Modelling framework

Consider a group of individuals. Time is discrete. At each moment, each individual is characterised by an action x they take and by an attitude y towards possible actions (0 ≤ x, y ≤ 1). If there are just two possible actions, say, x = 0 and x = 1, then attitude y measures their relative subjective appropriateness with high y implying high appropriateness of choosing x = 1. If there is a continuum of possible actions, then y specifies the most appropriate action (i.e. a personal norm, Schwartz, Reference Schwartz1977) from the point of view of the individual. An example of the former case would be a decision on whether to participate or not in a mass protest. An example of the latter would a decision on a share of their endowment to donate to a recipient in the Dictator game. Variable y is related to the strength of norm internalisation in Gavrilets and Richerson (Reference Kuran2017). Our approach can also be viewed as a special case of a more general framework of Gavrilets (Reference Gavrilets2021). Below when talking about attitude y, we will occasionally refer to it as ‘belief’ or ‘preference’.

As a result of action x the individual gets material utility (i.e. payoff) π(x), which can depend on the average action $\bar{x}$ of their group, and psychological utility (i.e. normative value) V (x, y), which depends on their action x and attitude y but does not depend on their social environment. We assume the existence of an authority promoting action x = G. We are agnostic about the source of power of the authority. Rather our model captures its effects by a single parameter F measuring efficiency of its messaging which can depend, for example, on the effort or legitimacy of (or trust in) the authority. We postulate that in deciding on the action the individual attempts to maximise utility function u:

(1)$$u = \underbrace{{\pi ( x) }}_{{{\rm material\ payoff}}} + \underbrace{{V( x, \;y) }}_{{{\rm normative\ value}}}-\underbrace{{k_1{( y-x) }^2}}_{{{\rm cognitive\ dissonance}}}-\underbrace{{k_2{( \bar{x}-x) }^2}}_{{{\rm conformity\ with\ peers}}}-\underbrace{{k_3{( G-x) }^2F, \;}}_{{{\rm conformity\ with\ authority}}}$$

where k ₁, k ₂ and k ₃ are individual-specific non-negative parameters measuring psychological costs owing to sensitivity of decision-making to the corresponding forces. These parameters can depend on specific characteristics of individuals, e.g. cognitive or cultural. For example, individuals reacting negatively to perceived attempts to restrict their freedom (that is, showing psychological reactance) would have low values of k ₂ and k ₃. Utility function (1) implies that material benefit π is traded off against psychological costs owing to cognitive dissonance (i.e. a mismatch between the action x and attitude y; Festinger, Reference Festinger1957) and mismatch of their action with the average action of their peers, $\bar{x}$, and action G promoted by the authority. Note also that eqn (1) implies that individuals condition their actions on the behaviour of their peers (Fischbacher and Gächter, Reference Fischbacher and Gächter2010). The conformity term is similar to those emerging in some models of coordination in economics (e.g. Brock and Durlauf, Reference Brock and Durlauf1999, Reference Brock and Durlauf2001; Kuran and Sandholm, Reference Kuran and Sandholm2008). In general, G can change, e.g. if the authority updates its messaging (Almagro and Andrés-Cerezo, Reference Almagro and Andrés-Cerezo2010), or the authority could also produce certain goods (e.g. Verdier and Zenou, Reference Verdier and Zenou2018). However for simplicity we will focus exclusively on messaging and treat G as a constant.

We will further postulate that after taking an action, the individual's attitude y can change owing to cognitive dissonance and conformity:

(2)$${y}^{\prime} = y + \underbrace{{\alpha \lpar x-y\rpar }}_{{{\rm cognitive\ dissonance}}} + \underbrace{{\beta \lpar \bar{x}-y\rpar }}_{{{\rm conformity\ with\ peers^{\prime}\ action}}}-\underbrace{{\lpar G-y\rpar F\comma \;}}_{{{\rm conformity\ with\ authority}}}$$

where α, β and γ are individual-specific non-negative parameters measuring sensitivity of attitudes to the corresponding forces. Here the cognitive dissonance term acts to align individual attitude y with a previously taken action x (Festinger, Reference Festinger1957; Rabin, Reference Rabin1994; Kuran and Sandholm, Reference Kuran and Sandholm2008; Calabuig et al., Reference Calabuig, Olcina and Panebianco2018). The two conformity terms act to align y with the average group behaviour $\bar{x}$ and the action G promoted by the authority, respectively. These terms have a simple linear form standard in models of social influence (DeGroot, Reference DeGroot1974; Watts, Reference Watts2002; Centola et al., Reference Centola, Willer and Macy2005; Friedkin et al., Reference Friedkin, Proskurnikov, Tempo and Parsegov2016) and are also related to those used in cognitive neuroscience (Olsson et al., Reference Olsson, Knapska and Lindstr¨om2020). Note that while individual actions x affect material payoffs directly, individual thoughts/beliefs y do so only indirectly via subsequent actions.

The change in attitude y described by eqn (2) leads to a change in the utility function u which in turn can cause subsequent changes in actions taken. We assume that individuals take their actions and update their attitudes synchronously. Between-individual variation in parameters k ₁, k ₂, k ₃, α, β and γ reflects differences in psychology, culture, personal history, information received, etc. Table 1 summarizes model variables and parameters. Note that we do not study the evolutionary origins of this variation (see e.g. Gavrilets and Richerson, Reference Gavrilets and Richerson2017) but rather postulate that it is already present and then explore its consequences on relatively short time-scales of individual life spans.

Table 1. Main variables, functions, and parameters

Next we consider three different applications of this model. In the first application, the ‘authority’ will be elders and prestigious individuals in a small-scale society promoting food sharing. In the second case, the authority will be a revolutionary leader promoting mass protests against the government. The third model will consider an imaginary authority – the identity group and its stereotypical behaviour.

For these three cases, we will present analytical approximations capturing the effects of different parameters on the average characteristics of the population at equilibrium. We will use agent-based simulations to illustrate the dynamics of x and y in time and the resulting distributions of y in the population.

Food sharing in small-scale societies

Consider a group of hunter–gatherers some of whom occasionally succeed in hunting and come into possession of some food items. Other group members express a desire to get a share from successful hunters who can oblige (i.e. choose x = 1) and lose food of value c or not oblige (i.e. choose x = 0). Individuals who share get normative value vy, where v is a non-negative parameter. Within the group there is an informal authority (e.g. elders who cannot hunt anymore themselves or some other accomplished and prestigious figures) inculcating food sharing among group members (i.e. G = 1); the efficiency of inculcation is controlled by parameter F. Let p be the frequency of sharing. Individuals assume that their peers who do share will disapprove of them if they do not share. Because everybody benefits from sharers, they do not suffer any cost of disapproval. Individuals differ in their attitude y towards sharing and in their sensitivity to social influence by peers and authority. We note that in the well-studied Hadza and the Ju’/hoansi there is a lot of bickering and shaming caused by a failure to share (Marlowe, Reference Marlowe2009; Wiessner, Reference Wiessner2014), which can be interpreted as reflecting individual variation in y. This model is described by a special case of eqns (1) and (2) (see the Supplementary Material).

We show in the Supplementary Material, that the deterministic version of this model predicts evolution to a line of equilibria different in p and in the distribution of attitudes y. At each equilibrium, a proportion of individuals have relatively high values of y and share food while remaining individuals have relatively low values of y and do not share food. The average difference in attitudes between individuals who share and those who do not is equal to the average of ratio α/(α + β + γ), which measures the relative importance of cognitive dissonance in attitude updating process. There can also be equilibria where everybody or nobody shares food. In particular, the state of universal sharing (p = 1, y = α + β + γ) is stable if the joint effect of normative value, cognitive dissonance, conformity with peers and conformity with authority on decision-making is larger than the benefit lost (v + ∑ k_i > c). The state of no sharing (p = 0, y = γ) is stable if the effects of authority both on actions (k ₃) and attitudes (γ) are small enough. All model parameters control the location and stability of the line of equilibrium (see the Supplementary Material).

The existence of multiple equilibria in general and of lines of equilibria in particular has two important implications. First, initial conditions become very important, so one needs to study their effects in detail. Second, models with lines of equilibria are structurally unstable, so one needs to study their behaviour under some perturbations.

Below we illustrate the model's behaviour using agent-based simulations which introduce additional features aiming to make the model more realistic. First, we allow for stochasticity in decision-making and in the process of updating the attitudes. The former will be characterised by precision parameter λ (large λ mean higher precision). The latter is characterised by a standard deviation σ of a random perturbation of attitude y during the update process (see the Supplementary Material for details). In our numerical simulations, the initial frequency of sharing is set to 0 but we vary the mean $\bar{y}$ and standard deviation σ of the distribution of attitude y in the population. Second, we will assume that there is not one but a large number of relatively small groups randomly exchanging migrants at rate m (Wright, Reference Wright1931). This assumption fits many hunting and gathering societies such as the Hudza in north-central Tanzania (Hill et al., Reference Hill, Walker, Bǒzǐcević, Eder, Headland, Hewlett and Wood2011). Since our main focus is on the effects of messaging, we choose high enough dispersal rates to reduce the effects of genetic relatedness on the model behaviour.

In agent-based simulations, we set c = 1 but allow for most other parameters to differ between individuals. Specifically, for each individual values of v, k and d are drawn randomly from a ‘broken-stick’ distribution [0, 1]. Similar procedure is used for assigning individual values of parameters α, β and γ. Besides its intuitive appeal, the advantage of the ‘broken-stick’ distribution is that, similarly to the uniform distribution, it does not have any parameters.

Our simulations show that the population quickly evolves to a stochastic equilibrium where a proportion of individuals share food and have high attitudes towards it. The average attitude $\bar{y}$ towards sharing is typically smaller than the actual frequency of sharing p (because the latter is augmented by the authority's influence). Providing the efficiency of inculcation F is high enough, a large proportion of individuals will share food. At the same time, the population as a whole can exhibit a substantial variation in underlying attitudes. Figure 1 illustrates these observations. If authority weakens its messaging, the model predicts that food sharing will be maintained for some time (because it is now internalised) but then it will get weaker and eventually disappear.

Figure 1. Predictions of the food sharing model for different efficiencies F of inculcation promoting sharing (G = 1). Left: the dynamics of the average attitude y (blue curves) and the frequency of sharers p (red curves) for 10 different independent runs for each value of F. Right: the initial (white bars) and final (blue bars) distributions of attitudes for one run. The blue and red vertical lines mark the mean attitude $\bar{y}$ and the frequency p of sharers respectively. Log–normal distribution of initial values of y with mean 0.2 and standard deviation 0.1. Parameters: benefit lost to sharing c = 1, 100 groups of size n = 10, precision in decision-making λ = 10, standard deviation of error in attitude updating σ = 0.05, probability of attitude updating u_y = 0.5, dispersal rate m = 0.1, probability of successful hunting s = 0.15. Individual parameters k ₁, k ₂, k ₃ and α, β, γ are chosen from broken stick distributions on [0, 1] while parameter v is set to 1.

Political protests

Here we will adapt and extend a classical model proposed by Kuran (Reference Kuran1989) in his study of unanticipated political revolutions.

Assume that each individual can publicly express the discontent with the current government (e.g. by joining a protest), x = 1, or not, x = 0. Let y be the attitude towards joining the protest (0 ≤ y ≤ 1) and let $\bar{x}$ be the perceived frequency of people who are protesting. Kuran (Reference Kuran1989) called the variables x, y and $\bar{x}$ ‘public preference’, ‘private preference’ and ‘collective sentiment’, respectively. He ignored material payoffs (i.e. set π = 0) and normative value (i.e. set V = 0) but assumed that individuals get an ‘integrity utility’ as well as a ‘reputational utility’. The former depended on the mismatch between the attitude and the action chosen; its effect on utility function is analogous to that of the ‘cognitive dissonance’ term in our eqn (1). The latter was proportional to the frequency of people choosing the same action; its effect is analogous to that of the ‘conformity with peers’ term in our eqn (1).

Let us generalise Kuran's model by postulating the existence of an authority promoting action G and assuming that individuals pay certain psychological costs by acting against its message as specified by the last term in eqn (1). If the focal authority is the government, then G = 0. If the focal authority is the opposition, then G = 1. Parameter F measuring the efficiency of the authority's messaging (propaganda) depends on the perceived trust/legitimacy of the authority and on its effort. Kuran (1989) assumed that individual attitudes y were constant in time. In contrast, we postulate that after taking the actions and observing peer behaviour, individuals update their attitudes according to eqn (2).

This model is similar to the one describing food sharing (except that here b = 0 and that we consider a well-mixed population with individuals knowing the frequency of protesters $\bar{x}$). We illustrate its behaviour using agent-based simulations which allow for stochasticity in decision-making and in updating attitudes. Figure 2 shows that the propaganda by a revolutionary leader can change the distribution of attitudes in the population and cause, if sufficiently strong, widespread protests. In contrast if we assume that attitudes do not change, then for the initial distribution of y and parameter values used in Figure 2, the frequency of people joining the protests will remain very low for all values of propaganda efficiency F considered in spite of a significant hidden discontent. Figures S1 in the Supplementary Material shows that the system can have multiple equilibria so that the eventual outcome of the dynamics can depend on the initial distribution of attitudes.

Figure 2. Predictions of the political protests model for different intensities of propaganda F promoting protests (G = 1). Left: the dynamics of the average attitude y (blue curves) and the frequency of protesters p (red curves) for 10 different independent runs for each value of F. Right: the initial (white bars) and final (blue bars) distributions of attitudes for one run. The blue and red vertical lines mark the mean attitude $\bar{y}$ and the frequency p of protests respectively. Log–normal distribution of initial values of y with mean 0.2 and standard deviation 0.1. Parameters: population size n = 1000, precision in decision-making λ = 10, standard deviation of error in attitude updating σ = 0.05, probability of attitude updating u_y = 0.5.

We can also consider the case when there are multiple alternative authorities in the society. For example, let one authority promote protests (i.e. G = 1) with efficiency F ₁ and another promote acquiescence (i.e. G = 0) with efficiency F ₀. This model is mathematically equivalent to that with a single authority promoting the value of G = F ₁/(F ₀ + F ₁) with efficiency F = F ₀ + F ₁. Figure S2 in the Supplementary Material shows that if the propaganda efficiency of the government matches that of the opposition (i.e. F ₀ = F ₁ so that F = 0.5), the protests are suppressed. This may explain why even regimes capable of severe repression nonetheless devote substantial resources to persuasion.

Social identity and altruism

Numerous studies have shown that priming social identity affects human decisions and preferences. For example, making ethnic or racial identity more salient affect time and risk preferences (Benjamin et al., Reference Benjamin, Choi and Strickland2010). Making religious identity more salient decreases contributions to public goods of Catholics but increases it for Protestants, and increases agnostic/atheist risk aversion (Benjamin et al., Reference Benjamin, Choi and Fisher2016). Chang et al. (Reference Chang, Chen and Krupka2019) used a series of Dictator games described with either politically charged tax or neutrally framed language. They observed that subjects’ political identities interact with these frames, causing changes in both norms and choices. In particular, framing made Democrats prefer equalised outcomes, and Republicans reluctant to redistribute payments even when it leaves them disadvantaged. Using surveys conducted in UK and US, Pickup et al. (Reference Pickup, Kimbrough and de Rooij2021) showed that individuals were aware of the norms attached to their political identities, were knowingly choosing norm-compliance at personal cost, and that their costly expression of political identity varied with norm salience and strength.

Within our framework, the effects of social identity can be formalised as emerging from messaging of an imaginary authority – the identity group. Parameter F then can be interpreted as measuring the salience of social identity while G as a stereotypical behaviour associated with the focal social identity group. To be specific, consider the Dictator game which is often used to study cooperative and altruistic behaviour with identity considerations (e.g. Chang et al., Reference Chang, Chen and Krupka2019). Let a focal individual make a decision on a share x of their endowment b to transfer to a partner. Then the material payoff term in eqn (1) is π = b(1 −x). Note that in this model all interactions are dyadic rather than are happening in a group setting. As above, let y be the individual's attitude, i.e. the most appropriate, from their personal perspective, proportion of the benefit to be shared under given circumstances. Let us ignore the ‘conformity with peers’ terms in eqns (1) and (2) because subjects often were not informed about peer behaviour. This complete the formulation of the model.

We show in the Supplementary Material that, given y, the best response action/donation is a weighted sum of the personal norm y and the action G promoted by the authority:

(3)$$x = \displaystyle{{\varepsilon _xy + (1-\varepsilon _x)G} \over {\tilde{c} + 1}}.$$

where parameter $\tilde{c} = c/(k_1 + k_3F)$ is the strength of material payoffs relative to that of cognitive factors in the utility function and parameter ɛ_x = k ₁/(k ₁ + k ₃F) measures the relative strength of the effects of cognitive dissonance on actions. Computing the derivative of x with respect to F, one can see that donation x to the partner increases with identity salience F if

$$G > \displaystyle{{k_1} \over {k_1 + c}}y$$

(that is, if stereotypical donation G is large enough relative to individual attitude y) but decreases with F otherwise. Thus, as argued by Benjamin et al. (Reference Benjamin, Choi and Strickland2010), this model predicts that priming people experiencing strong cognitive dissonance (i.e. with large v) can cause negative effect on x.

However one can also expect that personal norms y can change as a result of psychological processes. Without priming (i.e. if F = 0), eqn (2) shows that y will evolve to match x after which x will evolve to zero so that individuals will not transfer any funds to the partner. In contrast, with priming F, at equilibrium,

(4a)$$x^{\ast} = G-\displaystyle{{\tilde{c}} \over {1-\varepsilon _x\varepsilon _y}}, \;$$

(4b)$$y^{\ast} = G-\displaystyle{{\varepsilon _y\tilde{c}} \over {1-\varepsilon _x\varepsilon _y}},$$

where parameter ɛ_y = α/(α + γF) measures the relative strength of the effects of cognitive dissonance beliefs (see the Supplementary Material). Equations (4) show that x^∗ and y^∗ are both smaller than the group's standard G with the personal norm y being closer to G than x^∗ (so that G − y^∗ = ɛ_y(G − x^∗)). Increasing the salience F of social identity decreases $\tilde{b}$, ɛ_x and ɛ_y moving x^∗ and y^∗ close to the identity standard G. Increasing the cost of expressing identity $\tilde{b}$ or the strengths of cognitive dissonance in decision-making, ɛ_x, and belief formation, ɛ_y, have opposite effects.

The derivatives of x^∗ and y^∗ with respect to F are always positive meaning that priming always has positive effects on actions and beliefs. This conclusion is different from that above for the model where beliefs were constant (also see Benjamin et al., Reference Benjamin, Choi and Strickland2010). On a short time-scale, some people with a very high personal norm y and experiencing significant cognitive dissonance (large c) may donate money to the partner even in the absence of priming (i.e. if F = 0). However on larger time scales, our model predicts that without a continuous priming y will evolve to match x and then contributions will disappear.