1 Introduction

Public opinion is formed as a result of interrelated changes in individual opinions. Through a large number of social interactions, humans are greatly influenced by the opinions, attitudes, and behavior of others. The dynamics resulting from these interactions have been studied in various models [1–9]. One of the most appealing and widely studied representations is the voter model due to its simplicity [10–14]. The paradigmatic voter model (which we will call the Basic Voter Model, BVM) represents the opinions of individuals as a binary variable in a single opinion dimension. At each update, an individual adopts the opinion of one of his acquaintances. In the BVM, every individual is assumed to express his inner conviction.

In practice, however, there might be a difference between an individual’s publicly revealed and internal opinion. In the present paper, we focus on hypocrisy (i.e., individuals exhibit “self-censored” opinions in public [15]) and the abundance of hypocrites (i.e., individuals whose external and internal opinions differ). We study their frequency in the group and their role in the consensus-finding process.

Hypocrisy is abundant in all aspects of human social life. Voters withhold their true opinions in opinion polls, causing difficulties for the prediction of true outcomes [16–19]. Humans adjust their views and opinions to the circle in which they are present [20]. Social desirability is a general driver of expressing an opinion [21–23]. Even in the absence of direct influence or persuasion, individuals adjust their revealed opinion if they are exposed to the attitudes of others [24]. Actual behavior can fall very far from self-reports that are subject to social desirability bias, for instance in dietary intake [25], household labor [26,27], or physical activity [28]. Asch's classic experiments [29], in which he studied the influence of group pressure on individual opinions, show significant levels of conformity: participants of the experiments adopted the group's false opinion in a third of the trials, and more than half of the participants adopted a false group consensus at least once. Haun and Tomasello [30] demonstrated that, already at a young age, children try to say what they are expected to say without changing their own “real” judgment of the situation.

Many people attend church services without true faith, express popular opinions in discussions and meetings, follow a fashion trend they do not fancy, or adhere to regulations they disagree with. They show off with conspicuous consumption or by mowing their lawn while they have better things to do. People support morals they are not engaged with privately, have flexible virtues, and, following others, easily absolve themselves of moral responsibility [31–33].

Individuals might even openly support norms or legitimize a political system they privately do not sympathize with [34,35]. For example, Jiang and Yang [36] report quantitative evidence for “preference falsification” after a political purge in Shanghai in 2006. Laboratory experiments and computational models show that humans might punish norm violators in order to conform to group pressure against their own inner conviction [37,38]. The willingness to conform to social expectations might even lead to self-destructive behaviors such as binge drinking, shoplifting, and smoking among adolescents [39]. In extreme cases, this kind of hypocrisy can support a witch hunt, ostracism, or the public condemnation of scapegoats.

Hypocrisy is also a general feature of organizational behavior where reputation and promotions are at stake [40–42]. Within organizations, language use is often adjusted to the audience. For instance, the tone of political correctness depends on the expectations of receivers [43]. According to Noelle-Neumann [44], a significant proportion of people try to avoid isolation in the case of public questions. She posits that social relations and acceptance in their own environment are more important to people than revealing their own views in public.

All instances of hypocrisy create cognitive dissonance [45], which can be reduced mainly in two ways: internalization or externalization. In the case of internalization, the individual accepts the belief or opinion that he has expressed publicly [46]. Adjusting the internal opinion is an important last step in the process of socialization that is well described in the classic sociological and social psychological literature [47,48]. Opinions and individual attitudes are the products of contact with other members of the group that creates an internal and external conflict “with its resolution based on the internalization of external norms” [49]. Internalization is distinct from compliance as it covers the private acceptance of the norm or attitude [46]. As a result, the individual can be said to be depersonalized and fully assimilated to the group [50,51].

In the case of externalization, the previously concealed opinion becomes publicly expressed. In practice, this requires courage or expressiveness. The “coming out” necessarily results in public discomfort, but often produces relief, a positive cognitive and emotional state. Acting and producing a poker face for the inhibition of inner opinions is certainly costly [52], and the individual is liberated from these costs by expressing his internal belief. It depends on the empirical context to which extent it is feasible to reduce cognitive dissonance by internalization or externalization. For instance, externalization is particularly problematic for political opinions in oppressive regimes, but strongly encouraged when diverse standpoints need to be revealed such as in critical academic debates.

The objective of this article is to analyze the role that hypocrisy, internalization and externalization play in opinion formation. As a more realistic alternative to the Basic Voter Model (BVM), we study the Concealed Voter Model (CVM), in which the publicly expressed (external) opinion can differ from the internal one [53]. Before we give a detailed definition of both models in Section 2, let us first state and motivate their underlying assumptions. The BVM and CVM share the following simplifications:

1. There are only two kinds of potential opinions (which we call “red” and “blue”) on a particular issue, and these are mutually exclusive.
2. In the external layer, individuals interact in pairs.
3. The group is homogeneous in the sense that each rate is the same for all individuals and at all times.
4. Red and blue opinions have the same transition rates. That is, their competition is neutral.

Most versions of the voter model in the literature apply these simplifying assumptions (for reviews, see [12,14]). Extensions have relaxed some of the assumptions. For example, (A) was relaxed in the model by Vazquez and Redner [54], and (B) by Lambiotte and Redner [55]. Assumption (C) was relaxed by introducing zealots who never change their opinion [56,57] and (D) was relaxed in the biased voter model [58]. The CVM keeps (A)-(D) and adds an internal layer [53] (Fig 1). Some previous versions of the voter model have also considered a duality of opinions similar to our juxtaposition of external versus internal layer (e.g., [59,60]), but the internal opinions were static. A model that distinguishes between private and expressed opinions, but assumes continuous instead of binary opinions, was introduced by Ye et al. [61] and by Huang and Wen [62]. The CVM is the first discrete-state voter model in which the opinions in the internal layer can vary in response to the external layer within the individual, but not as a result of links to other internal opinions. In other multilayer voter models, nodes are connected in a multiplex network whose layers contain the same set of nodes with partially overlapping edges [63–67]. However, each layer in these multiplex models contains intra-layer edges so that none of the layers plays the role that the internal layer plays in the CVM: it contains private, hidden opinions that are unknown to other individuals.

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Fig 1. The Basic Voter Model (BVM) versus the Concealed Voter Model (CVM).

(A) In the BVM, every individual is represented by a node in a single layer, which is “external” in the sense that all of its neighbors can see and copy his opinion. The tension that is caused by the disagreement between neighbors (for example between the blue individual 1 and the red individual 2) can be released by adopting the neighbor’s opinion. Such copies happen with a rate c. (B) In the CVM, every individual is represented by two nodes: one in the external and another in the internal layer. For example, let us consider individual 1, outlined by the green rectangle. His external/internal opinion is marked by a blue/red circle. Because these opinions differ, we call individual 1 a “hypocrite”. The tension between the external versus internal opinion can be released either by externalization or internalization (with the corresponding rates e and i). None of the individuals knows the internal opinion of any other individual. Hence, there are no edges within the internal layer.

https://doi.org/10.1371/journal.pone.0218729.g001

Real-life examples for assumption (A) include those issues that are at stake in a social group and whose answer must be one of exactly two possibilities (e.g., yes/no or false/true). Many US elections, for instance, come down to a choice between only the Democrats or the Republicans. A recent example with exactly two alternatives on the ballot was the 2016 Brexit referendum, where voters could only choose between “remain” or “leave”.

Assumption (B) implies that external opinions can spread only from person to person so that the role of mass media is negligible. Mass media tend to portray the majority opinion as socially desirable and, hence, may increase the number of hypocrites in the population. Issues that are less influenced by mass media comply more with (B), for example newly trending issues (such as new slang expressions or underground music), shameful issues (e.g., norms of toilet use), or topics that are only of local or personal importance and interest (e.g., issues of taste, norms or habits in a small group or community).

Assumption (C) ensures an equal opportunity structure for the individuals in our model. It excludes the possibility that some of them can react to social influence, externalize, or internalize at a different rate from others. Assumption (D) guarantees that the dynamics does not give an intrinsic bias to one of the opinions. A typical asymmetry in real opinion formation may arise, for example, when one of the opinions provides more benefits in economic or social terms for the individuals than the other. We wish to exclude this kind of bias in order to study the process of spreading in a pure form. Thus, the BVM and the CVM can be used as neutral references to which more complex models or real-life systems can be compared.

Our previous results in the CVM have shown that, on average, the processes of internalization and externalization delay the consensus [53] because a minority opinion may still linger in the internal layer even when it has temporarily disappeared in public. In this paper, pursuing this path further, we ask fundamental questions about the role of hidden opinions and hypocrisy in the consensus formation process. We examine the proportion of hypocrites in the group over time and their effect on the dynamics of consensus formation. Our goal is to reveal how hidden opinions influence the diversity of public opinions over time and how they shape the dynamics leading to eventual unanimity.

2 Methods

In the Basic Voter Model (BVM), the opinion of each individual is one node in a single, external network layer (Fig 1A). By contrast, the Concealed Voter Model (CVM) represents each individual by two nodes: one opinion in the external and another in the internal layer, which do not need to be identical (i.e., hypocrisy is possible). In the BVM and CVM, we model the external layer as a complete graph. We include a link from each node to itself. The models with and without the self-link are in fact merely reparameterized versions of each other (see S1 Appendix). We include the self-link because it simplifies the notation of the equation-based analysis in the S1 Appendix. While the external nodes are thus maximally connected with each other, every node in the internal layer is linked only to the external node representing the same individual (Fig 1B).

We call the alternative opinions “red” or “blue” and denote the opinions in the external versus internal layer by capital versus lower case letters (R: external red, r: internal red, B: external blue, b: internal blue). In the BVM, one letter is sufficient to characterize the state of every individual: R or B. We denote the fraction of agents in these states as ρ_R and ρ_B = 1−ρ_R. The group has reached a consensus when ρ_R = 1 or ρ_B = 1. In the CVM, we need two letters to describe the state: Rb (externally red hypocrite), Br (externally blue hypocrite), Rr (red frank) or Bb (blue frank). We denote the fraction of individuals in these states by ρ_Rb, ρ_Br, ρ_Rr, and ρ_Bb respectively. The state of the whole group can be described by the combination (ρ_Rb, ρ_Br, ρ_Rr). We do not need to explicitly include ρ_Bb because it is uniquely determined by ρ_Bb = 1−ρ_Rb−ρ_Br−ρ_Rr. The abundances of internal and external opinions can be represented by the 2×2 matrix shown in Table 1. The proportion of hypocrites is ρ_Rb+ρ_Br. The group is in a consensus when ρ_Rr = 1 or ρ_Bb = 1. That is, we define a consensus as a state in which only one of the opinions exists; the alternative opinion has completely disappeared even from the internal layer.

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Table 1. Contingency table showing the fraction of individuals with different combinations of external and internal opinions.

https://doi.org/10.1371/journal.pone.0218729.t001

For both models, the consensus time T_cons is defined as the duration between the initial state and the consensus. (Its synonyms are exit time, hitting time, or first-passage time; cf. [58,68]).

An agent-based Monte Carlo simulation of the BVM with N individuals proceeds according to the following algorithm.

1. Initialization: We initialize the opinions such that a given fraction ρ_R of the nodes are red. All other nodes are blue. We initialize time: t←0.
2. Iteration:
   1. We choose a “focal” individual f uniformly at random from all of the N individuals.
   2. We pick a neighbor n of the focal individual uniformly at random from all of its neighbors. (On a complete graph with self-links, all individuals are neighbors of each other so that n can be any individual in the group.)
   3. f adopts n’s opinion. That is, if f and n have different opinions, we change f’s opinion to that of n. Otherwise the system remains unchanged.
   4. We increment the time by a random number Δt drawn from an exponential distribution with mean , where c is a positive number: t←t+Δt. We can interpret c as the rate with which individuals copy opinions of any of their neighbors.
   5. If all nodes have the same color (i.e., the individuals have reached a consensus), we set and terminate. Otherwise we go back to step (i).

Compared to the BVM, the CVM adds two options to step (ii): besides copying a neighbor’s external opinion, individuals may also externalize or internalize (Fig 1B) to reduce cognitive dissonance as described in Section 1. We model copying, externalization and internalization as independent Poisson processes with rates c, e and i, respectively, according to the following agent-based algorithm.

1. Initialization: We initialize the opinions so that
   • a given fraction ρ_Rb of the individuals is externally red and internally blue,
   • a given fraction ρ_Br is externally blue and internally red,
   • a given fraction ρ_Rr is red in both layers,
   • the rest is blue in both layers.
     We also initialize time: t←0.
2. Iteration:
   1. We choose a “focal” individual f uniformly at random from all of the N individuals.
   2. We generate a random number u that is uniformly distributed between 0 and c+e+i. We now distinguish between three cases.
      Case (A): Copying. If u<c, then f adopts the opinion of a random neighbor and we immediately go to step (iii).
      Case (B): Externalization. Otherwise, if c≤u<c+e, then f externalizes. That is, if the current state of f is Rb, we change its state to Bb. If the current state is Br, we change f’s state to Rr. For all other current states, the system does not change. We subsequently go immediately to step (iii).
      Case (C): Internalization. Otherwise f internalizes. That is, if the current state of f is Rb, we change its state to Rr. If the current state is Br, we change f’s state to Bb. In all other cases, the system does not change.
   3. We increment the time (i.e., t←t+Δt) by a random number Δt drawn from an exponential distribution with mean . We can interpret c+e+i as the rate with which individuals are active because they copy, externalize, or internalize.
   4. If all nodes have the same color (i.e., all internal and external opinions are identical), we set and terminate. Otherwise we go back to step (i).

We show snapshots of two illustrative sample runs of the CVM algorithm in Fig 2.

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Fig 2. Two illustrative simulations in a relatively small group (N = 16 individuals).

Each panel shows the state of the group at a given time from t = 0.0 to 100.0. (We write time with a decimal point because it is a continuous variable in our model.) The update rates are c = 1,e = 1/4, and i = 1/16. In each snapshot, the upper/lower nodes show the external/internal opinions (red or blue). In the external layer, the nodes are linked in a complete graph. We do not show the links, only indicate full connectedness by a gray box. (The reflexive self-link is also included. Therefore, every individual is linked with N = 16 individuals within the external layer.) The link between each individual’s external and internal opinion is represented by a vertical line. Arrows mark those individuals who are hypocrites. The color of the arrow shows the external opinion. Initially, both groups contain 12 red and 4 blue opinions in the external layer. The difference is in the occurrence of hypocrisy: in the left/right column, every individual is initially hypocritical/frank. By t = 100.0, one of the opinions has reached a significant majority in each simulation (i.e., the group is close to a consensus).

https://doi.org/10.1371/journal.pone.0218729.g002

The agent-based BVM and CVM algorithms that we have presented in this section make the relation between the models and individual opinion formation explicit. However, in practice, the corresponding Gillespie algorithms [69], based on the transition rate matrices given in [53], produce numeric results for both models more efficiently and are mathematically equivalent. We have, therefore, implemented our simulations with the Gillespie algorithms. The source code is available from GitHub [70].

3 Consensus time in the BVM and CVM

The BVM has been studied by numerous authors during its 45-year history (for reviews, see [12,14]), while the CVM is relatively new [53]. Here we review the main results concerning the consensus time in complete graphs. Both the BVM and the CVM lead to a consensus within a finite time, provided that the number of individuals N is finite (cf. [71] concerning the BVM and [53] about the CVM). The only exception occurs when i = 0 in the CVM and both r and b are present in the internal layer. In that special case, r and b persist for an infinitely long time.

In the BVM, the mean consensus time in the complete graph is [58] (1) where N is the number of individuals, c is the copying rate, and ρ_R is the proportion of the red opinion in the external (i.e., only) layer at time t = 0. Note that is symmetric under the exchange of the red and blue opinions because ρ_B = 1−ρ_R. For obtaining the mean consensus time in the CVM, it is worth introducing a new variable, (2) where ρ_R = ρ_Rb+ρ_Rr is the fraction of externally red individuals and ρ_r = ρ_Br+ρ_Rr is the fraction of internally red individuals (Table 1). We can interpret m as the overall “strength” of the red opinion because it characterizes the actual weight of the red opinion in the group, as e and i express the flow of information between the two layers. The expected value of m remains constant throughout the process of consensus formation; that is, m(ρ_R,ρ_r) is a martingale [53]. As a consequence, m(ρ_R,ρ_r) is equal to the probability that the consensus in a group with initial conditions (ρ_R,ρ_r) is red. (We will show numerical evidence for this equality in Section 4.3.)

The mean consensus time in the CVM can be directly obtained from that in the BVM, (3) where (4)

On one hand, because τ(c,e,i)>1, the mean consensus time is always longer in the CVM than in the BVM, given the same number of individuals N. Thus, the existence of hypocrisy in a group increases the consensus time. On the other hand, the probability m of a red consensus and the mean consensus time are fully determined by c, e, i, n, ρ_R, and ρ_r (see Eqs 2–4), and thus do not depend on the amount of hypocrites ρ_Rb or ρ_Br. To resolve this paradox, it is necessary to better understand the transient behavior of the CVM, particularly the role of hypocrites in the consensus-finding process. This observation motivated the present study.

4 Results

4.1 Consensus formation in the CVM

We study the changes in the state (ρ_Rb,ρ_Br,ρ_Rr) over time. Table 2 and Fig 3 present results from simulations for four different initial conditions. All these simulations were performed with the same parameters N = 400, c = 1, e = 1/4, and i = 1/16. We have also carried out simulations for various other parameter combinations and report representative results in the S2 Appendix. In Table 2, we denote by F the fraction of simulations in which the consensus is red. (In the table, we also list the equilibration time T_equal that we define and discuss in Section 4.2.) Table 2 underlines the excellent agreement between theoretical prediction and numeric simulations. We also observe agreement for other parameters. We support this claim by showing the results for another parameter combination in the S2 Appendix.

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Fig 3. Changes in the composition of the group over time in four realizations of the CVM.

Each point belongs to one “snapshot” between the initial state (large, filled symbols) and the consensus (large, open symbols). The parameters for all simulations are N = 400, c = 1, e = 1/4, i = 1/16. We show only a single simulation for each initial condition, but we note that repeated simulations from the same initial state can end in either of the absorbing states unless the initial state is already a consensus. The time intervals between the plotted points change along the trajectories: during the transient, we plot every 100^th time step until the 2000^th time step. Afterwards we plot only every 10000^th time step because the spatial density of data points is higher along the attractor. (A) Abundances of the two kinds of hypocrites (ρ_Rb and ρ_Br) and the proportion of frank red individuals (ρ_Rr). These variables completely describe the state of the system. The theoretical attractor (Eq 5) is shown as a black curve. (B) Equalization of the two kinds of hypocrites in the same realizations. In a consensus, there are no hypocrites (i.e., ρ_Rb = ρ_Br = 0). Therefore, the open symbols at the consensus overlap. The attractor appears as a straight line because it is a two-dimensional projection of the curve embedded in three dimensions. The line is exactly on the diagonal. (C) Equalization of the abundance of the red opinion in the external (ρ_R = ρ_Rb+ρ_Rr) and internal layer (ρ_r = ρ_Br+ρ_Rr). The attractor (black line) is exactly on the diagonal.

https://doi.org/10.1371/journal.pone.0218729.g003

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Table 2. The four cases of CVM simulations that are displayed in Figs 3 and 4.

https://doi.org/10.1371/journal.pone.0218729.t002

We show in Fig 3A and 3B (and, for another parameter combination, in the S2 Appendix) that the process of consensus formation in all realizations takes place in two stages. The first stage is relatively short and is dominated by a directional change in the group’s composition: the trajectories go toward an attractor, which is an arch-shaped curve (plotted as a black curve in Fig 3). The S1 Appendix shows that the curve is given by (5)

The second stage of consensus formation is characterized by a random walk along this curve. The deviations of the trajectories from the theoretical curve are caused by the finite size of the simulated system (N = 400). The walk ends by reaching one of the two consensus states (i.e., ρ_Rb = ρ_Br = 0 and either ρ_Rr = 0 or ρ_Rr = 1). The direction of each step in the walk is randomly selected. However, unlike in conventional unbiased random walks, the frequency of steps is decreasing as the consensus is approached because most copying events are between like-minded individuals. Moreover, near the consensus, there are only few hypocrites left so that externalization and internalization are also unlikely to change the state.

It is interesting to observe the wide range of the walk in the second CVM stage. For example, the trajectory indicated by purple diamonds in Fig 3C ends in a consensus in which the blue opinion wins (i.e., ρ_Rb = ρ_Br = ρ_Rr = 0); nevertheless, the trajectory goes through some points in which ρ_R>0.5, (i.e., the red opinion is temporarily the majority in the external layer). The probability of a reversal from an external red majority ρ_R(t)>0.5 at time t to a blue consensus is equal to 1−m[ρ_R(t),ρ_r(t)], where m is given by Eq 2. We note that this probability does not depend on the system size because m is independent of N.

4.2 Equalization of the number of hypocrites

There are two kinds of hypocrites: Rb (i.e., externally red) and Br (externally blue). We denote the difference in the abundances by D = ρ_Rb−ρ_Br. In Fig 3A, we see that the abundances of the two kinds of hypocrites tend toward equality in the first stage of consensus formation (i.e., , where the overline denotes the mean, averaged over different simulations). A projection of the data onto the (ρ_Rb,ρ_Br) plane (Fig 3B) shows this tendency even more clearly. The black line (i.e., the projection of the attractor) is exactly on the diagonal.

To examine this phenomenon in more detail, we plot the absolute value of from t = 0.0 to 10.0 (Fig 4). The data points were obtained from 1000 independent repetitions. As we derive in the S1 Appendix, we expect (6)

(A rigorous proof is given in [53].) The straight lines in Fig 4 indicate the theoretical predictions, which are an excellent fit to the numeric data.

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Fig 4. Equalization of the two kinds of hypocrites over time.

The parameter values are the same as in Fig 3. The initial conditions are also the same for the corresponding shapes and colors. The vertical axis shows the absolute value of the mean difference on a logarithmic scale. The mean was obtained from 1000 independent realizations. The error bars are shown only where they are larger than the symbol sizes. The solid lines are theoretical predictions under the assumption that decays exponentially with rate e+i (Eq 6).

https://doi.org/10.1371/journal.pone.0218729.g004

In Fig 3, we illustrate that the equalization process is relatively fast compared to the subsequent random walk on the attractor. Let us define the equalization time T_equal as the time t satisfying , (7)

In the logarithmic plot of Fig 4, T_equal is proportional to the inverse of the slope. The plotted examples and the last two rows of Table 2 confirm that T_equal is independent of the initial conditions. Table 2 also shows that T_equal is much shorter than the mean consensus time . For the chosen rates (c = 1, e = 1/4, i = 1/16), is longer than T_equal by around three orders of magnitude. The precise ratio of to T_equal depends on the initial conditions and the parameters c, e, i and N. In particular, increases linearly with N (see Eqs 1 and 3), but T_equal is independent of N. So, even for only moderately large group sizes, the CVM spends most of the time during consensus formation in the second, random walk stage. Simulations with a different parameter combination (S2 Appendix) provide further evidence for the generality of this result.

The number of hypocrites is not the only observable that equalizes in the first stage. We show in Fig 3C that the abundance of the red opinion becomes equal in the public (external) and in the concealed (internal) layer (i.e., ). The time of equalization is exactly the same as in the case of hypocrites above. The reason is that, because of ρ_Rb = ρ_R−ρ_Rr and ρ_Br = ρ_r−ρ_Rr, we obtain (8)

Consequently, T_equal applies as much to the equalization of the abundances between the two layers as it applies to the equalization of hypocrite types.

It is important to note that the ratio of red to blue is the same in both layers during the second stage, which is the predominantly longer stage during the process of consensus formation. This observation suggests that estimating the abundance of an opinion in the public layer gives a good prediction for the abundance in the concealed layer. (We consider the implications of this result in Section 5.1.) The statement refers, of course, to the averages in the population. The individuals who bear these opinions can be different at different times. Furthermore, Eq 6 neither predicts ρ_Rb nor ρ_Br, only the mean difference . At some parameter combinations, we temporarily find a considerable amount of hypocrites before reaching a consensus, while for other combinations most of the individuals are frank.

4.3 The (un)importance of hypocrisy

As we pointed out earlier, the opinion dynamics on a complete graph must end in either a red or blue consensus (provided that the number of individuals N is finite). How do the initial conditions and the rates c, e and i influence which opinion is going to win (i.e., becomes the consensus opinion)?

Fig 5 shows theoretical predictions together with simulation results. We show in Fig 5A that the frequency F of red winning is equal to the initial value of m, defined in Eq 2. Since the formula for m does not contain c, the probability of winning does not depend on the communication at the external layer, but only on the interaction between the external and internal layer (expressed by e and i). Another noteworthy result is that the abundances of the hypocrites ρ_Rb and ρ_Br do not appear in Eq 2 either. They are implicitly present in the equation because ρ_R = ρ_Rb+ρ_Rr and ρ_r = ρ_Br+ρ_Rr. However, to predict the eventual winner, it is sufficient to know only two variables, namely ρ_R and ρ_r. In fact, after the attractor (given by Eq 5; black curve in Fig 3) has been reached, we do not even need to know ρ_r any longer. By that time, the abundances in both layers are, on average, equal (see Fig 3C). So the probability of a red victory is directly given by the abundance ρ_R of red in the external layer, which is easier to observe in real life than the internal abundance ρ_r.

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Fig 5. The consensus distribution and consensus time in the CVM.

Numeric results confirm that these properties of the CVM depend only on m (defined in Eq 2), the initial strength of the red opinion. (In the plot, we have deliberately added jitter in the horizontal direction to make individual data points visible. Otherwise the overlap would obscure that there are multiple points on top of each other.) Each symbol shows the mean of 1000 simulations. All simulations use e = 1/4 and i = 1/16. N, c, and the initial conditions vary. (N is symbolized by the size, c by the shape, and the initial condition by the color; see legend.) (A) Proportion of simulations in which the red opinion wins (F) as a function of the initial strength of red [m(t = 0)]. The diagonal line indicates F = m. The overlapping symbols exemplify that simulations with different initial conditions produce the same F if the initial m is the same, despite different abundances of hypocrites. (B) The mean consensus time as a function of m. The gray curves are theoretical predictions from Eq 9, which is derived under the assumption that N is large [53]. The simulations confirm that Eq 9 is a good approximation even for moderately large N. For different N and c, the values of fall on different curves. However, for given N and c, depends only on m, but not on any further details of the initial conditions.

https://doi.org/10.1371/journal.pone.0218729.g005

Unlike F, the mean consensus time does depend on c and N. This statement follows from the combination of Eqs 1, 3 and 4, (9)

We confirm this result with Fig 5B: data points for different N and c fall on different curves. Still, neither ρ_Rb nor ρ_Br appear explicitly in Eq 9. Consequently, m is all we need to know about the initial condition in order to predict . After we have reached the attractor, m is equal to ρ_R so that we can, at that stage, make a prediction for the mean consensus time simply by observing the abundance of the red opinion in the external layer.

5 Discussion

5.2 How to shorten or prolong the consensus time

A long consensus time may be disadvantageous, for example, when a coordinated action is needed in a group. For instance, this is the case in jury decision making, which has been studied extensively [72–76]. Conversely, delaying a consensus may be desirable in those situations, in which the diversity of opinions is valuable. We have pointed out in Section 4.2 that increases linearly with N so that one strategy to delay a consensus simply is to expand the group size. The linear scaling of the consensus time with N, however, is not specific to the CVM. We find the same scaling relation in the BVM too (see Eq 1). Besides changing the group size, which properties of the concealed layer influence the consensus time?

We know from Eq 4 that the delay τ depends only on the copying rate c, the externalization rate e, and the internalization rate i. If these rates are positive, we have τ>1 so that the CVM needs more time to reach consensus than the BVM with the same value of c [53]. For example, for the parameter combination used in Table 2 (c = 1, e = 1/4, i = 1/16), we obtain τ≈12.8; that is, the CVM needs on average around one order of magnitude longer to reach a consensus. We are not aware of any empirical study that suggests specific values for e or i. To investigate the effect of varying e and i, we carry out simulations in the S2 Appendix where we swap their value compared to Table 2 (i.e., c = 1, e = 1/16, i = 1/4). In that case, the consensus time is shorter, as predicted by Eq 3.

It is instructive to consider the limiting cases of the CVM if the rates e or i are extremely fast or slow. On one hand, in the limit i→∞ we would have simply retrieved the BVM. The prediction of Eq 4 is, accordingly, τ→1 in this limit. On the other hand, increasing e and decreasing i can enhance the consensus time to enormous values. Thus, to maintain a longer coexistence of alternative opinions, one could aim to increase e, for example by encouraging self-expression [77–79], whereas decreasing i is more difficult to accomplish in real life because the institutions of socialization (school, work, the state, and the media) are well established. In the limit i→0, goes to infinity. The reason for this divergence is that, in this case, the internal layer does not change at all so that any disagreement in the internal layer remains frozen in time.

The zealot voter model, introduced by Mobilia et al. [57], applies the limit i→0 together with e→∞ for a certain fraction of the group. These “zealots” never change their internal opinion and always express this opinion openly. Some other models in the literature can also be “translated” into two-layer voter models with fixed internal states (i.e., i→0). For example, there are models with “inflexible minorities” [80], “stubborn agents” [81], or “partisans” [59,60], in which the individuals have a fixed and innate preference for one of the (external) states. To the best of our knowledge, the CVM is the first model in which the internal layer can change so that the interaction between the two layers is bidirectional.

The winning opinion is not necessarily a majority opinion; for instance, as Nemeth and Wachtler [82] expressed, consistent minority views can affect other members of a jury whilst reaching their verdict in fictional courts. Minority opinions can also have an impact on public discourse in social questions such as feminism, the death penalty, or homosexuality [79,83,84].

The interplay between the layers slows down the CVM compared to the BVM, but, interestingly, the number of hypocrites is not directly responsible for the deceleration. As we pointed out above, the hypocrite abundances ρ_Rb and ρ_Br do not appear explicitly in the formulas for m and (see Eqs 2 and 9). In other words, both m and are determined by the marginal frequencies in Table 1 (i.e., ρ_R and ρ_r) alone, which do not uniquely determine the joint frequencies ρ_Rr, ρ_Br, ρ_Rb, and ρ_Bb. For example, a group with ρ_R = 1/2 and ρ_r = 1/2 may consist purely of hypocrites (ρ_Rb = ρ_Br = 1/2, ρ_Rr = ρ_Bb = 0) or purely of frank individuals (ρ_Rr = ρ_Bb = 1/2, ρ_Rb = ρ_Br = 0). The consensus time is the same in both cases.

In this sense, the CVM is slower than the BVM because of the mere existence of the concealed layer and, thus, because of the opportunity for hypocrisy, but not because of the amount of hypocrisy. The concealed layer serves as a pool of opinions. Hypocrisy necessarily emerges in such a system as a path from one opinion to another, but the amount of hypocrisy does not drive the consensus process.

6 Summary and outlook

This study presented and analyzed the Concealed Voter Model (CVM) as a relatively simple model of opinion dynamics with the possibility of hypocrisy (i.e., internal and publicly expressed individual opinions might differ). The study demonstrated that the process of consensus formation in the model takes place in two stages. The first stage is relatively short and is characterized by an equilibration in the number of hypocrites of both types (ρ_Rb≈ρ_Br) or, equivalently, an equilibration in the opinions in both layers (ρ_R≈ρ_r). The second stage is a random walk along an attractor to which the opinion distributions quickly converge. We investigated the time needed to reach a consensus and compared to the consensus time in the Basic Voter Model (BVM), in which hypocrisy is non-existent.

An important result is that hypocrisy always prolongs the mean time needed to reach unanimity of opinions. This result calls for policy measures that could eliminate or decrease the second stage of the dynamics in jury and committee decision making where consensus and fast decisions are highly desired, for instance by requiring a critical (high) threshold instead of unanimity [72,73].

As a second important result, the study showed that the number of both kinds of hypocrites equalize. This is good news for opinion pollsters who suffer from bias induced by hidden opinions. The model demonstrates that, in a neutral scenario, the numbers of both kinds of hypocrites balance each other out–so any empirical observation of bias is due to external factors or heterogeneous externalization and internalization rates that correlate with party preference.

As a third important result, the study demonstrated that in the CVM, just like in the BVM, the consensus time increases linearly with population size. This result is in line with observations that coordination in a larger group is more difficult, but not disproportionately more difficult than in a smaller group [72,74].

The simple CVM can be extended in multiple directions. So far, we have assumed that the links between individuals constitute a complete graph. It would be interesting to consider whether permitting different graph structures would change the results qualitatively. Previous work highlighted that the network structure is important for opinion dynamics in general [85]. In the BVM, networks with broad degree distributions shorten the time till consensus [58]. In a model similar to the BVM, Axelrod [86] has shown that minority opinions can survive and consensus may never be reached if network routes of influence are dissolved so that opinions are cut out completely from interactions with opposing opinions. Minority opinions are also likely to survive in models where network ties of influence are updated according to opinion homophily so that there is a co-evolution of networks and opinions [87–90].

Another way to increase realism is to consider nonbinary opinions. The previous literature has highlighted substantial differences between opinions measured on binary, continuous or nominal scales [91]. From an empirical perspective, it would also be desirable to study non-symmetric competition between the opinions. Asymmetry could be introduced, for instance, by a “media” effect that causes bias in one direction (e.g., in social influence or externalization) [92–94] or by heterogeneous internalization rates that correlate with voter preferences. Beyond a constant media bias in one direction, one could study exogenous stimuli that alter the direction of the bias. Such an extension could potentially explain the occurrence of cyclical patterns that are not emergent from the current model, but are observable, for example, in fashion trends and public sentiments [95,96].

Despite the limitations, we believe that studies that permit concealed opinions have a lot of potential in understanding opinion dynamics because they incorporate an important feature of real-life opinions: not everything is expressed openly.

Supporting information

Acknowledgments

B. Oborny and M. Gulyás are grateful to Ferenc Mészáros for his helpful comments on the model.