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Chapter 2. How an Authoritarian Power AttainsIts Goals and Objectives: A Mathematical Perspective

Published onFeb 22, 2024
Chapter 2. How an Authoritarian Power AttainsIts Goals and Objectives: A Mathematical Perspective

It seems fitting to begin this chapter with a well-worn but apt quote from Galileo: “Must we not confess that geometry is the most powerful of all instruments for sharpening the wit and training the mind to think correctly?” It is no coincidence that the great minds of the past found themselves thinking about the foundation on which serious knowledge should stand. A thorough understanding and accounting of almost any process, whether of nature or society, needs to involve a variety of mathematical tools: we are not aware of any other ways to describe the world so that the description has both predictive and explanatory power. Even in high school, the methods of algebra and geometry prove useful in understanding the world as it is viewed by the natural sciences, especially physics and chemistry. More elaborate examination also requires a more advanced toolkit, and for the analysis of social and political processes that arise from the actions of many individuals, one cannot do without the methods of probability theory and mathematical statistics, to which we will now turn.

Results of elections are neither the mean of the will of the public, nor even the mean of the will of the electorate. At the end of the day, elections reflect the will of voters who have gone to the polls. There can be all sorts of reasons for going: because the heart or civic duty calls, because the boss told you to, because “everyone goes, and I go,” because there’s music and pastries at the polling station, because somebody paid you, because you could not stay silent anymore… The reasons are many, but one thing is certain: election results are always a combination of turnout and expressed will. The presence of “dark matter” voters who do not go to the polls (between 35 and 65% in the Russian federal elections from 2000 to 2021) gives rise to numerous speculations about their political preferences: whether they are completely conformist and agree with the current government, and thus do not consider it necessary to cast their ballots; or, on the contrary, they are in radical opposition to it and do not want to “participate in a farce.” However, evidence does not support such an extreme difference between the political preferences of the “silent” and voting populations.1

Sophisticated vote counting procedures mandated by law are designed to ensure votes are cast and tallied correctly. In turn, neglect and legislative erosion of those procedures create favorable conditions for manipulation of votes (especially when turnout is low) and falsification of results. But even under an authoritarian regime that has reshaped the electoral system to fit itself, so as to assure a parliamentary majority in advance, the election results can be analyzed and their integrity defended.

In this chapter, we want to demonstrate a simple idea: it is possible to use the methods of mathematics to understand the results of elections, and in particular their reliability. Moreover, these methods allow us not only to pinpoint “anomalies,” but often to identify and build models of their causes, the probable mechanisms for their implementation, and (within certain limits) to restore the real picture even if the voting was adulterated.

We shall start with a general question: how can we study “from the outside” what is happening within a large system that does not provide comprehensive access to its internal workings? The electoral system is just that. We only get to look at rather particular aggregate outputs: turnout, its hourly dynamics, and the total number of voters for the candidates, all broken down by polling station. At the same time, based on the reports of independent observers, there is reason to believe that these outputs may be significantly distorted in many precincts (but not all of them). It is important that the distortions are not universal: the presence of a certain proportion of polling stations with authentic results provides, on one hand, a basis for assessing the true outcome, and on the other, grounds for identifying fraudulent stations. It is also important that we have access to data detailed enough to see what is going on at polling places—that is, at the level where the actual fraud may be perpetrated (or not). This is precisely what allows us to separate “clean” results from “tainted” ones (those that were subjected to administrative intervention) based on their statistical characteristics.

Before proceeding to a detailed analysis of the problem, it is necessary to make several stipulations. First, numerical results alone cannot prove anything, since it is always necessary to establish a causal link. For example, even if we discover a correlation between weather fluctuations on Mars and the progression of US presidents, there is clearly no causal relationship between them. Second, large-scale statistical analysis, as a rule, does not allow for conclusions about specific violations at a particular polling station. Deviations from the mean (including significant ones) are always possible, but (as we will explain below) these deviations cannot be entirely arbitrary. Roughly speaking, if among a hundred similar polling stations (for example, ones located in residential areas of similar socioeconomic status across a city) the turnout ranges from 40 to 50%, and at one of them we observe a turnout of 38%, this may well be a random fluctuation. But a polling station with a turnout of 10% is already an anomaly and calls for some sort of explanation. It is with the help of mathematics that we can understand which values are abnormal, and which are, so to say, within reason. Normally, mathematics plays the role of an expert witness, pointing out anomalies, and a court would decide whether there had been fraud in any particular case. (For example, mathematical modeling serves as a standard argument in gerrymandering cases in the United States).2 However, even in the absence of a functioning independent judiciary, statistical methods make it possible to assess the overall extent of falsification, and in some cases prove its presence with utmost certainty.

Instead of delving into electoral mathematics immediately, let us start with an unrelated example that has become extremely important to the layman in the last few years: the effectiveness of COVID-19 vaccines. To see for yourself that most of the vaccines in use throughout the world (Pfizer, Moderna, Sputnik-V and others) do work and provide a reasonably high level of protection, you do not have to read complicated articles in medical journals. It is indeed quite enough to independently (!) study readily available public data.

We, of course, do not mean anecdotes and rumors about “an acquaintance” getting sick after being vaccinated, or about media personalities who got ill and died after getting the vaccine. In this regard we want not only to recite the name of the age-old fallacy, “post hoc ergo propter hoc,” but also to note that this way of reasoning implicitly ignores the larger picture. If a person in an accident is seriously injured while wearing a seat belt (or even by a seat belt!), this does not mean that seat belts are harmful: using a belt is significantly safer. Almost always in an accident, an unbelted person will be more injured than a belted one. The same with vaccination: if a person died after receiving a vaccine, this does not mean that they would have survived the disease without it.

The correct question about the effectiveness of vaccines can be stated as follows: “Does the presence of vaccination reduce the risk of catching the disease or, for those who have nevertheless caught it, the risk of higher severity?” Meanwhile, it is completely meaningless to compare the number of those vaccinated and unvaccinated who ended up in hospital, since such a comparison ignores the percentage of vaccinated people in the general population.

Applying conditional probabilities (Bayes’ theorem) immediately shows that, since the percentage of vaccinated persons “in the hospital”3 is less than the percentage of those vaccinated in “total,” the vaccine is effective in reducing the likelihood of being admitted to the hospital, that is, the vaccinated need medical attention less often, confirming that vaccination is effective.

Let us examine this question in more detail. Recall Bayes’ rule:

P(AP(AB)=P(BA)P(A)P(B)B)=P(BA)P(A)P(B)P(A|P(A|B)=\frac{P(B|A)P(A)}{P(B)}B)=\frac{P(B|A)P(A)}{P(B)}

Here P (A | B) denotes the conditional probability, that is, the probability for the event A to occur provided that event B occurred. Suppose that A denotes “going to the hospital” (that is, its probability is the proportion of those who ended up in hospitals), and event B is the person being vaccinated (its probability being the proportion of those vaccinated). In this case, the probability of being admitted to the hospital while vaccinated, (B), is calculated by knowing the proportion of those vaccinated in the hospital, (A), the proportion of those who fell ill, (A), and the proportion of those vaccinated, (B). Thus we are led to divide (B | A) by the proportion of those vaccinated, which is somewhat counterintuitive. It is easy to confirm using public data that (B), the proportion of hospitalizations among the vaccinated, is markedly less than (A), the overall proportion of cases. It is telling that the data necessary to study the effectiveness of the Russian-made Sputnik-V vaccine has to be obtained not from the Russian Ministry of Health (which essentially does not provide any), but from the ministries of health of Argentina, Hungary and other countries that use it.

Hypothetically, such a difference could be explained by factors other than pharmacology. For example, the generally more responsible vaccinated population could also be using other protective measures (masks, distancing). However, there is no evidence pointing towards any alternative causes of this sort. Therefore, according to Occam’s razor, one has to conclude that vaccination is effective as a way to reduce the likelihood of a severe progression of the disease.

Another question that regularly occurs in the public space is the percentage of deaths and the alleged immunity from or propensity for infection of various subgroups of the public. There is, for example, the notion that smokers are less likely to get sick than non-smokers. On the other hand, there were numerous references to mortality rates of almost 30–40% in nursing homes and among the elderly. Without playing down the gravity of the virus, we note that without further clarification, this sort of reasoning is unsound, since the elderly are in any case a higher-risk population, while an incorrect sample had been used to confirm the alleged immunity among smokers. However, the “hospital-average mortality rate” is also a deceptive statistic. The actual probability of complications for an elderly, smoking, obese diabetic will in any case be many times higher than for a young person who exercises regularly. As the saying goes: “I eat cabbage; the lord, meat; so on average, we eat cabbage rolls.”

The above considerations must be taken into account not only when analyzing vaccines, but also when studying any opaque system, including elections. In general, we would very much like to convince our readers that statistics and probability theory are an essential tool for understanding the processes taking place in the modern world.

Getting back to electoral procedures, we shall first discuss a few illustrative examples. Consider for the moment only the voter turnout. That is, for now we will concern ourselves with only one question: whether the voter came to the polling place or not. In this situation, each voter can be assigned a random variable that takes on the values one (came) and zero (did not come). The turnout is the ratio between the overall number of voters who came to polling stations (or, equivalently, the sum of the corresponding random variables) and the number of voters on the rolls. If 1000 people are registered at the polling station, 600 of them came and 400 did not, we get 600/1000, that is, a turnout of 60%.

This is in fact a Bernoulli process. We flip a +1 with probability p and a 0 with probability 1 – p, where p is numerically equal to the average turnout (by the definition of expected value). Assuming that people vote independently, we get that the probability of the turnout reaching 100% on a given precinct is p1000 (with the exponent equal to the number of registered voters). For any value of p that is even slightly below unity (not only for a realistic p = 0.60, as in the example above, but also for, say, p = 0.99), this probability will be indistinguishable from zero. Meanwhile, in the official results of Russian elections, we routinely encounter polling stations with thousands of voters and a turnout of 100%—for examples the reader can turn to the results of the 2016 federal parliamentary elections in Kemerovo Oblast. At the same time, nearby precincts inside the same settlement have a turnout of, say, 80 or 60%. A difference of this magnitude can have one of two explanations: either the quantity p, which characterizes the general propensity of the local electorate to go to the polls, does in fact vary that much across neighboring precincts, or the tallies at one of the polling stations are inaccurate. Based on a combination of general common-sense considerations (for example, the knowledge that borders of precincts within a city are drawn fairly randomly and arbitrarily) and concrete evidence (for example, video recordings of ballot box stuffing), it is the second option that seems most plausible.

Let’s move on. Within the formal framework proposed above, where a Bernoulli random variable is assigned to each voter, the turnout at a polling station is also a random variable: a binomial one, rescaled to the range from zero to one (from 0 to 100%). Given that a typical precinct in an urban area has around 1500–2500 people registered, such a random variable is indistinguishable from a Gaussian one with a mean of p and a variance on the order of 1–2 percentage points. If we consider a set of polling stations having roughly the same size and serving similar populations (that is, with similar values of the parameter p)—within the same city, for example,—then turnout can still be described by a random variable distributed as a mixture of narrow Gaussians with somewhat different means and variances; but in the absence of strong factors that would influence the value of p in a wildly varying (and preferably discrete) fashion in different precincts, the turnout distribution still retains its bell shape and remains pretty narrow.4 Therefore, if we see an average turnout of about 55 to 65%, for example, while at a particular nearby precinct we observe a turnout of, say, 5%, that is already an anomaly that demands an explanation. What happened? A localized election boycott, an epidemic, a natural disaster, or a procedural error? In this case, grounds for doubting the veracity of the election results arise when the results are analyzed using probability theory, which gives us the necessary tools: variance, expectation, the Chebyshev inequality, and the various limit theorems.

In the above example, we can calculate the variance directly, so the Chebyshev inequality gives us an upper bound on the probability that the turnout will deviate from the average, inversely proportional to the size of the deviation divided by the variance. Specifically, the inequality is as follows:

P(Xμ)>a)<σ2a2P(|X-\mu|)>a)<\frac{\sigma^2}{a^2}

Here σ denotes the standard deviation of the random variable (the square root of its variance). From this inequality, we get that the probability of a fluctuation of more than three root-mean-square deviations (that is, when a ≥ 3σ) is about 10% at most. Accordingly, in our example, a precinct with a 5% turnout can only appear with a vanishingly small probability (in the mathematical sense of the word), since it would be much farther away than three standard deviations. Could this happen? Theoretically, it could, but the probability is extremely low. As small, for example, as the probability of a plane crash.

Of course, comparing results this way requires caution. For example, with an average voter turnout of 60%, the presence of a limited number of polling stations with a turnout of 100% could have a reasonable explanation: there do exist small polling stations serving only a few people, perhaps in rural areas where “the whole kolkhoz votes;” there is voting at polar and research stations, etc. However, if we examined the polling stations located across the residential areas of a large city, excluding any “exceptional” ones (hospitals, prisons, etc.), and we saw that all of them had a turnout of 60%, except that one has 5%, that would be a substantial red flag.

Note that it is possible for the voting patterns in different regions to be polar opposites. For example, half of the country could be boycotting the election while the other half participates enthusiastically. Such cases, in theory, could well produce bimodal distributions and other oddities.5 However, “miracles” of this kind must have a rational basis in sociology. For example, a switch from a historically very narrow to a very wide (though still unimodal) distribution of polling stations by turnout occurred after 2015 in Venezuela, when voters opposed to the Maduro regime (concentrated in what were once the more prosperous areas) decided to boycott the election. But it is unlikely that the distribution in the presence of an observer is completely different from the distribution in the absence of one in the same region. A good example of such a difference, and not even with human, but with “mechanical” observers, namely KOIBs (a kind of optical ballot scanner), took place in Moscow during the 2007 parliamentary and the 2008 presidential elections.6 For instance, in 2007 the average turnout on polling stations using KOIBs was around 55%, and the “bell” of the distribution was localized in the range between 40 and 70%. Accordingly, a significant excess of polling stations without KOIBs with a turnout of more than 70% would seem to be an anomaly. That anomaly becomes even more curious when the turnout distribution among polling stations with KOIBs is more or less symmetrical, while in polling stations with ordinary ballot boxes it is miraculously asymmetric and skewed specifically towards higher turnouts. For a more rigorous argument, one can take the initial turnout data, calculate the expected value and the variance of the corresponding random variables, then apply the Chebyshev inequality mentioned above. Ultimately, we come to the conclusion that either the theory of probability does not work on Russian soil, or there has been large-scale stuffing during the voting. Observe that in the 2008 presidential election, the difference between polling stations with and without KOIBs becomes even more extreme. At the same time, after the Bolotnaya Square protests of 2011–2012, the distribution of turnout in Moscow reverted to full compatibility with mathematical laws.

Let us dwell in more detail on the idea of describing the behavior of an individual voter (hence the polling station as a whole) by a random variable. The justification of this thesis is to be found in sociology rather than mathematics. Upon analyzing elections in countries of all kinds, with very different political situations, one clearly sees that always and everywhere, with the exception of specific, easily explained cases, the distribution of polling stations by turnout largely fits a universal mold: a smooth, perhaps slightly asymmetric bell curve.7 Nowhere except in Russia do we see anything like the infamous “Churov’s saw”8 (a plot with huge peaks on “pleasing” integer turnout values of 60, 65, 70, 75, etc. percent). So one has to choose between two possibilities: either statistics in Russia does not work the same way as in the other countries of the world, or the elections are rigged. Claiming that electoral statistics do not follow mathematical laws is akin to a child claiming that a neighbor’s window was broken because of a meteorite, and not because the child hit it with a ball. In principle, that is possible, but it requires extraordinary evidence.

I hope that we have succeeded in illustrating more or less convincingly that statistical analysis can be adequately applied to election results. We reiterate that in a normal political system, reasoning along these lines should be admitted as evidence by the courts whenever election results are called into question. However, when legal roadblocks have been deliberately erected in the way of direct election observation, sometimes to the point of making it virtually impossible, the use of mathematical statistics becomes practically the only tool allowing us to catch the crooks stealing our votes red-handed and to assess the scale of the theft.

Anomalies in statistics of Russian elections and estimates of falsifications

Taking into account the above broad considerations on the applicability of statistical methods to elections, let us proceed to an analysis of the Russian electoral realities in particular. In addition to general mathematical and statistical ideas, the analysis heavily relies on the specific features of the Russian electoral system and on the available data, as well as on the experience and knowledge accumulated during the post-Soviet period by election analysts, observers, independent members of election commissions and volunteers watching live streams from polling places.

The election data available to us is multidimensional in an essential way: for each polling station there is data on the number of registered voters, the number of ballots cast for each candidate, the number of invalid ballots, as well as the approximate turnout reported throughout the election day. Each precinct is also inextricably embedded among its neighbors (spatial context), has a record of previous and simultaneous elections (temporal context), etc. This multidimensionality, on one hand, facilitates the identification of possible anomalies, and on the other, makes it necessary to pre-process the data for visualization in order to provide an overview of the voting before getting into the details. In our analysis, we visualize polling station data as two “projections” presented in a two-panel chart. An example of such a chart for the vote on amendments to the Constitution of the Russian Federation is shown in Fig. 1 (below).

The right panel shows a scatterplot of polling stations with the turnout on the horizontal axis and the results of the candidates on the vertical axis. Each precinct corresponds to as many points as there are candidates (in this case, there are three “candidates”—“Yes,” “No,” and an invalid ballot). The choice of these axes is not accidental: both of these parameters are important for reporting and therefore indicate the extent of administrative influence. The left panel shows the votes for each candidate, binned into 1%-wide intervals by the final turnout at the polling station; the shaded part shows the difference in shape at high turnouts between the vote distributions of the candidate with administrative support (“administrative” or “A-candidate”—in this case, the “Yes” candidate) and of the other candidates (in this case, “No” and an invalid ballot).

The A-candidate part of the scatterplot in the right panel looks like a “comet” consisting of two fundamentally different parts: a compact “nucleus” localized around a turnout value of about 45% and a result of about 65% and a “tail” extending towards 100% turnout and result. At the same time, the tail, unlike the nucleus, exhibits a distinctive grid pattern: precincts tend to concentrate around integer percentages, especially multiples of 5%. Another important detail is that the final official values of turnout and result (shown by the crosshairs) fall “into the middle of nowhere”—between the nucleus and the tail, where the density of polling stations is significantly less than in either the former or the densest part of the latter. That is, it turns out that there are uncharacteristically few “country-average” polling stations, much fewer than polling stations with higher or lower values of turnout and result.

All this suggests that two different mechanisms are responsible for forming the distribution of precincts by turnout and result: one for the nucleus, and another, different one for the tail. The key to understanding the nature of the tail is the grid pattern—the increased density of polling stations around percentage values that are “pleasing” from a human point of view. As discussed in the previous section, the turnout at a polling station, being the sum of decisions of independent voters, is a random variable whose spread is naturally bounded from below by the width of a binomial distribution for the typical turnout and voter population, which in any realistic situation will be on the order of one percentage point. Meanwhile, the observed pattern corresponds to the polling stations concentrating in narrow intervals some tenths of a percent wide—as if we were able to draw a millimeter grid on paper with a paint brush. An article9coauthored by one of the authors, using the data from the Russian federal election campaigns of 2000–2012, shows that the probability for such patterns to arise in the course of free voting is astronomically small; however, as we can see, they continue to appear.

What mechanism could provide such fine positioning of results and turnouts around “pleasing” percentages? It is obvious that this mechanism is somehow related to the decimal system, because in other number systems the “pleasing” percentages are not in any way noteworthy. For example, if seven-fingered aliens using base-7 notation looked at the diagram on the right, they could conclude that it was produced by beings who ascribe special significance to the numbers 5 and 10. And we know of such beings: humans. That is, the “decimalized” structure is caused by human influence. At the same time, as already mentioned, no influence at the level of freely voting voters (such as a voter mobilization campaign) could achieve the desired totals with the requisite accuracy of tenths of a percent, due to the statistical nature of those totals. We are led to the conclusion that the grid pattern in the tail of the distribution is a consequence of an influence affecting the turnout and result already at the polling-station level, that is, falsification.

The general shape of the tail in the diagram also fits into the falsification paradigm: the result of the A-candidate increases with turnout, and the results of other candidates decrease accordingly. This corresponds to the simplest possible approach to falsification: adding extra votes for the desired candidate, either as real ballots or in the final protocols. We are quite convinced that this does occur, both by observers on the ground and especially by live streams from polling stations. When ballots are added for the administrative candidate, two things happen: the polling station moves to the right on the turnout axis (with all its “points” on the scatterplot, for all candidates), while the vote percentage of the administrative candidate increases (because votes were added for him) and the results of the other candidates decrease (because the extra votes for the A-candidate increased the total number of votes—the denominator of the vote percentage). Thus, the points of the A-candidate move up and to the right, and the points of the other candidates move down and to the right, which is indeed what we observe in the right panel.

Under this assumption, it becomes clear how the “pleasing” results and turnouts emerge: the polling station staff will add votes not until an arbitrary threshold, but until the turnout or result of the administrative candidate reaches a value that seemed desirable to them (or perhaps was mandated from above). And here another important property of statistics of large collections of numbers appears: even if the share of stations which make up figures this way is small and they all operate independently, at the country level the pattern becomes visible.

It is important that the increased density of polling stations at integer percentages is solely a property of the tail of the “comet” in the right panel, and not of its nucleus. This justifies our assumption that the nucleus and the tail in the diagram are formed in different ways. Moreover, this is not the only anomaly exclusive to the precincts in the tail. Their protocols also contain abnormally many numbers ending in 0 and 5, which is likewise typical of numbers that were made up and not of numbers that were obtained from a random process, such as voting. In addition, there are other patterns in the tail that are impossible with free voting. For example, the cluster of points around 63% turnout and 78% result of “Yes” (next to the black crosshairs) accounts for essentially all the polling stations in the city of Kazan, with the exception of a few that were covered by independent observers (turnout there ranged from 32 to 40%) and several others that reported turnout just under 100%. As in the case of the grid discussed above, this cluster is too tight to have been formed as a result of free voting at polling stations, which means that we are witnessing a pervasive falsification of results on the scale of a city of a million inhabitants.

Let us now turn to the left panel of the diagram. It contains a histogram of votes for candidates binned by precinct turnout (i.e. the number of votes cast for candidates in the precincts, grouped into 1% intervals of turnout at closing time, spanning from an integer percentage to the next one; the turnout value of 100% is counted as its own interval). In effect, this is the weighted projection (marginal distribution) of the two-dimensional distribution on the right panel onto the turnout axis. The thin dotted line shows the distribution of “No” votes and invalid ballots, scaled to match the distribution of “Yes” votes at low turnouts. We see that in the range of turnouts that correspond to the position of the nucleus on the right panel, the shapes of these two distributions coincide, yet they diverge in the tail region. If we assume that the tail consists of the precincts where vote stuffing in favor of the administrative candidate took place, then under fairly general assumptions, the number of fraudulent votes for the A-candidate is given by the area of the shaded region between the two histograms—the A-candidate one (red line) and the scaled one (grey line).10 This area is indicated in the legend of the left panel (26,929 thousand anomalous votes). If the fraud is of a more complex nature (for example, the turnout and the results are simply made up), the number of anomalous votes calculated in this way ceases to be a good quantitative estimate of the amount of fraud, but remains a useful index for the extent of it and is suitable, for example, for comparative and historical analysis, to which the next section will be devoted. In addition, we note that the grid pattern on the right turns on the left into a “Churov’s saw”—a jagged distribution with peaks at turnout values proportional to 5%.

Remark 1. From a mathematical point of view, the analysis of election data containing falsified (distorted) values can be considered as a problem of robust (to distortions of input data) statistics. Such problems arise when processing data which may have been partly distorted—for example, by measurement errors, external factors, or contamination by data of a different nature. Well-known examples of robust estimators are the median and the interquartile range, which provide robust replacements for the mean and the standard deviation, respectively. The method that is used here to estimate the number of anomalous votes (and, accordingly, the election results sans falsifications) can be viewed as a specialized offshoot of robust statistics that takes into account a priori knowledge on the nature of the expected distortions.

Remark 2. One can ask whether we are being too bold when we declare the entire tail in the right panel of the diagram to consist of fraudulent precincts, even though statistical anomalies like the percentage grid, however rigorously proved, only affect a small part of those stations. When doing so, we take into account a variety of circumstances in addition to the presence of statistical anomalies. First, unlike the nucleus, the tail can appear and disappear at the regional level depending on political circumstances: for instance, in Moscow it was observed from 2007 to 2011 and disappeared overnight after the mass protests in the winter of 2011–2012 (reappearing only in the 2020 constitutional plebiscite). In the Komi Republic, the tail disappeared after the arrest of Governor Gaiser and, by way of collateral damage, the chairman of the republican election commission. In Khabarovsk Kray, the distributions lost their tails after Governor Furgal won the election against the administrative candidate. Conversely, in Samara Oblast, the size of the tails increased sharply after Governor Merkushkin arrived with his team from Mordovia. Across the municipalities of Moscow Oblast, tails appear and disappear in a haphazard way as the municipal governments change. The list of examples can be continued, and all of them will indicate that the tail is affected by the government, not society as a whole. Second, turnouts and results characteristic of the tail are nowhere to be seen when independent observers are present. Third, in the rare cases of court-determined fraud at polling stations, we see how the fraudulent voting results from the tail turn upon correction into real ones in the nucleus.11 Fourth, the high turnouts of the tail precincts are contradicted by live video streams where those were recorded.12 In accordance with the principles of “duck typing” (if it walks like a duck and quacks like a duck, then it must be a duck) and “the thirteenth stroke of the clock” (the thirteenth stroke of a clock casts doubt on the previous twelve), all these points (and still others that have gone unmentioned here) make it possible to classify as fraudulent not only those parts of the tail for which the presence of falsifications has been proven—statistically or otherwise—but also their “neighbors” on the diagram, with similar values of turnout and result. Of course, with this approach, it is possible that the purportedly fraudulent polling stations will in fact include a certain number of “honest” ones where the turnout and the result have, for some reason, strongly deviated from the normal values, but given the insufficient coverage by independent observation and the absence of judicial protection, with statistics remaining the only way to analyze election results, this is the best approach we have.

History of federal election fraud from a statistical viewpoint

In this section, we provide a brief overview of fraud in the federal elections of 2000–2021 in terms of the anomalous vote index described above, which allows us to quantify the evolution of administrative interference in the electoral system. The federal voting data of 2003–2021 are officially available on the Web at izbirkom.ru; the data of the elections of 2000 posted there are unlisted, but can be located using search engines. All datasets are available from the authors upon request.

1. 2000 Elections of the President of the Russian Federation

The distribution of the results of the administrative candidate (Vladimir Putin) at the polling stations in the turnout-result axes is a compact cluster around the official result (black crosshairs), while the distribution of votes for the administrative candidate is shaped almost the same as the distribution of votes for other candidates. The number of anomalous votes is minimal by Russian standards—less than 3 million—although subtracting it from the number of ballots and the number of votes for Putin gets the latter dangerously close to a runoff (Fig. 2).

2. 2003 Elections to the State Duma of the 4th convocation

In the parliamentary elections of 2003, a “comet’s tail” appears near the result cluster of the administrative candidate (United Russia), pointing in the direction of increasing turnout and result, and the official result migrates from the center of the comet’s nucleus to its edge. In turn, from a certain point the distribution of votes for United Russia by turnout deviates upward from the distribution of votes for other parties. Such a picture corresponds to extra votes being added for the A-candidate on some of the polling stations—as a result, the corresponding points of the A-candidate on the left diagram move to the right (the turnout grows) and upwards (the result of the A-candidate grows). The number of anomalous votes is approximately 4 million out of the 23 million total cast for United Russia (Fig. 3).

3. 2004 Elections of the President of the Russian Federation

In the presidential elections of 2004, the tail grows and develops structure: an increased concentration of data points appears at “pleasing” turnouts, that is multiples of 5%. On the plot on the left, this corresponds to the “teeth,” later dubbed “Churov’s saw” after the head of the Central Election Commission from 2007 to 2016. The number of anomalous votes for the A-candidate (Vladimir Putin) reaches 8 million.

The appearance of “Churov’s saw” marks an important turning point in the evolution of the electoral system: the emergence of centralized demand for a “pleasing” election result. Since then, this phenomenon has persisted in one form or another in all federal elections. The same is borne out by another phenomenon apparent in the right panel—a sharp jump in the density of precincts between 49 and 50% (in fact even between 49.9 and 50.0%): it is evident that polling stations systematically “pulled” the turnout over the 50% mark, even though this quantity had no legal significance for an individual precinct: the elections would have been valid if the turnout had merely exceeded 50% in the country as a whole (Fig. 4).

4. 2007 Elections to the State Duma of the 5th convocation

In the parliamentary elections of 2007, the “tail” grows further, and the number of anomalous votes for United Russia reaches 12 million. It was these anomalous votes that provided United Russia with a qualified majority in the new parliament (Fig. 5).

5. 2008 Elections of the President of the Russian Federation

In the presidential elections of 2008, the number of anomalous votes exceeds 14 million, and “Churov’s saw” manifests not only in turnout, but also in the result of the A-candidate (Dmitry Medvedev)—the tail becomes checkered. This federal campaign was the first to see mass manipulation of votes in Moscow (with the number of anomalous votes around 1 million) (Fig. 6).

6. 2011 Elections to the State Duma of the 6th convocation

Until recently, the parliamentary elections of 2011 shared with the presidential elections of 2008 the title of the most “anomalous.” The number of anomalous votes once again exceeded 14 million, and “Churov’s saw,” having been subject to some public discussion, weakened in the distribution of turnout, but grew stronger in the distribution of the result of United Russia (horizontal lines in the “comet’s tail”). The anomalous votes provided United Russia with a simple majority in parliament (Fig. 7).

7. 2012 Elections of the President of the Russian Federation

After the mass protests in the autumn and winter of 2011, the intensity of administrative manipulation slightly decreased in the 2012 presidential elections, and the number of anomalous votes decreased by a third. From that moment up until 2020, electoral fraud in Moscow was almost completely absent (Fig. 8).

8. 2016 Elections to the State Duma of the 7th convocation

During the 2016 elections, in the wake of the protests of 2011–2012, electoral manipulation was moved out of the cities, farther away from observers. As a result, the “two-humped Russia” was born—United Russia received a good half of its votes not in the main “nucleus” of the polling stations, but in the “tail.” As a result, half of the votes for United Russia in party-list proportional representation came from polling stations amounting to 23% of the total voting population of Russia, and the other half, from the rest of the polling stations with the remaining 77% of voters. The official election result ended up far outside the main cluster. Anomalous votes provided UR with 50% of the proportional-representation seats (Fig. 9).

9. 2018 Elections of the President of the Russian Federation

In the presidential elections of 2018, vote manipulation was again toned down, and the number of anomalous votes decreased to 10 million (Fig. 10).

10. 2020 Vote on amendments to the Constitution

The 2020 nationwide vote was held on a newly introduced multi-day schedule, and in principle one would expect some new statistics. For example, easier access to ballot boxes could have widened the distribution of turnout without changing the balance between the supporters and opponents of the amendments, and this has indeed been observed in some regions. However, at the level of the entire country, the picture ended up familiar qualitatively, though it surpassed everything seen to date quantitatively. Judging from the overall picture and reports from the field, week-long advance voting was used simply as a convenient way to add “yes” votes without any interference.

For the first time in the history of federal campaigns, the volume of the nucleus, measured in registered voters, turned out to be less than the volume of the tail. If we take the 57% turnout mark (to be on the safe side) to be the border between the nucleus and the tail, then approximately 34% of registered voters fall into the nucleus and approximately 66% into the tail. The results of the voting are: in the nucleus, the turnout is 44%, with 65% voting for the amendments; in the tail, the turnout is 80%, with 82% voting for the amendments (Fig. 11).

This is when the so-called “Moscow voting standard,” which had originated as a response to the protests of 2011–2012, “broke down”: in about a third of the city’s polling stations, the tallies were falsified.

11. 2021 Elections to the State Duma of the 8th convocation

The elections of deputies to the State Duma in 2021 were held on a three-day schedule. They were accompanied by a widespread deployment of electronic voting, which is subject to a whole number of concerns beyond the scope of this analysis—only “paper” voting is considered here. The number of anomalous votes for United Russia approached 14 million and accounted for more than half of all paper votes cast for United Russia. Thanks to this, United Russia yet again received half of the proportional-representation seats, although in reality it could have counted on a third.

As far as the turnout sans falsifications, 38%, this vote revisited the record low of 2016, and as far as the proportional-representation result of United Russia, 33%, it surpassed the previous low of 34% from 2011. Taken together, those mean that UR received the smallest absolute number of votes in the history of federal voting. The official turnout and UR result (black crosshairs in the right panel) ended up well off into the wilds: there are hardly any polling stations in the vicinity. On the whole, these results show a full divorce between the official election results and political reality (Fig. 12).

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