=Paper=
{{Paper
|id=Vol-2012/paper_03
|storemode=property
|title=Polarization and Bipolar Probabilistic Argumentation Frameworks
|pdfUrl=https://ceur-ws.org/Vol-2012/AI3-2017_paper_3.pdf
|volume=Vol-2012
|authors=Carlo Proietti
|dblpUrl=https://dblp.org/rec/conf/aiia/Proietti17
}}
==Polarization and Bipolar Probabilistic Argumentation Frameworks==
https://ceur-ws.org/Vol-2012/AI3-2017_paper_3.pdf
Polarization and Bipolar Probabilistic
Argumentation Frameworks
Carlo Proietti
Lund University
Abstract. Discussion among individuals about a given issue often in-
duces polarization and bipolarization effects, i.e. individuals radicalize
their initial opinion towards either the same or opposite directions. Ex-
perimental psychologists have put forward Persuasive Arguments Theory
(PAT) as a clue for explaining polarization. PAT claims that adding novel
and persuasive arguments pro or contra the debated issue is the major
cause for polarization. Recent developments in abstract argumentation
provide the tools for capturing these intuitions on a formal basis. Here
Bipolar Argumentation Frameworks (BAF) are employed as a tool for
encoding the information of agents in a debate relative to a given issue
a. A probabilistic extension of BAF allows to encode the likelihood of
the opinions pro or contra a before and after information exchange. It
is shown, by a straightforward example, how these measures provide the
basis to capture the intuitions of PAT.
1 Introduction
The work by [9] unveiled the phenomenon of risky shift, i.e. the tendency of a
group to take decisions that are more extreme than the average of the individ-
ual decisions of members before the group met. Subsequent research in social
psychology showed that a similar pattern applies, more generally, to change of
attitude and opinion after debate. This phenomenon goes under the name of
Group polarization. A side effect of polarization are bipolarization effects, i.e.
the tendency of different subgroups to radicalize their opinions towards opposite
directions. One explanation of polarization is provided by Persuasive Arguments
Theory, PAT for short, developed by social psychologists in the 1970s [12]. PAT
assumes that individuals become more convinced of their view when they hear
novel and persuasive arguments in favor of their position.1
1
PAT is not the only explanation for polarization. A relevant tradition in social psy-
chology has regarded polarization and bipolarization as a byproduct of social com-
parison [8]: to present themselves in a favorable light in their social environment,
individuals take a position which is similar to everyone else but a bit more extreme.
Both social comparison and PAT seem to play an important role in different po-
larization contexts. It is beyond our purposes to adjudicate between the two. We
only remark that PAT is fundamental in many such scenarios. To stress this point,
if social comparison was the only decisive factor, we should expect to find more pro-
nounced polarizing effects in face-to-face discussion rather than, e.g., web-mediated
exchange. But there is conflicting evidence thereof.
Some recently developed tools in Dung-style abstract argumentation [3] allow
to capture this phenomenon on a formal ground. Indeed, Bipolar Argumentation
Frameworks (BAF) ([1, 2]) allow to encode the argumentative knlowledge base
of an agent and the arguments both in favor and against a debated issue a.
Moreover, a probabilistic extension of BAF allows to encode the likelihood with
which opinions pro and contra the debated issue are entertained by the agent.
The probabilistic extension of BAF will be named PrBAF and is defined along
the same lines of [5] and [4]. In Section 4 PrBAF are employed to validate PAT.
There it is shown, by an example, how the exchange of arguments in a debate
modifies the likelihoods of pro and contra opinions. Furthermore, it is indicated
how polarization and bipolarization can be encoded in a way that tracks the
change of likelihood of pro and contra opinions.
2 Bipolar Argumentation Frameworks
Bipolar Argumentation Frameworks (BAF for short) [1] are defined as follows
Definition 1 (BAF). A Bipolar Argumentation Framework G is a triple (A, Ra , Rs )
where A is a finite and non-empty set of arguments and Ra , Rs ⊆ A × A
Here Ra and Rs are binary relations over A, called the attack and the sup-
port relation. aRa b means argument a attacks argument b, while aRs b means a
supports b.
d
b c
a
Fig. 1. An example of BAF. Labelled nodes represent arguments. Relations of sup-
port between arguments are indicated with a plain edge, while relations of attack are
indicated with a barred one.
The following definition provides some important relations between argu-
ments in a BAF.
Definition 2 (Defeat, defence, support and neutrality).
(i) a defeats b if there is a path aR1 . . . Rn b, Ri ∈ {Rs , Ra } ∀i ≤ n, n ≥ 1 such
that the path contains an odd number of attacks.
(ii) a defends b if there is a path aR1 . . . Rn b, Ri ∈ {Rs , Ra } ∀i ≤ n, n ≥ 1
such that the path contains an even number k of attacks with k > 0.
(iii) a supports (resp. weakly supports) b if there is a path aR1 . . . Rn b, Ri = Rs
∀i ≤ n, such that n ≥ 1 (resp. n ≥ 0).
(iv) a is neutral w.r.t. b if there is no path (including paths of length 0) between
a and b.
The BAF of Figure 1 helps to illustrate these notions. Argument b supports a,
while c defeats a and d defends a.
Fact 1. Provided that there is a unique path from a to b, we have either that a
defeats b, or that a defends b, or that a (weakly) supports b, and these options
are mutually exclusive.
Opinions held by the participants are represented as sets of arguments. The
opinions we are interested in are those which are either pro or contra a given
issue a. Based on Definition 2 we identify the positions pro a, P(a), and contra
a, C(a) as the following families of sets.
Definition 3 (Pro and contra opinions).
– a set S belongs to P(a) iff S 6= ∅ and ∀b ∈ S either b defends a or b weakly
supports a.
– a set S belongs to C(a) iff S 6= ∅ and ∀b ∈ S b defeats a.
In this framework, the acceptability of an argument depends on its member-
ship to some set, usually called extensions (or solutions). Solutions should have
some specific properties. The basic ones among them are conflict-freeness and
collective defense of their own arguments. Intuitively, conflict-freeness means
that a set of arguments is coherent, in the sense that no argument attacks an-
other in the same set.2 The second condition (defending all its elements) encodes
the fact that for an opinion to be fully acceptable it should be able to rebut all
its counterarguments.
Definition 4 (Properties of extensions).
(i) A set S is conflict-free if there is no a, b ∈ S s.t. a defeats b.
(ii) A set S defends collectively an argument a if for all b such that b defeats a
there is a c ∈ S s.t. c defeats b.
(iii) A set S is admissible if it is conflict-free and defends all its elements.
Admissibility is a key to define the likelihood measures of P(a) and C(a) in the
next section.
3 Probabilistic BAFs
One Argument b may be more or less persuasive in support or against a debated
issue a. Persuasivity can be factorized into two main components: (a) the intrinsic
2
A stronger notion of coherence is also provided in [1] under the name of ‘safety’.
However, we only need to introduce conflict-freeness for our present purposes.
strength of b which supports or defeats a and (b) the strength of the link between
b and a. Both these components are taken into account by the approach of [5]
and can be straightforwardly extended to the bipolar case. For simplicity, and
without any loss for our point, we restrict our attention to component (a) as
done by [4].
Definition 5 (PrBAF). A Probabilistic Bipolar Argumentation Framework G
is a tuple (A, Ra , Rs , p) where (A, Ra , Rs ) is a BAF and p : A → [0, 1] is a
probability function over arguments.
0 0 0
G0 = (A , Ra , Rs ) is a spanning subgraph of G = (A, Ra , Rs , p), denoted as
0 0
G0 v G, if A0 ⊆ A, Ra = Ra ⊗ A0 and Rs = Rs ⊗ A0 , where R ⊗ A = {(a, b) ∈
R | a, b ∈ A}.
Definition 6. Let G = (A, Ra , Rs , p) be a PrBAF and G0 v G. The probability
of G0 , denoted p(G0 ) is
Y Y
p(G0 ) = ( p(a))( (1 − p(a)))
a∈A0 a∈A\A0
p(G0 ) = 1 and therefore p defines a probability
P
It is a standard result that
G0 vG
assignment over the set of spanning subgraphs.
Let G be a PrBAF, G0 v G, and S ⊆ A a set of arguments. G0 |= S denotes
that S is admissible in G0 , for short G0 entails S (see [4]). We can now provide
the measures of the likelihood of the pro and contra opinions relative to a.
Definition 7 (Measures of likelihood).
p(G0 ) such that G0 |= S and S ∈ P(a)
P
1. p(P(a)) =
0 vG
GP
2. p(C(a)) = p(G0 ) such that G0 |= S and S ∈ C(a)
G0 vG
4 Change of likelihood, polarization and bipolarization
Figure 2 serves to illustrate the change of likelihood of P(a) and C(a) from an
initial knowledge base of an agent (Figure 2(a)) where no information is available
except for the debated issue a. At the subsequent stages arguments are added
with different configurations (Figure 2(b), Figure 2(c) and Figure 2(d)).
In Figure 2(a) the spanning subgraphs G0 v G are G1 = ({a}, ∅, ∅) and
G2 = (∅, ∅, ∅), and both have probability 0.5. Here G1 |= {a}, and {a} ∈ P(a).
On the contrary G2 entails nothing else than the empty set ∅, which does not
belong to P(a). Therefore p(P(a)) = 0.5. None of G1 and G2 instead entails any
set in C(a) and therefore p(C(a)) = 0.
When we add b to the debate the argumentative knowledge base transforms
into that of Figure 2(b), where the spanning subgraphs are G1 = ({a}, ∅, ∅),
G2 = (∅, ∅, ∅), G3 = ({b}, ∅, ∅), and G4 = ({b, a}, ∅, {(b, a)}). Here p(G1 ) =
c = 0.5
b = 0.5 b = 0.5 c = 0.5 b = 0.5
1 1 1
a = 0.5 a = 0.5 a = 0.5 a = 0.5
(a) (b) (c) (d)
Fig. 2. Change of likelihood for pro and contra after new arguments are added.
p(G2 ) = p(G3 ) = p(G4 ) = 0.25. Since all of G1 , G3 and G4 entail a set in P(a),
it follows, by the same reasoning as before, that p(P(a)) = 0.75, while we still
have p(C(a)) = 0. Therefore, our measures capture the intuition that adding one
argument in favor of a augments the likelihood of the opinion pro a.
By adding c, as in Figure 2(c), we obtain eight spanning subgraphs G1 =
({a}, ∅, ∅), G2 = (∅, ∅, ∅), G3 = ({b}, ∅, ∅), G4 = ({b, a}, ∅, {(b, a)}), G5 =
({c}, ∅, ∅), G6 = ({b, c}, ∅, ∅), G7 = ({c, a}, {(c, a)}, ∅) and G8 = ({b, c, a}, {(c, a)}, {(b, a)}).
All of them have probability 0.125. It can be seen that G1 , G3 , G4 , G6 and G8
entail some set in P(a) and therefore p(P(a)) = 0.625. By the same process we
see that G5 , G6 , G7 and G8 entail some set in C(a) and therefore p(C(a)) = 0.5.
Here we see that adding an argument against a not only increases the likelihood
of C(a), but also decreases that of P(a) and therefore acts as a balance.
Most interestingly, we can see that the configuration of the arguments in
the graph influences strongly the likelihoods of P(a) and C(a). Indeed if we
add the defeater c as in Figure 2(d), its impact changes w.r.t. the case of Fig-
ure 2(c). There are always eight spanning subgraphs, but now it can be checked
that p(P(a)) = 0.375 and p(C(a)) = 0.5. This helps to appreciate how differ-
ent argumentative strategies (attack of a supporter vs. direct attack) may have
a strong impact on opinion formation. Although the likelihoods of P(a) and
C(a) are correlated, it is often the case that p(P(a)) + p(C(a)) > 1 (see exam-
ple of Figure 2(c)). Therefore, adding new arguments in a debate may increase
the likelihood of both P(a) and of C(a), thus providing more grounds for both
opinions.
The question remains open how to understand polarization and bipolariza-
tion in this framework. According to PAT, polarization is to be viewed as a
change of the degree of our attitude towards a after an exchange of informa-
tion. In this framework the likelihoods p(P(a)) and p(C(a)) are the essential
components of such degree. The latter can be encoded by a weighting func-
tion w whose arguments are p(P(a)) and p(C(a)). Many such functions are
possible candidates. The most natural is the simple arithmetic difference, i.e.
w(p(P(a)), p(C(a))) = p(P(a)) − p(C(a)). Another option is provided by the
measures defined by [6]. It is also likely that different subjects i and j have dif-
ferent ways of weighting, i.e. wi and wj . Following this thread, polarization of an
individual i is measured as the change of the value of wi (p(P(a)), p(C(a))) after
information exchange with others. Analogously, we have a bipolarization effect
between two subjects i and j when we register a change of wi (p(P(a)), p(C(a)))
and wj (p(P(a)), p(C(a))) towards opposite directions after information exchange.
Some general considerations are in order to conclude. Group polarization is a
widespread phenomenon which, to some extent, speaks against theories attribut-
ing to collective discussion a positive role in improving the global performance
of a group. Two most notable cases are Surowiecki’s wisdom of crowds [11] and
Mercier and Sperber’s argumentative theory of reasoning [7]. However, the con-
trast is often only apparent. For example Surowiecki stresses very clearly that
the wisdom of crowds applies mostly to cognition, coordination and cooperation
tasks (see [11], introduction p. XVIII). Now, many contexts in which polarization
arises, like political debate [10], dont fall under these categories. Furthermore,
Surowiecki stresses three necessary conditions for a crowd to be wise, namely di-
versity, independence and decentralization. As he himself recognizes ([11], chapter
9) polarization often arises where diversity of opinion fails. The set of conditions
suggested by Surowiecki provide interesting tests for future research.
References
1. C. Cayrol and M.C. Lagasquie-Schiex. On the Acceptability of Arguments in
Bipolar Argumentation Frameworks. Lecture Notes in Computer Science, 3571:
378–389, 2005.
2. C. Cayrol and M.C. Lagasquie-Schiex. Bipolarity in Argumentation Graphs: To-
wards a better Understanding. International Journal of Approximate Reasoning,
54(7): 876–899, 2013.
3. P.M. Dung. On the acceptability of arguments and its fundamental role in non-
monotonic reasoning, logic programming and n-person games. Artificial Intelli-
gence 77 (2): 321–357, 1995.
4. A. Hunter. Some foundations for probabilistic abstract argumentation. COMMA
2012: 117–128, 2012.
5. H. Li, N. Oren and T.J. Norman. Probabilistic Argumentation Frameworks Lecture
Notes in Computer Science 7132: 1–16, 2011.
6. P. Matt and F. Toni. A Game-Theoretic Measure of Argument Strength for Ab-
stract Argumentation. Lecture Notes in Computer Science 5293: 285–297, 2008.
7. H. Mercier and D. Sperber. Why do humans reason? Arguments for an argumen-
tative theory. Behavioral and Brain Sciences 34: 57–111, 2011.
8. G.S. Sanders and R.S. Baron Is social comparison irrelevant for producing choice
shifts? Journal of Experimental Social Psychology 13: 303–314, 1977.
9. J.A. Stoner. A comparison of individual and group decision involving risk MA
thesis, Massachusetts Institute of Technology, 1961.
10. C. Sunstein. Why societies need Dissent. Cambridge, Harvard University Press,
2003.
11. J. Surowiecki. The Wisdom of Crowds. New York: Anchor Books, 2005.
12. A. Vinokur and E. Burnstein. Effects of partially shared persuasive arguments on
group-induced shifts, Journal of Personality and Social Psychology 29 (3): 305–15,
1974.