Rational populists: the social consequences of shared narratives

Abstract

We study a simple opinion dynamic model where a number of influencers have the possibility of conditioning it by supporting one of two alternative narratives. Influencers choose the narrative to support in order to optimize their return. They can therefore choose to support a narrative that is socially dysfunctional with which they need not agree with, insofar as the environmental conditions make it convenient. We show in particular under what conditions the dynamic leads to social polarization, that is, eventual takeover of one narrative over the other. The critical factors in this regard are the persuasion strength of the narratives and the number of influencers who are active. Stronger persuasion and a larger number of influencers both favor the takeover of one narrative over the other. In particular, simulation results suggest that even small changes in persuasion strength may cause major changes in the social dynamic and sudden regime shifts. We discuss the policy implications of these results, with an eye to current trends in online media in reinforcement of persuasion strength, e.g. by deploying artificial bots that reverberate one narrative at the expense of the others.

Appendices

Mathematical Appendix A: the dynamic system

The representative influencer has to solve the following optimization problem:

$$ \begin{array}{@{}rcl@{}} &&\underset{G\in \lbrack 0,1]}{MAX}\int\limits_{0}^{+\infty }\left\{ G^{\alpha }\left[ \overline{G}(N-1)\right]^{\beta }n^{\gamma }+(1-G)^{\alpha }\left[ (\overline{1-G})(N-1)\right]^{\beta }(1-n)^{\gamma }\right.\\ &&\qquad\qquad\quad\left.+\delta n+\varepsilon (1-n)\vphantom{\left[ \overline{G}(N-1)\right]^{\beta }}\right\} e^{-rt}dt \end{array} $$

(11)

subject to the dynamic constraint (2). We can write the current value Hamiltonian function as:

$$ \begin{array}{@{}rcl@{}} H(n,G,\lambda ) \!&=&\!G^{\alpha }\left[ \overline{G}(N-1)\right]^{\beta }n^{\gamma }+(1-G)^{\alpha }\left[ (\overline{1-G})(N-1)\right]^{\beta }(1-n)^{\gamma }\\ &&+\delta n+\varepsilon (1-n)+\lambda \left[ n(1-n)(a+bn)+c \left[ G+\overline{G}(N\!-1)\right] (1\!-n)\right.\\ &&\left.-c \left[ 1-G+(\overline{1-G})(N-1)\right] n\right] \end{array} $$

where λ is the multiplier associated to the n variable. The conditions that are derived from the Maximum principle are:

• 1. The value of G is chosen in order to maximize H(n, G, λ), given the values of n and λ. By computing the partial derivative of H(n, G, λ) with respect to G (remind that, by assumption, α + β = 1) one has:

  $$ \begin{array}{@{}rcl@{}} \frac{\partial H(n,G,\lambda )}{\partial G} &=&\alpha (N-1)^{\beta }\left[ n^{\gamma }-(1-n)^{\gamma }\right] +c \lambda (1-n)+c \lambda n \\ &=&\alpha (N-1)^{\beta }\left[ n^{\gamma }-(1-n)^{\gamma }\right] +c \lambda \end{array} $$

  It results that $\frac {\partial H(n,G,\lambda )}{\partial G}=0$ along the curve:

  $$ \lambda =\widetilde{\lambda }(n):=\frac{\alpha (N-1)}{c}^{\beta }\left[ (1-n)^{\gamma }-n^{\gamma }\right] $$

  (12)

  where $\widetilde {\lambda }^{\prime }(n)<0$, $\widetilde {\lambda }(0)=\frac { \alpha (N-1)}{c}^{\beta }>0$, $\widetilde {\lambda }(1)=-\frac {\alpha (N-1)}{c }^{\beta }<0$, and $\widetilde {\lambda }(1/2)=0$. The (12) curve separates, in the plane (n, λ) (see Fig. 1), the region where the representative influencer chooses G = 1 (above the curve) from the region where she chooses G = 0 (below the curve).

• 2. The dynamics of n and λ are given by (remind that, ex post, one has $\overline {G}=G$ and $\overline {1-G}=1-G$):

  $$ \begin{array}{@{}rcl@{}} \overset{\cdot }{n}&=&\frac{\partial H(n,G,\lambda )}{\partial \lambda } =n(1-n)(a+bn)+cN(G-n) \end{array} $$

  (13)
  $$ \begin{array}{@{}rcl@{}} \overset{\cdot }{\lambda }&=&r\lambda -\frac{\partial H(n,G,\lambda )}{ \partial n}\\ &=&r\lambda -\left[ \gamma (N-1)^{\beta }Gn^{\gamma -1}+\gamma (N-1)^{\beta }(1-G)(1-n)^{\gamma -1}+\delta -\varepsilon \right]\\&&-\lambda \left[ (1-2n)(\alpha +\beta n)+\beta n(1-n)-cN\right]\\ &=&\lambda \left\{ r+3bn^{2}+2\left( a-b\right) n-a+cN\right\}\\ &&-\gamma (N-1)^{\beta }\left[ Gn^{\gamma -1}+(1-G)(1-n)^{\gamma -1}\right] -\delta +\varepsilon \end{array} $$

  (14)

Moreover, the usual transversality condition has to be satisfied:

$$ \underset{t\rightarrow +\infty }{\lim}\lambda (t)n(t)e^{-rt}=0 $$

Here, it is always satisfied along the trajectories that lead to an equilibrium.

Mathematical Appendix B: basic mathematical results

Above the curve (12) the representative influencer chooses G = 1, and the system (13)-(14) becomes:

$$ \overset{\cdot }{n}=n(1-n)(a+bn)+c N(1-n) $$

(15)

$$ \overset{\cdot }{\lambda }=\lambda \left[ r+3bn^{2}+2\left( a-b\right) n-a+cN \right] -\gamma (N-1)^{\beta }n^{\gamma -1}-\delta +\varepsilon $$

(16)

while below the curve (12) the representative influencer chooses G = 0, and the system (13)-(14) becomes:

$$ \overset{\cdot }{n}=n(1-n)(a+bn)-cNn $$

(17)

$$ \overset{\cdot }{\lambda }=\lambda \left[ r+3bn^{2}+2\left( a-b\right) n-a+c N\right] -\gamma (N-1)^{\beta }(1-n)^{\gamma -1}-\delta +\varepsilon $$

(18)

The systems (15)-(16) and (17)-(18) are triangular, that is, $\overset {\cdot }{n}$ only depends on the value of n, whereas $\overset {\cdot }{\lambda }$ depends on both n and λ. Such structure is not anomalous for this type of control problems (bang-bang control) where the control variable (G in our case) jumps between two alternative states (0 and 1 in our case). Obviously, the value of λ plays an important role anyway in the dynamic of n, as the separatrix (12) draws the border between the region where the state (n, λ) is such that the representative influencer chooses G = 1 (above the curve), and the region where the opposite happens.

Moreover, the equilibrium value $\overline {n}$ of n is determined by the equation $\overset {\cdot }{n}=0$. Furthermore, to this equilibrium value there will correspond an equilibrium of the dynamic system if the equation $\overset {\cdot }{\lambda }=0$ is met (given $\overline {n}$) for a value $\overline {\lambda }$ of λ that belongs to the ‘right’ region. More specifically, it is required that:

• 1) If $\overline {n}$ is the value of n at which the right hand side of equation (15) equals zero, and $\overline {\lambda }$ is the value of λ at which (given $\overline {n}$) the right hand side of equation (16) equals zero, then the state ($\overline {n}, \overline {\lambda }$) is a stationary point of the system (15)-(16) only if it lies above the separatrix curve (12).

• 2) Likewise, if $\overline {n}$ is the value of n at which the right hand side of equation (17) equals zero, and $\overline {\lambda }$ is the value of λ at which the right hand side of equation (18) equals zero, then the state ($\overline {n},\overline { \lambda }$) is a stationary point of the system (17)-(18) only if it lies below the separatrix curve (12).

Taking into account the above, the state n = 1 corresponds to a stationary state of the system (15)-(16) if posing n = 1 in equation $\overset {\cdot }{\lambda }=0$, the latter is satisfied for a value of λ larger than $\widetilde {\lambda }(1)$, that is:

$$ \frac{\gamma (N-1)^{\beta }+\delta -\varepsilon }{r+b+a+cN}>-\frac{\alpha (N-1)}{c}^{\beta } $$

(19)

Condition (19) is satisfied if (ceteris paribus) the value of the influencers’ ‘persuasion strength’ c and that of the difference δ − ε are high enough, whatever the signs of the parameters a and b (which characterize the natural dynamic of narratives (1)). Notice that (19) can be satisfied even if δ − ε < 0; that is, even if the diffusion of Narrative 1 is undesirable in terms of the influencers’own inclinations.

Analogously, the state n = 0 corresponds to a stationary state of the system (17)-(18) if posing n = 0 in the equation $\overset {\cdot }{\lambda }=0$, the latter is satisfied for a value of λ smaller than $\widetilde {\lambda }(0)$, that is:

$$ \frac{\gamma (N-1)^{\beta }+\delta -\varepsilon }{r-a+cN}<\frac{\alpha (N-1) }{c}^{\beta } $$

(20)

Condition (20) is satisfied if the value of c is high enough and that of the difference δ − ε is low enough, whatever the signs of the parameters a and b. Condition (20) can be satisfied even if δ − ε > 0; that is, even if the diffusion of Narrative 2 is undesirable in terms of the influencers’own inclinations.

1.1 B.1 Dynamic with G = 1 (above the curve (12))

Equation (15) can be written as:

$$ \begin{array}{@{}rcl@{}} \overset{\cdot }{n}&=&n(1-n)(a+bn)+c N(1-n)\\ &=&(1-n)\left[ n(a+bn)+c N\right] \end{array} $$

(21)

where $\overset {\cdot }{n}=0$ for n = 1 and for:

$$ n=-\frac{c N}{a}\text{ \ }if\text{ \ }a\neq 0\wedge b=0 $$

$$ n=-\frac{1}{2b}\left( a-\sqrt{a^{2}-4b c N}\right) ,\text{ }-\frac{1}{2b} \left( a+\sqrt{a^{2}-4b c N}\right) \text{ \ }if\text{ }b\neq 0 $$

(22)

It may be useful to consider the problem of the existence of the equilibrium values of n by means of a geometrical approach. To this purpose, we split the expression in square brackets in the equation (21) in two parts: n(a + bn) and cN. The graph of the former is a parabola (with upward concavity if b > 0) that meets the n axis at n = 0 and n = −a/b. By adding the (positive) constant cN we operate an upward translation of the parabola (the more upward, the stronger the ”persuasion capacity” c and the number of agents N). This causes in turn a shift of the zeros of $\overset {\cdot }{n}$ in different directions according to the sign of b (they move further if b < 0, and get closer if b > 0). Remembering that n = −a/b is the internal steady state value of the natural dynamic (1), where both narratives coexist at equilibrium, we then notice that the cN term is the shift that is operated by the persuasive impact of Narrative 1 via the influencers’action. Moreover, if b < 0, we will have at most a single value of n ∈ (0, 1) at which $\overset {\cdot }{n}$ goes to zero, whereas if b > 0 we could have two such values of n. If the value of cN is high enough, then no equilibrium value of n can belong to the interval (0,1).

Let us now analyze the stability properties of the equilibria of system (15)-(16). The Jacobian matrix of system (15)-(16) is:

$$ \begin{array}{@{}rcl@{}} J_{G=1}(n,\lambda )\!\!&=&\!\!\left( \begin{array}{cc} \frac{\partial \overset{\cdot }{n}}{\partial n} & \frac{\partial \overset{ \cdot }{n}}{\partial \lambda } \\ \frac{\partial \overset{\cdot }{\lambda }}{\partial n} & \frac{\partial \overset{\cdot }{\lambda }}{\partial \lambda } \end{array} \right)\\ \!\!&=&\!\!\left( \begin{array}{cc} (1-n)(a+2bn)-\left[ n(a+bn)+c N\right] & 0 \\ \lambda \left[ 6bn+2\left( a-b\right) \right] +\gamma (1-\gamma )(N - 1)^{\beta }n^{\gamma -2} & r + 3bn^{2} + 2\left( a - b\right) n - a + c N \end{array} \right) \end{array} $$

with eigenvalues:

$$ \frac{\partial \overset{\cdot }{n}}{\partial n}=(1-n)(a+2bn)-\left[ n(a+bn)+c N\right] \text{ \ in direction of the }n-\text{axis} $$

(23)

$$ \frac{\partial \overset{\cdot }{\lambda }}{\partial \lambda } =r+3bn^{2}+2\left( a-b\right) n-a+c N $$

Note that by evaluating (23) at the equilibrium with n = 1 (when existing) we get:

$$ \frac{\partial \overset{\cdot }{n}}{\partial n}=-a-b-cN<0\ \ if\ \ c>-\frac{ a+b}{N} $$

(24)

while evaluating it at an equilibrium with $n=\overline {n}<1$we get:

$$ \frac{\partial \overset{\cdot }{n}}{\partial n}=(1-\overline{n})(a+2b \overline{n}) $$

From (24) we can conclude that when c and/or N are large enough, n = 1 becomes reachable as the associated eigenvalue is surely negative.

1.2 B.2 Dynamic with G = 0 (below the separatrix (12))

Equation (17) can be written as:

$$ \begin{array}{@{}rcl@{}} \overset{\cdot }{n}&=&n(1-n)(a+bn)-c Nn\\ &=&n\left[ (1-n)(a+bn)-c N\right] \end{array} $$

(25)

where $\overset {\cdot }{n}=0$ for n = 0 and for:

$$ n=1-\frac{c N}{a}\text{ \ }if\text{ \ }a\neq 0\wedge b=0 $$

$$ n=-\frac{1}{2b}\left( a - b + \sqrt{(a + b)^{2} - 4b c N}\right) ,\!\text{ \ }\frac{1}{ 2b}\left( -a + b + \sqrt{(a + b)^{2} - 4b c N}\right) \text{ \ }if\text{ }b\!\neq\! 0 $$

(26)

By following, as above, a geometrical approach, we can split the expression in square brackets of the equation (25) into the sum of two elements: (1 − n)(a + bn) and − cN. The graph of the former is a parabola (with upward concavity if b < 0, the opposite of what occurs for the case with G = 1), that meets the n axis at n = 1 and n = −a/b. By subtracting the (positive) constant cN we get a downward translation of the parabola, which causes a shift of the zeros of $\overset {\cdot }{n}$ in different directions according to the sign of b. If b < 0, we will have at most a single value of n ∈ (0, 1) at which $\overset {\cdot }{n}$ goes to zero, whereas if b > 0 we could have two such values of n. If the value of cN is high enough, then no equilibrium value of n can belong to the interval (0, 1).

The stability properties of the equilibria of system (17)-(18) are determined by the Jacobian matrix of system (17)-(18):

$$ \begin{array}{@{}rcl@{}} J_{G=0}(n,\lambda )\!\!&=&\!\!\left( \begin{array}{cc} \frac{\partial \overset{\cdot }{n}}{\partial n} & \frac{\partial \overset{ \cdot }{n}}{\partial \lambda } \\ \frac{\partial \overset{\cdot }{\lambda }}{\partial n} & \frac{\partial \overset{\cdot }{\lambda }}{\partial \lambda } \end{array} \right)\\ \!\!&=&\!\!\left( \begin{array}{cc} \left[ (1 - n)(a + bn) - c N\right] + n\left( b - a - 2bn\right) & 0 \\ \lambda \left[ 6bn + 2\left( a - b\right) \right] + c(1 - c )(N - 1)^{\beta }(1 - n)^{c -2} & r + 3bn^{2} + 2\left( a - b\right) n - a + cN \end{array} \right) \end{array} $$

The eigenvalues are:

$$ \frac{\partial \overset{\cdot }{n}}{\partial n} = \left[ (1-n)(a+bn)-c N\right] +n\left( b-a-2bn\right) \text{ \ in direction of the }n-\text{axis} $$

(27)

and:

$$ \frac{\partial \overset{\cdot }{\lambda }}{\partial \lambda } =r+3bn^{2}+2\left( a-b\right) n-a+c N $$

Note that by evaluating (27) at the equilibrium with n = 0 (when existing), we get:

$$ \frac{\partial \overset{\cdot }{n}}{\partial n}=a-c N<0\ \ if\ \ c>\frac{a}{N } $$

(28)

while evaluating it at an equilibrium with $n=\overline {n}>0$ we get:

$$ \frac{\partial \overset{\cdot }{n}}{\partial n}\mathbf{=}\overline{n}\left( b-a-2b\overline{n}\right) $$

Again as above, we find that strong persuasion and/or a high number of influencers, by ensuring that the eigenvalue evaluated at the steady state n = 0 in (28) is certainly negative, guarantees the reachability of the equilibrium where only Narrative 2 is present.