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- Permanent Link:
- https://ufdc.ufl.edu/UFE0049573/00001
## Material Information- Title:
- Analysis and Design of Tradable Credit Schemes under Uncertainty
- Creator:
- Shirmohammadi, Nima
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2015
- Language:
- english
- Physical Description:
- 1 online resource (86 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:
- WASHBURN,SCOTT STUART
- Committee Members:
- SRINIVASAN,SIVARAMAKRISHNAN
ELEFTERIADOU,AGELIKI LAWPHONGPANICH,SIRIPHONG - Graduation Date:
- 12/18/2015
## Subjects- Subjects / Keywords:
- Credit control policy ( jstor )
Credit ratings ( jstor ) Market prices ( jstor ) Price ceilings ( jstor ) Prices ( jstor ) Pricing ( jstor ) Tolls ( jstor ) Transportation ( jstor ) Travel demand ( jstor ) Travelers ( jstor ) Civil and Coastal Engineering -- Dissertations, Academic -- UF congestion -- credit -- pricing -- scheme -- tradable - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Civil Engineering thesis, Ph.D.
## Notes- Abstract:
- Tradable mobility credit schemes have been recently received much attention inspired by its successful experience in pollution control programs. In a typical tradable credit scheme, the transportation authority defines a quantity based control target, issues and distributes mobility credits among eligible travelers to achieve the target. Travelers are subsequently charged certain number of credits to access the transportation facilities. The credits are also allowed to be traded among travelers in a market set up for this purpose, and the credit price emerges as a result of the interaction between network and market equilibrium. To investigate the potential of tradable credit in a more realistic setting, this dissertation investigates the efficiency of tradable credit under uncertainty. To have a benchmark to evaluate the performance of tradable credit schemes, their efficiency are compared with congestion pricing. In a deterministic setting, we demonstrate that there is a one by one correspondence between congestion pricing and tradable credit to achieve the desired control target. However, we show numerically when there is uncertainty associated with demand side and(or) supply side of transportation system, the identity between two schemes falls apart. Essentially, tradable credit, as a quantity based instrument, preserves the quantity target but leaves the price to fluctuate. In contrast, congestion pricing, as a price based instrument, cannot guarantee achieving to the quantity target, because the toll cannot be adjusted automatically. We demonstrate that the volatility of credit price might undermine the public acceptability of tradable credit schemes, because the credit price might be too high in some cases. Hence, a safety-valve policy is proposed to impose a ceiling on the credit price. We show how the price ceiling can be designed appropriately to provide a trade off between public acceptability and success rate in achieving the control target. We incorporate the impact of travelers heterogeneity in the analysis and design of tradable credit scheme under uncertainty, by focusing on a specific tradable credit scheme that targets the temporal distribution of travel demands. We also investigate the challenges associated with implementation of tradable credits in real-world. ( 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, 2015.
- Local:
- Adviser: YIN,YAFENG.
- Local:
- Co-adviser: WASHBURN,SCOTT STUART.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2016-06-30
- Statement of Responsibility:
- by Nima Shirmohammadi.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Applicable rights reserved.
- Embargo Date:
- 6/30/2016
- Classification:
- LD1780 2015 ( lcc )
## UFDC Membership |

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PAGE 1 ANALYSIS AND DESIGN OF TRADABLE CREDIT SCHEMES UNDER UNCERTAINTY By NIMA SHIRMOHAMMADI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2015 PAGE 2 Â© 2015 Nima Shirmohammadi PAGE 3 To my parents PAGE 4 4 ACKNOWLEDGMENTS There simply are not wor d to convey the gratitude I feel for the guidance and support I rec eived from Dr. Yafeng Yin, who has given me a vast degree of encouragement through my Ph . D . studies at University of Florida. He has been beyond what is required of an advisor and made a huge difference in my knowledge and experience in both professional and personal life. I would like to express my deepest appreciation to my committee members Dr. Lily Elefteriadou, Dr. Sirphong Lawphongpanich, Dr. Srinivasan Sivaramakrishnan, and Dr. Scott Washburn for their valuable time, helpful suggestion s and great suppor t on my dissertation. My special thanks are given to all my friends in University of Florida Transportation Inistitue, especially Mahmood Zangui, Roosbeh Nowrouzian, Seckin Ozkul, Zhibin chen, and Danial Davarnia for making my life at UF happier through th ese years Finally, I would like to thanks my parent s , Fereshteh and Naser , for their endless support and encouragement. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 4 LIST OF TABLES ................................ ................................ ................................ ........................... 7 LIST OF FIGURES ................................ ................................ ................................ ......................... 8 ABSTRACT ................................ ................................ ................................ ................................ ..... 9 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .................. 11 1.1 Background ................................ ................................ ................................ ....................... 11 1.1 Dissertatio n Objectives ................................ ................................ ................................ ..... 15 1.2 Dissertation Outline ................................ ................................ ................................ .......... 15 2 LITERATURE REVIEW ................................ ................................ ................................ ....... 16 2.1 Methodological Extensions ................................ ................................ .............................. 17 2.2 Operational Extensions ................................ ................................ ................................ ..... 18 2.3 Others Applications ................................ ................................ ................................ .......... 20 3 IDENTITY OF CONGESTION PRICING AND TRADABLE CREDIT IN PERFECT CERTAINTY ................................ ................................ ................................ .......................... 22 3.1 Identity in Spatially Aimed Control Targets ................................ ................................ .... 22 3.1.1 Notation ................................ ................................ ................................ .................. 23 3.1.2 VMT as Control Target ................................ ................................ .......................... 24 3.1.3 Traffic Demand into an Area as Control Target ................................ ..................... 26 3 .1.4 Link Level Congestion as Control Target ................................ .............................. 28 3.2 Identity in Temporally Aimed Control Targets ................................ ................................ 31 3.2.1 Bottleneck Model ................................ ................................ ................................ ... 31 3.2.2 Zero Queue Length as Control Target ................................ ................................ .... 34 3.2.3 Maximum Queue Length as Control Target ................................ ........................... 36 3.3 Summary ................................ ................................ ................................ ........................... 41 4 CONGESTION PRICING VS. TRADABLE CREDIT UNDER UNCERTAINTY ............. 42 4.1 Spatially Aimed Quantity Control ................................ ................................ .................... 42 4.1.1 Monte Carlo Simulation ................................ ................................ ......................... 43 4.1.2 Sensitivity Analysis of Market and Network Equilibrium ................................ ..... 45 4.1.3 Design of Credit Schemes under Uncertainty ................................ ........................ 49 4.2 Temporally Aimed Quantity Targets ................................ ................................ ............... 55 4.2.1 Multi Step Toll Scheme under Demand Uncertainty ................................ ............. 55 PAGE 6 6 4.2.2 Tradable Credit Scheme under Demand Uncertainty ................................ ............. 58 4.2.3 Tradable Credit Scheme with Price Ceiling ................................ ........................... 59 4.2.4 Monte Carlo Simulation ................................ ................................ ......................... 60 4.3 Summary ................................ ................................ ................................ ........................... 63 5 EVALUATION OF TRADABLE CREDIT SCHEME WITH USER HETEROGENEITY ................................ ................................ ................................ ............... 64 5.1 Analysis of Proportional Heterogeneity ................................ ................................ ........... 64 5.2 Analysis of Heterogeneity ................................ ................................ ............................. 66 5.3 Design of Tradable Credit Scheme under Heterogeneity ................................ ............. 69 5.4 Analysis of the Modified Tradable Credit under Demand Fluctuation ............................ 71 5.5 Summary ................................ ................................ ................................ ........................... 73 6 CONCLUSIONS ................................ ................................ ................................ .................... 74 6.1 Findings ................................ ................................ ................................ ............................ 74 6.2 Discussion on Real World Implementation ................................ ................................ ...... 77 6.2.1 Ini tial Distribution of Credits ................................ ................................ ................. 77 6.2.2 Credit Charging ................................ ................................ ................................ ...... 79 6.2.3 Credit Market ................................ ................................ ................................ .......... 79 6.3 Concluding Remarks ................................ ................................ ................................ ........ 81 LIST OF REFERENCES ................................ ................................ ................................ ............... 82 BIOGRAP HICAL SKETCH ................................ ................................ ................................ ......... 86 PAGE 7 7 LIST OF TABLES Table page 3 1 Link parameters for the six node network ................................ ................................ ........ 30 4 1 Simulation with sample size of 1000 ................................ ................................ ................ 44 4 2 Iterations of the bi section procedure. ................................ ................................ .............. 55 4 3 Average (standard deviation) of prices in the proposed schemes ................................ ..... 62 PAGE 8 8 LIST OF FIGURES Figure page 3 1 A six node network ................................ ................................ ................................ ............ 29 3 2 Equilibrium arrival and departure pattern without any regulation ................................ ..... 34 3 3 Equilibrium queue pattern without any regulation ................................ ............................ 34 3 4 The charging scheme and marginal cost toll (zero queue length as control target) ........... 36 3 5 Equilibrium departure and arrival patterns (zero queue length as control target) ............. 36 3 6 Schematic queuing pattern of the proposed multi step toll for the design demand ........... 39 4 1 Realized VMT under two schemes. ................................ ................................ ................... 44 4 2 Volatility of credit price. ................................ ................................ ................................ .... 45 4 3 VMT and credit price under a safety valve policy. ................................ ........................... 52 4 4 Cumulative density function of VMT with respect to price. ................................ ............. 53 4 5 Schematic queuing pattern of the proposed multi step toll for the realized demand ......... 56 4 6 Volatility of the credit prices under demand uncertainty ................................ ................... 62 4 7 Volatility of the credit price and queue length under hybrid scheme ................................ 63 PAGE 9 9 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ANALYSIS AND DESIGN OF TRADABLE CREDIT SCHEMES UNDER UNCERTAINTY By Nima Shirmohammadi December 2015 Chair: Yafeng Yin Major: Civil Engineering Tradable mobility credit scheme ha s recently received much attention inspired by its successful implementation in pollution control programs. In a typical tradab le credit scheme, the transportation authority de termine s a quantity based control target . To achieve the target , mobility credits are issued and distributed among eligible travelers. Travelers are subsequently charged a certain number of credits to acces s the transportation facilities. Additionally, the credits are allowed to be traded among travelers in a market established for this purpose and the credit price emerges as a result of the interaction between network and market equilibrium s . To investigate the potential of tradable credit scheme in a more realistic setting, this dissertation investigates the ir efficiency under uncertainty. As a benchmark to evaluate the performance of tradable credit scheme, its efficiency is compared with congestion pricin g. In a deterministic setting, we demonstrate that there is a one to one correspondence between congestion pricing and tradable credit scheme in achiev ing a desired control target. However, as shown numerically , the two schemes no longer mirror each other when there is uncertainty associated with the demand side and/or supply side of the transportation system . As a quantity based instrument, tradable credit scheme pre serves the quantity target but allows the price to fluctuate. I n contrast, congestion prici ng as a price based instrument cannot guarantee achieving PAGE 10 10 the quantity target because the toll cannot be adjusted in response to revealed scenarios . T he volatility of credit price might undermine public acceptab ility of tradable credit scheme because the c redit price might be too high in some cases. Hence, a safety valve policy is proposed that impose s a ceiling on the credit price. It has been shown how the price ceiling can be designed appropriately to provide a trade off between public acceptability and success rate in achieving the control target. Finally, w on analysis and design of tradable credit scheme under uncertainty by focusing on a specific tradable credit scheme aiming the tempora l distributio n of travel demand . PAGE 11 11 CHAPTER 1 INTRODUCTION 1.1 Background Traffic congestion as one of the major c hallenges in metropolitan areas imposes monetary and non monetary cost s on society , threatening quality of life and economic prosperity . According to the Texas Transpo rtation Insti tute U rban M obility R eport (Schrank et al. 2012) , total travel delay in U.S. is estimated to be 5.5 billion hours, which imposes 121 billion dollars cost on the n ation. Traditionally , the solution to the rising level of congestion has been add ing new capacities to existing ones . Yet, such an approach is increasingly subject to spatial and financial constraints. More importantly, as new capacities usually induce new demands, the congestion relief brought by them is often short lived (Duranton an d Turner 2011) . Consequently, market based instruments have gained acknowledgment as more effective and cost efficient solutions to the congestion problem . Market based instruments for congestion mitigation are generally classified into two classes, i.e., price and quantity based. The former, widely known as congestion pricing, has been advocated by transportation economists since the seminal work by Pigou (1920). The idea behind the congestion pricing is to charge travelers the marginal external costs th at their trips impose to the society to reduce traffic congestion or increase social welfare. Fo r recent reviews on methods and technologies for congestion pricing, see, e.g., Yang and Huang (2005), Tsekeris and Voss (2009), de Palma and Lindsey (2011) . C ongestion pricing has been implemented across the world in variety of forms, such as cordon pricing in Stockholm, London, Singapore, and high occupancy vehicle/toll lanes implementation across US. However, the penetration of congestion pricing looks relati vely low , despite its theoretically proven potential in tackling PAGE 12 12 congestion. Indeed, congestion pricing has deficiencies act ing as barriers again st its extensive implementation. Optimal design of congestion pricing such that meet s the objective s defined b y the transportation authority ( here after referred to as authority prior knowledge of the characteristics of demand side and supply side of the transportation system. More specifically, to evaluate the consequences of a congestion pricing sche me the analyst should have full knowledge of the service function of the infrastructures . Among them, access to a reasonable demand function is the greatest challenging (Yang et al 2004). Congestion pricing sets a price on a serv ice that used to be free (Small 1992). Hence, the public may perceive it as another fo rm of taxation on their travel in addition to the gas tax. The opposition from the public causes political resistance aga inst the implementation of congestion pricing. There are concerns regarding the distributional effect of congestion pricing. The general perception among the public is that congestion pricing favor s high income people, and forces low income travelers to war d l ess desirable travel options (Jaensirisak et al. 2005) . While numerous studies are conducted to address the issues associated with congestion pricing, quantity based in struments are drawing attention . In contrast to price based instruments that use pric e to influence travel choices , quantity based instruments control the quantity target directly. Different quantity based mechanisms hav e been proposed thus far. Among them, the tradable credit scheme has recently received much attention. The concept of tr adable credit was initially proposed by Dales (19 68 ) as a mechanism to control water pollution. Since then, it has PAGE 13 13 been tested and implemented in several pollution control program s such as the Acid Rain Program to control the total SO 2 emission in US, NO x Trading P rogram to regulate NO x emission (Tietenbe g 2005) . Inspired by its successful implementation in pollution c ontrol, tradable credit is considered a potential solution for congestion mitigation . Verhoe f et al. (1997) identified several potential appl ications of tradable credit in regulation of transportation externality . The first theoretical attempt in this direction is conducted by Yang and Wang (201 1 ) who proposed a mathematical framework to study tradable credit by capturing the market and network equilibrium simultaneously. A typical tradable mobility credit scheme designed to mitigate congestion can be described as follows: The authority , as the responsible organization to manage mobility across the network, determines a quantity based cont r ol ta rget that guarantees a desired level of service. To achieve the control target, mobility credits are issued by the authority and distributed among eligible travelers . Each traveler wish ing to access transportation facilit ies is required to pay a certain nu mber of credits . Moreover, a virtual market will be created in which the credits can be traded among travelers through free market trading, and the credit price will subsequently emerge . If the initial allocation is not enough to fulfill the ir credit needs , travelers can purchase credits from other travelers in possessing surplus credits . Conceptually, t ravelers who place more value on using the facility fulfill their travel needs by purchasing credit s from those who value less . Some promising features of t radable credit are discussed below : In tradable credit schemes identify eligible travelers and distribute the credits among them. The schemes are essentially revenue neutral, and no wealth is transferred from the public to the authority. Therefore, it is expected to receive less opposition from the public and subsequently politicians. PAGE 14 14 The travelers displaced to less desirable modes, routes or departure times can sell their unused credits to oth er travelers, and receive compensation . The initial allocation of credits among the travelers can be adjusted by the authority to implementation of the scheme. However, simi lar to any other instrument, tradable credit has its own drawbacks and technical issues which need to be addressed before any real implementation . Below are some among others : The initial allocation of credits can be a challenge for the authority. How are the eligible travelers identified? More importantly, how many credits should be assigned to each eligible traveler? Even though the initial allocation might not alter the efficienc y of the instrument (Montgomery e cannot be ignored. Transaction cost associated with credit trading might suppress the efficiency of the scheme. With a higher transaction cost, the concern s regarding the initial allocation would be even more. Mobility credits are expected to be issued p eriodically. An important issue is then whether the credits are allowed to be transferred to the next periods. The transferability allowance will raise questions regarding the transferability rules, and potential banking and borrowing of credits and their effects on the efficiency of the scheme. Travelers might not be aware of the exact number of credits they need beforehand, which implies the risk in their decision reflected in analysis and design of tradable credit scheme . Uncertainty in both travel demand and supply can deteriorate the efficiency of tradable credit schemes. Hence, a robust design that enables the scheme to handle the unforeseen fluctuations is of great importance. The above mentione d issues have been investigated in the existing literatures. Among them, the impact of uncertainty on the efficiency of tradable cr edit is of particular interest. To the best of our knowledge, no major research is conducted to explore the consequences of d emand side or supply side uncertainty on the capability of the tradable credit scheme to manage congestion. This dissertation is specifically aimed to fill the gap in the literatures. We will show how recurring uncertainties associated with travel demand a nd supply affect the efficiency of PAGE 15 15 tradable credit scheme. More specifically, we compare the robustness of tradable credit to achie ve the design target with congestion pricing, and demonstra te how the outcomes of these two schemes differ under uncertainty. By highlighting negative impacts of uncertainty on the efficiency of tradable credit scheme , we will discuss how the design of tradable credit scheme should be modified to suppress unfavorable outcomes. In addition, we demonstrate how eity will affect the performance of tradable credit scheme under uncertainty, and how it should be designed in response. 1.1 D issertation O bjectives This dissertation is aimed to study the efficiency of tradable credit scheme under recurring uncertainty. We ar e going to achieve the following objectives in this dissertation: E xamine tradable credit schemes under a more realistic setting E xplore the pros and cons of tradable credit schemes, in the light of comparison with congestion pricing Identify the circumst ances to admit tradable credit as a worthy instrument Suggest appropriate designs of tradable credit under uncertainty 1.2 Dissertation O utline Chapter 2 , reviews the relevant literature of tradable credit. Chapter 3 formally establishes one by one correspon dence between tradable credit and congestion pricing in a deterministic setting. Chapter 4 investigates how tradable credit and congestion pricing fall apart when uncertainty is introduced into the problem , and how to design tradable credit schemes under u tradable credits, and how the design of tradable credit should be adjusted in response. Chapter 6 summarizes and concludes the dissertation. PAGE 16 16 CHAPTER 2 LITERATURE REVIEW This chapter provides a brief review of relevant researches on the application of the tradable credit scheme in congestion mitigation. The idea of tradable credit scheme has be en essentially borrowed from pollution control. Firstly , Dales (19 68 ) proposed the co ncept of tradable credit scheme under terminology of transferable rights to maintain the quality of water resources. Later, Montgomery (1972) proved that tradable credit schem e is the most cost efficient mechanism to achieve a pollution control target. Tr adable credit scheme is a relatively novel proposal i n congestion mitigation context and most of its related studies are relatively recent. The potential of tradable credit scheme in managing traffic congestion is firstly realized by Verhoef et al. (1997). They identified several p otential applications of tradable credit in managing vehicle ownership, driving day, vehicle miles, and fuel consumption ( Verhoef et al. 1997). The first theoretical attempt to analyze tradable credit scheme was conducted by Yang and Wang (201 1 ) who proposed a mathematical framework which is able to simultaneously capture the network and market equilibrium. In their proposed framework , the charging scheme is at link level, i.e. travelers need to pay certain number of credit s to be allowed to access a specific link. Yang and Wang (201 1 ) showed the link level charging scheme can be designed to decentralize the system optimum flow distribution. Yang and Wang (201 1 ) that each focused on a specific aspect of tradable credit scheme . Generally speaking, the extension s to Yang and Wang (201 1 ) can be fallen in to three categorie s . The first category , referred as the met hodological extensions, include studies that are enhancing the framework to analyze tradable credit scheme under more realistic assumption s . The s econd category, referred PAGE 17 17 as operational extensio ns, address es the concerns associated with implementation of tradable credit scheme . The studi es falling to the third categor y explor e other potential application s of tradable credit scheme in transportation management. In the following, each of these categorie s will be reviewed in separate sections. 2.1 Methodological E xtensio ns Wang et al. (2012) lifted the assumption of homogenou s travelers, by allowing discrete heterogeneity in value of travel time. Compared to the case with homogenous travelers, they argued that tradable credit scheme provides more appealing distributional effec t. The heterogeneity also increase s the wealth tran sfer from high income groups to low income groups. The resulting link flow distribution and credit price , however, are not necessarily unique under Nonetheless, Wang et al. (2012) provide the sufficient conditions for designing tr adable credit scheme which guarantee s the uniqueness of link flows and credit price. In the same line, Zhu et al. (2015) extend the methodology to capture the continuous heterogeneity in value of travel time. Yang and Wang (2011 based on the assumption that travelers are infinitesimal players. I n reality, however, there are Cournot Nash (CN) players such as logistic com panies who manage their fleet to minimize their overall travel costs. The existence of CN pla yers would change t he network equilibrium to mixed network equilibrium . He et al. (2012) studied the analysis and design of tradable credit scheme under mixed equilibrium behavior. I n the absence of transaction costs, t hey designed a tradable credit scheme with anonymous cha rging scheme that decentralizes the system optimum flow distribution. Another assumption in Yang and Wang (201 1 neutral. In other words, for a given number of credits being traded the magnitude of utility that trave ler s gain by selling their credits is equal t he magnitude of utility that the y lose by PAGE 18 18 purchasing the same number of credits . i with respect to g aining and losing are different . Re alizing this issue, Bao et al. Dynamic evolution of network flows and credit price are investigated by Ye and Yang (2013) under finite and infinite time horizon. In their model, travelers cont inuously adjust their price. The evolution of credit price at any moment depends on the current credit price and the remain ing credit supply. They formalize d the necessary and sufficient conditions to find the equilibrium state. 2.2 Operational Extensions As we discussed in C hapter 1, one of the drawbacks of the marginal cost p ricing is its dependency to hav e prior knowledge on demand function that is subjected to practical and methodological constraints. The same issue exists in designing tradable credit scheme when the control target is to decentraliz e the system optimum flow distribution. More specifically, to determine the charging scheme , the authority should h ave full knowledge of demand function. To overcome the drawback , Wang and Yang (2012 ) propose d a trial and error implementation of tradable credit scheme for a single link with unknown demand function. In their method ology , the auth ority would periodically adjust the charge rate on the link and the total number credits, through continuous monito r ing of the link flow and credit price. They suggested a bisection based algorithm to stabilize the observed link flow on the unknown system optimum. Subsequently, W ang et al. (2014) modified the trial and error algorithm to derive the unknown system optimum flow distribution for a general network . In th eir case , there is an unknown demand function associated with each origin destination. The authority can adjust t he link specific credit charge s and the total number of cre dits based on the signals received from the PAGE 19 19 observed link flows and the credit price . Realizing non convergence nature of bisection method, Wang et al. (2014) used method of successive average (MSA) t o update the design variables. Ano ther concern associated with implementation of tradable credit scheme is the impact of transaction cost s . Considering the mobility credit s as the y, there are transaction cost s associated with credit tra ding similar to transferring of any other properties . In fact, t o be involve d in the market traveler s should learn credit market rules , and know how many credits are needed to fulfill their travel need s . Either as a seller or buyer, the traveler s should fi nd and negotiate with a trading partner . To facilitate these processes , the authority might establish brokerage facilities and the transaction cost s can be paid in term of brokerage fee s . In addition, there might be some cost s associated with monitoring an d enforcement of the market ( S tavins 19 95). The negative impacts of t ransaction cost are twofold: First ly , the trading amount s are suppressed. Secondly, it imposes extra cost burden s directly on the travelers. Nie (2012 a ) studied the effect of transaction cos t on two types of credit market : auction and negotiated market. In both market s , the authority monitor s the market through selling extra credit. Nie (2012 a ) showed that if transaction cost is less than the credit price obtained in the absence of transa ction cost s , the efficiency of the tradable credit scheme would not be affected in the presence of transaction cost. In the case of negotiated market, however, the efficiency of tradable credit scheme would be deteriorated in t he essence of transaction cos t, independent to its magnitude. According to Nie (2012 a ), the degree of efficiency loss due to the transaction cost depends on the magnitude of transaction cost, the initial allocation, and the characteristics of transportation network . He et al. (2013) a lso analyzed tradable credit with mixed equilibrium in presence of transaction cost. Their finding confirms that transaction cost affects the trading volume. However, w hile the PAGE 20 20 concerns regarding the negative impacts of transaction cost on tradable credit efficiency is valid, high transaction costs are not expected due to the huge market size ( S tavins 19 95) . The other issue is how the mobility credits should be distributed, and more importantly, how the eligible travelers should be identified. To address t his issue, Nie (2012b) proposed a tradable credit scheme in which the mechanism for identifying the distribution are embedded in the scheme . The proposed scheme charges the travelers passing the bottleneck within the peak charging window uniformly , and rewards credit to the travelers who are avoid ing the bottleneck during the peak charging window. Nie and Yin (2013) extended ewarding to travelers who are shifting to alternative route or mode, in addition to those who are travel ing during off peak periods. Wu et al. (2012) studied the equity aspect of tradable credit scheme . A modeling framework is proposed to design a tradable credit scheme for a mul timodal transportation network that incorporate s income equ ity as well as social welfare. Compared to the optima l pricing design, the optimally designed tradable credit scheme is more progressive from the equity aspect, thanks to its flexibility in initial allocation of credits. In a hypothetical study, Mamun et al. (2014) compared the socioeconomic impacts of tradable credit scheme , gasoline tax and mileage fee. Compared to the other instrument s, tradable credit scheme achieves the control target with the least adverse effects on social welfare and has the best p osition from the equity perspective. 2.3 Others Applications Implementation of the tradable credit scheme can potentially promote transit ridership. Tian et al. (2013) studied the application of tradable credit scheme in a bi modal bottleneck model with travel ers heterogeneity in value of time. The scheme is essentially designed to clear all the queues behind the bottlenecks, while dictating the social optimum modal share. PAGE 21 21 Zhang et al. (2011) acquired tradable credit scheme as a mechanism to control congestion in a bi modal many to one network with parking capacity limitation. In their model, parking per mit distributed among travelers and travelers may purchase extra parking permits through trading. Their numerical example show ed that the efficiency of tradable parking permit is close to system optimum. PAGE 22 22 CHAPTER 3 IDENTITY OF CONGESTION PRICING AND TRADABLE CREDIT IN PERFECT CERTAINTY Economists have pointed out that the use of prices or quantities as management instruments would achieve the same level of efficiency in a n idealized envi ronment of perfect certainty (Weitzman 1974) . In this chapter, we show that there is a one to one correspondence between congestion pricing and tradable credit schemes in managing mobility for any specific control target . Traffic congestio n is a result of spatial and temporal concentration of travel demand. Managing travel demand through spreading in term s of time and space could potentially reduce the congestion le vel. The authority may define spatially or temporall y aimed quantity contro l targets corresponding to a desired level of service in the transportation system. A variety of control targets will be discussed through this chapter, and the corresponding tradable credit scheme and conge stion pricing will be designed by assuming a perf ect certainty in demand and supply side of the transportation system. The reminder of this chapter is organized as follows. Section 3.2 establishes the identity between tradable credit scheme and congestion pricing in achiev ing control target s through man aging the sp atial distribution of travel demand . Section 3.3 demonstrates such identity for control targets aim ing the temporal distribution of travel demand. 3.1 Identity in Spatially A imed Control Targets Different spatial distribution may be desired from th e transportation perspective and each of them can be expressed in term of specific quantities. One authority may be more concern ed on the total vehicle mileage traveled across the network and is looking for controlling it to not exceed a critic al threshold. The other one may be interested to manage the entering flow s into central business district (CBD) area. In a finer control target , the authority PAGE 23 23 might be look ing for controlling the spatial distribution of flows over every single link of the network. In this section, we will demo nstrate mathematically that each quantity based control target can be decentralized by an appropriately designed tradable credit scheme or congestion pricing. The analysis follows the mathematical fram ework proposed by Yang and Wang (201 1 ) . 3.1.1 Notation Let denote a road network, where and are the sets of nodes and links in the network , respectively. We denote a link as or the pair of its starting and ending nodes, i.e. . Let repres ent s the flow on link a and be a vector of link flows. Associated with each link, there is a travel time or link performance function, , which is continuous and monotonically increasing, and is a vector of these funct ions. In addition, denotes the length of link and > 0 for all . For travel demand, denotes the set of origin destination (OD) pairs and is the demand for OD pair . Associated with each OD pair, there is an inverse d emand function, , which is continuous and monotonically decreasing. Additionally, and are vectors of these demands and their inverse functions, respectively. (Hereinafter, the bold typeface indicates vecto rs of variables or To satisfy demands, we use to denote the set of all possible paths for OD pair . Then, represents the number of vehicles using path and is a vector of these path flows. Then, the set of all feasible flow demand triplets, , can be described as follows: where if arc is on path and otherwise. PAGE 24 24 Below we establish the identity between congestion pricing and tradable cre dit schemes for each specific regulation target. 3.1.2 VMT as C ontrol T arget Suppose that the traffic authority attempts to control the total vehicle mile travelled (VMT) across a network to be less than a threshold, say K . A tradable credits scheme will be simp ly to allocate K credits to all eligible travelers and then collect one credit for each mile traveled. Given such a credit scheme, the equilibrium conditions in the network and credit market can be written as: (3 1) (3 2) (3 3) (3 4) In ( 3 1), denotes the set of utilized paths with respect to . Si milarly, in ( 3 2) is the set of unutilized paths. In words, ( 3 1) and ( 3 2) state that, at equilibrium, all utilized paths have the same generalized travel cost that equals to the value of and the costs of those not utilized cannot be lower than . The generalized cost consists of two parts: travel time and the credit cost. The credit cost for using a particular link equals the length of the link, i .e., the number of credits consumed, , multiplied by the unit credit price p . To simplify the notations, the credit price in this paper is represented in the unit of time. Equations ( 3 3) and ( 3 4) represent PAGE 25 25 the credit market clearing conditions, wh ich stipulate that the equilibrium credit price is positive only when all the issued credits are consumed. The above equilibrium flow and price can be obtained by solving the following convex program, called as tradable credit problem (TCP): TCP (3 5) It is easy to verify that the first order optimality conditions of TCP are equivalent to (3 1) (3 4) with the marke t clearing credit price, p , being the Lagrangian multiplier associated with (3 5). With assumed properties of and , it is also easy to prove that the above program yields unique optimal link flows and realized demands. Proposition 3 1. The market clearing credit price determined from solving TCP is unique. Proof. Multiplying ( 3 1) by the corresponding path flow and summing it over all utilized paths for all OD pairs, and then applying the link flow definition condition, i.e., and the flow balance condition, i.e., , we have the following: Because is unique, the market clearing credit price, p , is unique. On the other hand, the target VMT can be also achieved by implementing a mileage fee that equals the above credit price p . This can be easily verified by solving the following tolled user eq uilibrium problem or TUE: PAGE 26 26 (3 6) 3.1.3 Traffic D emand into an A rea as C ontrol T arget We now suppose that the traffic authority wishe s to control the total number of vehicles into a certain area within a certain period to be less than a threshold, again, K , with a slight abuse of notation. Similarly, a credit scheme is to allocate K credits to travelers and collect one credit whenever a vehic le enters the area. Let be partitioned into two subsets, and , where the former contains links inside the control area and the later consists of those outside. By definition, and . Note that links in need not be connected. S imilarly, is divided into two subsets: and . The former consists of paths containing links in and using these paths requires paying one credit, even if the path enters the tolled area more than once. In general, paths in contain links in both and to connect the origin of OD pair to its destination. On the other hand, paths in contain no link in and are thus credit free. Given such a credit scheme, the equilibrium conditions can b e similarly written as: (3 7 ) (3 8 ) (3 9 ) (3 10 ) PAGE 27 27 (3 1 1 ) (3 1 2 ) The equilibrium flow distribution and credit price can be obtained by solving the following convex program: (3 1 3 ) The equivalence can be similarly established and the equilibrium credit price is the Lagrangian multiplier associated with (3 1 3 ). It is also easy to prove that the above program also yields unique optimal link flows and realized demands. Proposition 3 2. The credit price determined from solving the above program is unique. Proof. Multiplying both sides of ( 3 7 ) and ( 3 9 ) by and then summing over and for all OD pairs respectively, we have the following closed form expression for the credit price: When , , the uniqueness of will imply the uniqueness of the c redit price. Interpreting the Lagrangian multiplier as an entry fee, the target demand into the area can be achieved by implementing area based pricing. This can be easily verified by solving the following tolled user equilibrium problem or TUE PAGE 28 28 TUE: The above formulation is path based, which can be reformulated to be an equivalent link based formulation, although the travel cost is not link wise additive ( Lawphongpanich and Yin 20 12 ) 3.1.4 Link L evel Congestion as Control T arget Regulating congestion at the li nk level may be the finest level of control. It is assumed that the traffic authority attempts to control the number of vehicles travelling on each link of the network to replicate a particular flow distribution that, e.g., maximizes social welfare. Design of a tradable credits scheme to achieve this objective involves solving the following user equilibrium with side constraints: (3 1 4 ) where is the target flow at link a . In addition to the original feasibility conditions, the optimality conditions of the above program also include the following: (3 1 5 ) (3 1 6 ) (3 1 7 ) PAGE 29 29 (3 1 8 ) where is the Lagrangian multiplier associated with ( 3 1 4 ). From (3 1 5 ) (3 1 8 ), it is evident that a tradable credit scheme for controlling link level congestion can be constructed from the optimal solution of the above program. More specifically, the traffic authority needs to allocate to eligible travelers and then collect credits for traveling on link a. With such a scheme, the market clearing credit price is 1. If one scales up , the price will be driven down by the same scale. It can be proved that the total mar ket value of all cre dits is constant (Yang and Wang 201 1 ). In correspondence, a congestion pricing strategy is to directly charge at each link. Note that there may exist multiple pricing strategies that achieve the same co ntrol objective (Hearn & Ramana 1997). Each of them will have a tradable credit mirror. Below we use a numerical example to show the one to one correspondence between congestion pricing and tradable credit schemes in managing network mobility. Consider a network depicted in Figu re 3 1 with one OD pair between nodes 1 and 6, and the demand function is assumed to be , where is the equilibrium OD travel cost. The corresponding link parameters are presented in Table 3 1 and link travel time function is assumed to be the BPR function. Without any intervention, the total VM T of the network is 1859 and the demand into the central area (links 5 and 6) is 68.4. Figure 3 1. A six node network PAGE 30 30 Table 3 1 . Link p arameters for the s ix n ode n etwork Link ID Start E nd nodes Free F low T ravel T ime (min) Length (mi) Capacity 1 2 4 7 7 22.5 2 3 5 3 3 22.5 3 1 2 3 3 45 4 1 3 5 5 45 5 2 5 6 6 22.5 6 3 4 5 5 22.5 7 2 3 1 1 22.5 8 5 4 1 1 22.5 9 5 4 3 3 45 10 4 6 5 5 45 Below are three specific control targets that can be achieved by using either congestion pricing o r tradable credit scheme. a) Reducing VMT to be less than 1000 A tradable credit scheme allocates 1000 credits to all eligible travelers and collects one credit for each mile traveled. The market price for each credit is 1.503. The corresponding congestion pr icing implements a mileage fee of 1.503 and the resulting total VMT in the network will be 1000. b) Reducing the number of vehicles into the central area to be less than 50 A tradable credit scheme allocates 50 credits to all eligible travelers and collects o ne credit for each trip entering the central area. The market credit price is 6.055. The corresponding pricing scheme implements an area based charging of 6.055 and consequently total trips entering the central area will be 50. c) Maximizing social benefit A tradable credit scheme allocates 1523 credits to all eligible travelers and charges 5.002, 6.577, 4.739, 3.957, 4.995, 5.784, 0.018, 0.007, 4.745 and 3.951 credits respectively at links 1 to 10. The market credit price is 1. The corresponding pricing sche me charges the marginal cost PAGE 31 31 pricing toll at each link as follows: 5.002, 6.577, 4.739, 3.957, 4.995, 5.784, 0.018, 0.007, 4.745 and 3.951. 3.2 Identity in Temporally A imed Control Targets In the absence of any regulation, travelers choose their departure time such that their resulting arrival time to be as close as possible to their desired arrival time to their destination. On the other hand, workings activities in metropolitans are concentrated in CBD, and the working hours of employer s are almost the same. These imply that times during the morning peak are pretty similar. As a result, the demand to use transportation infrastructures would exceed their capacities , and therefore, queue formation and delay are inevitable. Market based instruments can also be acq uired to achieve more efficient temporal distribution of travel demand. More specifically, either quantity or price based instrument can be designed to suppress the queue lengths through spreading e s . Similar to the spatial distribution, the identity between congestion pricing and tradable credit in a dynamic setting is demonstrated . To do so, Vickery is used which is the simplest dynamic model to describe the queue for mati on and dissipation . Despite its simplicity, the bottleneck model is a powerful tool to study the implications of policies aimed to influence the tempora l distribution of travel demand . In the following, the bottleneck model will be introduced briefly , and it will be show n how the corresponding tradable credit and tolling schemes can be designed to achieve certain quantity targets expressed in term of queue length s . 3.2.1 Bottleneck M odel The morning commute problem was first introduced by Vickery (1969) to descr ibe the temporal distribution of morning commutes. Vickrey argued that, in addition to travel time, the PAGE 32 32 departure time choice . In fact, travelers who want to av oid traffic delay depart relatively early or late. On the other hand, travelers who arrive closer to the desired arrival time incur more travel delay. Smith (1984) and Daganzo (1985) proved, respectively, the existence and uniqueness of the equilibrium arr ival pattern for a single bottleneck. For recent comprehensive reviews on the morning commute probl em, see , e.g., Arnott et al. (1995 ) and de Palma and Fosgerau (2011) . In bottleneck model, we assume homogenous individuals are commuting every morning from their home (origin) to their workplaces located in the CBD (destination) . Without loss of generality, the free flow trav el time from home to work place and the work start time are assumed to be zer o. T he capacity of the highway connecting the origin to the destination is denoted by . Obviously, not all travelers can arrive at their workplace on the work start time when arrival rate exceeds the capacity . The generalized travel cost of a traveler wh o arrives at time follows the equation ( 3 1 9 ) (3 1 9 ) where is travel delay , is value of travel time, and are unit monetary cost of early and late arriv al, respectively . The arrival time of the first and last traveler are denoted by and , respectively . Travelers want to choose departure time s minimiz ing their own travel cost. Therefore, equilibrium can be defined as the state in which no tr aveler can reduce his/her travel cost by unilaterally changing his/her arrival time, which implies the travel cost is the same for all arrival time between and . Obviously, the bottleneck operates with full capacity during the congestion per iod, i.e. , from to , otherwise some travelers may better off by changing their PAGE 33 33 arrival time . From the equilibrium definition, we have , . Taking derivative from (3 1 9 ) and equalizing to zero yields to , (3 20 ) Moreover, q ueuing delay s associated with the first and last traveler are zero. Otherwise, they can reduce their travel time by departing a little bit earlier or later, respectively. Using (3 1 9 ), tr avel cost of the first and last travelers can be obtained as, (3 2 1 ) Additionally, because all travelers would arrive between and and bottleneck is fully utilized, we have: (3 2 2 ) Combining (3 2 1 ) and (3 2 2 ) is resulted in: (3 2 3 ) where . The equilibrium departure and arrival patterns are depicted in Figure 3 2 and the corresponding queuing pattern is p resented in Figure 3 3. Clearly, travelers who arrive closer to work start time and enjoy less schedule delay incur more queuing delay, and travelers who arrive farther away incur less queuing delay. PAGE 34 34 Figure 3 2. Equilibrium arrival and departure pattern without any regulation Figure 3 3. Equilibrium queue pattern without any regulation an economic loss and the queue length acts as a measure of its severity. To remedy the loss, the authori ty may define a variety of control targets. Among them, we focus on two of them: the ideal g oal of clearing all the queues and the more practical target of limiting the maximum queue length. It will be discuss ed how tradable credit and congestion pricing c an be designed to achieve these targets. 3.2.2 Zero Queue Length as Control Target The most ideal target which requires the finest level of market intervention is remov ing all the queue s behind the bottleneck. To achieve this target using tradable credit scheme , credits are distributed among travelers and a dynamic charging scheme would be implemented according to Figure 3 4 . Note that the start and end of congestion period are matched with and Cumulative # of travelers Arrival time Queuing delay Arrival time PAGE 35 35 of the case without any policy int ervention. The charging amount increases with rate of up to and then decreases with rate of . Hence, the traveler who arrives on time would be charged the most. In fact, the dynamic charging scheme w ould replace the queuing delay cost with the credit cost . It can be easily shown that the equilibrium credit price is one by comparing the travel cost of the first traveler and the travelers who arrives on time . T he proposed tradable credit scheme decentralizes the s ystem optimum departure pattern, b ecause all the queues would be removed and the resulting congestion period minimizes total schedule cost The corresponding tolling scheme that removes all the queues and decentral izes the system optimum is known in the literature as marginal cost or fine t oll (Arnott et al 1990), and follows the pattern of Figure 3 4. Under the marginal cost toll ing scheme , each traveler would pay the difference between his/her marginal cost imposed on the society and his/her own private cost . Hence , the queuing delay cost will be replaced with the toll, and the d eparture and arrival patterns would be as depicted in Figure 3 5. Dealing with zero queue length as the control target would require implementing a continuously variable credit charging or tolling schemes which are expected to be un pleasant for the travelers , and therefore, their implementation are subjected to practical restraints (Arnott et al 1990). In S ection 3.3.3 , we will study how to control the maximum queue length behind the bottleneck as the single target . PAGE 36 36 Figure 3 4. The charging scheme and marginal cost toll (zero queue length as control target) Figure 3 5. Equilibrium departure and arrival patterns (zero queue length as control target) 3.2.3 Maximum Q ueue L ength as C ontrol T arget Recently, Liu et al. (201 5) proposed a reservation system to manage bottleneck capacity. In their proposal, the traffic authority sets several time points around a reference time, e.g., work start time, and travelers then either choose a time point before the reference time and sh ould arrive before it or choose a time point after the reference point and arrive after it. To control the queue length, a limited number of travelers are allowed to choose a time point in the first come first serve manner. Liu et al. (2015) showed that by increasing the number of arrival time points, the system can reach a higher efficiency. Ultimately, when the number of time points increases to infinity, each arrival time is essentially reserved for one traveler and thus queuing will be eliminated. The p roposed reservation system resembles the tradable permit scheme proposed by Akamatsu et al. (2006) , in which the authority issues tradable permits specific to each arrival Number of Charge credits Arrival time Cumulative # of travelers Arrival time PAGE 37 37 time and any traveler who wishes to pass the bottleneck should obtain the specific p ermit associated with his/her arrival time. The scheme can be considered as a quantity based analogy However, despite theoretically appealing, the scheme is not very practical as the permits are time specific and change continuously . In the same spirit as Akamatsu et al. (2006) and Liu et al. (2015) , a multi step toll scheme will be designed to maintain the maximum queue length behind the bottleneck below a pre specified threshold, denoted by , for a maximum travel demand, denoted by . While designing based on the maximum travel demand is conservative and may lead to inefficient use of the bottleneck capacity, however, it can guarantee meeting the queue length requirement for all realize d travel demand. Let assume the number of charging windows or periods for early and late arrival are and , respectively. For early arrival, the th charging period starts at and ends at , where . For late a rrival, the th charging period starts at and ends at , where . The t oll level associated with the th early / late charging period is denoted by / . Among different possible design scheme s that can meet t he design target , we require the design to be such that the queue at the start of each early tolling period and at the end of each late tolling period be zero. This would lead us to a minimum number of toll steps . In addition, the design would be such that the first and last travelers do not need to pay toll. Because the first and last travelers pass the bottleneck free of charge, the congestion period, i.e. , , is similar to the basic model . In early arrival, the queue length with respect to arrival time increases with the rate of within each toll period. Therefore, the maximum queue length of toll period is associated with the travel er who arrives on . By imposing , we have: PAGE 38 38 ( 3 2 4 ) implies that ( 3 2 5 ) Defining , where , will be determined as ( 3 2 6 ) In late arrival, the queue length decreases with the rate of as we deviate from the work start time. Therefore, the maximum queue length of c harging period occurs at the start of the period, i.e. , . Again , by imposing , we have: ( 3 2 7 ) By setting one can obtain that ( 3 2 8 ) Denoting , will be determined as ( 3 2 9 ) Comparing ( 3 2 6 ) and ( 3 2 9 ), it is clear that the number of toll period s for early and late arrival is the same. The toll level should be determined following the definition of the departure time equilibrium. The equilibrium travel cost is . Hence, the toll level can be determined as: PAGE 39 39 ( 3 30 ) ( 3 3 1 ) Since , the toll level s at th early and late toll period are the same, and one can combine them and refer th e combined one as th toll period. Therefore, the multi step toll scheme can be characterized by toll window s, in which the th period imposes within . We a lso consider the free of charge periods as ( ) th period and set and . The corresponding queuing pattern is depicted in Figure 3 6 . Consistent with earlier studies on step toll s (Lindsey et al. 2012) , two types of discontinu ity can be identified in F igure 3 6 , immediately befo re and after each toll period . One type of discontinuity happens at , when the arrival time changes from period to period . Because the last traveler arriving within period +1 has the same arrival time as the first traveler arriving within period , and pays a lower toll at the same time, the equilibrium conditions require the traveler experiencing more queuing delay. Therefore, there should not be any departure for a period of time af ter her/his departure. Figure 3 6 . Schematic queuing pattern of the proposed multi step toll for the design demand The other type of discontinuity is associated with . Similarly, the first traveler who arrives within period +1 enjoys a lower t oll rate but arrives at the same time as the last traveler Queue length PAGE 40 40 arriving within period , and thus should experience more queuing delay. Liah (1994) justified the discontinuity by assuming the formation of an imaginary queue behind the bottleneck parallel to t he main queue. According to Liah model, referred as the separated waiting model, some travelers attempt to avoid higher toll and are competing for the arrival spot immediately after . Hence, the first traveler who arrives after is the first traveler in the separated queue and therefore experiences more queuing delays compared to the first traveler arriving within the previous charging period. In the corresponding tradable credit scheme, the authority determines multiple charging period s and issues a specific type of credits for each charging period. Then, all the credits are distributed among eligible traveler s. A ny traveler who wishes to pass the bottleneck within a charging period should have one credit for the specific charging perio d. The credits of each period can be traded at a market, and its credit price will be emerged eventually through free trading. In equilibrium, if there is any unused credit in a credit market, the market is not cleared and its credit price would be zero. S imilar to the basic model and multi step toll, the queue s form at the rate of during the earl y arrival periods and dissipate at the rate of during the late arrival periods. The scheme is designed for a travel demand of and would be such that the first and last travelers will not be charged. Therefore, the start a nd end of congestion period do not change. It can be easily verified that the number of early and late charging periods is the same and can be determined either by ( 3 2 6 ) or ( 3 2 9 ). The boundar ies of early and late charging periods can be also determined u sing ( 3 2 5 ) and ( 3 2 7 ). In this situation, the queuing pattern would be similar to Figure 3 6 . The authority may consider the early and late charging period as two separate or one combined credit type. Note that because the schedule delays at the start of early charging period and the PAGE 41 41 end of late charging period are the same, even if they are treated as two separate credit types, their credit prices w ould be the same. However, the number of credit markets with combined credit types is a half of the case with separate types. To summarize , the tradable credit scheme can be described as follows: t he authority charges one credit type on each traveler ar riving within . The total number of credits of type is . 3.3 Summary In this c hapter, the one to one correspondence between tradable credit and congestion pri cing is formally established. We showed that the authority can achieve the desired quantity based control target by implementing either tradable credit scheme or congestion pricing. The quantity target aimed to manage the travel demand either spatially or temporally. For the spatial distribution purpos e, three control targets have been studied : VMT control target, area based control target and link level control target. For each of the targets, a tradable credit scheme and its corresponding congestion prici ng are designed. For the temporal distribution purpose, the bottleneck model is used as the analysis tool, and the queue length is considered as the quantity measure. Two control targets have been i nvestigated: zero queue length and controlling the maximum queue length below a threshold . The equivalent tradable credit and tolling schemes that guarantee achieving the control target s are designed. PAGE 42 42 CHAPTER 4 CONGESTION PRICING VS. TRADABLE CREDIT UNDER UNCERTAINTY Chapter 3 showed there is a one to one correspondence between tradable credit scheme and congestion pric ing in a deterministic setting. However, the identity between the use of congest ion pricing and tradable credit scheme in managing network mobility falls apart when there is uncertainty associated with tra nsportation supply or demand. Uncertainty will cause congest ion pricing and tradable credit scheme behave differently and lead to divergent welfare consequence. Intuitively, in uncertain conditions, congestion pricing does not guarantee that a quantity tar get will be achieved while tradable credits ensure to meet the quantity target but leave the market credit price uncertain. The contribution of Chapter 4 is twofold. First, we conduct Monte Carlo simulations to show how the use of tradable credit s and con gestion pricing result in different outcome in uncertain conditions for both spatially and temporally aimed control targets. Second, safety valve policies are designed to impose a ceiling on the credit price. The remainder of C hapter 4 is organized as fo llows. Section 4.2 is assigned to spatially aimed control target s . Monte Carlo simulation is used to contrast the difference between tradable credit scheme and congestion pricing in managing the transportation network spatially, follows by a sensitivity an alysis of joint network and market equilibrium and design of safety valve policy. Section 4.3 highlights the difference between tradable credit scheme and congestion pricing in a dynamic setting using the bottleneck model, and shows how a safety valve poli cy can be designed. Finally , S ection 4.4 concludes C hapter 4 . 4.1 Spatially A imed Quantity Control In S ection 3.2.2, tradable credit and congestion pricing are designed to control VMT across the network. Hereinafter, in analyzing and designing of spatially aim ed credit schemes PAGE 43 43 under uncertainty, we limit our attenti on on the scheme that control s VMT across a n etwork. Below a numerical example is used to demonstrate how congestion pricing and tradable credit schemes behave in a transportation network with stocha stic capacity and demand. 4.1.1 Monte Carlo Simulation Consider the same network of Figure 3 1 and now suppose that link capacities are subject to stochastic degradation and follow a uniform distribution between the nominal capacity in Table 3 1 (denoted as ) and the lower bound , where for all links in the example. The O D demand function is assumed to be: where is a uniformly distributed random variable between t o reflect the stochastic nature of travel demand. In order to regulate VMT to less than 1000, we examine the following two schemes: Tradable credits scheme. The scheme is simply to allocate 1000 credits to travelers and collect one credit for each mile tra veled. Mileage fee scheme. The mileage fee is 1.464 obtained by solving TCP with average link capacities, i.e., and average demand function, i.e., A Monte Carlo simulation is conducted to evaluate the above two schemes. For simplicity, we assume that travelers are risk neutral and fully adapti ve in changing their route choices to each unfolded scenario. Therefore, for each sample of link capacities and demand, we solve TCP to estimate the resulting credit price and VMT. The mileage fee policy is evaluated by solving TUE. The comparison statisti cs are presented in Table 4 1 . Figures 4 1 and 4 2 depict the volatility in VMT and credit price, respectively. As expected, the price varies significantly, from 1.091 to 1.773, in the credit market associated with the stochastic network. The average PAGE 44 44 price is 1.439, lower than the mileage fee of 1.464. However, in 47.2% of cases, the credit price is higher than its pricing counterpart. On the other hand, the mileage fee policy yields a lower average VMT simply because it charges more in average. However, th e regulation objective is only achieved with a 52.7% success rate. If the government aims at a higher success rate, a higher mileage fee is needed. Alternatively, the fee needs to be constantly adjusted to adapt to the realized capacities and demands, whic h is difficult, if not impossible, to do in practice. In contrast, the uncertainty is accommodated by the market via spontaneousl y varying credit price. Table 4 1. Simulation with s ample s ize of 1000 Scheme VMT Mean(SD) Price Mean(SD) Success rate of achi eving control target Credit 1000(0) 1.439(0.167) 100% Pricing 985.4(90.6) 1.464(0) 52.7% Figure 4 1. Realized VMT under two schemes. PAGE 45 45 Figure 4 2 . Volatility of credit price. 4.1.2 Sensitivity Analysis of Market a nd Network Equilibrium The quantity polic y is more appealing to a risk averse traffic authority since it ensures the success of the regulation. However, one of its limitations is the volatility of credit prices. It is thus critical to characterize the volatility. As a first step toward this goal, below we conduct sensitivity analysis of the coupled network and market equilibrium to predict how credit price varies with respect to the perturbation associated with the supply or demand. The analysis is based on the approach developed by Tobin and Frie sz (1988) and Yang (1997). Oth er approaches, e.g., Patriksson (2004) and Cho et al. (2000) are also applicable here. We consider a general situation that there are perturbed parameters in the link performance and demand functions, namely, and , where is the vector of perturbations. The functions are once continuously differentiable with the perturbations. Additionally, denote the link path and OD path incidence matrices by and , respectively, and the link length vector by . The general perturbed network equilibrium with tradable credits can be written as the fo llowing variational inequality: Find such that where: PAGE 46 46 The restriction approach proposed by Tobin and Friesz is to select a path flow solution to the above problem when The path flow solution is required to be a nondegenerate extreme point of the equilibriu m path flow set. Assuming that is such a nondegenerate extreme point, there exists a so lution to the following system: where ; is the vector of multipliers associated with the nonnegativity condition of path flows, and is the vector of OD travel cost. We further assume that the credit price is strictly positive, i.e., at . One can expect that the path flows and credit price remain strictly positive in the vicinity of . Therefore, the above system can be reduced to the following: w system with respect to , we have: In the above Jacobian matrix (see, e.g., Hazewinkel 2001 ), denoted as , we have: PAGE 47 47 Consequently , the gradient of link flows, price, and OD travel times with respect to the perturbation vector can be explicitly written as follows: In the following, we app ly the above sensitivity analysis approach to estimate the changes of credit price with respect to perturbations associated link capacities and travel demand for the network in Figure 3 1. Between the OD pair 1 6, there are nine paths, and the link path in cidence matrix, , is: We consider the following perturbed travel time and demand function: where and are corresponding perturbed parameters. When , the link flows are 15.72 , 37.85 6, 57.144, 24.609, 2.049, 26.078, 39.324, 25.569, 67.368, and 14.385. A nondegenerate path flow solution can be obtained as PAGE 48 48 . Therefore, the corresponding reduced li nk path and OD path incidence matrices become: The Jacobian matrix, , can be es timated as follows: T he gradients of path costs with respect to the perturbation parameters are Finally, the gradient of credit price , , can be estimated as follows, The above derivative information provides insights on the direction and magnitude that the credit price may change with respect to demand or capacity perturbations. It can be seen that the changes in capacities of links 3 and 7 have more substantial impact on the credit price, but in PAGE 49 49 a different direction. In ad dition, the price is insensitive to the capacity of link 5, due to its low traffic volume and uncongested traffic condition. To see how the derivative information is accurate in predicting the credit price, suppose that the capacities of links 3 and 7 degr ade by 10% and the perturbation of the demand function is 0.05. The price obtained by solving TCP is 1.569. The predicted price based on the above derivative information is as follows: Note that the above sensitivity analysis approach is only applicable for studying marginal impacts of demand or supply perturbations. It is more desirable and also more challenging to characterize the probability distribution functions of the credit price under stochastic demands or capacities. Yet, with more restrictive assumptions on the types of uncertainty and route choice behaviors, it remains feasible to estimate the distribution functions. Another simple way is to conduct a Monte Carlo simulation to estim ate their moments and then use them to approximate the functions. 4.1.3 Design o f Credit Schemes u nder Uncertainty The key issue in designing credit schemes under uncertainty is to confine the volatility of credit price, because it can be too high to be acceptab le for the public. One remedy is to implement a price ceiling. More specifically, if the market price achieves the ceiling, the government will intervene in the market by selling additional credits at the ceiling price. The idea, dubbed as safety valve, wa s initially suggested by Roberts and Spence (1976) and later developed in the con text of climate policy by Pizer (2002). The conditions for network and market equilibrium under such a safety valve policy can be described as follows: (3 1) (3 2) PAGE 50 50 (4 1) (4 2) (4 3) (4 4) where is the additional credits that the government needs to provide at the ceiling pr ice . In the above, ( 3 1) and ( 3 2) represent the network equilibrium conditions; ( 4 1 ) and ( 4 2 ) imply the market clearing conditions, and ( 4 3 ) and ( 4 4 ) ensure that if the credit price reaches the ceiling, additional credits need to be provided to t he credit market. The solution to the above system, i.e., the equilibrium flow, price and number of credits, can be obtained by solving the following tradable credit problem with price ceiling or TCPC: (4 5) The equivalency can be easily established by examining the first order optimality conditions of the above convex program, with the market clearing credit price being the Lagrangian multiplier associated with ( 4 5). Proposition 4 1. The VMT under a safety valve policy is non increasing with respect to its price ceiling. Proof. Suppose that and are optimal solutions to TCPC when the price ceiling is and respectively. Note that is feasible to TCPC with being PAGE 51 51 the price ceiling and vice versa. Denoting we have : Summing the above inequal ities yields which implies that if then We now consider the numerical example in Section 4.2. 1 and conduct the same Monte Carlo simulation to evaluate the credit scheme with a price c eiling equal to 1.464, the mileage fee. For each sample of capacities and demand, TCPC is solved to estimate the resulting VMT and credit price. The results are presented in Figure 4 3 . It can be observed that the safety valve policy is essentially a hybri d of the price and quantity approaches. When the credit price is below the ceiling, the system acts as a credit scheme with the VMT fixed but the price left to adjust. When T he ceiling price is reached, the system behaves like the mileage fee policy, fixin g the charge but leaving VMT to adjust. The hybrid system yields the same success rate of 52.7% as the mileage fee scheme, but a much lower average price of 1.380, as compared to the mileage fee of 1.464. Therefore, such a safety valve policy appears more amenable to the public acceptance. If the government agency attempts to ensure a higher success rate, a higher price ceiling should be imposed. On the other hand, a lower price celling will confine the volatility of credit price, but leaving VMT less contr olled. The ceiling price is reached, the system behaves like the mileage fee policy, fixing the charge but leaving VMT to adjust. The hybrid system yields the same success rate of 52.7% as the mileage fee scheme, but a much lower average price of 1.380, a s compared to the mileage fee of 1.464. Therefore, such a safety valve policy appears more amenable to the public PAGE 52 52 acceptance. If the government agency attempts to ensure a higher success rate, a higher price ceiling should be imposed. On the other hand, a lower price celling will confine the volatility of credit price, but leaving VMT less controlled. F igure 4 3. VMT and credit price under a safety valve policy. Indeed, the price ceiling provides a means for the authority to make a tradeoff between the re gulation success and price volatility/public acceptability. It is thus of critical importance to investigate how to determine an appropriate price ceiling. Intuitively, it should be the lowest price that guarantees a targeted success rate, because it both confines the price volatility and meets the control objective as much as possible. The problem for finding such a value can be formulated as the following chance constrained price ceiling or CCPC I problem: (4 6) where is the (stochastic) flow at link a under stochastic demand or capacity, with being the price ceiling, and is the targeted success rate. Define . In words, the above problem fin ds the minimum price ceiling that ensures the control target to be met with a probability more than . Although appearing simple, the problem is PAGE 53 53 difficult to solve, because is a stochastic variable whose cumulative distribution function, , is implicitly decided by the price ceiling, it is difficult to obtain a closed form analytical expression for the function. However, the function possesses an interesting property. Consider two price ceilings and with . For any instance of stochastic demand or capacity, we always have due to Proposition 4 1 . This implies that has a first order stochastic dominance over , i.e. for any z . Their relationship is depicted in Figure 4 4 where M is a sufficiently large value. In Figure 4 4 , when the price ceiling is M , the credit price never reaches to its ceiling and thus the VMT will not exceed the control target K with 100% probability. In contrast, if the price ceiling is 0, the authority will issue credits as many as needed and the credit scheme has no effect on VMT, an uncontrolled scenario. The situations wit h and are in between, and . F igure 4 4. Cumulative density function of VMT with respect to price. PAGE 54 54 CCPC I essentially seeks for a minimum price ceiling with . In view of the above stochastic dominance relationship, the price can be found within an interval by a simulation based bi section procedure as follows: Step 0. Initialization. Set and . Step 1. Compute the price ceiling . Step 2. Conduct a Monte Carlo simulation to estimate the regulation success rate, i.e., . Step 3. If , ; Otherwise . Step 4. If ( where is sufficiently small number, go to step 1. Otherwise, return as the solution. In the above procedure, a Monte Carlo simulation is conducted at each iteration to estimate the success rate. In the simu lation, for each sample of stochastic capacities and demand, the resulting VMT is obtained by solving TCPC with the current price ceiling , and is then compared with the control target. If the VMT is no larger than the target, we consider it as a succe ss event, i.e., ; otherwise . Clearly, the success rate is , whose estimate can be calculated in a recursive manner during the simulation. The simulation stops when the standard deviation estimate of is sufficiently small. Consider th e numerical example in Section 3. Suppose that we are interested in finding a price ceiling to ensure a 90% success rate in regulating VMT less than 1000. Table 4 2 reports the implementation of the above bi section procedure with . The procedure f inds the optimal price ceiling to be 1.656 in 12 iterations. PAGE 55 55 Table 4 2 . Iterations of the b i s ection p rocedure. Iteration 1 0.000 2.000 1.000 0.00 2 1.000 2.000 1.500 0.59 3 1.500 2.000 1.750 0.99 4 1. 500 1.750 1.625 0.83 5 1.625 1.750 1.688 0.95 6 1.625 1.688 1.656 0.90 7 1.625 1.656 1.641 0.86 8 1.641 1.656 1.648 0.87 9 1.648 1.656 1.652 0.88 10 1.652 1.656 1.654 0.89 11 1.654 1.656 1.655 0.89 12 1.655 1.656 4.2 Temporally A imed Quantity Tar gets A s we discussed above, the multi step toll and tradable credit scheme s are both designed for the maximum possible travel demand . In reality, however, the realized travel demand can be less than its maximum and is subject to day to day fluctuation s . In what follows , the efficiency of the proposed schemes are analyzed under demand fluctuations . We start with the multi step toll and then will proceed to the tradable credit scheme . 4.2.1 Multi Step Toll Scheme under Demand Uncertainty Because the realized travel demand is lower than what the toll periods are designed for, the toll levels might be perceived relatively high compared to the actual level of congestion. Such disproportionality between the realized travel demand and the designed toll level s are intensi fied as the realized demand deviates from its maximum . Hence , travelers would avoid arriving at the times that the traffic delay reduction do es not offset the extra burden of tolls . In early periods, the idle intervals would be at the beginning of the toll periods , whe n arriving immediately after the toll increment would result in higher travel cost , even with experiencing zero queue delay. The first traveler at the toll period arrives at , and no arrival would occur PAGE 56 56 within . In the late periods, the idle periods would be at the end of toll periods, when travelers would be better off by defer ring their arrival to the next toll period and pay ing lower toll. The last traveler at the toll period arrives at and no one arrives during . The schematic queuing pattern when the realized travel is less than its maximum is depicted in Figure 4 5 . To have and , the number of travelers at the toll period is needed. P ropositions 4 2 to 4 4 would guide us to obtain these unknowns. Figure 4 5 . Schematic queuing pattern of the proposed multi step toll for the realized demand Proposition 4 2 . Unde r the proposed multi step toll scheme and independent of the realized travel demand, the same number of travelers arrive at each toll period, and the ratio of early to late arrivals is . Proof. Consider the end of early stage at th and th toll period s , i.e. and . The difference of early schedule penalties would be , that is equal to the difference of the toll level at these periods. Therefore, at equilibrium, the queue length at these moments should be the same , imply ing the same number of travelers at consecutive early periods. Similarly, we can show the number of travelers at the late stage of two consecutive toll periods is equal. Moreover, because the schedule delay at and within th toll period is the same, the queue length associated with them should be equal. Th is requires that the ratio of early to late arrival at the toll period to be . Queue length PAGE 57 57 Proposition 4 3 . For any realized demand, arrival would oc cur at all toll periods. Proof. From P roposition 4 2 , it is clear that if arrival occurs at any of the toll periods, it would occur at all other s . Now consider the situation that the travelers only arrive at the free period s . Then, any traveler can be bett er off by arriving within th toll period at or , because the reduction in schedule delay is greater than . Proposition 4 4 . If the travel demand drop s below , no traveler would arrive at free period. Proof. The travelers are better off by arriving inside the toll periods rather than the free periods as long as , ( 4 7 ) According to P ropositions 1 and 2, . By plugging into ( 4 7 ), and replacing and with ( 3 25 ) and with ( 3 30 ), we can derive . Now we can obtain the number of travelers at each period. Denote the number of travelers at the early and late stag e of toll period with and , respectively. Then, based on P ropositions 4 2 to 4 4 , we have: ( 4 8 ) Subsequently, and can be derived as, ( 4 9 ) PAGE 58 58 In addit ion, the start and end of the congestion period can be determined as, ( 4 10 ) 4.2.2 Tradable Credit Scheme under Demand Uncertainty Under the tradable credit scheme, the credit prices can adjust themselves based on the realized demand. Lower travel demand would result in milder co ngestion, and hence lower credit prices. In fact, the credit markets would adjust the credit prices to ensure operation of the bottleneck with full capacity. P ropositions 4 5 and 4 6 discuss how the credit price of each credit market can be estimated for a realized travel demand. Proposition 4 5 . The credit price of the markets associated with the first/last travelers is zero. Proof. Three cases with respect to arrivals of the first/last traveler can be identified. In the first case, the first/last traveler arrives during the free of charge periods. It is evident that they would not be charged at all. In the second case, the traveler arrives inside of a charging period. In this situation, the credit market is not cleared and thus its credit price would be ze ro. In the last case, the traveler arrives at the boundaries of a charging period, i.e. the start of a charging period in early arrivals or the end of a charging period in late arrivals. In this situation, although the market is cleared but the credit pric e would be zero. To see this, assume that the credit price is not zero. Then the traveler can be better off by shifting his/her arrival time to a little bit earlier or later in early or late stages, respectively, and arriving within a period that has an un cleared market or is free of charge. Then, the original credit market is not cleared anymore and its credit price would be driven to zero. In view of P roposition 4 5 and considering the first and last travelers always experience zero queue delay, we can conclude that , implying the first and last travelers arrive within PAGE 59 59 the same charging periods. All the other travelers arrive between and and there is no idle interval in between. Therefore, and Proposition 4 6 . Assume , where . Then the credit price of types is zero. The credit price of types is . Proof. If , then the credit price of types would be zero. Substituting and with their defin itions in ( 3 25 ), and plugging them into the boundaries of the condition, we can derive that . For , by comparing the travel cost of the first traveler arriving at th charging period with the tra vel cost of the first traveler, we can reach . 4.2.3 Tradable Credit Scheme with Price Ceiling The design of the proposed tradable credit scheme is characterized to be robust in meeting the queue length requirement, even for the hi ghest possible travel demand. As shown in Section 4.3.2 , the credit price is adaptive to the realized demand so that the bottleneck capacity is efficiently utilized to ensure the success in achieving the queue length requirement. Designing based on the max imum travel demand is also advantageous from the information perspective. However, the resulting credit prices might be too high to be acceptable by the public. As a remedy, the authority may consider imposing a ceiling for credit prices, denoted by . Whenever credit prices reach to the ceiling, the authority allows passing the bottleneck by paying a toll rate equal to the ceiling price. In this situation, the scheme operates as a hybrid between tradable credits and toll. Credit prices increase as the charging periods get closer to the work start time. Therefore, the inner charging periods always reach the price ceiling before the outer charging periods. All PAGE 60 60 the charging periods whose prices reach the price ceiling would be essentially operating as a single step toll. The remaining charging periods with credit prices below the price ceiling would operate as before. To analyze the bottleneck under the hybrid scheme, we first need to derive the demand thresholds that drive each credit price into the price ceiling. From Proposition 4 5, it can be found that . Therefore, the demand threshold that triggers the price ceiling at charging period , denoted by , can be obtained as ( 4 11 ) Whenever exceed , the charging periods would operate as a single step toll that charges over [ In this situation, the queue length associated with arrival times between can be obtained as follows, ( 4 12 ) Note that under the hybrid scheme, the queue length requirement cannot be necessarily maintained. In fact, the hybrid s cheme provides a tradeoff between the success in achieving the control target and the public acceptance. Imposing a higher price ceiling increases the chance of success in maintaining the queue length requirement. 4.2.4 Monte Carlo Simulation In this section, we use a numerical example to evaluate the above three schemes. The bottleneck capacity is assumed to be 1000 veh/hr and the maximum queue length requirement is 400 vehicles for a maximum travel demand of 4000. All vehicles are assumed to be single PAGE 61 61 occupant. The corresponding , , and are $6.4, 3.9, and 15.21 per hour , respectively (Nie and Yin 2013) . The resulting multi step toll scheme is comprised of four toll periods as follows Period 1: 9.86$ at [ 0.656, 0.168) Period 2: 7.30$ at [ 1.313, 0.656) [0.168, 0.337) Period 3: 4.74$ at [ 1.969, 1.313) [0.337, 0.505) Period 4 : 2.18$ at [ 2.626, 1.969) [0.505,0.673) Alternatively, in the tradable credit scheme, 824 credits of each type are issued and distributed amon g eligible travelers. The charging periods would be similar to the toll periods. A Monte Carlo simulation is conducted to evaluate the performance of the proposed schemes under demand uncertainty. We assume the travel demand is uniformly distributed betwee n 2,000 and 4,000, and for simplicity, travelers are fully adaptive in changing their departure time, and hence their arrival time, in response to each unfolded scenario of travel demand. In the simulation, 1000 scenarios of travel demand are generated. Fo r each scenario, the credit prices of the four credit market are obtained. Figure 4 6 depicts the volatility of credit prices, compared to the corresponding toll levels. Clearly, the capability of tradable credit to adjust prices based on the realized dema nd result in lower prices. The average prices are presented in Table 4 3 . It can be observed that in Period 1 with the highest price among all periods, the credit price is 69% of the corresponding toll level. The highest reduction in price belongs to Perio d 4, in which many cases with low realized demand have zero credit price. We also demonstrate the performance of the hybrid scheme by imposing a price ceiling of 6.76$, that is the average credit price at Period 1 in the credit scheme. It is clear from Fig ure 4 6 that credit prices never exceed 6.76$. However, confining the credit price to 6.76$ violates the queue length requirement whenever the credit price reaches the price ceiling, as depicted in Figure 4 6 A . PAGE 62 62 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 0 20 40 60 80 100 price($) # of replication Period 1 credit price toll average credit price 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0 20 40 60 80 100 price($) # of replication Period 2 credit price toll average credit price 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0 20 40 60 80 100 price($) # of replication Period 3 credit price toll average credit price 0.0 1.0 2.0 3.0 0 20 40 60 80 100 price($) # of replication Period 4 credit price toll average credit price Table 4 3. Average (standard deviation) of pr ices in the proposed schemes Scheme Period 1 Period 2 Period 3 Period 4 Toll scheme 9.86$ (0.00$) 7.30$ (0.00$) 4.74$ (0.00$) 2.18$ (0.00$) Credit scheme 6.76$ (1.79$) 4.20$ (1.79$) 1.81$ (1.55$) 0.38$ (0.63$) Hybrid scheme 5.98$ (1.01$) 4.17$ ( 1.76$) 1.81$ (1.55$) 0.38$ (0.63$) A B C D Figure 4 6. Volatility of the credit prices under demand uncertainty PAGE 63 63 Figure 4 7 . Volatility of the credit price and queue length under hybrid scheme 4.3 Summary In this chapter , we evaluated the performance of tradable credit schemes under uncertainty with emphasis on their difference with congestion pricing. We showed numerically how the identity between tradable credit and congestion pricing falls apart under uncertainty. Tradable credit guarantees success in achieving to the quantity target, but at the c ost of having no control on the price. We demonstrate d that a safety valve policy can be used to manage the upper bound of credit price. 200 400 600 800 1000 1200 1400 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0 10 20 30 40 50 60 70 80 90 Queue length Price ($) # of replication credit price max queue length PAGE 64 64 CHAPTER 5 EVALUATION OF TRADABLE CREDIT SCHEME WITH USER HETEROGENEITY So far, travelers are assumed to be homogenous in their value of time and unit schedule penalties. We now evaluate the capability of the proposed tradable credit scheme to maintain the queue length requirement under traveler heterogeneity in their scheduling parameters. While it is desirable to conduct such a nalysis under a general heterogeneity in which travelers may have any value of , , and , for tractability we limit our attention to two well known types of user heterogenei ty in the literatures (Arnott et al 1994, v an den Berg 2014 ): proportional het erogeneity and heterogeneity. The former represents the situation where all scheduling parameter s vary among travelers but their ratios are assumed to be fixed, while the latter casts the situation where only the consequence of late arrival is different among the travelers, i.e. the travelers have the same and , but different . 5.1 Analysis of P roportional H eterogeneity We assume and , where , , and are the scheduling parameters that the tra dable credit scheme is originally designed upon. Because the ratio of the scheduling parameters is the same, their distributions are equivalent and is characterized with the density function of and cumulative density function of . Th erefore, there are travelers with parameters of , , and . Under the proportional heterogeneity, travelers with higher , and therefore higher and , arrive closer to the work start time, and t raveler s with lower arrive farther away. Let represent the traveler who is indifferent between arriving within th and th charging period. s depend on the distribution of scheduling parameters and the realized travel dem and and can be obtained as PAGE 65 65 ( 5 1 ) where is the inverse of cumulative density function, and is defined in P roposition 4 6 . So, all travelers with between and arrive at the charging period ; portion of them arriving at the early stage of charging period and portion of them arriving at the late stage of charging period . It is also worth mentioning that all travelers at the char ging period will travel jointly, and their position within period will be random. This can be justified regarding the fact that the queue length increases at the rate of within the early stage of charging period and decreases at a rate of within the late stage of charging period . Therefore, no matter what position the traveler is at, his/her travel cost would be the same within the charging period . Proposition 5 1 . The queue length at the start and end of charging period is zero. Proof. It is evident for because there is no arrival at and . If , firstly note that the queue length at and should be the same. Otherwise, the traveler arriving at the time with longer queue length can shift to the time with shorter queue length, and because schedule penalties at and are the same, the traveler would be better off. Moreover, because only travelers with credit type can arrive within the charging period , the q ueue length would be zero. In view of Proposition 5 1 , the rate of queue formation and dissipation, and the length of early/late stage of charging periods, it can be concluded that the proposed tradable credit scheme originally designed for homogenous tr avelers can still maintain the queue length requirement. PAGE 66 66 To obtain the credit prices, note that all the charging period are not cleared and their credit prices are zero. The credit price of the first charging period before can be determined by eq ualizing the travel cost of the traveler with at th and th charging periods, ( 5 2 ) For , the credit price of charging period can be derived recursi vely by comparing the travel cost corresponding to the traveler with at th and th charging periods, ( 5 3 ) In the homogenous case, we found that the difference in credit prices of two consec utive charging periods is fixed. Under the proportional heterogeneity, in contrast, such difference increases as we get closer to work start time, due to the increase of . 5.2 Analysis of H eterogeneity Here, travelers are assumed to have the same and , as the homogenous case, but is distributed with the density function of and the cumulative density function of . The lower and upper bounds of are denoted by and , respectively. With heterogeneity, trav elers arrive in a decreasing order of . In other words, travelers with tighter schedule and higher arrive at the early stage of charging periods and travelers with more flexibility in arrival time and lower arrive at the late stage of charging peri ods. With such an ordering, the traveler with unit late arrival penalty of is indifferent between arriving early or late. Hence, travelers with arrive early while those with arrive late. Because the arrival time of earl y arrival travelers is governed solely by and , the travel cost of all travelers with is the same, and they are traveling jointly. PAGE 67 67 Under the proposed scheme of Section 3. 3.3 early stage o f th charging period and the last traveler arrives at the late stage of th charging period, where { }. Also, consider as the minimum of and . Then, credit prices at all charging period are zero. In addition, the bottleneck should operate with full capacity within . Otherwise, if there is any idle interval, the credit price associated with the charging period containing the intervals would be zero. Therefore, travelers from the outer charging periods can get better off by switching their arrival time to the idle interval. Clearly, the bottleneck is fully utilized within and . Because all travelers with arrive early, we have ( 5 4 ) So, the travel cost of travelers with would be . For , the travel cost corresponding to travelers with would be ( 5 5 ) Clearly, the travel cost reduces as decreases. The traveler with would arrive at and its travel cost would be , equal to the travel cost obtained from ( 5 5 ). This would result in Equation ( 5 6 ) that can be used to obtain , ( 5 6 ) As mentioned earlier, late arrival travelers ar rive in a decreasing order of represent the unit late arrival penalty of the traveler who is indifferent between arriving at the late stage of th and (i+1) th charging period. Given , we can derive PAGE 68 68 ( 5 7 ) Let denote the queue length at the start of the charging period , and stand for the queue length at the end of charging period . Now, we provide the set of equilibrium conditions . The equilibrium of the traveler arriving at and the first travelers arriving at the early stage of the charging period requires that ( 5 8 ) The travel cost of the last traveler arriving at the early stage of charging period should be the same as the first traveler arriving at the early stage of charging period . This implies ( 5 9 ) The indifference of the traveler with of arriving between the late stage of th and (i+1) th charging periods requires ( 5 10 ) Finally, because the traveler with should be indifferent between arriving early and late, the queue length should be continuous at . Hence, ( 5 11 ) Note that the queue length increases at the rate of from to , which guarantees the travel cost of the last traveler arriv ing in the early stage of charging period to be equal to the travel cost of the first trav eler. Therefore, by constructing E quation ( 5 9 ), E quation ( 5 10 ) for PAGE 69 69 is redundant. To summarize, we have independent linear equation s from ( 5 8 ) (5 11 ) and independent unknowns. By solving the system of linear equations ( 5 8 ) ( 5 11 ) , we can obtain all the unknowns to describe the equilibrium. However, the proposed tradable credit scheme originally designed for homogenous travelers may not be able to meet the queue length requirement. For the purpose of illustration, consider the sche me designed in the numerical example of Section 4.3.4 . realized demand is 3,000 and is distributed uniformly between $15 and 17 per hour, and all the parameters are the same as before. From ( 5 6 ), would be 15.408 and we can then obtain 15.296 from ( 5 7 ). The maximum queue length at the late stage of Period 1 would be 3000 +403, greater than the queue length requirement of 400. The failure of the proposed tradable credit scheme in satisfying the queue length requirement can be attributed to the limit ed information about user heterogeneity during the design process. In the next sections, we will show if more information associated with heterogeneity is available at the design stage, a more robust scheme can be designed . 5.3 Design of T radable C redit S ch eme under H eterogeneity Now, we assume the distribution of is known according to the specification of S ection 5.3 and we will modify the design in S ection 3.3.3 , based on the newly available information . As discussed earlier, under heterogenei ty, the travelers arrive in decreasing order of . The traveler with is indifferent between arriving early and late. Travelers with are forced to to arrive early to evade from the late arrival penalty. The design of tr adable credit scheme is to ensure full utilization of the bottleneck. Therefore, can be obtained from Equation ( 5 6 affected by demand fluctuation. PAGE 70 70 o f the travelers would arrive early, and because the travelers are homogenous with respect to and , the procedure of designing the charging periods in early arrivals would be similar to what have been discussed in Section 3.3.3 . Therefore, the start of the early charging periods, i.e. s, can be determined from (3 25) . However, the number of early charging periods would be different and are determined as ( 5 12 ) In contrast, because of the heteroge neity, the iso cost curves are nonlinear and the procedure of determining would be different. At the end of late charging period , there is a traveler with who is indifferent between arriving within th and ( ) th late charging periods. To have the minimum number of charging periods, we design such that queue at the beginning of late charging period would be and at the end of charging period reaches to zero. This implies that ( 5 13 ) where . Given , can be obtained by solving Equation ( 5 13 ). Subsequently, can be determined recursively as ( 5 14 ) where . However, because is not known in advance, an iterative procedure should be implemented, which add s one charging period at each iteration , up to a point when the resulting from ( 5 13 ) is less than . This indicates tha t no additional charging period is necessary . The PAGE 71 71 queue length at is that does not exceed . The following algorithm formalizes the steps to design the late charging periods: Initialization Let i=0 , , . Step 1 i=i+1 . Obtain from ( 5 13 ). Step 2 If , . Determine from ( 5 14 ) , and return to step 1. Otherwise report and terminate the algorithm. Note that in this design, the credits of th early and late charging periods are not the same and we cannot combine them as a single credit market. Therefore, there are sepa rate credit markets. In the next section, we analyze the modified tradable credit scheme under demand fluctuation to see how robust the scheme is in satisfying the queue length requirement. 5.4 Analysis of the M odified T radable C redit under D emand F luctuation Now, assume the realized demand is less than the maximum possible travel demand, i.e. . As discussed earlier, is independent of and would be obtain from ( 5 6 ). The start and end of congestion are also determined from ( 5 4 ). In addition, assume and are the charging intervals that contain and , res pectively. Because only travelers having credit type can pass the bottleneck, the queue length at is zero for all . With the same reasoning, the queue length at is zero for all . For each late charging period , the traveler with is indifferent between arriving at th and th late charging period. Given , can be determined from ( 5 7 ). Now, the queue length at the start of late charging period can be determined as . PAGE 72 72 Proposition 5 2 . Under the modified tradable credit scheme, the maximum queue length within charging periods does not exceed . Proof. The proof for early charging period is straightforward. For charging period , the queue length at the start of charging period is zero, and then increases at a rate of and reaches at the end of charging period. At charging period , the maximum queue length at most reaches . Fo r charging period , there is no arrival and therefore no queue. The proof for late arrival is more involved. Let assume is a function that returns at time for the travel demand of . Because are sorted in a de creasing order, we have: ( 5 15 ) From (34) it is clear that as decreases would not increase. The maximum queue length at the late charging period can be obtained as . On the other hand, the modified design of tradable credit scheme ensures that . Because, for any realized demand , we can conclude , and this completes the proof. We now obtain the credit prices of the modified tradable credit scheme. The credit price of th early charging period, denote by , can be determined as follows, ( 5 16 ) The credit price of th early charging period, denote by , is zero for . The credit price of c an be determined by considering the indifference of the traveler with between arriving at th and th late charging periods. This yields: PAGE 73 73 ( 5 17 ) where and .Therefore, the recursive equation ( 5 17 ) can be simplified as ( 5 18 ) 5.5 Summary In this chapter , the effect s are investigated on the efficiency of tradable credit scheme . Two widely accepted variations of heterogeneity have been studied: proportional heterogeneity and heterogeneity. The scheme that was originally designed for homogenous case can still maintain the queue le ngth below the threshold under proportional heterogeneity. However, under heterogeneity, the original scheme cannot guarantee achieving the control target. We showed i f extra information provided, the design can be revised to accommodate heterogeneit y. PAGE 74 74 CHAPTER 6 CONCLUSION S 6.1 Findings This dissertation explores the potential of tradable credit schemes in congestion mitigation across the transportation networks with emphasis on their performance under uncertainty . Inspired by successful implementation of the tr adable credit scheme in pollution control, the tradable credit scheme is increasingly considered as a viable instrument for congestion mitigation. Among issues associated with the tradable credit scheme , this dissertation investigates how their efficiency can be affected when there is uncertainty associated with the demand or supply of the transportation system. The pros and cons of tradable credit scheme can be understood in light of a comparison with its price based mirror, i.e. , conges tion pricing. In C hapter 3, we demonstrated systematically that there is one to one correspondence between congestion pricing and tradable credit scheme. Furthermore , any flow distribution that resulted from congestion pricing implementation can be replicated by an appropri ately designed tradable credit scheme, and vice versa. The existence of such equivalency is demonstrated for spatially and temporally aimed control targets. For the purpose of spatially aimed control target s , we started by designing a tradable credit sch eme to limit total VMT across the network as a network wide control target . T hen , we proceeded to finer control targets : controlling total flow into CBD area and flow s on the link level . It has been shown that as the authority defines finer control target s the charging scheme of tradable credit would be come more complicated. For the purpose of temporally aimed control target s , the bottleneck model is used as the modeling framework. Here, the primary objective is defined to control maximum queue length . PAGE 75 75 Be cause of the nature of control target the charging scheme of tradable credit would be time de pendent. In the most ideal case, where there is no queue behind the bottleneck , the charging scheme varies continuously. However, when the objective is set to limi t the max imum queue length a multi step charging scheme can be designed to ensure the maximum queue length does not exceed the pre defined threshold. Under specification of the proposed scheme, each specific charging interval requires its specific type of credit. Therefore, while the number of credit s being charged remains constant , the type of credits would be different. As the result, multiple markets should be established and ea ch credit type would find its market equilibrium price . The one to one corre spondence between congestion pricing and tradable credit is under assumption of perfect certainty. In fact, if there is uncertainty associated with either supply or demand side of transportation system, tradable credit scheme and congestion pric ing would no longer mirror each other . In this situation, the t radable credit scheme ensures the achievement of the control target but has no control on the resulting credit price. In contrast, the congestion pricing preserve s the price but cannot guarantee achievin g to the control target. The results obtained from conducting Monte Carlo simulations in C hapter 4 are in accordance with th is intuition. The Monte Carlo simulation revealed that while the capability of the tradable credit scheme to ensure success in achi eving the control target is appealing, its lack of control on the resulting credit price could be restrictive. Due to the uncertainty, the credit price might be too high to be accepted by travelers and it might create political resistance against the imple ment ation of the tradable credit scheme . To overcome such drawback, a safety valve mechanism is proposed to confine price volatility. Under this mechanism, t he authority imposes a ceiling for credit price. Whenever the credit price reaches to its ceiling l evel, the authority PAGE 76 76 would intervene in the credit market through sell ing extra credits at the ceiling price to ensure the credit price would not go beyond its ceiling . Tradable credit scheme with safety valve is essentially a hybrid scheme of tradable cred it scheme and congestion pricing. When the credit price is below the price ceili ng the scheme would operate as tradable credit scheme . Whenever the credit price reaches to the ceiling price, the scheme would operate as a congestion pricing scheme. The Mont e Carlo Simulation showed that t he hybrid scheme can achieve higher success rate in achieving the control target compared to pure congestion pricing, with a lower average price. When there are multiple credit markets, as the case for the temporally aimed control target, not all markets reach the ceiling price simultaneously. Therefore, som e credit markets might reach the price ceiling and then operate as a tolling scheme , while other credit markets have the credit price below the ceiling price and are oper ating as a tradable credit scheme. In C hapter 5, the importance of traveler heterogeneity in the analysis and design of tradable credit scheme has been demonstrated. We discussed how the existence of user heterogene ity might affect the efficiency of trad able credit scheme. More specifically, it can be shown that the performance of tradable credit scheme which is originally designed under homogenous assumption might not be as expected under user heterogeneity. We further showed that if information associ can be modified to be more robust in achieving the control target. In this dissertation , tradable credit scheme is introduced as an alternative to the traditional congestion pricing in managing network mobility. In practice, c ongestion pricing has been implemented in variety of forms. As a result of its implementation, pros and cons of different variations of congestion pricing have been identified by researchers and practitioners, and PAGE 77 77 suggestio n s have been made to overcome their deficiencies and amplify their strengths . In contrast, the tradable credit scheme is a relatively new concept in congestion mitigation and there is no real world instance of it s implementation. While some findings from t he application of the tradable credit scheme in pollution control, urban development and fishing control may be applicable , however, the design, implementation, and maintenance of the tradable credit scheme in travel demand management is more challenging because its number of player s are much higher than the previously examined contexts . In the follow ing, some thoughts associated with implementation of the tradable credit scheme in real world will be presented . 6.2 D iscussion on R eal W orld Implementation From the implementation perspective, a tradable credit scheme can be characterized by the way s the credits are initially allocated , traded , and charged. Below, each of these elements will be discussed in detail. 6.2.1 Initial D istribution of C redit s After determinin g of the total number of credits required for achiev ing a specific control target in the design stage , the authority should issue credits and distribute them among eligible travelers. It is necessary to have a precise definition of which migh t vary from one scheme to other. One might consider all taxpayers within the study area eligible, while the other one defines eligibility as having registered car within study area. The eligibility might be also defined based on work location. Each eligibl e traveler is required to open an account in an electronic platform. The initial allocation will be transferred in the account, and the subsequent charging will be deducted from it . The account might be connected to an E trading plat form and would be used in credit trading. The initial allocation of credit can be conducted in several ways. One way of credit allocation is grandfathering, where certain number of credits are allocated to each eligible PAGE 78 78 travelers at no cost. The credits might be distributed unif ormly, in which all eligible travelers rec eive the same number of credits; or non uniformly, in which the distributed number of credit varies among different group s of travelers. As discussed earlier, the authority might use the initial allocation of credi ts to enhance the equity and reduce the potential public resistance against its implementation. The number of allocated credit s might be determined based on origin destination, vehicle class, etc. However, as Nie (2012a) highlighted, the initial allocatio n based on geographical locat ions is difficult to implement because O D pairs in reality are not as well defined as in the models . The initial allocation of credit might not be free. The authority might decide to se ll the credits at a base price which is not necessarily the market equilibrium price. Any unused credits will be bought back at the base price at the end of period. Alternatively, the credits might be sold in auction. When the initial distribution of credits is not free of charge, however, trada ble credit scheme would lose its revenue neutral property that is highly emphasized by its advocators. The author ity might also embed a reward mechanism into the structure of the tradable credit scheme to reward credits to the travelers contributing in con gestion mitigation through changing their route, mode , or departure time. A reward mechanism similar to the one proposed by Nie and Yin (2013) may solve the issues associated with the initial allocation of credits. A hybrid scheme can be implemented on pre determined tolled road s during charging window s . The travelers using the toll road s within the charging window are required to either pay a certain number of credits or a premium toll at a higher rate. The travelers who avoid the toll roads by either trav eling outside the charging window or switching to alternative mode or route are rewarded certain number of credits. Subsequently, the travelers can use the rewarded credits for their own trips or trade them in the credit market . The reward mechanism is exp ected to provide PAGE 79 79 fairer distribution of benefits among t ravelers. However, the reward rate s should be designed such that it does not encourage traveling by itself . 6.2.2 Credit C harging The number of credit s being charged would be determined in the design stage . As demonstrated i n C hapter 4, the credit charge s might be location specific and /or time dependent. The credit charge s may also depend on the vehicle class es . The authority can offer lower charge rate s for specific vehicle classes such as electric vehicle s to promote their usage. The same technologies that are already utilized to implement the congestion pricing can be acquired for credit charging. A smart card or on board unit might be attached to the vehicle autom atically charges certain number of credits on the account. 6.2.3 Credit M arket In order to facilitate credit trading, t he authority has to create an artificial market. Establishing such a market is techn ically feasible and would not be cost intensive with respe ct to the vast development of electronic trading platforms. While the authority should not interfere in the credit market, it should monitor and enforce the market. In fact, continuously monitoring of the market is of quite importance to avoid illegal manipulation such as black marketing. With advancement in learning science , the fraud in the market can be readily detected. Because of the uncertainty associated with credit price, the authority might intervene in the market when the credit price is unrea sonably high. In this situation, the authority can sell additional credit s in a pre defined ceiling p rice to limit the credit price. One of the important issues that might affect the efficiency of tradable cre dit scheme is the presence of t ransaction cost s which can arise in any transfer of property rights, including PAGE 80 80 mobility credits. In tradable credit scheme, transaction cost may occur in different stages of legislation, design, or operation (Crals 2005). T raveler s should learn th e rules of credit market, and know how many credit s are need ed to fulfill their travel needs. Then, they should find and negotiate with a trading partner . In addition, there may be some cost associated with monitoring and enforc ing the credit market (Stavins 1995 ). Transaction cos t s can be fix ed or depend on the trading quantity with a constant, decreasing or increasing marginal rate . Constant marginal transaction cost is related to a fixed brokerage fee per trad ing unit . Decreasing transaction cost can be justified in two way s : on e is when the brok er offers discount s over quantity and the other emerge s in the long run when traveler s get more experience d in the market and are able to fulfill their travel needs with less searching and information costs (Cason and Gangadharan 2003 ). I ncreasing marginal transaction cost can be justified by considering the fact that if a traveler needs more credit, they might assign more effort to search and find a better trading partner and as a result its corresponding transaction cost will be increase d ( Kerr and Mare 1998 ). However , in creasing marginal transaction cost cannot be sustainedn because the trader s can reduce their transaction cost s by dividing them into smaller portions. However, Stavins (1995) discussed that by combining a proper fixed tran saction cost with marg inal increasing transaction cost the above mentioned problem can be avoided. Nevertheless , it should be noted that most of markets have constant or decreasing marginal transac tion cost (Cason and Gangadharan 2003 ). Transaction costs m ay reduce the efficiency of those optimal credit schemes. Transaction costs affect the trading and travel behaviors of travelers. More specifically, transaction costs suppress the trading of credits because they impose direct financial burden to credit buy ers and reduce the profit of credit sellers. When transaction costs are sufficiently high, travelers with PAGE 81 81 unused credits will stop selling them. Consequently, those travelers will not consider credit charges in making their traveling decisions. At the same time, travelers w ith insufficient credits will reduce their purchases. These forces cause the equilibrium flow pattern to deviate from the target state. However, transaction costs do not necessarily increase individual travel costs because travelers are a ble to change their route or departure time choices to adapt. The concerns associated with the potential negative impacts of transaction cost on the efficiency of tradable credit scheme are mostly stemmed from the pollution control experiences. However, because the number of potential traders in mobili ty credit market is much higher its transactions costs are expected to be low. Therefore, while transaction costs reduce the effici ency of tradable credit schemes, the reduction is unlikely substantial . 6.3 Conc luding R emarks In summary , the tradable credit scheme has great potential to be considered as a travel demand management strategy . However, with regards of the challenges assoc iated with its implementation, a pure tradable credit scheme implementation is n ot expected. In fact , a hybrid scheme that incorporates both congestion pricing and tradable credit scheme has higher chance of successful implementation. T radable credits can be utilized as an add on to congestion pricing to address its equity concerns an free initial allocation, the free accessibility to tolled facilities can be provided to some extent for all travelers. Therefore, the public concern that congestion pricing is in favor of high income trav elers can be partially addressed. PAGE 82 82 LIST OF REFERENCES Akamatsu, T., Sato, S., Nguyen, L. X., 2006 . Tradable time of day bottleneck permits for morning commuters. JSCE Journal of Infrastructure Planning and Management 62(4), 605 620. Arnott, R., De Palma, A., Li ndsey, R., 1990 . Economics of a bottleneck. Journal of urban Economics 27 (1) , 111 130. Arnott, R., De Palma, A., Lindsey, R., 1995 . Recent developments in the bottleneck model (No. 305 ). Boston College Department of Economics. Arnott, R., de Palma, A. , Lindsey, R., 1994 . The welfare effects of congestion tolls with heterogeneous commuters. Journal of Transport Economics and Policy 28 (2) , 139 161. Bao, Y., Gao, Z., Xu, M., Yang, H., 2014 . Tradable credit scheme for mobility management cons Transportation Research Part E: Logistics and Transportation Review 68 , 138 154. Cason, T.N., Gangadharan, L., 2003. Transactions costs in tradable permit markets: an experimental study of pollution market designs. Journal of Regulatory Economics 23(2), 145 165. Cho, H.J. , Smith, T.E., Friesz, T.L., 2000 . A reduction method for local sensitivity analysis of network equilibrium arc flows. Transportation Research Part B 34 (1), 31 51 . Crals, E., 2005. Tradable rights and trans action costs: A comparative analysis of alternative policy instruments for emissions, road use and public deficits, PhD dissertation, University Erfurt. Daganzo, C. F., 1985 . The uniqueness of a time dependent equilibrium distribution of arrivals at a sing le bottleneck. Transportation science 19(1) , 29 37. Dales, J. H., 1968 . Land, water, and ownership. The Canadian Journal of Economics/Revue canadienne d'Economique 1 (4) , 791 804. de Palma, A., Fosgerau, M. , 2011. Dynamic traffic modeling. In: de Palma, A. , Lindsey, R., Quinet, E., Vickeman, R. (Eds.), Handbook in Transport Economics . Cheltenham, Edward Elgar, UK, 29 37. de P alma, A., Lindsey, R., 2011. Traffic congestion pricing methodologies and technologies. Transportation Research Part C 19(6) , 1377 139 9. Duranton, G., Turner, M. A., 2012 . Urban growth and transportation. The Review of Economic Studies 79 (4), 1407 1440. Han, D., Yang, H., Wang, X., 2010 . Efficiency of the plate number based traffic rationing in general networks. Transportation Research Part E: Logistics and Transportation Review 46 (6) , 1095 1110 . PAGE 83 83 Hazewinkel, M., 2001. Encyclopedia of Mathematics , Springer. He, F., Yin, Y., Shirmohammadi, N., Nie, Y. M., 2013 . Tradable credit schemes on networks with mixed equilibrium behaviors. Transpor tation Research Part B: Methodological 57 , 47 65. Hearn, D. W., Ramana , M. V ., 1997. Toll p ricing m odels for t raffic n etworks . Proceedings of 8th IFAC Symposium on Transportation , Chania, Greece. Jaensirisak, S., Wardman, M., May, A. D., 2005 . Explaining variations in public acceptability of road pricing schemes. Journal of Transport Economics and Policy 2 (39) , 127 153. Kerr, S., Mar e , D., 1999. Transaction costs and tradable permit markets: the United States lead phasedown (Manuscript, Motu Economic and Public Policy Research, New Zealand). Laih, C. H., 1994 . Queueing at a bottleneck with single and multi step tolls. Transportation Research Part A: Policy and Practice 28 (3), 197 208. Lawphongpanich S., Yin, Y., 2012 . Nonlinear pricing on transportation n etworks. Transportation Research Part C 20 (1) , 218 235. Lindsey, C. R., van den Berg, V. A., & Verhoef, E. T., 2012 . Step tolling with bottleneck queuing congestion. Journal of Urban Economics , 72 (1), 46 59. Liu, W., Yang, H., Yin, Y., 2015 . Efficiency o f a highway use reservation system for morning commute. Transportation Research Part C: Emerging Technologies 56, 293 308 . Mamun, M. S., Michalaka, D., Yin, Y., Lawphongpanich, S., 2014 . Comparison of Socioeconomic Impacts of Market Based Instruments for M obility Management. International Journal of Sustainable Transportation . Montgomery, W. D., 1972 . Markets in licenses and efficient pollution control programs. Journal of economic theory 5 (3), 395 418. Nie, Y. M., 2012 a . Transaction costs and tradable mob ility credits. Transportation Research Part B: Methodological 46 (1) , 189 203. Nie, Y. M., 2012 b . A New t radable c redit s cheme for the m orning c ommute p roblem. Networks and Spatial Economics , 1 23. Nie, Y. M., Yin, Y., 2013 . Managing rush hour travel choic es with tradable credit scheme. Transportation Research Part B: Methodological 50 , 1 19. Patriksson, M. , 2004. Sensitivity analysis of traffic equilibria , Transportation Science 38, 258 281. Pigou, A.C., 1920. The Economics of Welfare , MacMillan. PAGE 84 84 Pizer, W. A., 2002. Combining price a ' nd quanity controls to mitigate global climate change. Journal of Public Ecnomics , 85: 409 434. Roberts, M.J. and Spence, M., 1976 . Effluent charges and licenses under uncertainty. Journal of Pubic Economics 5: 193 208. Schrank, D., Eisele, B., Lomax, T., 2012 . urban mobility report. Texas A&M Transportation Institute. The Texas A&M University System, 2012. Small, K. A., 1992 . Using the revenues from congestion pricing. Transportation 19 (4), 359 381. Smith, M. J., 1984 . The exis tence of a time dependent equilibrium distribution of arrivals at a single bottleneck. Transportation science 18 (4), 385 394. Stavins, R. N., 1995 . Transaction costs and tradeable permits. Journal of environmental economics and management 29 (2) , 133 148. Tian, L . J., Yang, H., Huang, H. J., 2013 . Tradable credit schemes for managing bottleneck congestion and modal split with heterogeneous users. Transportation Research Part E: Logistics and Transportation Review 54 , 1 13. Tietenberg, T., 2005 . Tradable pe rmits in principle and practice. Penn State Env ironmental Literature Review 14 , 251. Tsekeris, T., Voss, S., 2009. Design and evaluation of road pricing: state of the art and methodological advances. NETNOMICS: Economic Research and Electronic Networking 1 0 , 5 52. Tobin, R.L., Friesz, T.L., 1988. Sensitivity analysis for equilibrium network flow. Transportation Science 22 , 242 250 . van den Berg, V. A., 2014 . Coarse tolling with heterogeneous preferences. Transportation Research Part B: Methodological, 64, 1 23. Verhoef, E., Nijkamp, P., Rietveld, P., 1997 . Tradeable permits: their potential in the r egulation of road transport externalities. Environment and Planning B 24 , 527 548. Vickrey , W. S., 1969 . Congestion theory and transport investment. The American Economic Review , 251 260 Wang, X., Yang, H., Zhu, D., Li, C., 2012. Tradable travel credits for congestion management with heterogeneous users. Transportation Research Part E: Logistics and Transportation Review , 48 (2), 426 437. Wang, X., Yang, H., 2012 . B isection based trial and error implementation of marginal cost pricing and tradable credit scheme. Transportation Research Part B: Methodological 46 (9), 1085 1096. PAGE 85 85 Wang, X. , Yang, H., Han, D., Liu, W., 2014 . Trial and error method for optimal tradable cre dit schemes: The network case. Journal of Advanced Transportation 48 (6), 685 700. Weitzman, M.L., 1974. Prices vs. quantities. Review of Economic Studies 41 (4) , 477 491 . Wu, D., Yin, Y., Lawphongpanich, S., Yang, H., 2012 . Design of more equitable conges tion pricing and tradable credit schemes for multimodal transportation networks. Transportation Research Part B: Methodological , 46 (9), 1273 1287. Yang, H., 1997. Sensitivity analysis for the elastic demand network equilibrium problem with applications, T ransportation Research Part B 31 (1) , 55 70. Yang, H. and Huang, H.J., 2005. Mathematical and Economic Theory of Road Pricing , Elsevier. Yang, H., Wang, X., 2011 . Managing network mobility with tradable credits. Transportation Research Part B: Methodologic al 45 (3), 580 594. Yang, H., Meng, Q., Lee, D. H., 2004 . Trial and error implementation of marginal cost pricing on networks in the absence of demand functions. Transportation Research Part B: Methodological 38 (6), 477 493. Ye, H., Yang, H., 2013 . Contin uous price and flow dynamics of tradable mobility credits. Transportation Research Part B: Methodological 57 , 436 450. Zhang, X., Yang, H., Huang, H. J. , 2011 . Improving travel efficiency by parking permits distribution and trading. Transportation Research Part B: Methodological , 45 (7), 1018 1034. Zhu, D. L., Yang , H., Li, C. M., Wang, X. L., 2014 . Properties of the m ulticlass t raffic n etwork e quilibria u nder a t radable c redit s cheme. Transportation Science (in press) PAGE 86 86 BIOGRAPHICAL SKETCH Nima Shirmohamma di admitted into the PhD program in the Civil Engineering D epartment at the University of Florida in 2012. He received his B.Sc. in c ivil e ngineering from University of Tehran, Iran and M.Sc. in t ransportation e ngineering from Sharif University of Tech nology, Iran. He received his Ph.D. in civil engineering and his M.Sc. in i ndustrial a nd s ystems engineering from University of Florida in December 2015 . |