11institutetext: Cardiff University, Cardiff, UK
11email<EMAIL_ADDRESS>
# A Logic of Expertise

Joseph Singleton

###### Abstract

In this paper we introduce a simple modal logic framework to reason about the
_expertise_ of an information source. In the framework, a source is an expert
on a proposition $p$ if they are able to correctly determine the truth value
of $p$ in any possible world. We also consider how information may be false,
but true after _accounting for the lack of expertise_ of the source. This is
relevant for modelling situations in which information sources make claims
beyond their domain of expertise. We use non-standard semantics for the
language based on an _expertise set_ with certain closure properties. It turns
out there is a close connection between our semantics and S5 epistemic logic,
so that expertise can be expressed in terms of _knowledge_ at all possible
states. We use this connection to obtain a sound and complete axiomatisation.

###### Keywords:

Expertise Modal logic Information

##  1 Introduction

Many scenarios require handling information from non-expert sources. Such
information can be false, even when sources are sincere, when sources make
claims regarding topics on which they are not experts. Accordingly, the (lack
of) expertise of the source must be taken into account when new information is
received.

In this paper we introduce a modal logic formalism for reasoning about the
expertise of information sources. Our logic includes operators for expertise
($\mathsf{E}\varphi$) and _soundness_ ($\mathsf{S}\varphi$). Intuitively, a
source $s$ has expertise on $\varphi$ if $s$ is able to correctly determine
the truth value of $\varphi$ in _any possible world_. On the other hand,
$\varphi$ is sound if it true _up to the limits of the expertise_ of $s$. That
is, if $\varphi$ is logically weakened to ignore information beyond the
expertise of $s$, the resulting formula is true. This provides a crucial link
between expertise and truthfulness of information, which allows some
information to derived from false statements.

The related notion of _trust_ has been well-studied from a logical perspective
[3, 7, 4, 8, 6]. Despite some similarities, trustworthiness on $\varphi$ is
different from expertise on $\varphi$. Firstly, whether $s$ is trusted on
$\varphi$ is a property of the _truster_ , not of $s$. In contrast, we aim to
model expertise objectively as a property of $s$ alone. Secondly, we interpret
expertise _globally_ : whether or not $s$ is an expert on $\varphi$ does not
depend on the truth value of $\varphi$ in any particular state. In this sense
expertise is a counterfactual notion, in that it refers to possible worlds
other than the “actual” one. This is not necessarily so for trust; e.g. we may
not want to trust the judgement of $s$ on $\varphi$ if we know $\varphi$ to be
false in the actual world.

Contribution and paper outline. Our main conceptual contribution is a modal
logic framework for reasoning about expertise and soundness of information.
This framework is motivated via an example in Section 2, after which the
syntax and semantics are formally introduced. Section 3 goes on to establish a
connection between our logic and _S5 epistemic logic_ , which provides an
alternative interpretation of our notion of expertise in terms of S5
knowledge. Our main technical result is a sound and complete axiomatisation,
given in Section 4.

##  2 Expertise and Soundness

The core notions we aim to model are _expertise_ and _soundness_ of
information. We illustrate both with a simplified example.

###### Example 1

Consider an economist reporting on the effects of COVID-19 vaccine rollout,
who states that widespread vaccination will aid economic recovery ($r$), but
that the vaccine can cause health problems ($p$). Assume that the economist is
an expert on matters to do with the economy ($\mathsf{E}r$), so that they only
provide correct information on proposition $r$, but is not an expert on
matters of health ($\neg\mathsf{E}p$). For the sake of the example, suppose
economic recovery will indeed follow, but there are _no_ health problems
associated with the vaccine. Then while the economist’s report of $rp$ is
false, it is true on the propositions on which they are an expert.
Consequently, if one ignores the parts of the report on which the economist
has no expertise, the report becomes true. We say that $rp$ is _sound_ , given
the expertise of the source on $r$ but not $p$.

###  2.1 Syntax

We introduce the language of expertise and soundness. Let $\mathsf{Prop}$ be a
countable set of propositional variables. The language $\mathcal{L}$ is
defined by the following grammar:

$\varphi::=p\mid\neg\varphi\mid\varphi\varphi\mid\mathsf{E}\varphi\mid\mathsf{S}\varphi\mid\mathsf{A}\varphi$

for $p\in\mathsf{Prop}$. Note that formulas of $\mathcal{L}$ describe the
expertise of a _single_ source. The language can be easily extended to handle
multiple sources by adding modalities $\mathsf{E}_{s}$ and $\mathsf{S}_{s}$
for each source $s$, but we do not do so here.

We read $\mathsf{E}\varphi$ as “the source has expertise on $\varphi$”, and
$\mathsf{S}\varphi$ as “$\varphi$ is sound for the source to report”. We
include the universal modality $\mathsf{A}$ [5] for technical reasons to aid
in the axiomatisation of Section 4; $\mathsf{A}\varphi$ is to be read as
“$\varphi$ holds in all states”. Other Boolean connectives ($\vee$,
$\rightarrow$, $\leftrightarrow$) and truth values ($\top$, $\bot$) are
introduced as abbreviations. We denote by $\hat{\mathsf{E}}$,
$\hat{\mathsf{S}}$ and $\hat{\mathsf{A}}$ the dual operators corresponding to
$\mathsf{E}$, $\mathsf{S}$ and $\mathsf{A}$ respectively (e.g.
$\hat{\mathsf{E}}\varphi$ stands for $\neg\mathsf{E}\neg\varphi$).

###  2.2 Semantics

Formulas of $\mathcal{L}$ are interpreted via non-standard semantics on the
basis of an _expertise set_.

###### Definition 1

An _expertise frame_ is a pair $F=(X,P)$, where $X$ is a set of states and
$P\subseteq 2^{X}$ is an _expertise set_ satisfying the following properties:

  1. (P1) $X\in P$

  2. (P2) If $A\in P$ then $X\setminus A\in P$

  3. (P3) If $\\{A_{i}\\}_{i\in I}\subseteq P$, then $\bigcap_{i\in I}A_{i}\in P$

An _expertise model_ is a triple $M=(X,P,v)$, where $(X,P)$ is an expertise
frame and $v:\mathsf{Prop}\to 2^{X}$ is a valuation function.

Intuitively, $A\in P$ means the source has the expertise to determine whether
or not the “actual” state of the world, whatever that may be, lies in $A$.
Implicitly in this interpretation we assume that the expertise of the source
does not depend on this “actual” state. This makes our semantics for expertise
formulas a special case of the _neighbourhood semantics_ [9], where the
neighbourhoods of all states in $X$ are the same (i.e. $N(x)\equiv P$).

(P1) simply says that the source is able to determine that the state lies in
$X$, i.e. that the source has expertise on tautologies. (P2) says that $P$ is
closed under complements. This is a natural requirement, given our intended
interpretation of $P$: if the source can determine whether the actual state
$x$ lies inside $A$ or not, the same clearly holds for $X\setminus A$. Note
that (P1) and (P2) together imply that $\emptyset\in P$. Finally, (P3) says
that $P$ is closed under (arbitrary) intersections, which implies expertise is
closed under conjunctions. Together with (P2), this implies $P$ is also closed
under (arbitrary) unions, and thus expertise is also closed under
disjunctions. We come to the truth conditions for $\mathcal{L}$ formulas with
respect to expertise models.

###### Definition 2

Let $M=(X,P,v)$ be an expertise model. The satisfaction relation between
points $x\in X$ and formulas $\varphi\in\mathcal{L}$ is defined inductively as
follows:

$\begin{array}[]{lll}&M,x\vDash p&\iff x\in v(p)\\\ &M,x\vDash\neg\varphi&\iff
M,x\not\vDash\varphi\\\ &M,x\vDash\varphi\psi&\iff M,x\vDash\varphi\text{ and
}M,x\vDash\psi\\\ &M,x\vDash\mathsf{E}\varphi&\iff\|\varphi\|_{M}\in P\\\
&M,x\vDash\mathsf{S}\varphi&\iff\text{ for all }A\in P,\
\|\varphi\|_{M}\subseteq A\text{ implies }x\in A\\\
&M,x\vDash\mathsf{A}\varphi&\iff\text{ for all }y\in X,\
M,y\vDash\varphi\end{array}$

where $\|\varphi\|_{M}=\\{x\in X\mid M,x\vDash\varphi\\}$. We write
$M\vDash\varphi$ if $M,x\vDash\varphi$ for all $x\in X$, and $\vDash\varphi$
if $M\vDash\varphi$ for all expertise models $M$; we say $\varphi$ is _valid_
in this case. Write $\varphi\equiv\psi$ if $\vDash\varphi\leftrightarrow\psi$.

The clauses for propositional variables and propositional connectives are
standard, and the clause for $\mathsf{A}\varphi$ is straightforward. The
clause for $\mathsf{E}\varphi$ follows the intuition highlighted above:
$\mathsf{E}\varphi$ is true iff the set of states $\|\varphi\|_{M}$ at which
$\varphi$ is true lies in the expertise set. Note that the truth value of
$\mathsf{E}\varphi$ does not depend on the state $x$. For $\mathsf{S}\varphi$
to hold at $x\in X$, we require that all supersets of $\|\varphi\|_{M}$ on
which the source is an expert must contain $x$. That is, any logical weakening
of $\varphi$ is true at $x$, whenever the source has expertise on the weaker
formula. We illustrate the semantics by formalising Example 1, and conclude
this section by noting some (in)validities that follow directly from
Definition 2.

###### Example 2

Consider a model $M=(X,P,v)$, where $X=\\{a,b,c,d\\}$, $v(r)=\\{a,c\\}$ and
$v(p)=\\{a,b\\}$, and $P=\\{\emptyset,\\{a,c\\},\\{b,d\\},X\\}$. Then $a$
satisfies $rp$, $b$ satisfies $\neg rp$, $c$ satisfies $r\neg p$, and $d$
satisfies $\neg r\neg p$.

In Example 1 we assumed the economist had expertise on $r$. Here we have
$\|r\|_{M}=\\{a,c\\}\in P$, so $M\vDash\mathsf{E}r$ as expected. We also
claimed $rp$ was sound when $r$ is true and $p$ is false, i.e.
$M,c\vDash\mathsf{S}(rp)$. Indeed, $\|rp\|_{M}=\\{a\\}$, and the supersets of
$\\{a\\}$ in $P$ are $\\{a,c\\}$ and $X$. Clearly both contain $c$, so
$M,c\vDash\mathsf{S}(rp)$. This situation is also depicted graphically in Fig.
1.

$a$$b$$c$$d$$\|p\|_{M}$$\|r\|_{M}$ Figure 1:  Graphical depiction of the model
in Example 2. The red boxes show the sets in $P$ (with $\emptyset$ and $X$
omitted). While the report $rp$ is false at $c$ (i.e. $c$ is not contained in
the blue box), all red boxes containing the blue box do contain $c$, so
$M,c\vDash\mathsf{S}(rp)$.

###### Proposition 1

For any $\varphi,\psi\in\mathcal{L}$ and any model $M$,

  1. 1.

$\mathsf{E}\varphi\equiv\mathsf{E}\neg\varphi\equiv\mathsf{A}\mathsf{E}\varphi$

  2. 2.

Either $M\vDash\mathsf{E}\varphi$ or $M\vDash\neg\mathsf{E}\varphi$

  3. 3.

$\vDash\mathsf{E}\top\mathsf{E}\bot\mathsf{E}\mathsf{E}\varphi$

  4. 4.

$\vDash(\mathsf{E}\varphi\mathsf{E}\psi)\rightarrow\mathsf{E}(\varphi\psi)$

  5. 5.

The distribution axiom
$\mathsf{E}(\varphi\rightarrow\psi)\rightarrow(\mathsf{E}\varphi\rightarrow\mathsf{E}\psi)$
is not in general valid111 For a counterexample, consider $X=\\{a,b,c\\}$,
$P=\\{\emptyset,\\{a\\},\\{b,c\\},X\\}$ and $v$ such that $v(p)=\\{a\\}$,
$v(q)=\\{b\\}$. Then $\|p\|_{M}=\\{a\\}\in P$, $\|q\|_{M}=\\{b\\}\notin P$ and
$\|p\rightarrow q\|_{M}=\\{b,c\\}\in P$. Then we have
$M\vDash\mathsf{E}(p\rightarrow q)\mathsf{E}p\neg\mathsf{E}q$.

  6. 6.

$\vDash\varphi\rightarrow\mathsf{S}\varphi$

  7. 7.

If $\vDash\varphi\rightarrow\psi$ then
$\vDash(\mathsf{S}\varphi\mathsf{E}\psi)\rightarrow\psi$

##  3 Connection with S5 Epistemic Logic

In this section we show that, despite the non-standard semantics for
$\mathsf{E}\varphi$ and $\mathsf{S}\varphi$, expertise and soundness can be
equivalently defined by the standard relational semantics [1] in the language
$\mathcal{L}_{\mathsf{K}\mathsf{A}}$ with knowledge and universal modalities
$\mathsf{K}$ and $\mathsf{A}$. It will be seen that the accessibility relation
for $\mathsf{K}$ in the relational model $M^{*}$ corresponding to an expertise
model $M$ is in fact an equivalence relation, so that $M^{*}$ is an _S5 model_
[1, §4.1]. S5 represents an “ideal” form of knowledge, which satisfies the KT5
axioms: knowledge is closed under logical consequence (K:
$\mathsf{K}(\varphi\rightarrow\psi)\rightarrow(\mathsf{K}\varphi\rightarrow\mathsf{K}\psi)$),
all that is known is true (T: $\mathsf{K}\varphi\rightarrow\varphi$), and if
$\varphi$ is not known, this is itself known (5:
$\neg\mathsf{K}\varphi\rightarrow\mathsf{K}\neg\mathsf{K}\varphi$). First, let
us define the relational semantics for $\mathcal{L}_{\mathsf{K}\mathsf{A}}$.

###### Definition 3

A _relational model_ is a triple $M^{\prime}=(X,R,v)$, where $X$ is a set of
states, $R\subseteq X\times X$ is a binary relation on $X$, and
$v:\mathsf{Prop}\to 2^{X}$ is a valuation. Given a relational model
$M^{\prime}$, the satisfaction relation between points $x\in X$ and formulas
$\varphi\in\mathcal{L}_{\mathsf{K}\mathsf{A}}$ is defined inductively by

$\begin{array}[]{lll}&M^{\prime},x\vDash\mathsf{K}\varphi&\iff\text{ for all
}y\in X,xRy\text{ implies }M^{\prime},y\vDash\varphi\\\
&M^{\prime},x\vDash\mathsf{A}\varphi&\iff\text{ for all }y\in X,\
M^{\prime},y\vDash\varphi\end{array}$

where the clauses for propositional connectives are as in Definition 2.

Say a relational model $M^{\prime}=(X,R,v)$ is an _S5 model_ if $R$ is an
equivalence relation. In the context of S5, $R$ is an _epistemic accessibility
relation_ : $xRy$ means the source considers $y$ as a possible state if the
actual state is $x$. The source then ‘knows’ $\varphi$ at $x$ if $\varphi$ is
true in every state accessible from $x$.

The following result shows how one can form a unique S5 model from an
expertise model, and vice versa.

###### Lemma 1

Let $X$ be a set. Let $\mathcal{P}$ denote the set of expertise sets over $X$
– i.e. the set of all $P\subseteq X$ satisfying (P1), (P2) and (P3) – and let
$\mathcal{E}$ denote the set of equivalence relations over $X$. Then there is
a bijection $P\mapsto R_{P}$ from $\mathcal{P}$ into $\mathcal{E}$ such that,
for all $A\subseteq X$,

$A\in P\iff A\text{ is a union of equivalence classes of }R_{P}$ (1)

###### Proof

Given an expertise set $P\in\mathcal{P}$ and $x\in X$, write

$A_{x}=\bigcap\\{A\in P\mid x\in A\\}$

Note that $A_{x}\in P$ by (P3), so $A_{x}$ is the smallest set in $P$
containing $x$.222Also note that $X\in P$ by (P1), so there is at least one
$A\in P$ containing $x$. Set $\Pi=\\{A_{x}\mid x\in X\\}\subseteq P$. We claim
that $\Pi$ is a partition of $X$. It is clear that $\Pi$ covers $X$, since
each $x$ lies in $A_{x}$ by definition. We show that any distinct
$A_{x},A_{y}\in\Pi$ are disjoint. Without loss of generality,
$A_{x}\not\subseteq A_{y}$. Hence $x\notin A_{y}$ (otherwise $A_{y}$ appears
in the intersection defining $A_{x}$ and we get $A_{x}\subseteq A_{y}$). That
is, $x\in A_{x}\setminus A_{y}=A_{x}\cap(X\setminus A_{y})$. But this
difference lies in $P$ by (P2) and (P3). Since $A_{x}$ is the smallest set in
$P$ containing $x$, we get $A_{x}\subseteq A_{x}\setminus A_{y}$. In
particular, $A_{x}\cap A_{y}=\emptyset$.

Let $R_{P}$ be the equivalence relation defined by the partition $\Pi$, i.e.
$xR_{P}y$ iff $A_{x}=A_{y}$. We show Eq. 1 holds. First suppose $A\in P$. Then
$A=\bigcup_{x\in A}A_{x}$; the left-to-right inclusion is clear since $x\in
A_{x}$ for all $x$, and the right-to-left inclusion holds since $x\in A$
implies $A_{x}\subseteq A$ for $A\in P$. Since the $A_{x}$ form the
equivalence classes of $R_{P}$, we are done.

Now suppose $A$ is a union of equivalence classes of $R_{P}$, i.e.
$A=\bigcup_{x\in B}{A_{x}}$ for some $B\subseteq X$. Since each $A_{x}$ lies
in $P$ and $P$ is closed under unions by (P2) and (P3), we have $A\in P$.
Hence Eq. 1 is shown.

It only remains to show that the mapping $P\mapsto R_{P}$ is bijective.
Injectivity follows easily from Eq. 1, since $P$ is fully determined by
$R_{P}$. For surjectivity, take any equivalence relation $R\in\mathcal{E}$ on
$X$. For $x\in X$, let $[x]_{R}$ denote the equivalence class of $X$. Let $P$
consist of all unions of equivalence classes, i.e.

$P=\left\\{\bigcup_{x\in B}{[x]_{R}}\mid B\subseteq X\right\\}$

We need to show that $P\in\mathcal{P}$ – i.e. (P1), (P2) and (P3) hold – and
that $R_{P}=R$. For (P1), taking $B=X$ gives $X\in P$. For (P2), suppose
$A=\bigcup_{x\in B}{[x]_{R}}\in P$. It is easily verified that $X\setminus
A=\bigcup_{y\in X\setminus A}{[y]_{R}}\in P$, so (P2) holds. (P3) follows from
(P2) and the fact that $P$ is closed under unions, which is evident from the
definition. Finally, it follows from the definition of $P$ and Eq. 1 that a
set $A\subseteq X$ is a union of equivalence classes of $R$ if and only if it
is a union of equivalence classes of $R_{P}$. Since distinct equivalence
classes are disjoint, this implies that the equivalence classes of $R$
coincide with those of $R_{P}$, and $R=R_{P}$ as required. ∎

On the syntactic side, define a translation
$t:\mathcal{L}\to\mathcal{L}_{\mathsf{K}\mathsf{A}}$ inductively by $t(p)=p$,
$t(\neg\varphi)=\neg t(\varphi)$, $t(\varphi\psi)=t(\varphi)t(\psi)$,
$t(\mathsf{A}\varphi)=\mathsf{A}t(\varphi)$, and

$t(\mathsf{E}\varphi)=\mathsf{A}(t(\varphi)\rightarrow\mathsf{K}t(\varphi));\quad\quad
t(\mathsf{S}\varphi)=\neg\mathsf{K}\neg t(\varphi)$

We then have that $\varphi\in\mathcal{L}$ is true in an expertise model
exactly when when $t(\varphi)$ is true in the induced S5 model $M^{*}$.

###### Theorem 3.1

Let $M=(X,P,v)$ be an expertise model. Then $M^{*}=(X,R_{P},v)$ is an S5
model, and

$M,x\vDash\varphi\iff M^{*},x\vDash t(\varphi)$

Before the proof, note that since the mapping $P\mapsto R_{P}$ is a bijection
into the set of equivalence relations on $X$ (by Lemma 1), any S5 model
$M^{\prime}=(X,R,v)$ has an expertise counterpart $M=(X,P,v)$ such that
$M^{*}=M^{\prime}$. In this sense, the converse of Theorem 3.1 also holds.

######  Proof (Theorem 3.1)

Let $M=(X,P,v)$ be an expertise model. By Lemma 1, $R_{P}$ is an equivalence
relation and $M^{*}$ is indeed an S5 model. Let $\Pi$ denote the partition of
$X$ corresponding to $R_{P}$, and as in Lemma 1, let $A_{x}\in\Pi$ denote the
cell of $\Pi$ containing $x$, i.e. the equivalence class of $x$ in $R_{P}$. By
Eq. 1 in Lemma 1, $A\in P$ iff $A$ is a union of cells from $\Pi$.

We show the desired semantic correspondence by induction on formulas. The
cases for the Boolean connectives and $\mathsf{A}$ are straightforward.
Suppose the result holds for $\varphi$ and $M,x\vDash\mathsf{E}\varphi$. Then
$\|\varphi\|_{M}\in P$, so $\|\varphi\|_{M}=\bigcup\mathcal{A}$ for some
collection $\mathcal{A}\subseteq\Pi$. Now suppose $y\in X$ and $M^{*},y\vDash
t(\varphi)$. Suppose $yR_{P}z$. By the inductive hypothesis,
$\|t(\varphi)\|_{M^{*}}=\|\varphi\|_{M}$, so $y\in\bigcup\mathcal{A}$. Since
$A_{y}$ is the unique set in $\Pi$ containing $y$, we must have
$A_{y}\in\mathcal{A}$. Consequently, $yR_{P}z$ implies $z\in
A_{y}\subseteq\bigcup\mathcal{A}=\|t(\varphi)\|_{M^{*}}$. That is,
$M^{*},z\vDash t(\varphi)$. This shows $M^{*},y\vDash
t(\varphi)\rightarrow\mathsf{K}t(\varphi)$ for arbitrary $y\in X$, and so
$M^{*},x\vDash\mathsf{A}(t(\varphi)\rightarrow\mathsf{K}t(\varphi))$, i.e.
$M^{*},x\vDash t(\mathsf{E}\varphi)$.

Conversely, suppose
$M^{*},x\vDash\mathsf{A}(t(\varphi)\rightarrow\mathsf{K}t(\varphi))$. We claim
that $\|\varphi\|_{M}=\bigcup_{y\in\|\varphi\|_{M}}A_{y}$. The left-to-right
inclusion is clear since $y\in A_{y}$ for each $y$. For the reverse inclusion,
let $y\in\|\varphi\|_{M}$ and $z\in A_{y}$. Then $yR_{P}z$. By the inductive
hypothesis, $M^{*},y\vDash t(\varphi)$. Since
$t(\varphi)\rightarrow\mathsf{K}t(\varphi)$ holds everywhere in $M^{*}$ by
assumption, we have $M^{*},y\vDash\mathsf{K}t(\varphi)$. Hence $M^{*},z\vDash
t(\varphi)$, so $M,z\vDash\varphi$ and $z\in\|\varphi\|_{M}$. This shows
$\|\varphi\|_{M}=\bigcup_{y\in\|\varphi\|_{M}}A_{y}$, i.e. $\|\varphi\|_{M}$
is a union of cells of $\Pi$. Hence $\|\varphi\|_{M}\in P$ and
$M,x\vDash\mathsf{E}\varphi$ as required.

Next we take the $\mathsf{S}\varphi$ case. We prove both directions by
contraposition. First suppose $M,x\not\vDash\mathsf{S}\varphi$. Then there is
some $A\in P$ with $\|\varphi\|_{M}\subseteq A$ and $x\notin A$. Suppose
$xR_{P}y$. Then $A_{x}=A_{y}$. If $y\in A$ we would get $A_{x}=A_{y}\subseteq
A$, since $A_{y}$ is the smallest set in $P$ containing $y$, but this
contradicts $x\notin A$. Hence $y\notin A$. In particular,
$y\notin\|\varphi\|_{M}$. By the inductive hypothesis,
$y\notin\|t(\varphi)\|_{M^{*}}$, so $M^{*},y\vDash\neg t(\varphi)$. This shows
$M^{*},x\vDash\mathsf{K}\neg t(\varphi)$, i.e.
$M^{*},x\not\vDash\neg\mathsf{K}\neg t(\varphi)$ as required.

Finally, suppose $M^{*},x\not\vDash\neg\mathsf{K}\neg t(\varphi)$. Take
$A=\bigcup_{y\in\|\varphi\|_{M}}A_{y}$. Since each $A_{y}$ is in $P$ and $P$
is closed under unions by (P2) and (P3), we have $A\in P$. Clearly
$\|\varphi\|_{M}\subseteq A$. Suppose for contradiction that
$M,x\vDash\mathsf{S}\varphi$. Then $x\in A$, i.e. there is
$y\in\|\varphi\|_{M}$ such that $x\in A_{y}$. Consequently $xR_{P}y$, and
$M^{*},x\vDash\mathsf{K}\neg t(\varphi)$ implies $M^{*},y\vDash\neg
t(\varphi)$. But this means $y\notin\|t(\varphi)\|_{M^{*}}=\|\varphi\|_{M}$ –
contradiction. ∎

Note that, in the case of a propositional formula $\varphi$, the translation
$t$ takes $\mathsf{S}\varphi$ to $\neg\mathsf{K}\neg\varphi$, and
$\mathsf{E}\varphi$ to $\mathsf{A}(\varphi\rightarrow\mathsf{K}\varphi)$. The
semantic correspondence in Theorem 3.1 therefore shows that the soundness
operator $\mathsf{S}$ is just the dual of an S5 knowledge operator: $\varphi$
is sound iff the source does not _know_ $\neg\varphi$. Similarly,
$\mathsf{E}\varphi$ holds iff for _all_ possible states, if $\varphi$ were
true then the source would know it. Moreover, the equivalence relation used to
interpret $\mathsf{K}$ is uniquely derived from the expertise model which
interprets $\mathsf{E}$ and $\mathsf{S}$, by Lemma 1. This gives a new
interpretation of expertise and soundness which refers directly to the
source’s epistemic state via the $\mathsf{K}$ operator.

Theorem 3.1 also allows $\mathsf{E}\varphi$ be expressed solely in terms of
$\mathsf{A}$ and $\mathsf{S}$:

$\mathsf{E}\varphi\equiv\mathsf{A}(\mathsf{S}\varphi\rightarrow\varphi)$

i.e. the source has expertise on $\varphi$ iff, in every possible state,
$\varphi$ is sound only if it is in fact true. This can be seen by recalling
that $\mathsf{E}\varphi$ is equivalent to $\mathsf{E}\neg\varphi$ (by
Proposition 1), and noting that $t(\mathsf{E}\neg\varphi)$ is equivalent to
$t(\mathsf{A}(\mathsf{S}\varphi\rightarrow\varphi))$. Similarly, we can _lack_
of expertise in terms of $\mathsf{S}$ and the dual operator
$\hat{\mathsf{A}}$:

$\neg\mathsf{E}\varphi\equiv\hat{\mathsf{A}}(\mathsf{S}\varphi\neg\varphi)\equiv\hat{\mathsf{A}}(\varphi\mathsf{S}\neg\varphi)$

##  4 Axiomatisation

Theorem 3.1 demonstrates a close semantic link between the logic of expertise
and S5. Accordingly, we can obtain a sound and complete axiomatisation of the
validities of $\mathcal{L}$ by adapting any axiomatisation of S5 (although
some care is required to handle the universal modality). Let $\mathsf{L}$ be
the extension of the propositional calculus containing the axioms and
inference rules shows in Table 1.

Table 1: Axioms and inference rules for $\mathsf{L}$. (K$\mathsf{S}$) | $\mathsf{S}\varphi\neg\mathsf{S}\psi\rightarrow\mathsf{S}(\varphi\neg\psi)$
---|---
(T$\mathsf{S}$) | $\varphi\rightarrow\mathsf{S}\varphi$
(5$\mathsf{S}$) | $\mathsf{S}\neg\mathsf{S}\varphi\rightarrow\neg\mathsf{S}\varphi$
(K$\mathsf{A}$) | $\mathsf{A}(\varphi\rightarrow\psi)\rightarrow(\mathsf{A}\varphi\rightarrow\mathsf{A}\psi)$
(T$\mathsf{A}$) | $\mathsf{A}\varphi\rightarrow\varphi$
(5$\mathsf{A}$) | $\neg\mathsf{A}\varphi\rightarrow\mathsf{A}\neg\mathsf{A}\varphi$
(ES) | $\mathsf{E}\varphi\leftrightarrow\mathsf{A}(\mathsf{S}\varphi\rightarrow\varphi)$
(Inc) | $\mathsf{A}\varphi\rightarrow\neg\mathsf{S}\neg\varphi$
(MP) | From $\varphi$ and $\varphi\rightarrow\psi$ infer $\psi$
(Nec$\mathsf{A}$) | From $\varphi$ infer $\mathsf{A}\varphi$
(R$\mathsf{S}$) | From $\varphi\leftrightarrow\psi$ infer $\mathsf{S}\varphi\leftrightarrow\mathsf{S}\psi$

Here (K$\mathsf{A}$), (T$\mathsf{A}$) and (5$\mathsf{A}$) are the standard KT5
axioms for $\mathsf{A}$, which characterise S5. (K$\mathsf{S}$),
(T$\mathsf{S}$) and (5$\mathsf{S}$) are reformulations of the KT5 axioms for
the dual operator $\hat{\mathsf{S}}=\neg\mathsf{S}\neg$; we present them in
terms of $\mathsf{S}$ rather than $\hat{\mathsf{S}}$ to aid readability and
intuitive interpretation of the axioms. (ES) captures the interaction between
expertise and soundness; the validity of this axiom was already shown as a
consequence of Theorem 3.1. Note that the necessitation rule (Nec$\mathsf{A}$)
for $\mathsf{A}$ and (Inc) imply necessitation for $\hat{\mathsf{S}}$ by (MP).

###### Theorem 4.1

$\mathsf{L}$ is sound333Soundness of the logic $\mathsf{L}$ for expertise
frames should not be confused with the notion of soundness inside the
language. and complete with respect to expertise frames.

Soundness is immediate for most of the axioms and inference rules. We give
details only for (K$\mathsf{S}$) and (5$\mathsf{S}$).

###### Lemma 2

Axioms (K$\mathsf{S}$) and (5$\mathsf{S}$) are valid in all expertise frames.

###### Proof

Let $M=(X,P,v)$ be an expertise model. For (K$\mathsf{S}$), suppose
$M,x\vDash\mathsf{S}\varphi\neg\mathsf{S}\psi$. Take any $A\in P$ with
$\|\varphi\neg\psi\|_{M}\subseteq A$. Then
$\|\varphi\|_{M}\setminus\|\psi\|_{M}\subseteq A$, so
$\|\varphi\|_{M}\subseteq A\cup\|\psi\|_{M}$. Since
$M,x\not\vDash\mathsf{S}\psi$, there is $B\in P$ with $\|\psi\|_{M}\subseteq
B$ and $x\notin B$. Now, $\|\varphi\|_{M}\subseteq A\cup\|\psi\|_{M}\subseteq
A\cup B$, and $A\cup B\in P$ since $P$ is closed under unions. Since
$M,x\vDash\mathsf{S}\varphi$, any superset of $\|\varphi\|_{M}$ in $P$ must
contain $x$. Hence $x\in A\cup B$. But $x\notin B$, so we must have $x\in A$.
This shows $M,x\vDash\mathsf{S}(\varphi\neg\psi)$.

For (5$\mathsf{S}$), suppose $M,x\vDash\mathsf{S}\neg\mathsf{S}\varphi$. It
can be seen from Definition 2 that
$\|\neg\mathsf{S}\varphi\|_{M}=\bigcup\\{X\setminus B\mid B\in
P,\|\varphi\|_{M}\subseteq B\\}$. It follows from (P2) and (P3) that
$\|\neg\mathsf{S}\varphi\|_{M}\in P$. Consequently,
$\|\neg\mathsf{S}\varphi\|_{M}$ is itself a set in $P$ containing
$\|\neg\mathsf{S}\varphi\|_{M}$. Since
$M,x\vDash\mathsf{S}\neg\mathsf{S}\varphi$ we get
$x\in\|\neg\mathsf{S}\varphi\|_{M}$, i.e. $M,x\vDash\neg\mathsf{S}\varphi$ as
required. ∎

The completeness proof requires some more machinery, and we use ideas found in
[2, 5]. Let $\mathcal{L}_{\mathsf{S}\mathsf{A}}$ denote the fragment of
$\mathcal{L}$ without the $\mathsf{E}$ modality, and let
$\mathsf{L}_{\mathsf{S}\mathsf{A}}$ denote the logic of $\mathsf{L}$ for
$\mathcal{L}_{\mathsf{S}\mathsf{A}}$ without axiom (ES). For a frame
$F=(X,P)$, let $R_{P}$ denote the corresponding equivalence relation on $X$
from Lemma 1. An _augmented expertise frame_ is obtained by adding to any
frame $F$ an equivalence relation $R_{\mathsf{A}}$ on $X$ such that
$R_{P}\subseteq R_{\mathsf{A}}$ (c.f. [2]). An augmented model $N$ is an
augmented frame equipped with a valuation $v$. We define a satisfaction
relation $\vDash^{\text{aug}}$ between augmented models and
$\mathcal{L}_{\mathsf{S}\mathsf{A}}$ formulas, where

$N,x\vDash^{\text{aug}}\mathsf{A}\varphi\iff\text{ for all }y\in
X,x{R_{\mathsf{A}}}y\text{ implies }N,y\vDash^{\text{aug}}\varphi$

and satisfaction for other formulas is as in Definition 2. That is,
$\mathsf{A}\varphi$ is no longer the universal modality, and is instead
interpreted via relational semantics.

###### Lemma 3

$\mathsf{L}_{\mathsf{S}\mathsf{A}}$ is complete for
$\mathcal{L}_{\mathsf{S}\mathsf{A}}$ with respect to augmented frames.

######  Proof (sketch)

First note that from (K$\mathsf{S}$), (T$\mathsf{S}$), (5$\mathsf{S}$) and
(R$\mathsf{S}$), one can prove as theorems of
$\mathsf{L}_{\mathsf{S}\mathsf{A}}$ the usual KT5 axioms for the dual operator
$\hat{\mathsf{S}}$ – that is,
$\vdash_{\mathsf{L}_{\mathsf{S}\mathsf{A}}}\hat{\mathsf{S}}(\varphi\rightarrow\psi)\rightarrow(\hat{\mathsf{S}}\varphi\rightarrow\hat{\mathsf{S}}\psi)$,
$\vdash_{\mathsf{L}_{\mathsf{S}\mathsf{A}}}\hat{\mathsf{S}}\varphi\rightarrow\psi$
and
$\vdash_{\mathsf{L}_{\mathsf{S}\mathsf{A}}}\neg\hat{\mathsf{S}}\varphi\rightarrow\hat{\mathsf{S}}\neg\hat{\mathsf{S}}\varphi$
– where $\hat{\mathsf{S}}$ is an abbreviation for $\neg\mathsf{S}\neg$. As
remarked before, we also have the necessitation rule for both
$\hat{\mathsf{S}}$ and $\mathsf{A}$ by (Nec$\mathsf{A}$) and (Inc).

By the standard canonical model construction [1], we obtain the canonical
relational model $(X,R_{\hat{\mathsf{S}}},R_{\mathsf{A}},v)$, where $X$ is the
set of all maximally $\mathsf{L}_{\mathsf{S}\mathsf{A}}$-consistent subsets
(MCS) of $\mathcal{L}_{\mathsf{S}\mathsf{A}}$, $R_{\hat{\mathsf{S}}}$ and
$R_{\mathsf{A}}$ are accessibility relations for $\hat{\mathsf{S}}$ and
$\mathsf{A}$ respectively, and $\Delta\in X$ satisfies $\varphi$ under the
relational semantics iff $\varphi\in\Delta$, for any
$\varphi\in\mathcal{L}_{\mathsf{S}\mathsf{A}}$ (this fact is known as the
_truth lemma_). Moreover, $R_{\hat{\mathsf{S}}}$ and $R_{\mathsf{A}}$ are
equivalence relations by the KT5 axioms for $\hat{\mathsf{S}}$ and
$\mathsf{A}$ respectively, and (Inc) implies $R_{\hat{\mathsf{S}}}\subseteq
R_{\mathsf{A}}$. By Lemma 1, there is an expertise set $P$ such that
$R_{P}=R_{\hat{S}}$. Consequently, we obtain an augmented model
$N=(X,P,R_{\mathsf{A}},v)$. Applying the link between expertise-based and
relational semantics established in Theorem 3.1, one can adapt the truth lemma
to show that $N,\Delta\vDash^{\text{aug}}\varphi$ iff $\varphi\in\Delta$ for
any MCS $\Delta\in X$ and $\varphi\in\mathcal{L}_{\mathsf{S}\mathsf{A}}$.
Completeness now follows by contraposition. If
$\varphi\in\mathcal{L}_{\mathsf{S}\mathsf{A}}$ is not a theorem of
$\mathsf{L}_{\mathsf{S}\mathsf{A}}$, then $\\{\neg\varphi\\}$ is
$\mathsf{L}_{\mathsf{S}\mathsf{A}}$-consistent, and so there is a MCS $\Delta$
containing $\neg\varphi$ by Lindenbaum’s Lemma [1]. Consequently
$\varphi\notin\Delta$, so $N,\Delta\not\vDash^{\text{aug}}\varphi$ and
$\varphi$ is not valid in augmented frames. ∎

Completeness of $\mathsf{L}_{\mathsf{S}\mathsf{A}}$ for (non-augmented)
expertise frames follows by considering _generated sub-frames_ of augmented
frames.

###### Lemma 4

$\mathsf{L}_{\mathsf{S}\mathsf{A}}$ is complete for
$\mathcal{L}_{\mathsf{S}\mathsf{A}}$ with respect to expertise frames.

######  Proof (sketch)

Suppose $\varphi\in\mathcal{L}_{\mathsf{S}\mathsf{A}}$ is not a theorem of
$\mathsf{L}_{\mathsf{S}\mathsf{A}}$. By Lemma 3, there is an augmented model
$N=(X,P,R_{\mathsf{A}},v)$ and a state $x\in X$ such that
$N,x\not\vDash^{\text{aug}}\varphi$. Let $X^{\prime}\subseteq X$ be the
equivalence class of $x$ in $R_{\mathsf{A}}$. Consider the generated sub-model
$N^{\prime}=(X^{\prime},P^{\prime},R^{\prime}_{\mathsf{A}},v^{\prime})$, where
$P^{\prime}=\\{A\cap X^{\prime}\mid A\in P\\}$,
$R^{\prime}_{\mathsf{A}}=R_{\mathsf{A}}\cap(X^{\prime}\times X^{\prime})$ and
$v^{\prime}(p)=v(p)\cap X^{\prime}$. It can be shown that for all
$\psi\in\mathcal{L}_{\mathsf{S}\mathsf{A}}$ and $y\in X^{\prime}$, we have
$N,y\vDash^{\text{aug}}\psi$ iff $N^{\prime},y\vDash^{\text{aug}}\psi$.444
This is clear by induction on formulas, except for the case $\mathsf{S}\psi$.
Here we use the fact that $R_{P}\subseteq R_{\mathsf{A}}$ to show
$X^{\prime}=\bigcup_{z\in X^{\prime}}[z]_{R_{P}}$ – where $[z]_{R_{P}}$ is the
equivalence class of $z$ in $R_{P}$ – which implies $X^{\prime}\in P$. Using
the inductive hypothesis it is then straightforward to show that
$N,y\vDash^{\text{aug}}\mathsf{S}\psi$ iff
$N^{\prime},y\vDash^{\text{aug}}\mathsf{S}\psi$.  Hence
$N^{\prime},x\not\vDash^{\text{aug}}\varphi$.

Now, note that the relation $R^{\prime}_{\mathsf{A}}$ was obtained by
restricting $R_{\mathsf{A}}$ to one of its equivalence classes $X^{\prime}$.
It follows that $R^{\prime}_{\mathsf{A}}$ is in fact the universal relation
$X^{\prime}\times X^{\prime}$ on $X^{\prime}$. Consequently,
$N^{\prime},y\vDash^{\text{aug}}\mathsf{A}\psi$ iff
$N^{\prime},z\vDash^{\text{aug}}\varphi$ for all $z\in X^{\prime}$, i.e.
$\mathsf{A}$ is just the universal modality for $N^{\prime}$.

Writing $M$ for the non-augmented model obtained from $N^{\prime}$ by simply
dropping the $R^{\prime}_{\mathsf{A}}$ component, we see that $M,y\vDash\psi$
iff $N^{\prime},y\vDash^{\text{aug}}\psi$, for all $y\in X^{\prime}$ and
$\psi\in\mathcal{L}_{\mathsf{S}\mathsf{A}}$. In particular,
$M,x\not\vDash\varphi$, so $\varphi$ is not valid in all expertise frames. ∎

The completeness of $\mathsf{L}$ for the validities of the whole language
$\mathcal{L}$ now follows. Indeed, let
$g:\mathcal{L}\to\mathcal{L}_{\mathsf{S}\mathsf{A}}$ be the natural embedding
of $\mathcal{L}$ in $\mathcal{L}_{\mathsf{S}\mathsf{A}}$, where
$g(\mathsf{E}\varphi)=\mathsf{A}(\mathsf{S}g(\varphi)\rightarrow
g(\varphi))$.555…and $g(p)=p$, $g(\neg\varphi)=\neg g(\varphi)$,
$g(\varphi\psi)=g(\varphi)g(\psi)$,
$g(\mathsf{S}\varphi)=\mathsf{S}g(\varphi)$ and
$g(\mathsf{A}\varphi)=\mathsf{A}g(\varphi)$. In light of earlier remarks and
axiom (ES), we have both $\varphi\equiv g(\varphi)$ and
$\vdash_{\mathsf{L}}\varphi\leftrightarrow g(\varphi)$ for all
$\varphi\in\mathcal{L}$. Consequently, $\vDash\varphi$ implies $\vDash
g(\varphi)$ and thus $\vdash_{\mathsf{L}_{\mathsf{S}\mathsf{A}}}g(\varphi)$ by
Lemma 4; since $\mathsf{L}$ extends $\mathsf{L}_{\mathsf{S}\mathsf{A}}$ we
have $\vdash_{\mathsf{L}}g(\varphi)$, and
$\vdash_{\mathsf{L}}g(\varphi)\rightarrow\varphi$ implies
$\vdash_{\mathsf{L}}\varphi$ by (MP). This shows (weak) completeness, and
Theorem 4.1 is proved.

##  5 Conclusion

This paper introduced a simple modal language to reason about the expertise of
an information source. We used the notion of “soundness” – when information is
true after ignoring parts on which the source has no expertise – to establish
a connection with S5 epistemic logic. This provided alternative interpretation
of expertise, and led to a sound and complete axiomatisation.

There are many possible directions for future work. For instance, it may be
unrealistic to expect the expertise set of a source is fully known up front.
Methods for _estimating_ the expertise, e.g. based on past reports [4], could
be developed to reason about expertise approximately in practical settings.
The “binary” notion of expertise we consider may also be unrealistic: either
$\mathsf{E}\varphi$ holds or $\neg\mathsf{E}\varphi$ holds. Enriching the
language and semantics to handle _graded_ or _probabilistic_ levels is a
natural generalisation which would allow a more nuanced discussion of
expertise.

One could also investigate the relation between expertise and _trust_. For
example, can the trustworthiness of a source on $\varphi$ be _derived_ from
expertise on $\varphi$? The language $\mathcal{L}$ could be extended with a
trust operator $\mathsf{T}$ to model this formally in future work.

Also note that in this paper we only consider _static_ expertise. In reality,
expertise may change over time as new evidence becomes available and as the
epistemic state of the information source evolves. One could introduce
_dynamic operators_ , as is done in Dynamic Epistemic Logic, to model this
change in expertise in response to evidence and other epistemic events. When
it comes to the interaction between expertise and evidence specifically,
_evidence logics_ [11, 10] may be highly relevant. These logics use
neighbourhood semantics to interpret the evidence modalities, which is
technically (and perhaps also conceptually) similar to our semantics for the
expertise modality. We save the detailed analysis and comparison for future
work.

## Acknowledgements

We thank the anonymous ESSLLI 2021 student session reviewers, whose insightful
comments and suggestions have greatly improved the paper.

## References

  * [1] Blackburn, P., Rijke, M.d., Venema, Y.: Modal Logic. Cambridge University Press (2001)
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  * [3] Booth, R., Hunter, A.: Trust as a Precursor to Belief Revision. JAIR 61, 699–722 (2018)
  * [4] Dastani, M., Herzig, A., Hulstijn, J., Van Der Torre, L.: Inferring trust. In: CLIMA. pp. 144–160. Springer (2004)
  * [5] Goranko, V., Passy, S.: Using the universal modality: Gains and questions. Journal of Logic and Computation 2(1), 5–30 (1992)
  * [6] Herzig, A., Lorini, E., Hübner, J.F., Vercouter, L.: A logic of trust and reputation. Logic Journal of the IGPL 18(1), 214–244 (2010)
  * [7] Liau, C.J.: Belief, information acquisition, and trust in multi-agent systems – a modal logic formulation. Artificial Intelligence 149(1), 31–60 (2003)
  * [8] Lorini, E., Jiang, G., Perrussel, L.: Trust-based belief change. In: Proc. ECAI. pp. 549–554 (2014)
  * [9] Pacuit, E.: Neighborhood semantics for modal logic. Springer International Publishing (2017)
  * [10] van Benthem, J., Fernández-Duque, D., Pacuit, E.: Evidence and plausibility in neighborhood structures. Annals of Pure and Applied Logic 165(1), 106–133 (2014)
  * [11] van Benthem, J., Pacuit, E.: Dynamic logics of evidence-based beliefs. Studia Logica 99(1), 61–92 (2011)


11institutetext:  Université Clermont-Auvergne, Inrae, UR LISC, France
22institutetext: Université Clermont-Auvergne, LAPSCO, France 33institutetext:
Institute for Advanced Study in Toulouse, France 44institutetext: Karlsruhe
Institute of Technology, Germany 55institutetext: Corresponding author.
<EMAIL_ADDRESS>
# Regular access to constantly renewed online content favors radicalization of
opinions

Guillaume Deffuant  112255 Marijn A. Keijzer  33 Sven Banisch  44

###### Abstract

Worry over polarization has grown alongside the digital information
consumption revolution. Where most scientific work considered user-generated
and user-disseminated (i.e., Web 2.0) content as the culprit, the potential of
purely increased access to information (or Web 1.0) has been largely
overlooked. Here, we suggest that the shift to Web 1.0 alone could include a
powerful mechanism of belief extremization. We study an empirically calibrated
persuasive argument model with confirmation bias. We compare an offline
setting—in which a limited number of arguments is broadcast by traditional
media—with an online setting—in which the agent can choose to watch contents
within a very wide set of possibilities. In both cases, we assume that
positive and negative arguments are balanced. The simulations show that the
online setting leads to significantly more extreme opinions and amplifies
initial prejudice.

###### Keywords:

Opinion dynamics Online media Confirmation bias Web 1.0 Biased processing

##  1 Introduction

Political polarization—whether affective, ideological or network
polarization—seems to be growing in many countries over the last decades. For
instance, the divide between liberals and conservatives in the U.S. or between
globalists and populists in Europe appears deeper today than ever before. As a
result, we have seen massive violent protests and increased institutional
instability. It is thus important to better understand the causes of
polarization.

It is often suggested that increased access to online media that took place
during the same period could have contributed polarization. The popularity of
the Internet has revolutionized the way we consume and share information. Web
1.0 (i.e., cognition-oriented) technologies have liberated access to
information, Web 2.0 (communication-oriented) has democratized information
sharing and Web 3.0 (cooperation-oriented) have made it possible to more
easily coordinate actions [16]. While all three technological waves have
impacted opinion formation processes, the majority of the scientific work
linking political polarization to the popularity of the internet has focused
on Web 2.0 aspects of, mainly, online social media. In particular, on
communication in digital “echo chambers” created by algorithmic “filter
bubbles” in which large groups of users sharing similar views about a topic
reinforce each other in their views [4, 22]. Moreover, the unprecedented
competition for attracting the attention of users also favors extreme and
negative content, particularly in messages of limited size (like on Twitter),
which may also lead to opinion radicalization and entrenchment, without
algorithmic selection.

In this paper, we focus on the Web 1.0 aspects of the information revolution.
How did regular access to a wide diversity of constantly renewed contents
about any topic affect the opinion formation of individual users? Before the
emergence of online media, the means to access news was limited to a small set
of newspapers or TV channels, that strongly frame debates. The situation
changed radically, as the new media offer an almost infinite variety of
comments and viewpoints, constantly changing, among which the users navigate.

In order to address how Web 1.0 affects individual opinions, we consider a
recent, empirically-grounded model of opinion dynamics under biased processing
[8]. Here, an individual’s opinion is the aggregate of attitudes towards a
series of known arguments, in line with Persuasive Argument Theory (PAT) [31].
Typically, computational models of social influence consider the state
variable to be the opinion of the agents, and this opinion evolves according
to assumed rules during interactions [14]. In contrast, in the model
considered in this paper, the state of the agent is defined by its beliefs
about different arguments and these beliefs in association with the valence of
the related arguments (for or against a topic) define the agent’s opinion [6,
28].

The assumption that individuals engage in biased processing is by no means a
stretch as it is well-grounded in literature on confirmation bias [32].
Confirmation bias predicts that given the same argument, two individuals may
react differently as a function of their currently held belief. Specifically,
arguments that are closer to the currently held belief are considered stronger
than those farther away [21, 35]. Despite the prominence of confirmation bias,
only very few opinion dynamics models take it into account [8, 10, 14, 36],
the bounded confidence family of models [11, 19] being a prominent exception.

Comparing the mechanisms of biased processing between online and offline
contexts is particularly interesting for two main reasons. First, it might be
the case that biased processing and the availability of more information due
to Web 1.0 alone is enough to explain differences in polarization between
online and offline contexts with the calibrated model from Banisch and Shannon
[8]. Here, we investigate whether this is the case by looking at how opinions
are affected as a function of low (offline) or high (online) availability of
information. Second, many contributions in psychology from recent years have
shown how consumption of information online, in particular on online social
media, affects biased processing. There is ample evidence showing that online,
confirmation bias for the selection and processing of information is strong
[25] and may be stronger online than offline [33]. Some argue that motivated
reasoning—here referring to explicitly seeking out information that confirms
currently held beliefs [26]—could account for this effect [29]. But when it
comes to actual attitude change (as opposed to mere selection of information),
it appears that this effect obtains mostly because people tend to respond
differently when overwhelmed with information, rather than through motivated
reasoning [34]. Information overload thus strengthens individual tendencies to
process new information in light of what they know, rather than in relation to
the quality of this information [18]. This implies that online, people might
not only be confronted with more options to find information that suits their
beliefs best, but that their way to consider content is modified by the
abundance of signals. Here, we investigate the effect of strength of
confirmation bias in both the online and offline context.

In what follows, we define stylized situations that respectively correspond to
offline and online information landscapes and we compare the results obtained
in computer simulations with the model for different levels of confirmation
biases in these two different contexts. Our work speaks to seminal
experimental work on confirmation bias [27] which has shown that individuals
may “draw undue support for their initial positions from mixed or random
empirical findings” (p. 2098). The model leads to testable hypotheses about
how individuals are expected to change their opinion in different information
settings characterized more (online) or less (offline) information diversity.

##  2 Biased dynamics of argument endorsement

###  2.1 The general model

In this simple setting, we consider a single agent that has a prior belief
$\alpha\in[-1,1]$ about a given topic. The agent has access to $N_{a}$
arguments about the topic. We assume that each argument
$i\in\\{1,...,N_{a}\\}$ is either in favor of (then $v_{i}=1$) or against the
topic (then $v_{i}=-1$).

The state of the agent is characterized by two vectors:

  * •

The knowledge vector $(k_{i})_{i\in\\{1,...,N_{a}\\}}$. For each argument $i$,
if the agent knows the argument $i$, then $k_{i}=1$ otherwise, $k_{i}=0$.

  * •

The belief vector $(b_{i})_{i\in\\{1,...,N_{a}\\}}$. For each argument $i$, if
the agent believes the argument $i$, then $b_{i}=1$ otherwise, $b_{i}=0$. If
the agent does not know the argument, then $b_{i}=0$.

Then, the balance of beliefs $\overline{b}$ of the agent about the considered
topic is the sum of its prior belief and the beliefs multiplied by the values
of the arguments, for the arguments known by the agent:

$\displaystyle\overline{b}=\alpha+\sum_{i=1}^{N_{a}}b_{i}v_{i}.$ (1)

The dynamics rules the determination of beliefs about new arguments. Let
$\overline{b}(t)$, $k_{i}(t)$ and $b_{i}(t)$ be respectively the balance of
beliefs, the knowledge and the beliefs of the agent at time $t$. Assume that
at time $t+1$ the agent accesses to $2q\leq N_{a}$ arguments
$j_{1},..,j_{2q}$. Then, for $j\in\\{j_{1},...,j_{2q}\\}$:

  * •

If $k_{j}(t)=1$, the agent knows the argument and does not change its
knowledge or belief. Thus $k_{j}(t+1)=k_{j}(t)$ and $b_{j}(t+1)=b_{j}(t)$.

  * •

If $k_{j}(t)=0$:

    * –

The agent acquires the knowledge about the argument: $k_{j}(t+1)=1$.

    * –

The agent determines its belief about the argument. $b_{j}(t+1)=1$ with
probability :

$\displaystyle P(b_{j}(t+1)=1)=\frac{1}{1+\exp(-\beta\overline{b}(t)v_{j})}.$
(2)

Otherwise, $b_{j}(t+1)=0$. The probability to believe the item depends thus on
parameter $\alpha$ and $\beta$. Parameter $\alpha$ is a prior belief of the
agent about the issue. Parameter $\beta$ rules the confirmation bias in the
beliefs. The higher is $\beta$ the higher the tendency to believe items in the
direction of the current balance of beliefs $\overline{b}$.

Moreover, for $i\notin\\{j_{1},..,j_{2q}\\}$ $k_{i}(t+1)=k_{i}(t)$,
$b_{i}(t+1)=b_{i}(t)$. Finally, the attitude of the agent about the topic is
updated:

$\displaystyle\overline{b}(t+1)=\alpha+\sum_{i=1}^{N_{a}}b_{i}(t+1)v_{i}.$ (3)

We also define the attitude $a(t)\in[-1,1]$ of the agent, as follows :

$\displaystyle
a(t)=2\left(\frac{1}{1+\exp(-\beta\overline{b}(t)})-\frac{1}{2}\right).$ (4)

The attitude aims at quantifying how the agent is favorable or unfavorable to
the issue at stake. In the model this attitude is expressed by the propensity
of the agent to accept arguments from one side or the other. When the attitude
is positive, its value is the probability that the agent believes a positive
argument. When it is negative, its opposite is the probability that the agent
believes a negative argument. When the attitude is close to 1 or -1, this
corresponds to an extreme attitude, as the agent is completely close to the
opposite side and systematically accepts any argument on their own side. As a
result, for the same balance of beliefs, two agents having different
confirmation biases (ruled by parameter $\beta$) have different attitudes
towards the issue.

###  2.2 Offline setting

The typical situation that we aim to represent here is an individual consuming
TV or newspaper reports in which the pro and contra arguments about the topic
are balanced, but limited. In this offline setting, we assume that, at each
time step, the agent is exposed to $2q\leq N_{a}$ arguments $j_{1},..,j_{2q}$
such that the overall valence of the argument is neutral:

$\displaystyle\sum_{i=1}^{2q}v_{j_{i}}=0$ (5)

Hence, we have $q$ arguments of each sign.

Once the agent knows all the arguments, its attitude remains fixed. Let $T$ be
the time when this situation is reached. In the agent simulations, we start
with all knowledge $(k_{i})$ is equal to 0. We run the model for a number of
steps $T$ and compute the final distribution of opinions for different values
of the total number of arguments $N_{a}$ and the number exposed $2q$ at each
time step, .

We do not have to repeat computer simulations in order to evaluate the average
results, as the model can be represented as a Markov chain ruling the
probabilities of being in states defined by four integer values. Here, a state
of the Markov chain is indeed defined by the vector
$(K_{+},B_{+},K_{-},B_{-})$ of, respectively, the number of known positive
arguments, the number of believed positive arguments, the number of known
negative arguments, and the number of believed negative arguments.

$\displaystyle K_{+}$ $\displaystyle=\sum_{i=1}^{n_{a}}k_{i},\mbox{
}K_{-}=\sum_{i=n_{a}+1}^{N_{a}}k_{i},$ (6) $\displaystyle B_{+}$
$\displaystyle=\sum_{i=1}^{n_{a}}b_{i},\mbox{
}B_{-}=\sum_{i=n_{a}+1}^{N_{a}}b_{i}.$ (7)

As the number of believed arguments is lower or equal to the number of known
arguments, the total number of states is
$N_{S}=\frac{1}{4}(n_{a}+1)^{2}(n_{a}+2)^{2}$.

Considering an agent in the state $S=(K_{+},B_{+},K_{-},B_{-})$, we can
express the probability $P(S,S^{\prime})$ that the agent goes to the state
$S^{\prime}=(K^{\prime}_{+},B^{\prime}_{+},K^{\prime}_{-},B^{\prime}_{-})$
after being exposed to a random neutral set of $q$ arguments. Starting with a
probability 1 of being in the state $S_{0}=(0,0,0,0)$, we can evolve the
probabilities of the different states after each step. In this case, as the
number of arguments is limited, when the number of steps increase
indefinitely, the model converges to a fixed probability distribution.
Appendix 0.A.1 provides the details of this model.

###  2.3 Online setting

In the online setting, agents choose the arguments among a much wider set of
signals than in the offline case. Again, we assume that the platform proposes
a balanced set of pro and contra arguments.

Consider an agent who selects to consider $2q$ arguments about the topic of
interest at each point in time. The choice of the agent is based on the
valence (pro or against the considered topic) of the signals. We assume that
the selection of arguments to engage with by the agent is biased—which is in
line with previous empirical work [3, 29]—governed by $\gamma$. It can be
understood as a parameter that governs a first confirmation bias or possibly
the degree of motivated reasoning agents engage in [26]. The probability of
$c(t,j)$, that the agent chooses to watch content $j$ of valence $v_{j}$ at
time $t$, is:

$\displaystyle P(c(t,j))=\frac{1}{1+\exp(-\gamma\overline{b}(t)v_{j})}.$ (8)

After choosing what content to engage with, the agent updates its beliefs with
the same rules as before. In this case, the agent’s opinion potentially
changes all the time, as the agent always considers new arguments. Therefore,
we set a time horizon $T$ after which we stop the simulation.

The main difference with the offline setting is that the agent gets more
opportunities to choose the content it wants to engage with, and its initial
choice is itself subject to confirmation bias. We will consider in particular
the case of users who are careful to balance their selection of news (i.e. for
whom $\gamma=0$) in order to clearly distinguish between the effects of
selection and belief bias.111While motivated reasoning and confirmation bias
can both apply to conscious or less-conscious processes, motivated reasoning
is generally understood to be former and [26] confirmation bias to be latter
[32]. Therefore, it is possible that the selection of information includes
some motivated reasoning as well as being subject to confirmation bias.

We can actually represent the state of the agent by the balance of its beliefs
minus the prior belief: $\overline{b}-\alpha=B_{+}+B_{-}$. For a time horizon
$T$, the state $S$ is an integer in : $\\{-2qT,2qT\\}$. For any state in
$\\{-2q(T-1),2q(T-1)\\}$, we can compute the transition probabilities to all
the states in $\\{-2qT,2qT\\}$. Then, starting with the system being the state
$S=0$ with probability 1, we can compute the probability distribution over all
the states at any step $t<=T$. A detailed description of the Markov Chain can
be found in Appendix 0.A.2.

##  3 Simulation results

###  3.1 Increased extremism in the online setting

####  3.1.1 Indicator.

The indicator of extremism $E$, is defined at $t$ on the probability
distribution $\mu_{t}(\overline{b})$ of the agent holding balance of beliefs
$\overline{b}\in\\{-B,...B\\}$ as the average of the absolute value of the
attitude $a(\overline{b})$ (see Eq. 4):

$\displaystyle
E=\sum_{b=-B}^{B}\mu_{t}(\overline{b})\left|a(\overline{b})\right|,$ (9)

Indeed, if this indicator is close to 1, ti means that the agent’s attitude is
very likely to be 1 or -1, in which case the agent accepts all arguments from
their side and discards any argument from the opposite side, which fits the
usual view of an extreme behaviour.

####  3.1.2 Examples of comparisons between online and offline settings.

Figure 1 provides examples of comparison of the evolution of normalized
opinions between offline and online settings. In these examples, the number of
considered arguments at each iteration is 4 in both cases. In the offline
setting, two arguments of each sign are chosen at random at each iteration. In
the online setting, the agent chooses 4 arguments at random, with no selection
bias $\gamma=0$. In the offline model, the total number of arguments is
$N_{a}=12$. In the online, the number of arguments increases of 4 at each
iteration, as they are assumed new. The bias in the choice of the arguments to
consider is $\gamma=0$.

In these examples, polarization is much stronger in the online setting. This
is reflected by the extremism indicator which is lower than 0.5 in the offline
examples and higher than 0.7 in the online examples.

Figure 2 shows the evolution of the logarithm of the distribution of balance
of beliefs for the same examples as Figure 1. Indeed, after a few iterations,
especially in the online case, without the logarithm, the values of the
distribution are very small and difficult to distinguish. These figures show
that, in the online case, the agent has a high probability to reach balances
of beliefs which are higher than 20 or lower than -20, whereas in the online
case, the opinions are most likely between -3 and +3. In the online case, the
distribution is steadily enlarged at each iteration, which increases the
likelihood of extremism. As the number of arguments is fixed in the offline
case, the tendency to become more extreme comes only from the discovery of
unknown arguments.

|  | Offline | Online
---|---|---|---
$\beta=0.25$ |  Attitude |  |
|  | Iterations $t$ | Iterations $t$
|  | $E(10)=0.21$ | $E(10)=0.73$
$\beta=0.5$ |  Attitude |  |
|  | Iterations $t$ | Iterations $t$
|  | $E(10)=0.46$ | $E(10)=0.96$
Figure 1: Evolution of the distribution of attitude in Online and offline Markov models. In both models, the number of arguments considered at each iteration is $2q=4$ and the initial bias $\alpha=0$. The total number of arguments in the offline model is $N_{a}=12$. In the online model, the choice bias is $\gamma=0$. In each case, the extremism indicator $E$ is computed at the last iteration. |  | Offline | Online
---|---|---|---
$\beta=0.25$ |  Balance of beliefs |  |
$\beta=0.5$ |  Balance of beliefs |  |
|  | Iterations | Iterations
Figure 2: Evolution of the logarithm of the distribution of the balance of
beliefs $\overline{b}(t)$ in Online and offline Markov models. The parameters
are the same as on Figure 1. In both models, the number of arguments
considered at each iteration is $2q=4$ and the initial bias $\alpha=0$. The
total number of arguments in the offline model is $N_{a}=12$. In the online
model, the choice bias is $\gamma=0$. In each case, the extremism indicator
$E$ is computed at the last iteration.

####  3.1.3 Effects of different parameters on extremism.

Figure 3 shows how the extremism indicator evolves with the number of
iterations $t$, for different values of the belief confirmation bias parameter
$\beta$. For the considered values of the parameters, in the offline case,
extremism remains almost constant after 4 or 5 iterations. On the contrary, in
the online case, extremism continues to strongly increase until the last
iterations for all values of $\beta$. This increase gets smaller when the
extremism reaches values that are close to 1, like for $\beta=0.5$.

This constant increase of the extremism in the online setting can be explained
by the shape of the distribution of balances of beliefs over time as shown on
Figure 2, on the left panels. Indeed, at each iteration, there is a high
probability to get a stronger balance of beliefs by adding new positive or
negative beliefs. In the offline setting, this increase is constrained by the
limited number of items.

Figure 4 shows the effect of $\gamma$, the bias in the choice of the items
(online setting), on the evolution of the extremism for different values of
the bias in the belief of items $\beta$. All the other parameters being the
same as on Figure 1. The result shows a substantial increase of the extremism
for $\gamma=0.2$ and even more for $\gamma=0.4$.

The effect of parameter $\gamma$ is actually very similar to the effect of a
selection algorithm that could be implemented by the platform. It plays the
role of a filter prior to a more careful attention devoted to the item.

| Offline |  | Online $\gamma=0$
---|---|---|---
Iterations $t$ |  |  Iterations $t$ |
| $\beta$ |  | $\beta$
Figure 3: Extremism indicator during iterations. Left panel: offline. Right panel: Online. The parameters are the same as on Figure 1. | Online $\gamma=0.2$ |  | Online $\gamma=0.4$
---|---|---|---
Iterations $t$ |  |  Iterations $t$ |
| $\beta$ |  | $\beta$
Figure 4: Extremism indicator during iterations for online setting. Effect of
the bias in item choice parameter $\gamma$. The other parameters are the same
as on Figure 1.

###  3.2 Amplification of prejudice in online setting

####  3.2.1 Indicator.

Let $\mu_{t}(\overline{b})$ be the probability distribution of the balance of
beliefs $\overline{b}$, varying between $-B+\alpha$ and $B+\alpha$ at
iteration $t$.

The prejudice $P(t)$ is defined as the average of the attitude:

$\displaystyle
P(t)=\sum_{b=-B+\alpha}^{B+\alpha}a(\overline{b})\mu_{t}(\overline{b}),$ (10)

where $a(\overline{b})$ is the attitude associated with the balance of beliefs
as specified by equation 4. The value of this indicator is in $[-1,1]$ and
expresses the average tendency to believe positive or negative items. It is 0
when the distribution is symmetric, that is, when the probability to believe
negative or positive items are the same (as it is always the case when the
initial opinion $\alpha=0$). It is positive if the agent has a higher
probability to believe positive than negative items, it is negative in the
opposite case. Note that, at $t=0$, $P(0)$ is simply the attitude at $t=0$
because the distribution is all concentrated (Dirac distribution) at value
$\alpha$.

In the following, we measure the relative increase $I_{P}(t)$ of the prejudice
over time which is:

$\displaystyle I_{P}(t)=1-\frac{P(0)}{P(t)}$ (11)

In other words, $I_{P}(t)$ express to what extent the prejudice is amplified
over time.

####  3.2.2 Examples.

Figure 5 shows examples of model runs with an initial bias $\alpha=0.5$. In
this case the distribution is asymmetric. In both cases the initial prejudice
$P(0)=0.05$. However, at $t=10$, we have $P(10)=0.08$ for the offline setting
and $P(10)=0.18$ for the online setting. This example suggests that the
initial prejudice tends to increase more with the interactions in the online
setting.

|  | Offline | Online
---|---|---|---
$\beta=0.25$ |  Normalized opinions |  |
|  | Iterations $t$ | Iterations $t$
|  | $P(10)=0.08$ | $P(10)=0.18$
Figure 5: Initial bias $\alpha=0.5$. Evolution of the distribution of
normalized opinions in online and offline Markov models. In both models, the
bias on item belief is $\beta=0.2$. In each case, the prejudice indicator $P$
is computed at the last time step.

####  3.2.3 Systematic simulations.

Figure 6 shows the relative increase of prejudice $I_{P}(t)$ for different
values of $\beta$ and for a prior belief $\alpha=0.5$. In the online setting,
the figure shows the results for 3 values of the bias in the choice of the
items $\gamma\in\\{0,0.2,0.4\\}$.

When $\gamma=0$, the relative increase of prejudice is higher in the online
than in the offline setting for $0.1\leq\beta\leq 0.25$. However, for
$\beta>0.25$, the final relative increase of prejudice is higher in the
offline setting.

Overall, the relative amplification of the prejudice is even higher when the
bias on belief $\beta$ is small (for $\gamma=0.2$, and even more for
$\gamma=0.4$), while the opposite is true in the offline case. Moreover, the
amplification of the prejudice is higher online than offline for all values of
$\beta$, and the increase takes place more rapidly (after 6 or 7 iterations).

| Offline |  | Online $\gamma=0$
---|---|---|---
Iterations $t$ |  |  Iterations $t$ |
| $\beta$ |  | $\beta$
| Online $\gamma=0.2$ |  | Online $\gamma=0.4$
Iterations $t$ |  |  Iterations $t$ |
| $\beta$ |  | $\beta$
Figure 6: Relative increase of prejudice $I_{P}(t)=1-\frac{P(0)}{P(t)}$. The
non-specified parameters are the same as on Figure 1.

##  4 Discussion

The Web 1.0 revolution brought regular access to a wide diversity of
constantly renewed content about any topic to the billions of people connected
to the internet. Our model suggest that this access to online content
increases the effect of cognitive biases (i.e., confirmation bias and
motivated reasoning) which amplifies bi-polarization, extremization and
prejudice. Algorithmically-induced selective exposure (associated with Web
2.0) certainly increases these effects even more, but the role of the almost
infinite choice of contents offered by online environments should not be
neglected. This point resonates with effects of ‘globalization’ or network
size in other modeling traditions [1, 11, 12, 19, 20, 23, 24] and invites to
refrain putting the blame of polarization exclusively or even majorly on web
personalization [22].

The model introduced in this paper is based on a recent contribution that
experimentally calibrated the agents’ influence-response function after
exposure to a balanced set of arguments [8]. The approach relies on several
assumptions that deserve further investigation, in our view. First, all
beliefs are considered dichotomous. The agents either completely believe or
completely reject an argument. However, it is often the case that we believe
more or less an argument. It would be important to investigate if a model
introducing several levels of beliefs would show the same effects. Second, all
arguments hold the same weight when the aggregate opinion is constructed. How
opinions are precisely constructed from a set of arguments is an open
empirical question, but as some motivated reasoning scholars point to the
active construction of the same known arguments to reach different conclusions
[2, 13, 26], it is likely that there is some form of weighing of arguments
involved. Our model showed that different opinions can emerge given the same
information, if that information is evaluated based on some biased belief.
When weighing gets involved, different opinions can also emerge between agents
who believe the same information. Such a model could resonate well with the
reality of radically different interpretations accepting the same facts, which
we have seen during even the largest of societal challenges such as COVID-19
or in the climate change mitigation debate.

While enriching the model with more complex processes and calibrating those
processes to empirical micro and macro-level data could improve our
understanding of cognition and information consumption in different contexts,
we believe that the main conclusions from this paper are likely to hold up.
The discussion framed by traditional media focuses on a limited set of
arguments, because they should meet the interest of a wide audience.
Introducing very specific points or details introduces a high risk to loose
the average reader or spectator. With the freedom to browse on a virtually
infinite set of arguments, online information consumers can find large set of
items that fit perfectly their specific interest, dig deeper and deeper in
their own direction, ultimately forging a robust, likely extreme, opinion.

The work presented here fits into a broader tradition of social simulation
models that have tried to tackle the question of globalization from different
angles. Our findings resonate with those from the bounded confidence tradition
[11, 19] where confirmation bias can produce the empirical macro-patterns
observed online [12]. They echo results from nominal opinion models in the
tradition of Axelrod [1] with globalization causing fragmentation, not
consensus [24] and online communication creating large cleavages between those
who think differently [5, 23]. What is more, they fit the results from the
Construct-model tradition [9] that, much like PAT, considers the adoption of
facts based on information processing biases [30].

We propose a number of possible avenues for future theoretical and
experimental research based on the model discussed here, that stand between
this abstract model and the patterns we see at a macro-scale in reality [15].
Theoretically, the addition of information filtering systems can be an
interesting extension to the model. Web personalization can be implemented in
many ways—for instance, as network-based, popularity-based or cultural
distance-based filters—and the choice for a particular filtering procedure may
have profound impacts on the dynamics of opinions [22]. One could also imagine
to integrate this model with the model from Geschke et al. [17] who considered
individual, social and technological filters. In order to capture aspects of
the Web 2.0 the model should be enriched by incorporating social networks and
social influence components [7, 14]. Networks dictate how information spreads
through a population. Both social structure and the way in which communication
is structured vary between on and offline contexts and may impact model
dynamics [23]. Finally, including multiple opinions that co-evolve [6] could
create more complex agent profiles and possibly overcome amplification of
initial bias.

The research presented here also resulted in promising directions for
experimental research. While this model is built upon empirical work by
Banisch and Shannon [8], the previous work did not consider crowded (nor
infinte) information environments. What is more, the model here is extended
with a selection bias (modeled as $\gamma$ in Eq. 8). An extension of the
original experiment [35] could consider the information selection process
introduced here and control the amount of information that individuals can
choose to engage with. This would allow to empirically calibrate the selection
bias $\gamma$ and validate the conjectures of this paper.

The digital revolution has come with many challenges, of which increasing
polarization is just one. Theoretical explanations like the one outlined in
this paper help to understand the logic of assumed causal links, and point to
where the culprits may be. We have shown that we need not assume that
polarization arises from advanced Web 2.0 selective exposure, complex social
network topologies or heterogenous influence-response functions, but that a
cognitive processing suffices to generate extremization. Interpret these
findings at your own risk.

## Acknowledgements

Marijn Keijzer acknowledges IAST funding from the French National Research
Agency (ANR) under the Investments for the Future (Investissements d’Avenir)
program, grant ANR-17-EURE-0010.

##  Appendix 0.A Appendix: Details of the Markov models

###  0.A.1 Model with offline interactions

The model includes $N_{a}=2n_{a}$ arguments in total and during $T$ time
steps, at each time step, the agent receives information about $q$ positive
and $q$ negative arguments, chosen at random among the $N_{a}$ possible
arguments.

The different states of the agent are defined by four integers
$(k_{p},k_{n},b_{p},b_{n})$, where $k_{p}\in\\{0,...,n_{a}\\}$ is the number
of positive arguments known by the agent, $k_{n}\in\\{0,...,n_{a}\\}$ is the
number of negative arguments known by the agent, $b_{p}\in\\{0,...,k_{p}\\}$
is the number of positive arguments believed by the agent and
$b_{n}\in\\{0,...,k_{n}\\}$ is the number of negative arguments believed by
the agent. Let $\mathcal{S}$ be the total set of these possible states.

Let $\mu_{t}(k_{p},k_{n},b_{p},b_{n})$ be the probability that the agent is in
state $(k_{p},k_{n},b_{p},b_{n})$. We assume: $\mu_{0}(0,0,0,0)=1$ and for all
the other states, $\mu_{0}(k_{n},k_{p},b_{p},b_{n})=0$.

Then, at each step $t\in\\{1,...,T-1\\}$, we compute $\mu_{t+1}$ from
$\mu_{t}$. This is done as follows:

  1. 1.

We initialise the difference distribution $\delta(k_{p},k_{n},b_{p},b_{n})=0$,
for all $(k_{p},k_{n},b_{p},b_{n})\in\mathcal{S}$.

  2. 2.

For all states $(k_{p},k_{n},b_{p},b_{n})\in\mathcal{S}$ such that
$\mu_{t}(k_{p},k_{n},b_{p},b_{n})>0$:

    1. (a)

There are $n_{a}-k_{p}$ unknown positive arguments and $n_{a}-k_{n}$ negative
arguments. The maximum of new arguments received is thus
$m_{p}=\min(q,n_{a}-k_{p})$ positive and $m_{n}=\min(q,n_{a}-k_{n})$ negative
arguments. As the arguments are assumed randomly chosen, the probability that
the agent gets a new positive argument is $P_{kp}=\frac{m_{p}}{n_{a}}$ and the
probability to get $i_{p}\in\\{0,...,m_{p}\\}$ new arguments is (Bernoulli
formula):

$\displaystyle
P(k^{\prime}_{p}=k_{p}+i_{p})=\binom{i_{p}}{m_{p}}P_{kp}^{i_{p}}(1-P_{kp})^{m_{p}-i_{p}}.$
(12)

Similarly, the probability that the agent gets a new positive argument is
$P_{kn}=\frac{m_{n}}{n_{a}}$ and the probability to get
$i_{n}\in\\{0,...,m_{n}\\}$ new arguments is :

$\displaystyle
P(k^{\prime}_{n}=k_{n}+i_{n})=\binom{i_{n}}{m_{n}}P_{kn}^{i_{n}}(1-P_{kn})^{m_{n}-i_{n}}.$
(13)

    2. (b)

For each new positive argument among $i_{p}$, the probability to believe this
argument is:

$\displaystyle P_{bp}=\frac{1}{1+\exp(-\beta(b_{p}-b_{n}+\alpha))}.$ (14)

Hence, the probability to believe $j_{P}\in\\{0,...,i_{p}\\}$ positive
arguments among the $i_{p}$ new ones is:

$\displaystyle
P(b^{\prime}_{p}=b_{p}+j_{p})=\binom{j_{p}}{i_{p}}P_{bp}^{j_{p}}(1-P_{bp})^{i_{p}-j_{p}}.$
(15)

Similarly, for each new negative argument among $i_{n}$, the probability to
believe this argument is:

$\displaystyle P_{bn}=\frac{1}{1+\exp(\beta(b_{p}-b_{n}+\alpha))}.$ (16)

Hence, the probability to believe $j_{n}\in\\{0,...,i_{n}\\}$ negative
arguments among the $i_{n}$ new ones is:

$\displaystyle
P(b^{\prime}_{n}=b_{n}+j_{n})=\binom{j_{n}}{i_{n}}P_{bn}^{j_{n}}(1-P_{bn})^{i_{n}-j_{n}}.$
(17)

    3. (c)

For all considered values $i_{p}$, $i_{n}$, $j_{p}$, $j_{n}$, let:

$\displaystyle
dP=P(k^{\prime}_{n}=k_{n}+i_{n})P(k^{\prime}_{n}=k_{n}+i_{n})P(b^{\prime}_{p}=b_{p}+j_{p})P(b^{\prime}_{n}=b_{n}+j_{n}).$
(18)

We update $\delta$ as follows:

$\displaystyle\delta(k_{p},k_{n},b_{p},b_{n})$
$\displaystyle:=\delta(k_{p},k_{n},b_{p},b_{n})-dP$ (19)
$\displaystyle\delta(k_{p}+i_{p},k_{n}+i_{n},b_{p}+j_{p},b_{n}+j_{n})$
$\displaystyle:=\delta(k_{p}+i_{p},k_{n}+i_{n},b_{p}+j_{p},b_{n}+j_{n})+dP.$
(20)

  3. 3.

We update the distribution $\mu_{t}$. For all states:

$\displaystyle\mu_{t+1}(k_{p},k_{n},b_{p},b_{n})=\mu_{t}(k_{p},k_{n},b_{p},b_{n})+\delta(k_{p},k_{n},b_{p},b_{n}).$
(21)

Finally, we compute the distribution of probabilities to have a bias of
$2*\left(\frac{1}{1+\exp(-\beta(b+\alpha)}-\frac{1}{2}\right)$, as the sum of
$\mu(k_{p},k_{n},b_{p},b_{n})$ for all the couples $(b_{p},b_{n})$ such that
$b_{p}-b_{n}=b$.

###  0.A.2 Model with online interactions

In this case, the set of states is simpler: they correspond to all the
possible values of the balance of beliefs $b_{p}-b_{n}$, which are
$\mathcal{S}=\\{-Tn_{a},...,Tn_{a}\\}$, where $T$ is the number of iterations
in which the agent consults $2q$ items on the platform.

We define the probability distribution $\mu_{t}(b)$, for $b\in\mathcal{S}$.
Initially, $m_{0}(0)=1$, and $\mu_{0}(b)=0$ for $b\neq 0$.

Then, at each step $t\in\\{1,...,T-1\\}$, we compute $\mu_{t+1}$ from
$\mu_{t}$. This is done as follows:

  1. 1.

We initialise a difference distribution such that $\delta(b)=0$ for all
$b\in\mathcal{S}$,

  2. 2.

for each state $b$ such that $\mu_{t}(b)>0$:

    1. (a)

we compute the probability to choose a positive content:

$\displaystyle P_{c}=\frac{1}{1+\exp(-\gamma(b+\alpha))}.$ (22)

Then, for $c_{p}\in\\{0,...,2q\\}$ the probability to choose to consult
$i_{p}$ positive contents and $q_{a}-c_{p}$ negative contents is:

$\displaystyle
P_{cp}=\binom{c_{p}}{q_{a}}P_{c}^{c_{p}}(1-P_{c})^{q_{a}-c_{p}}.$ (23)

    2. (b)

Then, the probability to believe $j_{p}\in\\{0,...,c_{p}\\}$ contents among
the chosen $c_{p}$ is:

$\displaystyle
P(b^{\prime}_{p}=b_{p}+j_{p})=\binom{j_{p}}{c_{p}}P_{b}^{j_{p}}(1-P_{b})^{c_{p}-j_{p}},$
(24)

with:

$\displaystyle P_{b}=\frac{1}{1+\exp(-\beta(b+\alpha))}.$ (25)

Similarly, setting$c_{n}=q_{a}-c_{p}$, the probability to believe
$j_{n}\in\\{0,...,c_{n}\\}$ contents among the chosen $c_{n}$ is:

$\displaystyle
P(b^{\prime}_{n}=b_{n}+j_{n})=\binom{j_{n}}{c_{n}}P_{n}^{j_{n}}(1-P_{n})^{c_{n}-j_{n}},$
(26)

with:

$\displaystyle P_{n}=\frac{1}{1+\exp(\beta(b+\alpha))}.$ (27)

    3. (c)

For all the considered $(c_{p},c_{n},i_{p},i_{n})$, let:

$\displaystyle
dP=\mu(b)P_{cp}P(b^{\prime}_{p}=b_{n}+j_{p})P(b^{\prime}_{n}=b_{n}+j_{n})$
(28)

We update the distribution $\delta$ as follows:

$\displaystyle\delta(b)$ $\displaystyle:=\delta(b)-dP$ (29)
$\displaystyle\delta(b+j_{p}-j_{n})$
$\displaystyle:=\delta(b+j_{p}-j_{n})+dP.$ (30)

  3. 3.

We update the distribution $\mu_{t}$. For all states:

$\displaystyle\mu_{t+1}(b)=\mu_{t}(b)+\delta(b).$ (31)

Finally, like with the offline model, the distribution of probabilities to
have a bias of $2*\left(\frac{1}{1+\exp(-\beta(b+\alpha)}-\frac{1}{2}\right)$,
is $\mu(b)$.

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