Analysis and Design of Markets for Tradable MobilityCredit Schemes
Siyu Chen, Ravi Seshadri, Carlos Lima Azevedo, Arun P. Akkinepally, Renming Liu, Andrea Araldo, Yu Jiang, Moshe E. Ben-Akiva
AAnalysis and Design of Markets for Tradable MobilityCredit Schemes
Siyu Chen a, ∗ , Ravi Seshadri b , Carlos Lima Azevedo c , Arun P. Akkinepally d ,Renming Liu c , Andrea Araldo e , Yu Jiang c , Moshe E. Ben-Akiva a a Civil and Environmental Engineering Department, Massachusetts Institute of Technology,Cambridge, MA, US b Singapore-MIT Alliance for Research and Technology, Singapore c Department of Technology, Management and Economics, Technical University ofDenmark, Denmark d Caliper Corporation, United States e T´el´ecom SudParis - Institut Polytechnique de Paris, France
Abstract
Tradable mobility credit (TMC) schemes are an approach to travel demand man-agement that have received significant attention in the transportation domainin recent years as a promising means to mitigate the adverse environmental,economic and social effects of urban traffic congestion. In TMC schemes, aregulator provides an initial endowment of mobility credits (or tokens) to allpotential travelers. In order to use the transportation system, travelers need tospend a certain amount of tokens (tariff) that could vary with their choice ofmode, route, departure time etc. The tokens can be bought and sold in a marketthat is managed by and operated by a regulator at a price that is dynamicallydetermined by the demand and supply of tokens.This paper proposes and analyzes alternative market models for a TMC sys-tem (focusing on market design aspects such as allocation/expiration of credits,rules governing trading, transaction costs, regulator intervention, price dynam-ics), and develops a methodology to explicitly model the dis-aggregate behaviorof individuals within the market. Extensive simulation experiments are con-ducted within a departure time context for the morning commute problem to ∗ Corresponding author
Email address: [email protected] (Siyu Chen) Corresponding Author
Preprint submitted to Journal of L A TEX Templates January 5, 2021 a r X i v : . [ ec on . GN ] J a n ompare the performance of the alternative designs relative to congestion pric-ing and a no control scenario. The simulation experiments employ a day today assignment framework wherein transportation demand is modeled using alogit-mixture model and supply is modeled using a standard bottleneck model.The results indicate that when the actual network equilibrium does not deviatefrom the forecasted equilibrium (used by the regulator to design the toll struc-ture, which varies by time-of-day but is not day-to-day adaptive), the optimizedTMC system can achieve an identical social welfare as congestion pricing in theabsence of transaction costs (and only a marginally lower welfare in the pres-ence of transaction costs). On the other hand, when the forecasted and actualequilibria are different, the TMC system can yield efficiency gains over conges-tion pricing due to the price adjustment mechanism of the market (with andwithout transaction costs). The TMC system yields similar efficiency gains alsowhen step tolls are used (common in practice) and further, is more robust tosupply/demand shocks when the allocation of tokens occurs in continuous time.Finally, the results highlight the importance of transaction costs and decayingprices until expiration in mitigating undesirable behavior in the market.The paper addresses a growing and imminent need to develop methodologiesto realistically model TMCs that are suited for real-world deployments and canhelp us better understand the performance of these systems – and the impactin particular, of market dynamics. Keywords:
Tradable Mobility Credits; Demand Management; HumanBehavior; Traffic Management; Simulation
1. Introduction
Historically, transportation network inefficiencies such as congestion and ve-hicular emissions have been addressed through road pricing, which althoughused in several cities worldwide, is plagued by issues of inequity and political andpublic acceptability (Tsekeris and Voß, 2009; de Palma and Lindsey, 2011). Analternative approach to travel demand management that has received increas-2ng attention in the transportation domain in recent years is quantity control– in particular, tradable mobility credit (TMC) schemes (Fan and Jiang, 2013;Grant-Muller and Xu, 2014; Dogterom et al., 2017). Within a TMC system,a regulator provides an initial endowment of mobility credits to all potentialtravelers. In order to use a transportation system, users need to spend a certainnumber of permits (i.e.: tariff) that could vary with the conditions/performanceof the specific mobility alternative used. The permits can be bought and soldin a market that is monitored by the regulator at a price that is determined bydemand and supply interactions.In principle, TMC schemes are appealing since they offer a means of directlycontrolling quantity (important when the elasticity of demand to prices in theshort term may be low), they are revenue neutral in that there is no transfer ofmoney to the regulator, and they are viewed as being less vertically inequitablethan congestion pricing (de Palma and Lindsey, 2020). Despite these promises,several important questions remain with regard to the design and functioningof the market within TMC schemes, an aspect critical to the effective oper-ationalization of these schemes. For instance, how should the allocation andexpiration of tokens be designed? What rules should govern trading behaviorin the market so as to avoid undesirable speculation and trading (see Brandset al. (2020) for more on this), and yet ensure efficiency and revenue neutrality?How should the regulator intervene in the market in the presence of special ornon-recurrent events? What is the role and impact of transaction costs? Despitethe large body of literature on TMCs, issues of market design, market dynamicsand behavior of individuals in the market has received relatively less attention.This paper aims to address these issues and contributes to the existing lit-erature in several respects. First, we propose alternative market models (focus-ing on all aspects of market design including allocation/expiration of credits,rules governing trading, transaction costs, regulator intervention, price dynam-ics) for a TMC system, and develop a methodology that explicitly models thedis-aggregate behavior of individuals within the market. Second, we conduct ex-tensive simulation experiments within a departure time context for the morning3ommute problem to compare the performance of the alternative designs rela-tive to congestion pricing and a no control scenario. The simulation experimentsemploy a day to day assignment framework wherein transportation demand ismodeled using a logit-mixture model and supply is modeled using a standardbottleneck model. The experiments yield insights into market design and thecomparative performance of the TMC system relative to congestion pricing.The results indicate that when the actual network equilibrium does notdeviate from the forecasted equilibrium (used by the regulator to design thetoll structure, which varies by time-of-day but is not day-to-day adaptive), theoptimized TMC system can achieve an identical social welfare as congestionpricing in the absence of transaction costs (and a marginally lower welfare inthe presence of transaction costs). On the other hand, when the forecasted andactual equilibria are different, the TMC system can yield efficiency gains overcongestion pricing due to the price adjustment mechanism of the market (withand without transaction costs). The TMC system yields similar efficiency gainsalso when step tolls are used (common in practice) and further, is more robust tosupply/demand shocks when the allocation of tokens occurs in continuous time.Finally, the results highlight the importance of transaction costs and decayingprices until token expiration in mitigating undesirable behavior in the market.The TMC system that we propose (termed Trinity hereafter) comprises threemain components, an online bi-level optimization model, a market model anda smartphone app (see Figure 1). From the user’s perspective, Trinity is asmartphone app which includes (1) a personalized trip planner and (2) inter-faces for the user’s token account and (3) trading. Users are endowed with atoken budget, which is obtained through a subscription or free allocation bythe regulator. Prior to a trip, travelers open the trip planner, which presentsa menu with different travel alternatives along with their predicted attributes.Each alternative is also associated with a token tariff, which can be chargeddepending on the alternative-specific contribution to the system’s congestion.The Trinity app tracks and verifies realized trips for token charging. Moreover,Trinity learns individual user’s preferences from previous choices and presents4ersonalized menus, which increases the user’s benefit (Song et al., 2018).
Figure 1: Trinity Architecture Flowchart
The second component of Trinity is the bi-level optimization module, whichis responsible for setting the token charges or tariff for each travel alternative inreal time (‘system-level’ optimization) and providing personalized ‘user-optimal’menu’s to travelers (‘user-level’ optimization). The system-level optimizationutilizes a simulation-based predictive system that uses real-time data from themarket and from sensors in the transportation system (Araldo et al., 2019). Theoverall policy objectives for Trinity in terms of congestion, emissions, networkperformance, quality of service and sustainability is defined via the system-leveloptimization.The third component of Trinity is the Token Market in which users cansell or buy tokens. If a user chooses a travel alternative associated with acertain token amount and her/his token budget is insufficient, she/he can buythe remaining needed tokens. On the other hand, a user can sell excess tokensin her/his budget at any time. The token market price at which these exchangesoccur adjusts dynamically based on demand and supply of tokens. If demand5xceeds supply, the price increases and vice-versa. The market enables Trinityto achieve revenue neutrality, which means the system avoids taxes, user chargesor incentive funding programs. The operator can also intervene in the market,reducing or increasing the number of tokens available and thus allowing for abetter management of non-recurrent situations.As noted previously, our key focus in this paper is on the analysis and designof the market within Trinity. Future research will focus on other aspects of theTrinity system including the online bi-level optimization framework. The restof the paper is organized as follows.
2. Review of Literature
Although early work on the use of tradable mobility credits (TMCs; alsotermed TCS or Tradable Credit Schemes in the literature) in transportationdate back several years (Verhoef et al., 1997; Raux, 2007; Goddard, 1997), for-mulations of the market and network equilibrium for TMCs is more recent,pioneered by the work of Yang and Wang (2011) who proposed a user equilib-rium variant for a TMC. Their work, along with advancements in technologyand the widely recognized limitations of congestion pricing, has spurred interestin TMCs for transportation network management. Extensive reviews may befound in Grant-Muller and Xu (2014); Fan and Jiang (2013); Dogterom et al.(2017). We provide a brief summary of existing literature, limiting our atten-tion to that of mobility management (in the context of both entire networksand single bottlenecks) although applications may also be found in parking.In the model of Yang and Wang (2011), the regulator distributes a pre-specified number of credits to travelers, charges a link-specific credit tariff andallows trading of credits within a market. They demonstrate that for a given setof credit rates in a general network, the user equilibrium (UE) link flow patternis unique under standard assumptions and identify additional conditions (rela-tively mild) to ensure uniqueness of the credit price at the market equilibrium.Extensions to their model have been proposed to incorporate heterogeneity in6he value of time (Wang et al., 2012) and multiple user classes (Zhu et al., 2015)using variational inequality formulations to establish existence and uniquenessproperties of the network and market equilibrium. He et al. (2013) employ asimilar equilibrium approach considering allocations of credits to not just in-dividual travelers, but to transportation firms such as logistics companies andtransit agencies; the effect of transaction costs in a TMC scheme with two typesof markets (auction-based and negotiated) is considered by Nie (2012). In con-trast with the aforementioned TMC schemes, Kockelman and Kalmanje (2005);Gulipalli and Kockelman (2008) proposed a system of credit-based congestionpricing (termed CBCP) where credits are allowances used to pay tolls.While the studies discussed thus far have examined the application of TMCstypically in a route choice setting at the network level, several researchers haveproposed TMC schemes in the context of managing congestion at a single bot-tleneck (or simple two route networks) by achieving peak spreading. Nie andYin (2013) model a tradable credit scheme that manages commuters’ travelchoices and attempts to persuade commuters to spread their departure timesevenly within the rush hour and between alternative routes (see also Nie (2015))whereas Tian et al. (2013) investigate the efficiency of a tradable travel creditscheme for managing bottleneck congestion and modal split in a competitivehighway/transit network with a continuously distributed value of time. Alongrelated lines, Xiao et al. (2013) study a tradable credit system (consisting ofa time-varying credit charged at the bottleneck wherein the credits can betraded and the price is determined by a competitive market) to manage morn-ing commute congestion with both homogenous and heterogeneous users. Morerecently, Bao et al. (2019) studied the existence of equilibria under tradablecredit schemes using different models of dynamic congestion and Akamatsu andWada (2017) proposed a tradable bottleneck credit scheme where the regulatorissues link- and time-specific credits permitting passage through a certain linkor bottleneck in a pre-specified time period. They develop a model to describetime-dependent flow patterns at equilibrium under a system of tradable bot-tleneck permits for general networks and show that the equilibrium obtained7nder this system is efficient in that it minimizes the social transportation cost.In contrast with the previously described literature that largely focus onvariants of the standard user equilibrium under TMC schemes, a related streamof research examines the design of the TMC schemes/network design using bi-level optimization formulations in different contexts (Wu et al., 2012; Bao et al.,2017; Wang et al., 2014). On the other hand, the comparison of efficiencyproperties of tradable credits and congestion pricing has received relatively lesserattention. de Palma et al. (2018) performed a comparative analysis of the twoinstruments in a simple transportation network and showed that as long asthere is no uncertainty, price and quantity regulation are equivalent as in theregular market case studied by Weitzman (1974). In the presence of uncertaintyand strongly convex congestion costs, the TMC instrument outperforms thepricing instrument in efficiency terms. Akamatsu and Wada (2017) reachedsimilar conclusions (see also Shirmohammadi et al. (2013)), demonstrating theequivalence of the tradable permit system and a congestion pricing system whenthe road manager has perfect information of transportation demands. On thebehavior side, several stated preference studies have highlighted the importanceof key behavioral economics and cognitive psychology factors towards TMCDogterom et al. (2017) (see also Brands et al. (2020) for an interesting real-world experiment with tradable credits).In summary, despite the large body of research on TMCs, the modeling ofthe market has received little attention and almost all the studies employ anequilibrium approach to model the credit market (with the notable exceptionof Ye and Yang (2013) who model the price and flow dynamics of a tradablecredit scheme). Further, the literature has –to the best of our knowledge– thusfar not attempted to model realistically the disaggregate behavior of individualswithin the market that could enable the consideration of empirically observedphenomena such as loss aversion, endowment effects, mental accounting, day-to-day learning Dogterom et al. (2017). Finally, there is also the imminentneed to examine design apsects of the credit market including features such astoken allocation/expiration, trading, intervention, and transaction costs and the8mpact of these on behavior of individuals in the market. This paper aims toaddress some of these gaps.
3. Market design
Traditional road pricing which charges money for traveling (along certainroutes or at certain departure times) can be thought of in a TMC perspective,as a TMC system in which only buying is allowed. Specifically, instead ofcharging a time-of-day toll (in units of dollars), the regulator charges the sametoll but in electronic tokens and maintains a fixed token market price of $ $ r over entireday and each token has a lifetime L to avoid speculation. Let x dn ( t ) denotetraveler n ’s token account balance at time t on day d and a Bernoulli indicator I dn ( t ) denote whether the traveler n is at a full wallet ( F W ) state or not. Afull wallet state indicates that the number of tokens in the wallet has reached amaximum and, in the absence of travelling or selling, does not change since theacquisition of new tokens is balanced by an expiry of old tokens. Thus, a fullwallet implies that the oldest token in her account has an age L . In contrast,when the account is not in a full wallet state, it increases by an amount r ∆ t ina time interval ∆ t .In this study, in order to avoid quantity buildup and market manipulation,1) travelers can only buy tokens from regulator only at time of traveling forimmediate use if they are short of tokens, 2) they can sell tokens to regulator butthey have to sell all, and 3) buying and selling cannot happen at the same time,i.e. travelers can sell all tokens anytime except at time of traveling. Let T ( t )denote toll in electric tokens of travel alternative at time t and td dn representsdeparture time of traveler n on day d . At time t on day d , traveler n can performone and only one of the following actions:1. Perform a trip if t = td dn . • If a traveler n has enough tokens to perform the trip at time td dn ,i.e., x dn ( td dn ) ≤ T ( td dn ), she consumes T ( td dn ) from her account. Heraccount balance at time td dn + ∆ t is equal to x dn ( td dn + ∆ t ) and can10e written as: x dn ( td dn + ∆ t ) = x dn ( td dn ) − T ( td dn ) + r ∆ t (1) • Otherwise, she needs to buy T ( td dn ) − x dn ( td dn ) tokens at the price p d ( td dn ) plus additional transaction cost from the market. Her ac-count balance x dn ( t dn + ∆ t ) becomes: x dn ( t dn + ∆ t ) = r ∆ t (2)2. Does nothing. Her account balance x dn ( t + ∆ t ) becomes: x dn ( t + ∆ t ) = x dn ( t ) + I dn ( t ) r ∆ t (3)3. Sells all tokens x dn ( t ). Traveler n chooses to sell all her tokens and heraccount balance becomes: x dn ( t + ∆ t ) = r ∆ t (4)The buying price of tokens at time t on day d is equal to token market price p d ( t ). However, the selling price of tokens decays linearly as tokens expire inorder to account time values of tokens and avoid quantity buildup. In addition,there are two-part (fixed and proportional) transaction costs for both buyingand selling transaction. Let c s and c b denote proportional part of selling andbuying transaction costs; T C s and T C b denote fixed part of selling and buyingtransaction costs. Therefore, selling revenue of y tokens at time t can be writtenas S d ( y, t ) = min( y, F W ) p ds ( t ) − p ds y r − T Cs (5)where p ds ( t ) = p d ( t )(1 − c s ) and the quadratic term is to account for decayingtime values of tokens.Buying cost of y tokens at time t on day d can be written as B d ( y, t ) = yp db ( t ) + T C b (6)11here p db ( t ) = p d ( t )(1 + c b ).At time t on day d , assume traveler n has plan about upcoming travelalternative, she is able to calculate profit of selling now assuming no furtherselling until departure as follows prof it dn ( t ) = min( S d ( x n ( t ) , t ) , S d ( F W, t )) − I ( T ( td dn ) > F A ) B d ( T ( td dn ) − F A, t )(7)where
F A represents future allocation and is equal to min(( td dn − t ) r, F W ); T ( td dn ) represents toll in tokens of travel alternative at departure time td dn .Buying cost only occurs if toll is greater than traveler n ’s future allocation T ( td dn ) ≥ min(( td dn − t ) r, F W ).At time t on day d , traveler n considers selling only if profit value is positive( prof it dn ( t ) > I ( T ( td dn ) > F A ) = 1 (need to buy), it can be written as dprof it dn ( t ) dt = − rp db ( t ) x dn ( t ) = F Wrp ds ( t ) − x dn ( t ) p ds ( t )24 − rp db ( t ) otherwisewhich is always negative if p ds ( t ) ≤ p db ( t ). This implies that future profitis guaranteed to be less than current profit. She should sell now if profitis positive.2. When I ( T ( td dn ) > F A ) = 0 (no need to buy), it can be written as dprof it dn ( t ) dt = x dn ( t ) = F Wrp ds ( t ) − x dn ( t ) p ds ( t )24 otherwisewhich implies that current profit increases until reaching full wallet. Notethat profit function is not continuous so it will stop increasing once tollbecomes equal to future allocation.12n summary, selling strategy can be summarized as follows. Algorithm 1:
Selling Rule input: d, t, n, p d ( t ) , td dn , x dn ( t )Calculate prof it n ( t ); if prof it ( t ) > thenif T ( t d ) > min(( t d − t ) r, F W ) then Sell now; else if T ( t d ) < min(( t d − t ) r, F W ) thenif x ( t ) = F W then
Sell now; else
Do nothing; endelse
Sell now; endelse
Do nothing; end
The market place dictates the token price p d on day d and the policy toestablish it is defined a priori by the regulator. The price p d is modified dailywith a deterministic rule considering previous day’s regulator revenue R d − asfollows p d = p d − R d − ∈ [ − RBT D, RBT D ] p d − + ∆ p R d − < − RBT Dp d − − ∆ p R d − > RBT D (8)where ∆ p currently is a constant parameter representing price change amount. RBT D ensures that price will not fluctuate for small regulator revenue. Priceis ensured to be positive and below a certain cap p max as follows:13 d = max(0 , min( p d , p max )) (9)Although token price is constant within day for the most of time, regulatoris able to control token price for a period time in one day to accommodateunusual events. For example, if road capacity drops because of a large concertor extreme weather, regulator is able to increase token price in peak hour todiscourage travel in peak period. Numerical experiments are conducted to studythis in section 5.
4. Experiments
In this study, we conduct simulation-based experiments to assess three sce-narios: No Toll (NT), Congestion Pricing (CP), and Trinity. As we can seefrom Figure 2, for a given scenario, travelers use forecasted information, includ-ing travel time, departure time, and account balance, to make mobility decisionin the beginning of a day (Preday). For the sake of simplicity, travelers arehomogeneous and each one performs one morning trip per day without can-cellation allowed in this study. The network considered is a single OD pairconnected by a single link. Thus the mobility decision is departure time choiceonly, which is modeled by a discrete logit model. Determined departure timechoice will be simulated in the network within day along with rule-based tradingdecisions described in previous section. Congestion is modeled by a point queuemodel, in which a queue develops once flow exceeds capacity. The demand andsupply interaction is modeled by a dynamic process, which explicitly simulatesthe evolution of the system state considering day to day learning. Exponentialsmoothing filter is adopted to update forecasted information with realized costsif system state (e.g. travel time and flow) is not stationary across days. Wecompute social welfare at stationary state and compare it across scenarios.In next sections, we first describe formulations for a system model (travelersand network components in Fig.1) of commuters’ departure time choice in a14 tartScenario(NT/CP/Trinity)Forecasted info (e.g.travel time, departuretime, account balance)Preday mobility decisionNetwork; TradingStationary?Compute welfare Update forecasted infostopyes no
Figure 2: Flowchart of stochastic simulations. Travelers’s departure time choices determinedby discrete logit model using forecasted information in the beginning of the day are simulatedon a single link with a finite capacity along with trading activities. If system is not stationary(flow and travel time differ across days), forecasted information will be updated using realizedinformation. Social welfare will be computed at the end and compared across scenarios. single corridor and welfare computation for NT, CP and Trinity. Then, wepresent a variant Trinity in which regulator adopts a strict token quantity controlto control flow pattern. Finally, we propose different experiments to analyzeefficiency and robustness of Trinity. Important notation is shown in Table 1.15 able 1: Important notation
Variables Description h Time interval t Simulation time step d Day dt h Start time of interval h ∆ h Duration of time interval∆ t Time step α Value of time β E , β L Value of schedule delay early/late β D Value of delay due to postpone µ Scale parameter of random component (cid:15) d ( h ) Random utility component on day dp Fixed market price as $ h ˜ τ d ( h ) Forecasted travel time in h on day d ˜ ω d ( h ) Forecasted delay time in h on day d ˜ w d ( h ) Forecasted delay time in number of intervals on day dx n ( t ) Account balance of individual n at time t L Token lifetime r Token allocation rate t Free flow travel time t v ( t ) Waiting time in queue at tD ( t ) Number of drivers in queue at tτ d ( t ) Travel time at t on day dτ d ( h ) Averaged travel time of interval h on day dθ τ /θ ω /θ ψ Weights on previous day’s forecasts16 .2. System model
The setting we consider involves a single origin-destination pair connectedby a path containing a bottleneck of finite capacity. Unlike the classical bottle-neck model proposed by Vickrey (1969), users wish to arrive at the destinationwithin a certain “preferred arrival time window” in the morning, and can adjusttheir departure times to avoid congestion (similar to the model in Ben-Akivaet al. (1984), which is a dynamic extension of De Palma and Lefevre (1983)).In addition, the system is modeled using a stochastic process approach thatincorporates day-to-day and can be viewed as a simplification of the modelin Cascetta and Cantarella (1991), which considers the stochastic assignmentproblem in general networks. Day to day adjustment is modeled using suitablelearning and forecasting filters and a logit model is used to model within-daydeparture time decisions. We refer to Cantarella and Cascetta (1995) for anuanced discussion of terminology and a detailed description of deterministicand stochastic process models (with probabilistic assignment or a probabilisticmodel for users’ choice behavior). They propose conditions for existence anduniqueness of fixed-point attractors of the deterministic process, which extendresults for the traditional user and stochastic user equilibrium. In case of thestochastic process, conditions for regularity are proposed which ensure existenceand uniqueness of a stationary distribution of network states. It is noted thatthe model of Ben-Akiva et al. (1984) may be viewed as a deterministic processmodel with probabilistic assignment.The travel behavior model, network model, and demand-supply interactionsare discussed in detail next.
In this study, traveler’s only travel behavior —departure time choice— ata unit time interval h being chosen within a set of mutually exclusive possibledeparture time intervals { , ..., H } is modeled by a discrete logit model (similarto continuous logit departure time choice model considered in Ben-Akiva andWatanatada (1981)). According to random utility theory, let V d ( h ) be a sys-17ematic monetary utility (utility in unit of dollars) for departing at time interval h on day d and (cid:15) d ( h ) be independently and identically distributed as Gumbelrandom variables with zero mean and scale parameter µ . The probability ofdeparting in a time interval h on day d is written as P d ( h ) = exp( V d ( h ) µ ) (cid:80) Hh =1 exp( V ( h ) µ ) (10)in which a time interval h contains several simulation time steps t basedon simulation setting and travelers will randomly select a time point to departwithin a chosen time interval h .Systematic monetary utility V d ( h ) consists of four cost components thataffect the choice of departure time: forecasted travel time, schedule delay early,schedule delay late and forecasted toll (0 for NT). For a traveler, the marginalutility of an additional unit of travel time is α . For simplicity, we assumetravelers have common knowledge of forecasted travel time. Denote her desiredtime period for arrival as [ t ∗ − ∆ , t ∗ + ∆], where t ∗ represents the center of theperiod and ∆ represents arrival flexibility. If she arrives outside of desired timeperiod, she suffers a schedule delay. The marginal utility of an additional unitof schedule delay early is β E and an additional unit of schedule delay late is β L , which β E ≤ α ≤ β L according to empirical results (Small (1982)). UnderCP scheme, forecasted toll cost incurred at time interval h is equal to toll indollars charged at h , p CP ( h ). For Trinity, forecasted toll cost ˜ c dn ( h ) is basedon individual forecasted account balance on day d . Thus, the utility of anindividual n can be written as U dn ( h ) = V dn ( h ) + (cid:15) dn ( h ) (11)= − α ˜ τ d ( h ) − β E SDE ( h, t ∗ ) − β L SDL ( h, t ∗ ) (12) − ˜ c dn ( h ) + (cid:15) ( h ) (13)18here SDE ( h, t ∗ ) = max(0 , t ∗ − ∆ − ( h + ˜ τ d ( t h ))) (14) SDL ( h, t ∗ ) = max(0 , ( h + ˜ τ d ( t h )) − t ∗ − ∆) (15)˜ c dn ( h ) represents forecasted toll cost and depends on individual forecastedaccount balance as follows: c dn ( h ) = max( S d (T( h ) , t ) ,
0) ˜ x dn ( h ) ≥ T( h ) B d (T( h ) − ˜ x dn ( h ) , t ) + max( S d (˜ x n ( h )) , t ) ,
0) otherwise (16)If forecasted account balance is greater than toll in tokens (T( h )), the oppor-tunity cost of paying toll is equal to selling revenue of those tokens; otherwise,the opportunity cost is equal to selling revenue of current balance plus buyingcost of additional tokens. If selling price does not decay and transaction costsare zero, toll cost reduces to toll in tokens times token market price on day d ˜ c dn ( h ) = T( h ) p d . The network is assumed to be a single origin-destination pair connected bya single path containing a bottleneck of fixed capacity s (Arnott et al., 1990).A first-in-first-out (FIFO) queue develops once the flow of traveler exceeds s .The free flow travel time is t and the waiting time for a traveler at time step t is t v ( t ). Thus, the total travel time for a traveler at time t is: τ ( t ) = t v ( t ) + t (17)Let D ( t ) be the number of travelers in the queue at time t . The waitingtime at time t is derived from the deterministic queuing model as follows: t v ( t ) = D ( t ) s (18)19here D ( t ) = 0 and t v ( t ) = 0 when there is no congestion.Note that a time interval contains several time steps and travel time of atime interval is equal to averaged travel time of all travelers departing in thattime interval . Let d denote the index for the current day and τ d − ( h ) be the experiencedtravel time on day d − h . As we specified in mobility model,travelers are assumed to make their choices of departure time according toforecasted travel time ˜ τ d ( h ) from their memory and learning. In this study,we use an exponential smoothing filter, a type of homogeneous filter Cantarellaand Cascetta (1995), to model the learning and forecasting process by weightingactual and forecasted costs of previous day as follow:˜ τ d ( t ) = θ τ ˜ τ d − ( t ) + (1 − θ τ ) τ d − ( t ) , ∀ θ τ ∈ [0 ,
1] (19)where θ τ is learning weight for previous day’s forecasted travel time.We apply similar filter on individual forecasted departure time td dn on day d also as follows: ˜ td dn = θ td ˜ td d − n + (1 − θ td ) td d − n , ∀ θ td ∈ [0 ,
1] (20)With forecasted departure time, we can apply rule-based trading model toget forecasted individual account balance.
In addition to generic Trinity described in previous section, we also considera variant Trinity with token supply cap in this study, named as Trinity TSC. ForTrinity TSC, travelers are not able to sell tokens. Also, there is no token priceadjustment and price is fixed as $
1. However, regulator controls the maximumnumber of tokens can be bought —token supply cap— as another measure of20ontrol, in order to control the maximum number of vehicles on road (desiredflow cap). Regulator would set the token supply cap equal to the product ofthe desired flow cap and forecasted average number of tokens purchased. Asa result of token supply cap, if a traveler cannot buy extra tokens to travel,she has to postpone and wait in a queue to buy tokens to travel. Essentially,Trinity TSC manages travel flow with a strict quantity control. Changes of travelbehavior model, network model, and demand supply interactions are discussedin following sections.
Due to token supply cap controlled by regulator, some travelers have topostpone their departures if they cannot buy enough tokens to pay toll. Thus,we need to consider forecasted delay due to postponement ( ˜ w d ( h )) for eachtime interval h in mobility model. For a traveler, the marginal utility of anadditional unit of delay due to postponement is β D . Based on the intuitionthat travelers can spend delay due to postponement at home or doing work, weexpect β D ≤ β E . In addition, the effect of delay due to postponement on othercomponents (forecasted travel time, schedule delay and toll cost) should also beconsidered. With the fixed $ h + ˜ w d ( h ) is equal to T ( h + ˜ w d ( h )) times the fixedmarket price p = 1. Thus the utility of an individual n can be written as: U dn ( h ) = V dn ( h ) + (cid:15) d ( h ) (21)= − α ˜ τ d ( h + ˜ w d ( h )) − β E SDE ( h + ˜ w d ( h ) , t ∗ ) − β L SDL ( h + ˜ w d ( h ) , t ∗ )(22) − β D ˜ ω d ( h ) − T( h + ˜ w d ( h )) p + (cid:15) ( h + ˜ w d ( h )) (23)where forecasted delay time is converted to the number of time intervalsdelayed as ˜ w d ( h ) = (cid:98) ˜ ω d ( h )∆ h (cid:99) (24)21DE and SDL incorporating the effect of delay can be written as: SDE ( h, t ∗ ) = max(0 , t ∗ − ∆ − ( t h + ˜ w d ( h ) + ˜ τ d ( h + ˜ w d ( h )))) (25) SDL ( h, t ∗ ) = max(0 , ( t h + ˜ w d ( h ) + ˜ τ d ( h + ˜ w d ( h ))) − t ∗ − ∆) (26) For Trinity TSC, one additional FIFO queue develops once the number oftokens bought in a time interval exceeds the token supply cap set by regulator.In next time interval, delayed travelers will be processed first than those whoplanned to depart.
For Trinity TSC, let ω d − ( h ) be the experienced delay due to postponementon day d − h , and ψ d − ( h ) be the realized average number oftokens purchased at time interval h . As we specified in mobility model, travelersare assumed to make their choices of departure time according to forecastedtravel time ˜ τ d ( h ) and delay ˜ ω d ( h ) from their memory and learning. In addition,regulator sets token supply cap as the product of the desired flow cap andforecasted averaged number of tokens purchased. We apply similar exponentialsmoothing filters to model the learning and forecasting process as follow:˜ ω d ( h ) = θ ω ˜ ω d − ( h ) + (1 − θ ω ) ω d − ( h ) , ∀ θ ω ∈ [0 ,
1] (27)˜ ψ d ( h ) = θ ψ ˜ ψ d − ( h ) + (1 − θ ψ ) ψ d − ( h ) , ∀ θ ψ ∈ [0 ,
1] (28)where θ ω and θ ψ are learning weights on previous day’s forecasted delay andaverage number of tokens purchased. 22 .4. Simulation-based optimization In this study, the social welfare is adopted to measure the performance ofNT, CP and Trinity. For NT, the social welfare is equal to the consumer surplus,which is the sum of individual experienced utilities. For CP, the social welfareis equal to the sum of consumer surplus and regulator revenue. Regulator rev-enue is the sum of users’ out-of-pocket costs, which cancels out with toll costsconsidered in experienced utilities. Thus, the social welfare of CP is equivalentto the sum of travel time cost, schedule delay cost and unobserved attributes.The social welfare of Trinity is equal to the sum of consumer surplus, regu-lator revenue, user revenue and monetary endowment. Regulator revenue is thesum of users’ out-of-pocket payment minus cost of regulator paying for users’selling. User revenue is the sum of individual selling revenue. Monetary endow-ment is the sum of toll paid by endowment tokens because users get endowmenttokens for free. Because money transfer has no net impact on social welfare,welfare equals to the combination of experienced travel time cost, schedule delaycost, and unobserved attributes in the end. It can be written as SW d = N (cid:88) n =1 (cid:2) − ατ dn ( t dd,n ) − β E SDE ( t dd,n , t ∗ ) − β L SDL ( t dd,n , t ∗ ) + (cid:15) dn ( t dd,n ) (cid:3) (29)For Trinity with token supply cap, its welfare also includes experienced delaydue to postponement cost.Optimization is conducted to find the optimal toll (in dollars for CP and intokens for Trinity) which leads to best social welfare. To facilitate optimization,we represent toll profile by a (mixture) of Gaussian functions and optimize func-tion parameters instead. Noted that our optimization problem has no closed-form objectives for the stochastic dynamic simulation, but can be formulated asa simulation-based optimization problem. Our simulation captures the systemmodel presented in 4.2 including all detailed traveler and regulator states andactions along with the resulting network and market conditions.23o solve our simulation-based optimization, a Bayesian optimization (BO)approach is adopted as it can approximate the simulation-based objective func-tion using a few evaluations. Baysian optimization essentially has two iterative steps. First, a model thatapproximates a complex map from the input points (i.e., the parameters to beevaluated) to the output (i.e., objective function value) is updated by addingnew pairs of input and output. Second, the new input point is determined byoptimizing an acquisition function.In this paper, we assume that the objective function values with respect todifferent input points are joint distributed, and adopt a Gaussian Process (GP)to model our social welfare as follows, U ( X ) ∼ GP (cid:0) µ ( X ) , k ( X , X (cid:48) ) (cid:1) (30)where µ ( X ) is the mean function and k ( X , X (cid:48) ) is the covariance kernel functionof the GP. It is worth noting the mean function is defined as the mean of theinput points, and the Matern kernel is used. X is the input points which consistof vectors of parameters to be evaluated. The toll (or token) profile in CP(Trinity) is assumed to be a Gaussian curve with one peak, which has threeparameters: mean, variance and peak value. Specifically, each input point inCP, denoted as X P , consists of these three parameters, while the input point inTrinity, X T , has another (market) parameter ∆ p .Besides, upper confidence bound (UCB) is used as the acquisition function,which has a good performance in practice. To further improve the efficiency andreduce the number of runs of simulation, we use ’Latin Hypercube Sampling’ togenerate the initial sample points of X . In this study, we propose different sets of numerical experiments using sys-tem model described in previous section to assess three scenarios (NT, CP and24rinity) and demonstrate: 1) Trinity functions as expected, 2) the efficiency ofTrinity, 3) the robustness (adaptiveness) of Trinity and 4) market behavior ofTrinity with transaction costs and decaying selling price. In this section, wedescribe experiment settings and experiment results are reported in section 5.Simulation time step is set to 1-min and departure time interval considered is5-min. Choice-set of every individual is the same, which include 157 5-min timeintervals ranging from midnight to 1PM. Other parameter values we use areshown in Table 2.
Table 2: Simulation variables
Variables Description Values N Population 10 , α Value of time $ β E Value of schedule delay early $ β L Value of schedule delay late $ µ Scale parameter of mobility model 0.36 t Free flow time 15 mins t ∗ Center of on-time arrival period 8:15AM∆ Range of on-time arrivals 30 mins s Bottleneck capacity 95/min θ τ /θ ω /θ ψ Learning weights 0.9 L Token lifetime 1440 mins∆ t Simulation step 1 min∆ h Departure time interval 5 minFirst of all, we conduct simulations to inspect the functionality of Trinityand compare its performance with NT and CP. For NT, it does not need any tollprofile, while for CP, its toll profile is optimized through Bayesian Optimizationassuming actual conditions are the same as anticipated conditions. In otherwords, social welfare of CP with optimal toll profile is the best we can achieve.25egarding Trinity, it uses the same toll profile as CP but in unit of tokens. Withappropriate token allocation and functional price adjustment mechanism, tokenprice is supposed to reach $
5. Results and Discussion
In this section, we report results of extensive experiments we conduct. First,we examine efficiency of Trinity TSC and Trinity when 1) anticipated roadcapacity is the same as actual road capacity, 2) actual capacity is lower thananticipated capacity by 25% and 3) using 5-step toll profile (as commonly usedin practice) for Trinity only. Second, we evaluate robustness of Trinity undera sudden capacity reduction and usefulness of continuous allocation. Third, welook into undesired market behavior and analyze the role of the transactioncosts and decaying selling price.
To begin with, we inspects the functionality of Trinity TSC assuming antic-ipated road capacity is the same as actual road capacity. Recall that TrinityTSC utilizes a strict token quantity control which only allows buying tokens.Regulator is able to control token supply cap per time interval in order to con-trol desired flow cap (DFC) per time interval. Also, it uses the same toll profilein tokens as CP and has a fixed token price as $
1. Thus, it should perform thesame as CP when regulator does not control token supply cap and achieves thesystem optimum with toll profile based on anticipated road capacity. Regardlessof magnitude, allocation rate has no effect on social welfare as long as desiredflow cap is not constraining. However, it applies an effect once desired flowcap becomes constraining as shown in Figure 3. This is because flow is con-trolled through token supply cap, which depends on average number of tokenspurchased.Then, we inspects the efficiency of Trinity TSC assuming actual road capac-ity is lower by 25% than anticipated capacity in planning phase and regulator27 igure 3: Effect of AR and DFC on social welfare of Trinity TSC when anticipated capacity isthe same as actual capacity. When DFC is not constraining and AR is large enough, TrinityTSC performs the same as CP, which is the system optimum; the performance of Trinity TSCdecreases as DFC and AR decrease. does not want to update toll profile. CP with toll profile based on anticipatedcapacity does not perform the best and CP with toll profile based on actualcapacity is the system optimum we can achieve. For Trinity TSC with tollprofile based on anticipated capacity, among all flat desired flow caps we havetried, it cannot perform better than CP with toll profile based on anticipatedcapacity as shown in Figure 4. The best Trinity TSC can perform is the sameas CP with toll profile based on anticipated capacity when desired flow cap isunconstraining.In order to better understand why additional control, token supply cap,does not work, we assume regulator of Trinity TSC has some knowledge aboutoptimal flow pattern from CP with toll profile based on lower road capacity(SO) and sets token supply cap based on knowledge of it. Indeed, Trinity TSC28 igure 4: Effect of AR and DFC on social welfare of Trinity TSC when actual capacity is lowerthan anticipated capacity. When DFC is not constraining, Trinity TSC performs the same asCP with toll profile based on anticipated capacity, which is lower than system optimum; theperformance of Trinity TSC decreases as DFC decreases. achieves almost the same flow pattern as Benchmark. As shown in cumulativeflow pattern (Figure 5), cumulative departure and arrival of Trinity TSC overlapwith cumulative departure and arrival of Benchmark almost completely. How-ever, as shown in welfare plot (Figure 5), Trinity TSC has lower welfare thanthat of Benchmark. From components of welfare, we observe that Trinity TSChas similar experienced travel time cost and schedule delay cost as Benchmarkbecause of similar flow pattern and 0 value of delay due to postponement; butTrinity Step 1 has lower unobserved attributes values compared to Benchmark.This implies that travelers in Trinity Step 1 are constrained in decision makingto choose departure intervals with lower unobserved attributes and suffer fromless freedom of making their departure time choice.Assume anticipated capacity is the same as actual capacity and ignore trans-29 a) Cumulative flow of Benchmark andTrinity Step 1 (b)
Welfare of Benchmark and Trinity Step1
Figure 5: Cumulative flow and social welfare comparisons of Benchmark and Trinity TSC.Cumulative actual departure and arrival of Trinity TSC are the same as those of Benchmark.Shaded area is desired time period for arrival. Trinity TSC has lower welfare value becauseof restrictions in decision making. action costs and selling price decaying, Trinity with selling enabled and day today price adjustment can achieves the system optimum if using toll profile intokens based on anticipated capacity. As shown in Figure 6, different allocationrates lead to different equilibrium prices and social welfare. We are able to finda particular allocation rate, which is 78% of maximum toll in tokens charged,to achieve equilibrium price as $ $ a) Day to day price adjustment of differentallocation rates (b)
Equilibrium social welfare of differentallocation rates (c)
Equilibrium regulator revenue of differ-ent allocation rates
Figure 6: Equilibrium price, social welfare, regulator revenue of Trinity by allocation rates.Ignore transaction costs and selling price decaying, Trinity has a functional market place, inwhich users can buy and sell to regulator. Assume anticipated capacity is the same as actualcapacity, Trinity with toll profile based on anticipated capacity and a suitable allocation rateperforms the same as CP and achieves the system optimum. Trinity also achieves revenueneutrality because of price adjustment mechanism. neutrality.
Regarding evaluating the robustness of Trinity, we adopt a symmetric 5-step toll similar to that of Singapore. Without optimizing step-toll profile,Trinity with step toll can perform better than CP with step toll because ofprice adjustment as shown in Figure 8. With optimal step-toll, Trinity and CP31 a) Day to day price adjustment of differentallocation rates (b)
Equilibrium social welfare of differentallocation rates (c)
Equilibrium regulator revenue of differ-ent allocation rates
Figure 7: Equilibrium price, social welfare, regulator revenue of Trinity by allocation rates.Ignore transaction costs and selling price decaying, Trinity has a functional market place,in which users can buy and sell to regulator. Assume actual capacity is the lower thananticipated capacity by 25%, Trinity with toll profile based on anticipated capacity and asuitable allocation rate performs almost the same as CP with toll based on lower capacity(SO). Trinity also achieves revenue neutrality because of price adjustment mechanism. have similar performance as shown in Figure 9.Next, we look into the robustness and adaptiveness of Trinity assume there isroad capacity drop (because of weather or major accidents) from 6AM to 10AMon the 10th day when simulation has already reached an stationary state. Trin-ity regulator has flexibility to regulate token market price, allocation rate, andtransaction cost while users who have not traveled can update their plan accord-ing to new information. Users would also expect a travel time increase during32 a) Day to day price adjustment of differentallocation rates (b)
Equilibrium social welfare of differentallocation rates (c)
Equilibrium regulator revenue of differ-ent allocation rates
Figure 8: Equilibrium price, social welfare, regulator revenue of Trinity by allocation rates.Ignore transaction costs and selling price decaying, Trinity with a sub-optimal step-toll and asuitable allocation rate performs better than CP with the same step-toll. Trinity also achievesrevenue neutrality because of price adjustment mechanism. peak period between 6AM and 10AM. From optimization, the scale of traveltime increase is 1.4 and the period of travel time increase is from 6:40AM to9:35AM. For Trinity with continuous allocation without selling price decaying,we optimize token market price, allocation rate, and transaction costs that canbe controlled by regulator. It can perform much better than no intervention ex-cept broadcasting news, which makes travelers re-plan with increased perceivedtravel time for a period as shown in Figure 10. For Trinity with lump sum al-location, regulator can only regulates token market price. The optimum it can33 a) Day to day price adjustment of differentallocation rates (b)
Equilibrium social welfare of differentallocation rates (c)
Equilibrium regulator revenue of differ-ent allocation rates
Figure 9: Equilibrium price, social welfare, regulator revenue of Trinity by allocation rates.Ignore transaction costs and selling price decaying, Trinity with a optimal step-toll and asuitable allocation rate performs similar to CP with the same step-toll. Trinity also achievesrevenue neutrality because of price adjustment mechanism. achieve is worse than Trinity with continuous allocation as shown in Figure 10.In intuition, this is because toll cost term in utility specification of continuousallocation additionally depends on allocation rate and transaction costs, whichprovides regulator more degree of freedom to regulate.
Finally, we analyze the effect of transaction costs and decaying selling price.In this study, we define undesired transactions as selling tokens which would bebought back later. With proportional transaction costs of buying and selling34 igure 10: Social welfare plot of introducing an unusual event on 10th day. Trinity withcontinuous allocation can perform better than Trinity with lump sum allocation on day ofunusual event because regulator has more flexibility to regulate. fixed at 3% (commonly used by e-commerce platforms), we vary fixed transac-tion costs of buying and selling together and present results in Figure 11. Foreach case, we assume regulator does not want to change toll profile but canoptimize allocation rate. As we can see, small fixed transaction cost can reduceundesired transactions significantly and maintain social welfare similar to thesystem optimum. In addition, We find that although small proportional trans-action cost does not reduce social welfare too much, it does not reduce undesiredtransactions too. Last, we find introducing decaying selling price makes Trinityperforms worse. Instead of linearly decaying, we should try different decayingin future experiments. 35 a) Percentage of undesired transactionsover all selling transactions (b)
Average frequency of undesired trans-actions per individual (c)
Social welfare by different fixed transac-tion costs
Figure 11: The trade-off between social welfare and undesired transactions. Without changingtoll profile, the introduction of a small fixed transaction cost reduces the percentage of unde-sired transactions from 60% to almost 0% while Trinity still achieves similar social welfare.
6. Conclusions
This study presented a detailed formulation of a tradable mobility creditscheme, Trinity, with a focus on market design aspects including allocation/expirationof credits, trading rules, transaction costs, price dynamics and regulator inter-vention. We conducted extensive numerical experiments for a system model ofcommuters’ departure time choice in a single corridor. The system is modeledusing a stochastic process approach that incorporates day-to-day dynamics. Itconsists of discrete logit departure time choice model, rule-based trading model,deterministic queuing model, and exponential smoothing learning and forecast-36ng filter. We consider Trinity in congestion pricing applications, which usescongestion pricing toll profile in tokens and has a token market place to adjusttoken price daily. In addition, we consider a variant of Trinity using strict quan-tity control (Trinity TSC), which only allows buying and fixes market price at $
1. Regulator of Trinity TSC is able to control token supply cap to control actualflow. We formulated toll profile optimization as simulation-based optimizations,which were solved by a Bayesian optimization approach. We conducted simu-lation experiments of NT, CP, and Trinity assuming homogeneous commuters,which led to following insights:1. When actual capacity is the same as anticipated capacity, Trinity withtoll profile based on anticipated capacity is able to achieve the systemoptimum ignoring transaction costs and selling price decaying; TrinityTSC with unconstraining desired flow cap can also achieve the systemoptimum2. When actual capacity is lower than anticipated capacity by 25%, Trinitywith toll profile based on anticipated capacity is able to perform almostthe same as system optimum ignoring transaction costs and selling pricedecaying; Trinity TSC focusing on controlling flow pattern cannot performbetter than CP with toll profile based on anticipated capacity3. Ignoring transaction costs and selling price decaying, Trinity with sub-optimal step-toll performs better than CP with the same toll profile; Trin-ity with optimal step-toll performs similar to CP with the same step-toll4. Trinity with continuous allocation has more flexibility for regulator (throughregulating token price, allocation rate, and transaction cost) to accommo-date for unusual events better than Trinity with lump-sum5. Trinity with a small fixed transaction cost can reduce undesired transac-tions significantly while maintain system performance; Proportional trans-action costs do not prevent undesired transactions; Linearly selling pricedecaying makes Trinity have sub-optimal performance.Finally, more realistic market operation models, more investigations on mar-37et design and its dynamics, more tests on different decaying selling prices, in-cluding heterogeneity, examining adaptive toll profile, switching to large andreal network offer some interesting avenues for further work.
Appendix A. Stationarity Test
In statistics, a stochastic process is stationary if unconditional join prob-ability distribution does not change when shifted in time Gagniuc (2017). Inintuition, it means mean and variance do not change over time. In this study,since our experiments are stochastic, it is important to make sure simulationsbecome stationary as we compute social welfare. Otherwise, it is meaningless tocompare social welfare across different scenarios. The most basic methods forstationarity detection is to plot data and if there is any obvious violation. Morerigorous approach to detect stationarity is to perform statistic tests. In thisstudy, two statistical tests are used: Augmented Dickey Fuller (ADF) test andKwiatkowski-Phillips-Schmidt-Shin (KPSS) test. ADF test is used to determinethe presence of unit root in the series. The existence of unit root means 1 is aroot of the process’s characteristic equation and process is not stationary. Thenull hypothesis of ADF test is that the series has a unit root while the alternatehypothesis is that the series has no unit root. Therefore, if the null hypothe-sis is failed to be rejected, it provides evidence that the series is non-stationary.KPSS is another test for checking the stationarity. The null hypothesis of KPSStest is that the process is trend stationary while the alternate hypothesis is thatthe series has a unit root (non-stationary). The null and alternate hypothesisfor the KPSS are opposite that of the ADF test and KPSS test complementsADF test. The process is truly stationary if both tests conclude it is stationary.If KPSS indicates stationarity and ADF indicates non-stationarity, it meansthe process is trend stationary; if KPSS indicates non-stationarity and ADFindicates stationarity, it means the process is difference stationary.38 eferences
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