Asymmetric Contrarians in Opinion Dynamics

Abstract

Asymmetry in contrarian behavior is investigated within the Galam model of opinion dynamics using update groups of size 3 with two competing opinions A and B. Denoting x and y the respective proportions of A and B contrarians, four schemes of implementations are studied. The first scheme activates contrarians after each series of updates with probabilities x and y for agents holding respectively opinion A and B. Second scheme activates contrarians within the update groups only against global majority with probability x when A is the majority and y when B is the majority. The third scheme considers in-group contrarians acting prior to the local majority update against both local majority and minority opinions. The last scheme activates in-group contrarians prior to the local majority update but only against the local majority. The main result is the loss of the fifty-fifty attractor produced by symmetric contrarians. Producing a bit less contrarians on its own side than the other side becomes the key to win a public debate, which in turn can guarantee an election victory. The associated phase diagram of opinion dynamics is found to exhibit a rich variety of counterintuitive results.

Keywords: local and global majority; opinion dynamics; sociophysics; tipping points.

Figure 1

Variations of $p_A$ (left) and $p_B$(right) as a function of c are shown. The two attractors exist only in the range $0\le c\le 1$.

Figure 2

Using Equation (7) the expression $p_{t+1}$ is shown as a function of $p_t$ for $c=0$ (left) and $c=0.08$ (right). The arrows show the dynamics from $p_0=0.4$ for several iterations. Associated attractors are $p_A=0, p_B=1, p_t=0.50$ for the first case and $p_A=0.11, p_B=0.89, p_t=0.50$ for the second case.

Figure 3

Using Equation (7) the expression $p_{t+1}$ is shown as a function of $p_t$ for $c=0.2$ (left) and $c=0.5$ (right). The arrows show the dynamics from $p_0=0.1$ (left) for several iterations. For the second case, any initial value $p_0$ leads to the single attractor within one iteration. For both cases one single attractor $p_A=p_B=p_t=0.50$ drives the dynamics.

Figure 4

As soon as $c>0.5$ the dynamics becomes alternating. Using Equation (7) the expression $p_{t+1}$ is shown as a function of $p_t$ for $c=0.75$ (left) and $c=0.9$ (right). The arrows show the dynamics from $p_0=0.1$ (left) and $p_0=0.35$ (right) for several iterations. For the first case, any initial value $p_0$ leads to the single attractor $p_A=p_B=p_t=0.50$ but through jumps in respective supports for both opinions. For the second case, the iteration produces jumps around the tipping point $p_t=0.50$ but now alternating towards $p_A=0.15$ and $p_B=0.85$.

Figure 5

Using Equation (9) the expression $p_{t+1}$ as a function of $p_t$ is shown for $x=0.08,y=0.10$ (top) and $x=0.20,y=0.22$ (bottom). The associated fixed points are respectively $p_A=0.15,p_t=0.46,p_B=0.89$ and $p_A=p_t=p_B=0.57$.

Figure 6

Using Equation (11) the expression $p_{t+1}$ as a function of $p_t$ is shown for $x=0.20,y=0.18$ (top). The associated evolution from $p_0=0.26$ is exhibited (bottom). The attractor $p_t=\frac{1}{2}$ has split in two limiting points located at $p_t^a=0.4$ and $p_t^b=0.59$.

Figure 7

Using Equation (16) the expression $p_{t+1}$ as a function of $p_t$ is shown for $x=0.08,y=0.10$ (top) and $x=0.20,y=0.22$ (bottom). The associated fixed points are respectively $p_A=0.14,p_t=0.48,p_B=0.89$ and $p_A=p_t=p_B=0.54$.

Figure 8

Using Equation (20) the expression $p_{t+1}$ as a function of $p_t$ is shown for $x=0.08,y=0.10$ (top) and $x=0.20,y=0.22$ (bottom). The associated fixed points are respectively $p_A=0.13,p_t=0.47,p_B=0.90$ and $p_A=p_t=p_B=0.64$.

Figure 9

Fixed points of scheme 1 and 3 (left) and schemes 1 and 4 (right) as a function of x for $y=0.12$. The two dotted vertical lines show the area with a tipping point between two attractors. Outside the lines, the dynamics is driven by one single attractor.

Figure 10

Fixed points of schemes 1–4 as a function of x for $y=0$ (left) and $y=0.10$ (right).

Figure 11

Fixed points of schemes 1 and 3 and 4 as a function of x for $y=0.12$ (left) and $y=0.21$ (right).

Figure 12

Net balance between gain minus loss from contrarian behavior for scheme 4 and 3 when $x=0$ and $y=0.10$ (left) and $x=0.12$ and $y=0.10$ (right). More A contrarians for low values of p increases the net gain for A.

Figure 13

Amplitude of the negative contribution $-\frac{2}{3}x3p^2(1-p)$ (Loss) from A contrarians (proportion x) and the amplitude of the positive contribution $\frac{1}{3}x3p{(1-p)}^2$ (gain) for $x=0.20$. The positive contribution is seen to be bigger than the negative one for $p<\frac{1}{3}$.