[type=editor, orcid=0000-0001-5383-6459]

[1]

1]organization=Department of Mathematics and Computer Science, Jagiellonian
University, addressline=ul. Prof. Lojasiewicza 6, city=Krakow,
postcode=30-348, country=Poland

[orcid=0000-0002-9767-8883] URL]https://rafalkucharskipk.github.io/

2]organization=Doctoral School of Exact and Natural Sciences, Jagiellonian
University, city=Krakow, postcode=31-007, country=Poland

[cor1]Corresponding author

# Ride-pooling service assessment with rational, heterogeneous, non-
deterministic travellers

Michal Bujak<EMAIL_ADDRESS>[  Rafal Kucharski
<EMAIL_ADDRESS>[

###### Abstract

Ride-pooling remains a promising emerging mode with a potential to contribute
towards urban sustainability and emission reductions. Recent studies revealed
complexity and diversity among travellers’ ride-pooling aptitudes. So far,
ride-poling analyses assumed homogeneity and/or determinism of ride-pooling
travellers. This, as we demonstrate, leads to a false assessment of ride-
pooling system performance. We experiment with an actual NYC demand from 2016
and classify travellers into four groups of various ride-pooling behaviours
(value of time and penalty for sharing), as reported in the recent SP study.
We replicate their random behaviour to obtain meaningful distributions.

Unsurprisingly, results vary significantly from the deterministic benchmark:
expected mileage savings were lower, while the utility gains for travellers
were greater. Observing performance of heterogeneous travellers, we find that
those with a low value of time are most beneficial travellers in the pooling
system, while those with an average penalty for sharing benefit the most.
Notably, despite the highly variable travellers’ behaviour, the confidence
intervals for the key performance indicators are reasonably narrow and system-
wide performance remains predictable.

Such findings shed a new light on the expected performance of large scale
ride-pooling systems. We argue, that the policy recommendations shall be
revised to accommodate behavioural heterogeneity.

###### keywords:

ride-pooling behavioural heterogeneity shareability graph

We assess the excepted ride-pooling system performance assuming the
heterogeneous behaviour observed in recent studies.

We consider four groups of travellers varying with value of time and
willingness to share, and replicate the probabilistic ride-pooling problem to
obtain meaningful distributions.

Results differed significantly from the deterministic benchmark: expected
mileage savings were lower, while the utility gains for travellers were
greater.

Ride-pooling systems’ implementation shall take into account heterogeneous
groups of travellers.

System performance for low value-of-time travellers is substantially different
from the one used by high value-of-time travellers.

##  1 Introduction

Ride-pooling is a ride-hailing service which enables travellers to share a
ride. Two or more trip requests submitted to the ride-hailing platform (like
Uber, DiDi or Lyft) are pooled into a single vehicle, which picks consecutive
travellers at their origins and takes them to their respective destinations,
as illustrated in Fig. 1. Such pooled ride is typically longer (due to pick-up
delay and detour) and yields a sharing discomfort, which are compensated by a
lower ride fare.

Ride-pooling, despite being a promising emerging mode of urban mobility, still
struggles to gain momentum. Pre-pandemic it managed to reach critical mass and
become profitable for providers and attractive for travellers (Shaheen et al.
(2016)). Yet, in the post-pandemic landscape of urban mobility, it needs to
grow again. Ride-pooling services, when deployed successfully and on a bigger
scale, can substantially contribute towards sustainability transitions.
Offering an attractive alternative to private car usage or solo ride-hailing,
yet with greater occupancy. When designed carefully, it can complement the
public transport offering attractive first-last mile services, or being an
alternative in the areas and time-periods where high frequency mass transit is
unnecessary. This is challenging, specifically when the perspective of four
different parties need to be aligned: travellers (who want to arrive on time,
with high comfort and at the low price), drivers (who want to maximise their
earnings), platform (who wants to gain market share and benefit from high
commission rates) and the city (who wants to reach the objectives of its
transport policy). Such alignment is not trivial, specifically when we
acknowledge that individual agents are heterogeneous and have different
behavioural traits, leading to different perceptions of the new service. Here,
we contribute to that alignment and demonstrate how can we assess the ride-
pooling system’s performance with heterogeneous travellers.

Figure 1: Ride-pooling illustrated. Three travellers request to travel from
their origins (marked with $\times$) to destinations (marked with a
$\mathcal{r}$). Apart from offering them three separate solo-rides (marked
with grey), platform offers to pool them together and serve with a single trip
(red). This induces a discomfort (sharing trip with others), detours (route is
longer) and delay (three travellers need to coordinate their departure times)
which is compensated with a discounted trip fare. Will the travellers find the
pooled option attractive? This depends on their behavioural profiles. So far
travellers were assumed either captive, homogenous and/or deterministic. Here,
we utilise the recent findings on heterogeneous, non-deterministic ride-pooing
behaviours. Each of three travellers may now have different value-of-time or
willingness to share and decide to opt-out from the pool. We investigate how
such heterogeneity impacts the expected system performance.

While the early studies on ride-pooling focused on the computational
complexity and challenges of real-time operations, the perspective of
travellers was often neglected. State-of-the-practice algorithms Alonso-Mora
et al. (2017) successfully match travellers into high order groups, served in
real-time with ride-hailing vehicles. Though they often treat travellers
captive, insensitive to delay and detour of the ride-pooling services as long
as they fit into arbitrary time-windows. In reality, travellers are rational
utility maximisers, extremely sensitive to the service performance. Naturally
opting-out from the systems which they perceive not optimal. In that notion,
we built our previous ride-pooling algorithm ExMAS Kucharski and Cats (2020),
strictly limiting the ride-pooling to the rides attractive for the travellers,
for which the utility is greater than for the alternative (e.g. private ride-
hailing ride).

Here, we move further along this line and introduce the natural heterogeneity
into the picture. For new mobility modes, the perceptions may differ
dramatically, from enthusiasts to sceptics. This is no different in ride-
pooling where some travellers may avoid crowding, be afraid of infectious
diseases, aim to save time or save money. Unsurprisingly, such heterogeneity
was observed in empirical studies Alonso-González et al. (2021), but hitherto
not implemented into algorithms and experimental studies. As we demonstrate,
neglecting the fact that the decision to ride-pool is non-deterministic and
made by heterogeneous agents with substantially different attitudes, leads to
false assessments.

Ride-pooling needs a critical mass to gain momentum and take-off, which leads
to non-linearity in the system performance. However, not only the total number
of clients in the system, but also the presence and properties of some
individual travellers may be crucial. Thus, the assessment based on the mean,
average expected behaviour of ride-pooling individuals is likely to be not
representative for the performance of the whole system. When we replace the
average shareability of travellers (behavioural preferences regarding pooling)
with individual properties, we will observe on one hand isolated travellers
unwilling to share with anyone and on the other extremely shareable
individuals happy to join any other pooled group. This has a highly
asymmetrical impact on the structure of the underlying shareability graph and
thus on the ride-pooling performance, we observe it e.g. through the maximal
size of pooled rides, which increases when travellers become non-
deterministic.

Using findings from an empirical study Alonso-González et al. (2021), which
revealed four distinct classes of ride-pooling travellers and observed
variability in their ride-pooling decisions, we introduce two levels of non-
determinism into the ride-pooling modelling. First, we assume distinct
attitudes for ride-pooling, i.e. latent classes associated with different
perceptions and valuations of crucial ride-pooling service variables. Second,
we introduce an error term to the ride-pooling perceptions made by individual
travellers.

In the experimental part, we take the side of the operator which wants to
estimate the potential of introducing ride-pooling. We assume that the
operator can predict the demand for her services in terms of trip requests
(origin, destination and time), but she is not aware of the specific attitudes
of particular clients. She does not know their exact value-of-time and
willingness-to-share. Though she knows the system-wide composition of the
population (from empirical studies, like Alonso-González et al. (2021)) and
can decompose the population into distinct groups of more definite behavioural
traits.

Namely, we can tell that each of our clients belongs to a given behavioural
class (group) with a given probability and this class is associated with given
ride-pooling behaviour. This additional knowledge comes at the cost of non-
determinism, which we need to incorporate into our assessment. The final ride-
pooling decision of each individual is unknown for the provider and depends on
the class membership (which is random) and the value of the error term (which
is random again). To obtain a meaningful ride-pooling picture in such
probabilistic setting, we replicate this random process sufficiently enough to
estimate the distributions, which are not evident, as we report.

We run the experiment in New York, where we use the actual travel demand
observed for taxi trips. We use a 30-minute batch of $150$ trip requests and
match travellers into attractive pooled rides. We assume behavioural
parameters as they were revealed in the empirical stated-preference study by
Alonso-González et al. (2021), in particular the values of value-of-time and
willingness-to-share. After running 1000 replications of underling random
process, we obtain results that significantly enrich the deterministic
picture.

Such a perspective allows the provider for more meaningful estimation of the
system performance and its distribution. Our results, apart from the expected
values, reveal confidence intervals and tails. Estimates of performance (e.g.
vehicle kilometres travelled, fares collected) and perceived attractiveness
for travellers are now more meaningful and robust. Policymakers and system
operators may now understand better the different points of view and tailor
the system to the needs of varying target groups. Specifically important for
policymakers who consider subsidising ride-pooling services. With our method,
they can distinguish when the service is used primarily as the cheaper taxi
(and shall not be publicly subsidised in general) from the desired cases when
it complements the public transport and aligns with policy objectives. The
system deployed to serve the groups targeted by the policymakers (e.g. groups
with lower accessibility and equity) will have different performance and can
be tailored accordingly. Covering the specific needs of the target groups
increases the chances that the system reaches stability and sustainability.

###  1.1 Related work

Ride-pooling is environmental- and city-friendly service compared to standard
ride-hailing services offered by transport network companies (TNCs). Fagnant
and Kockelman (2014) argues that one shared vehicle can replace around eleven
privately owned vehicles. Benefits of partial adoption of on-demand vehicles
for ride-pooling service are discussed in Engelhardt et al. (2019). Martinez
et al. (2015) proposes a ride-pooling system which, while added to the
standard ride-hailing system, is beneficial for users. Zhang and Nie (2021)
discusses the up-sides and down-sides from travellers’ and operator’s
perspective. Martinez and Viegas (2017) provides an insight into CO2
reduction. The most common measure for environmental benefits is vehicle
kilometres reduction, widely reported in the studies, like Bilali et al.
(2020) which provides both theoretical estimation and simulation results of
ride-pooling efficiency in terms of reduced vehicle hours. Researchers
document different values (dependent upon demand and city structure, matching
algorithm): Zhu and Mo (2022) finds the total vehicle distance reduced by only
$8.21\%$, Chen et al. (2017) provides the value of $16\%$, while the seminar
paper by Santi et al. (2014) suggests savings exceeding $30\%$.

Ride-pooling can be analysed on many levels. Ramifications of spatiotemporal
demand structure are analysed for two major German cities by Zwick and
Axhausen (2022). Soza-Parra et al. (2022) analyses how ride-pooling
performance changes with the various synthetic demand distributions.
Operational challenges on the supply side are analysed in the paper by Zwick
et al. (2022). Zhang and Nie (2021) analyses pricing, finds market equilibrium
and proposes congestion tax policy. Competition in the duopoly ride-sourcing
market is analysed by Guo and Huang (2022). Li et al. (2022) analyses optimal
fare and fleet size with congestion effects.

Variety of algorithms were proposed to solve the challenging ride-pooling
problem. In the seminal work, Santi et al. (2014) introduces a real-time ride-
pooling algorithm and invaluable shareability networks, however limits his
solution to two sharing travellers. Alonso-Mora et al. (2017) proposed a
state-of-the-art real-time algorithm which allows a higher-degree rides and is
considered the benchmark for other approaches. Above authors, like many others
(Ke et al. (2021), Shah et al. (2020), Bilali et al. (2020), Wang et al.
(2021)) use fixed constraints, typically on maximal pick up delay and detour
time. The fixed constraints are irrespective of total travel time duration and
individual preference. An alternative approach is proposed by Kucharski and
Cats (2020). The proposed ExMAS algorithm (detailed in the Methodology
section) is an offline algorithm which has relative constraints posed by
travellers. The acceptable pick-up and detour time is determined by unified
behavioural parameters (such as value of time, penalty for sharing) and
travellers are pooling only if it is attractive for them.

To determine performance of the ride-pooling system, Ke et al. (2021) analyses
three aspects: fraction of passengers who effectively share a ride, average
detour of a traveller and average vehicle routing distance. Al-Abbasi et al.
(2019) introduces a deep neural network with the objective function aiming to
minimise four objectives: supply-demand mismatch, waiting time and detour
distance, travel time extension and number of vehicles and resources used.
Agatz et al. (2012) lists three system-wide performance indicators: vehicle
distance, travel time, fraction of pooled passengers.

Many studies have been conducted to understand ride-pooling preferences
(Krueger et al. (2016), Alonso-González et al. (2021), Lavieri and Bhat
(2019)). Chavis and Gayah (2017) examines preferences towards three services:
fixed, flexible and individual, and estimates alternative specific constants
of route type along with explanatory (travel time, cost, etc.), socioeconomic
and attitudinal variables. Krueger et al. (2016) introduces three
alternatives: non-shared ride-hailing, ride-pooling and public transit. The
primary variable is the value of time, which is further split into in-vehicle
time and waiting time. In a stated preference survey Geržinič et al. (2022)
assesses the demand for on-demand and pooled mobility services in urban areas.
Alonso-González et al. (2021) investigates in depth ride-pooling preferences
in ride-hailing services. They design a stated preference study and (based on
the sample of 1000 Dutch respondents) disentangle the sharing aspect from
related time–cost trade-offs (e.g. detours), investigate preference
heterogeneity and identify distinct market segments. The study, which results
we apply in our experiments, identifies four classes with distinct behavioural
characteristics, including value of time and willingness to share (penalty for
sharing).

###  1.2 Study objectives and approach

So far, the utility-based approaches for ride-pooling are scarce, the focus is
on real-time operations, assuming travellers are captive. In the rare
approaches, where the traveller is assumed to be a rational utility maximiser,
the decision process is simplified. Travellers are assumed deterministic
and/or homogeneous, which contradicts recent empirical observations in which a
strong heterogeneity and non-determinism of ride-pooling travellers was
revealed.

In this study, we fill this gap. We propose an off-line approach where we
assume the demand is known in advance. We pool travellers into attractive
shared rides and find the optimal matching (minimising the total mileage).
Unlike the real-time approaches, we do not rely on fixed time-window
constrains to determine if the pooling is attractive, but we explicitly treat
travellers as decision makers and reproduce their process of opting for the
subjectively optimal service.

While such approach better fits the reality, it is also challenging to
reproduce and to interpret the results. Adding two levels of noise into the
choice process makes the whole analysis less straightforward. First, the
practical experimental setting needs to be well-designed to represent the
actual and practical problem. Second, the underlying stochastic process needs
to be replicated to obtain meaningful results, which calls for rigorous and
non-trivial interpretation.

Eventually, by incorporating the notion of non-determinism and heterogeneity
into the ride-pooling we may address the following problem: What is the
expected performance of ride-pooling system for which we know the detailed
spatio-temporal demand pattern and the behavioural structure of the
population, yet the exact behaviour of individual clients remains latent? Such
problem statement setting, while clearly impractical for real-time operations,
significantly improves strategic assessment by providing meaningful estimates
of vehicle hours reductions, perceived attractiveness or profitability.

Our findings are instrumental for policymakers and platform decision makers
who want to enrich their strategic decisions with the notion of travellers’
heterogeneity. Policymakers may understand when and where the ride-pooling
system remains efficient, i.e. positively contribute towards sustainability
transitions. Platform executives may use it to make optimal business
decisions, which, in turn, make the ride-pooling operations profitable. Based
on such analysis, the platform provider may reasonably identify e.g. the
optimal discount rate at which ride-pooling service, for the expected demand,
is both reliable, attractive and profitable. This is particularly timely post-
pandemic, when reintroduced, ride-pooling services offer significantly lower
discounts for ride pooling than pre-pandemic (like Uber: comp. Lo and Morseman
(2018) and UberX (2023)). Other platform providers may want to identify what
kind of offer would be their best response. And this is crucial not only for
the benefits of a particular provider, but also system-wide - which we want to
benefit from ride-pooling.

##  2 Methodology

To reach the objectives of this study, we extend the previously proposed off-
line utility-based ride-pooling algorithm ExMAS. We introduce non-
deterministic utility formulas to integrate them with the previous
deterministic approach and solve the ride-pooling problem for heterogeneous
population. ExMAS solves the ride-pooling problem, i.e. for a given demand
(set of trip requests) it first identifies the attractive pooled groups
(rides) of any size (degree) and then optimally assigns each traveller to a
particular ride (pooled or private).

In the non-deterministic case, we first sample the behavioural profiles of
individual travellers, then we identify the attractive pooled groups and
identify matching that minimises the total mileage. From such solution we
estimate the central KPIs of the system: mileage reductions, perceived
attractiveness, passenger detour and profitability. In the experiments, we
replicate this process sufficiently enough to obtain meaningful distributions
of random variables (KPIs).

###  2.1 Ride-pooling problem solution with ExMAS

Exact matching of attractive shared rides (ExMAS, Kucharski and Cats (2020))
is a strategic, utility-driven ride-pooling algorithm. In contrast to e.g.
Alonso-Mora et al. (2017), ExMAS works offline, assuming a complete knowledge
of the demand and (since it takes the strictly demand-oriented perspective)
does not explicit consider individual vehicles. While such an approach is
clearly not suitable for real-time operations, it is invaluable for strategic,
demand-oriented analyses. Instead of fixed time windows or spatial constrains
in which travellers are pooled regardless of their preferences, the focus is
on the passengers’ utility.

More strictly, two or more passengers are assigned to a shared ride if the
pooled ride utility is greater than a solo-ride utility for each of them.
Utility is generically composed of travel times, costs and their relative
weights (behavioural parameters). In the original ExMAS, the (dis)utility of a
shared- (for passenger $i$ participating in ride $r_{k}$ denoted as
$U^{s}_{i,r_{k}}$) and of the non-shared ride (denoted $U^{ns}_{i}$) were the
following:

$\displaystyle\begin{split}&U^{ns}_{i}=-\rho l_{i}-\beta_{t}t_{i}\\\
&U^{s}_{i,r_{k}}=-(1-\lambda)\rho
l_{i}-\beta_{t}\beta_{s}(\hat{t}_{i}+\beta_{d}(\hat{t}^{p}_{i}))+\epsilon,\end{split}$
(1)

where $\rho$ stands for per-kilometre fare, while $\lambda$ denotes discount
for sharing a ride. Both are controlled by the operator. $\beta^{t}$,
$\beta^{s}$ $\beta^{d}$ are the behavioural parameters: value of time, penalty
for sharing and delay sensitivity, respectively. $t_{i}$ and $\hat{t}_{i}$
stand for travel time of non-shared and shared ride, respectively,
$\hat{t}^{p}_{i}$ is a pick-up delay. Travel time with the shared ride is
updated for each evaluated ride candidate and is typically greater than
$t_{i}$ due to both pooling detour (to pick-up and drop of other travellers)
and delay (to wait for others). $\epsilon$ is a random term. Hitherto, all
behavioural parameters ($\beta$’s) were assumed homogeneous among the
population and the problem was simplified by the assumption that $\epsilon=0$
which makes the problem purely deterministic. Consequently, a shared ride
$r_{k}$ was attractive by the $i$-th traveller if simply
$U^{s}_{i,r_{k}}>U^{ns}_{i}$.

ExMAS uses the above formulation to evaluate if any theoretically feasible
group of travellers is attractive. Starting with pairs of travellers and then
gradually increasing the degree (size) of the identified pooled rides. Sharing
discount is assumed to be flat, regardless of number of co-travellers, hence
attractive ride of higher order comprises attractive rides of lower order.
Applying the recursion and exploiting the shareability graph properties, ExMAS
identifies for potential attractive triples by combining pairs, later
quadruple based on triples, etc. The search is exhaustive, yet the filtering
for attractive rides only, prevents the search space explosion in practical
settings.

Identified rides may be represented with a so-called shareability networks,
one of the outcomes of ExMAS that we exploit in this study. Two networks are
generated: in the shareability network two nodes (travellers) are linked when
they can be attractively pooled, while in the matching network two nodes
(travellers) are linked if they actually travel together in the final solution
(bi-partite matching), as we illustrate in the results.

From the set of all feasible and attractive pooled rides we may identify the
optimal solution. Solution is found with a bi-partite matching problem
minimising the total vehicle hours with a constraint that each passenger is
served with exactly one ride (pooled or private). The output is a set of rides
with their schedules (subsequent pick-up and drop-off points and times served
along the ride) from which we can measure the ride-pooling system’s
performance.

Full details on the algorithmic steps in ExMAS are described in Kucharski and
Cats (2020). Notably, ExMAS is demand-driven and thus the vehicles (fleet) is
not considered explicitly with a reasonable assumptions that: a) the fleet
size needed to serve pooled services is not greater than serving individual
private rides and b) when the degree of attractive pooled rides starts
exceeding the capacity of a single vehicle (three travellers) the service
reached a critical mass advocating for bigger vehicles (e.g. 8-person vans).

###  2.2 Ride-pooling utility for non-deterministic heterogeneous travellers

In the study, we extend our previous approach to account for the heterogeneous
behavioural characteristics of individuals. Namely, any of behavioural
parameters ($\beta$’s in eq. 1) instead of being constants, can be now a
random variable. We assume that preferences of an individual are unknown,
while we know the distributions within the population. Moreover, to account
for additional uncertainty, we introduce a panel noise $\epsilon_{i}$
(traveller-specific) and noise $\epsilon_{i,r}$ (ride-specific for specific
traveller), leading to the following formulas:

$\displaystyle U^{ns}_{i}=\rho l_{i}+\mathring{\beta_{t}}t_{i}$ (2)
$\displaystyle U^{s}_{i,r_{k}}=(1-\lambda)\rho
l_{i}+\mathring{\beta_{t}}\mathring{\beta_{s}}(\hat{t}_{i}+\mathring{\beta_{d}}(\hat{t}^{p}_{i}))+\epsilon_{i}+\epsilon_{i,r},$
(3)

where all notions introduced in the eq. 1 still apply, yet $\mathring{\beta}$s
are now random variables. Consistently with a discrete choice theory, we
assume that the probability that a traveller $i$ finds a shared ride $r_{k}$
attractive is expressed with:

$P^{s}_{i,r_{k}}=\Pr(U^{s}_{i,r_{k}}>U^{ns}_{i}),$ (4)

which depending on the random variables distributions may become e.g. a logit
or a probit model. While in general any distribution of random variable can be
applied in the method, in the experiment we use multiple classes $\mathcal{C}$
and assume the random variables to follow a multimodal normal distribution
(unimodal within the class):

$\mathring{\beta_{j}}=\sum_{C\in\mathcal{C}}p_{C}X(\bar{\beta}_{j,C},\sigma_{j,C})\mathrm{\
for\ }j\in\\{c,t,s,d\\},$ (5)

where $p_{C}$ is the probability of belonging to the class $C$,
$X(\bar{\beta}_{t,C},\sigma_{j,C})$ is a random variable following normal
distribution with mean of $\bar{\beta}_{t,C}$ and a standard deviation of
$\sigma_{j,C}$ (coefficient- and class-specific).

The above formula may be applied for any behavioural parameter, yet in our
experiment only $\beta_{t}$ and $\beta_{s}$ are random variables. Which is in-
line with findings from recent studies on behavioural heterogeneity among
ride-pooling travellers’ (Krueger et al. (2016), Alonso-González et al.
(2021), Lavieri and Bhat (2019)). Value of time serves as a proxy between
longer travel time and monetary savings. Penalty for sharing reflects an
individual’s attitude towards sharing (including infection awareness), which
varies substantially. While VoT and PfS are collinear in $U^{s}$ formula, VoT
is the only one to impact $U^{ns}$, which disentangles the impact of the two
variables.

Mind that the two noise terms in eq. 2 are applied differently, the first-one
$\epsilon_{i}$ is the traveller-specific and is drawn once for each ride-
pooling problem. $\epsilon_{i}$ can be interpreted as an attitude towards
pooling, irrespective of the trip’s characteristics. While the traveller
remains a rational decision maker, there is always uncertainty associated with
choice which is not captured by the utility formula, expressed with
$\epsilon_{i,r}$ (drawn at each evaluation of potentially attractive pooled
ride).

On the final note, we assume that class-membership probability $p_{C}$ is an
independent random variable, yet this can be easily enhanced e.g. with
sociodemographic variables (when specific age, gender or income groups
constitute respective classes). In the practical scenario, the platform may
either successfully learn or predict the class membership of the specific
$i$-th traveller $\mathbf{P}({C}\text{\textbar}i)$, effectively improving
estimation. Unfortunately, such data was not included neither in the demand
set that we used, nor in the choice model estimation of Alonso-González et al.
(2021).

Subsequent to utility determination, the remainder of the ride-pooling
algorithm remains the same as summarised in Section 2.1 and detailed in the
Kucharski and Cats (2020).

###  2.3 Performance indicators

We measure the performance of ride-pooling solution with the four following
indicators. For the environment and city perspective we look at the vehicle-
kilometres saved due to pooling $\mathcal{D}$. For travellers, we observe
pooling costs (relative detours $\mathcal{T}$) and benefits (relative
improvement in utility $\mathcal{U}$). For the platform, we look at the
profitability of ride-pooling observations $\mathcal{P}$.

$\mathcal{D}$ expresses the relative mileage savings as follows:

$\mathcal{D}=\frac{D_{ns}-D_{s}}{D_{ns}},$ (6)

where $D_{ns}$ denotes distance in the non-shared scenario (e.g. if everyone
travelled alone) and $D_{s}$ \- total lengths of rides comprising the ride-
pooling solution.

Mind that here, we use only distance with travellers and do not include
deadheading (distance travelled without travellers) - which we assume do not
increase due to sharing (compared to private rides).

From the traveller’s perspective, we measure two quantities: trip detour and
utility gain. Similarly to $\mathcal{D}$, we introduce functions $\mathcal{T}$
and $\mathcal{U}$, given as:

$\displaystyle\mathcal{T}=\frac{T_{s}-T_{ns}}{T_{ns}},$ (7)
$\displaystyle\mathcal{U}=-\frac{U_{s}-U_{ns}}{U_{ns}},$ (8)

where $T_{ns}$ and $U_{ns}$ denote total travel time and utility of all the
travellers in non-shared scenario while $T_{s}$ and $U_{s}$ corresponding
values in ride-pooling scenario. While formally the travel time in shared
option is longer due to both detour (longer route due to pooling with others)
and delay (waiting for others), here, for brevity, we simply call it a detour.
Mind the negative sing in eq. 8, added since the utility (like usually in the
transportation) is the disutility (composed of monetary costs and time) and is
always negative.

We introduce the profitability $\mathcal{P}$, being greater than $1$ when the
provider’s mileage savings are greater than loss of revenues (due to sharing
discount). First, we introduce it at the ride level:

$\mathcal{P}_{r}=\begin{cases}\frac{(1-\lambda)\lambda_{p}\sum_{i\in
r}d^{ns}_{i}}{\lambda_{p}d_{r}}=\frac{(1-\lambda)\sum_{i\in
r}d^{ns}_{i}}{d_{r}}&\text{, if ride is pooled}\\\ 1&\text{,
otherwise,}\end{cases}$ (9)

where $d^{ns}_{i}$ denotes the direct trip length (traveller $i$ pays for the
distance that would be served with the private ride, not with the detour),
$d_{r}$ is a total mileage of the ride $r$ shared by travellers $i\in r$,
$\lambda$ stands for sharing discount and $\lambda_{p}$ is fare. The second
term reflects the travellers who were not pooled with anyone and paid the full
price.

Profitability $\mathcal{P}$ on the system-wide level is then:

$\mathcal{P}=\frac{\sum_{r\in\mathbf{R}}\mathcal{P}_{r}d_{r}}{\sum_{r\in\mathbf{R}}d_{r}},$
(10)

where $\sum_{i\in\mathbf{R}}d_{r}$ is a total length of all rides in the
solution.

The above indicators can be calculated not only at the system-level (summing
over all travellers), but also for individual travellers ($\mathcal{T}$ and
$\mathcal{U}$) or individual rides ($\mathcal{D}$ and $\mathcal{P}$). We will
use subscript $r$ or $t$ for reference to individual rides’
($\mathcal{D}_{r}$, $\mathcal{P}_{r}$) or traveller’s ($\mathcal{T}_{t}$,
$\mathcal{U}_{t}$) performance.

###  2.4 Assessment

Introduction of randomness when determining whether a pooling option is
attractive or not, results in a non-deterministic solution and thus a system’s
performance. Importantly, there are two intuitive perspectives on which one
can assess the variability. One is to look at the solution of a single ride-
pooling problem and apart from identifying the mean performance (e.g. utility
gains $\mathcal{U}$) we look at how this total is distributed across the
individual travellers $\mathcal{U}_{i}$. In this study, however, we replicate
the random process (due to non-deterministic behaviour) and see how the
performance vary across the replications (as we report e.g. in Fig. 4).
Moreover, two approaches can be joined when we look at individual travellers’
performance across the replications, like we report e.g. in Fig. 7.

##  3 Results

We illustrate how the proposed method enhances assessment of ride-pooling
services with the case-study of New York City. We reproduce the likely case
when the service provider can predict the demand and its behavioural
structure, yet the actual characteristics of individual behaviours remain
latent.

We start with parametrising the behaviour of four distinct traveller groups
revealed in empirical study in Section 3.2. Then, we visualise the underlying
shareability networks and show how they evolve in the non-deterministic
setting (Fig. 3). We show the distributions of key ride-pooling performance
indicators in Fig. 4 and compare with the deterministic benchmarks, further
stratified for four behavioural classes in Fig. 5. We report the asymmetrical
impact of individual willingness-to-share on system performance and we show,
in Fig. 6, how the maximal number of travellers riding together increases in a
non-deterministic case. Then, we report how varying behaviour of individuals
impacts their ride-pooling performance in Fig. 7. We conclude with reporting
how the variability of ride-pooling behaviour scales with the demand in Fig.
8.

###  3.1 Experiment details

We experiment with the trips actually requested in the New York City on
January $2016$. We use the set of about $150$ (exactly $147$) trips requested
in the $30$ minute batch (which is beyond the critical mass needed to induce
pooling). Each trip request links origin with destination at a given time (as
illustrated in Figure 2). We used the fare $\lambda_{p}$ of $1.5$ €/km
(converted from $, consistent with NYC Taxi & Limousine Commission (2022)),
sharing discount $\lambda$ is $30\%$ (within the range suggested by Shaheen
and Cohen (2019)). We assume $\beta_{d}=1$ while the rest of behavioural
parameters is detailed in the following section (3.2).

We assume that the service provider can reasonably estimate the travel demand
(trip requests) and knows the structure of travellers’ behaviour, yet cannot
predict which requests belong to which classes. To produce reliable estimates,
we fix the demand (requests) and replicate the experiment (assuming various
ride-pooling behaviour of individual travellers) for $1000$ times. Results are
available to reproduce on the public repository111Script to reproduce results
is on the branch probabilistic_topological of original ExMAS (direct link:
https://github.com/RafalKucharskiPK/ExMAS/blob/probabilistic_topological/Utils/Probabilistic_ExMAS_wrapper.py)..

Figure 2: Demand dataset for experiments: ca. $150$ trip requests from Jan
2016 in Manhattan. Blue dots are origins, purple destinations.

###  3.2 Behavioural component

Out of several empirical studies, we select the one which provides the richest
insights on heterogeneity of ride-pooling behaviour. In the stated-preference
survey on ride-pooling behaviour by Alonso-González et al. (2021), the Latent
Class Choice Model (LCCM) was estimated to fit the empirical results. It
indicates not only that people can be stratified into four distinct classes,
but there is also a significant behavioural variability inside each class.

Travellers were revealed to be divided into classes defined as “It’s my ride”
(C1, constituting $29\%$ of the sample), “Sharing is saving” (C2, $28\%$),
“Time is gold” (C3, $24\%$), “Cheap and half empty, please” (C4, $19\%$).
Participation in one of the classes is reflected in distinct behavioural
parameters, namely the value of time (VoT) and penalty for sharing (PfS)
varied among the classes.

The structure of the utility formulas in Alonso-González et al. (2021)
differed from the ExMAS utility formulas, thus, the following interpretations
were made to apply the results to our study. Firstly, we assumed that both PfS
and VoT in formula 2 follow a multimodal normal distribution given as a
mixture of four normal distributions (corresponding to each class), like in
eq. 5. The mean value of time is explicitly given in the study for respective
classes as $16.98,14.02,26.25,7.78$ (€/h) accordingly. Yet, the penalty
sharing needed to be derived from the reported willingness to share. We used
PfS distribution reported for four additional passengers for class C1 as a
reference (because such rides are admissible and appear in our experiments).
Corresponding distributions for other groups were not reported, hence we
retrieved them as scaled reference distribution. Applied proportions were
following proportions of reported fixed WtS penalty for $2$ additional
passengers.

Besides the mean values, we need to estimate their variances. Since this was
not reported explicitly, we derive it from reported parameters estimates and
z-values. To retrieve it, we apply the formula given by Seltman (2012)
assuming independent variables (covariance is null) and truncated normal
distributions:

$\textrm{Var}\left(\frac{X}{Y}\right)\approx\frac{\mu_{x}^{2}}{\mu_{y}^{2}}\left(\frac{\sigma_{x}^{2}}{\mu_{x}^{2}}+\frac{\sigma_{y}^{2}}{\mu_{y}^{2}}-2\frac{\textrm{Cov}(X,Y)}{\mu_{x}\mu_{y}}\right).$

Which led to the variances for PfS of $0.082$, $0.071$, $0.06$ and $0.076$ and
for VoT of $0.318$, $0.201$, $5.777$ and $13.063$ for the respective four
classes. We clipped the huge variance for the last class to $1$, to prevent a
class with practically random behaviour. Such big standard deviation
presumably resulted from a small sample in a study.

Eventually, we successfully obtained: class membership probabilities
($p_{C}$), mean values of value-of-time and penalty for sharing for four
different classes ($\beta_{s,C}$, $\beta_{t,C}$) and class-respective
variances ($\sigma_{s,C}$, $\sigma_{t,C}$) as reported presented in Table 1.

|  | class | C1 | C2 | C3 | C4
---|---|---|---|---|---|---
|  | name | “It’s my | “Sharing is | “Time is | “Cheap and
parameter |  |  | ride” | “saving” | gold" | “ half empty"
VoT | $\beta_{t,C}$ | mean | 16.98 | 14.02 | 26.25 | 7.78
st.dev. | 0.318 | 0.201 | 5.777 | 1∗
PfS | $\beta_{s,C}$ | mean | 1.22 | 1.135 | 1.049 | 1.18
st.dev. | 0.082 | 0.071 | 0.06 | 0.076
Share | $p_{C}$ |  | 29% | 28% | 24% | 19%

Table 1: Ride-pooling behavioural parameterization from Alonso-González et al.
(2021). Mean values and standard deviations of value of time (VoT) and penalty
for sharing (PfS) for four classes. ∗ for class 4 we arbitrarily clipped the
otherwise too wide st. dev. of VoT

The total variance is distributed in formula 2 into a random value of $\beta$
and user-specific and ride-specific Gaussian noises. Since such distinction
was not reported empirically, we assumed that standard deviation of the ride-
specific noise $\epsilon_{i,r_{k}}$ (drawn for every evaluation of
$U^{s}_{i,r_{k}}$) constitutes $10\%$ of standard deviation of user-specific
noise (drawn once per replication) which follows standard normal distribution.
Such noise is both consistent with empirical findings and mimics the
additional uncertainty of the probit model (Cf. Ortúzar and Willumsen (2011)).

###  3.3 Probabilistic shareability networks

To illustrate how the already complex shareability structures of ride-pooling
change in the non-deterministic setting, we present them as the graph
structures below. In Figure 3, we present shareability and matching structures
resulting from first, first two and all replications of ride-pooling problem.
In all networks, nodes represent each of $150$ travellers. Two travellers
(nodes) are linked in the shareability graphs (3(a), 3(b), 3(c)) if they can
be attractively pooled together, i.e. formula 2. In the matching networks
(3(d), 3(e), 3(f)), links represent the final realisation of rides, i.e.
solution of the bi-partite matching yielding minimal total mileage (see
details in Kucharski and Cats (2020)). Pair represents a ride comprising two
travellers, a triangle - three, etc. While figures 3(a), 3(b), 3(c) reveal the
algorithmic intermediate step of ExMAS, figures 3(d), 3(e), 3(f) visualise an
actual physical realisation of ride-pooling.

(a) Shareability 1 rep.

(b) Shareability 2 rep.

(c) Shareability 1000 rep.

(d) Matching 1 rep.

(e) Matching 2 rep.

(f) Matching 1000 rep.

Figure 3: Shareability and matching networks obtained in the first, two and
all replications of ride-pooling problem. Link widths (proportional to
weights) denote how frequent two travellers were connected, i.e. found
travelling together attractive (top row) or actually travelled together in the
optimal solution (bottom row).

Already replicating the random process twice (second column in Fig. 3),
reveals a high variability of resulting networks. Many links appear only in
one replication and travellers either do not find travelling together
attractive (in Fig. 3(b)), or they are not part of the solution minimising the
mileage (as on fig. 3(e)). This may be either due to their behaviour (high VoT
and high penalty for sharing (PfS)) or due to their spatio-temporal
incompatibility with others. The reason behind such variability lays in the
assumption that we cannot predict the class memberships of individual
travellers, thus an individual may be assumed to be a ride-pooling enthusiast
(C2) in one replication and sharing-averse (C1) in another. While this is not
true for individual travellers, this represents the perspective of the service
provider who is not aware of the behaviour of a particular traveller.

This trend evolves, however, as we replicate the random process and observe
diverse shareability and matching structures. We aggregate results from $1000$
replications and introduce the edges’ weights (link widths) as linearly
proportional to the number of occurrences among all replications. The weighted
structures are presented in Figure 3(c) and Figure 3(f). Comparing figures
3(c) and 3(f) to figures 3(a) and 3(d), we observe a significant increase in
the number of links. In the probabilistic shareability network (Fig. 3(c)),
only one traveller remained isolated after $1000$ replications (a quarter was
isolated in one replication). The number of links increased nearly tenfold
from about $600$ to $6000$. The matching network from a single replication has
$62$ links, while after $1000$ replications there are over $500$ links (Fig.
3(f)).

Above results confirm that the ride-pooling problem for non-deterministic and
heterogeneous travellers yield significantly different results. In the
deterministic setting, the links can be well predicted as we can reasonably
determine whether two travellers find it attractive to pool a ride. Yet, as
soon as we include their varying behaviour more and more links are formed, and
the structure becomes on one hand more dense and connected (as we aggregate
along the replications) yet also more variable (when we compare two
independent replications). It is substantially harder to predict whether two
(or more) travellers will be happy to share a ride together when we know they
may belong to classes of substantially different behaviours. That is why the
aggregated shareability network becomes less interpretable as the number of
feasible pairs significantly increases.

On the other hand, some of the links are more frequently observed than the
others. This is due to the fact that some trips are naturally shareable due to
their spatio-temporal similarity (their origins and destinations are close to
each other and the departure times are similar). Such similarity (or
dissimilarity) can well compensate behavioural penalty for sharing and value
of time. Disentangling the two factors (behaviour and spatio-temporal
similarity) is not trivial, as they both drive the attractiveness of pooling.

This initially chaotic representation becomes more rigid when we look at the
matching solutions. Mind that in the matching, we optimise for the distance
(which is deterministic), rather than user-varying random attractiveness. This
stabilises the shareability network and the link weights become more
pronounced. Some pairs of travellers are connected with high probability,
regardless of the varying behaviour of involved travellers. This demonstrates
some regularity and predictability of ride-pooling, despite highly variable
traveller behaviour, as we will demonstrate in more detail below.

###  3.4 Expected ride-pooling performance with non-deterministic travellers

In this section, we dive into details how the behavioural heteroscedasticity
impacts the ride-pooling performance. We assess the ride-pooling system’s
performance with the four indicators introduced in Section 2.3. Calculated
first for a consecutive replications (realisations of random variables) and
then accumulated over all 1000 replications (one value corresponds to one
realisation). Results are presented in Figure 4. Baseline for comparison is
the deterministic ExMAS (with VoT and PfS being weighted average of means
introduced in Table 1) which yields mileage reduced by $30\%$, $9.8\%$ detour,
utility increased by $4.5\%$ and profitability of $1.097$.

Regardless of behavioural variability, pooling always reduces the travel
distance ($\mathcal{D}$). $90\%$ of the observations range between $24.3\%$
and $29.7\%$ with the average of $27.1\%$, which is significantly worse than
30% reductions obtained in the deterministic setting (Figure 4(a)). While this
is a significant reduction, which should be hugely appreciated from the city’s
and environmental’s perspective, it also varies significantly, with
observations ranging from $20\%$ to $32.5\%$. The right tail seems to be
fatter, with occasional outliers greater than $30\%$ vehicle kilometres
reductions. The mean of average utility gain (which is user-subjective and
depends on behavioural profile) is approximately $6.1\%$, which, in turn, is
now greater than in the deterministic benchmark. The $90\%$ two-sided
confidence interval spans between $5.1\%$ and $7.2\%$ utility gains.

A more physical measure of pooling performance, from the user’s perspective,
is trip detour, which does not depend on traveller behaviour. Notably, the
acceptable detour (and delay), unlike the fixed time windows used in other
studies, depends on VoT and PfS and can vary significantly with the behaviour.

The mean travel time of pooled ride is approximately $11\%$ longer than for
the solo-rides only system (Figure 4(c)) and $90\%$ of the observations fit
between $8.6\%$ and $13.6\%$, which, again, is worse than in the deterministic
benchmark ($9.8\%$).

Regardless of the behavioural variability, the ride-pooling remains
profitable, with average profitability of $1.082$ and the $90\%$ confidence
interval from $1.05$ to $1.11$ (Fig. 4(d)). It is a very promising finding. It
means that in $95\%$ of scenarios, profitability gain was greater than $5\%$
and we did not observe any unprofitable replication, meaning that the platform
does not risk when offering a 30% discount for pooled rides.

(a) Mileage reduct. $\mathcal{D}$

(b) Utility gain $\mathcal{U}$

(c) Trip detour $\mathcal{T}$

(d) Profitability $\mathcal{P}$

Figure 4: Distribution of ride-pooling performance resulting from 1000
replications. Dotted lines show the deterministic benchmark.

###  3.5 Class dependent ride-pooling performance

Considering the importance of the behavioural preferences of individuals on
pooling performance, we analyse further the data with respect to individuals
and their classes. First, we show the cumulative distributions of individual
indicators observed in $1000$ replications. Along with overall distributions,
we present the profiles associated with four distinct classes. While the CDF
profiles serve for illustrative purposes mainly, we enhance them with tables
showing differences between classes in the four consecutive indicators as they
reveal intriguing patterns among the introduced behavioural classes.

Here, we aggregate all the rides, irrespective of in which replication they
appear. In Fig. 5, we present the cumulative distribution functions of the
utility gains and detours. For classes C1 and C3, the utility gains are below
average and detours are above average. For the class C4 the opposite is true,
while class C2 in on average with the mean performance.

(a) Utility gain ($\mathcal{U}_{r}$)

(b) Detour ($\mathcal{T}_{r}$)

Figure 5: Cumulative distribution functions of utility gains and detours for
the trips observed in 1000 replications, stratified into four classes of
behaviour.

In Table 2, we report the resulting trip detour for the passengers in the four
respective behavioural classes classes. Apart from mean and standard
deviation, we report values of $75$th, $90$th percentile and $95$th
percentile. First, we present the values obtained for all the rides and then,
for illustrative purposes, only for shared rides (excluding the travellers who
were not matched with anyone and travelled alone). While on average, pooled
rides are 8% longer than private rides, it varies from 5% for class C1 to 15%
for class C4. Notably, 5% of travellers in C4 decided to pool despite at least
61% longer travel time. Those trends are further pronounced when we restrict
the analysis to the rides which were successfully pooled (right columns).

| All rides | Shared rides
---|---|---
| Means | St.dev. | 75 | 90 | 95 | Means | St.dev. | 75 | 90 | 95
All | 0.08 | 0.16 | 0.11 | 0.27 | 0.39 | 0.16 | 0.19 | 0.22 | 0.39 | 0.52
C1 “It’s my ride” | 0.05 | 0.11 | 0.07 | 0.19 | 0.27 | 0.12 | 0.13 | 0.18 | 0.28 | 0.38
C2 “Sharing is saving” | 0.09 | 0.15 | 0.14 | 0.28 | 0.41 | 0.16 | 0.17 | 0.23 | 0.39 | 0.50
C3 “Time is gold” | 0.05 | 0.11 | 0.07 | 0.19 | 0.27 | 0.11 | 0.13 | 0.18 | 0.27 | 0.35
C4 “Cheap and half empty" | 0.15 | 0.23 | 0.23 | 0.45 | 0.61 | 0.23 | 0.25 | 0.33 | 0.56 | 0.69

Table 2: Statistical properties of trip detour (relative) $\mathcal{T}_{r}$
for travellers of four respective behavioural classes. Mean, variance and
significant percentiles are shown first for all the travellers (left) and then
(right) only for those who were pooled.

Similarly to travel time analysis in Table 2, we present data of relative
utility gain in ride-hailing system with ride-pooling (relative to system
without pooling). Surprisingly, the class which has the longest detours (C4)
has also the greatest utility gains (11% on average). For the class unwilling
to pool the benefits are only 4% (C1). For 10% of all the travellers ride-
pooling led to at least 25% increase in perceived utility.

| All rides | Shared rides
---|---|---
| Means | St.dev. | 75 | 90 | 95 | Means | St.dev. | 75 | 90 | 95
All | 0.06 | 0.10 | 0.09 | 0.18 | 0.25 | 0.11 | 0.12 | 0.15 | 0.25 | 0.33
C1 “It’s my ride” | 0.04 | 0.08 | 0.05 | 0.12 | 0.18 | 0.09 | 0.10 | 0.12 | 0.19 | 0.26
C2 “Sharing is saving” | 0.06 | 0.10 | 0.09 | 0.18 | 0.25 | 0.11 | 0.12 | 0.15 | 0.24 | 0.32
C3 “Time is gold” | 0.04 | 0.08 | 0.06 | 0.14 | 0.19 | 0.09 | 0.10 | 0.12 | 0.20 | 0.26
C4 “Cheap and half empty" | 0.11 | 0.14 | 0.16 | 0.28 | 0.37 | 0.16 | 0.15 | 0.22 | 0.33 | 0.42

Table 3: Statistical properties of travellers gains (in relative increase of
utility) $\mathcal{U}$ for travellers of four respective behavioural classes.
Mean, variance and significant percentiles are shown first for all the
travellers (left) and then (right) only for those who were pooling.

###  3.6 Asymmetrical impact of perceived sharing discomfort

Importantly, ride-pooling behaviour of an individual traveller may have
diverse impact on system performance. While travellers unwilling to share form
isolated nodes in the shareability graph, the travellers with high willingness
to share become super connected and can share with many other travellers and
join multiple groups. Thus, the importance of highly shareable travellers is
way more important for the ride-pooling performance than single isolated solo-
riders.

We illustrate impact of this this asymmetry with Figure 6, showing a relation
between the distance of the shared ride and its profitability. Colours show
the degree of a ride, i.e. number of travellers in the ride. In the
deterministic benchmark, the maximal observed degree was five, however in the
probabilistic experiment, rides composed of six and seven travellers were
observed. This is thanks to the asymmetrical impact of highly shareable
travellers on the shareability networks. Which is profound also for the ride-
profitability, as the strong positive trend between the ride degree and
profitability is observable in Fig. 6.

Figure 6: Relation between shared ride’s distance (x-axis, in kilometres) and
profitability (y-axis). Each dot denotes a ride observed in one of 1000
repliactions and is coloured accordingly to is degree (number of travellers).

###  3.7 Ride-pooling performance vs ride-pooling behaviour

In this section, we provide an insight into the relation between sampled
behavioural parameters (VoT, PfS) and performance of the indivudal travellers
($\mathcal{U}_{r}$, $\mathcal{T}_{r}$). In Figure 7, we scatter the values of
VoT and PfS (first and second row respectively) on $x$-axis against the
performance indicators, i.e. $\mathcal{T}_{r}$ and $\mathcal{U}_{r}$ (first
and second column) on $y$-axis. Each dot represent an individual traveller
coloured accordingly to her behavioural class.

(a) $\mathcal{T}_{r}$ as a function of VoT

(b) $\mathcal{U}_{r}$ as a function of VoT

(c) $\mathcal{T}_{r}$ as a function of PfS

(d) $\mathcal{U}_{r}$ as a function of PfS

Figure 7: Scatter plots of travellers’ behavioural parameters ($x$-axis)
against the resulting ride-pooling performance. Each dot denotes a single
traveller, coloured accordingly to the class assigned to her in the respective
replication. y-axis are scaled to represent relative change, e.g. $1$
corresponds to increase of $100\%$.

We can observe a clear trend of ride-pooling detours decreasing with an
increasing value of time (Fig. 7(a)), despite the longer detours for
travellers with low value of time, their benefits of pooling (relative
increase in utility) remains high (Fig. 7(b)), whereas for travellers with
high value of time utility gains are significantly lower. The three classes
(C1, C2 and C4) have relatively low variance and do not overlap, while for
class C3 the variance of VoT is very big. Mind that in our experiment each
traveller was assigned to different classes across replications, nonetheless
the class membership strongly correlate with the resulting ride-pooling
performance, despite the fixed spatiotemporal trip characteristics.

Those clear trends become blurred when we plot against PfS (Fig. 7(c) and
7(d)) where the Gaussian shape is observed yet class memberships are
indistinguishable. Both highest detours and utility gains are obtained for
intermediate values of PfS, while extreme cases have the lowest benefits. The
travellers with a low penalty for sharing (below 1 for a subset of class C3)
seem to be often exploited by the system and their utilities are sacrifices to
match preferences for travellers with lower preferences for sharing. Yet for
those with high penalty the benefits are also limited, since they can hardly
be pooled with others. The pooling seems to be best performing with the
penalties ranging from 1.1 to 1.2 where it can reach the greatest expected
benefits (Fig. 7(d)).

###  3.8 Scaling with the demand levels

The notion of critical mass is crucial to the ride-pooling and number of
studies reported how the ride-pooling grows non-linearly with the demand size.
Yet, up to now, those findings were reported in the deterministic setting
only. Here, apart from reporting how system performance improves with growing
demand, we provide insights on how the performance variability changes.

Demand set on which we run the experimental was selected to be slightly beyond
the critical mass needed to induce the effective pooling, we extend it here
from two sides: sub-critical ($99$ requests) and super-critical ($198$
requests). We report the present previously introduced performance measures
for three levels of demand in Figure 8. Unsurprisingly, we observe the
critical mass effect as the performance significantly increases when the
demand grows from 100 to 150 requests and then stabilizes when it reaches 200
requests (Fig. 8). Yet more importantly, we observe the trends with
variabilities, which are alternative: either narrowing with growing demand
(utility in Fig. 8(b)), widening (profitability in Fig. 8(b)), or remaining
roughly constant (travel times and mileage savings in Fig. 8(a) and Fig.
8(c)).

(a) Mileage save ($\mathcal{D}$)

(b) Utility gain ($\mathcal{U}$)

(c) Trip detour ($\mathcal{T}$)

(d) Profitability ($\mathcal{P}$)

Figure 8: Distributions of four performance indicators with the three levels
of demand: subcritical, critical and supercritical.

##  4 Discussion and conclusions

Most ride-pooling algorithms rely on the fixed constraints (usually time
windows). To more objectively assess the attractiveness of a shared ride from
the traveller’s perspective, we previously proposed the ExMAS algorithm. In
this study, we take the matter further. We no longer consider travellers
homogeneous and deterministic. Using recent empirical findings, we assume that
while we still do now know preferences of the individuals, we know the general
distributions.

To properly include this in our calculations, we introduced random variables
representing the value of time and perceived sharing discomfort. Such
reformulation allowed to reproduce the four distinct classes of travellers’
behaviour towards ride-pooling, as revealed in Alonso-González et al. (2021).
Conducting sufficient number of replications, we obtain reliable estimations
of the ride-pooling problem solution, which is no longer deterministic. Apart
from mean values, we can estimate the lower and upper bounds of the
performance indicators, which substantially differ from the deterministic
benchmark.

The additional probabilistic layer introduces a variability at the level of
individuals and at the system performance. We find that despite the high
variability of individuals, pooling performance remains stable and within
reasonably narrow confidence intervals. Notably, probabilistic results are
shifted compared to the deterministic benchmark. While the primary objective
of minimising mileage is better met in the deterministic scenario, we observe
a much higher satisfaction with the service in the heterogeneous setting.

Our study also provides an insight into how ride-pooling performs for
travellers of certain preferences. We analyse the impact of the value of time
and perceived sharing discomfort on trip detour and satisfaction with the
service. We find that people with low value of time can be considered both the
most flexible and the most beneficial travellers in the pooling system.
However, those with intermediate penalty for sharing not only benefit more
than those with high penalties (who does not want to share in general), yet
also more than those with low penalties (willing to share with anyone).

Similarly to the deterministic case, we observe the critical mass effect and
pooling becomes effective only when the demand levels reach the so-called
critical mass. Now we enrich this notion with findings on variability, which
may either decrease (in terms of utility gains) or increase (in terms of
profitability for the provider) with growing demand levels.

The proposed method is general and can be easily applied to new cases, both
for general demand patterns and different behavioural models. Also, the
specific experimental setting used in this study may be reformulated, e.g.
when we know the class membership, the demand is not predicted properly or
when the behaviour is assumed fixed, but demand is varying (like in Kucharski
et al. (2021)). In the future studies, those additional dimensions of
variability may be included for even richer assessments.

Finally, the proposed method may be valuable in the pandemic-analyses, when
virus-averse behaviour drives the pooling behaviour of individuals. With the
methodology that we propose, one may better understand the performance of
ride-pooling system under dynamically changing behaviour, which becomes highly
variable during disruptive changes. This can be instrumental in the event of
the new pandemic. One can now assess the potential of ride-pooling and balance
the performance with the spreading contribution. Notably, the COVID-19 risk-
awareness was heterogeneous and non-deterministic in the population. We argue
that the penalty-for-sharing, with which we experiment in this study, may be
interpreted as the virus-awareness and assess with the proposed method.

#### Acknolwedgements

This research was funded by National Science Centre in Poland program OPUS 19
(Grant Number 2020/37/B/HS4/01847)

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