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Mathematical tools for delegitimizing elections. Report

Mathematical tools for delegitimizing elections. Report

Mathematical tools for delegitimizing elections. Report

Report of the Russian Public Institute for Electoral Law (RPIEL)

On September 3, 2020, a report  "Mathematical tools of delegitimization of elections" was published on the website of the Russian Public Institute of Election Law (ROIEL). The authors were: Borisov I. B., Candidate of Legal Sciences, Honored Lawyer of the Russian Federation, Chairman of the Council of the ROIPL; Zadorin I. V., head of the research group of CIRCON; Ignatov A.V., Ph.D., executive director of the ROIPI; Marachevsky V.N., Ph.D., professor of St. Petersburg State University; Fedorov V.I., analyst of the Center for the Study of Political Transformations.

The author's team came to the conclusion that "with the help of existing methods of mathematical analysis it is impossible to describe and evaluate electoral behavior and voting results" and reduced everything to "political struggle". How convincingly he managed to do it, and what arguments his opponents used and what vocabulary they used, we leave it to the reader to see for himself;


Russian Public Institute of Election Law (RPIL)
Mathematical Tools for Delegitimizing Elections


Contents of the report:

Introduction and conclusions

1.    Scientific works on the application of normal distribution methods of voting results for their evaluation

2.    Correlation of election results and conclusions of «electoral mathematicians»

2.1.    Mathematical distributions on voting results and comments on them

2.2.    Examples of real situation in the territories

2.3.    Correlation of observers' estimations with plots of normal mathematical distribution

3.        Refutation of the hypothesis of mathematical definition of throw-ins by sociological research

4.       Elections abroad in the context of mathematical models

5.    Features of mathematical modeling of social processes

6.    Other mathematical models of interpretation of election results

7.    Manipulative technologies

8.       Conclusion and Recommendations

Appendix: RPIIP Policy Brief on the 2016 U.S. Presidential Election Survey of Minnesota and Wisconsin polling places.

Introduction and Conclusions

Ideas of sociological correlations of voting results with a normal mathematical distribution is nowadays a rather popular topic of post-election assessments, which, among other things, may be caused by the existing demand in society for renewal, including in the electoral sphere.

The attempts to apply the Gauss curve, derived by the German mathematician for application to homogeneous physical phenomena, to complex social processes, including voting at elections do not stop.

However, this formal-mathematical approach has never found meaningful justification in jurisprudence, sociology, mathematics, or other applied sciences.

No national or international normative act correlates «mathematical anomalies» with legally significant actions

  • the process of ascertaining the validity of the will of the voters and determining the final results.

There are no arguments known to modern science to justify the hypothesis

  • that a certain kind of sociological distribution must obey a normal distribution. «As numerous studies have shown, almost all distributions of real data do not belong to any of the known parametric families» – emphasized in his works mathematician, economist, statistician and sociologist Alexander Orlov2. The idea of normal distribution in sociology is unfounded.

The only mathematical component of these approaches is the similarity of justification of using normal distribution to describe the electoral dependence of the number of votes cast at polling stations on the percentage of turnout. But the principle of constructing this mathematical model remains unclear and de facto unexplained. If deviations in mathematical models would indicate fraud, then the majority of elections in the countries of the so-called «old democracy» should be recognized as rigged and «drawn».

As noted by V.N. Marachevsky, Professor of St. Petersburg State University, Doctor of Physical and Mathematical Sciences, a theoretical description of the voting process using a single Gaussian function is possible only in the hypothetical case if the probabilities of voting «for» in all cities and regions take the same value, and the same number of voters comes to each polling station in the country.

Back in 2016, a PhD thesis «Electoral anomalies in the post-socialist space: experience of statistical analysis» was defended at St. Petersburg State University.

N.E. Shalaev in his scientific study comes to an unambiguous conclusion: the analysis of the distribution of numbers within the framework of known methods is not applicable to the search for electoral anomalies.

  • In fact, some of the empirical distributions may be close to normal, but this may occur not due to regularity, but due to variability of distributions, based on the theory of large numbers..

The normal distribution works with random variables. It can perfectly illustrate some physical processes, but not the conscious choice of millions of people who differ in their place of residence, wealth, life priorities, nationality and other attributes.

Sociological research conducted to study the territorial peculiarities of voting confirms the high social and cultural diversity of electorates of individual polling stations, as well as the high differentiation of living conditions of citizens and the exercise of their electoral rights.

The factual substantiation of deviations of the voting results from the regional average, as well as the data of observers from polling stations presented in the report, testify to the lack of correlation between the conclusions of the chart compilers and the real situation on the ground.

These conclusions are supported by empirical and statistical studies of voting results in the Russian Federation and abroad, the results of which are presented in this report.

The authors of the report reached the following conclusions:

  1. With the help of existing methods of mathematical analysis it is impossible to describe and make an assessment of electoral behavior and voting results.
  2. Deviations from the normal mathematical distribution of the voting results are not a neomonical sign of violations. Different mathematical models of distribution of voting results testify to objective peculiarities of voters' preferences in separate polling stations and territories and their expression during the secret ballot.
  3. The results of elections in the so-called «old democracies» (Great Britain, Germany, Israel, USA) testify to the disobedience to the laws of normal mathematical distribution of the results of voting and to the change of the percentage of votes for a particular candidate (party) when the turnout increases/decreases.
  4. 2/3 of all estimates of the authors of graphs of mathematical distributions about «abnormality» of this or that territory do not correlate with the data of observers about emerging problems on the ground, which may indicate the random rather than regular nature of the determination of «abnormal» zones appearing on the graphs, as well as the lack of causal relationship of conclusions based on mathematical models with the specific situation in the region.
  5. There is a tendency to spread under the guise of «reliable information» and «proven facts» certain doubts in the absence of violations, which are derived from the probabilistic assessment of numerical data mathematical distributions not applicable to social processes. These speculations interpreting unusual voting results at specific polling stations as «falsifications» are usually part of manipulative political technologies used to delegitimize election results.
  6. It is not necessary to speak about «bona fide delusion» of the authors of «mathematical theories» of electoral behavior estimations, who disseminate unreliable information, in this case, because repeatedly and in detail a wide range of scientists (from mathematicians and physicists to sociologists and lawyers) explained the inapplicability of these methods to the description of social behavior models. But from election to election in mass media, social networks, various blogs, etc., voting data with far unscientific (false) conclusions are spreading. There is a high probability of deliberate actions of authors-compilers of graphs and comments to them, aimed at forming a negative public opinion towards democratic processes. 
  7. Today we can talk about the formation of a new direction of political struggle based on modern information and communication technologies and related to the deliberate undermining of confidence in the results of elections and delegitimization of elected bodies of state power and local self-government through the mass dissemination of inaccurate information, which requires, in the interests of ensuring and protecting the electoral (political) rights of citizens, the adoption of appropriate response measures. 

The full content of the report can be found at on the ROIPP website or download from the link below in PDF format. 

Unfortunately, unlike in our Lab, the authors did not provide you with the original data and, therefore, the opportunity to see for yourself what their conclusions are based on. But our Lab collects data from different countries, and you have the opportunity to see for yourself how elections in other countries differ from those in Russia, what distributions are similar, what anomalies do not exist at all.

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