Unanimity, Coexistence, and Rigidity: Three Sides of Polarization

Abstract

Political polarization is perceived as a threat to democracies. Using the Galam model of opinion dynamics deployed in a five-dimensional parameter space, I show that polarization is the byproduct of an essential hallmark of a vibrant democratic society, namely open and informal discussions among agents. Indeed, within a homogeneous social community with floaters, the dynamics lead gradually toward unanimity (zero entropy). Polarization can eventually appear as the juxtaposition of non-mixing social groups sharing different prejudices about the issue at stake. On the other hand, the inclusion of contrarian agents produces a polarization within a community that mixes when their proportion x is beyond a critical value xc=16≈0.167 for discussing groups of size three and four. Similarly, the presence of stubborn agents also produces a polarization of a community that mixes when the proportion of stubborn agents is greater than some critical value. For equal proportions of stubborn agents a along each opinion, ac=29≈0.22 for group size four against ac=14=0.25 for group size three. However, the evaluation of the proportion of individual opinion shifts at the attractor 12 and indicates that the polarization produced by contrarians is fluid with a good deal of agents who keep shifting between the two opposed blocks (high entropy). That favors a coexistence of opposite opinions in a divided community. In contrast, the polarization created by stubborn agents is found to be frozen with very few individuals shifting opinion between the two opinions (low entropy). That yields a basis for the emergence of hate between the frozen opposed blocks.

Keywords: contrarians; opinion dynamics; polarization; prejudices; sociophysics; stubbornness.

Figure 1

Contrarians produce four distinct regions with different behaviors as a function of their proportion. A tipping point dynamic with $p_{c,x}=\frac{1}{2}$ prevails in the range $0\le x<x_c$ (region 1). The two associated attractors feature a stable coexistence of a majority and a minority. In the range $x_c\le x\le \frac{1}{2}$, the dynamics turn into a one-attractor dynamic located at $\frac{1}{2}$. Any initial support $p_0$ moves monotonously towards $\frac{1}{2}$ with repeating local updates (region 2). There, both opinions coexist in a perfect overall balance. When $\frac{1}{2}<x<\frac{5}{6}$ the dynamics are still monitored by one attractor at $\frac{1}{2}$, but the convergence towards it becomes oscillatory (region 3). The fourth region extends in the range $\frac{5}{6}<x\le 1$. The dynamic returns to a tipping point one but with oscillatory dynamics between the two attractors (region 4).

Figure 2

Illustration of the dynamics of opinions in each one of the four regions produced by contrarians with respectively $x=0.10, p_0=0.40$ (upper left), $x=0.25, p_0=0.20$ (upper right), $x=0.70, p_0=0.20$ (lower left), $x=0.90, p_0=0.35$ (lower right). All cases have $k=\frac{1}{2}$. The respective values of the attractors and tipping points are indicated (in red).

Figure 3

Given $x=0.20$ (region 2, left part) and $x=0.65$ (region 3, right part), the Figure shows the variations of $S_{A,p,x_{-}},S_{A,p,x_{+}},S_{A,p,x},S_{A,p,0}$ (upper part), $S_{A,p,x},S_{B,p,x},S_{T,p,x},S_{T,p,0}$ (middle part), $S_{A,p,x}-S_{B,p,x},S_{A,p,0}-S_{B,p,0}$ (lower part), as a function of p.

Figure 4

The upper part shows $S_{T,\frac{1}{2},x}$ with $\frac{5+6x}{16}$ for size 4 (in blue), $\frac{1+3x}{8}$ for size 4 with no change at ties (in dotted blue), $\frac{1+2x}{4}$ for size 3 (in red) and $S_{T,\frac{1}{2},a}=\frac{5}{8}(\frac{1}{2}-x)$ for stubborn agents studied below (in magenta). The middle part shows $S_{A,p,x_{-}},S_{A,p,x_{+}},S_{A,p,x},$ for size 3 (in red, blue, green) and 4 (in dashed red, dashed blue, dashed green) at respectively $x=0.20$ and $x=0.65$. The quantities at $x=0$ are added in both cases (in black for size 3 and dashed black for size 4).

Figure 5

Attractors and tipping points as a function of a for a given b from Equation (12). Four cases are exhibited with $b=0$, $b=0.15$, $b=0.20$, and $b=0.25$.

Figure 6

The two distinct regimes produced by stubborn agents as a function of their proportion for $a=b$. A tipping point dynamic with $p_{c,x}=\frac{1}{2}$ prevails in the range $0\le a<a_c$. The two associated attractors feature a stable coexistence of a majority and a minority. In the range $a_c\le a\frac{1}{2}$, the dynamic is driven by one single attractor located at $\frac{1}{2}$. Any initial support $p_0<1-a$ moves monotonously towards $\frac{1}{2}$ with repeating local updates. There, both opinions coexist in a perfect balance.