# Competition and Cooperation of Autonomous Ridepooling Services: Game-Based
Simulation of a Broker Concept ††thanks: Submitted to: Frontiers in Future
Transportation

Roman Engelhardt, Patrick Malcolm, Florian Dandl, and Klaus Bogenberger
Chair of Traffic Engineering and Control
Technical University of Munich
Arcisstraße 21
80333 Munich, Germany
Corresponding Author<EMAIL_ADDRESS>

###### Abstract

With advances in digitization and automation, autonomous mobility on demand
services have the potential to disrupt the future mobility system landscape.
Ridepooling services in particular can decrease land consumption and increase
transportation efficiency by increasing the average vehicle occupancy.
Nevertheless, because ridepooling services require a sufficient user base for
pooling to take effect, their performance can suffer if multiple operators
offer such a service and must split the demand. This study presents a
simulation framework for evaluating the impact of competition and cooperation
among multiple ridepooling providers. Two different kinds of interaction via a
broker platform are compared with the base cases of a single monopolistic
operator and two independent operators with divided demand. In the first, the
broker presents trip offers from all operators to customers (similar to a
mobility-as-a-service platform), who can then freely choose an operator. In
the second, a regulated broker platform can manipulate operator offers with
the goal of shifting the customer-operator assignment from a user equilibrium
towards a system optimum. To model adoptions of the service design depending
on the different interaction scenario, a game setting is introduced. Within
alternating turns between operators, operators can adapt parameters of their
service (fleet size and objective function) to maximize profit. Results for a
case study based on Manhattan taxi data, show that operators generate the
highest profit in the broker setting while operating the largest fleet.
Additionally, pooling efficiency can nearly be maintained compared to a single
operator. With the resulting increased service rate, the regulated competition
benefits not only operators (profit) and cities (increased pooling
efficiency), but also customers. Contrarily, when users can decide freely, the
lowest pooling efficiency and operator profit is observed.

_K_ eywords Ridepooling $\cdot$ Mobility-On-Demand $\cdot$ Competition $\cdot$
Cooperation $\cdot$ Agent-Based Simulation

##  1 Introduction

With the increased availability of mobile internet, mobility-on-demand (MOD)
services have become increasingly popular over the last decade. In times of
urbanization, they can represent an alternative to private vehicles that
offers a similar convenience. MOD services result in a higher temporal
utilization of vehicles which, if replacing private vehicle trips, can
potentially free up urban space that would otherwise be used for parking.
Furthermore, ridepooling services have the potential to increase the average
vehicle occupancy during trips, thereby resulting in more spatially efficient
utilization of the road. The probability of finding and pooling similar trips
increases with demand density. Therefore, the pooling potential increases with
the scale of supply and demand. As a consequence, fragmentation of the
ridepooling market into multiple independent competitors can be expected to
decrease the efficiency of each competitor. Mobility-as-a-service (MaaS)
platforms represent a possibility to break the stark independence of
competitors, as offers from multiple mobility service providers are collected
in one place for travelers.

Compared to current MOD services, automation can change the cost structure
significantly Boesch et al. (2018). When these cost reductions from the fleet
operation with autonomous vehicles are translated into cheaper fares for
users, disruptions of the transportation systems as we know them are possible.
Hence, city authorities are confronted with the questions of whether and how
autonomous mobility-on-demand (AMOD) systems and competition between multiple
providers should be regulated.

This paper studies how interaction between AMOD operators can counteract the
effects of competition and fragmentation. To this end, the concept of an AMOD
broker is introduced which is a (possibly regulated) platform — similar to a
MaaS platform — for multiple AMOD operators. As illustrated in Figure 1), the
broker collects trip offers from multiple AMOD providers and forwards them to
the customers. In addition, the broker can be regulated to adapt the offers to
align the platform with city goals. The adaptation can range from sorting the
offers in a certain order or manipulating prices, to the suppression of
certain offers which are in conflict with city goals. The effect of the
regulating measures can be compared with moving the dynamic traffic assignment
from a user equilibrium towards the system optimum.

The goal of this study is to compare different types of AMOD provider
interaction and their impact on the providers. More specifically, we
investigate the following scenarios with the help of simulations: a
monopolistic AMOD service, independent AMOD providers, and the two forms of
broker systems (unregulated and fully regulated).

Figure 1: Illustration of an AMOD Broker System

###  1.1 Literature Review

Several studies deal with the operation of AMOD fleets and its impacts. Even
without pooling, a single AV can replace a significant number of private
vehicles Fagnant et al. (2015) or carsharing vehicles Dandl and Bogenberger
(2019). Optimization of request and repositioning assignments based on demand
estimations can further improve fleet performance Hyland and Mahmassani
(2018); Dandl et al. (2019c); Hoerl et al. (2019a). However, without pooling,
these vehicle reductions only affect stationary traffic, i.e. parking space.
To observe improvements to traffic flow, ridepooling is required Engelhardt et
al. (2019a); Ruch et al. (2020). The optimization of ridepooling assignments
is a challenging problem, which can be addressed with graph-based approaches
Santi et al. (2014); Alonso-Mora et al. (2017) and heuristics based on them
Simonetto et al. (2019); Hyland and Mahmassani (2020). For ridepooling
services, positive scaling properties, i.e. a higher efficiency for higher
levels of demand, are observed in both analytical and simulation models Tachet
et al. (2017); Bilali et al. (2020). Two effects play into this scaling
behavior for ridepooling systems: (i) a higher density of vehicles means that
the approach becomes shorter (similarly to in the ridehailing case), and (ii)
the probability of finding trips that can be matched with only minor detours
increases with demand.

These operational studies assume a fixed exogenous demand, and the problem is
to serve this demand as efficiently as possible. To study the impact of AMOD
on transportation systems, the integration of AMOD into demand models is
necessary. Open-source software packages like SimMobility Nahmias-Biran et al.
(2020); Oke et al. (2020), MATSim Hoerl et al. (2019b); Kaddoura et al.
(2020), Polaris Gurumurthy et al. (2020), and mobiTopp Wilkes et al. (2021),
as well as commercial software solutions already have capabilities to model
AMOD supply and demand interactions. Most of these demand models utilize a
pre-day assignment of AMOD demand, be it by iterative learning or a mode
choice model. Wilkes et al. (2021) developed a within-day mode choice model,
which is based on real-time information of the fleet and thereby relevant for
modeling MaaS platforms.

Most of the previously mentioned references study a single AMOD operator; a
generalization to multiple independent operators has been implemented in Dandl
et al. (2019a), and an operator with multiple service offers (hailing and
pooling with different vehicle sizes) was investigated in Atasoy et al.
(2015); Liu et al. (2019). For ridehailing, the impact of multiple AMOD
operators in the same market is analyzed with a theoretical model by Séjournè
et al. (2018) and data-driven models with simulation by Kondor et al. (2022).
Séjournè et al. (2018) show that demand patterns are crucial when it comes to
the division of the market and find two phases. The first is denoted
“fragmentation resilient” and describes a system where the price of sharing
the market decreases with the size of the market; in the other phase, denoted
“fragmentation affected”, a division of the market generates much higher costs
regardless of the size of the market because empty vehicle repositioning is
required to balance supply and demand. Kondor et al. (2022) derived a
mathematical equation for the cost of non-coordinated market fragmentation and
run simulations to find the coefficients for various cities. For ridepooling,
Pandey et al. (2019) analyzed three models of multi-company market models —
competitive, cooperative, and centralized — and derived approaches to address
the resulting problems with linear assignment problems.

There are several studies examining the even more complex market dynamics for
multiple ridehailing services with drivers. For instance, Jiang and Zhang
(2018) study the effect of “double-apping” in a (human driven) ridehailing
market with two competitors, where both drivers and customers have the
possibility to use both ridehailing apps. They observed that users and drivers
can benefit, but without any contract or guarantee that the other ridehailing
provider will do the same, a ridehailing operator does not benefit from
drivers serving customers of both providers. Qian and Ukkusuri (2017) study
the competition between a traditional taxi and a ridehailing provider in a
game, where passengers are the leaders and the two mobility providers are the
followers. They find that fleet size and pricing policy significantly impact
the outcome. In another study, Xi et al. (2021) propose a “name-your-own-price
auction” in a MaaS platform, where travelers and a wide range of mobility
service providers (with ridehailing being one of them) can submit a bid. They
also use a leader-follower formulation, with the MaaS platform being the
leader and the mobility service providers and the travelers being the
followers.

AMOD providers do not just compete against each other. They can compete with,
but also complement public transport, depending on their service designs. In
most demand models, AMOD systems are treated as a separate competing mode.
Positive and negative effects mainly depend on the number of users that are
attracted from private vehicle or public transport modes. Additionally, AMOD
systems can be utilized as feeder systems to increase intermodality and
improve public transport Liang et al. (2016); Wen et al. (2018). To avoid
competition, AMOD routes can also be restricted/designed to complement the
existing public transport system Dandl et al. (2019b), or AMOD and public
transport can be designed jointly Pinto et al. (2019). With a growing market
share of today’s MOD services, negative externalities of user-centric
ridehailing can be observed Henao and Marshall (2019); Schaller (2021).
Therefore, the regulation of MOD Li et al. (2019); Zhang and Nie (2019) – with
part of the focus on the regulatory protection of drivers – and of AMOD
services Simoni et al. (2019); Dandl et al. (2021); Mo et al. (2021) is
becoming increasingly relevant. Simoni et al. (2019) study various congestion
pricing models in the presence of an AMOD system. Dandl et al. (2021) consider
an AMOD ridepooling service which is regulated such that it cannot offer
guaranteed single-passenger rides. Moreover, they introduce a regulatory tri-
level framework optimizing a congestion-based road toll, parking fees, public
transport frequency, and an AMOD fleet limit, where the reaction of an AMOD
provider to changed regulatory settings is taken into account. Mo et al.
(2021) investigate how regulatory measures like fleet size limitations and
public transport subsidies can steer the competition between AMOD and line-
based public transport. The equilibrium state is found with an iterative
approach, in which the AMOD operator is updated every iteration – representing
a day – and the public transport service every month. These time scales should
reflect the frequencies with which AMOD and public transport operators are
likely to modify their service.

A collaboration of mobility services can help to create a better combined
service offer, which could reduce private vehicle ownership and be beneficial
to the service providers. MaaS platforms are one form of such collaboration.
Typically, they at least collect information of multiple providers, offer the
possibility to book mobility services, and provide a common method for payment
Smith and Hensher (2020). The design and possible regulation of a MaaS
platform, e.g. by pricing and bundling of services, can affect user decisions
Feneri et al. (2020) and ultimately help in reaching sustainability objectives
Muller et al. (2021).

###  1.2 Contribution

This paper contributes several new aspects to the literature. While most
previous studies focused on the ridehailing market, this paper evaluates the
losses resulting from fragmented ridepooling demand. Moreover, the effects of
different interactions between multiple operators and a central platform are
compared. The potential benefits of a broker which selects between the offers
of different providers, thereby representing the most extreme form of
regulation on this platform, is examined and compared to a platform where
customers select the offers by themselves. To the authors knowledge, this is
the first study that additionally evaluates the adoption of the service design
to optimize the operators profit for a given interaction scenario within a
game setting. The case study shows the significant impact that fleet size and
the operator objectives have on the level of service and overall
transportation system.

##  2 Methodology

This section describes the agent-based simulation environment, which is used
to study different operator interactions. First, the simulation’s agents and
process flow are introduced, and the representation of different AMOD
operators is explained. Then, the operator module with the task to assign
customers to vehicles is described in detail. Lastly, an iterative simulation
to model possible service adaptations to the studied operator interactions
(independent, unregulated and regulated broker) is presented.

###  2.1 Agent-Based Simulation Flow

The simulation environment consists of three or four main agents: (1)
customers, who request trips from AMOD operator(s) and choose their travel
mode; (2) operators, who provide the mobility service by operating a fleet of
vehicles with the tasks to create mobility offers for customer requests and
fulfill these offers in case customers accept them; (3) vehicles controlled
and dispatched by an operator which specifies which, where, and in which
sequence customers have to be picked up and dropped off; and (4) a broker,
which makes decisions to regulate the platform in the broker scenarios.

Customer and vehicle agents move on a network $G=(N,E)$ with nodes $N$ and
edges $E$ connecting these nodes. A customer request is defined by the tuple
$(i,t_{i},x_{i}^{s},x_{i}^{d})$ with a request id $i$, a time of request
$t_{i}$, the travel origin node of the request $x_{i}^{s}\in N$ and the travel
destination node of the request $x_{i}^{d}\in N$. Operators receive these
travel requests and, based on their current fleet state, try to find best
possible solutions to serve them and formulate offers for the service as a
reply. Offers from operators are defined as tuples of parameters defining the
course of the customer’s trip in case the offer is booked. In this study,
parameters defining the offers can be categorized into user parameters
$u_{i,o}$ and system parameters $s_{i,o}$, which influence the decision
process of users and the broker, respectively. We define user parameters as
parameters that users of the service are sensitive to when they have to decide
for or against the service. These parameters can include fare, expected
waiting time, and expected travel time for example. The broker on the other
hand is sensitive to the system parameters. These parameters describe measures
for the possible impact on the traffic system. In this study, the additional
distance, which the AMOD fleet has to drive in order to serve a customer, is
used. In case the operator is not able to serve a customer (i.e. no feasible
solution is found to serve a customer within time constraints for pick-up and
drop-off), no offer is made.

In so-called interaction scenarios, this study distinguishes several decision
processes defining which specific operator is booked by a customer. The four
different interaction scenarios implemented in this study are as follows:

#### 1) Single Operator:

In this scenario, only a single monopolistic AMOD operator is offering a
ridepooling service, and therefore no interaction between operators is
implemented. Customers requesting a trip from this operator always book a trip
if they receive an offer, and if not, they leave the system unserved.

#### 2) Independent Operators:

In this scenario, multiple AMOD operators are offering a ridepooling service,
but no direct interaction between them is assumed. Customers only request a
trip from one of these operators, and they always book a trip if they receive
an offer from this operator. If they don’t receive an offer, they leave the
system unserved. From the simulation point of view, this scenario is
equivalent to the Single Operator scenario, but with the demand for AMOD being
split between the operators.

#### 3) User Decision:

In this scenario, multiple AMOD operators are offering the mobility service
over a central platform, here referred to as a “broker”. Instead of
interacting directly with one of the operators, customers request a trip from
the broker, which forwards the request to each of the operators. The operators
then each send an offer to the broker, which presents these options to the
customer. The customer then chooses the offer with the highest user utility
$\phi^{user}(u_{i,o})$. If the broker does not receive an offer from either of
the operators, the customer leaves the system unserved. A flowchart of this
scenario is shown in Figure 2.

#### 4) Broker Decision:

In this scenario, multiple AMOD operators are also offering the mobility
service via a central broker. Customers send their requests to the broker,
which then forwards them to each of the operators, who send their offers back
to the broker. In contrast to the User Decision scenario, however, rather than
allowing the customer to choose their preferred offer, the broker chooses the
offer which it deems best for the transportation system by evaluating the
highest system utility $\phi^{broker}(s_{i,o})$. Therefore, a broker decision
aims towards a system-optimal state, whereas the user decision reflects a
quasi-user optimal scenario. A flowchart of this scenario is shown in Figure
2.

Figure 2: Flowchart of the User Decision and Broker Decision scenarios. The
only difference between the scenarios is the criteria used to choose an offer,
highlighted in yellow.

###  2.2 Fleet Operator Model

The main tasks of each operator are (i) to create offers for customers (or a
broker) which serves as their basis to decide for or against the service, (ii)
to assign and schedule its vehicles to customers who have booked their service
, and (iii) distribute idle vehicles according to expected demand by assigning
repositioning tasks.

The assignment of customers to vehicles and their corresponding schedules is
modeled as a solution of a dynamic vehicle routing problem. With the set of
vehicles $V_{o}$ of operator $o$, we define a schedule
$\psi_{k}(v,R_{\gamma})$ as the $k$-th feasible permutation of stops for
vehicle $v\in V_{o}$ serving the subset of requests $R_{\gamma}\subset R$ of
all currently active requests $R$. Hereby, stops refer to origin and
destination locations of requests in $R_{\gamma}$ where boarding and alighting
processes of the corresponding customers are performed. In this study, a
schedule is called feasible if

  1. 1.

for each customer, the alighting stop succeeds the boarding stop,

  2. 2.

at no point in time the number of on-board passengers exceeds the vehicle
capacity $c_{v}$,

  3. 3.

each customer $i\in R_{\gamma}$ has been or is scheduled to be picked up
before a maximum waiting time $t_{max}^{wait}$ has elapsed after the request
time $t_{i}$, and

  4. 4.

the in-vehicle travel time of each customer $i\in R_{\gamma}$ is not increased
by more than $\Delta$ compared to the direct travel time between $x_{i}^{s}$
and $x_{i}^{d}$.

To compare different schedules, each schedule $\psi_{k}(v,R_{\gamma})$ is
rated by an objective function $\rho_{\alpha}$ which we define in this study
by

$\rho_{\alpha}\left(\psi_{k}(v,R_{\gamma})\right)=c_{\alpha}^{dis}\cdot
d\left(\psi_{k}(v,R_{\gamma})\right)+c_{\alpha}^{vot}\cdot\left(\sum_{i\in
R_{\gamma}}t_{i}^{arrival}\left(\psi_{k}(v,R_{\gamma})\right)-t_{i}\right)-N_{R}\cdot|R_{\gamma}|.$
(1)

$d\left(\psi_{k}(v,R_{\gamma})\right)$ is the distance vehicle $v$ has to
drive when executing the schedule $\psi_{k}(v,R_{\gamma})$, and
$t_{i}^{arrival}\left(\psi_{k}(v,R_{\gamma})\right)$ is the expected arrival
time of customer $i$ according to this schedule. $N_{R}$ is a large assignment
reward to prioritize serving as many customers as possible. $c_{\alpha}^{dis}$
and $c_{\alpha}^{vot}$ are cost factors reflecting the distance cost and the
value of time for customers, respectively. The goal of the optimization is to
assign schedules to vehicles that minimize the sum of the objective functions
of all assigned schedules. Depending on the operational strategy $\alpha$, the
weights $c_{\alpha}^{dis}$ and $c_{\alpha}^{vot}$ can be set to prioritize low
passenger service times or low fleet mileage. If $c_{\alpha}^{dis}\gg
c_{\alpha}^{vot}$ this objective would favor schedules that require only small
distances to drive for vehicles. On the other hand, if $c_{\alpha}^{dis}\ll
c_{\alpha}^{vot}$, schedules are favored that serve customers as fast as
possible.

If all feasible schedules $\psi_{k}(v,R_{\gamma}))$ for all vehicles $v$ and
all request bundles $R_{\gamma}$ can be found, an optimization problem can be
solved to assign the currently best schedules to the vehicles. By defining a
V2RB (vehicle-to-request-bundle) $\Psi(v,R_{\gamma})$ as the set of all
feasible permutations $k$ of schedules of vehicle $v$ serving $R_{\gamma}$
with

$\rho_{v,\gamma}^{\alpha}=\rho_{\alpha}\left(\Psi(v,R_{\gamma})\right)=\min_{k}\rho_{\alpha}\left(\psi_{k}(v,R_{\gamma})\right),$
(2)

being the objective function value of this V2RB, an integer linear problem
(ILP) can be formulated:

minimize
$\displaystyle\qquad\sum_{v}\sum_{\gamma}\rho_{v,\gamma}^{\alpha}\cdot
z_{v,\gamma}$ (3) s.t. $\displaystyle\qquad\sum_{\gamma}z_{v,\gamma}\leq
1\qquad\forall v\in V_{o}$ (4)
$\displaystyle\qquad\sum_{v}\sum_{\gamma\in\Omega_{i}}z_{v,\gamma}=1\qquad\forall
i\in R_{a}$ (5)
$\displaystyle\qquad\sum_{v}\sum_{\gamma\in\Omega_{i}}z_{v,\gamma}\leq
1\qquad\forall i\in R_{u}\qquad.$ (6)

Equation (3) tries to select schedules with cost $\rho_{v,\gamma}^{\alpha}$ of
vehicle $v$ to serve the bundle of requests $\gamma$ that minimizes the total
cost. Thereby, $z_{v,\gamma}\in\\{0,1\\}$ is the decision variable taking the
value $1$ if schedule with cost $\rho_{v,\gamma}^{\alpha}$ is assigned and $0$
otherwise. Equation (4) ensures that only one schedule can be assigned to each
vehicle. Equation (5) ensures that each customer $i$ from the set of already
assigned customers $R_{a}$ has to be assigned to exactly one vehicle again.
Here, $\Omega_{i}$ corresponds to all request bundles that include customer
$i$. In the same way, Equation (6) ensures that each customer that has not
been assigned yet (set $R_{u}$) can be assigned to at most one vehicle.

Within the simulation, customers can request trips in every simulation time
step of 60 seconds. Depending on the scenario, the customers or the broker
decide for or against the service depending on the respective offers sent by
the operators. In this study, an immediate decision process is assumed, i.e.
the operators are informed that a customer is either booking a trip or
declines an offer before the next customer can request a trip. The operator
creates offers based on a solution (assigned vehicle schedule including the
request) of the optimization problem of Equation (3). If no solution can be
found, the request is declined by the operator. Since solving the optimization
problem for each customer would be computationally intractable, a multi-step
approach is applied. In a first step (offer phase), each time a customer
requests a trip, a heuristic is applied to find an initial solution for
Equation (3). This initial solution is used to create an offer. If the
customer books the service, the solution (schedule) is assigned to the
vehicle, otherwise the solution is discarded. In a second step, after all
customer requests in one time step have been processed, a global re-
optimization is performed by solving optimization problem (3) for all
currently scheduled or on-board requests. These two steps are described in
more detail in the following.

In the offer phase, an insertion heuristic is applied to find the initial
solution from which the offer is created. In this heuristic, new feasible
vehicle schedules are constructed by inserting customers into the currently
assigned vehicle schedules. Because a schedule can only be feasible if this
new customer can be picked up within $t_{max}^{wait}$, an insertion need only
be tested for vehicles that can reach the customer’s origin within this time
interval. Let $\psi_{k}(v,R_{\gamma})$ be a feasible insertion of customer $i$
into the current solution of vehicle $v$ and $\psi_{l}(v,R_{\gamma\setminus
i})$ be the current solution of vehicle $v$. The offer is based on the
solution of the local optimization problem

$\min_{v,k}\rho_{\alpha}\left(\psi_{k}(v,R_{\gamma})\right)-\rho_{\alpha}\left(\psi_{l}(v,R_{\gamma\setminus
i})\right)\qquad\forall\text{ feasible }v,k\text{ .}$ (7)

For re-optimizing the vehicle schedules once all new customers within the
current simulation step have been processed, an algorithm based on that of
(Alonso-Mora et al., 2017) is applied in this study. A high level description
of the implementation is presented here, while the reader is referred to
Engelhardt et al. (2019b) for details. The idea of the algorithm is to find
all feasible schedules first and solve the ILP (Equations (3) - (6)) based on
these schedules afterwards. Since an exhaustive search is intractable for the
ridepooling assignment problem, a guided search is applied. This guided search
can be divided into the following three steps: In a first step, all feasible
vehicle-customer combinations are searched. These combinations are defined as
feasible for all vehicles theoretically able to reach the origin of the
customer request within $t_{max}^{wait}$. In a second step, all feasible
customer-customer combinations are searched. A customer-customer combination
is defined as feasible if a feasible schedule for a hypothetical vehicle can
be found which serves both customers (shared or one-after-the-other). In a
third step, the first two algorithm steps are exploited to create all feasible
V2RBs (schedules) sorted by their grade, which we define as the number of
customers that are served by the corresponding schedules. A V2RB of grade one
for vehicle $v$ serving customer $i$ can only exist if the corresponding
vehicle-customer combination from the first step is feasible. A V2RB of grade
two can only exist if both vehicle-customer combinations between vehicle and
customers are feasible and additionally the customer-customer combination is
feasible. And finally, a V2RB of grade $n$ can only exist, if all V2RBs of
grade $n-1$ exist, where one of the $n$ customers is removed. That is, for a
V2RB $\Psi(v,R_{\gamma}=\\{1,2,3\\})$ to exist, the feasibility of V2RBs
$\Psi(v,R_{\gamma\setminus\\{3\\}}=\\{1,2\\})$,
$\Psi(v,R_{\gamma\setminus\\{1\\}}=\\{2,3\\})$ and
$\Psi(v,R_{\gamma\setminus\\{2\\}}=\\{1,3\\})$ is necessary. All feasible
schedules can now be created iteratively by increasing the grades of the
V2RBs.

To adjust the spatial distribution of vehicles for upcoming demand, a
repositioning strategy is applied periodically. Every $T^{repo}$, a parameter-
free rebalancing strategy based on (Pavone et al., 2012) is applied. After
estimating available vehicles and expected demand for each taxi zone, a
minimum transportation problem, which aims to minimize the travel costs to
reach some zone supply and demand balance constraints, is solved.

Figure 3: Illustration of the game. Operators take turns playing the “active”
role. In each turn, the active operator tests all of its possible service
parameters $O_{m,n}$ against the passive operator’s fixed parameters. The
parameters that bring the highest effective profit for the active operator are
then used in the next turn, where the roles are reversed.

###  2.3 Game

The different interaction scenarios introduced in the beginning of this
section describe different external environments in which the operators offer
their service. Depending on these environments operators will adapt their
service design to maximize profit. In this study, the adaptation of the
service of the operators is modeled as a turn-based game (illustrated in
Figure 3). Each operator starts with specific service parameters based on the
scenario without interaction. In each turn, one operator has the active role,
while the other one has the passive role. These roles are exchanged every
turn. The active operator explores different sets of service parameters (with
exhaustive search), while the service parameters of the passive operator
remain constant. At the end of each turn, the active operator adopts the
service parameters that resulted in the highest profit.

The profit $P$ is calculated by the difference of revenue $R$ and costs $C$
after each simulation.

$\displaystyle P$ $\displaystyle=R-C$ (8) $\displaystyle R$
$\displaystyle=\sum_{i\in C_{served}}d^{direct}_{i}\cdot f$ (9) $\displaystyle
C$ $\displaystyle=N_{v}\cdot C_{v}+d^{fleet}\cdot c_{dis},$ (10)

with $C_{served}$ being the set of all served customers, $d^{direct}_{i}$
their corresponding direct travel distances, and $f$ a distance-dependent fare
the customers have paid. $C_{v}$ is the fixed cost per vehicle, $d^{fleet}$ is
the driven distance of the vehicle fleet and $c_{dis}$ is the distance-
dependent vehicle cost.

To optimize this profit $P$ for only a few days of simulation, operators would
choose small fleets to increase overall vehicle utilization (including off-
peak times). However, the service rate would suffer strongly leading to an
unreasonable large number of customers that have to be rejected. Since such an
unreliable service is improbable to survive on the long run, another term
penalizing the number of requests, which did not receive an offer $N_{C,no}$
(within the given service quality constraints) during the simulation, should
be considered. Hence, we define the effective profit $P_{eff}$ to be maximized
within the game:

$P_{eff}=P-N_{C,no}\cdot p_{no}.$ (11)

Thereby, $p_{no}$ is a penalty cost for each request without offer. $p_{no}$
and $f$ will be determined within calibration simulations in the results
section.

Alternating turns with operators maximizing their effective profit are
repeated until equilibrium is reached. All operators adopting the same
parameter set is one of the equilibrium states the game can converge in. In
this case no operator has an advantage in changing their parameter sets
anymore. Generally, it is not guaranteed to find such equilibrium states.
Depending on the overall setting, it is for example feasible that operators
with different market penetrations and therefore different fleet sizes also
reflect a stable system. Nevertheless, it turns out that as long as certain
symmetries between the operators are assumed as described in the following
case study, these symmetric equilibrium states can be found in all scenarios
tested.

##  3 Case Study

We test the model on a case study for the publicly available taxi data set of
Manhattan, NYC. The simulation inputs are described in the following.

#### Network and Demand:

Figure 4 shows the street network and the operating area of the simulated
ridepooling services. All operators offer the service in the same operating
area in this study. The street network $G=(N,E)$ has been extracted from
OpenStreetMap data using the Python library OSMnx Boeing (2017). Initially,
edge travel times are assigned according to free flow speeds from the
OpenStreetMap data. In order to replicate realistic travel times, edge travel
times are scaled after every 15 min simulation time according to actual trip
travel times within the NYC taxi trip data. Shortest (travel time) paths are
computed using a combination of the classical Dijkstra algorithm and
preprocessing origin-destination node pairs in lookup tables.

As demand for the ride pooling service, NYC taxi trips that are starting and
ending within the operating area of Manhattan are used. Trip requests are
created for the week from 2018/11/11 to 2018/11/18. Trip origins and
destinations are matched onto the closest intersection nodes that are only
connected to roads with classes “living street”, “residential”, “primary”,
“secondary”, and “tertiary”. Presumably defective trip records with average
travel times below $1$ m/s or above $30$ m/s are removed from the data set.
Overall 1511476 trips remain in the data set. To decrease overall
computational time, this set is subsampled to generate the requests for the
ridepooling services: For each trip a random number between $[0,1[$ is drawn.
If this random number is smaller than $0.1$, the trip is transferred into the
set of ridepooling requests resembling a $10$% market penetration of the
simulated ridepooling services. Using different random seeds, three set of
request sets are generated and used within the simulations.

The rebalancing algorithm is called every $T^{repo}=15$ min. Demand and supply
forecasts are aggregated to the corresponding taxi zones. For simplicity, trip
forecasts, i.e. the average number of incoming and outgoing trips within a
time interval of $15$ min per zone, are created by counting the overall trips
in the data and multiplying the counts with the market penetration of $10$%.
In the case of multiple operators sharing the demand, it is assumed that all
operators rebalance the vehicle fleet based on the same spatio-temporal
forecast distribution. Therefore the average counts are additionally divided
by the number of operators.

Further details on network and trip data processing can be found in (Syed et
al., 2021).

Figure 4: Street network of Manhattan, NYC used in the case study.

#### Scenario Specification:

We evaluate a system with a maximum of two ridepooling operators. It is
assumed in this study that both operators offer a similar service quality.
Namely, the operators employ vehicles with maximum traveler capacity
$c_{v}=4$. Additionally, they only offer trips to customers that do not exceed
(i) a maximum waiting time of $t_{wait}^{max}=6$ min, and (ii) a relative
increase in travel time $\Delta=40$% compared to the duration of a direct
trip. Because a similar service is offered by the two ridepooling providers,
we additionally assume that: (i) customers do not have an inherent preference
towards any particular operator, and (ii) due to market pressure, both
operators synchronize their fares and offer their service for the same prices.
Therefore, price sensitivity of customers is not explicitly modeled.

With respect to the different interaction scenarios, the inputs for these
scenarios are the following:

  1. 1.

Single Operator: A single operator with the specified attributes serves the
whole demand.

  2. 2.

Independent Operators: The demand is split evenly between two operators. Each
customer can only request a trip from the corresponding assigned operator.

  3. 3.

User Decision: The broker forwards customer requests to both operators. In
case a customer $i$ receives offers from both operators, the decision to book
with operator $o_{i}$ is made based on the evaluation of

$o_{i}=\operatorname*{arg\,max}_{o}\phi^{user}(u_{i,o})=\operatorname*{arg\,min}_{o}t^{arr}_{i,o},$
(12)

with the arrival time $t^{arr}_{i,o}$ offered by operator $o$.

  4. 4.

Broker Decision: The broker requests trips for the customers from each
operator. In this study, the system costs are measured by the additional
driven distance to accommodate a new request. Hence, in case the broker
receives offers from both operators, the decision to book customer $i$ with
operator $o_{i}$ is made based on the evaluation of

$o_{i}=\operatorname*{arg\,max}_{o}\phi^{broker}(s_{i,o})=\operatorname*{arg\,min}_{o}\delta
d_{i,o},$ (13)

with the additional driving distance $\delta d_{i,o}$ required for operator
$o$ to serve customer $i$.

The parameters defining the objective function for each operator are set to
$c_{\alpha}^{dis}=0.25$€/km and $c_{\alpha}^{vot}=16.2$€/h, corresponding to
the estimated values in Boesch et al. (2018) and Frei et al. (2017),
respectively.

#### Game:

The goal of the game is to model operators’ adaptation of their service within
different environments (interaction scenarios) to maximize their profit. While
there are many different parameters for operators to adapt, in this study we
allow the operators (i) to change their fleet size and (ii) to modify their
objective function for assigning offers and vehicle plans. Fleet sizes $N_{v}$
can be changed initially in steps of 20 vehicles around the initial fleet size
to be defined in the following calibration step. In the “Single Operator”
scenario, one operator has to serve double the amount of requests; hence,
fleet size step sizes are doubled accordingly. Possible parameter options
$(c_{\alpha}^{dis},c_{\alpha}^{vot})$ for setting the objective function from
Equation (1) are ($0.0$€ /km, $16.2$€/h), ($12.5$€/km, $16.2$€/h),
($25.0$€/km, $16,2$€/h), ($25.0$€/km, $8.1$€/h), and ($25.0$€/km, $0.0$€/h)
initially. With these options, the objective function can be adapted quite
smoothly between purely minimizing the driven distance to purely minimizing
customer arrival times.

Once an equilibrium with the initial parameter step sizes can be observed, the
step sizes for fleet size and objective parameters are decreased for the
remaining steps of the game to increase the resolution quality of the
equilibrium state. Thereby, parameter steps are adopted by halving the step
size, setting the currently found optimum within the new parameter table to be
observed. This procedure is repeated until no clear symmetric equilibrium can
be found anymore, which is interpreted as the maximum solution quality
possible with respect to stochastic variations within the simulations. In the
conducted simulations, alternating jumps between neighbouring cells in the
parameter table are observed indicating the best possible resolution quality
of the optimal parameter set.

Parameters for calculating the costs in Equation (8), i.e. the fix cost per
vehicle $C_{v}$ and the distance-dependent cost $c_{dis}$, are set to $25$€per
day and $c_{dis}=25.0$€/km, respectively, according to Boesch et al. (2018).

The fare $f$ to calculate the profit in Equation (8) and the penalty cost for
requests without offer $p_{no}$, which are required to calculate the effective
profit in Equation (11), will be determined within calibration simulations in
the next chapter.

##  4 Results

In this section, results of the simulations are presented. Firstly, the
calibration is described to determine the initial fleet size as well as the
parameters $p_{no}$ and $f$. Secondly, the results after performing the game
are presented and lastly, fleet key performance indicators (KPIs) are compared
before and after the game and between the different interaction scenarios.

###  4.1 Calibration

Since the envisioned autonomous ridepooling services are not yet operating,
the values for $f$ and $p_{no}$ cannot be found empirically. Instead, we use
the interaction scenario of two independent operators as calibration scenario,
where we choose $90$% served customers as a target service rate. Conducting
simulations for fleet sizes ranging from 75 to 250 reveals 190 vehicles are
needed for each operator to achieve this service rate. The distance dependent
fare $f$ is chosen to create a break even profit using 190 vehicles resulting
in $f=43$ ct/km (see blue curve in Figure 5). The goal of calibrating the
penalty cost parameter $p_{no}$ for unserved requests is to create a maximum
for effective profit $P_{eff}$ at the target service rate of $90$%. A value of
$p_{no}=46$ ct accomplishes this target (see orange curve in Figure 5) and is
used for further simulations.

Figure 5: Calibration of break even fare and unserved customer penalty. The
break even fare is chosen to achieve $0$ € Profit at $90$% served customers,
while the unserved customer penalty is set to result in a maximum for the
Effective Profit at $90$% served customers as shown in the left figure. $190$
vehicles are needed for each operator to served $90$% customers as depicted in
the right figure.

###  4.2 Game

Figure 6 shows the development of operator service parameters over the course
of the game for the broker scenario. Within each turn, the active operator
explores 6 by 6 different possibilities for fleet size and objective function
parameters, respectively, while the parameters of the passive operator remain
fixed. During the course of the game the differences between neighboring
explored parameter possibilities (in the region of the optimum in the rougher
grid) become smaller to increase accuracy. This is illustrated by grey fields
in Figure 6 as yet unexplored combinations. After each turn, the active
operator takes over the parameter set resulting in the highest effective
profit indicated by the orange boxes. For all interaction scenarios an
equilibrium can be observed by no later than 6 turns. As indicated in Figure 6
no clear symmetric equilibrium is observed after increasing the step accuracy,
which is also the case for the user decision interaction scenario. Instead
alternating jumps between neighbouring cells are observed, which can likely be
attributed to the dynamic and stochastic nature of the agent-based simulation
model. In the shown example of the broker scenario, simulations until turn 10
reveal alternating jumps within the cells (Fleet Size = $210$ veh,
$c_{\alpha}^{vot}=2.025$ €/h, $c_{\alpha}^{dis}=0.25$ €/km) and (Fleet Size =
$215$ veh, $c_{\alpha}^{vot}=2.025$ €/h, $c_{\alpha}^{dis}=0.25$ €/km). For
reasons of symmetry and because only jumps between neighboring cells occur,
symmetric operator parameters are assumed for further evaluation. The
parameter set after the first jump to neighbouring cells is applied for both
operators. In the case of the example in Figure 6, this leads to a final
parameter set of (Fleet Size = $210$ veh, $c_{\alpha}^{vot}=2.025$ €/h,
$c_{\alpha}^{dis}=0.25$ €/km) in turn 6.

Figure 6: Illustration of the development of operator parameters over the
course of the game for the broker interaction scenario. Blue arrows indicate
parameter settings of the passive operator. Orange arrows indicate the
parameter selection resulting in the highest effective profit of the active
operator. A first equilibrium can be observed at turn 4, when OP2 adjusts its
parameters to the same parameters as OP1. After turn 6 alternating behavior is
revealed once the step size is decreased further in turn 5.

Table 1 shows operator parameters before and after the game. Initially all
operators start with a fleet size of 190 vehicles, or 380 vehicles in the case
of a single operator, and an objective function parameterized by
$c_{\alpha}^{vot}=16.2$ €/h, $c_{\alpha}^{dis}=0.25$ €/km. For a single
operator, the game breaks down to a single round with one optimization table
for each parameter set to be explored. As a result of the optimization, the
single monopolistic operator decreases the fleet size and the weight of
$c_{\alpha}^{vot}$. Due to scaling effects of ridepooling services the larger
demand can be served more efficiently, and the cost reduction of operating a
smaller fleet (viewed relatively) thereby exceeds the overall penalty of
unserved customers. Additionally, costs for driven distance can be reduced
without direct competition by decreasing the weight of $c_{\alpha}^{vot}$,
thereby putting more focus on decreasing fleet mileage and increased pooling
efficiency rather than fast customer pickup and delivery. Similar behavior for
adjusting the objective function can be observed for two independent
operators. Nevertheless, they even have to slightly increase their fleet size
because the fleet can be used less efficient when demand is shared between the
operators. Within the user decision scenario, operators are in direct
competition against each other. For a customer to book a ride with a specific
operator, the operator has to offer the smallest combined waiting and travel
time. Therefore, operators select the highest value for $c_{\alpha}^{vot}$ in
this scenario to assign routes with small customer arrival time. In the broker
decision scenario, operators are also in direct competition to each other, but
the decision for a customer to book with one of the operators is based on the
offer with the smallest additional driven distance. Similarly to the scenario
with independent operators, the value for $c_{\alpha}^{vot}$ is decreased
which puts a higher relative weight on the distance cost factor
$c_{\alpha}^{dis}$ for assigning routes. Compared to the other interaction
scenarios, operators tend to have the highest fleet sizes in the case of the
broker decision. A higher density of vehicles will lead to shorter pick-up
trips (on average) and seems preferable in this scenario.

| Single Operator Scenario | Multi-Operator Scenario
---|---|---
| Initial | Final | Initial | Final
Parameter |  | Single Op |  | Indep. Ops | User Dec. | Broker Dec.
Fleet Size [veh] | 380 | 310 | 190 | 195 | 195 | 210
$c_{\alpha}^{vot}$ [€/h] | 16.2 | 2.025 | 16.2 | 2.025 | 6.075 | 2.025
$c_{\alpha}^{dis}$ [€/km] | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25
Table 1: Operator service parameters before (Initial) and after the game
(Final).

###  4.3 Fleet KPIs

Figure 7 shows the fraction of served requests before and after the game for
each interaction scenario. In all cases around $90$% of all requests could be
served as targeted within the calibration. Before the game the overall fleet
size in the system is set the same for all interaction scenarios to illustrate
the price of non-coordination. Therefore, most customers could be served
within a single monopolistic operator setting, because the fleet can be
controlled most efficiently having full access to all customers. On the
contrary, with completely independent operators fewest customers can be served
before the game due to effects of market fragmentation. Because customers have
access to both operators and can choose the other operator in case the first
cannot serve them, in the broker and user decision scenarios the fraction of
served customers lies in between. After the game, the single operator
decreases its fleet size resulting in the lowest fraction of served customers.
Most customers are served in the broker decision scenario because operating
larger vehicle fleets is profitable in this case indicating an advantage also
for customers in this regulated scenario.

Figure 7: Served Customers before and after the game for the different
interaction scenarios.

In Figure 8 the effective profit and the actual profit before and after the
game is illustrated. Before the game the effective profit is dominated by the
penalty for unserved customers resulting — similar to Figure 7 — in the
highest value for the single operator and the lowest one for independent
operators. The highest combined actual profit can be obtained within the
broker decision scenario. The operator assignment process of selecting the
operator with the lowest additional driven distance is here in line with the
distance-dependent operating cost. The profit for independent operators is
close to zero because this scenario is chosen in the calibration step to
define the break even fare.

Figure 8: Effective Profit (top) and Profit (bottom) before and after the game
for the different interaction scenarios.

After the game, operators could increase their effective as well as their
actual profit in all interaction scenarios. The highest gain is obtained for
the single operator who could decrease both fixed costs by decreasing fleet
size and distance-dependent costs by changing the objective to select shorter
routes without losing many customers by competition. The smallest gain is
observed in the user decision case. Because of pressure due to competition,
operators have to focus on assigning routes with low waiting and detour time
for customers which results in in a trade-off to higher fleet mileage and
therefore in higher costs. Within all scenarios with more than one operator,
operators achieve most actual as well as effective profit in the broker
decision setting after the game. On the one hand, assigning customers to
operators with the smallest additional driven distance is equal to the option
that produces the lowest costs for the operator. On the other hand operators
can additionally change their objective to putting more focus on assigning
short routes without the market pressure from customers deciding for fastest
trips.

The effectiveness of pooling can be measured by the relative saved distance
$rsd$, which is plotted in Figure 9 and defined by

$rsd=\frac{\sum_{i\in C_{served}}d^{direct}_{i}-d^{fleet}}{\sum_{i\in
C_{served}}d^{direct}_{i}},$ (14)

with the direct distance $d^{direct}_{i}$ of each served customer $C_{served}$
and the fleet driven distance $d^{fleet}$. The higher this quantity is, the
higher the fraction of fleet driven distance that has been shared between
customers. However, in contrast to simply evaluating the average occupancy,
unreasonable detours with multiple passengers on board do not improve this
performance indicator. Before the game the saved distance of all operators is
below or close to zero for all interaction scenarios indicating that the fleet
would actually drive more distance than if customers would drive on a direct
trip on their own. The main reason is that before the game the objective
weight of $c_{\alpha}^{vot}$ is very high compared to after the game resulting
in the preference towards direct trips contrarily to pooled trips. After the
game the relative saved distance could be improved for all interaction
scenarios mainly because all operators decreased their objective weight
$c_{\alpha}^{vot}$. The highest value for $rsd$ is measured for the single
operator scenario where most pooling can be realized with a centralized option
for optimization. The lowest value is observed in the case of user decisions.
Here, the operators are forced to keep a rather high value for
$c_{\alpha}^{vot}$. Additionally, if multiple options for a trip are
available, customers tend to choose trips without pooling because these trips
would in many cases result in longer waiting and detour times. The pooling
efficiency in the broker decision scenario is nearly as high as with a single
operator. With a combination of operators adjusting their parameters
accordingly (low value of $c_{\alpha}^{vot}$ and higher fleet size) and the
broker preferring pooled ride options, the pooling efficiency lost due to
market fragmentation can nearly be restored.

Figure 9: Saved Distance before and after the game for the different
interaction scenarios.

Lastly, Figure 10 shows customer waiting and detour times. Before the game the
average relative detour per customer is rather low indicating few pooled
trips, in line with the evaluation of the relative saved distance of Figure 9.
While the change in customer waiting times comparing before and after the game
are minor in all scenarios, a large increase in detour times can be observed
especially in the single operator and broker decision scenario. In these
scenarios also the relative saved distance increases most, showing the trade-
off between customer travel time and efficiency of sharing rides.
Nevertheless, the average relative detour of up to $15$% is still acceptable
as it is limited by constraints to $40$%. Comparing the scenarios after the
game, the lowest customer waiting and detour times can be observed for the
user decision scenario. Here, customers pick offers with the smallest waiting
and travel times while operators additionally put more focus on assigning
routes that minimize these parameters.

Figure 10: Customer waiting time (top) and relative detour time (bottom)
before and after the game for the different interaction scenarios.

##  5 Conclusion

###  5.1 Summary and Implications

This study evaluates and quantifies the negative operational impacts of
fragmenting AMOD ridepooling demand. Moreover, the concept of an AMOD broker
is introduced to counteract these negative aspects. Two extreme forms of this
broker, in which (i) the broker only collects the offers of multiple operators
and the user selects the operator, and (ii) the broker selects the operator
better suited from a system viewpoint, are evaluated in a case study for taxi
trips in Manhattan, NYC. The evaluation is performed with agent-based
simulations in a static setting with constant fleet size and operator
parameters, as well as in a game setting allowing the operators to adapt their
service to maximize profit.

After operators adopted their service parameters in the game setting, the
cumulative AMOD fleet size increased to 390 vehicles in the user decision
scenario and 420 vehicles in the broker decision scenarios, compared to 310
vehicles in the single-operator system. These increased fleet sizes
correspondingly resulted in higher service-rates when competition is present.
In most interaction scenarios, operators increase their weight on minimizing
fleet mileage to save costs. Only in the scenario where customers choose their
AMOD service, operators are forced to offer trips with fast customer arrival
times to succeed in competition. Correspondingly, in the user decision
scenario pooling efficiency measured by the relative saved distance is reduced
by around 14% compared to a single-operator system. This result indicates,
that operators might prioritize offering non-shared trips when competition
based on customer decision is present. Contrarily, the broker successfully
shifts the operators objectives to decrease fleet mileage resulting in only 2%
in relative saved distance loss compared to the single-operator system. Due to
the correlation of fleet mileage and operating costs, most profit could be
made when the broker regulates the competition of multiple operators.

All in all, it can be observed that regulating the broker platform can be
beneficial for everyone: the transportation system has to supply fewer driven
kilometers, the operators can save operational costs, thereby offsetting the
fixed costs of adding more vehicles to the fleet, and the users experience a
better level-of-service from the additional AMOD vehicles.

###  5.2 Future Work

Several open questions have to be addressed before a real-world application
makes sense:

  1. 1.

Who should operate a broker platform?

  2. 2.

How will users respond to a platform making the decision for them about which
AMOD provider serves them?

  3. 3.

Are the shown financial benefits enough motivation for AMOD providers to join
a broker, or will municipalities have to enforce it?

The components of the broker objective could be more sophisticated than just
additional driven distance, and additionally have to be traceable, i.e. allow
a live tracking of fleet KPIs. The complexity of operating such platform is
rather high and might be challenging for municipalities. However, as the
broker objective and regulations should be aligned with public goals, a
private platform provider at least should be paid for and commissioned by
municipalities. To avoid cherry-picking, the AMOD services should likely
receive a certain level-of-service / share of served request goal, e.g. by
adding penalties for requests for which they make no offer. Moreover, an
integration into existing public transportation services will be studied in
the future.

With respect to the second question, behavioral studies have to be employed.
In addition to the extreme regulatory measure of the broker choosing the AMOD
operator, some intermediary levels of regulation can be studied in future
work, in which the broker does not impose the choice of operator, but rather
merely manipulates the offers (e.g. fares) to influence traveler behavior.
Within this context, the symmetry between AMOD service levels should also be
relaxed. That is, a broker should also be able to make valuable decisions in
case one operator offers a service with high customer convenience, higher
fares and lower occupancy compared to a service with the opposite strategy. To
quantify such systems, more advanced mode choice models will be required.

When demand is modeled as price-sensitive, it also makes sense to integrate
competitive pricing into the game framework. Moreover, the effect of different
repositioning strategies can affect results significantly. It will be
interesting to evaluate whether users benefit from competition as trade-offs
are likely: users likely experience cheaper fares from competitive pricing,
but might also suffer from lower service quality due to reduced ridepooling
efficiency resulting from market fragmentation. Moreover, future work could
also include studies with more than 2 operators. Asymmetric service design or
even strongly asymmetric initial conditions can hint at whether the AMOD
market will steer towards monopolies or a shared market with broker platforms.
Nevertheless, the applied game framework might not be suited to investigate
asymmetric final operator states. Therefore, the application limits of the
current game setting have to be studied, and possibly, new methods have to be
developed to study these effects.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

Study conception and design: RE, PM, FD, KB; data collection: RE, PM; analysis
and interpretation of results: RE, PM, FD; draft manuscript preparation: RE,
PM, FD. All authors reviewed the results and approved the final version of the
manuscript.

## Funding

The German Federal Ministry of Transport and Digital Infrastructure provides
funding through the project “EasyRide” with grant number 16AVF2108H.
The authors remain responsible for all findings and opinions presented in the
paper.

## References

  * Alonso-Mora et al. [2017] Javier Alonso-Mora, Samitha Samaranayake, Alex Wallar, Emilio Frazzoli, and Daniela Rus.  On-demand high-capacity ride-sharing via dynamic trip-vehicle assignment.  _Proceedings of the National Academy of Sciences_ , 114:462–467, 2017.  doi: 10.1073/pnas.1611675114.
  * Atasoy et al. [2015] Bilge Atasoy, Takuro Ikeda, Xiang Song, and Moshe E. Ben-Akiva.  The concept and impact analysis of a flexible mobility on demand system.  _Transportation Research Part C: Emerging Technologies_ , 56:373–392, 2015.  ISSN 0968090X.  doi: 10.1016/j.trc.2015.04.009.
  * Bilali et al. [2020] Aledia Bilali, Roman Engelhardt, Florian Dandl, Ulrich Fastenrath, and Klaus Bogenberger.  Analytical and agent-based model to evaluate ride-pooling impact factors.  _Transportation Research Record: Journal of the Transportation Research Board_ , page 036119812091766, 2020.  ISSN 0361-1981.  doi: 10.1177/0361198120917666.
  * Boeing [2017] Geoff Boeing.  Osmnx: New methods for acquiring, constructing, analyzing, and visualizing complex street networks.  _Computers, Environment and Urban Systems_ , 65:126–139, 2017.  ISSN 0198-9715.  doi: 10.1016/j.compenvurbsys.2017.05.004.  URL https://www.sciencedirect.com/science/article/pii/S0198971516303970.
  * Boesch et al. [2018] Patrick M. Boesch, Felix Becker, Henrik Becker, and Kay W. Axhausen.  Cost-based analysis of autonomous mobility services.  _Transport Policy_ , 64(11):76–91, 2018.  ISSN 0967070X.  doi: 10.1016/j.tranpol.2017.09.005.
  * Dandl and Bogenberger [2019] Florian Dandl and Klaus Bogenberger.  Comparing future autonomous electric taxis with an existing free-floating carsharing system.  _IEEE Transactions on Intelligent Transportation Systems_ , 20(6):2037–2047, 2019.  ISSN 1524-9050.  doi: 10.1109/TITS.2018.2857208.
  * Dandl et al. [2019a] Florian Dandl, Klaus Bogenberger, and Hani S. Mahmassani.  Autonomous mobility-on-demand real-time gaming framework.  In _2019 6th International Conference on Models and Technologies for Intelligent Transportation Systems (MT-ITS)_ , pages 1–10. IEEE, 2019a.  ISBN 978-1-5386-9484-8.  doi: 10.1109/MTITS.2019.8883286.
  * Dandl et al. [2019b] Florian Dandl, Bernhard Grueber, Hanna Friese, and Klaus Bogenberger.  Design and simulation of a public-transportation-complimentary autonomous commuter shuttle.  _Transportation Research Procedia_ , 41:240–250, 2019b.  ISSN 23521465.  doi: 10.1016/j.trpro.2019.09.043.
  * Dandl et al. [2019c] Florian Dandl, Michael Hyland, Klaus Bogenberger, and Hani S. Mahmassani.  Evaluating the impact of spatio-temporal demand forecast aggregation on the operational performance of shared autonomous mobility fleets.  _Transportation_ , 46(6):1975–1996, 2019c.  ISSN 0049-4488.  doi: 10.1007/s11116-019-10007-9.
  * Dandl et al. [2021] Florian Dandl, Roman Engelhardt, Michael Hyland, Gabriel Tilg, Klaus Bogenberger, and Hani S. Mahmassani.  Regulating mobility-on-demand services: Tri-level model and bayesian optimization solution approach.  _Transportation Research Part C: Emerging Technologies_ , 125(1):103075, 2021.  ISSN 0968090X.  doi: 10.1016/j.trc.2021.103075.
  * Engelhardt et al. [2019a] Roman Engelhardt, Florian Dandl, Aledia Bilali, and Klaus Bogenberger.  Quantifying the benefits of autonomous on-demand ride-pooling: A simulation study for munich, germany.  In _2019 22nd IEEE Intelligent Transportation Systems Conference (ITSC)_ , pages 2992–2997. IEEE, 2019a.  ISBN 978-1-5386-7024-8.  doi: 10.1109/ITSC.2019.8916955.
  * Engelhardt et al. [2019b] Roman Engelhardt, Florian Dandl, and Klaus Bogenberger.  Speed-up heuristic for an on-demand ride-pooling algorithm.  2019b.  URL http://arxiv.org/pdf/2007.14877v1.
  * Fagnant et al. [2015] Daniel J. Fagnant, Kara. Kockelman, and Prateek Bansal.  Operations of shared autonomous vehicle fleet for austin, texas market.  _Transportation Research Record: Journal of the Transportation Research Board_ , 2536:98–106, 2015.  ISSN 0361-1981.  doi: 10.3141/2536-12.
  * Feneri et al. [2020] Anna-Maria Feneri, Soora Rasouli, and Harry J.P. Timmermans.  Modeling the effect of mobility-as-a-service on mode choice decisions.  _Transportation Letters_ , 12(2):1–8, 2020.  ISSN 1942-7867.  doi: 10.1080/19427867.2020.1730025.
  * Frei et al. [2017] Charlotte Frei, Michael Hyland, and Hani S. Mahmassani.  Flexing service schedules: Assessing the potential for demand-adaptive hybrid transit via a stated preference approach.  _Transportation Research Part C: Emerging Technologies_ , 76:71–89, 2017.  ISSN 0968-090X.  doi: 10.1016/j.trc.2016.12.017.  URL http://www.sciencedirect.com/science/article/pii/S0968090X1630273X.
  * Gurumurthy et al. [2020] Krishna Murthy Gurumurthy, Felipe de Souza, Annesha Enam, and Joshua Auld.  Integrating supply and demand perspectives for a large-scale simulation of shared autonomous vehicles.  _Transportation Research Record: Journal of the Transportation Research Board_ , page 036119812092115, 2020.  ISSN 0361-1981.  doi: 10.1177/0361198120921157.
  * Henao and Marshall [2019] Alejandro Henao and Wesley E. Marshall.  The impact of ride-hailing on vehicle miles traveled.  _Transportation_ , 46(6):2173–2194, 2019.  ISSN 0049-4488.  doi: 10.1007/s11116-018-9923-2.
  * Hoerl et al. [2019a] S. Hoerl, C. Ruch, F. Becker, E. Frazzoli, and K. W. Axhausen.  Fleet operational policies for automated mobility: A simulation assessment for zurich.  _Transportation Research Part C: Emerging Technologies_ , 102:20–31, 2019a.  ISSN 0968090X.  doi: 10.1016/j.trc.2019.02.020.
  * Hoerl et al. [2019b] Sebastian Hoerl, Milos Balac, and Kay W. Axhausen.  Dynamic demand estimation for an amod system in paris.  _IEEE Intelligent Vehicle Symposium_ , 2019b.  doi: 10.1109/IVS.2019.8814051.
  * Hyland and Mahmassani [2018] Michael Hyland and Hani S. Mahmassani.  Dynamic autonomous vehicle fleet operations: Optimization-based strategies to assign avs to immediate traveler demand requests.  _Transportation Research Part C: Emerging Technologies_ , 92:278–297, 2018.  ISSN 0968090X.  doi: 10.1016/j.trc.2018.05.003.
  * Hyland and Mahmassani [2020] Michael Hyland and Hani S. Mahmassani.  Operational benefits and challenges of shared-ride automated mobility-on-demand services.  _Transportation Research Part A: Policy and Practice_ , 134:251–270, 2020.  ISSN 09658564.  doi: 10.1016/j.tra.2020.02.017.
  * Jiang and Zhang [2018] Weiwei Jiang and Lin Zhang.  Evaluating the effects of double-apping on the smartphone-based e-hailing service: A simulation-based study.  _IEEE Access_ , 6:6654–6667, 2018.  ISSN 2169-3536.  doi: 10.1109/ACCESS.2018.2797207.
  * Kaddoura et al. [2020] Ihab Kaddoura, Gregor Leich, and Kai Nagel.  The impact of pricing and service area design on the modal shift towards demand responsive transit.  _Procedia Computer Science_ , 170:807–812, 2020.  ISSN 18770509.  doi: 10.1016/j.procs.2020.03.152.
  * Kondor et al. [2022] Dániel Kondor, Iva Bojic, Giovanni Resta, Fábio Duarte, Paolo Santi, and Carlo Ratti.  The cost of non-coordination in urban on-demand mobility.  _Scientific reports_ , 12(1):4669, 2022.  doi: 10.1038/s41598-022-08427-2.
  * Li et al. [2019] Sen Li, Hamidreza Tavafoghi, Kameshwar Poolla, and Pravin Varaiya.  Regulating tncs: Should uber and lyft set their own rules?  _Transportation Research Part B: Methodological_ , 129:193–225, 2019.  ISSN 01912615.  doi: 10.1016/j.trb.2019.09.008.
  * Liang et al. [2016] Xiao Liang, Correia, Gonçalo Homem de Almeida, and Bart van Arem.  Optimizing the service area and trip selection of an electric automated taxi system used for the last mile of train trips.  _Transportation Research Part E: Logistics and Transportation Review_ , 93:115–129, 2016.  ISSN 13665545.  doi: 10.1016/j.tre.2016.05.006.
  * Liu et al. [2019] Yang Liu, Prateek Bansal, Ricardo Daziano, and Samitha Samaranayake.  A framework to integrate mode choice in the design of mobility-on-demand systems.  _Transportation Research Part C: Emerging Technologies_ , (105):648–665, 2019.  ISSN 0968090X.  doi: 10.1016/j.trc.2018.09.022.  URL https://arxiv.org/abs/1805.06094.
  * Mo et al. [2021] Baichuan Mo, Zhejing Cao, Hongmou Zhang, Yu Shen, and Jinhua Zhao.  Competition between shared autonomous vehicles and public transit: A case study in singapore.  _Transportation Research Part C: Emerging Technologies_ , 127:103058, 2021.  ISSN 0968090X.  doi: 10.1016/j.trc.2021.103058.
  * Muller et al. [2021] Mark Muller, Seri Park, Ross Lee, Brett Fusco, and Correia, Gonçalo Homem de Almeida.  Review of whole system simulation methodologies for assessing mobility as a service (maas) as an enabler for sustainable urban mobility.  _Sustainability_ , 13(10):5591, 2021.  doi: 10.3390/su13105591.
  * Nahmias-Biran et al. [2020] Bat-hen Nahmias-Biran, Jimi B. Oke, Nishant Kumar, Carlos Lima Azevedo, and Moshe Ben-Akiva.  Evaluating the impacts of shared automated mobility on-demand services: an activity-based accessibility approach.  _Transportation_ , 2020.  ISSN 0049-4488.  doi: 10.1007/s11116-020-10106-y.
  * Oke et al. [2020] Jimi B. Oke, Arun Prakash Akkinepally, Siyu Chen, Yifei Xie, Youssef M. Aboutaleb, Carlos Lima Azevedo, P. Christopher Zegras, Joseph Ferreira, and Moshe Ben-Akiva.  Evaluating the systemic effects of automated mobility-on-demand services via large-scale agent-based simulation of auto-dependent prototype cities.  _Transportation Research Part A: Policy and Practice_ , 140:98–126, 2020.  ISSN 09658564.  doi: 10.1016/j.tra.2020.06.013.
  * Pandey et al. [2019] Venktesh Pandey, Julien Monteil, Claudio Gambella, and Andrea Simonetto.  On the needs for maas platforms to handle competition in ridesharing mobility.  _Transportation Research Part C: Emerging Technologies_ , 108(2):269–288, 2019.  ISSN 0968090X.  doi: 10.1016/j.trc.2019.09.021.
  * Pavone et al. [2012] M. Pavone, S. L. Smith, E. Frazzoli, and D. Rus.  Robotic load balancing for mobility-on-demand systems.  _The International Journal of Robotics Research_ , 31(7):839–854, 2012.  ISSN 0278-3649.  doi: 10.1177/0278364912444766.  URL http://asl.stanford.edu/wp-content/papercite-data/pdf/Pavone.Smith.ea.IJRR12.pdf.
  * Pinto et al. [2019] Helen K.R.F. Pinto, Michael F. Hyland, Hani S. Mahmassani, and İ. Ömer Verbas.  Joint design of multimodal transit networks and shared autonomous mobility fleets.  _Transportation Research Procedia_ , 38:98–118, 2019.  ISSN 23521465.  doi: 10.1016/j.trpro.2019.05.007.
  * Qian and Ukkusuri [2017] Xinwu Qian and Satish V. Ukkusuri.  Taxi market equilibrium with third-party hailing service.  _Transportation Research Part B: Methodological_ , 100(3):43–63, 2017.  ISSN 01912615.  doi: 10.1016/j.trb.2017.01.012.
  * Ruch et al. [2020] Claudio Ruch, ChengQi Lu, Lukas Sieber, and Emilio Frazzoli.  Quantifying the efficiency of ride sharing.  _IEEE Transactions on Intelligent Transportation Systems_ , pages 1–6, 2020.  ISSN 1524-9050.  doi: 10.1109/TITS.2020.2990202.
  * Santi et al. [2014] Paolo Santi, Giovanni Resta, Michael Szell, Stanislav Sobolevsky, Steven H. Strogatz, and Carlo Ratti.  Quantifying the benefits of vehicle pooling with shareability networks.  _Proceedings of the National Academy of Sciences of the United States of America_ , 111(37):13290–13294, 2014.  ISSN 0027-8424.  doi: 10.1073/pnas.1403657111.
  * Schaller [2021] Bruce Schaller.  Can sharing a ride make for less traffic? evidence from uber and lyft and implications for cities.  _Transport Policy_ , 102(3):1–10, 2021.  ISSN 0967070X.  doi: 10.1016/j.tranpol.2020.12.015.
  * Séjournè et al. [2018] Thibault Séjournè, Samitha Samaranayake, and Siddhartha Banerjee.  The price of fragmentation in mobility-on-demand services.  _Proceedings of the ACM on Measurement and Analysis of Computing Systems_ , 2(2):1–26, 2018.  ISSN 2476-1249.  doi: 10.1145/3224425.
  * Simonetto et al. [2019] Andrea Simonetto, Julien Monteil, and Claudio Gambella.  Real-time city-scale ridesharing via linear assignment problems.  _Transportation Research Part C: Emerging Technologies_ , 101:208–232, 2019.  ISSN 0968090X.  doi: 10.1016/j.trc.2019.01.019.
  * Simoni et al. [2019] Michele D. Simoni, Kara M. Kockelman, Krishna M. Gurumurthy, and Joschka Bischoff.  Congestion pricing in a world of self-driving vehicles: An analysis of different strategies in alternative future scenarios.  _Transportation Research Part C: Emerging Technologies_ , 98:167–185, 2019.  ISSN 0968090X.  doi: 10.1016/j.trc.2018.11.002.
  * Smith and Hensher [2020] Goeran Smith and David A. Hensher.  Towards a framework for mobility-as-a-service policies.  _Transport Policy_ , 89(1):54–65, 2020.  ISSN 0967070X.  doi: 10.1016/j.tranpol.2020.02.004.
  * Syed et al. [2021] Arslan Ali Syed, Florian Dandl, Bernd Kaltenhaeuser, and Klaus Bogenberger.  Density based distribution model for repositioning strategies of ride hailing services.  _Frontiers in Future Transportation_ , 2:13, 2021.  ISSN 2673-5210.  doi: 10.3389/ffutr.2021.681451.  URL https://www.frontiersin.org/articles/10.3389/ffutr.2021.681451/full#B6.
  * Tachet et al. [2017] R. Tachet, O. Sagarra, P. Santi, G. Resta, M. Szell, S. H. Strogatz, and C. Ratti.  Scaling law of urban ride sharing.  _Scientific reports_ , 7:42868, 2017.  doi: 10.1038/srep42868.
  * Wen et al. [2018] Jian Wen, Yu Xin Chen, Neema Nassir, and Jinhua Zhao.  Transit-oriented autonomous vehicle operation with integrated demand-supply interaction.  _Transportation Research Part C: Emerging Technologies_ , 97:216–234, 2018.  ISSN 0968090X.  doi: 10.1016/j.trc.2018.10.018.
  * Wilkes et al. [2021] Gabriel Wilkes, Roman Engelhardt, Lars Briem, Florian Dandl, Peter Vortisch, Klaus Bogenberger, and Martin Kagerbauer.  Self-regulating demand and supply equilibrium in joint simulation of travel demand and a ride-pooling service.  _Transportation Research Record: Journal of the Transportation Research Board_ , page 036119812199714, 2021.  ISSN 0361-1981.  doi: 10.1177/0361198121997140.
  * Xi et al. [2021] Haoning Xi, Didier Aussel, Wei Liu, S. Travis Waller, and David Rey.  Single-leader multi-follower games for the regulation of two-sided mobility-as-a-service markets.  2021\.  URL http://arxiv.org/pdf/2106.02151v2.
  * Zhang and Nie [2019] Kenan Zhang and Marco Nie.  To pool or not to pool: Equilibrium, pricing and regulation.  _SSRN Electronic Journal_ , 114(3):18, 2019.  ISSN 1556-5068.  doi: 10.2139/ssrn.3497808.