Distribution of Votes and a Model of Political Opinion Formation for Majority Elections

Abstract

We study the behavior of the number of votes cast for different electoral subjects in majority elections, and in particular, the Albanian elections of the last 10 years, as well as the British, Russian, and Canadian elections. We report the frequency of obtaining a certain percentage (fraction) of votes versus this fraction for the parliamentary elections. In the distribution of votes cast in majority elections we identify two regimes. In the low percentiles we see a power law distribution, with exponent about −1.7. In the power law regime we find over 80% of the data points, while they relate to 20% of the votes cast. Votes of the small electoral subjects are found in this regime. The other regime includes percentiles above 20%, and has Gaussian distribution. It corresponds to large electoral subjects. A similar pattern is observed in other first past the post (FPP) elections, such as British and Canadian, but here the Gaussian is reduced to an exponential. Finally we show that this distribution can not be reproduced by a modified "word of mouth" model of opinion formation. This behavior can be reproduced by a model that comprises different number of zealots, as well as different campaign strengths for different electoral subjects, in presence of preferential attachment of voters to candidates.

General Features of the q-XY Opinion Model

“…Firstly, the power law distribution predicted by the preferential attachment mechanisms is theoretical, in practice different distribution have been observed. Prenga and Ifti (2012) [25] proposed an ad hoc modification in the formula by including local field effects aiming in reproducing electoral distribution. The amended preferential attachment rule reads 𝑝 𝑖.𝑗 ~ℎ𝑗 (𝑘 𝑗 /𝑠𝑢𝑚(𝑘 𝑗 )).…”