# Mind the hubris: complexity can misfire

Arnald Puy  School of Geography, Earth and Environmental Sciences, University
of Birmingham, Birmingham B15 2TT, United Kingdom. E-mail<EMAIL_ADDRESS>Centre for the Study of the Sciences and the Humanities (SVT), University of
Bergen, Parkveien 9, PB 7805, 5020 Bergen, Norway. Andrea Saltelli  Centre for
the Study of the Sciences and the Humanities (SVT), University of Bergen,
Parkveien 9, PB 7805, 5020 Bergen, Norway. Barcelona School of Management,
Pompeu Fabra University, Carrer de Balmes 132, 08008 Barcelona, Spain.

###### Abstract

Here we briefly reflect on the philosophical foundations that ground the quest
towards ever-detailed models and identify four practical dangers derived from
this pursuit: explosion of the model’s uncertainty space, model black-boxing,
computational exhaustion and model attachment. We argue that the growth of a
mathematical model should be carefully and continuously pondered lest models
become extraneous constructs chasing the Cartesian dream.

This is a draft of a chapter that has been accepted for publication by Oxford
University Press in the forthcoming book Views on Mathematical Modelling,
edited by Andrea Saltelli and Monica di Fiore and due for publication in 2023.

Keywords: Complexity, Uncertainty, Sensitivity Analysis, Climate Change,
Global Models, Curse of Dimensionality, Cartesian Dream, Platonism

## Mathematics and tales

Possibly the most famous quote of the French philosopher [22, 51] is that of
man as “master and possessor of nature”, an individual that uses the power of
mathematical reason to decode the natural order and improve human condition.
Technically, Descartes saw more clearly than others how applying algebra to
solve geometrical problems would open the door to the solution of practical
challenges. This vision was received by Descartes in a dream and has fueled
the extraordinary development that science and technology has experienced over
the last three centuries. In fact, Descartes’ dream has been so successful
that “it would not be a mistake to call our age and all its scientific
aspirations Cartesian” [20, 260].

Descartes’ vision of a mechanical world dominated by mathematical laws prone
to be deciphered through scientific inquiry was also shared by philosophers
such as Galilei, Leibniz, Laplace or Condorcet [32]. Underlying this premise
there is the platonic idea of a mathematical truth existing “out there” that
provides structure to the universe independently of the individual and its
cognitive abilities. Some significant defenders of this view in the 19th–20th
century were [30] and [34], who argued against mathematics being a product of
the human psyche and endorsed the platonistic view as the “only one tenable”.
Years later, Benacerraf inspired one of the most famous epistemological
objections to mathematical platonism by arguing that platonists cannot explain
mathematical beliefs because mathematical objects are abstract and hence not
causally active –nothing connects the knowledge holder with the object known
[10, 28]. For [4, 176], the debate between mathematical platonists and anti-
platonists may not be solvable given that both sides hold convincing arguments
and are perfectly workable philosophies of mathematics.

And yet the assumption that mathematical entities objectively exist for us to
appraise through scientific inquiry seems to have sidelined the alternative in
the field of mathematical modeling. This is apparent in the trend towards
ever-complex models characterizing natural science fields such as hydrology or
climate studies: models get bigger to accommodate the newly acquired knowledge
on previously hidden mechanisms that had been brought to light by scientific
methods and state-of-the-art technologies111This is especially apparent if one
observes the evolution of climate, hydrological and epidemiological models
over the last 80 years. For instance, the simple general circulation models of
the 1960’s have become exhaustive atmosphere-ocean general circulation models
[36, 73]. The classic bucket-type models of the 1970’s have turned into global
hydrological models that simulate the whole water cycle and the impact of
humans in it [46, 13]. In epidemiology, the first compartment models in the
1930s–1940s had just a few parameters [42]; the covid-19 model of the Imperial
College London had c. 900 [27]. See [62] for further details.. Uncertainties
are regarded as mostly epistemic and prone to be overcome or significantly
abated once we dig deeper into the mathematical structure underlying the
process under study222The design of finer-grained models is regarded as a
critical step towards that aim. In climate prediction, more detailed models
are assumed to solve convective cloud systems and their influence upon the
reflection of incoming solar radiation [58, 74, 31]. In hydrology, finer-
grained detail is thought to provide better representations of human water
demands, land-atmospheric interactions, topography and vegetation and soil
moisture and evapotranspiration processes [13, 83]. See [11, 12] for a
critique of the drive towards model hyper-resolution in hydrology.. Unless it
embraces the idea that humans are capable to eventually decipher the main
mathematical intricacies of the universe, this current implicitly assumes that
the quest for larger and more descriptive models may not have an end.

If an anti-platonic understanding of mathematics is equally defensible,
however, we must concede that our mathematical representations of physical
phenomena may simply be metaphors without any objective proof. The success of
mathematics in describing these phenomena does not necessarily demonstrate its
existence333The assumption that there is a mathematical truth because
mathematics is essential to explain the success of science is known as the
Quine-Putnam indispensability argument for mathematical realism [60], and has
been undermined among others by [45] and [76].: [29] and [3] showed that
Newtonian physics and quantum mechanics can be precisely nominalised and
described without mathematics, pointing towards the scientific usefulness of
mathematics as being mostly pragmatic, not idealistic –mathematics just makes
calculations simpler. Whatever the fit there is between a mathematical
representation of a regularity and the regularity itself may hence occur in
the mind of the observer and not in the real world. According to [43, 81],
this makes mathematics a human (and not divine) phenomenon444The application
of mathematics to the social sciences is also the subject of concern, both for
its possible dangerous styles of thinking imposed on the social issue under
study and for the performative, non-neutral role of mathematics as an element
of education and regimentation [24]. The mathematization of economics is
lamented by [23, 53, 64, 66]. See also Drechsler in this volume for a critique
of a “quantitative-mathematical social science”.:

> (…) it follows from the empirical study of numbers as a product of mind that
> it is natural for people to believe that numbers are not a product of mind!

Mathematical models in this current would be meta-metaphors given their
higher-order abstraction of a set of already abstract concepts. By assuming
the absence of transcendent mathematics, this view ultimately regards models
as figurative representations of an allegory and thus as intrinsically
uncertain constructs –their design, scope and conceptualization is not defined
by the underlying set of mechanisms modeled as much as by the subjective
perspective of the modeler(s), which inevitably reflects their methodological
and disciplinary configuration [71]. The addition of model detail as a mean to
get closer to the “truth” may lead a modeler astray in the pursuit of a
“truth” that only exists as such in the modeler’s mind.

The predominance of the platonic view in mathematical modelling is a
distinctive tenet of modernity and its willingness to unfold universal,
timeless physical truths over local, timely principles –what [80] refers to as
the triumph of Descartes’ certainty over Montaigne’s doubt. The progressive
formalization of reality into numbers and equations of increasing
sophistication reflects the thirst for developing more objective, rational
solutions to the social-environmental challenges that our society faces. A
conspicuous example of this ethos is the Destination Earth Initiative launched
in 2022 by the [26], which aims at creating a digital twin of the Earth (“a
highly accurate model”) to “monitor, model and predict natural and human
activity”, ultimately helping to “tackle climate change and protect nature”.
The same can be said of the willingness to simulate global water flows at a 1
km resolution and every 1-3 h by a section of the hydrological community in
order to “adequately address critical water science questions” [83]. This
ambition resembles [17]’s fictional tale of a nation that developed a map with
a scale of a mile to the mile, or [15]’ story of the map of the Empire whose
“size was that of the Empire, and which coincided point for point with it”. In
both tales, the map ends up being discarded as useless.

Carrol’s and Borges’ tales do not differ much from what models are in the
anti-platonic view of mathematics given their abstraction of ideas with a
likeness to the empirical world. And yet their figurative meaning does not
diminish their capacity to produce reflection and wisdom: we all get what the
insight is and how the fictional stories connect with our experiences. The
tales are not “out there” but they convey an understanding about a complex,
specific feature of human behavior in a plain and simple way. The addition of
narrative detail may or may not nuance the story and hence it is not an end in
itself. If models can philosophically be thought of as meta-metaphors in a
legitimate way, then their metaphysical status may be indistinguishable from
that of any literary work –mathematical units such as $\pi$ would be
indistinguishable from fictional characters such as Sherlock Holmes [14, 345].

The philosophical validity of the anti-platonic view of mathematics opens the
door to important questions with regards to our overreliance on the use of
models as tools for prediction, management and control. It also suggests
caution in the quest towards ever-complex models so as to achieve sharper
insights. Our use of the Greek word “hubris” in the title denotes the self-
assured arrogance that comes with the lack of attention to these
epistemological issues. For [38, 238], mathematical models are “technologies
of hubris” when they aim at predicting phenomena to facilitate management and
control in areas of high uncertainty. The addition of conceptual depth to a
model with unexplored ambiguities makes the hubris problem worse. In the next
pages we discuss four specific issues derived from this problem: explosion of
the uncertainty space, blackboxing, computational exhaustion and model
attachment. We conclude by offering some reflections on why mathematical
models may benefit from relaxing the pursuit of the Cartesian dream.

## Explosion of the uncertainty space

In the environmental, climate or epidemiological sciences the addition of
model detail often involves increasing the spatial/temporal resolution of the
model, adding new parameters and feedbacks or enlarging the model with new
compartments linked in a causal chain: for instance, a model simulating how an
aquifer gets polluted may be extended with compartments that model how this
pollution reaches the surface, how it is metabolized by crops and finally how
it impacts the health of the affected population [35]. During this model
expansion process new parameters need to be estimated with their error due to
natural variation, measurement bias or disagreement amongst experts about
their “true” value. There might also be ambiguity as to which is the best way
to mathematically integrate the new parameters into the model. All this often
makes the model output uncertainty increase, and not decrease, at each model
upgrade stage. Scholars refer to this phenomenon as the “uncertainty cascade”
effect, where uncertainties add up and expand with model complexification [37,
81].

Why do more detailed mathematical models tend to produce more uncertain
estimates? To understand this paradox we draw attention to the concept of the
model’s uncertainty space, which is the space formed by the number of
uncertain parameters $k$. When $k=1$ the uncertainty space can be represented
as a segment, when $k=2$ as a plane, when $k=3$ as a cube and when $k>3$ as a
$k$-dimensional hypercube. Now let us assume that all parameters are
distributed in the interval $[0,1]$. Ten points suffice to sample the segment
with a distance of 0.1 between points; if we want to keep the same sampling
density to sample the plane, the cube and the $k$-dimensional hypercube, we
will need 100, 1,000 and $10^{k}$ points respectively. Note how the number of
sampling points increases exponentially with every addition of a new parameter
(Fig. 1a).

Fig. 1: The curse of dimensionality. a) Increase in the number of points
needed to evenly sample an uncertainty space with a distance of 0.1 between
adjacent points as a function of the number of model parameters $k$. b)
Fraction of the space occupied by the corners of the hypercube along $k$
dimensions. c) Number of possible interactions between parameters as a
function of $k$.

The more parameters we add in our model, the larger the proportion of the
uncertainty space (and hence the number of sampling points) that will be
located in the corners and edges of the hypercube. To illustrate this fact,
let us think about the ratio of the volume of an hypersphere of radius 1/2 to
the volume of a unit hypercube of dimension $k$ [68]. The center of the
uncertainty space would equal the hypersphere whereas the corners would equal
the space of the hypercube outside the hypersphere. For $k=2$ and $k=3$, these
geometrical forms can be pictured as a circle inscribed in a plane and as a
sphere within a cube. Consider how the volume of the space occupied by the
corners increases exponentially with the addition of parameters: for $k=2$ it
amounts to 21% of the space (Fig. 2a); for $k=3$ it corresponds to 47% of the
space (Fig. 2b). By the time we reach $k=10$, the corners already form 99.7%
of the volume of the hypercube (Fig. 1b).

Fig. 2: Increase in the corners of the uncertainty space as a function of the
number of parameters $k$. a) $k=2$. b) $k=3$.

It is in the corners and edges of the hypercube where high-order effects,
e.g., interactions between parameters whose effect on the model output
uncertainty is larger than their individual effects [72], tend to occur. Non-
additive operations such as multiplications, divisions or exponentials are
enough to promote interactions, whose number grows as $2^{k}-k-1$. For a model
with three parameters $f(\bm{x}),\bm{x}=(x_{1},x_{2},x_{3})$, there might be
four interactions up to the third-order, e.g., three two-order interactions
($x_{1}x_{2}$, $x_{1}x_{3}$, $x_{2}x_{3}$) plus one three-order interaction
($x_{1}x_{2}x_{3}$). The addition of one and two extra parameters rises the
number of possible interactions to 11 and 26 and up to four and five-order
effects respectively. With ten parameters, the number of possible interactions
up to the tenth-order effect rises to 1,024 (Fig. 1c).

This explosion of the uncertainty space with every addition of a parameter
means that finer-grained models have uncertainty spaces with disproportionally
larger corners and potential higher-order interactions, a consequence of the
curse of dimensionality [9]. If active, these high-order effects tend to boost
the model output uncertainty. And the order of the highest-order effect active
in the model tends to be higher in models with a larger number of uncertain
parameters and structures [62]. This explains why the addition of model detail
may not necessarily increase accuracy, but swamp the model output under
indeterminacy. This becomes apparent if and only if the model’s uncertainty
space is thoroughly explored. If it is not, its estimates will be spuriously
accurate. And highly-detailed models tend to be so computationally demanding
to run that the exploration of their uncertainty space often becomes
unaffordable.

## Blackboxing

The accumulation of parameters, feedbacks and interlinked compartments creates
models whose behavior may not be easy to grasp intuitively. It is customary
for models in the climate, environmental or epidemiological sciences to be
formed by thousands of lines of code added up over decades and prone to
include bugs, undesired behaviours and deprecated or obsolete language
features. Global climate models, for instance, are on average composed of
500,000 lines of code in Fortran (with the model CESM of the Department of
Energy of the National Science Foundation peaking at c. 1,300,000 lines of
code) (Fig. 3a), and include both obsolete statements and important cyclomatic
complexities (Fig. 3b) [51]. The model of the Imperial College London
underpinning the response of the UK against covid-19 had c. 15,000 lines of
code written over $\sim$13 years with several dangerous coding constructs,
undefined behaviors and violations of code conventions [84]. If we consider
that in software development industry [whose quality control practices are
more stringent than those of academia [40]] there may be on average 1-25
errors per 1,000 lines of code [49], we may get to appreciate the number of
undetected errors potentially hiding in the code of big scientific models.
Lubarsky’s Law of Cybernetic Entomology (“There is always one more bug”) looms
larger in the quest for ever-exhaustive simulations.

Fig. 3: Code of Global Climate Models (GCM). a) Lines of source code. b)
Fraction of subprograms in each GCM with cyclomatic complexities, defined as
the number of independent paths through a program unit [48]. Softwares with a
higher cyclomatic complexity are harder to test and riskier to modify without
introducing faults in the code, and are more prone to include bugs [18]. The
plots are based on the data provided by [51] for 16 GCM.

Even under an optimistic scenario (e.g., only 10% of the bugs are capable to
meaningfully affect the results without the analyst noticing; only 50% of the
model is executed; 10 errors per 1,000 lines of code), the chance of obtaining
wrong results with large models, say with a model formed by 20,000 lines of
code, may become certainty [77].

To increase quality and open up the model’s black box several authors have
requested modeling teams to give open source software licenses to their code
and post it in repositories for inspection, error detection and reuse [55, 6].
The addition of comments and a user’s manual may also improve usability and
help pinpointing undesirable behaviors due to programming mistakes. These
initiatives address the technical problems that arise when writing and
upgrading code to improve the model’s descriptive capacity, but fall short in
handling another consequence of model hubris: the expansion in the number of
value-laden assumptions embedded in the code [70]. Such features escape the
formal checks used to locate and correct code defects and yet their impact in
the simulations can easily be substantial: a bug in the code may offset the
output of an algorithm that informs hurricane forecasts by 30% [59]; the
assumption that farmers prioritize long rather than short-maturing maize
varieties can lead an algorithm to leave several million people without water
insurance payouts [39].

A paradigmatic example of how the expansion of a model comes along with the
addition of potentially problematic assumptions can be found in the use of
Impact Assessment Models (IAMs) to guide policies against climate change. When
the 1.5º C goal became the target of climate action after the 2015 Paris
agreement, IAMs were upgraded with modules that allowed the achievement of
this target by assuming a wide and intensive use of negative emission
technologies (e.g., bioenergy, carbon capture and storage) [8, 2]. Such vision
implied that a technology still in its infancy is deployed to fight climate
change at a global scale and delivers as expected 15 years later. Many other
questionable assumptions are likely to hide behind IAMs –-in fact, their
number may be as large as to be unmanageable in the context of scientific
publishing practices [75]. This effectively places IAMs beyond the reach of
peer-review [67].

## Computational exhaustion

Until 2004–2006, the development of increasingly finer-grained models was
fueled by computational advances that allowed to double the number of
transistors in a chip while keeping the power requirements per unit of area
constant. In other words: the speed of arithmetic operations could grow
exponentially without significantly increasing power consumption. The doubling
of the number of transistors and the scaling between chip area and power use
are known as Moore’s law and Dennard’s scaling respectively [54, 21], and are
two key trends to understand the evolution of computing performance and
mathematical modeling over the last 50 years. Among others, they facilitated
the onset of our current numerical, physics-based approach to model planetary
climate change by sustaining the development of a “hierarchy of models of
increasing complexity”, as put by computeer pioneer and meteorologist Jule G.
Charney [5].

After 2006 it became apparent that this trend was over (Fig. 4). Basically,
the doubling in the number of transistors in a chip could no longer be matched
by a proportional boost in energy efficiency. This resulted in chips with
increased power density and more heat generated per unit of area, which had to
be dissipated to prevent thermal runaway and breakdown. A solution was to keep
a fraction of the chip power-gated or idled –the part known as “dark silicon”
[79]. The higher the density in the number of transistors, the larger the
fraction of the chip that has to be underused, a limitation that tapers off
the improvement in clock frequencies and single-thread performances (Fig. 4a).
Since higher chip densities are being required to match the demands of machine
learning, big data and mathematical models, the fraction occupied by “dark
silicon” may soon be larger than 90% [79, 41] (Fig. 4b). Adding extra cores to
improve performance is unlikely to sort this issue out due to its meagre
computational returns [25].

Fig. 4: Computational limits. a) Microprocessor trend data up to the year
2020, retrieved from Karl Rupp in https://github.com/karlrupp/microprocessor-
trend-data (Accessed 25th of May 2022). b) Fraction occupied by “dark silicon”
as a function of the technology ($x$ axis). Modified from Fig. 1.1 in [41].

Power consumption hence imposes a severe constraint to the addition of model
resolution and suggests that, under our current computational paradigm, the
quest towards ever-detailed models is computationally unsustainable. And the
problem is not only one of arithmetic performance, but also of data storage.
The 100 models participating in phase 6 of the Coupled Model Intercomparison
Project currently produce c. 80 petabytes of data [1 petabyte (PB) = 1,000
terabytes (TB)]; if they increase resolution at a $\sim$1km, the output may
reach 45,000 PB [74], $\sim$2,200 times more data than that stored in the
Library of Congress in 2019 ($\sim 20$ PB) [78]. To tackle the data avalanche
derived from going hyperresolution some authors have suggested to just save
the simulation setup and re-run the simulation on demand [74], yet this
strategy may breach the FAIR data principles whereby data should be findable,
accessible, interoperable and replicable [82], and re-computing may be even
more costly than archiving [7].

## Model attachment

The pursuit of finer-grained models requires the training of specialists able
to set, calibrate, upgrade and analyze these models and pull appropriate
insights for scientific and policy purposes. The acquisition of this expertise
is time-consuming and usually demands several years of education at a
university or research institution. As with any other learning process, this
situation sets the ground for the development of a domain-specific knowledge
that endows its possessor with the capacity to efficiently put the learned
skills to use in exchange for a higher risk of suffering cognitive biases such
as [47]’s hammer –“If the only tool you have is a hammer, it is tempting to
treat everything as if it were a nail”. In mathematical modeling this
phenomenon takes place when a model is used repeatedly regardless of purpose
or adequacy (a model of “everything and everywhere”), a bias that has been
attested for hydrological modeling: based on a sample size of c. 1,500 papers
and seven global hydrological models (GHM), [1] observed that in 74% of the
studies the model selected could be predicted based solely on the
institutional affiliation of the first author.

Here we extend the work by [1] to 13 new global (hydrological, vegetation and
land surface) models across c. 1,000 papers [CLM, CWatM, DBHM, GR4J, H08,
JULES-W1, LPJmL, MATSIRO, MHM, MPI-HM, ORCHIDEE, PCR-GLOBWBW and WaterGAP; see
[61] for the methods and the references], with the same results: the
institutional affiliation of the first author anticipates the model used in
$\sim 75$% of the publications (Fig. 5a). In fact, several institutions
display a very high level of attachment. This is the case of the INRAE (100%,
GR4J, 15 papers), Osaka University (100%, H08, 11 papers), Utrecht University
(91%, PCR-GLOBWB, 33 papers), the Potsdam Institut Fur Klimafolgenforschung
(93% LPJmL, 54 papers), the University of Exeter (96%, JULES-W1, 23 papers) or
Udice –Universités de Recherche Françaises (96%, ORCHIDEE, 28 papers) (Fig.
6a). The attachment of these institutions to their favourite model is also
very consistent through time (Fig. 5b).

Fig. 5: Institutional attachment to global models. a) Strength of the
institutional attachment to a favorite model. A value of 1 means that 100% of
the studies published by a given institution rely on the same global model.
Only institutions with more than 5 articles published are considered. b)
Strength of the attachment for some specific institutions through time. Fig.
6: Institutional attachment to global models. Only the 20 most productive
institutions are displayed. a) Stacked barplot with the number of publications
per institution. b) Fractioned stacked barplot.

The case of global models illustrates the extent to which model selection may
be influenced by path-dependencies that do not necessarily match criteria of
adequacy, but rather convenience, experience and habit [1]. This inertia might
be seen as an instance of Einstellung effect, the insistence on using familiar
tools and frames to tackle a problem even if they are sub-optimal or
inadequate [44]. Very large models such as global models may become
institutionalized to offset in the mid-term the costs derived from setting up
a more efficiency ecosystem in terms of writing of code, training of
personnel, selection of modeling methods, calibration algorithms, upgrading,
etc. Once a given modeling ecosystem is in place, the workflow is streamlined
and the researcher has “to do nothing but to press a few enters and then [the
modeling pre-processement] is done” [50, 15]. Switching to other models resets
this process and moves the watch back for the institution and the researcher.
Larger models may be more prone to promote model attachment and become models
of “everything and everywhere” given the much higher costs involved in the
first stages of their institutionalization.

## Concluding remarks

In this chapter we have suggested moderating our Cartesian thirst to unfold
the secrets of the natural world via increasingly detailed mathematical
models. This pursuit may lead to overlook important discussions over
mathematical Platonism and the nature of mathematical knowledge. It may also
promote black-boxing, computational exhaustion and model attachment. There
exists in mathematical modeling a “political economy” whereby larger models
command more epistemic authority and offer to their developer(s) better
opportunities to defend them against criticisms. Thus the resistance of some
modelers to come to terms with the full uncertainty of their work may have
motivations such as that of “navigating the political” [8], i.e., defending
the role of modeling work in policy-relevant settings, where epistemic
authority needs to be preserved [65].

Modeling is but one among many instances of quantification where the issue of
overambition, in the words of [63, 36], is a recurring theme. In that sense,
sociology of quantification offers powerful instruments to dissect the
normative implications of our ambitions to box reality behind numbers. There
are instances of quantification where contestation comes more natural, such as
in the use of statistical indicators in socio-economical domains, discouted by
statactivists [16, 52]. Even when it comes to rating and ranking there is a
rich literature condemning the excesses of quantification [56, 57], and a
recent success of activism in this field is the fight against the Work Bank’s
Doing Business index [19]. The use of statistical rituals in sociology is
observed by [33] and the mathematization of the economy is discussed by Lukas
and Drechsler in another chapter of this book. As argued elsewhere [69],
mathematical modelling tends to be shielded from more stringent forms of
critical activism by the barrier that the complexity of the mathematical
constructs and coding pose to experts and non-experts alike.

## Data availability

The code to reproduce the results and the figures of this study can be found
in [61] and in GitHub (https://github.com/arnaldpuy/model_hubris).

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11institutetext: School of Computing, University of Leeds, Leeds, UK

# Consensus-based Participatory Budgeting for Legitimacy: Decision Support via
Multi-agent Reinforcement Learning

Srijoni Majumdar  Evangelos Pournaras

###### Abstract

The legitimacy of bottom-up democratic processes for the distribution of
public funds by policy-makers is challenging and complex. Participatory
budgeting is such a process, where voting outcomes may not always be fair or
inclusive. Deliberation for which project ideas to put for voting and choose
for implementation lack systematization and do not scale. This paper addresses
these grand challenges by introducing a novel and legitimate iterative
consensus-based participatory budgeting process. Consensus is designed to be a
result of decision support via an innovative multi-agent reinforcement
learning approach. Voters are assisted to interact with each other to make
viable compromises. Extensive experimental evaluation with real-world
participatory budgeting data from Poland reveal striking findings: Consensus
is reachable, efficient and robust. Compromise is required, which is though
comparable to the one of existing voting aggregation methods that promote
fairness and inclusion without though attaining consensus.

###### Keywords:

participatory budgeting reinforcement learning consensus legitimacy social
choice decision support collective decision making digital democracy

##  1 Introduction

Participatory budgeting (PB) is a bottom-up collective decision-making process
with which citizens decide how to spend a budget of the local municipality
[16, 4]. Citizens initially submit proposals for implementation of various
project ideas, i.e. public welfare amenities. These are evaluated by the city
officials and finally, a subset is put for voting. Citizens then express their
preferences using different input voting methods such as approval or score
voting [9]. Finally, voting aggregation methods are applied to select the
winner projects [4].

The selection of the winner projects depends on both input and aggregation
methods [2]. As preferences via approvals or scores are based on self-
interest, voting outcomes may yield different satisfaction levels, under-
representation, and poor legitimacy. For a more stable, conclusive, shared and
legitimate voting outcome, a form of systematic and scalable deliberation is
missing among citizens so that individual preferences are exchanged, debated
and compromised in a viable way to reach consensus [6]. This challenge is
addressed in this paper.

A new multi-agent reinforcement learning approach (MARL-PB) is introduced to
model a novel iterative consensus-based PB process. In the proposed approach,
consensus emerges as a result of (i) reward-based learning based on project
ideas proposed and selected in the past and (ii) decentralized voter
communication that supports information exchange and deliberation.

MARL-PB is implemented as a decision-support system that finds applicability
in three use cases by three beneficiaries as shown in Figure 1: (i) _Citizens_
: digital assistance to communicate, deliberate and reach a common ground for
which projects to implement. This is expected to increase the participation,
satisfaction and legitimacy in participatory budgeting. (ii) _Policy-makers_ :
digital assistance to filter out projects during the project ideation phase
with the aim to put for voting a reasonable and legitimate number of projects
that results in informed and expressive choices during voting without
informational overload. (iii) _Researcher_ : digital assistance for the
assessment of fair and inclusive voting aggregation methods (e.g. equal
shares, Phragmen) via comparisons with a fine-grained consensus-based model
such as the one of MARL-PB.

Figure 1: Consensus-based participatory budgeting using a multi-agent
reinforcement learning (MARL-PB). A decision-support framework is designed for
three different use cases and beneficiaries: citizens, policy-makers and
researchers.

MARL-PB is extensively assessed using state-of-the-art real-world
participatory budgeting datasets [15] from Poland. The following three
research questions are addressed: (i) _How effective multi-agent reinforcement
learning is to assist voters reach consensus in participatory budgeting?_ (ii)
_What level of flexibility is required by voters to compromise and reach
consensus in participatory budgeting?_ (iii) _How efficient and robust a
consensus-based participatory budgeting is by using multi-agent reinforcement
learning?_. The quality of consensus, its efficiency and robustness are
studied, along with how they are influenced by factors such as the following:
(i) number of possible consensus bundles, (ii) in-degree of the communication
network, (iii) number of voters, (iv) districts, (v) voting aggregation
methods and (vi) project attributes.

The contributions of this paper are summarized as follows: (i) The multi-agent
reinforcement learning approach of MARL-PB to model and implement an iterative
consensus-based PB process. (ii) A decision-support framework based on MARL-PB
to digitally assist three use cases by three beneficiaries. (iii) An extension
of the reward-based learning strategy with a gossip-based agents communication
protocol for decentralized information exchange and consensus building. (iv) A
compilation of metrics that characterize and assess the legitimacy of the
consensus-based PB process. (v) Practical and revealing insights about the
nature of the achieved consensus: requires compromises comparable to the ones
of the voting aggregation methods that promote fairness and inclusion. (vi) An
open-source software artifact of MARL-PB for reproducibility and encouraging
further research in this niche research area 111 https://github.com/DISC-
Systems-Lab/MARL-PB (last accessed: July 2023).

This paper is outlined as follows: Section 2 reviews related work. Section 3
introduces the consensus-based approach. Section 4 illustrates the empirical
results and findings. Section 5 concludes this paper and outlines future work.

##  2 Related Literature Review

This section provides an overview of related literature, with a focus on
iterative reward-based learning for collective decision-making processes.

Social dilemma games such as Prisoners Dilemma has been studied in the context
of reward-based learning agents [12] with two agents and discrete rewards,
i.e. punishment (0) or no punishment (1), to explore the compromises that two
agents make to reach consensus. Using multiple agents, this experiment
provides insights on how learning can stabilize using limited voters and
deterministic rewards. This provides a relevant direction for dealing with
voting for social choice and collective preferences.

Airiau et al. [1] model an iterative single-winner voting process in a
reinforcement learning setup to analyze the learning capabilities of voting
agents to obtain more legitimate collective decisions. The proposed framework
provides a new variant of iterative voting that allows agents to change their
choices at the same time if they wish. The rank of the winner at every stage
in the preference order of voters is used as a reward for the agents to learn
and re-select. The proposed work by Liekah et al. [11] additionally calculates
the average satisfaction among voters in every iteration based on the winner
and individual preferences.

Table 1: Comparison of this work with earlier multi-agent reinforcement
learning approaches for collective decision making.

Aspect |  Macy et al. [12] |  Airiau et al. [1] |  Liekah et al. [11] | Proposed Approach (MARL-PB)
---|---|---|---|---
Outcome | Single Winner | Single Winner | Single Winner | Multiple Winners
Rewards | Deterministic |  Stochastic$*$ |  Stochastic$*$ | Deterministic (project attributes)
| 4 discrete values |  |  | Stochastic (from communication)
Execution | Centralized | Centralized | Centralized | Shared aggregate rewards, decentralized
Action Space | 4 | 5 | 5 | till 100
$*$Rank of winner in the preference order of the voter (within an iteration).

Prediction of complete PB ballots using machine learning classification is
recently studied as a way to decrease information overload of voters using
partial ballots [10]. This approach could complement MARL-PB to speed up the
consensus process.

Existing approaches (see Table 1) do not incorporate inter-agent communication
for large-scale information exchange in multi-winner voting systems. The
rewards are fixed in centralized settings and do not model the preferences of
the voters. Moreover, the feasibility of reaching a consensus via
communication with other voters has not been studied. This is relevant to the
problem of scaling up and automating deliberation in collective decision-
making to reach more legitimate voting outcomes. These are some of the gaps
addressed in this paper.

##  3 Consensus-based Iterative Participatory Budgeting

In this section, an iterative participatory budgeting process is introduced
modeled by a multi-agent reinforcement learning approach. The voters (agents)
maximize their self-interest but also compromise to reach a consensus in a
multi-agent system, where the choices of others are initially only partially
known.

###  3.1 Multi-armed Bandit Formulation

In a participatory budgeting process, voters collectively choose multiple
projects subject to a constraint that the total cost of the projects is within
the total budget. To incorporate this knapsack constraint, a combinatorial
model [2] is designed to formulate bundles from the available list of
projects. So for three projects, there are seven possible bundles, out of
which a subset fulfills the budget constraints. These are referred to as valid
knapsack bundles and they constitute the possible actions in a multi-arm
bandit formulation (see Figure 2).

Figure 2: Calculation of bundles: They represent all possible combinations
from the listed projects where the total cost of all the projects in the
bundle is within the budget. In the example, three projects with their
corresponding costs are listed for voting in a participatory budgeting
process. The total budget is 700. Five out of the seven possible bundles are
valid and satisfy the budget constraint.

The bundles encode all possible multi-winner preferences the voters can
collectively have. Learning valid knapsack bundles instead of individual
project selections prevents early terminations by budget violations [3].

The iterative version of the PB process introduces a partial voters’
communications at every iteration to exchange preferences with the aim they
converge (compromise) to the same bundle (same preferences). This process
models a large-scale automated deliberation process. The selection of an
action by a voter depends only on the rewards associated with the bundles,
hence, the problem is modeled as a multi-armed bandit reinforcement learning
approach [14].

The multi-arm bandits are defined in the form of a tuple $<A;R>$, where

$A$ represents actions that are the possible bundles and $R_{A}=P[r|A]$ is the
probability distribution of rewards over the bundles (actions).

Actions: For a participatory budgeting process with a set of projects and a
total budget, the actions comprise of the valid knapsack bundles formed from
the projects and their associated costs.

Rewards: The preference modelling in the form of rewards plays an important
role in reaching consensus in a large action space with multiple agents. The
aggregate preferences of the voters over past and the current year are encoded
for a region in the form of rewards that signify how the needs for public
amenities evolve over the years and thus can help to predict the collective
preference for the current participatory budgeting process. These are modeled
and calculated as deterministic rewards for each bundle.

To reach a consensus, voters explore the action space of each other via
information exchange. This exchange models a large-scale and automated
deliberation process, which voters use to learn, compromise and adjust their
choices.

Deterministic Rewards: A project is related to a type of public welfare
amenities222It is assumed that preferences for such projects persist over the
passage of time, in contrast to infrastructure projects that once they are
implemented, they may not be preferred anymore. such as urban greenery,
sports, culture, education, environmental protection etc., or a population
group that benefits such as elderly, families with children, etc. The
preference of citizens are mostly associated with these attributes and can be
used to estimate collective preferences for the population of a region.

The number of occurrence of such project attributes, which are put for voting
and selected in the past years of a region is used as reward utilities:

$R^{a}\ =\ \Sigma_{y\in Y}(\mathbb{C}(a)+\mathcal{C}(a)),$

where $a$ is a specific project attribute, $\mathbb{C}$ and $\mathcal{C}$
signify the normalized total count of occurrence of the project attribute
across listed and selected projects respectively over $Y$ years of
participatory budgeting processes in a region.

The reward for a project is determined as follows:

$R_{p}=\sigma(\Sigma_{i=1}^{\mathcal{A}}(R^{a}_{i}))+\tanh(\frac{c_{p}}{\mathcal{B}}),$

where $\mathcal{A}$ is the total number of attributes associated with a
project, $c_{p}$ is the individual project cost and $\mathcal{B}$ is the total
budget of the PB process. The rewards for a bundle is the sum of the rewards
of each of its projects.

Rewards from inter-agent communication: At every iteration, we update a
dynamic random bidirectional graph using a decentralized process such as the
gossip-based peer sampling [8] for peer-to-peer communication. At each
iteration, the connected agents send the bundle that has received the highest
rewards (other randomized schemes are supported by the code), together with
the reward itself. As the neighbors are randomly decided, the accumulated
rewards from information exchange are stochastic. The stateless variant of the
Q learning approach is augmented to incorporate rewards obtained from
information exchange:

$Q(b_{t})\leftarrow Q(b_{t})+\alpha(r+\delta(\underset{\ \ \ \
b^{c}_{t}}{max}(Q(b_{t}^{c})-Q(b_{t}))),$

where $Q(b_{t}^{c})$ is the rewards obtained via agent communication for a
bundle b at time t. The introduced learning rate for rewards from information
exchange -$\delta$ is set empirically to 0.1. The discount factor $\gamma$ in
the Q learning is set to zero as future rewards are not considered. Algorithm
1 outlines the learning process.

Algorithm 1 Augmented Q-Learning for consensus in participatory budgeting.

1:Populate project list and cost for the current participatory budgeting
process

2:Initialize the fixed rewards for projects

3:Calculate rewards for all valid bundles

4:for each iteration $i$ $\geq$ $1$ do

5: for each voter $v$ $\in$ V do

6: if $i$ == 1 then

7: Assign the bundle with highest overlap to original individual preferences

8: Update random graph via the peer sampling service

9: Aggregate rewards of bundles from neighbors

10: Update total rewards for a bundle in the Q table

11: Select action (bundle) according to $\epsilon$-greedy policy,
$\epsilon$$\in$[0,1]

Initially, the agents select a bundle according to their preference (first
iteration) and then they start communication with other agents during which
they adjust their preferred bundle. The selection of the bundle at each
iteration is based on the cumulative sum of both rewards, which the agents
maximize using an $\epsilon$ greedy exploration strategy. For a low number of
projects and voters size, the initial preferences from the multi-winner
approvals of the voters may result in a reduced action space for exploration.

###  3.2 Assessment Model for Consensus

The following metrics are designed to assess the quality of the consensus
(legitimacy) based on popularity, representation and budget utilization that
can increase the satisfaction of the citizens, increase participation and
improve the quality of the overall PB process [2, 13]. The level of compromise
made by voters is assessed. These metrics also characterize how difficult it
is to reach a consensus in an iterative voting process for participatory
budgeting. The metrics that model the legitimacy are outlined as follows:

  * •

Compromise Cost: The mean non-overlap (1 - mean overlap) of projects between
the preferred bundle of the voters and the consensus bundle, calculated using
the Jaccard Index [5].

  * •

Unfairness: The coefficient of variation of the compromise cost over all
agents.

  * •

Popularity (fitness of consensus): The normalized ranking score of the
projects in the consensus bundle, calculated using the number of votes of each
project from the original voters’ preferences.

  * •

Budget Utilization: The cost of the projects in the consensus bundle divided
by the total available budget in the participatory budgeting process.

##  4 Experimental Evaluation

This section illustrates the results obtained from the evaluation of the
consensus-based participatory budgeting process, using real-world data. These
results shed light on the the efficiency of reward models, the communication
protocol and the exploration strategy to reach consensus.

Dataset: The pabulib PB dataset (http://pabulib.org/) is used for the
evaluation. It contains the metadata related to projects and voters along with
the voting records for multiple participatory budgeting instances, for various
districts and cities of Poland. Each project is associated with multiple
attributes such as urban greenery, education, relavance to children etc.,
along with information about the project costs. There are multiple
participatory budgeting instances for every district or city for multiple
years and different ballot designs such as k-approval, cumulative and score
voting. Furthermore, the winners are calculated using various aggregation
methods such as the method equal shares, phragmen, and utilitarian greedy [4]
to assess the quality of the consensus bundles compared to the ones calculated
by methods that promote fairness and inclusion. Three districts are selected -
Ruda, Ursynow and Rembertow, whose valid bundles vary from a smaller set (12
for Ruda) to a larger one (90 for Rembertow).

Table 2: Parameters for experimentation with each dataset. The ranges signify that experiments are performed incrementally, for instance, 5, 6, 7,….90 for the # of bundles selected randomly in Rembertow. The maximum number of valid bundles extracted from 20 projects is 90. The available data is for 4 years for each district. The experiments are performed for the latest year and the aggregate preference (rewards) are calculated using all years. The decay rate and learning rates are set to 0.1 after empirical investigation. Dataset | # of Projects | # of Bundles | In-degree | # of Agents
---|---|---|---|---
Rembertow | 20 | 5 to 90 | 2 to 26 | 50 to 100
Ursynow | 18 | 5 to 75 | 2 to 26 | 50 to 100
Ruda | 10 | 3 to 12 | 2 to 10 | 50 to 100

Design: The framework is tested using various settings such as the numbers of
combinations (bundles), number of agents, learning rate, decay rate, and the
in-degree of the random graph updated at every iteration (see Table 2). For
each of these settings, the projects selected in the consensus are analyzed
and compared with winners selected using other aggregation methods such as the
method of equal shares and greedy [4].

###  4.1 RQ1: Effectiveness to Reach Consensus

Quality of Consensus: The convergence to the consensus bundle depends on the
in-degree of the random graph and the number of bundles (action space). Figure
3 shows the budget utilization as a function of the number of bundles and in-
degree. Budget utilization of the consensus bundles (Figure 3) for Rembertow
is low (0.40 to 0.45) in most cases, which could be attributed to considerable
high costs for popular projects and fewer projects in the consensus bundle.

Figure 3: Total budget utilization of the consensus bundles for different
number of valid bundle combinations and in-degree. The budget utilization for
the consensus bundles for Ursynow is the highest.

Figure 4 shows the popularity index (fitness of the consensus) for different
number of bundles and in-degrees. In case of Ursynow, the consensus bundles
have more projects and a higher percentage of popular projects (0.65 to 0.70),
that also have medium costs, which results in higher overall budget
utilization. The percentage of popular projects is low for Rembertow (0.40 to
0.45) and selected popular projects have considerably high costs too, as the
overall budget utilization is also low (see Figure 3).

Figure 4: Popularity index (fitness of consensus) for different number of
valid bundle combinations and in-degrees.

Comparison with other aggregation methods: The overlap of projects between the
consensus bundle (using the maximum number of bundles for action space) and
the winners from the aggregation methods are calculated (see Table 3). When a
larger number of projects are listed, e.g. Rembertow, the overlap with
consensus bundle is higher with equal shares (0.62) and Phragmen (0.61).
Hence, these methods maximize fairness and representation and also produce
more legitimate winners. For a lower number of projects, e.g. n Ruda, greedy
has higher overlap (0.72) with the consensus bundle. The reward based
iterative learning with communication can reach a consensus, which has a
higher overlap with aggregation methods that promote fairness for a higher
number of listed projects.

Table 3:  The overlap between the projects in the consensus bundle and three aggregation methods: greedy, equal shares and phragmen. The highest possible number of valid knapsack bundles are used. G: Greedy, PG: Phragmen, ES∗: Equal Shares*, MARL-PB: Proposed approach Dataset | Size of Consensus Bundle | MARL-PB Overlap
---|---|---
| MARL-PB | ES | PG | G | ES | PG | G
Rembertow | 8 | 9 | 9 | 6 | 0.62 | 0.61 | 0.49
Ursynow | 8 | 8 | 8 | 6 | 0.72 | 0.75 | 0.73
Ruda | 7 | 8 | 8 | 5 | 0.62 | 0.66 | 0.72
$*$ phragmen completion method used for the method of equal shares.

Analysis of the reward modelling: The top-3 amenities associated with all
projects (listed and selected) over all years in Ursynow are public space
(22%), education(17%), environmental protection (12%), impact on children
(22%) adults (21.1%) and seniors (19%). Similarly, for Rembertow and Ruda the
most popular ones are public space (24%) and education (22.7 %). These project
attributes affect a large proportion of the population. Figure 5 shows the
amenities selected via MARL-PB and the aggregation methods, as well as how
they compare with the original aggregate preferences based on the past data.
The projects selected using equal shares and MARL-PB correspond to similar
public amenities (e.g. for Rembertow, projects related to public space,
education and culture are selected in higher proportion). This also signifies
that consensus projects in MARL-PB prioritize fairness and better
representation. The public amenities selected in the greedy method do not
correlate with the ones selected in the consensus for Rembertow and Ruda. It
can be observed for Ursynow that the selected projects for any aggregation
method and with consensus mostly conform. Collective preferences for these
projects remain stable over time in this region.

Figure 5: The attributes of the project selections with the different methods.

###  4.2 RQ2: Level of Flexibility to Compromise and Reach Consensus

Figure 6 compares the the mean voters’ compromise of MARL-PB with the one of
different aggregation methods. The mean cost of compromise among the three
districts for greedy is 0.56, while for MARL-PB is 0.68, which is close to the
one of equal shares and Phragmen with 0.66 and 0.62 respectively. These
results show the following: Consensus requires compromise that is not observed
in the standard greedy voting aggregation method, however, this compromise is
attainable and comparable to the one observed with consensus-oriented voting
aggregation methods.

Figure 6: Mean voters’ compromise cost of MARL-PB and the different voting
aggregation methods. The coefficient of variation (COV) measures unfairness,
which is how compromises spread within the voters’ population.

The mean unfairness of MARL-PB is 0.17, while greedy is 0.19. The equal shares
and Phragmen have an unfairness of 0.16 and 0.15 respectively. The case of
Ruda does not align with the results of the other districts and this is likely
an artifact of the lower number of projects.

###  4.3 RQ3: Efficiency and Robustness

Convergence: An increase in the number of agents increase the converge time
for any combination of in-degree or action (bundle) space (see Figure 7). The
iterations increase by 5% on average for an increase of 50 voters. Although
this increase is only based on the data from the three districts, the system
demonstrates to be scalable by converging within finite time as the number of
voters increases.

Figure 7: Convergence time increases as the number of agents increases.
Significance values (p) from t-test (iterations, agents): Rembertow: p = 0.04,
Ursynow: p = 0.04, Ruda: p = 0.04. The values here are averaged over the three
districts.

Figure 8 shows for each district the convergence time as function of in-degree
and bundles size. Smaller bundles with higher in-degrees improve the speed.

Figure 8: Convergence time decreases for smaller action (bundle) spaces and
higher in-degrees: In-degree vs. iterations (significance values (p) from
t-test): Rembertow: p = 0.001, Ursynow: p = 0.002, Ruda: p = 0.03). Bundle
size vs. iterations (significance values (p) from t-test): Rembertow: p =
0.01, Ursynow: p = 0.001, Ruda: p = 0.04.

Robustness: The influence of the randomness in the dynamic communication
network on the stability of convergence is assessed by repeating the learning
process multiple times, with different size of bundles. Figure 9 shows the
required number of repetitions for a stable convergence speed. More
repetitions are required for lower in-degrees due to limited information
exchange for deliberation. A higher action space (number of bundles) results
in a larger number of alternatives to explore and thus stability requires a
higher number of repetitions.

Figure 9: Number of repetitions (simulations) required to reach the same
consensus for a certain set of parameters. In-degree vs. simulations
(significance values (p) from t-test): Rembertow: p = 0.03, Ursynow: p = 0.03,
Ruda: p = 0.04. Action (bundle) space vs. simulations (significance values (p)
from t-test): Rembertow: p = 0.01, Ursynow: p = 0.02, Ruda: p = 0.04. The
number of projects in a bundle does not have significant impact on the
simulations (Rembertow: p = 0.09, Ursynow: p = 0.06, Ruda: p = 0.13).

##  5 Conclusion and Future Work

This paper concludes that a consensus-based participatory budgeting process,
with three use cases introduced, is feasible via a novel multi-agent
reinforcement learning approach. The consensus process actually models a more
systematic, large-scale and automated deliberation process, which has so far
remained decoupled from the collective choice of voting. The experimental
evaluation with real-world data confirms that that the studied consensus is
reachable, efficient and robust. The results also demonstrate that the
consensus in MARL-PB requires compromises from voters, which are though
comparable to the ones of existing voting aggregation methods that promote
fairness and inclusion.

This is a key result with impact and significant implications: voters may not
need in the future to rely anymore on a top-down arbitrary selection of the
aggregation method. Instead, communities will be empowered to institutionalize
and directly apply independently their own consensus-based decision-making
processes. Moreover, city authorities may use the proposed method to filter
out projects during the project ideation phase, which usually relies on
subjective criteria with risks on legitimacy.

As part of future work, the agent communication may expand to different
dynamic topologies that represent more closely social networks and proximity.
The expansion of the multi-agent reinforcement learning approach with other
preferential elicitation methods [7], beyond approval voting, is expected to
further strengthen the accuracy and legitimacy of consensus-based
participatory budgeting. A more advanced design of the rewards scheme will
further expand the applicability of this ambitious approach.

## Acknowledgements

This work is supported by a UKRI Future Leaders Fellowship (MR/W009560/1): ‘
_Digitally Assisted Collective Governance of Smart City Commons–ARTIO_ ’, and
the SNF NRP77 ‘Digital Transformation’ project “Digital Democracy: Innovations
in Decision-making Processes”, #407740_187249.

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