CONTINUOUS PRICE AND FLOW DYNAMICS OF TRADABLE MOBILITY CREDITS Hongbo YE and Hai YANG The Hong Kong University of Science and Technology 21/12/2012

Outline  Introduction  Continuous price and flow dynamics  Homogeneous value of time (VOT)  Fixed demand  Numerical example  Conclusion 2

Introduction 1.

Background Travel Demand Management  Rationing  Direct and expedient  May lose effectiveness in the long run  Pricing  Effective and efficient  Not equal among different income levels, government is not revenue-neutral 4

 Tradable Credit Scheme: Yang and Wang (2011)  The social planner initially distributes a certain number of credits to all eligible travelers set the expiration date of the current credits charges a link-specific number of credits from travelers using each link allows free trading of the credits among travelers  Effective, efficient, equitable, revenue-neutral  Unique equilibrium flow pattern and credit price Literature Review In practice, price and flow may fluctuate from day to day. 5 Static case

 Day-to-day Dynamics of Traffic Flows  Smith (1984)  Cascetta (1989)  Friesz et al. (1994)  Zhang and Nagurney (1996)  Watling and Hazelton (2003)  Cho and Hwang (2005)  Yang and Zhang (2009) Literature Review 6

Day-to-day Dynamics of Traffic Flows  Horowitz (1984)  Discrete-time dynamic process, stochastic UE, two-link network  Time-varying learning parameter would affect system’s stability through day-to-day evolution  Cantarella and Cascetta (1995)  A general framework of day-to-day traffic dynamics based on path flow’s demand and supply interaction  Existence, uniqueness and stability of equilibrium conditions  Watling (1999)  The stability of a general network taking a specific form in Cascetta and Cantarella (1995)  Bie and Lo (2010)  Stability and attraction domain Literature Review 7

Objective  How the traffic flow and credit price will impact each other and evolve together, considering  Travelers’ learning behavior on travel time  Travelers’ route choice behavior based on their perceived path travel cost  Price adjustment rule according to the fluctuation of credit demand and supply 8

Price and Flow Dynamics 2.

Notations 10

Notations 11

Model Assumptions ① Travelers’ learning behavior. Travelers update their perception of path travel times based on their previous perception and new traffic information. 12 Real Traffic InformationPerception >0

Model Assumptions ② Travelers’ route choice. Probabilities for travelers choosing routes depend on the perceived travel time on all the routes. 13

Model Assumptions ③ Credit price adjustment. The credit price depends on the expected daily excess credit demand, defined as the difference between the credits consumed on the current day and the average credits per day available during the rest of the period. 14

Model Assumptions ③ Credit price adjustment. 15 daily credit demand total available credtits remaing time daily credit supply

Continuous Evolution Model Combine the three assumptions with initial conditions 16

Trajectories of price and flows Credit Supply Credit Consumption 17

Existence of the Equilibrium Point Brouwer’s fixed point theorem. Every continuous function from a convex compact subset of a Euclidean space to itself has a fixed point. 18

Existence of the Equilibrium Point 19

Uniqueness of the Equilibrium Point 20

Existence and Uniqueness of Equil. Point 21

System Stability time-variant system time-invariant system 22

System Stability 23

System Stability 24

System Stability price adjustment function Link travel time function Logit model can satisfies (c) 25 credit charging scheme

Numerical Example 3.

Numerical Example 27

Numerical Example (1) Price evolution with different lengths of time horizon and different initial prices 28

Numerical Example (2)  Evolution of perceived travel time with different initial values 29

Numerical Example (3)  Sensitivity of equilibrium points with respect to different credit schemes 30

Numerical Example (3)  Sensitivity of equilibrium points with respect to different credit schemes 31

Numerical Example (4)  Influence of system parameters on price evolution 32

Conclusion 4.

Conclusion  A continuous-time model to describe the dynamics of price and perceived travel time under the tradable credit scheme based on fixed demand and homogeneous VOT considering travelers’ route choice and learning behavior price adjustment process with the variation of credit demand and supply  Some important property of the dynamic model Existence and uniqueness of the evolution trajectories Existence and uniqueness of the equilibrium point Stability and convergence when time horizon goes infinite 34

Conclusion  Numerical example The choice of time horizon of the credit scheme is critical for the system performance, especially the stability and convergence issues of the system. When time is long enough, the system with different initial conditions will eventually be stable and convergent. Choosing a proper credit scheme is also critical. 35

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