Votes and a half-binomial

AbstractIn two party elections with popular vote ratio pq, 12≤p=1 −q, a theoretical model suggests replacing the so-called MacMahon cube law approximation (pq)3, for the ratio PQ of candidates elected, by the ratio ƒk(p)ƒk(q) of the two half sums in the binomial expansion of (p+q)2k+1 for some k. This ratio is nearly (pq)3 when k = 6. The success probability gk(p)=(pa(pa+qa) for the power law (pq)a≐PQ is shown to so closely approximate ƒk(p)=Σ0k(r2k+1)p2k+1−rqr, if we choose a = ak=(2k+1)!4kk!k!, that 1≤ƒk(p)gk(p)≤1.01884086 for k≥1 if 12≤p≤1. Computationally, we avoid large binomial coefficients in computing ƒk(p) for k>22 by expressing 2ƒk(p)−1 as the sum (p−q) Σ0k(4pq)sas(2s+1), whose terms decrease by the factors (4pq)(1−12s). Setting K = 4k+3, we compute ak for the large k using a continued fraction πak2=K+12(2K+32(2K+52(2K+…))) derived from the ratio of π to the finite Wallis product approximation

Spatial correlations in vote statistics: a diffusive field model for decision-making

We study the statistics of turnout rates and results of the French elections since 1992. We find that the distribution of turnout rates across towns is surprisingly stable over time. The spatial correlation of the turnout rates, or of the fraction of winning votes, is found to decay logarithmically with the distance between towns. Based on these empirical observations and on the analogy with a two-dimensional random diffusion equation, we propose that individual decisions can be rationalised in terms of an underlying "cultural" field, that locally biases the decision of the population of a given region, on top of an idiosyncratic, town-dependent field, with short range correlations. Using symmetry considerations and a set of plausible assumptions, we suggest that this cultural field obeys a random diffusion equation.Comment: 18 pages, 5 figures; added sociophysics references

In Search of the Black Swan: Analysis of the Statistical Evidence of Electoral Fraud in Venezuela

This study analyzes diverse hypotheses of electronic fraud in the Recall Referendum celebrated in Venezuela on August 15, 2004. We define fraud as the difference between the elector's intent, and the official vote tally. Our null hypothesis is that there was no fraud, and we attempt to search for evidence that will allow us to reject this hypothesis. We find no evidence that fraud was committed by applying numerical maximums to machines in some precincts. Equally, we discard any hypothesis that implies altering some machines and not others, at each electoral precinct, because the variation patterns between machines at each precinct are normal. However, the statistical evidence is compatible with the occurrence of fraud that has affected every machine in a single precinct, but differentially more in some precincts than others. We find that the deviation pattern between precincts, based on the relationship between the signatures collected to request the referendum in November 2003 (the so-called, Reafirmazo), and the YES votes on August 15, is positive and significantly correlated with the deviation pattern in the relationship between exit polls and votes in those same precincts. In other words, those precincts in which, according to the number of signatures, there are an unusually low number of YES votes (i.e., votes to impeach the president), is also where, according to the exit polls, the same thing occurs.Comment: Published in at http://dx.doi.org/10.1214/11-STS373 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Between order and disorder: a 'weak law' on recent electoral behavior among urban voters?

A new viewpoint on electoral involvement is proposed from the study of the statistics of the proportions of abstentionists, blank and null, and votes according to list of choices, in a large number of national elections in different countries. Considering 11 countries without compulsory voting (Austria, Canada, Czech Republic, France, Germany, Italy, Mexico, Poland, Romania, Spain and Switzerland), a stylized fact emerges for the most populated cities when one computes the entropy associated to the three ratios, which we call the entropy of civic involvement of the electorate. The distribution of this entropy (over all elections and countries) appears to be sharply peaked near a common value. This almost common value is typically shared since the 1970's by electorates of the most populated municipalities, and this despite the wide disparities between voting systems and types of elections. Performing different statistical analyses, we notably show that this stylized fact reveals particular correlations between the blank/null votes and abstentionists ratios. We suggest that the existence of this hidden regularity, which we propose to coin as a `weak law on recent electoral behavior among urban voters', reveals an emerging collective behavioral norm characteristic of urban citizen voting behavior in modern democracies. Analyzing exceptions to the rule provide insights into the conditions under which this normative behavior can be expected to occur.Comment: Version 1: main text 19 pages, 13 figures, 2 tables; Supporting Information: 19 pages. Version 2: minor correction

Voting Power Derives from the Poll Distribution. Shedding Light on Contentious Issues of Weighted Votes and the Constitutional Treaty

Analysis of the Constitutional Treaty of the European Union shows that there is a serious discrepancy between the voting power gradient of Member States computed by the Shapley-Shubik and Banzhaf indices. Given the lack of compelling arguments to choose between these indices on purely axiomatic grounds, we turn to a probabilistic approach as pioneered by Straffin (1977) focusing on the probability distribution of voting poll outcomes. We present a unifying model of power indices as expected decisiveness, which shows that the defining feature of each approach is a particular distribution of the voting poll. Empirical evidence drawn from voting situations, in addition to a consideration of first principles, leads us to reject one of these approaches. The unified formulation allows us to develop useful related concepts of efficiency and blocking leverage, previously used solely by a 'Banzhaf' approach, for the case of Shapley-Shubik, and a comparison of results is shown.Voting power indices, Power gradient, Coefficient of representation, Expected decisiveness, Efficiency, Blocking leverage, Constitution of the European Union