Fig 1.
By employing the principle of maximum entropy under the constraints of a fixed number of voters and candidates, we derive the conditional probability p(v|mi), that a candidate i receives v votes, provided that i spends an amount of money mi in the campaign.
Since the amount of money spent usually differ from candidate to candidate, the final distribution of votes should depend on the distribution of money spent. A formalism based on Superstatistics [26] is then used to establish a relation between these two distributions.
Fig 2.
Empirical data for the 2014 and 2018 elections for federal deputies in Brazil, which counted more than 6000 candidates, roughly 140 million voters, and more than 280 million dollars of total investment in each campaign.
(A) The standard deviation as a function of the average number of votes per candidate at the state level, for the top four populated Brazilian states, namely, São Paulo (SP, circles), Rio de Janeiro (RJ, squares), Minas Gerais (MG, diamonds), and Bahia (BA, triangles), empty symbols are for 2018, and filled symbols are for 2014. For each state, candidates were grouped by the amount of money that they officially declared to have spent in their campaigns. The dashed line corresponds to a linear behavior. (B) Distribution of rescaled number of votes for São Paulo (orange circles) and the entire country of Brazil (blue triangles), where 〈v〉 and σv are the average and standard deviation of the number of votes received by the candidates in the same interval (logarithmic binning) of money spent. The black line corresponds to
, as predicted by Eq (5), if we assume Z(m) = 1/μ. Following the prediction given by Eq (5) from our theoretical approach, the distribution of votes for more than 99% of the candidates follows an exponential distribution. However, it is remarkable that the number of candidates with votes that deviate more than 6σv (highlighted region) from the average is higher than expected, suggesting the existence of outliers.
Fig 3.
Results for the 2018 election for federal deputies in the state of São Paulo, Brazil, with 1686 candidates, more than 33 million voters, and about 29 million dollars of total investment in the campaigns.
(A) Number of votes as a function of the money spent in the campaign for all candidates (gray circles) and the average value within each bin (orange circles). The (black-)solid line is obtained using Eq (7) with β = 0.609 (in units of inverse of money) estimated using the 2014 election (see Fig 4). The blue circles are outliers, which we defined as the candidates with a number of votes that deviate more than 6σv from the average. (B) Distribution of the number of votes per candidate. The (orange) circles were obtained from the data and the solid line was calculated from the distribution of money spent in the campaign. Precisely, the solid line is obtained by randomly assigning a number of votes v for each candidate from the distribution given by Eq (5), where m is the amount of money officially declared to have been spent in the campaign. The obtained curve is remarkably consistent with the empirical data over more than three orders of magnitude.
Fig 4.
Parameter β as a function of the ratio of total number of votes Nv and money spent R.
Results are obtained by fitting the data for the 2014 election for federal deputies in all states in Brazil with Eq (7). The solid line is a linear fit to the data obtained with the least-squares method.