# A Bayesian Agent-Based Framework for Argument Exchange Across Networks

Leon Assaad<EMAIL_ADDRESS>Munich Center for Mathematical
Philosophy, LMUGeschwister Scholl Platz 1MunichBavariaGermany80539 ,  Rafael
Fuchs  Graduate School of Systemic Neuroscience, LMUGroßhadernerstraße 2
Planegg-MartinsriedBavariaGermany82152 ,  Ammar Jalalimanesh  ,  Kirsty
Phillips  Birkbeck College, University of LondonMalet StreetLondonUnited
KingdomWC1E 7HX ,  Leon Schöppl  Munich Center for Mathematical Philosophy,
LMUGeschwister Scholl Platz 1MunichBavariaGermany and  Ulrike Hahn  Birkbeck
College, University of LondonMalet StreetLondonUnited Kingdomand Munich Center
for Mathematical Philosophy, LMUGeschwister Scholl Platz 1MunichBavariaGermany

(14 Nov, 2023)

###### Abstract.

In this paper, we introduce a new framework for modelling the exchange of
multiple arguments across agents in a social network. To date, most modelling
work concerned with opinion dynamics, testimony, or communication across
social networks has involved only the simulated exchange of a single opinion
or single claim. By contrast, real-world debate involves the provision of
numerous individual arguments relevant to such an opinion. This may include
arguments both for and against, and arguments varying in strength. This
prompts the need for appropriate aggregation rules for combining diverse
evidence as well as rules for communication. Here, we draw on the Bayesian
framework to create an agent-based modelling environment that allows the study
of belief dynamics across complex domains characterised by Bayesian Networks.
Initial case studies illustrate the scope of the framework.

Agent-Based Model; Argumentation; Bayesian Argumentation; Argument Exchange

††copyright: none

##  1\. Introduction

Human societies are based on information exchange, deliberation, and
negotiation. This means human societies rely fundamentally on argumentation.
As a result, argumentation –broadly construed– is a topic of active research
interest across a wide range of disciplines.

Some of these, such as research on argumentation (Walton, 2009) and research
on persuasion (Maio et al., 2018) have tended to focus on detailed
characteristics of individual arguments. Others, such as research in
computational social science (Lazer et al., 2009) studying large-scale debates
across online social platforms like Twitter or Facebook, have focussed in
detail on the spread of arguments (Hofman et al., 2021)–with the arguments
themselves subjected to far more coarse-grained analysis in terms of keywords
or sentiments (e.g., (Berger and Milkman, 2012)). Research focussed on spread
includes also agent-based modelling of belief- or opinion dynamics. Here,
‘arguments’ have been highly stylised -represented only by numbers or elements
of a vector (Mäs and Flache, 2013). Or they have been captured only implicitly
by their effects, as in approaches that model opinion dynamics as a kind of
‘contagion’.

This disconnect between research traditions focussed on individual arguments,
and research traditions focussed on dynamics of spread has left a fundamental
gap in the current understanding of how the exchange of arguments figures in
human society. Moreover, this gap encompasses pressing theoretical and
practical questions, for example concerning the impact of large online
platforms on democratic debate, and with it, the health of democratic
societies.

Bridging that gap will, arguably, require bringing together tools and theories
of research traditions that have focussed on individual arguments with those
concerned with characteristics of spread. For example, both argumentation and
persuasion research have historically focussed on dyads: communicating pairs
exchanging reasons for claims in an ongoing exchange that involves competing
and supporting arguments that combine in complex, often hierarchically nested
ways. Researchers have sought to understand both argument ‘quality’ and
persuasive ‘success’ in that dyadic frame of reference, developing both
procedural rules for engagement (Van Eemeren and Houtlosser, 2003; Van Eemeren
et al., 2013, 2015) and graphing techniques or ‘maps’ to aid argument
evaluation and production (Gordon et al., 2007). This scales only partly to
contexts with multiple, and possibly large numbers of, communicating agents
(see also (Bonevac, 2003; Lewiński and Aakhus, 2014)) and the historical focus
on dyads has left fundamental questions about argumentation and persuasion
unaddressed. Conversely, the insights that might be gained from analysing the
dynamics of the spread of arguments across large corpora of public debate are
restricted by the level of content analysis applied to the arguments
themselves.

The research presented in this paper aims to help bridge this gap.
Specifically, we introduce a new framework, NormAN –short for Normative
Argument Exchange across Networks– for agent-based modelling of argument
exchange across social networks. This framework, we argue, combines important
features of argumentation research on argument evaluation with agent-based
modelling. Specifically, its goal is to incorporate multiple fundamental
features of real-world communication: In discussing or debating a claim,
individuals exchange arguments (individual reasons) for believing that claim
to be true or false. Some of these may be arguments for, others arguments
against, and some of these arguments may be better than others. And while some
of this may be a matter of subjective evaluation, what arguments are available
and how strong they are is also constrained by the topic at hand. Finally,
communicative exchanges might take place in anything from small, tightly knit
groups exchanging in-depth information, to large networks involving only
fleeting exchange. This means understanding real-world arguments also requires
understanding the impact of who agents communicate with and what they choose
to exchange.

No single model, let alone a single investigation, will be able to give equal
focus to each of these fundamental features of real-world argumentation. As a
result, NormAN is designed as a framework. In this paper, we set out this
framework and introduce a basic model, NormAN version 1.0, that incorporates
each of these core features. Specifically, the paper proceeds in three main
parts. We first (2) briefly describe research across both sides of the ‘gap’
in order to situate NormAN in the context of current research and motivate our
specific design choices. We then introduce the framework and NormAN v. 1.0 in
more detail (3). Finally, we describe two case studies to illustrate the
features and benefits of the framework. In particular, we seek to show how
this framework –though still highly stylised in nature– affords both deeper
understanding of longstanding questions, and opens up new avenues for
research.

##  2\. Motivation and Background

The goal of NormAN is to bring together the detail of research on argument
evaluation and the simulation of multi-agent contexts in order to bridge the
current gap and enrich both research traditions. We next provide background on
the most important strands within these respective traditions.

###  2.1. Argument Quality and Dialectics

Traditional research on argumentation has not used agent-based simulations.
Rather, this highly interdisciplinary field has drawn on observation, formal
analysis, and behavioural experiments.

####  2.1.1. The Breadth of Argumentation Research

Philosophers have focussed on normative theories, that is, theories of how we
_should_ behave. The traditional standard has been formal logic, but more
recently, pragma-dialectical theories have focussed on the norms and
conventions governing argumentative process (e.g., (Van Eemeren et al., 2013,
2015; Walton, 1998; Walton and Godden, 2007)). Within psychology, ‘persuasion’
has been a central topic of social psychological research (e.g., (Eagly and
Chaiken, 1993)). This vast literature has identified many moderating variables
(e.g., speaker likeability, engagement, mode of presentation, fit with prior
beliefs) that affect the degree to which persuasive communication will be
effective. Developmental and education research has focussed on the way
children’s argumentation skills develop and examined ways in which critical
thinking and argument skills might be improved (e.g., (Felton and Kuhn, 2001;
Kuhn and Udell, 2003; Von Aufschnaiter et al., 2008)). Logicians and computer
scientists have sought to devise argumentation frameworks for dealing with
dialectical information, seeking to capture the structural relationships
between multiple theses, rebuttals, and supporting arguments for use in
computational argumentation system (Dung, 1995; Prakken and Vreeswijk, 2001;
Rahwan and Simari, 2009).

####  2.1.2. The Central Role of Normative Concerns

The sheer breadth of disciplinary perspectives, research questions, and
methods makes for a bewildering array of literatures and findings on
argumentation. Furthermore, many of these literatures have historically been
largely or even wholly disconnected from one another. There is, however, a
shared focal concern across most, if not all, argument research. This is the
issue of argument quality or ‘what makes a good argument?’, and, with that,
the question of how good arguments can be distinguished from bad ones. This
question is a normative, evaluative, question about what kinds of arguments
should convince us, and which are the appropriate normative standards against
which argument quality should be judged. Across fields and research interests,
this question features both as an explicit topic of study and as an implicit
concern.

It is of explicit interest within philosophy in research on human rationality
and the epistemological question of how we can arrive at secure knowledge of
the world ((Rescher, 1977; Dawid et al., 2015; Hartmann, 2021; Eva and
Hartmann, 2018; Godden and Zenker, 2018). In psychology, cognitive
psychologists study the quality of people’s argumentation (e.g., (Kuhn, 1991)
as part of a long tradition of research on reasoning, judgment and decision-
making (e.g., (Oaksford and Chater, 2009; Stanovich and West, 2000; Kahneman,
2011; Hahn and Oaksford, 2012)). And educational psychologists teaching or
improving argument skills and critical thinking (e.g., (Van Eemeren and
Houtlosser, 2003)) must clarify their intended target.

In other research, the question of argument quality is raised _implicitly_ by
research goals and methodological constraints. For example, argument quality
matters for logicians and computer scientists interested in argumentation as a
tool for artificial intelligence systems (e.g., (Jackson, 1986; Neapolitan,
1990)), because, to work well, such systems must adequately weigh and
aggregate information. So how can argument quality be measured? What normative
standards might be devised?

####  2.1.3. Standards for Argument Quality

A wide range of tools, from different disciplines, has historically been
applied to the question of what makes a ‘good’ argument:

  1. (1)

classical logic

  2. (2)

attempts to elucidate argument by mapping out structural relations between
arguments:

     * •

either informally by tagging them as ‘claims’, ‘warrants’, or ‘rebuttals’
(e.g., (Toulmin, 2003))

     * •

or in formal, computational frameworks (e.g., so-called ‘argumentation
frameworks’, (Dung, 1995))

  3. (3)

pragma-dialectical theories spelling out putative norms underlying
argumentative discourse, such as a ‘right to reply’ or ‘burdens of proof’ (Van
Eemeren et al., 2004)

While all of these are useful and aid interesting research questions in
different fields, they still miss much about argument quality.

Classical logic says nothing about most everyday informal arguments, other
than that they are not logically valid (Toulmin, 2003; Hamblin, 1970), and,
hence, it is too restrictive. 111At the same time, it is too permissive in
that it renders arguments strong that actually seem poor: For example, ‘A,
therefore, B or not B’, where A is wholly irrelevant rather than providing a
meaningful reason. Likewise, the quality of argument content cannot generally
be reduced to procedural rules or to systems that map out support, attack and
defeat relations. To illustrate: “the book is in the library…no it’s not,
because the moon is made of cheese” involves an (intended) counter-argument,
but is patently absurd (Hahn, 2020). Simply noting that an argument is
_offered_ as support or attack is itself a purely structural, syntactic
evaluation. A content-based measure of argument strength is still needed in
order to know whether intended ‘support’ or ‘defeat’ is successful. Likewise,
pragma-dialectic notions such as ‘burden of proof’ depend on the specific
content of an argument in order to determine whether or not a burden of proof
has actually been met (Hahn and Oaksford, 2007).

This means that normative standards in addition to classical logic, procedural
rules or merely syntactic renditions of the structural relations between
arguments are necessary in order to capture argument content adequately. This
has recommended a Bayesian approach to argumentation.

####  2.1.4. Bayesian Argumentation

The probability calculus is intensional (Pearl, 1988): the probabilities that
attach to propositions are determined by their specific content, not (just)
their logical form. The resulting ability of the probability calculus (or,
where decisions and utilities are involved, Bayesian decision theory) to
meaningfully capture normative questions about argument content is
demonstrated by its application to the catalogue of so-called fallacies of
argumentation. The fallacies are argument schemes such as ‘arguments from
ignorance’, ‘ad hominem arguments’ etc. that have long posed a challenge for
explanations of why exactly they are poor arguments (see (Woods, [n. d.];
Hamblin, 1970)). One central difficulty encountered here was that not all
instances of these schemes seem equally poor or fallacious, and a Bayesian
treatment has elucidated those differences (Hahn, 2020).

The Bayesian framework has also been applied to a broader set of schemes for
everyday argument from the informal logic literature that, unlike the
fallacies, are presumptively (but defeasibly) ‘good’ arguments. Specifically,
they provide reasonable, albeit defeasible, inferences for uncertain,
ampliative reasoning (which sets them apart from logical schemes such as the
classic set of syllogism or conditional reasoning schemes such as modus
ponens, modus tollens etc.). The literature on informal argument previously
catalogued 60+ such schemes (Walton et al., 2008) that identify recurring
structures that occur with varying content (and hence varying strength) in
everyday discourse. As (Hahn and Hornikx, 2016) seeks to show, the Bayesian
framework can provide a normative basis for these schemes. It can thus further
the long-standing goals of research on argument schemes, namely a
computationally explicit treatment with guidance for lay reasoners on when
particular instances of these schemes are weak or strong.

In the Bayesian framework, argument strength can be captured by considering
the extent to which an argument or piece of evidence rationally changes one’s
beliefs. The posterior, $P(C|A)$, is affected by the likelihood (i.e., the
sensitivity of the evidential test $P(A|C)$), and by the false positive rate
(i.e., $P(A|notC)$) as captured in the likelihood ratio (i.e.,
$P(A|C)/P(A|notC)$).

With the likelihood ratio, the Bayesian framework has a notion of
informational relevance. This helps with the fallacies, given that fallacies
are typically fallacies of relevance (Walton, 2004). It is also essential to
capturing argument quality in general, and elucidating the notion of relevance
in a formally satisfactory, non-question-begging, way has been a long-standing
challenge (see, (Sperber and Wilson, 1986; Hahn and Oaksford, 2006)). Finally,
the Bayesian framework has a well-developed normative foundation that links to
goals such as inaccuracy minimisation (on the link between ‘being Bayesian’
and inaccuracy minimisation see (Pettigrew, 2016), for discussion of normative
foundations for argumentation more generally, see (Corner and Hahn, 2013)).

Bayesian Argumentation has also been expanded to other features of argument
(e.g., such as ‘argument cogency’ or ‘degrees of justification’ (Zenker, 2012;
Godden and Zenker, 2018)). At the same time, work by Hartmann and colleagues
has extended the formal arsenal of Bayesian Argumentation in order to broaden
the scope of possible inferences (Eva and Hartmann, 2018; Eva et al., 2020)
and has provided detailed treatments of scientific inference schemes (e.g.,
(Dawid et al., 2015)) in a programme paralleling the treatment of everyday
schemes. Specifically, Hartmann and colleagues have shown how forms of new
‘evidence’ not amenable to Bayesian conditionalization may be captured through
the application of Kullback-Leibler divergence.

The body of work on Bayesian Argumentation arguably represents the state of
the art with respect to measuring argument quality, in that both a
quantitative measure and a well-developed normative basis is provided (see
also (Nussbaum, 2011)). It is for this reason that we adopt the Bayesian
framework for NormAN.

While a Bayesian perspective on argument quality has arguably been productive,
there are key features of argumentation –discussed next– that have been
neglected to date, not just by Bayesian argumentation, but by argumentation
research as a whole.

###  2.2. Beyond Dyads: Multi-Agent Models

As noted above, most work on argumentation has, at best, concerned itself with
dyads, that is, a proponent and an opponent engaged in direct exchange. Public
debate, however, has many actors choosing when to contribute and when not,
which arguments to repeat, which to ignore, which to address, and how. This
fundamental feature of real-world argument has remained largely outside the
view of argumentation research.

Even where argumentation research has concerned itself with large-scale
debates, it has either attempted to assimilate these into dialogue-oriented
models (Lewiński and Aakhus, 2014) or it has focussed exclusively on the
arguments themselves (e.g., in argument mapping approaches to large-scale
debates such as Kialo222See https://www.kialo.com; on such tools more
generally see e.g., (Benetos, 2023)). This obscures all sense of the
_dynamics_ of argument exchange in public debate and the many underlying
decisions by participants that give rise to those dynamics.

The dynamics of debate, however, have become a matter of research interest
with the advent of online social media. With online social media came the
ability to harvest and analyse large volumes of real-world debate. And the
rise of computational social science has seen the analysis of online data from
large platforms such as Twitter, Facebook or Reddit become a major topic of
research (Cioffi-Revilla, 2014; Lazer et al., 2009). At the same time,
putative negative impacts of large online platforms on misinformation,
polarization, extremism, and a weakening of democratic norms (Lorenz-Spreen et
al., 2020; Lewandowsky et al., 2020; Lorenz-Spreen et al., 2023) has fuelled
interest in belief and opinion formation across social networks. This has led
to a wealth of modelling research to help understand how opinions spread
across networks.

There remains, however, a significant gap: the analysis of real world data
from platforms such as Twitter has largely focussed on limited features of
such data, focussing either on spread by analysing retweets (Java et al.,
2007; Suh et al., 2010; Ten Thij et al., 2014; Cha et al., 2010) or analysing
content in very restricted ways such as sentiment analysis (Hutto and Gilbert,
2014), bags of words (Naveed et al., 2011; Brady et al., 2017; Storey and
O’Leary, 2022) and/or topic modelling (Zhao et al., 2011; Corti et al., 2022)
(but see also more recently e.g., (Visser et al., 2020)). This is a far cry
from the detailed analyses of content common within the research tradition
concerned with argument quality outlined in the previous section. And, as the
following sections will show, models of belief or opinion-dynamics are
arguably even more restrictive: At present, most ABMs do not involve the
communication of reasons for claims. In other words, they do not capture
argument at all.

####  2.2.1. Models of Opinion Dynamics

The modelling of opinion dynamics has seen multiple frameworks. Two of these
import concepts from other disciplines: contagion and social physics models.
Contagion-based models, in effect, treat the spread of opinions or behaviours
as a kind of “infection” (López-Pintado, 2008; Barash, 2011; Centola, 2018).
This allows models from epidemiology to be applied. Contagion based models
have been used to examine everything from basic contagion dynamics (Barash,
2011; Izquierdo et al., 2018), effects of network structure (Jackson and
Rogers, 2007; López-Pintado, 2008), the influence of word of mouth reports of
events on elections (Moya et al., 2017), extremism (Youngblood, 2020), to echo
chambers and viral misinformation (Törnberg, 2018). Methods have ranged from
standard analytic models within epidemiology (see e.g., (Kiss et al., 2017)),
through statistical models, to agent-based modelling.

The social physics approach draws on models developed within physics to
capture opinion dynamics (Castellano et al., 2009). In particular, methods
(e.g., mean field approximation) and models from statistical mechanics, such
as diffusion models and the Ising model (Dorogovtsev et al., 2008), have been
used to model issues such as opinion polarization (Macy et al., 2003) or the
spread of misinformation (Budak et al., 2011).

Finally, one of the earliest and one of the most influential models of opinion
dynamics was first put forward by statistician M. DeGroot (DeGroot, 1974). The
DeGroot model was proposed initially to shed light on how groups might use
opinion pooling to reach a consensus judgement. It is based on repeated
(weighted) belief averaging until beliefs converge. Iterated averaging also
underlies the Lehrer-Wagner (Lehrer and Wagner, 1981) model in philosophy that
has been used extensively to develop notions of rational consensus, and the
work of Hegselmann and Krause (e.g.,(Hegselmann and Krause, 2002)), which we
discuss further in the next section.

With respect to motivating NormAN, we note two main observations about models
of belief- and opinion dynamics discussed so far: First, they use an
unanalysed aggregate –the opinion or belief in question, typically represented
as a boolean or continuous variable– without the provision of reasons; this
limits the research focus of such models to the population dynamics regarding
that single quantity. This renders this body of work impossible to connect
meaningfully to the research on argumentation described in section 2.1 above.
Second, there is no ‘ground truth’ at stake in these models (but for an
addition of ‘ground truth’ to the DeGroot model see Golub and Jackson (Golub
and Jackson, 2010)). Hence many questions about knowledge and accuracy, of
either individual agents or the collective as a whole (Hahn, 2022), are
necessarily outside the scope of these models.

####  2.2.2. Agent Based Models in Social Epistemology

Questions of how knowledge comes about and how it comes about specifically in
social contexts are, however, the central concern of social epistemology.

Considerable research within social epistemology has utilised the DeGroot
model—either in the form of the Lehrer-Wagner (1981) or the Hegselman-Krause
model (Hegselmann and Krause, 2002, 2015). Hegselman-Krause added the idea of
weights reflecting differential ‘trust’ in other members of the collective
such that agents only listen to others who are sufficiently ‘close’ in their
estimates (giving rise to so-called convergence threshold models (Hegselmann
and Krause, 2015)). Work using these models has focussed on understanding when
networks do and do not converge (see for extensive analysis, (Krause, 2015)).

In order to connect better to the concerns of social epistemology, Hegselman
and Krause (Hegselmann et al., 2006) later also added to their model the idea
that a specific opinion value may be designated as ‘the truth’ (for other
extensions see (Douven and Riegler, 2010), including, outside of social
epistemology, toward greater psychological realism (Xu et al., 2023); for a
review, (Douven, 2019a)).

An influential further class of models in social epistemology is bandit models
(Zollman, 2007, 2010). These models use one- or multi-armed bandits (Slivkins
et al., 2019) to generate evidence about an underlying state of the world.
That evidence may be observed directly by individual agents or indirectly
through observation of other agents’ states based on aggregates of that
evidence, or received via communication. Used initially by economists to study
social learning across networks and its effect on economic outcomes (Bala and
Goyal, 1998, 2000), bandit-based models have been applied in social
epistemology to questions such as when (and when not!) communication is
beneficial to the progress of science (Zollman, 2010), the effects of
misinformation (O’Connor and Weatherall, 2018), and polarization (O’Connor and
Weatherall, 2018). Although they often involve Bayesian updating at the
individual agent level, bandit models have also been combined with DeGroot-
like averaging (Douven, 2019b). Conceptually, bandit models allow there to be
a model ground truth, and the evidence dispensed by the bandit provides at
least a very limited notion of ‘argument’.

A different model aimed at many of the same research questions is the model of
testimony first proposed by Olsson and colleagues (Olsson, 2011; Olsson and
Vallinder, 2013; Olsson, 2013; Angere and Olsson, 2017). The realisation that
much of what humans believe to know stems from the testimony of others (Coady,
1992), has fuelled research concerned with the conditions under which
testimony is reliable and a meaningful guide to truth. A significant
proportion of that work has drawn on probability theory to explicate those
conditions in formal models (Olsson, 2005; Bovens and Hartmann, 2003; Olsson
and Schubert, 2007) including agent-based simulations. The Olsson (2011) model
is such an agent-based model.

As it has inspired many of the features of the new framework presented in this
paper, we outline it in some detail here. In the model, there is a single
proposition (represented as a Boolean variable) at issue. Agents in the model
each occupy a particular position in a social network. At each time step,
there is a probability of acquiring a piece of evidence ‘from the world’, and
a probability of communication. Communication links are symmetrical, and
communicating agents will affirm that proposition C, the target hypothesis
under scrutiny in this artificial society, is true, whenever their degree of
belief in C exceeds a threshold of assertion (say, p = .8). When belief drops
below 1 minus the threshold of assertion, agents will assert not-C; on all
other occasions they will remain silent. This is designed to capture the fact
that real-world communication does not typically involve communication of
point values, but rather involves the assertion of claims (‘C is true’). The
agents in the model are Bayesian, using Bayes’ rule to revise both belief in
the target hypothesis and the reliability of their sources (including their
own inquiry into evidence from the world).333More precisely they are naive
Bayesian agents in that they make the simplifying assumption that evidence is
independent, (Ng and Jordan, 2002). For an examination of the consequences of
this assumption see (Merdes et al., 2021; Hahn, 2023).

In the Olsson model, agents have a belief (in the claim at issue) and there is
a ground truth. There is also a very simple type of ‘argument’ or evidence
which consists solely of assertion that the claim in question is true or
false.

The Olsson model has been used, among other things, to study the impact of
network topology on accuracy (Hahn et al., 2018a), polarization (Olsson, 2013,
2020; Pallavicini et al., 2021), the impacts of different strategies for
estimating the reliability of testimonial sources (Hahn et al., 2018b; Collins
et al., 2018) and the dependence created through communication (Hahn et al.,
2019). This includes the role of communication in the context of science,
specifically asking whether the overall progress of science is helped or
hindered by frequent communication between scientists (Angere and Olsson,
2017).

This latter question has also been studied in so-called epistemic landscape
models (Weisberg and Muldoon, 2009; Pinto and Pinto, 2018; Grim et al., 2013).
These models capture scientific exploration of a topic by agent-based probing
of a fitness landscape: The boundaries of the landscape represent the
boundaries of the topic; the coordinates of the landscape correspond to
different approaches scientists could be bringing to its study and the
topography of the landscape represents the relative ‘significance’ of the
resultant scientific work.

The recent argumentation-based ABM of (Borg et al., 2019) represents yet a
further attempt to study the same problem. In this model, agents seek to
explore an underlying argument map setting out the relevant arguments on the
issue. In effect, the ‘epistemic landscape’ is the argument map. This argument
map is formulated in the abstract argumentation framework of (Dung, 1995) and
agents exchange information about arguments they have encountered. This allows
investigation of the impact of different argument selection and communication
strategies by agents with respect to achieving full end-state knowledge of the
relevant state of affairs.

In both epistemic landscape models and the Borg et al. argumentation-based
ABM, there is a ‘ground truth’ of sorts implicit in the model. However, the
design of the underlying ‘landscape’ (whether the fitness landscape or the
argument map) is essentially arbitrary and unconstrained. To the extent that
properties of that landscape matter to model behaviour and findings, results
will remain somewhat difficult to interpret –in particular, with respect to
how they generalise to real-world situations. At the same time, however, there
is no explicit representation of agent beliefs regarding a particular target
hypothesis, which separates these models fundamentally from models of opinion
dynamics.

To fully join dyadic models of argument with opinion- and belief dynamics, a
modelling framework that distinguishes between arguments, aggregate beliefs or
opinions, and the social network across which communication is happening is
required (for related points see also (Grim et al., 2013)). We return to these
issues below.

####  2.2.3. Multi-Agent Models of Argumentation

Finally, the exchange of arguments between computational agents has been a
focal point for research on multi-agent-systems (for introduction and
overviews to multi-agent-systems see e.g., (Dorri et al., 2018; Van der Hoek
and Wooldridge, 2008)). Much of the modelling here has involved logical
formalisms of one kind or another (Calegari et al., 2021; Chesnevar et al.,
2000), though other argumentation frameworks such as Dung’s abstract
argumentation framework (Dung, 1991) and extensions thereof (e.g., (Bench-
Capon, 2002)) have also been used (see for an overview of relevant approaches
(Rahwan et al., 2003; Carrera and Iglesias, 2015)). And there has been some
(but comparatively limited) interest in the Bayesian framework (e.g., (Saha
and Sen, 2004; Nielsen and Parsons, 2007; Vreeswijk, 2004)).

Both the tools used for capturing argument and some of the research questions
asked have connections to traditional (dyad focussed) argumentation research
as described in section 2.1. Not only are there researchers that have
contributed to both communities, input has specifically been sought from non-
computational argumentation researchers (see for illustration of this point
e.g., (Rahwan and Simari, 2009)). However, the different focus of most
research on argumentation for autonomous agents means that this body of
research does not ultimately connect well to research on belief- or opinion
dynamics, or to research concerned with the spread of arguments (at least at
present). This stems from the fact that multi-agent-systems research typically
has in mind practical applications for which collections of agents provide a
potential computational solution. This makes a top-down approach to systems
involving many agents the natural focus. The goal is to solve complex
computing problems, and multi-agent systems, as a type of distributed
artificial intelligence, provide a potential tool. By contrast, most of the
research discussed thus far is interested in the bottom-up question of
dynamics and patterns that emerge naturally from groups of interacting agents.

###  2.3. The Value of Normative Models

The preceding discussion should have made clear that there is a particular
value to normative models in the context of argumentation research. In the
context of individual-focused, or (at best) dialectical research on
argumentation the explicit and implicit normative focus is clear (section
2.1.2 above). Not only have normative issues been of direct, explicit,
interest, but normative models have methodological value even in otherwise
purely descriptive endeavours.

Specifically, the new (to argumentation research) normative standard provided
by Bayesian probability not only addressed long-standing theoretical,
philosophical concerns (e.g., about fallacies of argumentation), it also
opened up novel paths for empirical, behavioural experimentation examining
laypeople’s reasoning (e.g., (Corner et al., 2011; Bhatia and Oaksford, 2015;
Corner and Hahn, 2009; Harris et al., 2012; Hornikx et al., 2018)). And the
specificity of research questions pursued in those studies goes considerably
beyond what was possible with normatively limited frameworks such as the
Toulmin framework (see also (Hahn and Tešić, 2023) for further discussion of
this point).

In the context of multi-agent models, the importance of normative concerns
should also be clear. Again, there is considerable interest in normative
accounts of collective discussion and information exchange in social
epistemology. Likewise, there is considerable interest, for example, in
improving online platforms, and for such endeavour an understanding of what is
possible, in the best case, is important. Finally, normative models are
important in making sense of descriptive data which all too often are simply
assumed to reflect bias and irrationality whenever data patterns seem
surprising (as illustrated, by the literature on shifts to extremity in group
discussions or polarization, both examined in more detail below, C.1 and C.2).

At present, however, there is a large gap in that there are no normative
models of argument exchange across collectives that would allow researchers to
address issues such as accuracy and link up to classic, individual- and dyad-
focussed research on argument.

Crucially, in order to achieve that, two components of the model need to have
a normative grounding: argument evaluation and aggregation on the one hand,
and, on the other, a grounding of the evidence distribution in a ground truth
world model, against which beliefs can be compared and scored.

Both the evaluation/aggregation rules used by agents and the distribution of
(in principle) available evidence will affect belief dynamics. Consequently,
making both of these aspects principled (and not ‘just so’) seems of
fundamental importance for connecting to real-world contexts in meaningful
ways.

This gives rise to the following desiderata for an agent-based model of
argument exchange. What is required is a model with (i) a ground truth world,
(ii) evidence linked to that ground truth world, giving rise to a principled
evidence distribution, and (iii) rational agents who form optimal beliefs
given the evidence they have. Furthermore, such a model should be easy to use
and extend. NormAN seeks to provide a general framework for creating just such
models.

##  3\. Introducing the NormAN Framework

The core conceptual components of the NormAN framework are illustrated in
Figure 1. It comprises the three core elements that make up a NormAN model: a
ground truth’ ‘world’, individual ‘agents’, and the social ‘network’ across
which these agents communicate. The ground truth world determines the true
state of the claim (hypothesis) at issue in the discussion, along with the
evidence for it that could be discovered in principle. Agents receive evidence
about that world (through inquiry) and may communicate that evidence to others
as arguments and receive it in turn.444The framework of Bayesian argumentation
elides the difference between evidence and arguments in as much as it models
arguments with the same basic machinery used to model other evidence (though
substantive differences may, of course, arise as a result of that
formalisation). For clarity, it can be helpful to restrict the term ‘evidence’
for information received in the model ‘from the world’ (whether through
initial assignment or subsequent inquiry), and ‘argument’ for communication of
that evidence. Agents aggregate all evidence/arguments that they have
encountered to form a degree of belief in the claim at issue. Communication,
finally, takes place across a social network (including a possible ‘null
network’ in which no communication takes place for comparison).

Figure 1. Illustration of the main components of NormAN. A model within the
NormAN framework specifies (1) a ground truth world, (2) a social network, and
(3) individual agents communicating across that network. Each of these three
components has core specifications. In addition, the square in the middle
“activity levels” refers to aspects that could variously be conceived of as
properties of world, network or agent. The assignment of model parameters in
the current version of NormAN (1.0) to these various model aspects is
described in the text.

This is an overview of NormAN. It shows the three core components ”world”
(blue box), ”network” (red box), and ”agent” (yellow box) along with the core
features/specifications of these components (green boxes).

NormAN sets out a general framework in as much as each of these core
components is inherently modifiable: users can modify the ‘world’, the
evidence received, aggregation rules, communication rules, and network
topology. Moreover, NormAN is designed this way because it is our contention
that all three components are essential in coming to a deep understanding of
argumentation, communication, opinion dynamics and deliberation. As a
consequence of this foundational assumption, even the initial release of
NormAN (version 1.0) already has sufficient flexibility to allow users to
define models selected from a broad space of models varying world, agent and
network characteristics.

Moreover, NormAN is freely available (see E) so as to allow users to readily
adapt and extend the code and create new models by modifying the core
components.

As outlined above (see 2.3), the absence of agent-based models capturing
argument exchange over a ground truth world is central to the current ‘gap’. A
key problem in developing NormAN was thus how to build a ground truth world.
Extant models such as the Olsson (2011) model or bandit models (e.g.,
(Zollman, 2007) ) utilise a simple binomial process to this end. The modeller
stipulates a particular ‘hypothesis’ (say ‘1’) to be true, and a binomial
process with probability $p$ –representing the accuracy of the evidence
source555Assuming $p(E|H)=p(E|notH)$, i.e., sensitivity and specificity of the
‘test’ are set to be equal, see also (Hahn et al., 2018b) for discussion of
the extent to which this does and does not limit generalisability of model
results.– produces a stream of 0s and 1s as ‘evidence’ which can then form the
basis of Bayesian updating on the part of the agents.

Rich argument exchange over a ground truth world requires extending this kind
of approach in appropriate ways. This means not just generating an evidence
distribution that is plausible vis a vis the real world, but also generating
evidence in such a way as to allow agents to form probabilistically coherent
beliefs (‘be Bayesian’) _at least in principle_. While one’s analytic
interests need by no means be limited to optimal agents (ours are not), the
mere possibility of implementing a rational agent exhibiting normatively
correct reasoning (or as close as possible to that) means one cannot simply
generate wholly unconstrained, arbitrary combinations of evidence, because
doing so may potentially generate evidence combinations over which no rational
belief is possible.

To satisfy the dual demands of a plausible world and meaningful evidence
distribution, NormAN adopts Bayesian Belief Networks (BNs) as a tool for
generating the world. BNs are graphical representations of multi-variable
relationships (Pearl, 1988, 2000; Korb and Nicholson, 2010; Scutari and Denis,
2021). They are widely used across many disciplines (ranging from computer
science, through statistics, philosophy, psychology and sociology, among
others) for both theoretical analysis and practical software and engineering
applications (e.g., (Kammouh et al., 2020)). Specifically, BNs summarise the
way variables do and, more importantly, do not influence one another in a
graphical representation that simplifies Bayesian calculations. BNs thus have
a normative (Bayesian) foundation and they connect to extant work on
argumentation (see section 2.1.2 above) including Bayesian models of argument
generation (Zukerman et al., 1998, 1999; Jitnah et al., 2000; Keppens, 2019;
Timmer et al., 2015). Furthermore, their use for statistical analysis (Salini
and Kenett, 2009) and decision-support systems (Fenton and Neil, 2018) mean
that there exist repositories of BNs (e.g., the bnlearn repository,
https://www.bnlearn.com/bnrepository/) that putatively capture real-world
structure within the respective application domain.

NormAN allows users to select a BN and use it to generate a suitable ground
truth world through a simple trick. A variable in the network is selected as
the target hypothesis or claim at issue; its value is set for the purposes of
one or more model runs to represent the ’true state of the world’ regarding
that claim. The resultant probabilities for the remaining variables (given
that true state) are then used to stochastically generate a body of evidence
that is available, in principle, in that ground truth world (for fuller
details see below). Agents may subsequently receive evidence from that world
and exchange what evidence they have received via communication. Conceptually,
this generating procedure encompasses the simple binomial processes used in
past models as a special case.

Finally, while the use of this generating procedure is an integral part of the
appeal or value of NormAN (at least to us), it should be noted that the
framework is general enough to allow incorporation of other ’world
representations’ beyond BNs. Agent-based models defined over arbitrary
argument graphs (such as (Borg et al., 2019), see Section 2.2.2), for example,
can readily be captured as a type of NormAN model that uses an argument graph
instead of a BN as an underlying world, and in which agents’ belief
aggregation is disabled.

The key feature of the basic NormAN agents (as implemented in version 1.0) is
that they optimally aggregate evidence via Bayes’ rule. To do so, they too,
draw on a BN, which in the basic version of the model is a veridical model of
‘the world’, that is, in essence, a matching (subjective) BN ‘in the head’
(future extensions of the model involve relaxing that requirement of veridical
match between world and model in the head and also the subjective models of
other agents). Crucially, agents must also communicate and NormAN version 1.0
implements a variety of different rules and constraints on what and when
agents communicate. This includes both rules motivated by past work (e.g.,
(Mäs and Flache, 2013)) and initial suggestions for other communication rules
in a multi-argument selection context.

NormAN also allows users to vary the structure of the communication network
across which the argument exchange takes place. In the current version, this
includes selecting from a range of network types, as well as basic network
parameters such as network size and link density (the number of connections
between agents).

Finally, the framework allows modellers to determine the relative balance
between evidence ‘from the world’ (both initial and as a result of ongoing
inquiry) and the amount of communication. This feature derives from the Olsson
(2011) model and is of importance because it has been shown to affect a
variety of outcomes from individual and collective accuracy (Angere and
Olsson, 2017; Hahn et al., 2019) to polarization (Hahn et al., 2023; Hahn,
2023).

We describe the framework and its current implementation in more detail next.
For a full technical description of NormAN 1.0 following the ODD protocol
(Grimm et al., 2010), see the Appendix (Section A).

###  3.1. Main Components

The core parameters of NormAN are shown in Table 1. We describe the world, the
agents, and the networks in turn.

Entity | Variable | Value range/Type | Description
---|---|---|---
World | causal-structure | Bayesian network | Determines evidence nodes
|  |  | and their causal relation to hypothesis node.
| hypothesis | Variable | The hypothesis proposition (truth values: true or false).
| hypothesis-probability | $0-1$ | Probability that hypothesis is true.
| evidence-list | List | Stores truth values of evidence nodes (values: true/false).
Agents | agent-evidence-list | List (dynamic) | Stores truth values of evidence nodes agents encountered.
| agent-belief | 0 - 1 (dynamic) | Belief in the hypothesis.
| initial-belief | 0 - 1 (dynamic) | Unconditional prior belief in hypothesis.
| update-list | List (dynamic) | Stores impact.
| recency-list | List (dynamic) | Governs communication via the recency rule.
| chattiness | $0-1$ | Probability of communication.
| curiosity | $0-1$ | Probability of inquiry (evidence gathering).
| conviction-threshold | $0-1$ | Conviction in claim required to join debate.
| max-draws | Integer | Determines number of inquiries agents can perform.
| share | Chooser | Determines agents’ communication rule. Values:
|  |  |  random, impact, recent.
Network | number-of-agents | 1 - 1000 | Determines the size of the network.
| social-network | Chooser | Network type: null, complete, small-world, wheel.
Table 1. The Core Parameters of NormAN governing world, agents, and social
network (for a complete table, cf. Appendix Fig 2).

####  3.1.1. The World

The world model consists of a BN comprising a set of nodes (variables) and the
probabilistic/causal links between them. In general, Bayesian networks consist
of a so-called directed acyclic graph and a matching probability distribution
that specifies the conditional probabilities of each variable in accordance
with the graph.666$B=\langle G,P\rangle$ is a directed acyclical graph
$G=\langle V,E\rangle$, with a set of nodes (variables) $V$ and edges $E$, and
a joint probability distribution $P$ over $V$ such that $G$ satisfies the
parental Markov condition together with $P$. As an example, consider the well-
known lung cancer/asia network (Lauritzen and Spiegelhalter, 1988b), as seen
in Fig. 2: a hypothetical network from the medical field that models the
causal (probabilistic) relationships between a patient’s habits (smoking,
visiting Asia) and their symptoms (dyspnoea, positive X-ray).

Figure 2. The original ‘Asia’ lung cancer network (Lauritzen and
Spiegelhalter, 1988a). The Asia BN model was accessed via the bnlearn Bayesian
Network Repository (https://www.bnlearn.com/bnrepository/discrete-small.html);
it is also one of the exemplar BNs used in the bnlearn package documentation
(https://www.bnlearn.com/documentation/bnlearn-manual.pdf). This BN was
constructed using a hypothetical case of qualitative medical knowledge to
illustrate the utility of Bayes’ rule for expert systems. The target
hypothesis is ‘Lung’ (whether or not lung cancer is true, shown as the blue
node), and there are seven observable evidence nodes (shown as the orange
nodes): Asia (recent visit to Asia); smoking; tuberculosis; bronchitis;
dyspnoea (shortness of breath); and, x-ray (chest x-ray). The likelihood of
lung cancer is increased when smoking, x-ray, bronchitis and dyspnoea are set
to true. Combinations of evidence lead to interactions. The ‘either’ node is a
deterministic node that is used in this network to represent the fact that
both tuberculosis and lung cancer can result in positive x-ray. Network
properties: Number of nodes = 8, Number of arcs = 8, Number of parameters =
18, Average Markov blanket size: 2.5, Average degree = 2 and Maximum in-degree
= 2.

In such a BN, the modeller identifies a variable as the hypothesis variable
(hypothesis), or H for short, (e.g., ‘lung cancer’) and chooses a subset of
the other nodes as evidence nodes
($\texttt{E}_{1},\texttt{E}_{2},\ldots,\texttt{E}_{n}$). In NormAN 1.0,
hypothesis nodes and evidence nodes must be two-valued, that is, they are
either true or false. The model assigns such a truth value to the hypothesis
(manually or probabilistically). The following procedure then determines the
values of the evidence nodes. The marginal conditional probability of the
evidence is calculated; and on initialisation, this chance stochastically
determines the truth value of each piece of evidence. For example, if it is
true that the patient has lung cancer, and $P(bronchitis|lungcancer)=0.2$,
then there is a 20% chance that the value of the variable bronchitis is true.
Since the evidence nodes are two-valued, this procedure yields a chain of
evidence in the form of $\neg E_{1},E_{2},E_{3},\ldots,\neg E_{n}$ (where
$\neg E_{i}$ denotes ‘$\texttt{E}_{i}$ is false’ and $E_{i}$ denotes
‘$\texttt{E}_{i}$ is true’). This list of indexed truth values is stored in
evidence-list.

Crucially, this evidence assignment determines what evidence, on a given run,
counts as evidence for or against the hypothesis. While the structure of the
BN determines the evidential impact of a piece of evidence (e.g., the degree
to which the presence of smoking [‘smoke’] increases belief in ‘lung cancer’),
it is the actual value assigned to that variable on a given run which
determines the evidence as for or against _in this world_ : if the value of
‘smoke’ is initialised to false, it provides evidence against the hypothesis
lung cancer as knowledge of that fact will lower degree of belief in ‘lung
cancer’ being true.

This means also that many possible combinations of evidence for and against
will be generated by a single BN world model, see Fig 3. And in the user
interface (UI) NormAN users can determine not only which BN they wish to work
with, but also whether or not the evidence is re-initialised on a given run.

Figure 3. Different instantiations of the ‘world’ defined by the ‘Asia’ lung
cancer network (Lauritzen and Spiegelhalter, 1988a). On a given model run, the
base net (a), can give rise to different ‘worlds’ with varying arguments ‘for’
(green) and ‘against’ (red), depending on the stochastic initialisation.

.

####  3.1.2. The Agents

Each agent is characterised by (a) their degree of belief in the hypothesis
(variable agent-belief), (b) their representation of the causal structure of
the world, and (c) a list of evidence they have already encountered. We go
through each feature in turn. First, each agent assigns a degree of belief to
the hypothesis (variable agent-belief). Second, they use a BN that connects
the evidence to the hypothesis as their representation of the world to compute
said belief. Third, they store the truth values of the evidence they have
already encountered (variable agent-evidence-list). These three aspects are
related in a dynamic, straightforward way. Suppose an agent $A$ stores the
following list of evidence at time $t$: $\texttt{agent-evidence-
list}_{A}^{t}=\\{E_{1},\neg E_{3}\\}$. In that case, they will use their
Bayesian network to compute $\texttt{agent-belief}_{A}^{t}=P(H|E_{1},\neg
E_{3})$ by using Bayesian conditionalization. Whenever agents encounter a new
piece of evidence (e.g., $E_{2}$), they update their degree of belief (e.g.,
$\texttt{agent-belief}_{A}^{t+1}=P(H|E_{1},E_{2},\neg E_{3})$). When the
agent’s agent-evidence-list is empty, that is, when they have not yet
encountered any evidence, their agent-belief is simply the base rate (marginal
probability) of the hypothesis node in their BN. This value is stored in the
agent-variable initial-belief as their agnostic, pre-evidence belief in the
hypothesis.

In the first version of NormAN presented here, we assume that each agent’s BN
simply corresponds to the world model’s network: that is, we assume that
agents represent the world correctly (on relaxing this assumption, see 4.3.1
below). This homogeneity of worldviews entails that whenever two agents have
access to the same evidence, they also have the same degree of belief in the
hypothesis. This assumption can be interpreted as fulfilling the uniqueness
standard of rationality, that is, the claim that for any body of evidence $E$
and proposition $P$, $E$ justifies at most one doxastic attitude toward $P$
(Feldman and Antony, 2011; White, 2019). This homogeneity also means that
disagreements are entirely the result of asymmetric information. Heterogeneity
in beliefs arises because agents may have access to different sets of
evidence.

####  3.1.3. The Social Network

The model places ‘number-of-agents’ agents on a grid and then specifies who is
connected to whom via undirected communication links. Agents can only
communicate with their link neighbours. NormAN provides a number of different
network structures that the user can select before initialisation (via the
chooser variable social-network), such as the complete network, a ‘wheel’ (cf.
(Zollman, 2010; Frey and Šešelja, 2020)) and small-world networks (also known
as Watts- Strogatz networks (Watts and Strogatz, 1998; Wilensky, 2005)). The
latter are a type of network structure found in many social and biological
networks. They are characterised by comparatively short paths between nodes in
the network (‘five degrees of separation’) and comparatively high clustering,
although the density of connections is relatively low (see Fig. 4 for a
visualisation).

Figure 4. Two groups of 50 agents connected in a social network: a ‘small-
world’ network on the left, and a complete network on the right. Green
triangles represent agents who currently support the hypothesis, and red those
who do not (cf. Section 3.2). Both network types are used in the case studies
(sections 4.1, 4.2). Parameters of the small-world network: rewiring-
probability=0.2, k=2.

###  3.2. Process Overview

Deliberation unfolds dynamically, in discrete time steps. At each step, agents
follow this protocol:

  * (1)

Collect evidence: agents may collect a new piece of evidence from the world.

  * (2)

Communication: agents may share one piece of evidence they have already
encountered with their link neighbours.

Collecting evidence facilitates the flow of information into the network, and
communication facilitates the flow of information through the network. This
subsection explains when and how agents perform each activity (a detailed
description and a flowchart of the protocol can be found in section A.3, Fig.
9).

To collect evidence (or ‘inquire’), agents randomly select an item from the
world model’s evidence-list that they have not yet encountered. They add this
truth value to their personal agent-evidence-list.777As an example, suppose
that at time $t$, agent $A$ stores the truth values $\texttt{agent-evidence-
list}_{A}^{t}=\\{E_{1},\neg E_{3}\\}$. Through inquiry, they may find that
$E_{2}$ is indeed true, thus extending their list to $\texttt{agent-evidence-
list}_{A}^{t+1}=\\{E_{1},E_{2},\neg E_{3}\\}$. Inquiry is therefore modelled
by ‘drawing’ from the world’s evidence-list. Learning any new piece of
evidence (be it via inquiry or communication) is modelled as learning that a
certain piece of evidence is true or false. Two agent variables govern
inquiry. First, agents have a fixed maximum number of inquiries (the variable
max-draws determines how many pieces they may each collect during one
simulation). Second, agents will not necessarily inquire every round. Rather,
their chance of inquiry is determined by a curiosity parameter. Hence, agents
only collect evidence if they are ‘curious’, and if they still have ‘draws’.

Next, in each round, agents may communicate and receive evidence through
communication. In NormAN 1.0, communication is modelled via a simple
transmission mechanism: the communicating agent $A$ chooses which piece of
evidence to transmit to their link neighbours. Each link neighbour, e.g., $B$,
then either adds this evidence to their agent-evidence-list, and computes a
new agent-belief, or ignores this evidence if they have already heard
it.888For instance, if $A$’s list is $\texttt{agent-evidence-
list}^{i}=\\{E_{1},\neg E_{3}\\}$, and $B$’s agent-evidence-list is
$\texttt{agent-evidence-list}_{B}^{i}=\\{\neg E_{3}\\}$, then $A$’s sharing
$E_{1}$ will enrich agent $B$’s list to $\texttt{agent-evidence-
list}_{B}^{i+1}=\\{E_{1},\neg E_{3}\\}$. Had $A$ chosen $\neg E_{3}$, $B$’s
list would have remained unchanged. In NormAN, agents recognise distinct
pieces of evidence and never ‘double count’ pieces of evidence.

Although this mechanism of informational ‘passing the parcel’ is simple in
that it avoids the complexity of testimony, it can be used to capture
distinct, complex styles of communication. In NormAN 1.0, three sharing rules
are examined:

  1. (1)

Random: Agents share a random piece of evidence from their agent-evidence-
list.

  2. (2)

Recency: Agents share the piece of evidence they most recently encountered.

  3. (3)

Impact: Agents share the piece of evidence that they hold to be the best piece
of evidence in favour of their current position.

Since the random rule is self-explanatory, we briefly explain how the recency
and impact rules work (for a detailed, technical explanation, see Section
A.3). Under the recency rule (loosely inspired by Maes and Flache’s model of
bi-polarization (Mäs and Flache, 2013)), agents are most likely to share the
last piece of evidence they heard. This is implemented by each agent’s
recency-list, which keeps track of the order of receipt.999Importantly, even
if agents receive a piece of evidence they have already encountered, this
piece is ‘popped’ to the top of the recency-list. With a high probability of
$x$ the agents share the last element of this list, but with a probability of
$1-x$ they share another random element from the list.101010In the base model,
$x=0.9$.

The impact-sharing rule provides a very basic implementation of the idea that
speakers seek to communicate what they consider to be (most) relevant. This
means sharing what they consider to be their best—strongest—piece of evidence.
In our simple impact rule, this is the piece of evidence which most convinces
agents of their current position. In NormAN, agents track the magnitude of the
belief update that evidence triggers, that is, its ‘impact’. To measure this,
for each evidence $E_{i}$, agents store the update magnitude
$P(H|E_{i})-P^{initial}(H)$, where $P^{initial}(H)$ marks the agent’s prior,
pre-evidence belief (initial-belief). That is, the impact of a piece of
evidence is measured by how far it moved (or would move) an agent’s belief
away from their agnostic prior. Each agent has an update-list recording the
impact of each piece of received evidence. If an agent currently supports the
hypothesis, they share the evidence with the highest update value (and they
share the evidence with the lowest, i.e., largest negative impact if they
currently oppose it). NormAN models this ‘support’ as follows: if the agent’s
agent-belief¿initial-belief, they support the hypothesis, and they oppose it
if agent-belief¡initial-belief (cf. Fig. 4). Hence, an agent’s position is
measured relative to their pre-evidence, agnostic prior.

Communication is regulated by a conviction threshold, a percentage value that
serves as a cut-off point for when an agent’s belief departs sufficiently from
their agnostic, pre-evidence prior (initial-belief) for them to jump into the
discussion. This threshold is set by the global variable conviction-threshold,
which determines a percentage by which the agent’s conviction needs to exceed
their initial, pre-evidence belief.111111Specifically, it defines a lower
bound and an upper bound for agent beliefs. The lower bound is computed as
($\texttt{initial-belief}-\texttt{initial-belief}\times\texttt{conviction-
threshold}$). The upper bound is computed as ($\texttt{initial-
belief}+(1-\texttt{initial-belief})\times\texttt{conviction-threshold}$). If
an agent’s agent-belief does not exceed the threshold (above or below), they
will not share. Note that if conviction-threshold is set to $0$, the sharing
condition is trivially met in most cases: agents will share whenever their
agent-belief $\not=$ initial-belief.121212As an example, if conviction-
threshold $=0$, initial-belief $=0.3$, and agents use the impact sharing rule,
they will share pieces of evidence ‘against’ $H$ if their current belief is
below 0.3 (and vice versa for agent-belief$>0.3$).

One last agent-variable co-determines the frequency of communication: agents’
chattiness $\in[0,1]$, that is, the chance that they will communicate with
their link neighbours on each round (determined by the global variable
chattiness). If an agent passes the conviction threshold and is chatty, they
will send an argument (item from to their agent-evidence-list) to their link
neighbours.

To summarise, in each time step, agents may first collect new evidence from
the world (if they are curious and still have ‘draws’). Then, if they cross
the threshold and are chatty, they share one of their pieces of evidence with
their neighbours (according to the sharing rule chosen by the model user).
Whenever they learn of a new piece of evidence, they compute their new belief
in the hypothesis.

###  3.3. Implementation and usage

In order to make the NormAN framework accessible to researchers from a broad
range of backgrounds we chose to implement it in NetLogo (Wilensky, 1999;
Wilensky and Rand, 2015). Designed initially to teach programming to
beginners, NetLogo is an accessible, well-documented, platform that has been
and continues to be widely used in agent-based modelling research (Gunaratne
and Garibay, 2021), including specifically for research on opinion dynamics
(Lorenz, 2017; Wang et al., 2022), belief dynamics (Hahn et al., 2018b, 2019,
2023), and social epistemology (Weisberg and Muldoon, 2009; Pinto and Pinto,
2018).

Its benefits lie in the fact that much of the machinery required for setting
up an ABM and running simulations with it is in-built, leading to very compact
code: the initial version of NormAN (version 1.0) has only 500 lines of code
(excluding the BN definitions).

Moreover, Netlogo has extensions for both R (Thiele et al., 2012) and Python
(Jaxa-Rozen and Kwakkel, 2018), that allow two-way interactions with the
extensive scientific computing resources of those platforms. For our initial
version of NormAN we chose the R extension (a version with Python is planned
in future). Specifically, NormAN draws on the R-package bnlearn (Scutari,
2009; Scutari and Denis, 2021) to handle all Bayesian belief updating over
BNs. NormAN 1.0 was developed using NetLogo version 6.2.1, which efficiently
implements the R extension (developed by (Thiele and Grimm, 2010)).

The combination of R (bnlearn) and NetLogo makes for a very flexible modelling
environment: to characterize the world BN, the modeller can load whole $R$
files into the NetLogo model, or simply copy and paste the lines of R code
into the indicated section of the NetLogo code. One can also use one of eight
preset BN’s (see B). The NetLogo interface handles the rest: sliders, inputs
and switches determine the size and shape of the social network, the agent
variables such as sharing styles, as well as the specification of which BN
nodes ought to count as the evidence and hypothesis.

####  3.3.1. Running and Evaluating Simulations

With respect to running and evaluating simulations, the use of the R extension
means that users have two routes for controlling ‘experiments’ and model
explorations: Netlogo’s built-in BehaviorSpace (Tisue and Wilensky, 2004) and
directly through R (Thiele et al., 2012).

While the use of a high-level language such as NetLogo does come at a
performance cost, we found simulations with NormAN version 1.0 to not only
(easily) be efficient enough for practical purposes in the range of network
sizes we think modellers will most likely wish to explore in detail (up to
around 100). It is also possible to run larger networks. We have, albeit
slowly, run networks with 100,000 agents in the NetLogo User Interface (on a
2020 MacBook Pro with 2 GHz Quad-Core Intel Core i5, and 16 GB of RAM).

It is thus possible, even in the current implementation, to check how findings
scale and whether ‘more is different’ for the target phenomenon of interest
(Anderson, 1972). Furthermore, much of the processing time for large networks
involves the construction of the social network (in particular, the small
world network), suggesting paths for scalable future versions (Railsback et
al., 2017).

While we consider the balance between accessibility and performance to be a
suitable one with respect to our current goals (see also (Burbach et al.,
2020; Railsback et al., 2017)), re-implementing the NormAN framework not just
with other extensions (Salecker et al., 2019; Gunaratne and Garibay, 2021),
but also within other platforms and languages is a goal for future work.

##  4\. Initial Case Studies

In the third main part of this paper, we seek to demonstrate the utility of
NormAN with two case studies. These have been chosen to illustrate the value
of its features and demonstrate the reasons for our basic design choices. In
particular, they serve to underscore the claims of section 2.3 above, that a
suitable model of argument exchange needs normative grounding both with
respect to the aggregation of evidence/arguments by individual agents and with
respect to the ground truth world. The case studies have been chosen also to
illustrate how NormAN, as a normative model, may contribute to extant
theoretical concerns across a range of disciplines.

###  4.1. Case Study 1: Shift to Extremity

The so-called ‘shift to extremity’ (Stoner, 1968) is the original
‘polarization’ phenomenon. Although the term ‘polarization’ has now become
associated with belief or opinion _divergence_ within a population, the term
was first used to describe the phenomenon whereby deliberating groups tended
to shift further in the direction of an initial opinion over the course of
deliberation (for a review see e.g., (Isenberg, 1986)). This shift to
extremity attracted considerable research spanning six decades to date and has
proved highly reliable (it has been observed with lab-based studies, mock
juries, deliberative polling (Myers and Lamm, 1976) and citizen debates
(Lindell et al., 2017) and with tasks as diverse as risk acceptance,
probability judgment, policy choice and negotiation (Lamm, 1988)), though it
is not observed in every group discussion. This interest has been fuelled not
just by the phenomenon’s practical relevance, but also the fact that it (at
least initially) seemed counter-intuitive and in need of explanation: after
all, one might expect a deliberation to surface both arguments for and against
a claim. This made it seem surprising that beliefs might shift regularly in
one particular direction.

Multiple lines of explanation were pursued in the literature, such as the idea
that the shift reflects social comparison processes (Sanders and Baron, 1977)
or social identity considerations (Abrams et al., 1990): people may become
comfortable expressing positions they initially feared others might view as
extreme or they may, as a matter of identity, seek to adopt attitudes
stereotypical of the group. A third account, by contrast, attributed the shift
to the arguments that surfaced within a debate. Burnstein and Vinokur’s
‘argumentative theory’ proposed that group members lean toward an initial
position because they have more (or stronger) arguments in favour of that
position and more (or stronger) arguments in favour of that position will
consequently be available for exchange in the deliberation (Burnstein and
Vinokur, 1977). Experimental research subsequently sought to distinguish these
competing (but ultimately not mutually exclusive) accounts (Lamm, 1988;
Sanders and Baron, 1977; Vinokur and Burnstein, 1978).

The argumentative theory was also supported through simulations in an agent-
based model by (Mäs and Flache, 2013). In this model, Maes and Flache
implement the substantive assumptions of the persuasive argumentation theory
and combine them with the modelling of homophily in order to understand bi-
polarization or belief divergence (an issue we turn to in our next case
study). In effect, their model of the latter combines the shift to extremity
afforded by persuasive argumentation with homophily-based segregation to
explain divergence. In their model, agents have a numerically valued opinion
(drawn from the interval -1 to 1) representing their stance on the issue in
question. Additionally, there is a set of arguments that address that issue.
The valence of an argument is expressed numerically (pro = 1, con = -1), and
all arguments carry equal weight. An agent’s current stance is based on the
average value of the arguments they are currently considering, and arguments
for communication are selected randomly from the agent’s current relevance set
–a subset of the encountered arguments determined by recency. Resultant
findings support the argumentative theory in as much as the positions of
agents within the homophily-driven clusters become more extreme.

One limitation of the model, however, is that both the generation and
evaluation of arguments lack a principled basis. And the initial distribution
of arguments in the population is essentially ‘just so’.

Most recently, it has been pointed out that both of these concerns may be
addressed by adopting a Bayesian perspective on argument (Hahn, 2023), as
described in section 2.1.4 above. From that perspective, multiple interlocking
components give rise to the shift to extremity: Group members’ pre-
deliberation beliefs are based on a sample of the arguments available ‘in the
world’. The available population of arguments for and against the claim is
likely to contain stronger arguments in one direction than the other. Pre-
deliberation beliefs based on samples will, on average, reflect those
characteristics. By the same token, combining samples via group discussion is
more likely to see individual members add stronger arguments supporting the
initial direction, which in turn will shift the group mean.

The core component that the expected distribution of arguments available _in
the world_ is skewed follows from the Bayesian conceptualisation of argument
–outlined in section 2.1.4– whereby an argument or piece of evidence is strong
or diagnostic to the extent that it is much more likely to be found if the
hypothesis is true than if it is false (expressed by the likelihood ratio
$P(e|H)/P(e|notH)$). This translates into an expected distribution of
arguments by virtue of the fact that how likely a piece of evidence is (its
so-called marginal probability) is determined by ‘total probability’:
$P(e)=P(e|H)\times P(H)+P(e|notH)\times P(notH)$. In other words, the
probability of a piece of evidence is determined by the probability of
obtaining it if the underlying hypothesis is true, weighted by the probability
of the hypothesis, plus the probability of a false positive, weighted by the
probability of the hypothesis being false. This means that (all other things
equal), strong evidence in favour of a hypothesis is more likely than equally
strong evidence against if the hypothesis is true (Hahn, 2023).

In short, the Bayesian framework helps make explicit the fundamental point
that –at least for claims about issues of fact– what arguments are available
in a domain is determined by the underlying structure of the world. And that
evidence distribution, in turn, will impact agents’ subsequent beliefs. The
shift to extremity is simply an initially counter-intuitive consequence of
that fundamental point. We should expect a group with prior evidence/arguments
to lean, on average, in a particular direction and expect that exchanging that
evidence will likely lead to a further shift in that direction.

Figure 5. Results from simulation using the ‘Big Net’ network. Shown are the
mean beliefs of agents in the target hypothesis and standard error bars,
across all 800 experiment runs, at the pre (0) and post-deliberation phase (in
this experiment after 25 exchanges/steps). Groups are split by the agents’
mean initial direction of belief for a given run.

This is the Shift to Extremity

Figure 6. The Figure shows the trajectories of mean group beliefs across time
(thin lines) and the average of those means (dashed lines) with standard error
(grey shaded area), split by whether the group initially leaned ’for’ (blue)
or ’against’ (red). The top row shows the simulation results for complete
networks of various sizes (10,50,100, and 500 agents). The bottom row shows
the same dynamics in small-world networks with corresponding numbers of
agents. Note that the ‘pre-evidence’ neutral point in these simulations is .5
(base rate of hypothesis), as indicated by the black dotted line.

This is the Shift to Extremity

In our first case study, we demonstrate these theoretical points through
simple simulations with the NormAN framework. To this end, we simply ran
NormAN using one of our default networks –the ‘Big Net’ network (see Fig. 15
in Appendix B below). This stylised network is useful for making the current
point because it is symmetrical by design: the structure contains three
arguments that, when true, are evidence for the claim, and three that, when
true, are equally strong evidence against. On average, instantiations of this
world will not, however, lead to equal numbers of arguments for and against
(though this will sometimes occur) for the reasons just outlined. As a result,
the shift to extremity is readily observed despite the balanced nature of this
stylised network. To show this, we simulated many model runs, graphing the
mean belief at 2 points in time: once after the initial argument-draw by
agents, and once at the end of the run. The former represents the pre-
communication, and hence the pre-deliberation state, the latter the end of
deliberation (for full details see Appendix C.1) Figure 5 shows the respective
pre- and post-deliberation means split by whether the group’s initial belief
(given their initial access to evidence prior to deliberation) leans ’for’ or
’against’ the claim in question. The second figure, Figure 6 shows the same
measurements but split by individual runs of the simulation. It plots the
belief dynamics across time for the individual runs, split again by whether
the group’s initial belief (given their initial access to evidence prior to
deliberation) leans ‘for’ or ‘against’ the claim in question (i.e., the target
hypothesis). As can be seen, the shift does not always happen but happens most
of the time.

The experimental literature on the shift typically considers fairly small
groups, under conditions where members typically all hear one another. In
network terms, participants constitute a complete network. These conditions
are captured by the top left panel Figure 6. But we can also explore how
increasing the group size influences the dynamics. To this end, the four
columns of row 1 in Fig. 6 show group sizes of 10, 50, 100 and 500 agents
respectively. These show (for a constant pool of available arguments) a
sharpening of the shift as a function of group size. This reflects the fact
that the available evidence enters the group discussion more quickly. The
bottom row of Fig. 6 shows the same information for a small world network
(Watts and Strogatz, 1998). One can see a dampening of the shift, due to the
slower diffusion of arguments among agents. Finally, to demonstrate that these
findings are not a unique feature of the particular BN ‘world’ selected, Fig.
18 in the Appendix section C.1 shows the same results for a different network
included with NormAN version 1.0, the ‘Asia network’ of Fig. 2 above. The
Appendix also contains the full simulation details, model parameters, and
further supplementary figures that elucidate model behaviour.

These simple simulations illustrate the different components of the ‘shift to
extremity’ that a Bayesian perspective helps unify and make explicit. In so
doing, they also illustrate how additional insight and refinement of
understanding becomes available as a result of moving from the argumentative
theories’ initial, purely verbal, formulation, through Maes and Flache’s
agent-based model (Mäs and Flache, 2013) to a normative framework. Most
importantly, however, these simple simulations highlight the important point
that evidence or argument distribution matters fundamentally to understanding
model behaviour. By the same token, it matters fundamentally to the
understanding of the kind of real-world behaviours these models are trying to
elucidate.

###  4.2. Case Study 2: Polarization versus Convergence

The goal of consensus as unanimous agreement is one of the key motivations for
deliberation in early theories of deliberative democracy (Landemore and Page,
2015). Conversely, polarization as (extreme) belief divergence is seen as a
threat to contemporary democracies worldwide (Sunstein, 2018). Under which
conditions can we expect a group to converge on consensus—and correct
consensus at that? And under which conditions does polarization emerge?
Computational models of deliberation have identified conditions undermining
consensus, that is, conditions that lead to non-convergence. While such
models, to date, have yielded a wealth of insight, in particular formal
insight (e.g., (Krause, 2015)), there are key features of the most popular
paradigms that significantly limit or distort insight into polarization as a
real-world phenomenon.

As discussed in section 2.2.1 above, most opinion dynamic or social-influence
models revolve around the notion that individuals communicate their opinions
and influence is typically implemented as opinion averaging (Hegselmann and
Krause, 2002; French Jr, 1956; Friedkin and Johnsen, 2011; Deffuant et al.,
2005). They thus abstract away entirely from supporting arguments themselves.
As a result, these models have several consequences that are strongly at odds
with what is empirically observed. For one, they typically exhibit an
inevitable drive to convergence (Abelson, 1964; Lorenz, 2006; Krause, 2015)
which has meant that other factors preventing convergence and giving rise to
polarization must additionally be imposed (e.g., (O’Connor and Weatherall,
2018)). As Maes and Flache (2013) note, many of these factors can be
understood as negative influence of one form or another (Baldassarri and
Bearman, 2007; Macy et al., 2003; Mason et al., 2007; Olsson, 2020) giving
rise to two competing forms of influence—positive influence from similar,
like-minded agents, and negative influence from dissimilar agents. One aim of
the Maes and Flache (2013) model introduced in the previous section is to
demonstrate how positive influence alone (in the form of persuasive
argumentation coupled with homophily) can give rise to divergence. At the same
time, models based on opinion averaging also fail to capture the shift to
extremity. Averaging implies that averaging agents will not adjust their
opinions when they interact with others with whom they already agree (Mäs and
Flache, 2013). This follows necessarily from the fact that the average of two
near-identical values will not only be similar to these values but will be
less extreme than the more extreme of the two. It is thus difficult to
generate the empirically observed dynamics whereby deliberation leads to views
more extreme than those of any of the participants prior to the interaction.
Yet this possibility has been observed widely in the empirical research on the
shift to extremity discussed above.

A model that is rich enough to include individual arguments thus seems
essential to fully understanding real-world divergence of beliefs. In this
context, the dynamics of NormAN help clarify a fundamental mechanism producing
non-convergence: namely, the incomplete sharing of information on a social
network. Whenever two agents start with the same prior degree of belief and
also interpret pieces of evidence in the same way, the acquisition of
different sets of evidence will drive their posterior beliefs apart. Hence, in
our model, whenever deliberation does not result in a state where all agents
have access to the same evidence, then final beliefs may be scattered as well.
This is a natural consequence of the uniqueness assumption implemented in
NormAN 1.0: for each set of evidence, there is only one permissible ‘doxastic
state’, that is, only one possible degree of belief. Other models identify a
different cause of non-convergence: differences in the interpretation of
evidence. Cook and Lewandowski (Cook and Lewandowsky, 2016) highlight how
Bayesian agents entertaining different BNs (in particular, different
conditional probability distributions) will exhibit belief divergence when
receiving the same evidence (an experimental finding dating back to (Lord et
al., 1979)). Indeed, relaxing the uniqueness requirement and allowing agents
to have different conceptions of the world’s causal structure will make
perfect convergence via deliberation (as characterized here by the exchange of
arguments/evidence) rather hard to attain: even if all agents have access to
the same evidence, they may still disagree.

Figure 7. The beliefs of a population over time, for three sharing rules:
random (top left), recency (top right) and impact (below). Each function
tracks the state of one agent’s agent-belief at each time step. Initial
conditions: causal-structure= Vole (see the Appendix, section B for an
explanation), chattiness=0.5, conviction-threshold=0, curiosity=0, initial-
draws=1, max-draws=1, social-network=complete, number-of-agents=50. The prior
initial-belief$\approx 0.3$.

When can we expect agents to end up with identical sets of evidence, and
therefore in consensus? In this brief case study, we use the NormAN framework
to exemplify how a common style of communication can undermine consensus:
impact-driven sharing (as explained in section 3.2). Instead of sharing all
their evidence, agents only share the piece of evidence that most convinces
them of their current position, that is, the argument or piece of evidence
that they find most impactful. Crucially, this type of incomplete, selective
sharing need not be an irrational or bad-faith communication strategy. Using
the impact rule simply means curating what one shares, leading to
communication of what one considers one’s best, most truth-conducive piece of
evidence. Given that both communicating and processing arguments is costly,
such a strategy may be pragmatically appropriate in many contexts. It is
intuitive that a communication style between agents who do not (or cannot)
share their entire evidence may lead to non-convergence: even if no piece of
evidence is completely unknown to the group, the impact rule, or any
incomplete sharing rule (i.e., rules where agents communicate only a subset of
their evidence), will make an end state of fully shared information less
likely. Instead, each agent’s access to evidence is filtered by their social
environment: If their neighbours believe that the hypothesis is true, they are
more likely to receive confirmatory pieces of evidence through communication
(and vice versa).

Figure 7 shows three exemplifying model runs of a population of 50 agents
connected in a complete network (the world model uses the ‘Vole’ network,
explained in the Appendix, Section B). The graphs track the agents’ agent-
beliefs in the hypothesis during a very simple deliberation process:
initially, each agent has one piece of evidence (i.e., their agent-evidence-
list contains the truth value of one randomly chosen evidence node). Subfigure
1 illustrates the evolution of beliefs resulting from deliberation following
the ‘random’ sharing rule. Convergence is virtually guaranteed as agents will
share all their evidence eventually (the random rule is a ‘complete’ sharing
rule). The recency rule (Subfigure 2) creates a similar dynamic: although
convergence takes longer to achieve, agents do, eventually, form a consensus.
Subfigure 3 illustrates non-convergence as a consequence of the ‘incomplete’,
selective impact sharing. This non-convergence is easily explained: meaningful
communication between two agents is interrupted when the sender has
communicated what they consider to be their best evidence (the sender may
repeat themselves, but the receiving agent, having already incorporated the
information, will now ignore this communication with respect to their belief
in the hypothesis). It is only resumed when the sender learns even stronger
information in favour of their position or when they change their mind (i.e.,
cross the conviction threshold). However, once agents find themselves in a
group of like-minded neighbours, they are unlikely to receive further evidence
that changes their minds. Consequently, communication ceases and divergent
beliefs stabilise.

Figure 8. Different deliberations tracked by (left) the number of
transmissions of each argument per round and (right) the degrees of belief of
the agents. As for the tracking of arguments (left), if the argument $S$ was
shared by 20 agents at time step $10$, the red line marks $\\#uttered=20$ on
the y-axis. On the right-hand side, we see a histogram of the agents’ agent-
beliefs at time step 50 (the beliefs have stabilized at that point). The top
figure shows a run driven by the random rule. The middle figure shows a run
controlled by the recency rule. The bottom figure shows a run controlled by
the impact rule. Initial conditions: causal-structure= Vole (see the Appendix,
section B for an explanation), chattiness=0.5, conviction-threshold=0,
curiosity=0, initial-draws=1, max-draws=1, social-network=complete, number-of-
agents=50. The prior initial-belief$\approx 0.3$.

This non-convergence result contrasts with models of homophily and echo
chambers where agents themselves (possibly unknowingly) select sub-optimal
information environments. The present simulations with NormAN reveal that
divergence may arise also as a function of good faith communication strategies
whereby agents seek simply to convey what they consider to be their ‘best
evidence’, without homophily or any other attempt to filter their (input)
sources, and also without any form of negative influence. These sample runs
also demonstrate a further important point. Paradigmatic models of
polarization typically showcase only one type of non-convergence: wholly
driving apart two subgroups, possibly to the extremes (e.g., 0, that is,
certainty that the central hypothesis is false and 1, certainty of truth)
(Olsson, 2013; Pallavicini et al., 2021; Bramson et al., 2017). Realistic
models of deliberation that connect meaningfully to real-world data, however,
plausibly ought to reconstruct non-convergence as a scattering of intermediate
beliefs across the unit interval as well.

As the model runs of NormAN show, scattering of beliefs can arise as the
result of incomplete sharing in a social network. In the simulations (as,
arguably, in the real world), exactly how much the group’s beliefs diverge,
will depend partly on the nature of the available evidence: situations where
there are strong pieces of evidence both in favour and against the hypothesis
will prove more divisive than more homogenous distributions. Furthermore,
using scarcer, more realistic network structures (as opposed to the complete
networks in the above sample run) will also exacerbate divergence effects:
using a small-world network slows down belief convergence induced by the
complete sharing rules (recency and random), and exacerbates the scattering
effect of the impact rule (cf. Fig. 19 in the Appendix.)

The case study simulations show how different communication rules will give
rise to different patterns. Needless to say, in the real world these factors
may additionally interact with other factors that have been shown to give rise
to polarization such as homophily or differences in trust. A richer underlying
model of communication that involves argument exchange thus opens up new
avenues for understanding the complex interplay of factors likely at work in
real-world settings.

A corollary of the fact that communication rules are so central is that it
highlights the need to understand in more depth _what has been communicated_.
Different belief distributions come about because agents end up being exposed
to different subsets of the evidence. To analyze the dynamics of argument
exchange, NormAN allows the modeller to track the sharing frequency of
particular pieces of evidence. Fig. 8 shows three model runs (with the same
initial conditions as in Fig. 7), with the left panels tracking the number of
transmissions of each piece of evidence per time step (on the right we see the
final distributions of the agent’s beliefs). Both the first (top) Subfigure,
showing a random run, and the second (middle), showing a run driven by the
recency rule, end in convergence. Note, however, the differences in the
frequency of argument sharing: While the frequencies of arguments sent remain
roughly the same when sharing is random (each argument is similarly popular),
the frequency of arguments governed by the recency rule rises and falls in
waves. Finally, in the Subfigure below, which shows a run governed by the
impact rule, we can see that two groups of disagreeing agents (one with agent-
belief¡initial-belief, the other with agent-belief¿initial-belief) each select
what they consider to be their ‘best’ evidence and stabilize on sharing those
arguments repeatedly. Therefore, less impactful evidence is not communicated
and, as can be seen on the right-hand side, the agents’ beliefs do not
converge.

Crucially, this illustrates also that NormAN can be used to study not only
belief dynamics but also _argument dynamics_. However, studying the dynamics
of what arguments are exchanged is not just a matter of supplementary
analysis. It deserves to be seen as a research topic in its own right. As
noted in 2.2.1 above, computational social science has seen large amounts of
research devoted to aspects of how particular messages spread across real-
world social networks. That analysis has, however, remained coarse-grained,
arguably, in good part because of a lack of models to help formulate theory-
driven questions and hypotheses. We would hope that NormAN simulations could
provide new impetus here.

###  4.3. Implications of the Case Studies and Future Directions

Both case studies illustrate how the NormAN framework may contribute to extant
research. While these case studies are intended to be of interest in their own
right, we think also that they illustrate the breadth of disciplines and
research questions that the NormAN framework could contribute to. Case Study 1
on the shift to extremity helps further illuminate an empirical phenomenon
that has exercised, in particular, psychologists and political scientists.
Case Study 2 on polarization contributes to a research topic that has
attracted interest across a broad range of disciplines, ranging from
psychology (e.g., (Fasce et al., 2023; Brown et al., 2022), communication
studies (Moore-Berg et al., 2020; Kubin and von Sikorski, 2021; Lee et al.,
2014), sociology (e.g., (Mäs and Flache, 2013)), epistemology (e.g., (Olsson,
2013)), political science (e.g., (Baldassarri and Bearman, 2007; Fiorina and
Abrams, 2008)), economics (e.g., (Fang et al., 2021)), as well as complex
systems researchers examining models inspired by perspectives as diverse as
epidemiology (e.g., (Vasconcelos et al., 2019)), and physics (e.g., (Galam,
2005)).

Furthermore, our case studies underscore key points that have shaped the
design of the framework. First, they serve to highlight why one cannot simply
study belief dynamics with representations that involve only a single
numerical opinion. As models of polarization have shown, this does not
generalise well. In particular, the difficulty of obtaining anything other
than convergence in the majority of extant models of opinion dynamics
illustrates that point. Enriching models with arguments or reasons in support
of beliefs is thus essential for a more realistic understanding.

Second, doing so highlights how the distribution of arguments that are, in
principle, available to agents—and from which the arguments they personally
have available are sampled—strongly influences belief dynamics. This opens the
door for a deeper understanding of extant theories, alternative mechanisms for
known phenomena, and novel predictions that can be brought to real-world data.

Third, the fact that ‘the world’ matters to observable belief dynamics
furthermore makes it important that the distributional assumptions about
underlying arguments or evidence are sufficiently grounded. The Bayesian
framework helps with this problem because BN models of domains can be learned
from real-world data (Heckerman, 2008).

Fourth, the rules by which agents evaluate arguments and form beliefs also
clearly matter. Hence it is important to supply agents with principled
argument evaluation and aggregation rules. This is not to claim that humans
are perfect Bayesians. They may approximate Bayesian norms more or less
faithfully in some contexts, however (Peterson and Beach, 1967; Chater et al.,
2006, 2010). The analytic value of considering ‘rational’ or optimal
aggregation, though, does not ultimately rest on the precise extent of that
approximation. Rather, consideration of optimal rules aids the identification
of deep structural challenges that cognitive agents face and the attribution
of ‘bias’ or irrationality is meaningful only against the foil of the
performance of an ideal agent (Hahn and Harris, 2014).

Fifth, even the simple simulations of our case studies highlight the
fundamental impact of agents’ communication rules, that is, what evidence they
choose to communicate and why. This makes clear how much of the final model
behaviour depends on a key component that is itself without a (clear)
normative basis. This reflects a more general gap in normatively oriented
research: normative accounts of _where evidence actually comes_ from are,
arguably, under-developed in this and other contexts (Meder et al., 2022).

In fact, early work in social epistemology emphasised how simulations might
aid the discovery of appropriate norms for communication (Olsson, 2011).
Beyond the question of the frequency or intensity of communication within
epistemic collectives (e.g., (Angere and Olsson, 2017; Zollman, 2010; Borg et
al., 2019; Hahn et al., 2019; Hahn, 2022) very little progress has been made
with respect to this question. Arguably, this is because extant models have
been too restrictive to support much in the way of variation in communication
rules. Even the simple BNs explored in this paper, however, are rich enough to
allow one to formulate key elements of the factors that plausibly influence
communication in real life, such as speaker confidence (influencing whether or
not to speak at all), own belief and perceptions of argument strength which
feed into the pragmatic rules or maxims governing cooperative exchange (Grice,
1969; Levinson, 1983), as well as deviations in non-cooperative exchange.

Sixth and last, communication rules shape the rise and fall of arguments.
Incorporating arguments into the model opens a new world of exploring argument
dynamics alongside belief dynamics. Exploring how argument dynamics are shaped
by communication rules should open up new avenues for linking argumentation
research to work in computational social science. For one, examining real-
world patterns of argument spread across online social media and comparing
this with models could inform the identification of underlying (real-world)
communication rules.

We conclude with some indication of future directions.

####  4.3.1. Future Directions

Crucially, NormAN is conceptualised not as a specific model, but as a
framework in which to develop concrete models by adjusting key components:
world, agent belief formation, communication rules, and network
characteristics.

As argued extensively above, the fact that our framework has normative
grounding is (to us) an essential requirement. That said, however, it is
entirely possible to strip away any such interpretation and simply treat the
Bayesian belief revision implemented for our agents as purely descriptive,
that is, as ‘just another update rule’. From that perspective, Bayesian belief
revision is simply another weighted averaging rule (Jones and Love, 2011).
This makes it an interesting research question how it compares, on both the
individual and collective level, to other popular rules in models of opinion
dynamics.

Second, our initial case studies highlight just how much model behaviour (and,
by the same token, real-world belief- and opinion dynamics) are shaped by
agents’ communication rules. This makes studying the impact of communication
rules a central topic for future research. Accordingly, future versions of
NormAN should implement richer notions of communication: for example, rules
that include a model of the listener (e.g., that listener’s knowledge states
(Levinson, 1983)). This would also enable agents to include strategic
considerations in their communications (Roth et al., 2007; Matt and Toni,
2008; Rahwan and Larson, 2009).

Third, as mentioned above, future NormAN models should allow agents to possess
subjective models of the world (BNs) that differ from the ground truth world,
and that differ across agents.

Fourth, richer agent models should incorporate notions of trust in order to
connect with the rich modelling literature on testimony (e.g., (Olsson, 2011;
Bovens and Hartmann, 2003; Shafto et al., 2012)) and with bounded confidence
models of opinion dynamics (e.g., (Hegselmann and Krause, 2002)).

With respect to ‘the world’, future research should involve systematic
exploration of the impact of different worlds, including richer models
involving many more variables.

Last, but not at all least, there should be a much deeper exploration of the
impact of network topology than the current version allows. In particular, it
will be important to study not just other types of networks (e.g.,
preferential attachment networks (Barabási and Albert, 1999)), and their
impact on argument dynamics. This should also include dynamic networks, in
which agents can change who they communicate with (Sekara et al., 2016); this
not only affords greater realism, but it will specifically allow the study of
the epistemological and argumentative impacts of homophily (McPherson et al.,
2001). Finally, this should include hierarchical networks (Ravasz and
Barabási, 2003).

##  5\. Conclusions

In this paper, we argued that there is currently a significant gap in the
research literature. On the one hand, traditional research on argumentation
does not connect well with the many real-world contexts that involve more than
two agents or competing perspectives. On the other, the wealth of research
trying to understand belief and opinion dynamics across social networks is
limited by the fact that it has not considered, or been able to consider
properly, individual arguments as the actual information exchanged that drives
those dynamics. In order to bridge that gap, agent-based models involving
argument exchange are required.

We argued further that a normative, Bayesian perspective provides a privileged
way to build such a model. We have sought to outline why normative models,
more generally, are relevant not just for research concerned with how we ought
to behave, but also for descriptively oriented research concerns. More
specifically, we have detailed how, within the argumentation literature, the
Bayesian framework allowed one to capture the content of arguments with
sufficient detail to advance long-standing research questions. We have
detailed also how the Bayesian framework allows one to capture belief dynamics
and evidence/argument aggregation. We have shown a novel application of the
Bayesian framework: namely how Bayesian Belief Networks, can be employed to
create a ground truth world and evidence distribution for agent-based
simulations.

These aspects are embodied in NormAN, a new framework for the study of
argument exchange across social networks. We have sought to illustrate with
case studies different ways in which NormAN models might benefit extant
research. It is hoped that NormAN will help bridge the current ‘gap’ and
support new research across the breadth of research on argumentation, opinion
dynamics, and communication discussed in this paper.

###### Acknowledgements.

The research reported in this paper was supported by the UK’s Arts and
Humanities Research Council grant AH/V003380/1, and the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) project number
455912038. L.A. was supported by a Konrad-Adenauer Stiftung scholarship.
Special thanks go to Michael Maes, Davide Grossi for many helpful discussions
and Borut Trpin for feedback on an initial draft of this manuscript.

###  5.1. CRediT statement

Conceptualization: L.A., R.F., U.H., A.J., and L.S. Data curation: K.P. Formal
analysis: K.P. Funding acquisition: U.H. Investigation: K.P. Methodology:
L.A., R.F., U.H., A.J., K.P., and L.S. Project administration: U.H. Resources:
K.P. Software: L.A., A.J., and L.S. Validation: L.A., U.H., K.P., and L.S.
Visualization: R.F., U.H., and K.P. Writing - original draft: L.A. and U.H.
Writing - review & editing: L.A., R.F., U.H., A.J., K.P., and L.S.

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