# Market Design for Tradable Mobility Credits

Siyu Chen  Ravi Seshadri<EMAIL_ADDRESS>Carlos Lima Azevedo  Arun P.
Akkinepally  Renming Liu  Andrea Araldo  Yu Jiang  Moshe E. Ben-Akiva  Civil
and Environmental Engineering Department, Massachusetts Institute of
Technology, Cambridge, MA, US  Department of Technology, Management and
Economics, Technical University of Denmark, Denmark  Caliper Corporation,
United States  Télécom SudParis - Institut Polytechnique de Paris, France

###### Abstract

Tradable mobility credit (TMC) schemes are an approach to travel demand
management that have received significant attention in recent years as a
promising means to mitigate the adverse environmental, economic and social
effects of urban traffic congestion. This paper proposes and analyzes
alternative market models for a TMC system – focusing on market design aspects
such as allocation/expiration of tokens, rules governing trading, transaction
fees, and regulator intervention – and develops a methodology to explicitly
model the dis-aggregate behavior of individuals within the market. Extensive
simulation experiments are conducted within a combined mode and departure time
context for the morning commute problem to compare the performance of the
alternative designs relative to congestion pricing and a no-control scenario.
The simulation experiments employ a day-to-day assignment framework wherein
transportation demand is modeled using a logit-mixture model with income
effects and supply is modeled using a standard bottleneck model.

The results indicate that small fixed transaction fees can effectively
mitigate undesirable behavior in the market without a significant loss in
efficiency (total welfare) whereas proportional transaction fees are less
effective both in terms of efficiency and in avoiding undesirable market
behavior. Further, an allocation of tokens in continuous time can be
beneficial in dealing with non-recurrent events and avoiding concentrated
trading activity. In the presence of income effects, despite small fixed
transaction fees, the TMC system yields a marginally higher social welfare
than congestion pricing while attaining revenue neutrality. Further, it is
more robust in the presence of forecasting errors and non-recurrent events due
to the adaptiveness of the market. Finally, as expected, the TMC scheme is
more equitable (when revenues from congestion pricing are not redistributed)
although it is not guaranteed to be Pareto-improving when tokens are
distributed equally.

###### keywords:

Tradable Mobility Credits; Demand Management; Traffic Management; Simulation

††journal: Elsevier

##  1 Introduction

Historically, transportation network inefficiencies and externalities such as
congestion and vehicular emissions have been addressed through road pricing,
which although used in several cities worldwide, is plagued by issues of
inequity and public acceptability [Tsekeris and Voß, 2009, de Palma and
Lindsey, 2011]. An alternative approach to travel demand management that has
received increasing 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 or
tokens to potential travelers. In order to use the road network or
transportation system, users need to spend a certain number of tokens (i.e.,
tariff) that could vary with the attributes or performance of the specific
mobility alternative used. The tokens can be bought and sold in a market at a
price determined by token demand and supply.

In principle, TMC schemes are appealing since they offer a means of directly
controlling quantity, they are revenue neutral in that there is no transfer of
money to the regulator, and they are viewed as being less vertically
inequitable than congestion pricing [de Palma and Lindsey, 2020]. Despite
these promises, several important questions remain with regard to the design
and functioning of the market within TMC schemes, an aspect critical to the
effective operationalization of these schemes. For instance, how should the
allocation and expiration of tokens be designed? What rules should govern
trading behavior in the market so as to avoid undesirable speculation and
trading (see Brands et 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 or non-recurrent events? What is the role and impact
of transaction fees? Despite the large body of literature on TMCs, issues of
market design, market dynamics and market behavior have received relatively
little attention although being critical to the successful real-world
deployment of a TMC scheme.

This paper aims to address these issues and contributes to the existing
literature in several respects. First, we propose alternative market models
(focusing on all aspects of market design including allocation/expiration of
credits, rules governing trading, transaction fees, regulator intervention,
and price dynamics) for a TMC system and develop a methodology that explicitly
models the dis-aggregate behavior of individuals within the market. Second, we
conduct extensive simulation experiments within a departure time and mode
choice context for the morning commute problem to compare the performance of
the alternative designs relative to congestion pricing and a no-control
scenario. The simulation experiments employ a day-to-day assignment framework
wherein transportation demand is modeled using a logit-mixture model (with
income effects) and supply is modeled using a standard bottleneck model. The
experiments yield insights into market design and the comparative performance
of the TMC system relative to congestion pricing.

The results indicate that small fixed transaction fees can effectively
mitigate undesirable behavior in the market without a significant loss in
efficiency (total welfare) whereas proportional transaction fees are less
effective both in terms of efficiency and in avoiding undesirable market
behavior. Further, an allocation of tokens in continuous time provides the TMC
system with additional flexibility (compared to a lump-sum allocation) and can
be beneficial in dealing with non-recurrent events. With regard to the
relative performance vis-a-vis congestion pricing, the results indicate that
the TMC scheme attains a marginally higher social welfare (under income
effects and a small fixed transaction fee). Further, the TMC scheme is more
robust in the presence of forecasting errors (during the optimization of the
toll profiles) and by adjusting token allocation, can achieve a higher welfare
than congestion pricing when actual demand and supply differs from the
anticipated demand and/or supply. Finally, as expected, the TMC scheme is more
equitable (when revenues from congestion pricing are not redistributed)
although it is not guaranteed to be Pareto-improving when tokens are
distributed equally.

The paper addresses a growing and imminent need to develop methodologies to
realistically model TMCs that are suited for real-world deployments and can
help us better understand the performance of these systems – and the impact in
particular, of market dynamics.

The rest of the paper is organized as follows. Section 2 presents a review of
the literature and identifies contributions of our paper. Section 3 and 4
propose a market design for TMCs and a framework for modeling market behavior,
respectively. Section 5 introduces the simulation model, including demand,
supply and day-to-day learning. Section 6 describes findings from extensive
simulation experiments and Section 7 provides concluding remarks and
directions for further research.

##  2 Review of Literature

Although early work on the use of tradable mobility credits (TMCs; also termed
TCS or Tradable Credit Schemes in the literature) in transportation date back
several years [Verhoef et al., 1997, Raux, 2007, Goddard, 1997], formulations
of the market and network equilibrium for TMCs is more recent, pioneered by
the work of Yang and Wang [2011] who proposed a user equilibrium variant for a
TMC. Their work, along with advancements in technology and the widely
recognized limitations of congestion pricing, has spurred interest in TMCs for
transportation network management. Extensive reviews may be found in Grant-
Muller and Xu [2014], Fan and Jiang [2013], Dogterom et al. [2017]. We provide
a brief summary of existing literature, limiting our attention to that of
mobility management (in the context of both entire networks and 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 and allows trading of credits within a market. They identify conditions
under which the network and market equilibrium are unique. Extensions to their
model have been proposed to incorporate heterogeneity in the value of time
[Wang et al., 2012] and multiple user classes [Zhu et al., 2015] using
variational inequality formulations to establish existence and uniqueness of
the equilibrium. He et al. [2013] employed a similar equilibrium approach
considering allocations of credits to not just individual travelers, but to
transportation firms such as logistics companies and transit agencies; the
effect of transaction costs in a TMC scheme with two types of markets
(auction-based and negotiated) is considered by Nie [2012]. In contrast with
the aforementioned TMC schemes, Kockelman and Kalmanje [2005], Gulipalli and
Kockelman [2008] proposed a system of credit-based congestion pricing (termed
CBCP) where credits are allowances used to pay tolls.

TMC schemes have also been studied in the context of managing congestion at a
single bottleneck (or simple two route networks) by achieving peak spreading.
Nie and Yin [2013] modeled a tradable credit scheme that manages commuters’
travel choices and attempts to persuade commuters to spread their departure
times evenly within the rush hour and between alternative routes (see also Nie
[2015]) whereas Tian et al. [2013] investigated the efficiency of a tradable
travel credit scheme for managing bottleneck congestion and modal split in a
competitive highway/transit network with heterogeneity. Along related lines,
Xiao et al. [2013] studied a tradable credit system (consisting of a time-
varying credit charge at the bottleneck wherein the credits can be traded and
the price is determined by a competitive market) to manage morning commute
congestion with both homogeneous and heterogeneous users. More recently, Bao
et al. [2019] examined the existence of equilibria under tradable credit
schemes using different models of dynamic congestion and Akamatsu and Wada
[2017] proposed a tradable bottleneck credit scheme where the regulator issues
link- and time-specific credits required for passing through a certain link or
bottleneck in a pre-specified time period. Liu et al. [2022] considered
distance-based token tariffs in a TMC scheme within a departure-time setting
and examined its performance using a trip-based MFD supply model.

In contrast with the previously described literature that largely focus on
variants of the standard user equilibrium under TMC schemes, a related stream
of research examines the design of the TMC schemes 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 efficiency properties of
tradable credits and congestion pricing has received relatively lesser
attention. de Palma et al. [2018] performed a comparative analysis of the two
instruments in a simple static transportation network (see also Seshadri et
al. [2021] for a within-day dynamic setting) and showed that as long as there
is no uncertainty, price and quantity regulation are equivalent as in the
regular market case studied by Weitzman [1974]. In the presence of uncertainty
and strongly convex congestion costs, the TMC instrument outperforms the
pricing instrument in efficiency terms (see also de Palma and Lindsey [2020]
for comparisons in the case with one route and time period under elastic
demand). Akamatsu and Wada [2017] reached similar conclusions (see also
Shirmohammadi et al. [2013]), demonstrating the equivalence of the tradable
permit system and a congestion pricing system when the road manager has
perfect information of transportation demands. On the behavior side, several
stated preference studies have highlighted the importance of key factors from
the perspective of behavioral economics and cognitive psychology towards
tradable credits [Dogterom et al., 2017].

To the best of our knowledge, Brands et al. [2020] is the only study thus far
to examine issues of market design for tradable credits. They conducted a lab-
in-the-field experiment in a parking context and examine performance of the
credit system empirically in terms of several criteria including undesirable
speculation, price stability and transaction costs.

In summary, despite the large body of research on TMCs, several gaps remain.
First, the modeling of the market has received little attention and almost all
the studies employ an equilibrium approach to model the credit market (with
the notable exception of Ye and Yang [2013] who model the price and flow
dynamics of a tradable credit scheme). The literature has – to the best of our
knowledge – thus far not attempted to realistically model the disaggregate
behavior of individuals within the market. This would enable the consideration
of empirically observed phenomena such as loss aversion, endowment effects,
mental accounting, day-to-day learning [Dogterom et al., 2017]. Second,
despite being a critical step towards real-world deployment, design aspects of
the credit market have received little attention. In particular, features such
as token allocation/expiration, trading, intervention, and transaction fees,
and their impact on efficiency and market behavior remain to be studied.
Finally, income effects, which impact both efficiency and equity, have
received relatively little attention (with the exception of Wu et al. [2012]
who consider it in a route choice setting). This paper aims to address these
gaps by proposing and analyzing alternative market designs of the TMC system
and investigating their performance relative to congestion pricing using
realistic models of traveler behavior (with heterogeneity and income effects)
and congestion.

##  3 Market Design

In this section, we focus on market design for a tradable credit scheme.
Within the TMC scheme, the regulator provides a token endowment to all
potential travelers. Although our design is generic, the application we
explore involves a daily commute context where in order to use the network at
a particular time-of-day (e.g., for a given departure time interval),
travelers have to pay a pre-specified toll in tokens that does not vary from
day to day. In other words, the toll in tokens is dynamic and varies by time-
of-day, but is fixed across days. The rationale for this assumption is that
modifying the toll in tokens from day to day would involve communicating the
tariff or toll structure on a daily basis, which is complicated, particularly
in large general networks (for instance, the electronic road pricing or ERP
scheme in Singapore includes dynamic tolls, which are revised only every three
months or longer). The market design we propose can be used for other
applications including parking management.

The regulator operates a market where tokens can be bought and sold at a
prevailing market price and may also levy pre-specified transaction fees for
buying and selling. The market price of the token varies across days and is
adjusted by the regulator to achieve revenue neutrality, considering the
demand and supply of tokens in the market. All transactions take place between
an individual and the regulator directly, who guarantees all buying and
selling requests. This central market with a regulator who acts as a price
setting intermediary is similar to the virtual bank in Brands et al. [2020],
who note that such a market can significantly reduce transaction costs
(associated with information acquisition, negotiation, finding a potential
buyer or seller etc.) compared to designs that involve consumer to consumer
trading (and over existing designs such as Dutch and English auctions, sealed-
bid auctions and Vickrey auction markets). The regulator may also intervene in
the token market within the day by controlling token market price, token
allocation, and transaction fees to manage non-recurrent events.

With regard to the token allocation or endowment, we adopt a ‘continuous time’
approach wherein tokens are acquired (provided by the regulator) at a certain
rate over the entire day and each token has a lifetime (i.e., it expires after
a certain period specified by the regulator). The expiration of tokens will
avoid undesirable consequences of the TMC system that can compromise public
acceptability such as speculative behavior and hedging in the market. The
‘continuous’ allocation avoids concentrated trading activities and excessive
trading near a boundary (a time period when a large amount of tokens expire at
the same time, such as for instance, in a lump-sum allocation). It also
provides more degrees of freedom for the regulator to intervene than that of a
‘lump sum’ allocation which distributes tokens at the beginning of each day. A
comparison between the two allocation approaches will be performed through
numerical experiments presented in Section 6.6.

As a result, each individual acquires tokens at a constant rate $r$ over the
entire day (credited into a wallet) and each token has a lifetime $L$ to avoid
speculation and hoarding. Let $x_{n}^{d}(t)$ denote traveler $n$’s token
account (or wallet) balance at time $t$ on day $d$. A full wallet state
indicates that the number of tokens in the wallet has reached a maximum
($Lr$), and in the absence of travelling or selling, does not change since the
acquisition of new tokens is balanced by an expiry of old tokens. Thus, a full
wallet implies that the oldest token in an individual’s account has an age of
$L$. In contrast, when the account is not in a full wallet state, it increases
by an amount $r\Delta_{t}$ in a unit time interval $\Delta_{t}$.

Several additional assumptions regarding market design are noteworthy – these
serve to avoid quantity buildup and market manipulation. First, travelers can
only buy tokens from the regulator at the time of traveling for immediate use,
i.e., only if they wish to travel and are short of tokens. Second, when they
sell tokens to the regulator, they have to sell all tokens in their wallet.
Third, buying and selling cannot happen at the same time, i.e. travelers can
sell all tokens anytime except at the time of buying. Note that the second
assumption differs from the design of Brands et al. [2020], who assume that
tokens can be traded per piece, and implications of this assumption warrant
more investigation, particularly when the market prices vary within-day. Since
a large part of our experiments do not involve within-day dynamic prices and
given that it considerably simplifies the modeling of selling behavior, we
defer the relaxation of this assumption to future research.

###  3.1 Account Evolution

Let $T(t)$ denote the toll in tokens to travel at time $t$,
$\tilde{t}_{n}^{d}$ represent the departure time of traveler $n$ on day $d$
and $D$ represent the duration of one day. Note that in the simulation
framework (Section 5), time will be discretized into intervals of a specified
size; for now, we treat it as continuous. Let $r$ denote the allocation rate,
$L$ denote token lifetime, and $x_{n}^{d}(t)$ denote traveler $n$’s token
account balance at time $t$ on day $d$. At time $t$ on day $d$, traveler $n$
can perform one and only one of the following actions:

  1. 1.

Perform a trip if $t=\tilde{t}_{n}^{d}$.

     * (a)

If $x_{n}^{d}(t)\geq T(t)$, she consumes $T(t)$ tokens. Her account balance at
$t+\Delta_{t}$, $x_{n}^{d}(t+\Delta_{t})$, can be written as:

$\displaystyle
x_{n}^{d}(t+\Delta_{t})=\textnormal{min}\left(x_{n}^{d}(t)-T(t)+r\Delta_{t},Lr\right),$
(1)

where the cap $Lr$ ensures that tokens with life greater than $L$ expire.

     * (b)

If $x_{n}^{d}(t)<T(t)$, she needs to buy $T(t)-x_{n}^{d}(t)$ tokens. Her
account balance $x_{n}^{d}(t+\Delta_{t})$ becomes:

$\displaystyle x_{n}^{d}(t+\Delta_{t})=r\Delta_{t}$ (2)

since all of $x_{n}^{d}(t)$ and the newly bought tokens are used to travel.

  2. 2.

Does nothing. Her account balance $x_{n}^{d}(t+\Delta_{t})$ becomes:

$\displaystyle
x_{n}^{d}(t+\Delta_{t})=\textnormal{min}\left(x_{n}^{d}(t)+r\Delta_{t},Lr\right)$
(3)

  3. 3.

Sells all tokens $x_{n}^{d}(t)$. Her account balance becomes:

$\displaystyle x_{n}^{d}(t+\Delta_{t})=r\Delta_{t}$ (4)

###  3.2 Buying and Selling

The token market price $p^{d}$ is fixed within day $d$ (in the absence of non-
recurrent events) and is only adjusted day to day. Details of the price
adjustment process are discussed in Section 3.3. We assume that the regulator
levies a two-part (fixed and proportional) transaction fee for both buying and
selling transactions. Let $F_{S}^{P}$, $F_{B}^{P}$ ($F_{S}^{P},F_{B}^{P}\geq
0$) denote the proportional part of selling and buying transaction fees (this
component of the transaction fee is proportional to the amount of the trade),
and $F_{S}^{F}$, $F_{B}^{F}$ ($F_{S}^{F},F_{B}^{F}\geq 0$) denote the fixed
part of selling and buying transaction fees. The effect of transaction fees on
market behavior and efficiency will be examined in Section 6.4.

The revenue obtained from selling $y$ tokens ($y\leq Lr$) with transaction
fees on day $d$ at time $t$ can be written as,

$\displaystyle S(y)=yp_{s}^{d}-F_{S}^{F},$ (5)

where $p_{s}^{d}=p^{d}(1-F_{S}^{P})$ is the token market price adjusted for
the proportional selling transaction fee. Transaction fees and price are not
expressed in function inputs for conciseness.

The cost of buying $y$ tokens ($y\leq Lr$) with transaction fees at time $t$
on day $d$ can be written as,

$\displaystyle B(y)=yp_{b}^{d}+F_{B}^{F},$ (6)

where $p_{b}^{d}=p^{d}(1+F_{B}^{P})$ is the token market price adjusted for
the proportional buying transaction fee.

###  3.3 Price Adjustment

The marketplace dictates the token price $p^{d}$ on day $d$, which is adjusted
according to an apriori rule established by the regulator to achieve revenue
neutrality. The price $p^{d}$ is modified daily with a deterministic rule
considering the regulator revenue $K^{d-1}$ (net revenue from all buying and
selling transactions of users) from the previous day as follows

$\displaystyle p^{d}=\begin{cases}p^{d-1}&\text{$K^{d-1}\in[-K_{t},K_{t}]$}\\\
p^{d-1}+\Delta p&\text{$K^{d-1}<-K_{t}$}\\\ p^{d-1}-\Delta
p&\text{$K^{d-1}>K_{t}$},\end{cases}$ (7)

where $\Delta p$ currently is a constant parameter representing the price
change. $K_{t}$ is a constant parameter representing a regulator revenue
threshold to adjust the price and ensures that price will not fluctuate for
small regulator revenues close to zero. Price is ensured to be positive and
below a certain cap $p_{m}$ as follows:

$\displaystyle p^{d}=\max\left(0,\min\left(p^{d},p_{m}\right)\right)$ (8)

Although token price is typically constant within a day, the regulator may
intervene in the market to adjust the market price during a day in the
presence of unusual events. For example, if road capacity drops because of an
accident, or if demand increases due to a concert, the regulator can
intervene, increasing token price in a certain period to discourage travel and
reduce congestion. Numerical experiments are conducted to study this in
Section 6.6.

Market elements discussed in this section are summarized in Table 1.

Table 1: Market elements for the tradable mobility credits system Elements | Design | Motivation
---|---|---
Allocation | Lump-sum | Simple; automated trading
Continuous |  Avoid concentrated trading; additional control
Expiration | Lifetime | Avoid quantity buildup
Transaction fee | Proportional | Avoid undesirable market behavior (e.g. frequent selling)
Fixed
Price adjustment |  Day to day constant adjustment | Balance demand and supply
Market rules governing trading

##  4 Market Behavior

As buying behavior is governed by the previously specified buying rule, this
section primarily discusses individual selling behavior. It is assumed that
the individual selling decision and mobility decision (departure time and
mode) are inter-dependent. In other words, selling decisions are made
conditional on a departure time/mode chosen at the beginning of the day, which
in turn is based on a forecast of the account balance over the entire day.
This forecast is based on historical travel and selling decisions of the user
and his/her past experience (described in more detail in Section 5.1.3). We
note that one could think of this selling behavior as a strategy or an
automated operation performed through (or programmed into) for example a
smartphone application, since in practice, it may be onerous to expect users
to constantly make these selling decisions ‘manually’. In this respect, one
may also view it as an element in the overall design of the tradable credit
scheme. However, note that our modeling framework does not preclude the use of
an actual behavioral model of selling in the market (in place of the selling
strategy we formulate next), which would require the collection of empirical
data; we defer this to future research.

From the perspective of simply maximizing profit (which is a reasonable
selling strategy), the decision to sell can be formulated as a dynamic
programming or optimal control problem, where the optimal selling strategy is
characterized by Bellman’s equation [Kirk, 2004]. However, this is
complicated, both from the standpoint of computational complexity and system
design, and instead, we derive a simpler heuristic approach to characterize an
individual’s selling strategy.

At time $t$ on day $d$, assume traveler $n$ has an upcoming planned trip at a
time denoted by $\tilde{t}_{n}$, where $\tilde{t}_{n}=\tilde{t}_{n}^{d}$ if
$t\leq\tilde{t}_{n}^{d}$, and $\tilde{t}_{n}=\tilde{t}_{n}^{(d+1)}$, if
$t>\tilde{t}_{n}^{d}$. Given the next trip, a conditional profit function
$\Pi_{n}^{d}(t)$, which represents the profit obtained by selling all tokens
at time $t$ (with no further selling until the next departure $\tilde{t}_{n}$)
can be written as follows,

$\displaystyle\Pi_{n}^{d}(t)$
$\displaystyle=S\left(x^{d}_{n}(t)\right)-\mathbb{I}\left(T(\tilde{t}_{n})\geq\hat{x}_{n}(\tilde{t}_{n})\right)\cdot
B\left(T(\tilde{t}_{n})-\hat{x}_{n}(\tilde{t}_{n})\right)$ (9)
$\displaystyle=x^{d}_{n}(t)p_{s}^{d}-F_{S}^{F}-\mathbb{I}\left(T(\tilde{t}_{n})\geq\hat{x}_{n}(\tilde{t}_{n})\right)\cdot\left(\left(T(\tilde{t}_{n})-\hat{x}_{n}(\tilde{t}_{n})\right)p_{b}^{d}+F_{B}^{F}\right)$

where $\hat{x}_{n}(\tilde{t}_{n})$ represents the expected account balance at
the time of the next trip $\tilde{t}_{n}$. Since it is assumed there will be
no further selling until the next departure $\tilde{t}_{n}$, it can be written
as,

$\displaystyle\hat{x}_{n}(\tilde{t}_{n})=\min\left[(\tilde{t}_{n}-t)r,Lr\right]$
(10)

For other notation in the conditional profit function $\Pi_{n}^{d}(t)$,
$T(\tilde{t}_{n})$ represents the toll in tokens of traveling at departure
time $\tilde{t}_{n}$. A buying cost is incurred only if the toll at
$\tilde{t}_{n}$ is greater than or equal to traveler $n$’s expected account
balance (i.e. $T(\tilde{t}_{n})\geq\hat{x}_{n}(\tilde{t}_{n})$), which is
represented by the indicator function. Note that in defining the profit
function above, we have made the critical assumption that if a decision to
sell at the current time is made, no further selling will occur until the next
trip. This simplification allows us to derive an optimal selling strategy
analytically and is partly justifiable given that we also assume that during
selling, an individual needs to sell all tokens in her wallet, and that prices
do not vary within-day. However, observe that the selling strategy we derive,
when applied, involves a decision made at every time point $t$, implying that
it does not preclude the possibility of an individual making multiple selling
decisions in the time period until the next trip if this is beneficial.

Under our assumptions, at time $t$ on day $d$, traveler $n$ will consider
selling tokens only if the profit value is positive, i.e., $\Pi_{n}^{d}(t)>0$.
If the profit value is positive, she may still decide to wait if the
derivative of the profit function is positive (meaning that the profit is
expected to increase if she defers the decision to sell). Therefore, the
selling strategy depends on both the profit function and its derivative, which
can be analyzed from the following three cases:

  1. 1.

$T(\tilde{t}_{n})<\hat{x}_{n}(\tilde{t}_{n})$ (no tokens need to be bought for
the next trip)

The profit function $\Pi_{n}^{d}(t)$ can be written as

$\displaystyle\Pi_{n}^{d}(t)$ $\displaystyle=x^{d}_{n}(t)p_{s}^{d}-F_{S}^{F}$
(11)

and the derivative can be written as

$\displaystyle\frac{d\Pi_{n}^{d}(t)}{dt}=\begin{cases}0&\text{$x_{n}^{d}(t)=Lr$}\\\
rp_{s}^{d}&\text{otherwise},\end{cases}$ (12)

which implies that profit will continue to increase until a full wallet is
reached. It does not make sense to wait longer at a full wallet because newly
acquired tokens simply replace expired tokens. Hence, selling should be at a
full wallet.

However, it is worth noting that, without fixed transaction fees, the selling
revenue at full wallet is the same as that obtained from selling every time
when one receives new tokens. In fact, as long as one avoids token expiration,
any selling strategy is equivalent in the absence of fixed transaction fees.
It is fixed transaction fees that prevent frequent selling.

  2. 2.

$T(\tilde{t}_{n})>\hat{x}_{n}(\tilde{t}_{n})$ (tokens need to be bought for
the next trip)

The profit function $\Pi_{n}^{d}(t)$ can be written as

$\displaystyle\Pi_{n}^{d}(t)$
$\displaystyle=x^{d}_{n}(t)p_{s}^{d}-F_{S}^{F}-\left(\left(T(\tilde{t}_{n})-\hat{x}_{n}(\tilde{t}_{n})\right)p_{b}^{d}+F_{B}^{F}\right)$
(13)

and its derivative can be written as

$\displaystyle\frac{d\Pi_{n}^{d}(t)}{dt}=\begin{cases}-rp_{b}^{d}&\text{$x_{n}^{d}(t)=Lr$
}\\\ rp_{s}^{d}-rp_{b}^{d}&\text{otherwise},\end{cases}$ (14)

which is always negative since $p_{s}^{d}<p_{b}^{d}$ given $F_{B}^{P}$ or
$F_{S}^{P}$ is greater than 0. This implies that profit obtained from waiting
and selling at any time in the future (until the next trip) is guaranteed to
be less than the profit from selling now. Hence, she should sell now if the
profit is positive.

Without transaction fees, the profit function $\Pi_{n}^{d}(t)$ can be written
as

$\displaystyle\Pi_{n}^{d}(t)$
$\displaystyle=x^{d}_{n}(t)p^{d}-\left(T(\tilde{t}_{n})-\hat{x}_{n}(\tilde{t}_{n})\right)p^{d}$
(15)

and its derivative can be written as

$\displaystyle\frac{d\Pi_{n}^{d}(t)}{dt}=\begin{cases}-rp^{d}&\text{$x_{n}^{d}(t)=Lr$
}\\\ 0&\text{otherwise},\end{cases}$ (16)

which means that as long as account balance is not full, it does not matter
whether one sells now or later. However, once we introduce fixed transaction
fees, it is better to sell at a full wallet to minimize the number of
transactions. With additional proportional transaction fees, it is better to
sell immediately and not worth waiting anymore as the derivative is always
negative.

  3. 3.

$T(\tilde{t}_{n})=\hat{x}_{n}(\tilde{t}_{n})$ (the expected account balance is
just enough to cover the toll of the next trip)

The profit function $\Pi_{n}^{d}(t)$ can be written as

$\displaystyle\Pi_{n}^{d}(t)$ $\displaystyle=x^{d}_{n}(t)p_{s}^{d}-F_{S}^{F}$
(17)

but its derivative does not exist because the conditional profit function is
discontinuous at $t$ due to the transaction fees of buying. To avoid any
buying transaction fees (either fixed or proportional), it is optimal to sell
immediately if profit $\Pi_{n}^{d}(t)$ is positive. Without transaction fees,
similarly, it does not matter whether one sells now or later as long as token
expiration is avoided.

Based on the analysis in this section, the effect of fixed transaction fees is
to prevent multiple transactions while the effect of proportional transaction
fees is to make one sell as soon as possible when the conditional profit is
positive (if tokens need to be bought for the next trip). The proportional
transaction fee is not preferable because it does not prevent frequent selling
but instead prevents selling at a full wallet. Numerical experiments in
Section 6 will provide further justification for the use of only a fixed
transaction fee from an efficiency perspective.

The selling strategy for an individual $n$ at any time $t$ on day $d$
considering positive transaction fees is summarized in Algorithm 1.

input :
$d,t,n,p^{d},\tilde{t}_{n},x_{n}^{d}(t),L,r,F_{S}^{P},F_{B}^{P},F_{S}^{F},F_{B}^{F}$

At time $t$ on day $d$, calculate $\Pi_{n}^{d}(t)$;

and expected account balance
$\hat{x}_{n}(\tilde{t}_{n})=\min\left[(\tilde{t}_{n}-t)r,Lr\right]$;

if  _$\Pi_{n}^{d}(t) >0$_ then

if  _$T(\tilde{t}_{n})\geq\hat{x}_{n}(\tilde{t}_{n})$_ then

Sell now;

else

if  _$x(t)=Lr$_ then

Sell now;

else

Do nothing;

end if

end if

else

Do nothing;

end if

Algorithm 1 Selling Rule

##  5 Simulation Framework

This section describes the modeling and simulation framework for evaluating
the performance of the designed instruments including a no-toll or no-control
benchmark, referred to as NT, congestion pricing, referred to as CP, and the
tradable mobility credit scheme termed TMC. The overall simulation framework
is shown in Figure 1.

$N$ travelers perform a daily commute between a single origin-destination
pair. For the sake of simplicity, each traveler performs a single morning trip
and a single evening trip. Only their morning commute trip will be explicitly
simulated and their evening trip is assumed to be a mirror of the morning
trip.

At the beginning of each day, every traveler uses forecasted information of
travel times, schedule delays and their account balance over the entire day to
make a pre-day mobility decision, which is the combination of a choice of mode
(between car and public transit, hereafter PT) and departure time (over a
individual set of departure time choices) for their morning commute trip.
Travelers who choose to drive may be subject to a time-of-day toll. For TMC,
the time-of-day toll profile is in units of tokens. Note that mobility credits
can only be used for toll road payment. The individual mobility decision is
modeled using a logit mixture model allowing for heterogeneity and non-linear
income effects.

Next, the mobility decisions along with trading decisions – which occur over
the entire day (i.e., are within-day) – are simulated on a simple network
connected by a single driving path and an alternative public transit (PT)
line. Congestion (for driving) is modeled by a point queue model (bottleneck
of finite capacity), in which a queue develops once flow exceeds capacity.
Travel time of PT is assumed to be constant.

Figure 1: Simulation Framework

Travelers’ day-to-day learning is modeled through an exponential smoothing
filter to update forecasts of travel time and account balance over the day.
The day-to-day framework in Figure 1 is used to simulate the evolution of the
system state (departure flows, travel times) until a measure of convergence
has been reached. The performance measures (overall welfare, distribution of
user benefits, congestion, and mode shares) at convergence are used to
evaluate the different instruments.

In the following sections, we first describe the models of demand, supply and
day-to-day learning, termed the system model, in more detail. Next, we discuss
social welfare computation and the simulation-based toll optimization problem
(to determine optimal tolls) for the different instruments. Relevant notation
is summarized in Table 2.

Table 2: Notation Variables | Description
---|---
$h$ | Departure time interval
$\bar{t}$ | Simulation time step
$d$ | Day $d$
$t_{h}$ | Start time of interval $h$
$\Delta_{h}$ | Duration of departure time interval
$\Delta_{t}$ | Duration of simulation time step
$\Delta_{a}$ | Size of desired arrival window
$n$ | Individual $n$
$\alpha_{n}$ | Value of time of individual $n$
$\beta_{En}/\beta_{Ln}$ | Value of schedule delay early/late of individual $n$
$\lambda$ | Coefficient of nonlinear income effect
$\gamma$ | Nonlinear income effect adjustment parameter
$\mu_{n}$ | Random component scale parameter of individual $n$
$\epsilon_{in}$ | Random utility component for mobility decision $i$ of individual $n$
$I_{n}$ | Disposable income of individual $n$
$H_{n}$ | Departure time choice set of individual $n$
$M_{n}$ | Mode choice set of individual $n$
$\hat{t}_{n}$ | Desired arrival time of individual $n$
$\eta$ | Departure time window size parameter
$p$ | Market price
$\mathrm{T}^{j}(h)$ | Toll of instrument $j$ in $h$
$\tilde{\tau}_{i}$ | Forecasted travel time of choice $i$
$\tilde{c}_{in}$ | Expected cost for mobility decision $i$
$x_{n}^{d}(t)$ | Account balance of individual $n$ at time $t$
$L$ | Token lifetime
$r$ | Token allocation rate
$t_{f}$ | Free flow travel time
$t_{v}(t)$ | Delay in queue at $t$
$Q(t)$ | Number of drivers in queue at $t$
$\theta_{\tau}/\theta_{t}$ | Weights on previous day’s forecasts
$(\cdot)^{j}$ | Variable associated with instrument $j$

###  5.1 System Model

As noted previously, the setting we consider involves $N$ users traveling
between a single origin-destination pair connected by a path containing a
bottleneck of finite capacity and a PT line. Users wish to arrive at the
destination within a certain “preferred arrival time window” in the morning,
and can choose between PT and car. If they decide to drive, they can adjust
their departure times to avoid congestion (similar to the model in Ben-Akiva
et al. [1984], which is a dynamic extension of De Palma and Lefevre [1983]).
The system is modeled using a stochastic process approach that can be viewed
as a simplification of the model in Cascetta and Cantarella [1991], who
consider the stochastic assignment problem in general networks. Day to day
adjustment is modeled using suitable learning and forecasting filters, within-
day departure time decisions and mode choices are modeled using a logit-
mixture model, and supply is modeled using a point queue model. We refer to
Cantarella and Cascetta [1995] and Watling [1999] for a nuanced discussion of
terminology and a detailed description of deterministic and stochastic process
models.

The mobility demand model, network model, and demand-supply interactions are
discussed in detail next.

####  5.1.1 Demand Model

The demand model (preday mobility decision) is a combined model of departure
time and mode choice. Unless otherwise specified, the discussion in this
subsection pertains to a specific day $d$ (we omit $d$ in all quantities for
notational convenience). The day is discretized into $h=1\ldots H$ time
intervals of size $\Delta_{h}$ (let the set of all time intervals in the day
be denoted by $\mathcal{H}=\\{1,\ldots,h,\ldots,H\\})$, and it is assumed that
each individual $n$ has a preferred/desired arrival time $\hat{t}_{n}$ (more
specifically, users are assumed to wish to arrive within a time window of size
$2\Delta_{a}$ centered around $\hat{t}_{n}$; this is discussed in more detail
later). The day is also discretized into smaller time intervals of size
$\bar{t}=1\ldots\bar{T}$ of size $\Delta_{t}$, which is the resolution of the
supply model and trading (selling) decisions.

The choice set of mode for individual $n$ is defined as $M_{n}=\\{C,PT\\}$,
where $C$ represents car and $PT$ represents transit. The choice set of
feasible departure time intervals $H_{n}\subset\mathcal{H}$ is individual-
specific and defined as
$H_{n}=\\{\tilde{t}_{0n}-\eta\Delta_{h},\tilde{t}_{0n}-(\eta-1)\Delta_{h},\dots,\tilde{t}_{0n}+\eta\Delta_{h}\\}$,
where $\eta$ is a parameter, and $\tilde{t}_{0n}$ represents the initial
departure time interval on day 0, which is computed based on the preferred
arrival time $\hat{t}_{n}$ and the free flow travel time. Thus, the departure
time choice set $H_{n}$ consists of $2\eta$ time intervals of size
$\Delta_{h}$ centered around the preferred departure time interval on day 0,
$\tilde{t}_{0n}$. Note that because we model income effects, the individual
departure time choice set is also subject to a budget constraint (i.e., an
individual cannot choose a departure time that is not affordable). Thus, we
define the set of feasible departure time intervals under instrument $j$
($j=NT,P,M$ for the No Toll scenario, congestion pricing, and the tradable
mobility credit scheme respectively), as $H_{n}^{j}\subseteq H_{n}$. Under the
No Toll scenario, $H_{n}^{NT}=H_{n}$. We will revisit the budget constraint
later when discussing income effects. Let $i=(m,h)$ represent an individual’s
mobility decision as a combination of mode and departure time choice
($i\in\\{m,h|m\in M_{n},h\in H_{n}^{j}\\}$).

Each individual is assumed to be rational and wishes to maximize her money-
metric utility from the choice situation. The utility of the mobility decision
$i$ for individual $n$ is denoted by $U_{in}$, which consists of two parts: a
systematic utility $V_{in}$ which is a function of observable variables and a
random utility component $\epsilon_{in}$ that represents the analyst’s
imperfect knowledge. $\epsilon_{in}$ is assumed to follow an i.i.d. extreme
value distribution with zero mean and individual specific scale parameter
$\mu_{n}$. It is also assumed that the individual random error component is
perfectly correlated across days and across instruments (i.e. remains the same
before the ‘change’ and after the ‘change’) assuming before and after periods
are not too far apart (e.g., McFadden [2001], de Palma and Kilani [2005]).
This assumption can be relaxed in future work (see for example Delle Site and
Salucci [2013], Zhao et al. [2008]).

The systematic money-metric utility for individual $n$ departing in time
interval $h$ by car under instrument $j$ is denoted by
$V_{in}(\boldsymbol{\boldsymbol{\tilde{\phi}}}_{i}^{j})$, where
$i\in\\{m=C,h|h\in H_{n}^{j}\\}$. $\boldsymbol{\tilde{\phi}}_{i}^{j}$ is a
vector of forecasted information in the systematic utility that affects the
choice of departure time interval for driving and consists of five components.
The first is forecasted/expected travel time $\tilde{\tau}^{j}_{i}$, which
determines the expected schedule delay early (second component) and schedule
delay late (third component). The fourth component is expected cost
$\tilde{c}^{j}_{in}$ which is explained in more detail next. The last
component is remaining income, which is equal to the disposable income for
transportation $I_{n}$ minus expected cost $\tilde{c}^{j}_{in}$.

The marginal utility of an additional unit of travel time for individual $n$
is denoted by $\alpha_{n}$. For simplicity, we assume travelers have common
knowledge of forecasted travel times (more on this in section 5.1.3). The
desired arrival time window for individual $n$ is defined as
$[\hat{t}_{n}-\Delta_{a},\hat{t}_{n}+\Delta_{a}]$, where $\hat{t}_{n}$
represents the center of the period and $\Delta_{a}$ represents arrival
flexibility. If she arrives outside of the desired time period, she incurs a
schedule delay. The marginal utility of an additional unit of schedule delay
early is $\beta_{En}$ and an additional unit of schedule delay late is
$\beta_{Ln}$, where $\beta_{En}\leq\alpha_{n}\leq\beta_{Ln}$ from empirical
evidence (e.g., Small [1982]).

The expected cost $\tilde{c}^{j}_{in}$ warrants additional discussion. Under
the No Toll (NT) scenario, it is equal to the operational cost $c_{f}$ (fuel
cost). Under pricing ($j=P$), it is equal to the toll in dollars charged for
departing in time interval $h$, $T^{P}(h)$, plus the operational cost $c_{f}$,
which can be written as

$\displaystyle\tilde{c}^{P}_{in}=T^{P}(h)+c_{f}$ (18)

Under the TMC ($j=M$) scheme, it depends on an individual’s expected
opportunity cost of tokens $\tilde{R}_{in}$ (which can be negative if one has
a net revenue from selling tokens) plus the operation cost $c_{f}$ as follows:

$\displaystyle\tilde{c}^{M}_{in}=\tilde{R}_{in}+c_{f}$ (19)

Recall that the selling revenue of $y$ tokens with transaction fees
($F_{S}^{F},F_{S}^{P}$) and token price ($p$) can be written as (selling
revenue function),

$\displaystyle S(y)=yp\left(1-F_{S}^{P}\right)-F_{S}^{F}$ (20)

and similarly, the buying cost of $y$ tokens can be written as (buying cost
function),

$\displaystyle B(y)=yp\left(1+F_{B}^{P}\right)+F_{B}^{F}$ (21)

Let $t_{h}$ be the start time of interval $h$, $\tilde{x}_{n}(t_{h})$ be the
expected account balance at time $t_{h}$, the beginning time of the time
interval $h$. If a traveler does not need to pay any toll, she can sell the
entire day’s token allocation completely. Hence, the opportunity cost (or
negative opportunity benefit) is equal to the negative of selling revenue of
the entire day’s allocation, $-S(Lr)$.

If a traveler needs to pay a toll $T^{M}(h)$ in $h$ but the expected account
balance $\tilde{x}_{n}(t_{h})$ is greater or equal to $T^{M}(h)$ (no buying),
her opportunity cost is equal to the negative of selling revenue of the one-
day allocation $Lr$ minus the toll in tokens $T^{M}(h)$, which can be written
as

$\displaystyle\tilde{R}_{in}=-S\left(Lr-T^{M}(h)\right)$ (22)

However, if she does not have enough account balance to cover the toll
$T^{M}(h)$, she has to buy additional tokens equal to
$T^{M}(h)-\tilde{x}_{n}(t_{h})$ in order to travel in $h$. The amount of
tokens she can sell for profit is equal to the one-day allocation $Lr$ minus
her expected account balance $\tilde{x}_{n}(t_{h})$ since all of her tokens
will be used for toll payment if she departs in $h$. The opportunity cost can
be written as

$\displaystyle\tilde{R}_{in}=-S\left(Lr-\tilde{x}_{n}(t_{h})\right)+B\left(T^{M}(h)-\tilde{x}_{n}(t_{h})\right)$
(23)

In summary, the expected opportunity cost $\tilde{R}_{in}$ of departing by car
in interval $h$ depends on an individual’s forecasted account balance
$\tilde{x}_{n}(t_{h})$, market price $p$, the toll in tokens $T^{M}(h)$ and
transaction fees as follows:

$\displaystyle\tilde{R}_{in}=\begin{cases}-S\left(Lr-T^{M}(h)\right)&\text{$\tilde{x}_{n}(t_{h})\geq\mathrm{T}^{M}(h)$}\\\
-S\left(Lr-\tilde{x}_{n}(t_{h})\right)+B\left(T^{M}(h)-\tilde{x}_{n}(t_{h})\right)&\text{otherwise}\end{cases}$
(24)

Note that if transaction fees are zero, the opportunity cost in Equation 24
reduces to the one-day allocation minus the toll in tokens times token price,
i.e., $\tilde{R}_{in}=-\left(Lr-\mathrm{T}^{M}(h)\right)p$. In the absence of
non-linear income effects, $Lrp$ can be ignored because it is a constant
(appearing in all alternatives) that does not affect the choice and the
expression reduces to $\mathrm{T}^{M}(h)p$, which is intuitive.

Regarding the income effect, the diminishing marginal utility of income
suggests that as an individual’s income increases, the extra benefit to that
individual decreases. It is thus natural to model this nonlinear effect of
remaining income by a quasiconcave function (as per McFadden [2017]). Hence,
we add the remaining income plus a natural log of the remaining income to the
systematic money-metric utility.

The utility of an individual $n$ driving and departing in time interval $h$
(choosing a mobility decision $i\in\\{m=C,h|h\in H_{n}^{j}\\}$) under
instrument $j$ can thus be written as,

$\displaystyle U_{in}\left(\boldsymbol{\tilde{\phi}}_{i}^{j}\right)=$
$\displaystyle
V_{in}\left(\boldsymbol{\tilde{\phi}}_{i}^{j}\right)+{\epsilon_{in}}$ (25)
$\displaystyle=$
$\displaystyle-2\alpha_{n}\tilde{\tau}^{j}_{i}-\beta_{En}SDE\left(h,\hat{t}_{n},\tilde{\tau}^{j}_{i}\right)-\beta_{Ln}SDL\left(h,\hat{t}_{n},\tilde{\tau}^{j}_{i}\right)$
$\displaystyle+I_{n}-2\tilde{c}^{j}_{in}+\lambda
ln\left(\gamma+I_{n}-2\tilde{c}^{j}_{in}\right)+{\epsilon_{in}},$

where

$\displaystyle
SDE\left(h,\hat{t}_{n},\tilde{\tau}^{j}_{i}\right)=\max\left(0,\hat{t}_{n}-\Delta_{a}-(t_{h}+\tilde{\tau}^{j}_{i})\right)$
(26) $\displaystyle
SDL\left(h,\hat{t}_{n},\tilde{\tau}^{j}_{i}\right)=\max\left(0,(t_{h}+\tilde{\tau}^{j}_{i})-\hat{t}_{n}-\Delta_{a}\right)$
(27)

Schedule delay of the evening trip is ignored because it is assumed to be more
flexible.

The systematic money-metric utility function of user $n$ who departs in time
interval $h$ by PT is denoted as $V_{in}(\boldsymbol{\tilde{\phi}}_{i}^{j})$,
where $i\in\\{m=PT,h|h=\lfloor\hat{t}_{n}-\tau_{PT}\rfloor\\}$. Since the
travel time and headway of PT are constant, we only need to consider one
departure time interval $h$, which has a corresponding arrival time closest to
the desired arrival time $\hat{t}_{n}$. For PT, the input vector
$\boldsymbol{\tilde{\phi}}_{i}^{j}$ for the systematic utility consists of
four components: PT travel time $\tau_{PT}$, expected waiting time $W_{PT}$,
expected PT cost $\tilde{c}^{j}_{in}$ and remaining income
$I_{n}-\tilde{c}^{j}_{in}$.

The marginal utility of an additional unit of PT travel time of individual $n$
is assumed to be the same as that of car travel time, $\alpha_{n}$. The
marginal utility of an additional unit of waiting time is $\beta_{Wn}$.

The expected PT cost $\tilde{c}^{j}_{in}$ is equal to the PT fare $c_{PT}$
under the No Toll (NT) scenario and pricing. Under the TMC scheme, it depends
on an individual’s expected opportunity cost of tokens $\tilde{R}_{in}$ and
the PT fare $c_{PT}$, where $\tilde{R}_{in}$ is equal to the negative of
selling revenue of a full wallet $Lr$ since travelers who choose PT can sell
all of their tokens acquired in one day for maximum return. It can be written
as $\tilde{R}_{in}=-S(Lr)$.

Hence, the expected PT cost $\tilde{c}^{j}_{in}$ under the TMC scheme can be
written as

$\displaystyle\tilde{c}^{M}_{in}=\tilde{R}_{in}+c_{PT}$ (28)

The utility of an individual $n$ using PT who departs in interval $h$
(choosing a mobility decision
$i\in\\{m=PT,h|h=\lfloor\hat{t}_{n}-\tau_{PT}\rfloor\\}$) can be thus written
as,

$\displaystyle U_{in}\left(\boldsymbol{\tilde{\phi}}_{i}^{j}\right)=$
$\displaystyle
V_{in}\left(\boldsymbol{\tilde{\phi}}_{i}^{j}\right)+\epsilon_{in}$ (29)
$\displaystyle=$
$\displaystyle-2\alpha_{n}\tau_{PT}-2\beta_{Wn}W_{PT}+I_{n}-2\tilde{c}^{j}_{in}+\lambda
ln\left(\gamma+I_{n}-2\tilde{c}^{j}_{in}\right)+\epsilon_{in}$

####  5.1.2 Supply Model

The network is assumed to be a single origin-destination pair connected by a
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 travelers
exceeds $s$. The free flow travel time is $t_{f}$ and the extra delay time for
a traveler departing from home at time $t$ is $t_{v}(t)$. Thus, the total
travel time for a traveler departing from home at time $t$ is:

$\displaystyle\tau(t)=t_{v}(t)+t_{f}$ (30)

Let $Q(t)$ be the number of travelers in the queue at time $t$. The delay at
time $t$ is derived from the deterministic queuing model as follows:

$\displaystyle t_{v}(t)=\frac{Q(t)}{s},$ (31)

where $Q(t)=0$ and $t_{v}(t)=0$ when there is no congestion.

Note that within our simulation, the capacity $s$ is defined for time
intervals $\bar{t}$ of size $\Delta_{t}$. The travel time for a given
departure time interval $h$ is obtained by averaging the travel times of all
travelers departing in $h$. Further, the exact time of departure of a traveler
within the supply model is randomly (uniformly) drawn within the chosen
departure time interval $h$.

The alternative PT line has a constant travel time $\tau_{PT}$. Its headway is
also constant, which is equal to twice the expected waiting time $W_{PT}$.

####  5.1.3 Day-to-Day Learning

Let $\tau^{d-1,j}_{i}$ denote the actual or experienced car travel time on day
$d-1$ of choice $i$ under instrument $j$, where $i\in\\{m=C,h|h\in
H_{n}^{j}\\}$. As we specified in the demand model, travelers are assumed to
make their choices of departure time according to forecasted car travel times
$\tilde{\tau}^{d,j}_{i},\;\forall h\in\mathcal{H}$ from their memory and
learning. We use an exponential smoothing filter, a type of homogeneous filter
[Cantarella and Cascetta, 1995], to model the learning and forecasting process
by weighting actual and forecasted costs of the previous day as follows:

$\displaystyle\tilde{\tau}^{d,j}_{i}=(1-\theta_{\tau})\tilde{\tau}^{d-1,j}_{i}+\theta_{\tau}{\tau}^{d-1,j}_{i}$
(32)

where $\theta_{\tau}\in[0,1]$ is a learning weight given to the previous day’s
realized travel time.

Under the TMC scheme, in order to obtain the individual forecasted account
balance on day $d$, denoted by $\tilde{x}^{d}_{n}(\bar{t})$, the individual
forecasted departure time $\tilde{t}^{d}_{n}$ is first computed by applying a
similar filter as follows:

$\displaystyle\tilde{t}^{d}_{n}=(1-\theta_{{t}})\tilde{t}^{d-1}_{n}+\theta_{{t}}{{t}}^{d-1}_{n}$
(33)

where $\theta_{\tilde{t}}\in[0,1]$.

Next, the trading model presented in Section 3.2 is applied using the
individual forecasted departure time to determine their forecasted account
balance over the entire day, which is used to compute the expected toll costs
under the TMC scheme through Equation 24.

###  5.2 Simulation-based optimization

The problem of determining the optimal toll in dollars for congestion pricing,
$T^{P}(h),\;\forall h\in\mathcal{H}$ and the optimal toll in tokens for the
TMC scheme, $T^{M}(h),\;\forall h\in\mathcal{H}$ can be formulated as a
simulation-based optimization problem with the objective of maximizing total
social welfare ($SW$). The social welfare of the CP and TMC instruments is
calculated relative to the NT scenario and is equal to the sum of user
benefits ($UB$) and regulator revenue ($K$). Under congestion pricing (P), the
regulator revenue is given by,

$\displaystyle K^{P}=\sum_{n=1}^{N}\sum_{i\in M_{n}\times
H^{P}_{n}}\hat{c}^{P}_{in}\mathbb{I}_{n}\left(i|\boldsymbol{T}^{P}\right),$
(34)

where $i$ is the mobility decision, which is a combination of mode and
departure time choice; $\mathbb{I}_{n}(i|\boldsymbol{T}^{P})$ is an indicator
if traveler $n$ chooses mobility choice $i$ given toll vector
$\boldsymbol{T}^{P}=\\{T^{P}(h)|h\in\mathcal{H}\\}$; $\hat{c}^{P}_{in}$ is
equal to the toll payment for driving ($T^{P}(h)$) or the PT fare payment for
PT ($c_{PT}$); and $H^{P}_{n}$ is the set of feasible departure time intervals
taking into account budget constraints.

Under the TMC scheme, the regulator revenue $K^{M}$ consists of two parts. The
first part is the sum of PT fare payments and the second part is the sum of
tokens bought (by individuals) minus tokens sold over one day. $K^{M}$ can be
written as,

$\displaystyle
K^{M}=\sum_{n=1}^{N}\left(c_{PT}\mathbb{I}_{n}\left(PT|T^{M}(h)\right)+\sum_{\bar{t}\in\\{1...\bar{T}\\}}\left(B\left(T^{M}\left(\bar{t}\right)-x_{n}(\bar{t})\right)\mathbb{I}^{B}_{n}\left(\bar{t}|\boldsymbol{T}^{M}\right)-S\left(x_{n}(\bar{t})\right)\mathbb{I}^{S}_{n}(\bar{t}|\boldsymbol{T}^{M})\right)\right),$
(35)

where $B(\cdot)$ is cost of buying function and $S(\cdot)$ is revenue of
selling function; $\mathbb{I}^{B}_{n}\left(\bar{t}|\boldsymbol{T}^{M}\right)$
and $\mathbb{I}^{S}_{n}\left(\bar{t}|\boldsymbol{T}^{M}\right)$ are indicators
of buying or selling at $\bar{t}$ given the toll in tokens
$\boldsymbol{T}^{M}$. Note that the price adjustment mechanism described in
Section 3.3 is designed to ensure that the regulator revenue of the TMC scheme
(second part in Equation 35) at equilibrium is close to zero, thereby
achieving revenue neutrality.

The user benefits, which are a measure of consumer well being can be computed
in three ways in the presence of non-linear income effects (we refer the
reader to McFadden [2017] for a detailed discussion): Market Compensating
Equivalent (MCE), Hicksian Compensating Variation (HCV), and Hicksian
Equivalent Variation (HEV). MCE is equal to the difference in indirect
utilities between a “but for” scenario and “as is” scenario, scaled to money
metric units by dividing by the marginal utility of income (MUI) at the “as
is” scenario. It differs from the commonly used Marshallian consumer surplus
(MCS) only in the MUI scaling factor. It can be easily computed when the
indirect utility function and its derivatives are known. HCV is equal to the
net decrease in the “but for” scenario income that equates utility in the two
scenarios while HEV is equal to the net increase in the “as is”” scenario
income that equates utility in the two scenarios.

A potential drawback of these three measures is that their ethical
implications are not defensible as pointed out by Blackorby and Donaldson
[1990]. Well-being measured in units of income treat increases in income as
equally socially valuable no matter who receives them. This is not the case
with net utility improvement since the nonlinear effect of income improvement
is captured by the income effect term in the utility specification (lower
income users have a higher marginal utility of income). Hence, we measure user
benefits ($Z^{j}$) under instrument $j$ as the sum of all users’ net
experienced utilities relative to NT denoted as $z^{j}_{n}$ (along the lines
of De Palma and Lindsey [2004]). Since the utilities adopted in this study are
money-metric, the net utility amount serves as a meaningful measurement of
improvement directly. An individual $n$’s net experienced utility is the
difference between maximum utility under instrument $j$ and under NT, which
can be written as,

$\displaystyle z^{j}_{n}=\max_{i\in M_{n}\times
H^{j}_{n}}\left(U_{in}\left(\boldsymbol{\phi}_{i}^{j}\right)\right)-\max_{i\in
M_{n}\times
H^{NT}_{n}}\left(U_{in}\left(\boldsymbol{\phi}_{i}^{NT}\right)\right),$ (36)

where $\boldsymbol{\phi}_{i}^{j}$ is a vector of experienced variables under
instrument $j$ and $\boldsymbol{\phi}_{i}^{NT}$ is a vector of experienced
variables under $NT$.

Hence, the user benefits $Z^{j}$ can be written as

$\displaystyle Z^{j}=\sum_{n=1}^{N}z^{j}_{n}$ (37)

As noted before, in the case of the CP scheme, we determine the toll in
dollars which maximizes social welfare $SW$, computed at the equilibrium (or
after convergence of the day-to-day model). For the TMC system, in addition to
the toll in tokens, which is optimized, other parameters like the allocation
rate $r$ and transaction fees are set exogenously (explained in more detail in
Section 6.3). This is formulated as,

$\displaystyle\max_{\boldsymbol{T}^{j}}$ $\displaystyle Z^{j}+K^{j}$ (38) s.t.
$\displaystyle
Z^{j},K^{j}=SM\left(\boldsymbol{T}^{j},\boldsymbol{\xi},\boldsymbol{\psi}\right)$
$\displaystyle\boldsymbol{T}^{j}=\\{T^{j}(h)|h\in\mathcal{H}\\}$
$\displaystyle\boldsymbol{T}^{j}\geq 0$

where $j$ can be either $P$ or $M$; the toll profile $\boldsymbol{T}^{j}$ is a
set of toll values over the entire day. $\boldsymbol{\xi}$ represents all
input data for simulation, such as individual income, preferred arrival time,
and choice attributes. $\boldsymbol{\psi}$ represents all model parameters,
such as demand model coefficients, bottleneck capacity, user learning weights,
and market parameters for the TMC scheme. The $SM(\cdot)$ function is the
system model discussed in Section 5.1. The toll function that we consider is a
step toll profile (of the kind implemented in Singapore and Stockholm), which
consists of five step toll values and six break points.

Clearly, our optimization problem has no closed-form since the objective
function for a given toll profile is the outcome of a simulation of the
stochastic process (a simulation-based optimization problem), or more
specifically, the system model presented in 5.1, which includes traveler
behavior, regulator states and actions, and the resulting network and market
conditions. In order to solve this simulation-based optimization problem, a
differential evolution (DE) algorithm is adopted as it is derivative-free and
performs well for global optimization problems of this kind [Storn and Price,
1997].

####  5.2.1 Differential evolution algorithm

In this section, we illustrate the application of the DE algorithm. Let
$\mathbf{X}$ represent the decision variables of the simulation-based
optimization problem (i.e., the parameters of the step toll profile). The DE
algorithm essentially has three iterative operators, mutation, crossover and
selection, to iteratively improve candidate solutions.

The mutation operator uses individuals from the current solution population to
generate variant vectors. The $g$th variable of vector $i$ at generation $k$,
$\mathbf{Y}_{g,i}^{k}$, is given by

$\displaystyle\mathbf{Y}_{g,i}^{k}=\mathbf{X}_{g,r1}^{k}+F\cdot(\mathbf{X}_{g,r2}^{k}-\mathbf{X}_{g,r3}^{k})$
(39)

where $r1,~{}r2,~{}r3\in[1,NP]$, $i\neq r1\neq r2\neq r3$, $F$ is a scale
factor and $NP$ is the solution population size.

Next, the crossover operator creates a trial vector $\mathbf{U}_{i}^{k}$ by
combining the variant vector and original vector as follows,

$\displaystyle\mathbf{U}_{g,i}^{k}=\begin{cases}\mathbf{Y}_{g,i}^{k}&\text{if}~{}rand(0,1)<CR~{}\text{or}~{}g=rg\\\
\mathbf{X}_{g,i}^{k}&\text{otherwise}\end{cases}$ (40)

where $CR\in[0,1]$ is the crossover rate, $rand(0,1)$ represents a random
uniformly distributed variable within $(0,1)$, and $rg$ is a random integer in
$[1,3M]$ ensuring at least one variable of the trial vector
$\mathbf{U}_{i}^{k}$ is from the variant vector $\mathbf{Y}_{i}^{k}$.

Finally, the selection operator produces the next generation of vectors by
comparing the original vector $\mathbf{X}_{i}^{k}$ and the trial vector
$\mathbf{U}_{i}^{k}$ in terms of social welfare as follows,

$\displaystyle\mathbf{X}_{i}^{k+1}=\begin{cases}\mathbf{U}_{i}^{k}&\text{if}~{}SW(\mathbf{U}_{i}^{k})>SW(\mathbf{X}_{i}^{k})\\\
\mathbf{X}_{i}^{k}&\text{otherwise}\end{cases}$ (41)

##  6 Numerical Experiments

This section describes numerical experiments that examine the performance of
the TMC scheme using the simulation framework described in Section 5. Data and
parameters of the demand model, supply model, and the day-to-day learning
model are introduced in Section 6.1. The parameters are calibrated based on
empirical evidence to represent a realistic base case. Selected parameters
will be varied in later experiments. The objectives of the experiments are to:
1) examine the overall performance of the TMC system under alternative market
designs – more specifically, the effect of transactions fees in avoiding
undesirable market behavior whilst retaining efficiency, 2) compare the
efficiency and equity of the TMC scheme (market design based on experiments
under 1) and CP under varying levels of congestion, heterogeneity, income
effects, and 3) examine the robustness (adaptiveness) of the TMC scheme with a
lump sum allocation versus a continuous allocation in the presence of unusual
events.

###  6.1 Data and Model Parameters

The demand model requires both choice attributes and individual
characteristics as input data. The individual characteristics include
disposable income $I_{n}$ and preferred arrival time $\hat{t}_{n}$. Recall
that disposable income $I_{n}$ in our context is defined as personal net
income after taxes and subtracting necessary living expenses (e.g. housing,
health, food). The individual pre-tax annual income is assumed to follow a
lognormal distribution and is fitted using the Integrated Public Use Microdata
Series (IPUMS) 2019 census data [Ruggles et al., 2021]. The cumulative
distribution function (CDF) of IPUMS data and fitted data are shown in Figure
2. Note that all annual incomes greater than 500 thousand dollars are grouped
together. As we can see, the lognormal distribution fits the income
distribution reasonably well.

Figure 2: Individual pre-tax annual income distribution

Individual daily income is computed as the annual income divided by 260
working days per year and the individual hourly wage rate is computed by
dividing daily income by 8 working hours per day. The minimum wage rate is set
to $7.25 per hour as per the U.S. Department of Labor. It is less
straightforward to obtain disposable income after taxes since necessary living
expenses could vary significantly based on income and disaggregate data on
this is sparse. Based on average data from the Bureau of Labor Statistics
[U.S. Bureau of Labor Statistics, 2019], we assume that each traveler’s daily
disposable income after taxes and necessary living expenses is equal to 60% of
their pre-tax daily income.

The preferred departure time distribution (preferred arrival time
$\hat{t}_{n}$ minus the free flow travel time) is based on a recent empirical
study of road users in Stockholm [Kristoffersson and Engelson, 2018], and is
shown in Figure 3. For simplicity, the size of the preferred arrival window
$\Delta_{a}$ is set to 0, which implies that individuals have a single
preferred arrival time as in the standard Vickrey model. The departure time
window size parameter $\eta$ is set to 30, which means the individual
departure time choice set $H_{n}$ ranges over a 60-minute interval.

Figure 3: Individual preferred departure times distribution

Empirical evidence (e.g. Small et al. [2005]) indicates that there is
significant heterogeneity in the value of time across drivers. We assume that
the individual value of time $\alpha_{n}$ is perfectly correlated with income
and is one third of the wage rate [White, 2016]. Note that this assumption
will be relaxed in some experimental scenarios to consider different levels of
heterogeneity.

Values of schedule delay early $\beta_{En}$ and late $\beta_{Ln}$ are also
likely to be distributed across individuals. Due to the lack of empirical data
on this, the literature on bottleneck models incorporate heterogeneity by
making assumptions on the ratios between values of schedule early/late to
values of time. Proportional heterogeneity (first considered by Vickrey
[1973]) assumes that values of time and schedule delays vary proportionally or
in other words, the ratio of the parameters is identical for all individuals
[Van den Berg and Verhoef, 2011]. Ratio heterogeneity assumes that the values
of schedule delays are fixed and only values of time are distributed [De Palma
and Lindsey, 2002]. As a result, ratios of parameters are distributions. Van
Den Berg and Verhoef [2011] considers a more general heterogeneity, assuming
the ratio of values of time to values of schedule delay early follows a
symmetric triangular distribution from 1 to 3 based on intuition and ratio of
values of schedule delay late to values of schedule delay early is a constant
3.9 based on Arnott et al. [1990]. Along similar lines, we assume that the
ratio of values of schedule delay early $\beta_{En}$ to values of time
$\alpha_{n}$ follows a triangular distribution from 0.1 to 1 with a mode at
0.5. The ratio of values of schedule delay late $\beta_{Ln}$ to $\alpha_{n}$
is assumed to follow a triangular distribution from 1 to 3 with a mode at 2
(the selection of the modes as 0.5 and 2 are based on Small [2012]). The
bounds are set based the empirical relationship
$\beta_{En}\leq\alpha_{n}\leq\beta_{Ln}$ [Small, 2012].

Further, as pointed out by Small [2012], waiting times are onerous compared to
in–vehicle times by multiples of two to three by most assessments. For
simplicity, the ratio of values of time $\alpha_{n}$ to values of waiting time
$\beta_{Wn}$ is assumed to be a constant equal to 3. With regard to the income
effect, $\gamma$ is set to 2 and $\lambda$ is calibrated to be 3 to have the
highest marginal utility of income to be less than 1.34 [Layard et al., 2008].

The scale parameter $\mu_{n}$ is known to be both confounded with the
systematic utility and inversely related to error variance within the choice
data [Ben-Akiva et al., 1985]. As pointed out in the literature (e.g. Louviere
and Eagle [2006]), the modeled heterogeneity can come from heterogeneity in
individual coefficients and scale heterogeneity that is shared across
coefficients. We assume the scale parameter follows a lognormal distribution.
The mean value is calibrated based on price elasticity (discussed
subsequently) and the coefficient of variation is set to 0.5 based on
judgement.

Next, we calibrate the mean value of the scale parameter to ensure that the
aggregate price elasticities of the mobility model (for departure time choice)
are reasonable and accord with empirical evidence. The price elasticity of
peak hour demand (7 AM - 8AM) is computed assuming there exists a flat toll
profile during the period from 6:30-9:30 AM. From the literature, the
aggregate elasticities of peak hour travel vary greatly from case to case as
they are dependent on the model structure, physical environment, activity
type, initial toll levels and many other factors. Ding et al. [2015] estimated
the elasticity of departing during the peak in Washington D.C. to be -0.0906
for driving alone. Sasic and Habib [2013] estimated that the elasticity of
departing by car in the AM peak for work trips in Toronto is between -0.067 to
-0.12. Holguin-Veras et al. [2005] found price elasticity of using crossings
(tunnels and bridges) in NYC ranges from -0.31 to -1.97 for weekdays depending
on the time-of-day. When there is no initial toll, the price elasticity simply
represents fuel price elasticity. Lipow [2008] estimated fuel price elasticity
as -0.17 and Gillingham [2014] estimated fuel price elasticity in California
as -0.15.

Table 3: Price elasticities across income groups by toll levels Toll | $\leq 25\%$ |  $25\%$ to $50\%$ |  $50\%$ to $75\%$ |  $75\%$ to $90\%$ | $90\%\leq$ | Total
---|---|---|---|---|---|---
0 | -0.34 | -0.29 | -0.12 | 0.00 | 0.00 | -0.19
2.5 | -1.14 | -0.59 | -0.10 | -0.04 | -0.03 | -0.38
5 | -1.57 | -1.07 | -0.20 | -0.09 | -0.06 | -0.53

From calibration, the mean of scale parameter is determined to be 0.5. The
corresponding price elasticities across different income groups and initial
toll levels are presented in Table 3. As we can see, low income users are more
sensitive to price than high income users, and when there is no toll, the
aggregate price elasticity is similar to the empirical fuel elasticity. As the
toll level increases, the aggregate price elasticity also increases and is
similar to empirical values found in Holguin-Veras et al. [2005].

Regarding the supply model, the free flow speed of car is set to be 45mph [Ali
et al., 2007] and the one way driving distance is assumed to be 18 miles (free
flow travel time of 24 minutes). The operational cost of driving is assumed to
be composed of only fuel cost, which is equal to $3.13 (driving distance times
1/23 gallon per mile times 4 dollars per gallon). For public transit, based on
the New York City MTA data, the fare is set to $ 2, average speed is 25mph,
and headway is 10 minutes. The PT distance is also assumed to be 18 miles, and
the resulting PT travel time is 43 minutes since both headway and travel time
of PT are constant. The expected waiting time is assumed to be 5 minutes.

Table 4: Model and simulation parameters Variables | Description | Values
---|---|---
$N$ | Population | $7,500$
$\Delta_{t}$ | Duration of a simulation time step | 1 min
$\Delta_{h}$ | Duration of a departure time interval | 5 min
$\Delta_{a}$ | Size of desired arrival window | 0 min
$\eta$ | Departure time window size parameter | 30
$\lambda$ | Coefficient of nonlinear income effect | 3
$\gamma$ | Nonlinear income effect adjustment parameter | 2
$s$ | Bottleneck capacity (per min) | 39
$t_{0}$ | Free flow travel time |  $24$ mins
$c_{f}$ | Operation cost of car | $3.13
$\tau_{PT}$ | PT travel time |  $43$ mins
$W_{PT}$ | Expected waiting time |  $5$ mins
$c_{PT}$ | Operation cost of PT | $2
$\theta_{\tau}/\theta_{\bar{t}}$ | Learning weights | 0.1

The bottleneck capacity $s$ is determined based on a calibration of the travel
time index (TTI), which represents the ratio between actual travel time and
free flow travel time. The base case capacity is determined to be 2340
vehicles per hour to have a reasonable level of congestion with a TTI of 1.68
[Chen, 2010] under the no toll (NT) scenario.

As described in Section 5.1.3, an exponential smoothing filter is adopted to
update travel time information and individual departure time. The greater the
learning weights are, the more unstable the system becomes [Cantarella and
Cascetta, 1995]. The learning weights $\theta_{\tau}$ and $\theta_{\tilde{t}}$
are assumed to be 0.1.

Recall that we focus on the morning commute and hence, we simulate half a day
(12 hours) with a simulation time interval $\Delta_{t}$ of 1-minute, yielding
720 time intervals, $\bar{t}=0\ldots 719$. The market elements (allocation,
expiration, and price adjustment) and trading behavior are also simulated for
the first half. The second half of a day is assumed to be a mirror of the
first half. The departure time interval ($\Delta_{h}$) is assumed to be 5
minutes and the population size $N$ is 7500. Descriptions and values of key
parameters are summarized in Table 4.

###  6.2 Existence and Uniqueness of Equilibrium

Existence and uniqueness of equilibrium in the bottleneck model have been
widely studied (e.g. Hendrickson and Kocur [1981], Hurdle et al. [1983], Smith
[1984], Daganzo [1985]), although heterogeneity is typically limited to the
desired arrival time. Lindsey [2004] considers a more general heterogeneity in
values of time, desired arrival times, and values of schedule delay and
establishes conditions for existence and uniqueness of a deterministic
departure time user equilibrium in the bottleneck model. De Palma et al.
[1983] examine a stochastic user equilibrium where travelers are assumed to
have identical systematic travel cost functions. They show that the
equilibrium departure rate is unique but their findings cannot be directly
applied to our model as we consider heterogeneity. Miyao and Shapiro [1981]
establish conditions for the existence, uniqueness and stability of
equilibrium for discrete choice models. Although their framework is relatively
general, it assumes individuals have the same choice set whereas in our case,
choice sets (for departure time choices) are individual specific. To the best
of our knowledge, there are no analytical results on uniqueness of the
equilibrium for our specific model. However, simulations with different
initial conditions suggests that the day-to-day model converges to the same
equilibrium solution. The tests on equilibrium and stationarity are summarized
in A.

Note that the travel time of driving is used as a representative measure of
the system state partly because it is central to the day-to-day learning
process of travelers (alternatively, departure flows could also be used). The
infinity norm (supremum norm) of day $d-1$’s travel time vector of driving and
day $d$’s travel time vector of driving is calculated and used as a measure of
convergence across days. This is given by,

$\displaystyle||\boldsymbol{\tau}^{d-1}-\boldsymbol{\tau}^{d}||_{\infty}=\sup{\\{|\tau^{d-1}_{i}-\tau^{d}_{i}|:i\in\\{m=C,h|h\in\mathcal{H}\\}\\}}$
(42)

where $\boldsymbol{\tau}^{d-1}$ is the vector of travel times of day $d-1$
defined over a set of time intervals $\mathcal{H}$.

(a) Convergence of the social welfare

(b) Optimal toll profile

Figure 4: Convergence of the social welfare and optimal time-of-day step toll
profile of Congestion Pricing for different initial values

###  6.3 Optimization of tolls in TMC and Congestion Pricing

As discussed in Section 5.2.1, the time-of-day step toll profile is optimized
using a type of metaheuristic algorithm known as Differential Evolution (DE).
Metaheuristic algorithms have been shown to work well for nonconvex and
nonlinear toll design problems (e.g. Shepherd and Sumalee [2004], Zhang and
Yang [2004]). The population size of the DE algorithm $NP$ is set to 15. Using
data and parameters discussed in Section 6.1, we first examine the performance
of the optimization algorithm for the congestion pricing (CP) instrument using
three different initial populations (with 15 candidates). Next, the
performance of the optimized TMC instrument and convergence properties are
examined.

For CP, the convergence of social welfare along with the optimal time-of-day
step toll profile with three different initial populations are shown in Figure
4. The algorithm converges to the same optimal welfare (within a tolerance of
$0.01) for three different initial populations. The optimal toll profiles are
near identical although there are minor differences at the locations when the
toll changes (‘steps’).

For the TMC instrument, we set the total daily allocation of tokens for each
individual as the per capita regulator revenue from congestion pricing (this
should in theory yield a 1 dollar equilibrium market price and a toll in
tokens that is identical to the optimal toll in dollars from CP, assuming
there are no transaction costs and income effects). Note that the daily
allocation can be set arbitrarily. Next, we optimize the toll profile in
tokens given the specified token allocation. We assume that tokens are
distributed uniformly to everyone and are allocated in continuous time. Every
traveler is assumed to have a random account balance at the beginning of the
simulation (to reflect different times of entry into the system; more on this
in Section 6.4). Recall also that we assume the evening period is a mirror of
the morning period, and hence we only simulate half a day, and assume the
lifetime $L$ of tokens is 720 minutes. For simplicity, in these experiments,
transaction fees are set to zero. Other parameters of the TMC are summarized
in Table 5.

Table 5: Market parameters for tradable mobility credits (base case) Variables | Description | Values
---|---|---
$r$ | Token allocation rate | 0.00285 per min
$L$ | Token lifetime | 720 mins
$F^{F}_{B}/F^{F}_{S}$ | Proportional transaction fee of buying/selling | $0
$F^{P}_{B}/F^{P}_{S}$ | Fixed transaction fee of buying/selling | $0
$p^{0}$ | Initial token price | $1
$\Delta p$ | Price change | $0.05
$K_{t}$ | Regulator revenue threshold | $300

The convergence of the optimized objective values (social welfare) along with
the optimal time-of-day step toll profile in tokens with three different
allocation rates ranging from 15% less than the baseline to 15% more than the
baseline are shown in Figure 5. As we can see, the different allocation rates
converge to the same social welfare (within a tolerance of $0.02) at the end
of 300 iterations. The token price of the baseline allocation rate is close to
$1 (as expected) while the lower allocation rate has a higher token price of
$1.1 and the higher allocation rate has a lower token price equal to $0.9.
This is consistent with our expectation that the lower allocation rate leads
to a higher token price due to less supply and vice versa. The optimal toll
profiles in tokens in Figure 5(b) show that the lower allocation rate leads to
the overall higher tolls in tokens and vice versa, again, as expected.
Interestingly, due to the non-linear income effects, the social welfare of the
TMC scheme is slightly greater than that of congestion pricing (we return to
this later).

(a) Convergence of the social welfare

(b) Optimal toll profile in tokens

Figure 5: Convergence of the social welfare, optimal time-of-day step toll
profile in tokens and in dollars of TMC for three allocation rates

Given that disaggregate market behavior is modeled, it is also worthwhile to
examine stability and convergence of the market prices. We examine this under
different allocation rates $r$ (where the toll in tokens is not varied and is
set to the optimal profile under the baseline allocation rate) and convergence
of social welfare, market price, regulator revenue, and travel time of driving
are shown in Figure 6. As we can see, since tolls are not re-optimized,
different allocation rates lead to different social welfare and price values.
The results also indicate stability of the credit market, and that the
variation of equilibrium price and welfare with allocation rate is intuitive.
The effect of various initial market prices on convergence of token price and
social welfare are also examined in B.

(a) Convergence of social welfare

(b) Convergence of token price

(c) Convergence of regulator revenue

(d) Convergence of travel time of driving

Figure 6: Convergence of social welfare, token price, regulator revenue, and
travel time of driving by allocation rate $r$

###  6.4 TMC design and market behavior

In the literature, a transaction fee has been used to prevent undesirable
market behavior like frequent selling. For example, Brands et al. [2020] apply
a small transaction fee of 0.01 euro to prevent frequent selling in their
experiment. However, it has also been shown that transaction fees could reduce
system efficiency [Nie, 2012]. Our analysis in Section 4 shows that a fixed
transaction fee has the effect of preventing multiple transactions while the
proportional transaction fees has the effect of making one sell as soon as
possible when the conditional profit is positive (if buying is required at the
time of the next trip). As an alternative to transaction fees, the regulator
may also impose a minimum threshold (in dollar amounts) below which
transactions are not permitted.

Numerical experiments in this section examine the effect of proportional and
fixed transaction fees on social welfare and undesirable behavior.
Specifically, undesirable behavior is defined as buying back tokens sold
previously. We deem this undesirable because we would like to have users
strictly being either sellers or buyers (not both). Sellers are the ones who
travel in the off peak (or by transit) and sell their tokens while buyers are
the those with a high willingness to pay to travel by car during the peak
period.

For simplicity, the fixed transaction fees of buying and selling are varied
together with the proportional transaction fees set to zero and vice versa.
The effects of fixed and proportional transaction fees on social welfare and
buyback behavior are shown in Figure 7. From the simulation experiments, a
small fixed transaction fee (5 cents in this study) is seen to be able to
eliminate buyback behavior in Figure 7(c) and reduce welfare only slightly in
Figure 7(a). On the other hand, there are higher social welfare losses in the
case of the proportional transactions fees in Figure 7(b), which is also less
effective in reducing buyback behavior in Figure 7(d). This is consistent with
the findings from the literature on efficiency losses from proportional
transaction costs [Nie, 2012].

(a) Effect of Fixed TF on SW

(b) Effect of Proportional TF on SW

(c) Effect of Fixed TF on Buyback

(d) Effect of Proportional TF on Buyback

Figure 7: The effect of fixed and proportional transaction fees on social
welfare and buyback behavior

Next, we examine the effect of full initial account balances and random
initial account balances on the transaction numbers and amount by time-of-day
at equilibrium. The plots are based on simulations with a particular random
seed because stochasticity can make the visualizations hard to interpret.

(a) Number of transactions: full wallets

(b) Avg. trading amount: full wallets

(c) Number of transactions: random wallets

(d) Avg. trading amount: random wallets

Figure 8: The effects of full and random initial account balances on the
transaction numbers and amount by time-of-day at equilibrium

In Figure 8(a), the numbers of buying and selling transactions as a percentage
of the corresponding total number by time-of-day at equilibrium for full
initial account balances are plotted. As we can see, buying transactions only
happen in the peak hour because travelers can only buy tokens at time of
traveling if they are short of tokens. In contrast, selling transactions
happen at the beginning of the day, in the early morning, and peak hour, which
can be explained by the plot of the average transaction amount by time-of-day
for full initial account balances in Figure 8(b). For travelers selling at the
beginning of the day, all of them sell at full wallets as shown in Figure 8(b)
(2.052 tokens as equilibrium token price is $1) because they travel in the off
peak and do not need to use tokens; for travelers that sell in the early
morning (around 2AM), their account balances at time of selling are not full
because their future token allocations until their departure times can cover
their toll and it is optimal for them to sell now; for travelers that sell in
the peak period, they sell at full wallets because their account balances
reach the full wallet after paying small toll charges. The selling behavior is
consistent with the derived selling strategy but the excessive trading at the
beginning of the day may be undesirable and avoiding this was in fact one of
the motivations of the continuous allocation.

In practice, it is plausible that travelers will register for the program at
different times in the day (one may think of the system as being implemented
via a smartphone app). As a result, their account balances at the beginning of
the day will be different, and we now assume the initial account balances are
distributed uniformly between 0 and the maximum account balance (2.052
tokens). As shown in Figure 8(c), the selling transactions are now spread
across the day with a relatively mild peak in the early morning (around 2AM).
Apart from these travelers who sell in the early morning not at full wallets,
other travelers sell only at full wallets as shown in in Figure 8(d). Under
this assumption, we see a much more desirable pattern of transactions over the
day, and the usefulness of the continuous allocation. However, the fact that
different initial account states lead to different patterns of selling
behavior at equilibrium is a problematic property of the system (despite the
fact that the optimal welfare and associated market prices and flows are
unique), and one that deserves further investigation.

###  6.5 Performance of the TMC Scheme under varying levels of congestion,
heterogeneity and income effects

Table 6: Factor levels for experiments Factor | Level 1 | Level 2 | Level 3
---|---|---|---
Capacity ($s$) | -15% | 0% | 15%
Income Effect ($\lambda$) | 0 | 3 | 6
Heterogeneity (c.o.v) | 0.2 | 0.9 | 1.6

We next examine the performance of the TMC scheme relative to congestion
pricing at varying levels of three important experimental factors: capacity,
income effect and heterogeneity. For the TMC scheme, fixed transaction fees
are set to $0.05 and proportional transaction fees are set to 0 based on the
experiments in the previous section. The factors are varied one at a time
across three levels as presented in Table 6. Values used in the base case are
highlighted in red (when varying a given factor, other factors are fixed at
the base level). With regard to capacity, bottleneck capacity $s$ is varied
from 15% less capacity than the baseline to 15% more capacity than the
baseline; for the income effect, the nonlinear income effect coefficient in
the utility specification $\lambda$ is varied from 0 to 6; for heterogeneity,
the coefficient of variation of value of time $\alpha_{n}$ is varied from 0.2
to 1.6. In the following discussion, the TMC system is denoted by MU (U
denotes the uniform allocation of tokens) and congestion pricing is denoted by
P$-$ to indicate that we do not assume a redistribution of toll revenues in
any form. The No-toll scenario is denoted by NT. For each scenario in the
experimental design, the two instruments and NT are simulated with five
different random seeds until convergence.

(a) Social welfare

(b) Gini coefficient

(c) PT share

(d) Travel Time Index

Figure 9: Variation of performance measures with capacity

####  6.5.1 Capacity

The comparative performance of the various instruments under varying levels of
capacity in terms of social welfare (relative to the NT scenario), Gini
coefficient, PT share and travel time index (TTI) are shown in Figure 9.
First, we observe that the overall welfare of pricing and the TMC scheme are
similar at all levels of capacity, and in fact, marginally higher for the TMC
scheme despite the small fixed transaction fees (due to the income effect).
This is in line with the general finding that both pricing and tradable
credits are equivalent in terms of efficiency under deterministic
demand/supply and in the absence of transaction costs and income effects (Yang
and Wang [2011], de Palma et al. [2018]). As expected, overall welfare gains
decrease as the capacity increases and congestion effects are less severe.

(a) Ordered by benefit

(b) Ordered by income

Figure 10: Variation of cumulative user benefits with capacity

Next, Figure 9(b) shows that when toll revenues are not redistributed, the
congestion pricing scheme is regressive, as seen by the increase in the Gini
coefficient or GC (computed based on individual disposable income $I_{n}$ plus
user benefit $z_{n}$, see Equation 37) relative to the NT case. Observe also
that the GC ($P-$) increases as capacity level decreases, which implies that
$P-$ becomes less equitable because as capacity decreases, the tolls increase
to deal with the increasing congestion leading to the greater losses of low
income users.

In contrast, the TMC scheme improves the GC relative to the NT case, due to
the free uniform allocation of tokens to all travelers (a uniform allocation
increases the proportion of cumulative benefits obtained by the travelers with
lower values of time). The regressive nature of the congestion pricing scheme
can also be observed in Figure 10(a) which plots cumulative user benefits
(normalized by population size) as a function of the user benefit percentile
and Figure 10(b) which plots cumulative user benefits (normalized by
population size) as a function of income percentile. Clearly, at all capacity
levels, one can observe that a large proportion of users are worse off from
pricing (negative user benefits) whereas in the case of the TMC scheme the
proportion of ‘losers’ is significantly smaller. Although not clearly visible
in the plots, there are still some ‘losers’ with small negative benefits in
the TMC scheme. Thus, under a uniform allocation of tokens, a tradable credit
scheme does not necessarily guarantee Pareto improvement. This conclusion
accords with the finding in Arnott et al. [1994] that under pricing, even with
a uniform revenue rebate, some users may still be worse off. Fan et al. [2022]
discuss the conditions under which Pareto Improvement is guaranteed for a
tradable credit scheme using the standard bottleneck model with homogeneous
users.

It should be pointed out that that the above discussion on the regressiveness
of pricing is premised on the assumption that value of time is correlated with
income and that there is a one-to-one relationship between VOT and income.
Empirical studies have identified other correlates of income and hence,
Verhoef and Small [2004] have cautioned against viewing the VOT distribution
as simply representing the income distribution (see also Lehe [2020] on this).
Moreover, as noted in Eliasson and Mattsson [2006], ultimately, the
distributional outcomes of congestion pricing depend largely on how the toll
revenues –which can be significantly larger than net user benefits– are used.
The ratio between tolls revenues and user benefits under pricing in our
experiments are in the range of the empirical values reported in Eliasson and
Mattsson [2006].

(a) Social welfare

(b) Gini coefficient

(c) PT share

(d) Travel Time Index

Figure 11: Variation of performance measures with income effect level

In terms of network performance, significant improvements in the TTI and
larger transit shares are observed under both instruments (relative to NT) at
all capacity levels (Figures 9(c) and 9(d) ).

####  6.5.2 Income Effect

Next we examine the impact of income effects, measured by the parameter
$\lambda$. As noted previously, non-linear income effects do impact behavior
and the relative efficiency of pricing and TMCs. This impact on behavior along
with the fact that the valuation of the token allocation is higher for lower
income users (who have a higher marginal utility of income - recall that our
measure of user benefit is directly the money metric utility) results in
increasing welfare differences as $\lambda$ increases (this can be seen in
Figure 11(a)). However, we caution that in terms of magnitudes, these
differences are still small. As expected, when income effects are absent, the
welfare of the TMC is marginally lower than P- due to transaction fees.
Observe also that the social welfare of $P-$ decreases as the income effect
increases since users are more sensitive to the tolls.

The impact on behavior can also be seen in the lower PT shares with increasing
$\lambda$ in Figure 11(c). In terms of distributional impacts, we observe
similar trends in the GC and patterns in user benefits as described in Section
6.5.1 (which show the regressiveness and larger proportion of ‘losers’ under
pricing; see the base capacity case in Figures 10(a) and 10(b)). There are
small variations across $\lambda$ and hence, for brevity, we omit plots of
user benefits.

(a) Social welfare

(b) Gini coefficients

(c) PT share

(d) TTI

Figure 12: Variation of performance measures with heterogeneity levels

(a) Lorenz curve of user benefits

(b) Distribution of user benefits by income

Figure 13: Distributional impacts of various instruments by heterogeneity
levels

####  6.5.3 Heterogeneity

As can be seen in Figure 12(a), as the extent of heterogeneity increases, the
social welfare of $P-$ increases, which is consistent with findings in the
literature (e.g. Van Den Berg and Verhoef [2011]). Neglecting heterogeneity
underestimates the benefits of pricing [Verhoef and Small, 2004]). The
relative performance of the two instruments does not change appreciably with
the extent of heterogeneity, with the TMC scheme having marginally higher
welfare and significantly better GC at all levels of heterogeneity. In terms
of distributional impacts, at low levels of heterogeneity (COV of 0.2), it can
be seen in Figure 13(a) that all users are worse off under P$-$ whereas most
of them are better off under the credit system.

###  6.6 Robustness

In practice, toll profiles may often be sub-optimal because of changing
conditions, forecast errors and uncertainty. Practically, it is difficult to
update these toll profiles (especially at the network level) regularly in
practice. For example, Singapore updates the ERP scheme once every three
months. In contrast, some market elements of the TMC scheme (e.g. allocation
rate) are easier to adapt and have the potential to influence travelers’
behavior (through the market price) to recover efficiency losses. In this
section, two scenarios of a sub-optimal toll profile are investigated,
including forecast error and non-recurrent events.

The first scenario is forecast error wherein actual road capacity is assumed
to be 15% less than the anticipated road capacity used to optimize the toll
profile. The social welfare of pricing and TMC with this sub-optimal toll
profile (based on anticipated road capacity) are plotted in Figure 14 and
denoted by $P-_{S}$ and $MU_{S}$. The social welfare of pricing and TMC with
the associated optimal toll profiles are also plotted and denoted as $P-_{O}$
and $MU_{O}$. As we can see, $MU_{S}$ (TMC with sub-optimal tolls) is able to
recover efficiency losses through a reduction in the allocation rate, which
reduces token supply and increases token price. The optimal allocation rate is
determined using a grid search and is found to be 15% lower than the original
allocation rate.

Figure 14: The social welfare of pricing and TMC with sub-optimal and optimal
toll profiles

The second scenario is a non-recurrent event. Specifically, it is assumed that
there is a sudden within-day capacity drop by 15% (e.g., due to an accident or
incident) from 7AM to 8:30AM on the 10th day after the system has reached an
equilibrium. The social welfare across days for the three instruments are
plotted in Figure 15. The first instrument is pricing without distribution and
denoted as $P-$. The second one is TMC with a lump-sum allocation and denoted
by $MU_{L}$. The third instrument is TMC with a continuous allocation and
denoted as $MU_{C}$.

Under $MU_{L}$, travelers receive the entire day’s token allocation at the
beginning of the day in the form of a ‘lump-sum’ allocation. This form of
token allocation is the standard design of TMC schemes in the literature (e.g.
Yang and Wang [2011], Brands et al. [2020]). Regarding trading, they can buy
additional tokens at the time of traveling for immediate use and redeem unused
tokens at the end of the day. Since trading is automated, there is no
transaction fee considered under the lump-sum allocation. The regulator has
three market parameters to control including within-day token price,
regulation starting time and ending time. It cannot control allocation rate as
all the tokens have already been allocated at the beginning of the day. Users
who have not traveled yet before the new token price takes effect can update
their plans according to the new information. We assume the same behavioral
model used in the pre-day decision applies (with updated token prices and
transaction fees). Using the DE algorithm, we determine that it is optimal for
the regulator to increase the token price to $\$1.8$ between 6:55AM and
9:15AM. As shown in Figure 15, it performs better than $P-$.

For $MU_{C}$, the regulator can control not only token price, regulation
starting and ending time, but also the allocation rate (which affects the
forecasted account balance of users over the rest of the day) and transaction
fees. We optimize the fixed transaction fees of buying and selling together.
Through optimization, between 7:05AM and 8:50AM, the regulator should set
token price equal to $\$1.25$, allocation rate $r$ equal to 0 and fixed
transaction fee equal to $\$0.5$. It performs better than the lump-sum
allocation $MU_{L}$ as shown in Figure 15. This is intuitive because the
travel behavior is impacted by both the allocation rate and transaction fees,
which provide the regulator more degrees of freedom to intervene. This
demonstrates the advantages of a continuous allocation of tokens over a lump-
sum allocation of tokens.

Figure 15: Social welfare of pricing and TMC with sub-optimal and optimal toll
profiles

##  7 Conclusions

This paper proposes and analyzes alternative market models for a tradable
mobility credit system focusing on allocation/expiration of tokens, rules
governing trading, transaction fees, and regulator intervention. We develop a
methodology to explicitly model the dis-aggregate behavior of individuals
within the market. Extensive simulation experiments are conducted within a
combined mode and departure time context for the morning commute problem using
a day-to-day assignment framework wherein transportation demand is modeled
using a logit-mixture model with income effects and supply is modeled using a
standard bottleneck model.

The results show that small fixed transaction fees can effectively mitigate
undesirable behavior in the market without a significant loss in efficiency
whereas proportional transaction fees are less effective both in terms of
efficiency and in avoiding undesirable market behavior. The market design we
adopt is shown to yield stable and convergent market prices and is revenue
neutral. We also show that an allocation of tokens in continuous time can be
beneficial in dealing with non-recurrent events. One undesirable property of
the system we do observe is that the selling behavior of individuals at
equilibrium is dependent on the initial account states (although optimal
welfare, flows and market prices are unique), and this deserves further
investigation. With regard to the comparative performance relative to
congestion pricing, in the presence of income effects, the TMC system yields a
marginally higher social welfare. Finally, the TMC scheme is more equitable
(when revenues from congestion pricing are not redistributed) although it is
not guaranteed to be Pareto-improving when tokens are distributed equally.

There are several promising directions for future research. First, given that
a uniform allocation of tokens does not guarantee Pareto-improvement, it is
clear that personalization is necessary to ensure there are no ‘losers’. The
design of personalized token allocation schemes is an important direction of
future research. Second, more systematic experiments on day-to-day variability
and its impact on the robustness of the TMC scheme and market are warranted.
Finally, large scale simulations on real-world networks with disaggregate
agent based models is a natural next step in studying market design and other
aspects of the tradable credit schemes towards real-world deployments.

## Acknowledgements

This research was funded by the U.S. National Science Foundation (Award
CMMI-1917891) and the NEMESYS project funded by the DTU (Technical University
of Denmark)-NTU (Nanyang Technical University) Alliance.

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##  Appendix A Existence and Uniqueness of Equilibrium

In this section, we examine convergence of the day-to-day model (Section 5).
Using data and parameters discussed in the Section 6.1, a No Toll (NT)
scenario with different sets of initial travel time information for the
driving alternatives is simulated. Five independent replications or draws of
the error terms in the utility (Equation 25) are performed. The four different
sets of initial travel time information of driving are plotted in Figure 16.
Note that the initial travel time information serves as the basis for the
departure time decisions in day 0 of the day-to-day simulation. The initial
travel time set 0 represents free flow travel times across the entire day; the
initial travel time set 1 consists of equilibrium travel times of driving
obtained from the simulation using the initial travel time set 0 for a
particular random seed; the initial travel time set 2 is 0.6 times the travel
time set 1; finally, the initial travel time set 3 represents a constant 30
minutes travel time across the entire day.

(a) Initial travel times

(b) Convergence of travel time of driving

Figure 16: Initial travel time information of driving and corresponding
convergence

The corresponding convergence of travel times of driving is plotted in Figure
16. The lines and bands in the plot represent averages and standard
deviations, respectively, across the five replications. As we can see from the
plot, the infinity norm (Equation 42) converges to within a threshold of 0.01
by about 50 days, indicating acceptable convergence of the day-to-day model.
In order to examine uniqueness, the travel time vectors and flow vectors of
driving at equilibrium are plotted in Figure 17. The results indicate that
simulations with different initial travel times of driving converge to the
same travel time and departure flow rate patterns at equilibrium. As only two
mode choices are considered and the travel time and headway of PT are
constant, once the travel time of driving converges, the resulting departure
flow of PT also converges.

(a) Travel time of driving at equilibrium

(b) Flow rate of driving at equilibrium

Figure 17: Travel time and flow of driving at equilibrium with different sets
of initial travel time information

(a) Social Welfare

(b) Token market price

(c) Regulator Revenue

(d) Travel time of driving

Figure 18: Convergence of various metrics with different initial prices
$p^{0}$

##  Appendix B Convergence of Market Prices

The effect of various initial market prices $p^{0}$ on convergence of token
market price and social welfare are examined in Figure 18. The price and
social welfare converge to values that are not statistically significantly
different at a significance level of 0.05 regardless of the initial price. The
regulator revenues under the three initial prices converge to be within the
regulator revenue threshold band (the black lines) as shown in Figure 18(c).
Travel times under the three initial prices converge too as shown in Figure
18(d). Similar experiments under a range of scenarios showed that the price
adjustment scheme converges (in terms of travel times, flows and token prices)
to the same equilibrium for different initial market prices.