Congestion tolling - Dollars versus tokens Within-day dynamics

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Introduction
Congestion is a pervasive problem in most transportation networks worldwide and the standard approach to address this issue has been to internalize congestion externalities through a toll in dollars (congestion pricing).Extensive reviews may be found in de Palma and Lindsey (2011) and Santos and Verhoef (2011).Pricing, however, has long been beset by issues of inequity, complexity, high infrastructure costs and public and political acceptability, notwithstanding the redistribution of toll revenues, whose benefits may take years to be realized (see also Jaensirisak et al. (2005) and de Palma and Lindsey (2020) for more on the acceptability of road pricing).In contrast, tradable credit schemes (or tolling in tokens; also called tradable permit schemes) are a form of quantity control typically characterized by the following features (Fan and Jiang, 2013): (1) a fixed total number of tokens (mobility credits) or 'quota' is pre-specified by the regulator, (2) an initial endowment allocates or distributes the tokens to a selected population (all individuals may not receive tokens), (3) individuals are allowed to buy and sell tokens in a market, (4) use of the road network requires tokens and can be differentiated by time of day, geography, vehicle type etc., and (5) enforcement is necessary to ensure valid trading/consumption of tokens.
Tradable credit schemes have several potential advantages over pricing.First, they are revenue neutral and hence, may not be faced with similar public opposition, more so if the tokens are handed out for free.Second, they are viewed as being less vertically inequitable than pricing (de Palma and Lindsey, 2020).Since the number of tokens each user receives may differ, any regressive effect (very well documented for congestion pricing), which may trigger political opposition, will not occur with tokens.In other words, lower income users who tend to travel less by car can obtain monetary gains by selling their excess credits.Third, implementation costs may be low given developments in information and communications technology.Finally, they provide the ability to directly control quantity, which may be beneficial in some situations, for example, when the price elasticity of demand in the short or medium term is low.Note that the distribution of tokens allows the regulator to cap the maximum number of travelers.This is a pure version of quantity control, which appears to be more flexible than conventional policies (such as the odd and even license plate policy, or reduced speed) to reduce environmental costs.Other forms of quantity control include ramp metering, perimeter control, and metering-based dynamic priority schemes (Lamotte et al., 2022).
Despite the large body of literature on Tradable Credit Schemes (TCS) in the transportation context, relatively little attention has been paid to the comparison of the price and quantity control instruments under uncertainty.In this paper, we consider a tradable credit scheme in a departure time context wherein users are provided an initial endowment of tokens by the regulator and incur a token charge (determined prior to all departures) to travel in a specific time period.Tokens can be bought and sold within a marketplace at a price determined endogenously by token demand and supply.Travel demand is modeled using a logit mixture model and supply consists of static congestion specific to each time period.
This paper contributes to the existing literature on tradable credit schemes and the comparative performance of price and quantity control instruments under uncertainty in several respects.First, from a methodological standpoint, we propose a smooth market clearing mechanism that can be used to model price dynamics across time intervals for a tradable credit scheme (wherein tokens are bankable across time intervals) using a tractable equilibrium approach.Selling decisions of users are explicitly modeled and determine the distribution of token supply in the market over the time of day (to the best of our knowledge, this has yet to be addressed in the literature).This enables us to study the impact of selling behavior on performance of the credit system.Although we use a logit model for convenience, in principle, any continuous model may be used.The proposed framework can also be used to incorporate more complex models of selling behavior (see for example, Dogterom et al. (2017)).
Second, in terms of numerical findings, we obtain new insights into the impact of selling behavior on the performance of a tradable credit scheme relative to pricing under uncertainty: (a) Welfare losses are present in a TCS system with irrational sellers, which has the effect of equalizing token supply across time intervals leading to a deterioration in the performance of the quantity instrument.However, despite these losses, the TCS with fixed token supply is still superior to pricing when congestion effects are severe; (b) In the case when token supply in the TCS can be adapted from day to day, the quantity control system is still not consistently superior to price control as one may expect (it can be inferior in cases with irrational selling behavior when congestion effects are less severe); (c) Even when toll revenues in the price instrument are equally redistributed, tolling in tokens (when tokens are equally distributed) is marginally more equitable in scenarios where congestion effects are more severe.These findings underscore the importance of examining disaggregate market behavior when designing tradable credit schemes.
Finally, we show that the under uncertainty, in a more complex setting involving within-day dynamics (with time dependent tolls, market prices and congestion), quantity control is superior to price control in cases where congestion effects are more severe (steep BPR functions, high demand, high day-to-day variability).The optimal network usage is relatively similar across states for quantity control whereas the optimal toll in dollar amounts varies significantly across states.In this regard, we generalize the findings of de Palma et al. (2018), de Palma and Lindsey (2020), showing that their results on the efficiency differences between the two instruments are robust (holding even in the presence of irrational sellers) and generalize to this more complex setting.These findings make a potential case for tolling in tokens.
The remainder of the paper is organized as follows.Section 2 reviews the existing literature on tradable credits and describes our contributions.Section 3 describes the basic model of supply, demand and equilibrium in the case of deterministic demand (and supply) for both instruments (dollars and tokens).The comparison of the two instruments in this case can be performed analytically.Following this, Section 4 describes the model for stochastic demand for which, the comparisons must be performed numerically.The numerical experiments and findings are described in Section 5. Finally, Section 6 provides concluding remarks and directions for future research.

Review of literature
Although discussions on the use of tradable credits in the transportation sector date back to Raux (2004), Verhoef et al. (1997) and Goddard (1997), they have received significant attention only in the recent past (detailed reviews may be found in Fan and Jiang (2013) and Grant-Muller and Xu (2014)).In particular, a variety of tradable credit schemes have been proposed (utilizing largely network and market equilibrium approaches) in the context of mobility management (at the network level) and bottleneck management, and these are discussed in turn.
In the context of mobility management, Yang and Wang (2011) propose a scheme wherein the social planner initially distributes a certain number of credits to all potential travelers, charges a link-specific number of credits for a given link, and allows trading of the credits among travelers.Supply is modeled using static congestion (separable link performance functions) and travelers are assumed to be homogeneous.They demonstrate that for a given set of tolls in tokens (or credit rates) in a general network, the user equilibrium (UE) link flow pattern is unique under standard assumptions, and the credit price at the market equilibrium is unique under some relatively mild additional assumptions (i.e., if all equilibrium path flow patterns contain at least two paths with different credit charges connecting the same O-D pair).The proposed network equilibrium formulation is a variant of the standard UE model with the additional network-wide credit feasibility constraint, which simply states that the total consumption of tokens at equilibrium is less than or equal to the total credit endowment.Extensions that consider heterogeneity in the value of time and multiple user classes are proposed in Zhu et al. (2015) and Wang et al. (2012), whereas He et al. (2013) consider allocations of credits to not just individual travelers, but to transportation firms such as logistics companies and transit agencies.In a similar vein, Pareto-improving credit based congestion management schemes on a general two mode network are investigated in Liu and Nie (2017).Multi-period tradable credit schemes have also been proposed where travelers can transfer credits to future periods within a planning horizon (Miralinaghi andPeeta, 2016, 2019).
In contrast with most studies which adopt an equilibrium perspective, Ye and Yang (2013) examine the price and flow dynamics under a tradable credit scheme using a continuous dynamic model within a finite time horizon, deriving conditions for asymptotic stability and convergence.Recognizing the practical challenges of optimally designing tradable credit schemes, which require knowledge of both demand and supply functions, Wang and Yang (2012), Wang et al. (2014c) propose trial and error methods (when demand functions are not known) for a single road and general networks, respectively.Both the credit tariffs and credit allocation are updated in an iterative fashion based on observed link flows and revealed credit prices.Nie (2012) examines the effect of transaction costs in a tradable credit scheme for two types of markets: an auction market in which users purchase all of the needed mobility credits through a competitive bidding process (they can also purchase directly from the government at a pre-specified price), and a negotiated market in which users initially receive certain amount of mobility credits from the government and trade with each other through negotiation to meet their needs.A brokerage service is built into both markets to facilitate transactions and accordingly, the users have to pay a commission fee proportional to the value of the trade.The modified UE formulation of Yang and Wang (2011) is extended to incorporate transaction costs for both the auction and negotiated markets.Based on numerical experiments on a toy network, it is shown that an auction market can achieve an equilibrium allocation of mobility credits if the government sets the price suitably (price at which users purchase tokens directly from the government) and the unit transaction cost is lower than the price that the market would reach in the absence of transaction costs.The work also highlights the fact that in case of the negotiated market, the initial allocation of mobility credits may affect the final equilibrium when marginal transaction costs are constant.
The design of tradable credit schemes for mobility management has also been explored using bi-level optimization formulations (mathematical programming problems with equilibrium constraints).For instance, Wang et al. (2014b,a) formulate the continuous network design problem with a tradeable credit scheme as a bi-level programming problem, where the decision variables for the upper level problem are capacity enhancements for selected links whereas the lower level problem determines equilibrium link flows and the credit price.Along similar lines, Wu et al. (2012) proposed a framework that considers decisions of mode/route choice and trip generation on a multimodal transportation network to design efficient and equitable congestion pricing and tradable credit schemes (considering a measure including both net social benefit and equity).They find that the Pareto frontier (with respect to the two aforementioned objectives) of the credit scheme strictly dominates that of congestion pricing although the two schemes achieve the same level of maximum net benefits.Further, their results suggest that tradable credit schemes can be progressive whereas congestion pricing schemes are largely regressive for the tested network.Finally, the literature on mobility management also includes a series of studies on credit-based congestion pricing (CBCP), where credits in CBCPs are allowances used to pay tolls (Kalmanje and Kockelman, 2004;Kockelman and Kalmanje, 2005).The studies involved the use of destination, mode, and departure time choice models to examine the potential impacts of using a CBCP scheme.
Tradable mobility credit schemes have also been proposed to manage bottleneck congestion and achieve peak spreading in an efficient manner.Nie and Yin (2013) developed an analytical framework to model a tradable credit scheme that manages commuters' travel choices in a simple transportation system consisting of two parallel routes.The scheme attempts to persuade commuters to spread their departure times evenly within the rush hour and between primary and alternative routes to mitigate traffic congestion.It defines a 'peak-time' window within which users are charged mobility credits to travel on the primary route and those that avoid either the peak-time window or the primary route may be rewarded with credits (see also Nie, 2015).Tian et al. (2013) investigate the efficiency of a tradable travel credit scheme for managing bottleneck congestion and modal split in a competitive highway/transit network with a continuously distributed value of time.They propose a tradable credit scheme which emulates bottleneck congestion pricing and transit subsidy in a revenue-neutral manner and demonstrate that both the modal split and credit charge at equilibrium are unique.Xiao et al. (2013) examined the efficiency of a tradable credit system in managing morning commute congestion with both homogeneous and heterogeneous users.They show that even in the presence of heterogeneity, an optimal credit scheme that eliminates the bottleneck queue always exists under the assumption that late arrival is not allowed.More recently, Akamatsu and Wada (2017) proposed a tradable bottleneck credit scheme where the regulator issues link-and time-specific credits permitting passage through a certain link or bottleneck in a pre-specified time period.They develop a model to describe time-dependent flow patterns at equilibrium under a system of tradable bottleneck permits for general networks and show that the equilibrium obtained under this system is efficient in that it minimizes the social transportation cost.Bao et al. (2019) show that the equilibrium with a tradable credit scheme may not be unique for particular models of traffic congestion, including the first-best solution for the conventional Vickrey's bottleneck model.Liu et al. (2020) examine properties of a TCS in a departure time context using a day-today modeling framework and a trip-based MFD model of traffic congestion, and Chen et al. (2021) examine market design aspects of the TCS, including allocation and expiration of tokens, price adjustment, regulator intervention, and the role of transaction fees.Finally, Brands et al. (2020) conduct an interesting lab-in-the-field experiment of tradable credit schemes with virtual mobility behavior and real financial incentives.They adopt a market design, which lets users trade with a price setting intermediary, termed a virtual bank.An incremental price adjustment scheme is adopted and their experiments suggest that it ensures that the price stays largely within the equilibrium range.Overall, their results are promising and indicate that tradable credits can be a viable alternative to pricing in a parking setting.
In contrast, comparisons of price and quantity control under uncertainty in transportation are relatively sparse (for other contexts see Weitzman (1974), Laffont (1977)).Note that in the emissions context, agents are not directly impacted by the externality they generate, while in the transportation context they are (as far as congestion is concerned).For this reason, their results on stochastic demand and stochastic supply are not directly applicable here, even if they are useful as a general guideline.Shirmohammadi et al. (2013) examine the performance of tradable credit systems under demand and supply uncertainty using a toy network.Specifically, they compare the performance of a link differentiated toll system (in dollars) and a mobility credit system that is differentiated by link.However, their analysis does not focus on measures of efficiency but rather examines performance relative to a given target volume of cars.They find strong variations in the credit prices are required to ensure demand matches the specified volume targets.de Palma et al. (2018) compare the performance (in efficiency terms) of congestion pricing and tradable mobility credit schemes under uncertainty using a simple road network in a stochastic route choice setting (including a public transit alternative).They find that when the tolls (in either dollars or tokens) cannot be adapted from day to day, the credit scheme performs better typically when the slope of the congestion function is steep.Further, when the token supply can be adapted from day to day, the token system always outperforms congestion pricing.More recently, de Palma and Lindsey (2020) ranked the efficiency of tradable permits and tolls for one route, one time period and elastic demand.They study additive and multiplicative demand and cost (capacity) shocks and show that these may lead to qualitatively different results.Their approach is analytical and considers linear and non-linear demand (showing the role of the convexity of demand).They also study the impact of the correlation between demand and supply on the ranking.Along similar lines, Lindsey et al. (2022) compare the allocative efficiency of tradable permits and tolls under uncertainty on a bimodal network and derive a general rule for ranking the efficiency of the two instruments.Rezaeinia et al. (2021) consider the comparison of tolls and permits in a radial network, within the Vickrey framework.Users select mode, departure time and route based on a Nested Logit continuous/discrete framework, and congestion is computed using the METROPOLIS software ( de Palma et al., 1997).Tolls and permits are independent of the time of the day.They find that tolls outperform permits if capacities shocks are perfectly correlated.However, the ranking is reversed if capacities shocks are independent.
In summary, literature on the comparative performance of the price and quantity control instruments under uncertainty is relatively sparse.Existing studies (Shirmohammadi et al., 2013;de Palma et al., 2018;de Palma and Lindsey, 2020) are limited to route choice and a single time-period (i.e., they are static models).This paper attempts to extend the literature by generalizing the work of (1) de Palma et al. (2018), who consider static congestion (with parallel routes) and one time period, and (2) de Palma and Lindsey (2020), who consider a single route with elastic demand and static congestion.Here instead, users decide when to travel within a multi-period setting (departure-time choice context), and we propose a smooth market clearing mechanism within each time-period to describe within-day market price dynamics of a TCS when tokens or credits are bankable across time intervals.This enhancement, which involves explicitly modeling selling behavior in the market (to the best of our knowledge, this has not yet been addressed in the literature) is a critical step in moving to large-scale simulation-based assessments of tradable credit schemes on real-world networks.It also enables us to study the impact of selling behavior on performance of the credit system.Given the complexity of our model, comparisons of the two instruments under uncertainty cannot be performed analytically.
Lastly, in comparison with Ye and Yang (2013), two key points are noteworthy.First, they propose a continuous-time evolution model to study price and flow dynamics of a TCS within a finite-length time period of interest (tokens expire at the end of this time period).In contrast, we consider a discrete-time setting with  time intervals within a day where tokens are bankable across these time intervals.The dynamics of market prices over the day are determined by a market clearing mechanism in each interval.Note that the numerical scheme we utilize to compute the market clearing price in each time interval is largely identical to their price evolution process.However, rather than the evolution process within each time interval, our focus lies in the equilibrium time-of-day prices and their impact on comparative efficiency of the TCS relative to the price instrument.Second, we consider day-to-day variability in the form of demand stochasticity (in contrast with the deterministic demand setting they study).

Multiperiod model: Deterministic demand
The transportation network of interest consists of a single origin-destination pair connected by a single route (this can be extended to multiple OD pairs and routes with no methodological difficulty).We consider commuting trips performed within a time period  , which is partitioned into  sub-periods  1 ,  2 , … ,   …   so that  1 ⋃  2 … ⋃   =  .Note that  1 ,  2 , … ,   may represent any  time periods within a day, and are not necessarily contiguous (the terms time period and time interval are used interchangeably when referring to   ).We denote the set of the  time intervals by .There are a total of  users who wish to travel, and each user performs a single trip or activity during the day in any one of the  intervals or chooses to stay at home (denoted by  0 ).A glossary of notation can be found in Appendix D.
We assume that the network is subject to time dependent congestion justifying the need for congestion control in the form of either a price instrument (tolling in dollars) or a quantity instrument (tolling in tokens).Under the price control instrument, users have to pay a toll in dollars   to travel in time period   ( = 1 … ).In case of the quantity control system, we make several assumptions.The regulator distributes a certain number of credits (tokens)  (known in advance) to each potential user at the beginning of the time period  .The tokens expire at the end of time period  , or in other words, their market value is zero at the end of time period  .Tokens cannot be banked or traded across days (i.e., across periods  ).This assumption is similar to that of Brands et al. (2020) and is used to mitigate undesirable speculation and hoarding of tokens.Further, the tokens cannot be transferred or used outside the market.Users have to spend a certain number of tokens to travel in time interval   ,  = 1 … , given by   (toll in tokens).Tokens can be bought and sold within a marketplace.The price of the token in time interval   is denoted by   , and is determined endogenously by the demand and supply of tokens in the market in time interval   .In order to avoid speculation, tokens can only bought if they are needed for travel.This will prevent users from buying tokens for the sole purpose of trying to make a profit in the market, an aspect of the tradable credit scheme that can hamper acceptability.Users must sell all their unused tokens in a single time interval   ,  = 1 … .Once again, this assumption minimizes excessive transactions for the sole purpose of making a profit.However, it should be pointed out that this assumption can potentially affect market efficiency (see Brands et al. (2020) for a detailed discussion).Nevertheless, we defer relaxation of this assumption to future research.Finally, to improve acceptability and ease of implementation, we assume that all transactions take place at the beginning of each time interval   .
Note that the regulator has the flexibility to institute any desired token allocation scheme including ones wherein users receive an unequal number of tokens.Given that user choices are unaffected by the token allocation (ignoring income effects and transaction costs), this implies that in principle any desired distribution of equity can be achieved through the initial token allocation (see Yang and Wang (2011)).
In this section, we consider the case of deterministic demand (i.e., the number of users  is deterministic and known).The transportation model is first described, followed by the two instruments (price and quantity) in turn and a comparison with respect to individual benefits, social benefits, and equity.

Transportation model: Demand, supply and equilibrium
The money-metric utility of an individual  traveling in a time period   ,  = 0 …  ( = 0 denotes the stay at home option) is given by, where   is the flow in time period   ,   is an i.i.d.Gumbel disturbance term and   ,    ,   are individual specific parameters with log(  ),    , log(  ) normally distributed.The alternative specific benefit    incorporates time-period specific scheduling preferences, or alternatively, time-period specific schedule delay costs.  and   are the value of time and scale parameter, respectively, for individual .
A standard BPR type function is assumed to model the travel time   (  ) in time period , where     is the free flow time in period ,   is a capacity associated with time period  and   ,   are function parameters.Note that we assume that the  time periods are independent in the sense that congestion does not spill over from one period to the next (in other words we have  static models).This is a time-sliced static traffic assignment common in planning applications (Nakayama and Connors, 2014), which admittedly may not realistically capture the dynamics of congestion and flow propagation.However, it suffices for the comparison of the instruments in our context.A more detailed treatment of the dynamics of congestion is beyond the scope of the paper although we note that the methodology proposed is generic and not restricted to any specific type of congestion model.
Further, let   = (  ,    ,   ) denote the vector of parameters for individual .The probability of individual  choosing to travel in time period   is given by, where   (  ,   ) is given by, The number of travelers traveling in the  periods   ( = 1 … ), given   ( = 1 … ) in the price system and     ( = 1 … ) in the quantity system, are obtained by solving the fixed-point problem (note that  0 =  − ∑ = =1   ), ,  = 1 … . (5) Since the set of demand feasible flows forms a closed and convex set, and the right hand side of Eq. ( 5) is a continuous function of flows, Brouwer's fixed-point theorem implies that a solution exists to the fixed-point problem in Eq. ( 5).On the conditions for solution uniqueness we refer the reader to Konishi (2004) and Lindsey (2004).

Tolls in dollars: Price control
In the price control system, the regulator is assumed to have knowledge of the demand  and sets the tolls in dollars   ( = 1 … ) to maximize total welfare (defined as the sum of consumer surplus and regulator revenue).Consumer surplus is calculated as the logsum measure (see Small and Rosen (1981) and De Jong et al. (2007)) and regulator revenue is simply the total toll revenue.In our context, the definition of welfare is straightforward since there are no income effects (Anderson et al., 1992).Thus, determination of the tolls can be formulated as the following optimization problem, The optimum welfare obtained by solving (6) above and the corresponding optimum welfare and tolls in dollars are denoted by  *  and  * = ( *  ,  = 1 … ) respectively.Note that the optimization problem in Eq. ( 6) is non-convex due to the non-linear equilibrium constraints.Solution approaches for this problem are discussed in Section 5.1.

Tolls in tokens: Quantity control
Recall that in the quantity control system, the regulator distributes a certain number of credits (tokens)  to each potential user at the beginning of a time period  , and the number of tokens required to travel in time interval   ,  = 1 …  is given by   .

Demand for tokens
For a given set of tolls in tokens and market prices   ,   ( = 1 … ), the total demand for tokens in period   is     and the total amount of tokens possessed by people traveling in period   is   , where  is the initial token endowment.Thus, the demand for tokens in period   ( = 1 … ), is given by, where   ( = 1 … ) are obtained from the solution to Eq. ( 5).Note that in the above we assume that the token endowment to each user is equal ().This assumption can be relaxed.

Supply of tokens
We assume that the decision to sell tokens is made after the mobility decision and hence, users sell all unused tokens.Note also that it is assumed that there is some scarcity in the system, namely that the number of tokens available is less than what would be consumed if the tokens were free.Formally, if X ( = 0 … ) denotes the equilibrium flows in the absence of tolls, we assume that ∑ =1... X   > .
In the case that there is no congestion (i.e.,   is constant), we have   = 0 ( = 1 … ), or equivalently,   <  ( = 1 … ), and hence, the price of tokens   ( = 1 … ) is zero and the tokens have no effect/value.We focus on the case where congestion effects are present, and hence, it should be the case that at least one of  1 ,  2 , …   is larger than  (note that  0 = 0), which implies that   > 0, for at least one period   .Let the subset of time periods where   >  be denoted by , then the subset of time periods where   <  is  ⧵ .
As noted before, we assume that tokens can only be bought if they are needed for travel.Further, we assume that the regulator requires the user to sell all her unused tokens in one time interval.This assumption is made from a practical standpoint to ensure simplicity of the system.Since the tokens are worthless at the end of the last time interval, no user will keep tokens unused.
Observe that sellers are individuals who have chosen to travel in an interval   ,  ∈  ⧵  whereas buyers are individuals who have chosen to travel in   ,  ∈ .We assume that the token price in interval   ,  ∈  perceived by a seller is given by, where   is the market clearing price of the token and ε is an i.i.d.Gumbel term with scale parameter μ.The rationale for the additive term is the following.There may exist day-to-day fluctuations around the average equilibrium price (long-run market clearing price that we have referred to as   thus far) for a given interval.These fluctuations are the result of the operation of the market and dynamic adjustments (they may also arise due to the intrinsically stochastic nature of traffic flows and other sources of uncertainty, see for example (Chen et al., 2002)).Travelers do not have exact knowledge of these fluctuations when making the travel decision at the beginning of the day and hence make their decision based on the long-run average prices (Eq.( 1)).The selling decisions on the other hand are made at the beginning of each interval, within the day.When the users make their selling decisions, they have additional information about the day including actual prices revealed in the intervals thus far.This could give rise to different perceptions of the actual prices.Thus, there is a distribution of perceived prices across the population (around the long-run average market clearing prices) that determines the selling decisions of users.Note that Eq. ( 8) permits an alternative interpretation, i.e., p is the utility obtained from selling in interval .The selling decision may be based on a number of factors apart from price that are subsumed into the error term.These could be the result of factors like different reservation prices, lack of attention satisfying behavior, endowment effects, loss aversion, risk aversion, multiple trips during the day (because of which users wish to sell at different intervals) and so on, all of which are not incorporated within the model.
For a seller, we model the choice of a selling interval using a simple logit model based on Eq. ( 8), where the systematic utility of selling in interval   ,  ∈  is based on the average market price in the interval, and given by p .Consider a user  who has decided to travel in period   ,  ∈  ⧵ .We assume that the amount of tokens to sell does not influence the time period of selling.The probability of the user  selling his/her token in interval   ,  ∈ ,   is assumed to be, The supply of tokens from users traveling in   ,  ∈  ⧵  who have decided to sell in period   ,  ∈  is given by, Or, The total supply of tokens is given by, We next study the market clearing condition.

Market clearing
The market clearing conditions in each interval   ,  ∈  imply, Or, Further, the total demand for tokens is given by, Thus, market clearing requires  = , or, The market clearing conditions in all periods is satisfied (  =   ,  ∈ ).However, the demand (or supply) for tokens in different intervals may be different, i.e., R. Seshadri et al.

Computation of market clearing prices
For a given vector of tolls in tokens (  ,  = 1 … ), the market clearing price in each time interval can be computed through the following iterative process, where the price in iteration  + 1 is given by, where ℎ ′ (.) > 0 and ℎ(0) = 0;    ,    denote the demand and supply of tokens in interval  and iteration , respectively.This is a standard cobweb adjustment process.It should be noted that the price adjustment process above is merely a numerical method to compute the market clearing prices and does not imply that the market actually operates in this manner.Otherwise, users could discover this and make use of the knowledge of this process strategically, which will lead to a complex game theoretic problem outside the scope of this paper (and possibly behaviorally unrealistic in any case).Alternatively, a simpler iterative process can be used to compute the market clearing prices (∀ ∈ ): where   > 0.More details on how this price adjustment scheme is used within the solution framework (to compute market clearing prices) are provided in Section 5.1.
Conjecture 1.The equilibrium prices for the  time intervals satisfy A discussion of this is provided in Appendix C.

Optimization
As before, let Ĩ denote the subset of time periods where   > .We assume that the regulator has knowledge of the demand  and sets the tolls in tokens for time periods 1, 2 … −1, +1 … , and the supply of tokens  (  is normalized to 1 token without loss of generality; the period  can be chosen arbitrarily, for example the morning peak hour) that maximizes total welfare.The welfare in the case of quantity control is simply the consumer surplus, since the regulator revenue is zero given that the instrument is revenue neutral.However, the calculation of consumer surplus differs slightly from the price control case.Recall that in the definition of the utility in Eq. ( 1), the entire token payment for each individual contributes to a real monetary disutility or monetary loss and does not account for the initial free token allocation or the monetary gains for the users who sell tokens.Therefore, this monetary loss has to be added back to the consumer surplus to reflect the free allocation from the regulator and the monetary gains for sellers.Note that monetary transfers between users and between the regulator and users have no impacts on overall welfare, provided the transfers do not alter or change behavior.Thus, the determination of the toll in tokens can be formulated as the following optimization problem, where the equilibrium prices  = ( 1 ,  2 , … ,   ) satisfy the market equilibrium conditions (and   is given by Eq. ( 9)), The optimum welfare obtained by solving (19) above and the corresponding optimal tolls in tokens are denoted by  *  and  * = ( *  ,  = 1 … ) respectively.The associated market clearing prices are denoted by  *  ,  = 1 … .We refer the reader to Appendix A for a discussion of the existence and uniqueness of a solution for the market and network equilibrium constraints in Eq. ( 19).Convergence of the price adjustment schemes in Section 3.3.4 is also discussed.

Comparison
In the deterministic case, the comparison of the two instruments is trivial and can be performed analytically.The two instruments, when optimally chosen, yield identical social welfare.This is shown in Proposition 1 below.
Proposition 1.Under deterministic demand and supply, the two instruments, price and quantity, when optimally chosen, are equivalent.
Proof.Refer Appendix B.

Multiperiod model: Stochastic demand
In the discussion thus far, transportation demand and supply were assumed to be deterministic.We now turn our attention to the case where the demand is stochastic.

Transportation model: Demand, supply and equilibrium
Assume that there exist two days or states of nature 1 and 2 (denoted by ,  = 1, 2), where the alternative specific benefit to travel for individual , varies across the days, taking values  ,1  for period  ( = 1, 2, … , ) with probability  and values  ,2  ,  = 1, 2, … ,  with probability 1 − .In other words, the source of day-to-day variability or stochasticity is on the demand side and arises due to fluctuations in the scheduling preferences of travelers (note that the total number of users is fixed).The variability in scheduling preferences may be due to special events, weather etc. leading to a higher number of users who wish to travel during the peak period.Stochasticity may also arise from external factors affecting supply such as incidents and accidents, or factors affecting both demand and supply.The methodological framework can be extended in a straightforward manner to model these cases as well.Note also that the framework can be extended to several states of nature or even for a continuum of states with no methodological difficulty.
The systematic utilities to travel in time period   on day ,  = 1, 2 (denoted by    ) are given by, where    and    are the token price and number of individuals traveling in time period    , respectively.Note that in case of the price system, the terms      are replaced by   .As before, for a given set of tolls in tokens (  ,  = 1...) and token prices (   ,  = 1...;  = 1, 2) -or tolls in dollars (  ,  = 1...) in the case of price control-,    for  = 1, 2 can be determined by solving the following fixed-point problem (note that   0 =  −

Tolls in dollars: Price control
In case of the price instrument, we assume that the regulator may not wish to change the tolls from day to day for reasons of acceptability and ease of implementation (or may not have knowledge of the specific realization of the state of nature).For instance, in the ERP system of Singapore, tolls are revised only once every few months and do not vary from day to day.Thus, in the case of stochastic demand, the regulator sets the tolls in tokens for the  time periods that maximizes expected total welfare, formulated as the following optimization problem, s.t

Tolls in tokens: Quantity control
We distinguish two configurations of the quantity control system.First, in the case of adaptive token supply, the supply of tokens can vary by day and is denoted  1 and  2 , whereas in the case of fixed token supply, it is assumed that the total supply of tokens is fixed across days i.e.,  1 =  2 = .From the standpoint of implementation, adapting the token supply is likely to be far easier than adapting the tolls in tokens (or dollars), which may involve communicating a complex tariff structure (in a general network) to commuters.
The market clearing conditions in Eq. ( 14) now apply in each time interval for both days and are given by (the same notation as before is used with the added superscript  to denote the day), In the case of fixed token supply, we assume that the regulator does not have knowledge of the specific realization of the state of nature (or day) and hence, sets the tolls in tokens for time periods 1, 2 …  − 1,  + 1 … , and the supply of tokens  (as before   is normalized to 1 token without loss of generality) that maximizes expected total welfare, formulated as the following optimization problem, s.t where the equilibrium prices   = (  1 ,   2 , … ,    ),  = 1, 2 satisfy the market equilibrium conditions: In the case of adaptive token supply, we assume that the regulator has knowledge of the specific realization of the state of nature (or day) and sets the tolls in tokens for time periods 1, 2 …  − 1,  + 1 … , and the supply of tokens  1 ,  2 (as before   is normalized to 1 token without loss of generality) that maximizes expected total welfare, formulated as the following optimization problem,

Comparison
In contrast with the deterministic case, when demand (or supply) is stochastic, the comparison of the price and quantity control instruments cannot be performed analytically.Hence, we perform the comparison numerically.

Experimental design
The two instruments are compared using a synthetic example across a wide range of demand and supply inputs.The setting we consider involves three time intervals (e.g., early morning, peak, off-peak), i.e.,  = 3, wherein there are  potential travelers who may choose to either travel in one of three time periods (  ,  = 1 … 3) or cancel their trip (option  0 ).The stochasticity or variability in demand -as noted in Section 4 -is modeled by varying the levels of the alternative specific benefit to travel (in time periods) on the two days,  ,  ,  = 1 … 3,  = 1, 2 (refer Eq. ( 20)).Thus, the source of day-to-day variability or stochasticity is on the demand side and arises due to fluctuations in the scheduling preferences of travelers, which may arise due to special events, weather etc. leading to a higher number of users who wish to travel.The mean and standard deviation of the alternative specific benefit to cancel trip ( , 0  = 1, 2) are normalized to zero, and the probability  is assumed to be 0.5.The values of the fixed factors are shown in Table 1.The capacities are set based on the range of demand values (varies with scenario, see Table 2) to yield a ratio of congested to free flow travel time (in the absence of tolls) in the range 1.25-2.5.The free flow travel time is set to be 13 minutes (assuming a free flow speed of 60 km∕hr, this corresponds to a trip length of 13 km, which is in the range of average trip lengths in typical urban transportation networks).The mean of the alternative specific benefit is assumed to be higher in period two to represent peaking effects and commute behavior (note that the table describes the distribution of the alternative specific benefits in the different time periods for day 1; the values on day 2 vary with the scenario, and are part of the experimental design, which is described later in the section).The coefficient of variation of the alternative specific benefit is assumed to be lower in period  2 reflecting a morning commute context where work start times are largely in this time interval.Further, we introduce some asymmetry in periods  1 and  3 through the BPR congestion function, which could potentially reflect choices of different routes in these periods.The mean and standard deviation of the value of time are assumed to be 0.33$ per min (around 20$ per hr) and 0.067$ per min (around 4$ per hr) respectively (refer Prato et al. (2014), Hess et al. (2005), Cirillo and Axhausen (2006) for empirical evidence; note that the literature reports a wide range of values for the coefficient of variation, we adopt a conservative value of 0.2).
In the experimental design, five factors are varied, which include the coefficient of variation (COV) of the scale parameter   in the mobility model (the mean of   is fixed at 1.5), the scale parameter of the selling model μ, total number of users , the congestion coefficient , and the benefit difference between the two days .The factor levels are shown in Table 2.A total of 432 test instances or scenarios (4 2 × 3 3 ) were simulated.
Several additional points are noteworthy.First, in all the scenarios wherein   is deterministic (in other words, COV of   is zero), the standard deviations of all other randomly distributed parameters (i.e.,   ;  ,  ,  = 1 … 3,  = 1, 2) are also set to zero.Thus, this subset of scenarios represents the setting with no heterogeneity in the mobility model.Second, in order to set the values of the alternative specific benefits for a given scenario with a benefit difference , the values of  ,1  ,  = 1 … 3, are first sampled (based on the mean and standard deviation in Table 1), and  ,2  ,  = 1 … 3 is given by,  ,2  =  ,1  + ,  = 1 … 3. Third, the scenarios with μ = 100 000 represent non-rational market behavior or a purely random selling model (i.e., μ → ∞).
The two instruments (tolling in dollars and tolling in tokens) are compared across the 432 test instances based on the optimum social welfare obtained by solving the optimization problems in Eqs. ( 22), ( 24) and ( 25).These are solved as bi-level problems; given the non-convexity of the problems, two approaches are adopted for solving the upper-level problem.
For price control (Eq.( 22)), in the first approach, the optimization problem at the upper-level (decision variable is the toll in dollars;   ,  = 1 … ) is solved using a genetic algorithm (GA).The GA is well suited to mathematical programs with equilibrium constraints (MPEC) (Zhang and Yang, 2004).A population size of 50 with a maximum of 300 generations and a crossover fraction of 0.8 is used (for more details see Deep et al. (2009)).This is implemented directly using the MATLAB ga routine.For each evaluation of the objective function for a given candidate solution of the toll in dollars   ,  = 1 … , the lower-level Stochastic User Equilibrium (SUE) problem is solved as a system of non-linear equations using the lsqnonlin routine to determine the flows (   ,  = 0 … ,  = 1, 2).In the second approach, the GA is replaced by the sequential quadratic programming algorithm, which has also been applied to non-convex MPEC problems (Meng et al., 2004).However, in this case, given the non-convexity, 25 randomly generated starting points are used for the optimization algorithm (the value of 25 was arrived at empirically based on preliminary experiments wherein it was found that increasing the number of starting points beyond 25 did not yield improvements in the objective value).This is implemented using the fmincon routine in MATLAB.For quantity control (Eqs.( 24) and ( 25)), in the first approach, the upper-level problem (decision variable is the toll in tokens;   ,  = 1 … ) is again solved using GA and implemented using the MATLAB ga routine.For each evaluation of the objective function for a given candidate solution of the toll in tokens   ,  = 1 … , we now need to compute both the equilibrium (SUE) flows (   ,  = 0 … ,  = 1, 2) and the market clearing prices for each interval (   ,  = 1 … ,  = 1, 2).Here, we use the iterative price adjustment process described in Section 3.3.4.We start with a candidate set of prices (   ,  = 1 … ,  = 1, 2), and solve the SUE problem (again using the lsqnonlin routine) to determine the flows (   ,  = 0 … ,  = 1, 2).Next, the demand and supply of tokens (   ,    ,  = 1 … ,  = 1, 2) are computed and the prices are adjusted using Eq. ( 18).This process is repeated until convergence of the prices, at which point we have the solutions of (   ,    ,  = 1 … ,  = 1, 2) for a given toll in tokens (  ,  = 1 … ), and we can compute the objective function.In this price adjustment process we use a simple bisection method where the value of   in Eq. ( 16) progressively decreases with successive iterations.In the second approach, once again, the Genetic algorithm is simply replaced by the SQP algorithm.
In all the 432 test instances (for all the instruments), both the GA and the SQP algorithm (with multiple starting points) converged to the same optimal welfare and toll values.

Results and discussion
The results from the numerical experiments and their implications are discussed in this section.After a description of the overall results in terms of optimum social welfare, the effects of congestion, extent of day to day variability, selling model, and heterogeneity are discussed in turn.

Welfare
Summary statistics (across the 432 scenarios) of the welfare differences between various instruments under stochastic demand are presented in Table 3.The following abbreviations are used: NT for the no-toll equilibrium, SP for the price system or tolling in dollars, SQ for the quantity system or tolling in tokens.We also include a benchmark (abbreviated ADP) in which the tolls (in either dollars or tokens) are adaptive across the two days and set by the regulator based on the realization of demand.Clearly, in this case, the price and quantity instruments are equivalent (as shown in Section 3.4), and this benchmark represents the maximum welfare that can be attained in case of stochastic demand.  denotes the number of scenarios.
The results show that in the case of the fixed token supply, neither instrument is consistently superior across all scenarios (column SQ-SP in Table 3).The quantity system is superior in around 81% of the tested scenarios, with the absolute welfare difference (SQ-SP) ranging between −545$ and 701$, and mean and median values of 127 $ and 90 $ respectively.To put these differences in context, the total welfare of the no toll equilibrium ranges between 4645$ and 9021$ while that of the benchmark ranges between 8024$ and 10336$; the total toll revenue in the benchmark system ranges between 4240$ and 7027$.The percentage difference in welfare (SQ-SP) relative to the welfare of the no-toll equilibrium ranges between −6.8% and 13.9%, with mean and median percentage differences of 2% and 1.2% respectively.Note that all welfare values (ADP, SQ, SP, NT) can be considered as being relative to a situation where all travelers stay at home (i.e for example, due to very large travel times), which will yield zero welfare due to the normalization of the utility of the cancel trip option.The percentage differences need to be interpreted in this context.
Moreover, an examination of the average welfare difference between the price and quantity instrument (column SQ-SP in Table 3) relative to the average welfare difference between the price instrument and the adaptive benchmark (column ADP-SP in Table 3) suggests that the added flexibility of the credit market allows us to recover a little over 57% of the welfare lost due to the tolls in dollars and tokens being fixed across days.This is also evident when looking at the welfare differences between the price and quantity instruments relative to the adaptive benchmark (column (SQ-SP)/ADP in Table 3), which ranges between −6.2% and 7.2% with a mean value of 1.4%.
The overall distribution of welfare differences (SQ-SP) is shown in Figs.1(a) and 1(b) (the kernel density is plotted assuming a normal kernel function), which as noted before indicate that when the supply of tokens is fixed across days, neither instrument is consistently superior in terms of efficiency.In order to gain more insights into the conditions under which the quantity instrument is superior, we next examine the impacts of the shape of the congestion function, selling behavior, the benefit difference across days and the extent of heterogeneity.

Effect of congestion function
We first examine the effect of the congestion function, and draw on recent theoretical insights from de Palma and Lindsey (2020), who study tradable permit schemes in a setting with homogeneous agents and a single congestible facility.They conjecture (and explore through simple numerical examples) that in the case of variable demand and a fixed, but nonlinear cost function, the performance of a quantity control system dominates that of a price control system when the cost function is more steeply curved.This relates to their general finding that a quantity control system is relatively efficient if optimal usage levels are similar across states whereas a congestion fee achieves high efficiency if the first-best fee varies little over states.
The results from our experiments support these findings.First, the BPR congestion coefficient has a statistically significant effect (level of significance  = 0.01) on the difference in total welfare between the quantity and price control systems.The average difference increases from $2.1 at  = 3 to $63.8 at  = 4, and $313.9 at  = 6.Moreover, the percentage of scenarios where the quantity control system is superior increases from 61.8% at  = 3 to 82.6% at  = 4 and 99.3% at  = 6.The effect of  on the welfare difference is shown in the box plot in Fig. 2(a) where it can be seen that the quantity control mechanism is superior in terms of total welfare typically when the congestion curve is steeper or more convex (high value of  = 4, 6).In the box plot, the lower and upper edges of the blue box represent the 25th and 75th percentile respectively, the red line represents the median, and the notch represents a 95% confidence interval for the median.
A similar trend of increasing and statistically significant ( = 0.01) welfare differences (SQ-SP) is observed as the demand level increases (under fixed capacity, i.e., congestion levels increase).The average difference increases from $26.9 at  = 1400 to $138.6 at  = 1550, and 214.4$ at  = 1700.Moreover, the percentage of scenarios where the quantity control system is superior increases from 50.7% at  = 1400 to 93.1% at  = 1550 and 100% at  = 1700.The effect of total demand (number of travelers) on the welfare differences between the quantity and price instruments is shown in the box plot in Fig. 2

(b).
In order to gain more insight into the effect of the BPR congestion coefficient and the total demand (number of users), we examine several illustrative scenarios.First, we compare two scenarios (referred to as 1 and 2) with  = 3,  = 1400,  = 5 and  = 6,  = 1400,  = 5, respectively.All other factors including the scale of the mobility model and selling model are the same.The welfare difference between the quantity and price instruments (SQ-SP) are −72.9$and 271.0$ in scenario 1 and 2 respectively.
Table 4 summarizes the flows in different time periods and the tolls in dollar amounts (note that for the price instrument this is directly the toll in dollars whereas for the quantity instrument it is the product of the toll in tokens and the token market price).First, observe that for both scenarios, as expected, under the price instrument (SP), the number of individuals traveling (total flow in periods  1 ,  2 ,  3 ) varies significantly across the days s1 and s2 (also evident from the number of travelers canceling trip, i.e., flow in  0 ) whereas the toll in dollar amounts is fixed.In contrast, under the quantity instrument, the number of travelers traveling is roughly the same across the two days whereas the toll in dollar amounts varies significantly.Next, we see that the optimal usage of the network (or number of people traveling) under the adaptive benchmark ADP varies more across the states S1 and S2 in scenario 1 than in scenario 2 (difference in the optimal flows for  0 across days is 319 − 118 = 201 for scenario 1 versus 389 − 263 = 126 in scenario 2).Conversely, looking at optimal tolls under the benchmark ADP, one can see that the toll difference across s1 and s2 is higher in scenario 2 compared to scenario 1 (in interval  2 , 7.06 − 4.30 = 2.76 for scenario 2 versus 6.36 − 3.96 = 2.40 for scenario 1).Thus, the results suggest that at higher BPR coefficients or steeper congestion functions, the optimal usage levels of the network are relatively more similar across states leading to superiority of the quantity instrument.More intuition for this is provided analytically by de Palma and Lindsey (2020) who look at the welfare losses of the two instruments using a single congested alternative under linear and non-linear demand.
Along similar lines, to examine the effect of the total number of travelers , we compare two different scenarios (referred to as 1 and 2) with  = 3,  = 1400,  = 5 and  = 3,  = 1700,  = 5 respectively.All other factors including the scale of the mobility model and selling model are the same.The welfare difference between the quantity and price instruments (SQ-SP) are −72.9$and 88.73$ in scenario 1 and 2, respectively.Table 5 presents the flows in different time periods and the tolls in dollar amounts for these two scenarios.We observe -as before in the case of higher  -that when the overall demand level is higher (scenario 2), the difference in optimal toll rates across s1 and s2 is higher compared to scenario 1 (in interval  2 , 7.55 − 4.57 = 2.98 for scenario 2 versus 6.36 − 3.96 = 2.40 for scenario 1).Thus, at higher demand levels (and hence, more severe congestion effects) the optimal toll rates vary more across states, once again leading to superior performance of the quantity control instrument relative to scenarios with lower demand levels.

Selling model
The explicit treatment of selling behavior is an important characteristic of the proposed model and allows us to examine the impact of the selling decisions on the performance of the quantity control system.Fig. 3(a) presents a box plot of the effect of the scale parameter of the selling model on the difference between the quantity and price instruments.The value of μ = 100 000, which corresponds to μ → ∞ represents a purely random selling model (or a non-rational market) and has the impact of equalizing the supply of tokens across the three time periods (consequently, the demand of tokens as well).As the results show, this has the effect  , performance of the quantity control does not vary substantially.However, interestingly, even within the case of a non-rational market, the quantity instrument remains superior to the price instrument in cases where congestion effects are severe (e.g. = 6 in Fig. 3(b)).In other words, the advantages of the quantity control system noted in Section 5.2.2 remain even if users are not perfectly rational in the selling market, although the extent of welfare difference is marginally lower.The impact of market behavior is illustrated in Fig. 4 for a few selected scenarios which yield the highest deterioration in welfare for μ = 100 000 compared to the corresponding scenarios with μ = 1, 1.5, 2. Each marker type or series represents scenarios where all parameters are identical except μ.Note that the interpolating lines between each point are not meant to be indicative of the actual trend but are used to simply make the figure more legible.As noted above, we observe that within the range μ ∈ [1, 2], the differences in welfare of the quantity instrument are negligible whereas the welfare deteriorates if users are more irrational in the selling market.Thus, the findings suggest that market design aspects of the quantity control instrument are important and can have effects on efficiency.This is explained in more detail next.
In order to gain more intuition into the effect of selling behavior in the market, we examine two illustrative scenarios, one with μ = 1 and the second with μ = 100 000 ( = 4,  = 1550,  = 3 in both scenarios).Thus, the two scenarios are identical in all respects except the scale parameter of the selling model.The welfare of the scenario with μ = 1 is higher than that with μ = 100 000 by  104.2$.Table 6 summarizes the flows in different time periods, the tolls in dollar amounts (note that is the product of the toll in tokens and the token market price) and the demand and supply of tokens for all time intervals and both days s1 and s2.First, note that non-rational selling behavior or μ = 100 000 has the effect of equalizing the probability of selling in all three time intervals and hence, equalizes the supply of tokens for all three intervals (last two rows in Table 6).Interestingly, this also causes the token supply to be equal on both days s1 and s2 (for all three intervals).Thus, we see that the optimal tolls (SQ with μ = 100 000) on days s1 and s2 differ by an additive constant of 3$ (which is exactly equal to the benefit difference between the two days,  = 3$) resulting in identical token supply and also, identical flows on both days s1 and s2.Moreover, the equal token supply results in lower flows in interval  2 (for example, 440 on s1 for μ = 1 versus 378 on s1 for μ = 100 000).This results in a significantly higher number of travelers canceling trip (choosing  0 ) in the scenario with μ = 100 000 (476) versus the scenario with μ = 1 (391 and 386) leading to a loss in welfare.In summary, we see that the quantity control instrument is robust with respect to selling behavior in the market and is still superior even with irrational sellers when congestion effects are more severe.However, there is a deterioration in welfare when the behavior of sellers is more irrational, which causes the equilibrium price of tokens to increase leading to less travel.

Difference between states of nature
The benefit difference between the two days or states of nature () is a measure of the extent of day to day variability, and the results indicate -similar to the congestion coefficient and demand -that it significantly affects the relative performance of the two instruments ( = 0.05).The average difference increases from $81 at  = 3 to $125 at  = 4, and 175.4$ at  = 5.Further, the variance in difference between the two instruments also increases as is evident from the boxplot in Fig. 5(a).This can be better understood by examining the interaction effects of the benefit difference with the BPR congestion coefficient.Thus, when the congestion curve is steeper, a larger degree of day to day variability results in a greater advantage for the quantity control system ( = 6,  = 5 versus  = 6,  = 3 in Fig. 5(b)).On the hand, when congestion effects are less severe, an increase in day to day variation ( = 3,  = 5 versus  = 3,  = 3 in Fig. 5(b)) results in a poorer performance of the quantity control system.
This can be explained once again by the fact that at higher levels of the benefit difference (or day to day variability) and BPR coefficient, the optimal toll rates (in dollar amounts) tend to vary significantly across the states s1 and s2, leading to superior performance of the quantity instrument.

Impacts of heterogeneity
The impact of the coefficient of variation (COV) of the scale parameter   in the mobility model (note that   is assumed to be lognormally distributed across the population of travelers) is shown in the boxplot in Fig. 6.First, it can be observed that as the COV increases from   = 0.2 to   = 0.5, the mean difference in welfare between the quantity and price instruments increases only marginally from 109.8 $ to 123.1 $ (statistically insignificant at  = 0.05).
In contrast, when we examine the effect of overall heterogeneity in the mobility model (recall that the scenarios with   = 0 represent the homogeneous case where all other parameters in the mobility model are also assumed to be deterministic), we see a significant effect ( = 0.05).The mean difference between the quantity and price control systems for the homogeneous scenarios is in fact higher at 158$ (scenarios with COV of   = 0 in Fig. 6) compared to 115$ when heterogeneity is considered (scenarios with COV of   = 0.2, 0.33, 0.5 in Fig. 6) .This has important implications and suggests that ignoring heterogeneity can potentially overestimate the benefits of the quantity control system.This contrasts with the findings in de Palma et al. (2018) where heterogeneity was found to slightly increase the average welfare difference between the two instruments.The intuition for these differences is hard to arrive at; one potential cause may be differences in the nature of variability, which arises from scheduling preferences in our case as opposed to the total number of users in their case.

Equity: winners and losers
Equity is a key consideration in the comparison between the price and quantity control instruments.In this section, we compare the two instruments using the Gini coefficient computed based on the logsum (a measure of user benefits).For a detailed discussion of measures of inequality and welfare in the transportation context, we refer the readers to Trannoy (2011) andDelle Site et al. (2021).We first discuss the computation of the Gini coefficient followed by a discussion of findings.
Consider the population of  travelers ( = 1 … ), and let  () denote the user benefit of individual  in $ amounts.In the case of the no toll equilibrium (denoted NT) and tolling in dollars with no redistribution of toll revenues (denoted SPN),  () is simply the logsum of individual .In the case of tolling in dollars with an equal redistribution of toll revenues (denoted SP),  () is the logsum of individual  plus the average toll revenue per individual.Finally, in the case of tolling in tokens (denoted SQ),  () is the logsum of individual  plus the market value of the initial token endowment (since tokens are distributed for free and no tokens are unused at the end of the day).Assume that drivers arranged in increasing order of their user benefit and let  = ∕.Define,  where () represents the Lorenz curve, which is the cumulative share of total user benefits (based on the logsum measure) obtained by the bottom  individuals in the population (note that  is an integer).The Gini coefficient of user benefits (denoted GC) is computed as, The Gini coefficient is a measure of equity and takes a value between 0 and 1; a value equal to zero implies total equity and a value of 1 indicates total inequity.The larger it is, the more inequitable is the policy.Table 7 summarizes the distribution of the Gini coefficient (across the 324 scenarios with heterogeneity) for the NT equilibrium (denoted by   ) and percentage differences between the Gini coefficient for the different instruments.First, observe that with tolling in dollars wherein toll revenues are not redistributed, in a majority of the scenarios (96.3%), the Gini coefficient increases (i.e., is more inequitable) relative to the No Toll equilibrium and is on average 28.7% higher (column three of Table 7).This is in line with the general observation that pricing is vertically inequitable and benefits the rich (here the individuals with high value of time) more than the poor.The scatter plots (Fig. 7) of logsum difference (between SP and NT) versus value of time (a proxy for income) corroborate this observation, where we see that the benefits clearly increase with an increase in value of time.The plots represent two illustrative scenarios for the s2 day and each point in the plot represents an individual.Interestingly, there are a small number of scenarios (3.7%),where the Gini coefficient reduces even when toll revenues are not redistributed.This occurs in scenarios where the congestion effects are the most severe (BPR coefficient of 6 and highest demand level).
Our second observation is that both in case of the quantity instrument and the price instrument (when toll revenues are equally redistributed), there is a significant improvement of equity, by an average of 43.7%, relative to the No Toll equilibrium (column four and five of Table 7).This is a key finding and implies that both instruments favor the poor (low value of time individuals) The results are summarized in Table 8 and as expected, indicate that the quantity control system with adaptive supply is superior to that with fixed supply in all scenarios.The mean welfare difference between the quantity system with adaptive supply (denoted SQ_ A) and the price system is 190 $ compared to 127 $ with fixed supply (refer columns SQ-SP and SQ_A-SP).Moreover, a comparison of these numbers against the mean difference of 222$ between the adaptive benchmark and the price system (ADP-SP) reveals the extent of welfare improvements that can be attained by adapting the token supply across days.Thus, while the quantity instrument with fixed token supply recovers a little over 57% of the welfare loss due to fixing the tolls (in dollars and tokens) across days, the quantity instrument with adaptive token supply recovers almost 86% of this welfare loss.
However, contrary to intuition, even with adaptive token supply, the quantity control system is still not consistently superior to the price control system although it yields a higher welfare in 94.9% of the tested scenarios.This is in contrast with the findings in de Palma et al. (2018) for a single period setting where the quantity control with adaptive token supply is consistently superior.A more detailed examination shows that the scenarios where the price control is superior are in fact all scenarios where the selling behavior is non-rational or completely random (i.e., μ → ∞), as shown in the box plot in Fig. 8.As shown in Section 5.2.3, this has the effect of forcing the token supply to be equal across time periods, reducing the efficiency of the quantity control system in a manner that is not redressed even with the adaptive token supply.This once again highlights the importance of market design in the efficiency of the quantity control system.

Conclusions and further research
This paper develops a methodology to compare price control (tolling in dollars) and quantity control (tolling in tokens) instruments in the context of a within-day setting with departure time choice.In the quantity control system, users are provided an initial endowment of tokens by the regulator and incur a token charge to travel in a specific time period.Tokens can be bought and sold within a marketplace at a price determined by a smooth market clearing mechanism in each time period.A key feature of the market model is that the dynamics of price adjustment and selling decisions of users are explicitly considered.
Numerical experiments across a wide range of scenarios with demand uncertainty yield the following key insights.First, when the tolls (in dollars and tokens) can be adapted from day to day, the two instruments are equivalent.Second, when the token supply is fixed across days or states of nature and the tolls (in dollars and tokens) are non-adaptive, the quantity control instrument is superior in welfare terms when congestion effects are more severe, i.e., steep congestion functions (realistic BPR models), high demand levels and high day-to-day variability.In these scenarios, the optimal network usage is relatively similar across states whereas the optimal toll in dollar amounts varies significantly across states.Third, non-rational selling behavior, which has the effect of equalizing token supply across time intervals leads to a deterioration in the performance of the quantity instrument.However, in general, the token system is robust (in welfare terms) with respect to selling behavior in the market.Fourth, when the token supply can be adapted from day-to-day, the quantity instrument is superior in all scenarios where selling behavior is rational.Finally, when toll revenues in the price instrument are equally redistributed (typically difficult to implement in practice) and tokens (in the quantity instrument) are equally distributed, tolling in tokens is marginally more equitable in scenarios where congestion effects are more severe.These findings make a potential case for quantity control.
Several points are however noteworthy.First, income effects and second-order effects on the use of toll revenues are not considered.Second, transaction costs associated with the trading of credits, the process of finding a buyer or seller, negotiating a price, etc. are ignored.These are likely to affect the overall welfare of the quantity control system (see Nie (2012)).However, as noted by Brands et al. (2020), transaction costs may be minimized through suitable market designs.For instance, they make use of a price setting intermediary with whom users trade, and point out that this can significantly reduce transaction and negotiation costs compared to designs that include consumer to consumer trading (and over existing designs such as Dutch and English auctions, sealed-bid auctions and Vickrey auction markets).Third, the public acceptability of tradable credits is not necessarily guaranteed and will depend on the initial allocation of credits and the extent of volatility in the credit market.
There are several avenues of further research including the use of more realistic network and congestion models (for example, the morning commute problem with  −  −  preferences (Lamotte and Geroliminis, 2021) and general networks), the consideration of both departure time and route/mode choice, joint modeling of the travel and selling decision, and the inclusion of income effects and transaction costs.Further, the model can be extended to allow for users to buy tokens and resell them later if they change their travel decision, a market feature that is clearly desirable.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.18)).

Fig. 4 .
Fig. 4. Effect of μ on welfare of quantity control (SQ) for selected scenarios (note: each series represents a given scenario with varying μ).

Fig. 5 .
Fig. 5. Welfare difference: Effect of benefit difference and interaction between benefit difference/beta.

Fig. 6 .
Fig. 6.Welfare difference: Effect of heterogeneity and COV of mobility scale.
R.Seshadri et al.

Table 1
Fixed factors.

Table 3
Summary statistics: Welfare differences.

Table 4
Illustrative scenarios: Effect of BPR congestion coefficient .

Table 5
Illustrative scenarios: Effect of demand.

Table 6
Illustrative scenarios: Effect of selling behavior.
Percentage difference in Gini coefficient (SPN-NT)/NT (SP-NT)/NT (SQ-NT)/NT (SQ-SP)/SP Subset of time intervals   where   >   Token endowment per traveler   Token endowment per traveler on day  ( = 1, 2)  Number of travelers   Token market price in time interval   ( = 1 … ) p Perceived token market price for time interval   ( = 1 … )    Token market price in time interval   ( = 1 … ) on day  ( = 1, 2)   Probability of selling in time interval   ,  ∈   State of nature or day ( = 1, 2)   Supply of tokens in time interval   ,  ∈      Free flow travel time in interval   ( = 1 … )   Congested travel time in interval   ( = 1 … )   Time interval  ( = 0 … )   (  ) Utility of time interval  for individual    (  ) Systematic utility of time interval  for individual     (   ) Systematic utility of time interval  on day  for individual    Flow in time interval   ( = 1 … )    Flow in time interval   ( = 1 … ) on day  ( = 1, 2)   Value of time of individual    BPR function parameter for interval   ( = 1 … )   BPR function parameter for interval   ( = 1 … )   Toll in tokens for time interval   ( = 1 … )   Error term in utility for time interval     Scale parameter of individual  (mobility decision) μ Scale parameter of selling model   Optimum welfare of price instrument   Optimum welfare of quantity instrument   Toll in dollars for time interval   ( = 1 … ) Alternative specific benefit for individual  in time interval   ( = 0 … )   Capacity in time interval   ( = 1 … )   Demand for tokens in time interval   ,  ∈   Set of time intervals (  ,  = 0 … ) 