Democratic Thwarting of Majority Rule in Opinion Dynamics: 1. Unavowed Prejudices Versus Contrarians

Abstract

I study the conditions under which the democratic dynamics of a public debate drives a minority-to-majority transition. A landscape of the opinion dynamics is thus built using the Galam Majority Model (GMM) in a 3-dimensional parameter space for three different sizes, r=2,3,4, of local discussion groups. The related parameters are (p0,k,x), the respective proportions of initial agents supporting opinion A, unavowed tie prejudices breaking in favor of opinion A, and contrarians. Combining k and x yields unexpected and counterintuitive results. In most of the landscape the final outcome is predetermined, with a single-attractor dynamics, independent of the initial support for the competing opinions. Large domains of (k,x) values are found to lead an initial minority to turn into a majority democratically without any external influence. A new alternating regime is also unveiled in narrow ranges of extreme proportions of contrarians. The findings indicate that the expected democratic character of free opinion dynamics is indeed rarely satisfied. The actual values of (k,x) are found to be instrumental to predetermining the final winning opinion independently of p0. Therefore, the conflicting challenge for the predetermined opinion to lose is to modify these values appropriately to become the winner. However, developing a model which could help in manipulating public opinion raises ethical questions. This issue is discussed in the Conclusions.

Keywords: attractors; contrarians; democratic balance; opinion dynamics; prejudices; sociophysics; thwarting; tipping points.

Figure 1

Update curves $p_{i+1,r,k}$ as a function of $p_{i,r,k}$ for $r=2$ (blue curves) and $r=4$ (green curves) at $k=0.2$ (left) and $k=0.80$ (right). Update curve for $p_{i+1,3}$ (r = 3) as a function of $p_{i,3}$ is also shown (red curves). The dotted lines represent the diagonal $p_{i+1}=p_i$. Arrows indicate update directions, solid circles indicate tipping points, solid circles with a border indicate attractors and solid squares with a border indicate either attractors or tipping points depending on r and k. Only $r=3$ yields a democratic balance, with $p_T={\displaystyle \frac{1}{2}}$.

Figure 2

Attractors and tipping points yielded by Equation (6) are shown on the upper left side for $r=3$ as a function of the proportion x of contrarians. Only the values between 0 and 1 are meaningful. Solid lines (in red, magenta, blue) are attractors while tipping points (in dotted blue) exist only for $x<{\displaystyle \frac{1}{6}}\approx 0.17$. When $x>{\displaystyle \frac{1}{6}}$, one unique attractor drives the dynamics toward perfect equality. For $x>{\displaystyle \frac{1}{2}}$, with more than half the population being contrarians, alternating dynamics is expected. The upper right side shows the completed dynamics with a single-attractor dynamics for ${\displaystyle \frac{1}{6}}<x<{\displaystyle \frac{5}{6}}$ and a dynamics with two alternating attractors when $x>{\displaystyle \frac{5}{6}}$. Update curves $p_{i+1,3,x}$ as a function of $p_{i,3,x}$ for $x=0,0.10,0.20,0.30$ and $x=0.70,0.80,0.90,1$ are shown in the lower part. The dotted red lines represent the diagonals $p_{i+1}=p_i$ and $p_{i+1}=1-p_i$. Arrows indicate update directions, red circles indicate attractors, and blue circles indicate alternating attractors. The $p_T={\displaystyle \frac{1}{2}}$ square in the middle is either a tipping point, when $x<{\displaystyle \frac{1}{6}}$ and $x>{\displaystyle \frac{5}{6}}$, or an attractor when ${\displaystyle \frac{1}{6}}<x<{\displaystyle \frac{5}{6}}$. The lower part shows $p_{i+1,3,x}$ as a function of $p_{i,3,x}$ for $x=0,0.10,0.20,0.30,0.70,0.80,0.90,1$. Arrows indicate the direction of the updates. Double arrows signal an alternating dynamics.

Figure 3

The upper left part shows the update Equation (6) for $x=0.105$, which has a tipping point at 0.50 with two attractors at 0.06 and 0.94. The dynamics is illustrated from $p_{0,3,x}=0.40$. The upper right part shows the case $x=0.25$, which is a single-attractor dynamics with one attractor at 0.50. The dynamics is illustrated from $p_{0,3,x}=0.05$. The lower part exhibits both updates $p_{i+1,3,x}$ in red and $p_{i+2,3,x}$ in blue as function of $p_{i,3,x}$. The left side shows the case $x=0.75$, which is an alternating single-attractor dynamics with one attractor at 0.50. The dynamics is illustrated from $p_{0,3,x}=0.20$. The right side shows the case $x=0.90$, which is an alternating dynamics with a tipping point at 0.50 and two alternating attractors at 0.15 and 0.85. The dynamics is illustrated from $p_{0,3,x}=0.40$.

Figure 4

The upper left part of the figure shows the attractors and tipping points driving the dynamics as a function of k when $x=0$ for groups of size 2. Blue color lines indicate attractors. Red lines correspond to tipping points. Arrows shows the direction of update dynamics. The upper right part shows the effect of a tiny proportion of contrarians $(x=0.01)$ turning the dynamics to single-attractor. The prejudice effect is simultaneously slightly reduced. A larger proportion of contrarians accentuates the reduction effect of prejudice, as seen in the middle left part with $x=0.15$. From $x\approx 0.30$ to $x=1$, $p_{B,2,k,x}$ is quasi-linear as a function of k for a given x, as illustrated in the middle right part and left and right lower part for $x=0.35,0.65,0.95$.

Figure 5

The upper part of the figure shows the domains with a single-attractor dynamics and an alternating-tipping-point dynamics for the update $p_{i+1,2,k,x}$. The left part shows them for x as a function of k and the right part for k as a function of x (TP = tipping point). The middle part shows the single fixed point $p_{B,2,k,x}$ for $x=0.95$ as a function of k; the left part when only the update $p_{i+1,2,k,x}$ is used; and the right part when the stability of the fixed point has been added using either $p_{i+2,2,k,x}$ or the derivative of $p_{i+1,2,k,x}$ with respect to $p_{i,2,k,x}$ at the fixed point. The lower part exhibits both $p_{i+1,2,k,x}$ and $p_{i+2,2,k,x}$ as a function of $p_{i+1,2,k,x}$ for $x=0.95$, with $k=0.7$ on the left and $k=1$ on the right. Arrows shows the evolution of $p_{i,2,k,x}=0.20$ for six successive updates for each case. Vertical ones show the effect of one update while horizontal ones show from where the next update takes place. Blue curves show $p_{i+1,3,x}$ and red curves show $p_{i+2,3,x}$ as a function of $p_{i,3,x}$.

Figure 5

The upper part of the figure shows the domains with a single-attractor dynamics and an alternating-tipping-point dynamics for the update $p_{i+1,2,k,x}$. The left part shows them for x as a function of k and the right part for k as a function of x (TP = tipping point). The middle part shows the single fixed point $p_{B,2,k,x}$ for $x=0.95$ as a function of k; the left part when only the update $p_{i+1,2,k,x}$ is used; and the right part when the stability of the fixed point has been added using either $p_{i+2,2,k,x}$ or the derivative of $p_{i+1,2,k,x}$ with respect to $p_{i,2,k,x}$ at the fixed point. The lower part exhibits both $p_{i+1,2,k,x}$ and $p_{i+2,2,k,x}$ as a function of $p_{i+1,2,k,x}$ for $x=0.95$, with $k=0.7$ on the left and $k=1$ on the right. Arrows shows the evolution of $p_{i,2,k,x}=0.20$ for six successive updates for each case. Vertical ones show the effect of one update while horizontal ones show from where the next update takes place. Blue curves show $p_{i+1,3,x}$ and red curves show $p_{i+2,3,x}$ as a function of $p_{i,3,x}$.

Figure 6

Four different regimes of the update $p_{i+1,4,k,x}$ are shown in red as a function of both k and x. The blue line shows $p_{i+2,4,k,x}$, which identifies the alternating attractors. The upper left part of the figure shows a tipping-point dynamics at a very low proportion of contrarians, with $x=0.05$ and $k=0.80$. The upper right part shows that already at $x=0.10$ the contrarians turn the dynamics to a single-attractor dynamics, with the attractor located at a very high value, with $p_{A,4,0.8,0.1}=0.89$. At high concentrations of contrarians, the dynamics stays single-attractor, as shown in the lower left part of the figure, with $x=0.75$ but with a lower value $p_{A,4,0.8,0.75}=0.47$. At $x=0.85$, the dynamics becomes alternating with two attractors, as exhibited in the lower right part of the figure.

Figure 7

Evolution of attractors and tipping points as a function of the proportion x of contrarians for $k=1$ (upper left part), $k=0$ (lower left part), $k=0.60$ (upper right part), and $k=0.40$ (lower right part). Closed curves represent tipping points and attractors while single curves denote attractors except when inside a closed curve. Arrows indicate the direction of the flow of opinion dynamics while double arrows signal an alternating dynamics.

Figure 8

Evolution of attractors and tipping points as a function of the proportion x of contrarians for $k=1,0.6,0.501$ (upper left part) and $k=0.499,0.4,0$ (upper right part). Closed curves represent tipping points and attractors while single curves denote attractors except when inside a closed curve. The lower part combines both upper cases. The asymmetry between $k<{\displaystyle \frac{1}{2}}$ and $k>{\displaystyle \frac{1}{2}}$ as well as between $x<{\displaystyle \frac{1}{2}}$ and $x<{\displaystyle \frac{1}{2}}$ is clearly seen.

Figure 8

Evolution of attractors and tipping points as a function of the proportion x of contrarians for $k=1,0.6,0.501$ (upper left part) and $k=0.499,0.4,0$ (upper right part). Closed curves represent tipping points and attractors while single curves denote attractors except when inside a closed curve. The lower part combines both upper cases. The asymmetry between $k<{\displaystyle \frac{1}{2}}$ and $k>{\displaystyle \frac{1}{2}}$ as well as between $x<{\displaystyle \frac{1}{2}}$ and $x<{\displaystyle \frac{1}{2}}$ is clearly seen.

Figure 9

Plots of the values of $x_c$, $x_{c>}$, $p_{B,k\le 0.5,x_c}$, $p_{A,k\ge 0.5,x_c}$, $p_{A,k\le 0.5,x_{x>}}$, and $p_{B,k\ge 0.5,x_{x>}}$ as a function of k for $r=4$ according to the values of Table 1.