Scaling and Universality in Proportional Elections

A most debated topic of the last years is whether simple statistical physics models can explain collective features of social dynamics. A necessary step in this line of endeavor is to find regularities in data referring to large-scale social phenomena, such as scaling and universality. We show that, in proportional elections, the distribution of the number of votes received by candidates is a universal scaling function, identical in different countries and years. This finding reveals the existence in the voting process of a general microscopic dynamics that does not depend on the historical, political, and/or economical context where voters operate. A simple dynamical model for the behavior of voters, similar to a branching process, reproduces the universal distribution.

Figure 1

(a) Scaling behavior of the distribution of votes received by candidates. Data refer to the Italian parliamentary elections in 1972, but we obtained very similar results from the analysis of each data set. The histogram $P(v,Q,N)$ only depends on the ratio $v_0=N/Q$, so $P(v,Q,N)=P_0(v,v_0)$. (b) The function $P_0(v,v_0)$ shown in (a) only depends on the ratio $v/v_0=vQ/N$.

Figure 2

Universality of the scaling function $F(vQ/N)$ across different countries and years. The lognormal fit, performed on the Polish curve, describes very well the data. The universal curve is well reproduced by our model, where the dynamics of the voters’ opinions reflects the spreading of the word of mouth in the party’s electorate.

Figure 3

Spreading of the word of mouth among voters. The candidate (right) convinces some of his or her contacts to vote for him/her. The convinced voters become “activists” and try to convince some of their acquaintances, and so on. Successful interactions are indicated by solid lines, unsuccessful interactions are displayed as dashed lines.