Opinion Dynamics with Confirmation Bias Armen E. Allahverdyan1) and Aram Galstyan2) 1)Yerevan Physics Institute, Alikhanian Brothers Street 2, Yerevan 375036, Armenia, 2) USC Information Sciences Institute, 4676 Admiralty Way, Marina del Rey, CA 90292, USA Background. Confirmation bias is a tendency to acquire new information via confirming one's preconceptions. It is omnipresent in psychology, economics, scientific practices etc. The previous theoretical research of this phenomenon focused on its economic implications possibly missing its potential connections with broader notions of cognitive science. Methodology/Principal Findings. We formulate a (non-Bayesian) rule for updating the subjective probabilistic opinion of an agent in the light of a persuasive opinion. The rule follows the ideas of semantic information theory: the agent does not react to persuasion that is either far from his current opinion (confirmation bias to unexpected information) or coincide with it (no new information). The model accounts for the basic phenomenology of cognitive dissonance and the social judgment theory. It allows to describe the boomerang effect as an extreme form of the confirmation bias. The model also displays the order of presentation effect: when consecutively learning two contradicting opinions, the preference is given to the last opinion (recency) or the first opinion (primacy) depending on the degree of confirmation bias. We also uncover several features of repeated persuasion process. Conclusions. A single model accounts for a spectrum of effects and relates to each other the confirmation bias, primacy-recency phenomenon, boomerang effect and cognitive dissonance. We point out several limitations of the model that should motivate its future development. I. INTRODUCTION ceived, the agent revises his opinion via the Bayes rule .X Pr(Ai) ! Pr(Ai;E) Pr(Ak;E) : (1) Confirmation bias is a tendency to acquire new infor- k mation in a way that confirms one's preconceptions and This is the normative theory of the rational behavior: 1 avoids information which contradicts prior opinions [29]. provided that Pr(Ak;E) are available, not behaving ac- Various manifestations of this bias have been reported in cording to (1) implies losses in certain types of econom- cognitive psychology [2, 42], social psychology [11, 31], ical actions [7]. The Bayes rule is by definition free of politics [24] and media economics [16, 28, 32, 45] 2. Re- any confirmation bias. Hence economists studied the cent evidence suggests that scientific practices too con- confirmation bias by reducing it to specific deviations tain a variety of confirmation biases [5, 21{23, 29], even from (1) [16, 28, 32, 45], e.g. when the joint probabilities though the imperative of avoiding precisely this bias is Pr(Ak;E) are available, but agents do not combine them frequently presented as a pillar of the scientific method. as in (1). Indeed, people rarely satisfy the Bayes rule, also because it is not always successful in the real world Here we are interested in the dynamics of opinion [17]. change in the presence of confirmation bias; see [6, 7, 29] However, we note that the confirmation bias is de- for reviews. veloped with respect to an essentially new information It is assumed that an agent's uncertain opinion (de- that is going to change the existing opinions; otherwise, the new information is accepted without problems and is fined on an exhaustive event set fAkg) is quantified by even given some priority. Then we expect that the full probabilities Pr(Ak) that describe his degree of confi- dence on the truth of these events [7]. The second as- probability Pr(Ak;E) may not be available [12]. sumption is done within the Bayesian approach to opin- Hence we abandon the premise of the Bayesian ap- proach on the availability of joint probabilities Pr(A ;E). ion revision: the agent has joint probabilities Pr(Ak;E) k Instead, our model assumes 3 inputs: the (subjective) for Ak and some evidence E. Once this evidence is re- probabilistic opinion of the target agent P, the opinion of a persuading agent Q, and the degree of confirmation bias shown by P. We proceed within the opinion com- 1 We use opinion instead of belief, though these terms are virtually bination approach developed in statistics and applied in interchangeable. We prefer the first term, because belief implies a many fields; see [10, 15] for reviews. certain commitment (based on cultural or personal faith, moral- We propose a set of conditions that define cognitive ity, or values), whereas opinion is vaguer, and possibly more aspects of confirmation bias and formalize it within no- flexible and subject to change. 2 The bias has several different names that underline its various tions of semantic information theory. The main message aspects: myside bias, affirmation bias, conservatism, disconfir- of these conditions is that P does not changes his opinion mation bias, overconfidence. if the opinion of Q is either very far or identical to his 2 opinion. Next we propose an opinion updating rule that method models the response of P to persuasion by Q. This rule describes several key effects of the social judg- dk = wP pk + wQqk; wP + wQ = 1; 1 ≤ k ≤ N; (4) ment theory that attempts to explain how people react to persuasion [6]. These effects include: separation of where wP and wQ are positive weights that quantify the opinion into different latitudes, the weighted average ap- importance of each agent for the decision maker, and the proach, change-discrepancy relations. logarithmic method The rule also produced new results: the recency ef- wP wQ fect is related to confirmation bias; repeated persuasions pk qk dk = P ; wP + wQ = 1; 1 ≤ k ≤ N: (5) N wP wQ are shown to hold certain monotonicity features, but do l=1 pl ql not hold the law of diminishing returns; the boomerang (backfire) phenomenon is related to confirmation bias These methods are different, they apply to different sit- and to the primacy effect. uations, because each one has its merits and drawbacks This paper is organized as follows. In Section II we dis- [10, 15] 3. As shown below, their specific combination is cuss the opinion representation via probabilities, define suitable for describing confirmation bias. our axioms and introduce the confirmation biased opin- ion combination rule. Section III relates our set-up to the social judgment theory. Next two sections show how B. Opinion combination rule our model accounts for two basic results of experimen- tal social psychology: opinion change versus discrepancy Let an agent P is persuaded by an agent Q, i.e. the and the order of presentation effect. Repeated persua- opinion p of P is going to change under influence of q sion is studied in section VII. Section VIII shows that 4; see (2). We propose the following conditions for the the boomerang effect|the agent changes his opinion not combination rule of p and q. towards the persuasion, but against it|can appear as 1. The final opinion pek of P reads a form of confirmation bias. Section IX shows how our / model formalizes some concepts of cognitive dissonance XN pek = F [pk; qk; ϵ] F [pl; ql; ϵ] ; (6) and outlines new scenarios of its emergence. We summa- l=1 rize and conclude in the last section. where F [x; y; ϵ] is a smooth function of 3 variables that change between 0 and 1, and II. OPINION COMBINATION VIA CONFIRMATION BIAS 1 > ϵ > 0; (7) A. Representation of opinions characterizes the degree of the confirmation bias of P, as well be seen below. For ϵ ! 1 the opinion does not e ≤ ≤ Consider two agents P and Q, and assume they quan- change: pk = pk for 1 k N. Hence tify their opinion via probabilities F [x; y; 1] = x for 0 ≤ x; y ≤ 1: (8) XN XN f gN f gN Eq. (6) means that P first evaluates the (non- p = pk k=1 and q = qk k=1; pk = qk = 1; (2) k=1 k=1 normalized) weight for the event k solely on the base of pk and qk, and applies the overall normalization at the respectively, on the same set of events k = 1; :::; N, e.g. end. k = (rain; norain), if these opinions are on a weather forecast. Note that k = x can be a continuous variable, if (for example) the forecast concerns the chance of having rain 3 Note that the opinion combination problem does not admit a or the amount of rain. Then the respective probability straightforward Bayesian representation [7]. Nevertheless, the densities are: Bayesian approach can be generalized to this case, although it Z Z requires more assumptions than usually; see e.g. [26]. 4 We assume that the opinion of Q is communicated to P. One p(x) and q(x); dx p(x) = dx q(x) = 1: (3) option is that the full probability is communicated. Another option is that only some moments of the real probabilistic opinion of Q are communicated (e.g. its average and dispersion) together Both opinions are subjective, both are based on incom- with the domain of events, where those real probabilities of Q plete information and take into account different back- are strictly non-zero. Then P can approximately reconstruct grounds of P and Q. Let a decision maker wants to com- the opinion of Q via the maximum entropy method, and now the opinion q of Q should refer to that reconstructed opinion. bine them together, hoping to get a more reliable opinion. In particular, the Gaussian density (17) can be regarded as the There are two basic methods of combining p and q into maximum-entropy reconstruction of a probability density from the opinion d of the decision maker [10, 15]: the linear its first and second moments.