Collective decision making with a mix of majority and minority seekers

We study a model of a population making a binary decision based on information spreading within the population, which is fully connected or covering a square grid. We assume that a fraction of the population wants to make the choice of the minority, whereas the rest want to make the majority choice. This resembles opinion spreading with “contrarian” agents but has the game theoretic aspect that agents try to optimize their own situation in ways that are incompatible with the common good. When this fraction is less than 1/2, the population can efficiently self-organize to a state where agents get what they want—the majority (i.e., the majority seekers) have one opinion, the minority seekers have the other. If the fraction is larger than 1/2, there is a frustration in the population that dramatically changes the dynamics. In this region, the population converges, through some distinct phases, to a state of approximately equal-sized opinions. Just over the threshold the state of the population is furthest from the collectively optimal solution.

Figure 1

Illustration of the behavior of the spatial version of the model for the extreme values of $\varphi$. The upper row shows the time evolution of the state variable $s_i$ for a situation when all agents are majority seekers. The lower row shows the situation with only minority seekers. The initial condition is $p=1/2$, i.e., $s_i$ is set to 1 or $-1$ with equal probability. The decision threshold $\theta$ is 10.

Figure 2

Illustration of the all-minority-seeker case with a larger decision threshold ($\theta =50$) which separates the time evolution into different phases. Panel (a) shows snapshots of the population for different times. Panel (b) shows the time evolution of the average opinion $q$ and the fraction of agents with fixated opinions $f$. The dots on the curves mark the snapshots in panel (a). Here we took $p=1$ so that initially all agents have $s_i=1$.

Figure 3

The average opinion $q$ depending on the decision threshold $\theta$. Panels (a)–(c) represent fully connected networks; panels (d)–(f) show the results on square grids. Panels (a) and (d) show $q$ for various values of $\varphi$ and $p=0.51$; panels (b) and (e) show the results for the population of only majority seekers ($\varphi =0$) and various values of $p$; panels (c) and (f) show the results for the population of only minority seekers ($\varphi =1$).

Figure 4

The ratio of satisfied agents $\sigma$ as a function of the decision threshold $\theta$. The panels correspond to the panels of Fig. 3.

Figure 5

The average opinion $q$ as a function of the fraction of minority seekers $\varphi$. Panels (a) and (b) show the results for fully connected networks; panels (c) and (d) are the corresponding panels for square grids. Panels (a) and (c) have $p=0.51$; panels (b) and (d) have $p=1$.

Figure 6

Here we investigate how $\sigma$ depends on $\varphi$. Panels (a) and (b) display the results for fully connected networks; panels (c) and (d) show the corresponding figures for square grids. (a) and (c) have $p=0.51$; (b) and (d) have $p=1$.

Figure 7

Plotting how $\sigma$ depends on $\epsilon$. Panel (a) shows the results for the fully connected topology, and panel (b) displays the results for the square grids. $\theta =2560$ was used for both panels. Data points for $\varphi \le 0.505$ are excluded as the theory applies to large $\varphi$. The solid line shows $\sigma =\epsilon$.