values representin g street hailing and ride sourcing technologies respectively. To represent a wide range of market conditions, three demand levels are designed for each r The fleet size also changes accordingly to achieve the same reservation profit across a ll simulation runs. In total, we have a 12 simulation runs. Table 6 1 provides the base demand used in the analysis. Three demand levels are obtained via multiplying by an adjusting parameter We thus have the low, median and high demand scena rios for each r The trip length matrix corresponding to the OD pairs is summarized in Table 6 2 which is assumed symmetric. All the other parameters used in the simulation are listed in Table 6 3 To be consistent with the real world operations, most of t he parameters are estimated based on the Uber/taxi data from the Taxi & Limousine Commission (TCL) at New York ( TLC, 2016 ) Not e that in our simulation, one unit distance and time approximately equals 0.06 mile and 1/6 min. 6.3 R esults and Discussions This section contains the discussion of the simulation output s which mainly demonstrate the market dynamics; and the regression analysis of the matching function. 6.3.1 Analysis of the Output Data 6.3.1.1 Equilibrium p roperty When a static equilibrium is achieved all variables of interest are asymptotically constant. Dividing Eq. ( 6 4 ) by Eq. ( 6 5 ) we obtain: ( 6 6 ) PAGE 137 137 Therefore, the ratio of to should equal one ideally. Note that all variables in the equation above are computed as the average across the network, as no spatial dimension is captured in characterizi ng this relationship. This ratio is plotted for each simulation run. As is given in Figure 6 3 all the p oints fluctuate closely around one confirming the equilibrium state. In fact, this ratio is monitored during the simulation and serves as a supplement ary criterion to determine the equilibrium state, a point when we terminate the simulation. The whole procedure that converges to the equilibrium state is demonstrated in the A ppendix H under the item 6.3.1.2 Matching f rictions In the spatial ride sourcing market, matching frictions always exist: both drivers and customers have to wait for certain amount of time to be served. Following Chapter 5, w e decompose the matching process into two stage s : 1) matching stage: the platform randomly matches a cust omer to her closet available vehicle within r when possible; 2) meeting stage: the vehicle comes and picks up the customer conditional on a successful matching Accordingly, c can be d ecomposed into two parts: ( 6 7 ) where is th e average time for the requesting customer being matched; represents the average travel time for the driver to pick up the requesting customer after a successful matching. is: ( 6 8 ) where is the averag e time for a vacant vehicle to be matched. PAGE 138 138 We now analyze the patterns of the matching frictions. Note that the average trip distances (and trip times) are approximately equal for the investigated scenarios. As vehicles are either occupied or vacant, the a verage searching time for the drivers must be about the same so that the same reservation profit level can be achieved. For all the investigated simulation scenarios, is around 4.2 min. iting time. Figure 6 4 displays the average waiting time under varying demand level s and matching radii. It is found that decreases as the increase of the demand for each matching radius. To explain, the increasing dem and attracts the same proportional drivers to enter which gener all y raises the number of vacant vehicles ( ) and customers ( ) at the equilibrium. The simultaneous increase of and will reduce the distance be tween the customers and drivers, which leads to lower market frictions For each demand level, decreases as the increase of r but the mechanism differs. For the lower matc hing radius (e.g., street hailing), the majority of the waiting time is spent on the matching time due to the high transaction cost (or more precisely the bilateral searching cost). Comparatively, the meeting time is negligible since the matched pair is qu ite close. At r = 1 (0.06 mile) and for example, the average value for the matching time and meeting time is around 1.8 min and 0.2 min, respectively. In contrast, for the case of high matching radius (e.g. ride sourcing), the m atching time is reduced significantly while the customer has to wait an increasing amount of time for a vehicle that can be farther away (but is still the closet available vehicle within the matching radius). The net outcome is the dominance of the r educti on of the matching time When r = 15 (0.9 mile) and the average value of the two portions are around 0.05 min and 0.6 min, respectively. PAGE 139 139 6.3.1.3 Success rate of zonal t ransition As has been discussed in the Zonal Choice Module, not all the vacant vehicles are able to transition to their ideal zones successfully. We provide the rate of successful zonal transition for the low demand case under r =1 and r = 15 in Table 6 4 The rate of success decreases as vehicles travel to zones f urther away. Besides, higher matching radius leads to lower success rate, i.e., ride sourcing vehicles are more likely to be matched than the traditional taxis when transitioning uccess rate and the difference between the two is nil. The specific matrixes are not provided due to the limited space. Overall, it is clear that the success rate is not high from our simulation. The assumption of perfect success rate in previous analytica l studies can be problematic even for the traditional street hailing market ( Yang and Wong, 1998 ; Yang et al., 2010b ; Nicholas, 2015 ) 6.3.2 Regression Analysis We calibrate the Cobb Douglas mat ching function assumed in Chapter 3 To be consistent, we regress the realized demand against t he number of vacant vehicles ( ) and waiting customers ( ) : ( 6 9 ) The parameters ( ) measures the efficiency of the matching technology. Take logarithm of both sides: ( 6 10 ) With observations of at the equilibrium, the associated parameters can be estimated using standard multivariate regression technique. PAGE 140 140 To generate enough sample s for the regression, we use the data at the zonal level. Therefore, a total 36 observations are utilized for each simulation. The estimated results are presented in Table 6 5 and the standard error for each estimate is given in the associated parenthesis. All the parameters are found significant at the 95% confidence interval. As the goodness of fit measure, the R 2 values are close to 1, indicating the explanatory variables capture the majority of the variance in the da ta. Two observations are worth mentioning. First, the estimated results confirm the increasing returns to scale property (i.e., ) for all matching radii. Second and perhaps more interesting, the increase of the matching radius is associated with a decreasing magnitude of the returns to scale but increasing intercept term The efficiency of the matching technology in large matching radius is mainly reflected by the parameter A To explain, the obstacle fo r the street hailing market is the high search cost due to the limited matching radius (e.g., 0.06 mile is assumed). Therefore, an increase of both and significantly increases the matching probability v ia reducing the distance between the customers and drivers This indicates a matching function of high returns to scale. On the other hand, the matching radius in the ride sourcing market has already been sufficiently large (e.g., 0.9 mile is assumed). The benefit in increasing the matching probability is thus marginal. However, both sides still enjoy the reduced distance because the matched pair can be realized more qu ickly ( i.e., economies of density ). This is why the re turns to scale is always above one To shed more light on the impact of the matching technology, we explicitly account for the matching radius as: ( 6 11 ) PAGE 141 141 The previous 144 observations are jointly used for the parameterization above. We are only interested in the value of which turns out to be 0.65 (with the standard error 0.03) and is significantly larger than 0. In the investigated matching ranges, an increase in the matching radius leads to larger otal transaction) 6.4 Summary We develop an agent based simulation for the ride sourcing market with special fo cus on demonstrating the market dynamics at equilibrium and calibrating the matching fun ction. A simulation test bed i s built in NetLogo while customers and d re specified in its Application Programming Interface (API). Thre e major modules ar e developed. The Basic Agent Module specifies the basic attributes and moving kinematics of the simulated agents. The Transaction Module reproduces the algorithms in the current matching technology of the ride sourcing applications. The Z onal Choice Module consists of the utility maximization rule of how a driver determines the location for seeking the cus tomers. Different scenarios are created via modifying the zonal demand levels and the matching radii. We conclude that: Our simulation demonstrate s the capability of delineating the evolution of the market dynamics as well as the equilibrium state often assumed in previous analytical studies ( He and She n, 2015 ; Wang et al., 2016 ) The rate of s uccessful zonal transitioning decreases as vehicles travel further away; a higher matching radius le a d s to a lower success rate. An increase i n the matching radi us in general le a d s to lower average waiting time for the customers. Ho wever, a trade off exists in the matching time and the meeting time : the reduced s earch cost significantly lowers the matching time while the increa sing possibility of matching a vehi cl e further away increases the meeting time The proposed Cobb Dou glas matching function accounts for the majority of the variance in the simulated data and the matching technologies exhibits increasing returns to scale. Lower matching radi us (e.g. street h ailing) implies higher returns to scale but smaller intercept term. That is, the technology with lower matching radius yields lower matching rate; however, a simultaneous increase of the number of vacant vehicles and waiting customers gives more significan PAGE 142 142 Figure 6 1. Simulation Configuration PAGE 143 143 Figure 6 2. Demand and Supply Equilibrium in the Ride Sourcing Market PAGE 144 144 Figure 6 3 Display of the Ratio of ( w v / w c ) to ( N v / N c ) f or Each Simulation Run PAGE 145 145 Figure 6 4 Radii PAGE 146 146 Table 6 1. O/D 0 1 2 3 4 5 6 7 8 9 10 11 0 30 30 30 30 30 30 30 30 30 30 30 30 1 30 30 30 30 30 30 30 30 30 30 30 3 0 2 30 30 30 30 30 30 30 30 30 30 30 30 3 30 30 30 30 30 30 30 30 30 30 30 30 4 30 30 30 30 30 30 30 30 30 30 30 30 5 30 30 30 30 30 30 30 30 30 30 30 30 6 30 30 30 30 30 30 30 30 30 30 30 30 7 30 30 30 30 30 30 30 30 30 30 30 30 8 30 30 30 30 30 30 30 30 30 30 30 30 9 30 30 30 30 30 30 30 30 30 30 30 30 10 30 30 30 30 30 30 30 30 30 30 30 30 11 30 30 30 30 30 30 30 30 30 30 30 30 PAGE 147 147 Table 6 2. Distributions of Trip Distances (mile). O/D 0 1 2 3 4 5 6 7 8 9 10 11 0 N/A 1.3 1.3 3.0 3.1 3.1 5.5 5.2 7.8 8.0 7.1 10.6 1 1.3 N/A 1.0 1.8 1.8 2.5 4.2 3.8 6.6 6.8 5.7 9.4 2 1.3 1.0 N/A 2.8 2.5 1.8 5.2 3.9 7.6 6.9 5.7 9.8 3 3.0 1.8 2.8 N/A 0.7 1.5 2.4 2.7 4.8 5.4 4.5 7.6 4 3.1 1.8 2.5 0.7 N/A 0.7 3.2 2.0 5.6 5.0 3.8 7.9 5 3.1 2.5 1.8 1.5 0.7 N/A 3.9 2.1 6.3 5.1 4.0 8.0 6 5.5 4.2 5.2 2.4 3.2 3.9 N/A 6.0 2.4 3.0 4.1 5.2 7 5.2 3.8 3.9 2.7 2.0 2.1 6.0 N/A 3.6 3.0 1.8 5.9 8 7.8 6.6 7.6 4.8 5.6 6.3 2.4 3.6 N/A 0.6 1.7 2.8 9 8.0 6.8 6.9 5.4 5.0 5.1 3.0 3.0 0.6 N/A 1.2 2.9 10 7.1 5.7 5.7 4.5 3.8 4.0 4.1 1.8 1.7 1.2 N/A 4.1 11 10.6 9.4 9.8 7.6 7.9 8.0 5.2 5.9 2.8 2.9 4.1 N/A PAGE 148 148 Table 6 3. Suggested Value of the Parameters. Parameter Description Value Matching radius Varies Reservation profit for the drivers $ 19.6/hr Average speed 13 mph Allowable angle of left and right rotation with each movement 15 Allowable angle of left and right rot ation when a vehicle encounters the boundary 180 Minimum distance required for intra zone trip 0.6 mile Average cost $ 11/hr Flag drop fee $ 6 Incremental fee $ 0.5/min Percentage charged by the platform from the final trip fare 20% 1 Amount of time a vehicle sea rches before deciding if it should leave 8.3 16.6 min Maximum waiting time 10 min PAGE 149 149 Table 6 4. Success Rate for Zonal Choice. A) r = 1 (0.06 mile) O/D 0 1 2 3 4 5 6 7 8 9 10 11 0 N/A 0.34 0.35 0.10 0.14 0.18 0.03 0.00 0.00 0.00 0.00 0.00 1 0.32 N/A 0.40 0.20 0.20 0.12 0.03 0.01 0.00 0.00 0.00 0.00 2 0.35 0.39 N/A 0.13 0.14 0.21 0.03 0.04 0.00 0.00 0.00 0.00 3 0.12 0.15 0.10 N/A 0.39 0.31 0.05 0.03 0.02 0.00 0.06 0.00 4 0.12 0.19 0.17 0.44 N/A 0.44 0.05 0.10 0.00 0.03 0.0 2 0.00 5 0.12 0.14 0.23 0.32 0.42 N/A 0.05 0.05 0.00 0.00 0.03 0.00 6 0.01 0.06 0.04 0.08 0.07 0.00 N/A 0.00 0.07 0.06 0.03 0.03 7 0.02 0.04 0.02 0.04 0.10 0.05 0.00 N/A 0.07 0.07 0.12 0.01 8 0.00 0.00 0.00 0.00 0.08 0.00 0.05 0.04 N/A 0.48 0.22 0.14 9 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.07 0.47 N/A 0.29 0.10 10 0.00 0.00 0.00 0.03 0.07 0.05 0.03 0.10 0.28 0.30 N/A 0.07 11 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.00 0.28 0.19 0.13 N/A B) r = 15 (0.9 mile) O/D 0 1 2 3 4 5 6 7 8 9 10 11 0 N/A 0.25 0.25 0.08 0.09 0.06 0.00 0.00 0.00 0.00 0.00 0.00 1 0.27 N/A 0.34 0.14 0.12 0.09 0.02 0.00 0.00 0.00 0.00 0.00 2 0.24 0.31 N/A 0.10 0.07 0.12 0.00 0.02 0.00 0.00 0.00 0.00 3 0.06 0.13 0.05 N/A 0.30 0.24 0.02 0.03 0.00 0.00 0.00 0.00 4 0.06 0.12 0.09 0.36 N /A 0.39 0.01 0.03 0.00 0.00 0.00 0.00 5 0.04 0.08 0.11 0.25 0.37 N/A 0.01 0.02 0.00 0.00 0.00 0.00 6 0.01 0.02 0.01 0.01 0.02 0.01 N/A 0.00 0.04 0.04 0.04 0.01 7 0.00 0.02 0.00 0.03 0.03 0.02 0.03 N/A 0.02 0.03 0.04 0.01 8 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.00 N/A 0.51 0.27 0.10 9 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.02 0.48 N/A 0.26 0.09 10 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.05 0.18 0.23 N/A 0.06 11 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.03 0.16 0.18 0.14 N/A PAGE 150 150 Table 6 5. Summary of Regressio n Results. r = 1 0.94 (0.46) 0.86 (0.02) 0.80 (0.09) 1.66 0.86 r = 5 2.76 (0.19) 0.58 (0.03) 0.81 (0.06) 1.39 0.94 r = 10 2.83 (0.18) 0.55 (0.03) 0.83 (0.06) 1.38 0.94 r = 15 2.86 (0.19) 0.69 (0.05) 0.60 (0.10) 1.29 0.93 PAGE 151 151 CHAPTER 7 CONCLUSION S AND FUTURE RESEARCH 7 .1 Conclusions In this dissertation, we develop both analytical and simulation tools for analyzing eme rging (on demand) ride sourcing markets. Our investigation focuses on the key issues that include dynamic pricing, platform competition and regulation. A n aggregate and static model is first introduced to understand the effect s of platform competition in t rip fares and the social welfare. An exogenous Cobb Douglas matching function is used to delineate the matching technology the platform offers to match the customers with the drivers. Possible regulation policies are then analytically explored and compared To explore the effects of dynamic pricing, we extend our modeling framework in two ways. First, we adopt a deductive approach to describe the matching process which features the imensions in the proposed framework. Accordingly we are capable of capturing driver decisions, the spatial variations of market frictions and the change of welfare under dynamic pricing. In addition, we present a simple regul ation scheme for dynamic pricing if market power is a concern. Lastly, a n agent based simulation is developed to validate the properti es of the matching function. It also demonstrates the potential for the applications i n other domains of the markets. Ove rall, the proposed methodologi es are characterized by different modeling focus es and resolutions, which can be used by regulatory agencies to better understand and manage the ride sourcing markets. The main results and policy i mplications are summarized a s follows: The matching of the ride sourcing services exhibits increasing returns to scale property: a simultaneous increase in the intensities of unmatched vehicles and requesting customers PAGE 152 152 leads to a reduction of Th e reason behind is that r ide sourcing technology benefits from the reduced distance between the matched pairs ( so the meeting time for a vehicle to pick up a requesting customer decreases ); given the sufficient matching radius however, the time for match i ng a vehicle to a requesting customer (matching time) is nearly unchanged. Although the matching in traditional street hailing is also incr easing returns to scale the mechanism differs. Due to the limited matching range, the reduction of the average waiti ng time is mostly f rom the matching time while the meeting time is negligible. In fact, more advanced matching technology (with larger matching radius) comes with lower returns to scale an observation verified by our simulation. The increasing returns to scale matching process implies that for the ride sourcing the number of ride sourcing companies probably better caters the preferences of the customers. Market friction s (average waiting and searching times) increase as well. If the increased friction is a dominant factor, competition can neither lower down the trip price nor improve the social welfare. The optimal decision for the regulatory agency is then to encourage the merger of the platforms and regulate them as a monopolist. The impact of dynamic pricing on the welfare of the involved market agents is mixed when compared with the optimal static pricing. In general, dynamic pricing benefits the platform and the dri vers due to the current revenue sharing structure (e.g., the platform takes 20% of the final fare as the commission while the remaining goes to the drivers). However, customers may be worse off during highly surged periods. Dynamic pricing has the potentia l to create a win win situation when trip fare is adjusted dynamically below its static counterpart. This phenomenon occurs mostly in geographic zones that are characterized by longer average trip duration. As trip price is proportional to trip duration (w hen traffic congestion is not considered) a revenue maximizing platform tries to es even at the price of lowering down the surge Lastly in case market power is a concern, we propose a simple regulation scheme that can further improve the efficiency of the market: capping the amount of commission from each transaction. Under the proposed regulation, a revenue maximizing platform has the i ncentive to maximize total transactions. The additional revenue from price surging completely goes to the drivers. We prove that such a regulation scheme can achieve the second best based on our static model; its performance is also numerically confirmed u nder dynamic pricing. Given the heterogeneity of the trip durations, the proposed scheme may be implemented as the distance based charge, time based charge or the combination of both. However, a proper choice of commission cap should fully explore the cost structure of sourcing companies, as we must balanc e a healthy profit margin of a ride sourcing company and the welfare of its users. PAGE 153 153 7 2 Future Research Despite the efforts we have made, there are tremendous opportunities to be explored. Future research can be done in at least the following directions. Calibration of the current framework. For example, the application of the temporal model in Chapter 4 exp ande d network, which is pos sible given data. Besides, the geographic matching framework applies for both street hailing and ride sourcing technologies It is therefore necessary to estimate the corresponding matching radius as well as the duration of a matching step Ba sed on this, a head to head comparison on market frictions will provide convincing evidences on the trade offs of both technologies Integrating with an event based dynamic pricing Our current formulations approximate the market behavior under recurrent d emand patterns (so that the platform knows driver s know exactly the market conditions ). In reality, there are scenarios which involve unexpected random shocks of demand. Besides, estimating a tempor ally and spatially differentiated demand function may be cumbersome. Therefore, incorporating the event trigger ed (e.g., based on the matching time) dynamic pricing may give a comprehensive evaluation. Exploring (near) Pareto optimal polic ies (where no ag ent is worse off compared with the optimal static pricing) Note that the revenue maximizing dynamic pricing may make customers worse off in highly surged periods. One possible strategy is a user specific dynamic pricing with the commission cap regulation initiated only for trips with short durations. S uch a hybrid policy synchronizes the observed win win features under dynamic pricing and the customer favored pricing under the proposed regulation Capturing the e ffect of traffic congestion. The level of tr affic congestion (either caused by the ride sourcing vehicles or conventional vehicles) may change some of the findin gs. For example, the increasing returns to scale matching process is only valid without severe traffic congestion; otherwise, the scaling o f either requesting customers or unmatched drivers will make the transportation network more congested which may reduce the returns to scale. PAGE 154 154 APPENDIX A RELAXATION OF P2 Let solves P1 and the optimal platform profit is Considering the equivalence of P1 and P2, we know solves P2 with the optimal joint profit Note by construction. Let solves the relaxation of P2 in which Eq. ( 3 26 ) is dropped and the optimal joint profit is It follows that Define Using the fact we can spell out as: (A 1) As is assumed to be nonnegative, we have: (A 2 ) w hich leads to: (A 3) Substituting the above inequality into Eq. (A1) we obtain Clearly, satisfies the constraints of P1 and thus is feasible. It follows that Therefore, the optimal solution of the relaxation of P2 solves P1. PAGE 155 155 APPENDIX B PROOF OF PROPOSITION 1 If then and from the complementary slackness conditions If then which conflicts with Therefore, PAGE 156 156 APPENDIX C VALUE OF TIME We can assume a continuous distribution of in the utility functions defined in Eqs. ( 3 5 ) ( 3 6 ) Generally and in the demand function does not present a linear relationship and thus can be written as: (C 1) where Substituting this demand function to all the investigated formulations (P1~P4), one can verify that the tangency condition for the monopoly, first best and second best scenarios are replaced respectively by: (C 2) (C 3) (C 4) where , and Note when calculating the consume we fix the average waiting time at the equilibrium level and integrate under a hypothetical market demand curve ( Cairns and Liston Heyes, 1996 ; Yang et al., 2002 ) If (the case of homogeneous value of waiting time), Eqs. (C 2) (C 4) will reduce to the tangency condition. PAGE 157 157 APPENDIX D SENSITIVITY ANALYSIS Expressing the Cob b Dou glas type matching function utilizing the tangency condition, we have: (D 1) where Further, at equilibrium ( D 2 ) Without loss of generosity, we assume the dispersion parame ter to be 1. Differentiating the above equations w.r.t A : ( D 3) ( D 4) The specification of the generalized cost depends on the investigated scenario. Case (1): Zero profit s econd best. Together with Eqs. ( D 3) and ( D 4): ( D 5) Re arranging the terms, we obtain: ( D 6) PAGE 158 158 It is straightforward to see that the numerator is negative. Next, we will sh ow the sign of the denominator is positive. Given the assumption that the joint profit for the monopoly solution is nonnegative: ( D 7) The asterisk denotes the monopoly solution. For the specified demand function Therefore we have: ( D 8) When the demand increases from the monopoly to the second best level, Therefore, at the second best: ( D 9) That is, which gives at the second best. This result still holds for other second best solutions with varying reservation profit levels. Case (2) Monopoly solution. Then Eq. ( D 4) can be rewritten as: ( D 10) Substituting as specified in Eq. ( D 3) into Eq. ( D 10) leads to: PAGE 159 159 ( D 11) The numerator of is positive. From Eq. ( D 8): ( D 12) Then ( D 13) which indicates at the monopoly solution. Following a similar procedure, one can verify the sign of PAGE 160 160 APPENDIX E EQUILIBRIUM MODEL WITH LONG TRIP D URATION We consider long trip duration in formulating the equilibrium model in this appendix. E.1 Model Development Due to the potential long trip duration, the specification of in Eq. ( 4 4 ) needs to be revised accordingly. With the additional assumption that demand is constantly realized over a time period, we have the following approximation: ( E 1 ) which can be spelled out as: ( E 2 ) When is far less than T When is close to T To facilitate the discussion, we denote the departure time as associated with a path The index of a link in is the same as that of its starting node. Take the network in Figure 4 1 as an 2, 2 24, 24 1. For a path we define links in that proceed a transition link as divergent links. Note that the definition of the divergent l ink is path dependent. For a path highlighted in Figure A 1, divergent links are 8 9 and 11 For vehicles at link that depart in period al ong path we denote a 0 1 variable that represents their availability and an auxiliary variable that represents the service/break time (ceiling valued to be consistent with th e modeling resolution). For since there is no customer demand. For is endogenously determined. Whenever a vehicle has been previously occupied bu t is unavailable PAGE 161 161 in the current period, (e.g., link 7 8 in Figure A 1). If a vehicle is available at a non divergent link, (e.g., links 6 7, 10 11). If at a divergent link, th e vehicle is assumed unable to provide service to ensure flow integrity but the corresponding is set to T (e.g., link 8 9). We also assume for on a feasible path (e.g., link 9 10 ). The specifications of help trace the availability of a vehicle. Without loss of generality, we assume We formally define the availability of a vehicle at a non divergent link as: ( E 3 ) where measures the cu mulative time from the departure hour. denotes the preceding link of on path of interval A vehicle is available at link a only when at a non divergent link is in return related to its availability: ( E 4 ) For a divergent link the issue of flow integrity is the dominant factor under consideration. The availability of a vehicle is explicitly determined by and i mplicitly affected by through Together, we have: ( E 5 ) ( E 6 ) The following equa tion delineates the relationship between link and path flows: PAGE 162 162 ( E 7 ) Unlike the previous two formulations, is endogenously determined via Eqs. ( E 3 ) ( E 6 ) and Eq. ( E 7 ) is thus non linear and non convex. E.2 Equilib rium definition and formulation For simplicity, drivers are assumed to behave as the neoclassical theory suggests, and thus the equilibrium conditions are the same as in Definition 1. When the availability of vehicles is explicitly traced, the average path profit is revised as: ( E 8 ) Similarly, the equilibrium condition c an be characterized by the following VI: (ME L) ( E 9 ) where However, the existence of a solution is not always guaranteed, as may not be continuous in due to the complexity of If exist, there may be multiple solut ions for path flows as strict monotonicity does not hold. A flow swapping algorithm can be adopted to solve the VI ( Huang and Lam, 2002 ; Lu et al., 2009 ) For a given path set, the algorithm consists of two loops. The inner loop aims to load the vehicles to the network and determines The outer loop adjusts the path flow from less profitable paths to those with maximal profit. E.3 Numerical Results To be consistent with our parameterization, we assume an average trip distance of 4.5 miles, and the period specific speed is given in Figure E 2. The corresponding minima l and maximal trip duration is 0.16 and 1.2 hours, respectively. PAGE 163 163 When the availability of driver s is explicitly traced, may not always be equal to The ceiling of service durations ( ) are two hours for periods 7:00 9:00 a.m. and one hour for the remaining periods. For a path used by a driver of Class 2 (O 1 2 3 4 5 6 7 8 D), she stops working after 8:00 a.m. and is not available on link 7 8. Generally, drivers are less li kely to be available on a divergent link or on a subsequent link of ones with long service duration. W e next compare the outcomes of the enhanced formulation (ME L) with the original one (ME N). We plot distributions of average searching tim es and waiting times in Figure E 3 and Figure E 4, respectively. A longer average searching time is found during peak hours (with longer trip time), because more drivers are attracted to those links so that they can enjoy a higher fare once occupied. The competition amon g the drivers raises the average searching time. Customers see a similar tendency in terms of the average waiting time yet at a smaller magnitude, which mainly results from the longer time for the vehicle to come and pick up the customer as the average spe ed d rops Despite a similar daily pattern ME N tends to overestimate average searching time and underestimate average waiting time, particularly at peak hours. This is because the available vehicle hours in ME N are generally larger than those in MN L due to Assumption 2. PAGE 164 164 Figure E 1. Display of A Sample Work Schedule with Corresponding Modeling Components. PAGE 165 165 Figure E 2. Distribution of Average Speed. PAGE 166 166 Figure E 3. Average Searching Time and Service Duration with Time Varying Trip Durations. PAGE 167 167 Figure E 4. Average Waiting Time with Time Varying Trip Durations. PAGE 168 168 APPENDIX F NOTATIONS Table F 1. Summary of Parameters. Parameter Description Value Area of the ride souring market ( mile 2 ) 300 Val $/hr ) See Chapter 4 vehicle travel time ( $/hr ) See Chapter 4 Demand sensitivity of generalized cost See Chapter 4 Link specific cost of driver class m ( $/hr ) See Chapter 4 Path dependent cost for a driver of of class m on path p ($) See Chapter 4 Total cost incurred by a driver of class m choosing path p ($) See Chap ter 4 Average trip time (hr) 0.3 Average speed of vehicles ( mile/hr ) 15 Unit cost of cumulative working hours of driver class m See Chapter 4 L evel of aversion to cumulative work hours of driver class m 2 Fleet size of driver class m (veh) 2000 Flag drop fee per trip ( $/trip ) 2 Time based charge per hour ( $/hr ) 40 Proportion charged by the platform per completed trip 20% Time dependent hourly base demand ( trip/hr ) See Chapter 4 Income target for driver class m ($) See Chapter 4 Constant in the reference dependent utility ($) 500 Degree of loss aversion for driver class m 0.2 Link path incidence element See Chapter 4 Cumul ative working hours of driver class m on path p (hr) See Chapter 4 Node link incidence matrix See Chapter 4 Column vector in shortest path finding See Chapter 4 Reservati on profit levels for driver class m ($) See Chapter 4 Initial tolerance used for solving bi level problems 1 PAGE 169 169 APPENDIX G EXISTENCE OF EQUILIBRIUM orem ( De la Fuente, 2000 ) We first construct a convex and closed set for the vector of demand Then we will argue there exists a continuous function mapping from to itself. We define and wh en A valid set for demand is defined as which is clearly closed and convex. Given Eqs. ( 5 20 ) ( 5 22 ) imply that Using Eq. ( 5 22 ) again, we solve for Clearly, and are continuous in is guaranteed to be greater than a positive c onstant if where Eq. ( 5 15 ) implies a possible way of choosing is via T here may be an identification problem when we inversely solve for given A graphic illus tration is presented in Figure G 1 where may intercept the curve twice (Point A, B in Figure G 1). In this case, we choose the right point (Point B) to make sure the continuity condition holds. Subsequently, we obtain via Eq. ( 5 17 ) and Eq. ( 5 18 ) re spectively. As shown in Figure G 2, is continuously decreasing in is then the sum of and both continuous in PAGE 170 170 Finally, we define the mappi ng: = where It is straightforward to see is continuous. G iven the way we chose and k together w ith the condition that we claim the nonlinear system admits at least one solution. PAGE 171 171 Figure G 1. Average Searching Time and Its Matching Time Portion. PAGE 172 172 Figure G 2. Matching Frictions. PAGE 173 173 APPENDIX H OVERVIEW OF THE SIMULA TION PACKAGE This section gives an overview of the simulation interface. The gener al outlook is presented F igure H 1 It contains 9 parts as been labeled. The functionality of each part is explained subsequently. 1. Setup /Go Setup: Restores the simulation to its initial state. Go once: Moves the simulation forward 1 time step. Go: Runs the simulation continuously when first pressed; stops the simulation when pressed again. 2. Sliders number vehicles: Adjusts the total number of vehicles to be used in the simulat ion. This number remains fixed throughout the simulation. communication range: Adjusts the matching radius over which customers can summon vehicles. 3. Simulation Space As discussed in Section 6 .1. 4. Simulation Summary Assemble the evolution of all the variable s of interest at the network level. At equilibrium, each plot remains approximately constant. 5. Revenue and Operating Cost Plots the average revenue gained by each vehicle (per time step) and the operating cost that each vehicle incurs (per time step), which guarantees the same reservation profit at the equilibrium. 6. Waiting Customers and Vacant Vehicles by Zone Plots the number of waiting customers and vacant vehicles in each zone. At the equilibrium, each plot remains approximately constant. 7. Average Search Time by Zone PAGE 174 174 Plots the average searching time for vehicles in each zone. At the equilibrium, each plot remains approximately constant. 8. Demand by Zone Plots the realized demand of customers in each zone. At the equilibrium, each plot remains approximately c onstant. 9. Average Waiting Time by Zone Plots the average waiting time for vehicles in each zone. At the equilibrium, each plot remains approximately constant. 10. Equilibrium check As discussed in Section 6.3.1 PAGE 175 175 Figure H 1. 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Transportation Research Part A: Policy and Practice 39, 17 40. Yang, T., Yang, H., Wong, S.C., Sze, N.N., 2014. Returns to Scale in The Production of Taxi Services: An Empirical Analysis. Transportmetrica A: Transport Sc ience 10, 775 790. PAGE 182 182 BIOGRAPHICAL SKETCH Liteng Zha obtained his B achelor of Science ( 2012), M aster of Science ( 2014) and Ph.D. ( 2017) degrees respectively from the Southeast University, Texas A&M University and the University of Florida, all majored in transportation engineering. Liteng s pri mary research interest is in the applications of the emerging technologies in the field of transportation. His work covers the safety applications of connected vehicle technologies at the signalized intersection s as well as the economic analysis of recent booming of the ride sourcing markets. Liteng won the System Optimum Award offered by the University of Florida Transportation Institute (UFTI) in 2015 Advancing Safety Performance Monitoring a t Signalized Intersections Using Connected Vehicle Technology was selected as the Young Researcher Award for the Safety Data, Analysis and Evaluation Committee at the 93rd Transportation Research Annual Meeting (TRB, 2014) He was the recipient of the pre stigious Eisenhower Fellowship from the Federal Highway Administration (FHWA) in 2013 |