Polarization and Bipolar Probabilistic Argumentation Frameworks

Abstract

Discussion among individuals about a given issue often induces polarization and bipolarization e ects, i.e. individuals radicalize their initial opinion towards either the same or opposite directions. Experimental 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.

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 individual 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 e ect of polarization are bipolarization e ects, i.e. the tendency of di erent 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 psychology has regarded polarization and bipolarization as a byproduct of social comparison [ 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 di erent polarization 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 nd more pronounced polarizing e ects in face-to-face discussion rather than, e.g., web-mediated exchange. But there is con icting 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 de ned 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 modi es 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 de ned as follows De nition 1 (BAF). A Bipolar Argumentation Framework G is a triple (A; Ra; Rs) where A is a nite 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 support relation. aRab means argument a attacks argument b, while aRsb means a supports b.

b a d c (i) a defeats b if there is a path aR1 : : : Rnb, Ri 2 fRs; Rag 8i n, n that the path contains an odd number of attacks. (ii) a defends b if there is a path aR1 : : : Rnb, Ri 2 fRs; Rag 8i n, n such that the path contains an even number k of attacks with k > 0. 1 such 1 (iii) a supports (resp. weakly supports) b if there is a path aR1 : : : Rnb, Ri = Rs 8i 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 De nition 2 we identify the positions pro a, P(a), and contra a, C(a) as the following families of sets.

De nition 3 (Pro and contra opinions).

{ a set S belongs to P(a) i S 6= ; and 8b 2 S either b defends a or b weakly supports a. { a set S belongs to C(a) i S 6= ; and 8b 2 S b defeats a.

In this framework, the acceptability of an argument depends on its membership to some set, usually called extensions (or solutions ). Solutions should have some speci c properties. The basic ones among them are con ict-freeness and collective defense of their own arguments. Intuitively, con ict-freeness means that a set of arguments is coherent, in the sense that no argument attacks another 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.

De nition 4 (Properties of extensions).

(i) A set S is con ict-free if there is no a; b 2 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 2 S s.t. c defeats b. (iii) A set S is admissible if it is con ict-free and defends all its elements. Admissibility is a key to de ne 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 con ict-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 ].

De nition 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.

G0 = (A0 ; R0a; R0s) is a spanning subgraph of G = (A; Ra; Rs; p), denoted as G0 v G, if A0 A, R0a = Ra A0 and R0s = Rs A0, where R A = f(a; b) 2 R j a; b 2 Ag.

De nition 6. Let G = (A; Ra; Rs; p) be a PrBAF and G0 v G. The probability of G0, denoted p(G0) is p(G0) = ( Y p(a))( Y (1

p(a))) a2A0 a2AnA0 It is a standard result that P p(G0) = 1 and therefore p de nes a probability

G0vG assignment over the set of spanning subgraphs.

Let G be a PrBAF, G0 v G, and S A a set of arguments. G0 j= 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.

De nition 7 (Measures of likelihood).

1. p(P(a)) = P p(G0) such that G0 j= S and S 2 P(a)

G0vG 2. p(C(a)) = P p(G0) such that G0 j= S and S 2 C(a)

G0vG 4

Change of likelihood, polarization and bipolarization 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 = (fag; ;; ;), G2 = (;; ;; ;), G3 = (fbg; ;; ;), G4 = (fb; ag; ;; f(b; a)g), G5 = (fcg; ;; ;), G6 = (fb; cg; ;; ;), G7 = (fc; ag; f(c; a)g; ;) and G8 = (fb; c; ag; f(c; a)g; f(b; a)g). 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 con guration of the arguments in the graph in uences 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 Figure 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 di erent 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 example 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 bipolarization 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 information. 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 function w whose arguments are p(P(a)) and p(C(a)). Many such functions are possible candidates. The most natural is the simple arithmetic di erence, i.e. w(p(P(a)); p(C(a))) = p(P(a)) p(C(a)). Another option is provided by the measures de ned by [ 6 ]. It is also likely that di erent subjects i and j have different 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 e ect 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 attributing 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 contrast 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 diversity, 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.

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