Comments on the Models

Having presented the models formally as dynamical systems, some brief commen- tary is now given on the models in light of their context in influence network mo- delling. These comments should be considered throughout the thesis, as they will be relevant to all works presented.

Representation of an Opinion: First, the representation of an opinion yi(t) as a real number is elucidated. Several applications exist in which the definition of yi as a real number is useful. For example, the social network may be discussing a topic which is defined by a question with a necessarily subjective answer, e.g. “is pasta tasty?” Then, negative and positive values of yi represent disagreement and agreement, respectively, while yi = 0 represents a neutral stance. The magnitude of yi(t) represents the intensity of the agreement or disagreement. Alternatively, one could consider the topic as a statement on an idea, e.g. “same-sex marriage should be legalised”, with yi representing individual i’s attitude towards the idea. Then, negative and positive values represent i opposing and supporting the idea, respectively. Depending on the model and problem context, it may be useful to scale the opinions so that y_i(0)∈ [−1, 1], withy_i = −1 and y_i = 1 representing the extreme opinions on the spectrum; a well-defined model (such as the DeGroot and Friedkin–Johnsen models) would then have the property that yi(t) ∈ [−1, 1]for all

t ≥ 0 . Yet other works consideryi(t) ∈ [0, 1]. Some works, e.g. [Yildiz et al., 2013; Nowak et al., 1990], consideryi(t)as a discrete variable (with one typical choice being binary 0, 1). These may be suitable for opinions that lead to actions being taken, e.g.

yi may represent the voting choice for individualiin a political election, or a choice on whether to buy the latest smart phone. This thesis elects to consideryi ∈ R as it better capturesdifferencesin opinions, such as the differences that can arise between an individual’s private and expressed opinion, as studied in Chapter 3 and 4.

Terminology: The terms “opinion”, “attitude”, “belief” are just a few of many that appear in the social science literature. There are some subtle differences, with dis- tinctions made difficult due to a lack of consistent and agreed upon definitions across different scientific communities. In this thesis, the author takes the view that an indi-

vidual’sbelief is his/her position on a statement which is provable to be true or false, e.g. “the Earth orbits around the Sun.” An individual’s opinionis his/her position on a subjective statement which cannot be proved to be true or false, e.g. “vanilla ice cream tastes better than chocolate ice cream.” This distinction is simply one choice of the definitions from the many possible versions in the literature. No further attempt is made to distinguish the terms, and unless stated otherwise, this thesis will exclu- sively use the term “opinion” when referring to yi(t). This is because the models considered in this thesis are general enough to cover many scenario applications.

Multiple Topics: If the individuals are discussing mindependent topics, then one

can define yi(t) = [y1i(t), . . . ,ymi (t)]> as individual i’s vector of opinions, with yik denoting i’s opinion on topic k ∈ {1, . . . ,m}. The Kronecker product is used to trivially extend existing results, e.g. Theorem 2.1 and 2.2. For example, the DeGroot compact form becomes

y= (W⊗Im)y(t), (2.8)

where y(t) = [y₁>, . . . ,y>_n]>. When the m topics become dependent on each other, new analysis is required, and Chapter 9 investigates one model’s method of captu- ring interdependence among the topics.

Interpretation of Parameters: For parameters wij, λi and the other parameters that will be introduced in later models, it is clear that their values depend on many factors, such as individual i’s personality, culture, upbringing and experience, or whether i is an expert on the topic of discussion. The magnitudes of wij can de- pend on level of friendship, status and rank (formal or informal) of the individuals in the network, etc. This thesis does not aim to identify these values for a given social network, or explain how or why the parameters may be different for different individuals. The works in thesis only postulate that the parameters exist, and that the individuals’ opinions evolve according to the models that will be later presented. It is not even assumed that individuals necessarily know the exact value themselves, or are aware that their opinions evolve as captured in the models. The key focus of this thesis is to considerhow the opinions evolve for a given set of parameters, and draw quantitative or semi-quantitative conclusions on the effects of the parameters on the opinion evolution, which may be used to gain high-level insight into interperso- nal influence networks.

Time-Scales: Last, it should be noted that the above models are typically suited for application on problems with short time-scales, e.g. a boardroom discussion lasting several hours or perhaps at a workshop over a week. Such models may not accurately reflect discussion over months or years, because almost certainlywij,λi, etc., would change over time (the precise nature of the time-variation depends on many factors). Chapters 5 through 8 do consider a model where the network discusses a sequence of issues, which may be appropriate for longer time-scales. However, that model assumes that each individual’s self-weight wii changes after discussion on a issue, following a social process called reflected self-appraisal. Thus, the interest is in the evolution ofwiiover the issue sequence, andyi is not the primary variable of interest.

How Differences in Private and

Expressed Opinions Arise

A Novel Model for Opinion

Dynamics Under Pressure to

Conform

Part Summary

Part I studies a novel opinion dynamics model proposed by the author, termed the Expressed–Private–Opinion (EPO) model. The model draws inspiration from some of the most classical results, and aims to develop a mathematical framework for describing social phenomena involving individuals who have different expressed and private opinions. In the EPO model, each individual is assumed to have a private and expressed opinion, and each individual’s expressed opinion is altered from the individual’s private opinion by a pressure to conform to the social norm. Chapter 3 introduces the EPO model and presents theoretical results on stability, and analyses the private and expressed opinions at equilibrium. Chapter 4 uses the model to revisit the seminal conformity experiments by Solomon Asch and investigate how pluralistic ignorance can arise due to the presence of stubborn extremists.

Chapter Summary

This chapter introduces the Expressed–Private–Opinion (EPO) model. A number of phenomena involving individuals with different private and expressed opinions have been recorded and studied in the social sciences via qualitative and experimen- tal methods, and these are discussed in the introduction immediately below. After introducing and explaining the motivation of the model, conditions on parameters of the influence network are obtained which guarantee that the opinions convergence to a steady state. Then, investigations are conducted to draw several interesting con- clusions on the effects of individual stubbornness, and resilience to the pressure to conform to a group majority, in generating differences in an individual’s expressed and private opinion.

3.1

Introduction

In much of the existing literature on agent-based opinion dynamics modelling, it is assumed that each individual has an opiniony_i(t)which is communicated to others in the network. Few models consider the possibility that an individual expresses an opinion different to his/her private opinion, even though the reader will almost certainly have been in a situation where this has occurred to them.

On the other hand, these situations are well studied in the social and political sciences. In [Waters and Hans, 2009], the authors found that over one third of jurors in criminal trials would have privately voted against the decision of the jury panel they were on. Large differences between the private and expressed opinions of the civilian population can generate discontent and tension, which might result inviolent

and unforeseen actions such as the Arab Spring movement [Goodwin, 2011] and the

fall of the Soviet Union [Kuran, 1989]. During the rise of Islamic State in 2014, a US led coalition readily expressed agreement to attack Raqqa, then the de-facto capital of Islamic State. Later, when deciding which troops were to lead the ground assault, theopposing private opinionsof the Turkish and Kurdish representatives in the coalition emerged and created a deadlock for almost two years [Mintz and Wayne, 2016]. Access to the public actions of individuals, without being able to observe their thought processes that led to the actions, can spark an informational cascade where

all successive individuals choose the wrong action[Bikhchandani et al., 1992]. This was used to help explain why farmers in Iowa refused to adopt hybrid seed corn for years, despite its benefits [Ryan and Gross, 1943]. Due to fears of social isolation and exposure, some individuals enforce social norms despite privately disliking the norms [Centola et al., 2005; Willer et al., 2009].

Naturally, there is interest in identifying what creates such differences or dis- crepancies between expressed and private opinions/actions. One commonly hypot- hesised reason is that such differences arise due to a pressure to conform to a group

standard or norm. Formal study of such pressure goes back many decades. In 1951,

Solomon E. Asch’s seminal paper [Asch, 1951] showed that individuals could re- act differently when their judgment about an indisputable fact was challenged by a unanimous majority. Some individuals could withstand the pressure, whereas the actions and judgments of other individuals were heavily affected. A variety of other studies have been reported, and they generally establish that such pressures not only generate different expressed and private opinions/actions, but can also have other consequences. In some instances, high productivity factory workers were pressured to lower their production rate to match factory averages [Coch and French Jr, 1948]. Peer punishment is often dealt to individuals who deviate from group norms, such as in gangs [Thrasher, 1963]. This occurs even if the norm is destructive or unhealthy for the group itself [Abbink et al., 2017]. The pressure exerted on an individual to conform may change over time, or depend on his/her opinions and/or the opinion- s/actions of others in the group [Waters and Hans, 2009; Asch, 1951; Schachter, 1951; Gorden, 1952].

analysis of the key factors in each model that determine specific dynamical proper- ties (see Chapter 1). Despite this, existing agent-based models have failed to provide a thorough account of phenomena involving (i) differences in private and expressed opinions, and (ii) the effects of a pressure to conform to the group norm (both of which have been well-studied in the social and social psychology literature). There is therefore significant need and motivation to examine these interesting social phe- nomena from the perspective of agent-based models, and investigate the precise me- chanisms that drive said phenomena. The aim is to, for the first time, provide a mathematical framework for the study of opinion evolution under pressure to con- form. This chapter will introduce the EPO model, drawing inspiration from the established Friedkin–Johnsen model. In the proposed EPO model, each individual has both an expressed and a private opinion, and the expressed opinion is altered from the private opinion due to a pressure to conform; this is a key departure from most existing works. This chapter will focus on development and analysis of the model from a systems and control perspective, including the establishing of conver- gence results and drawing of semi-quantitative conclusions that give insight into how stubbornness and resilience to pressure to conform affect the expressed and private opinions of individuals in an influence network. In the subsequent Chapter 4, Solo- mon E. Asch’s seminal experiments are revisited using the proposed model, and the well-studied phenomenon ofpluralistic ignoranceis explained using the model.