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An NPR story on the upcoming Russian presidential election mentioned that 5% of polling sites would be equipped with new electronic ballot boxes that would reject attempts to submit multiple ballots. There was "concern" that this would not do enough to stop widespread ballot box stuffing (which was the topic of most of the story). Ignoring for a moment the efficacy of the special ballot boxes, wouldn't a "known accurate" sample of 5% of voters be more than enough to determine if the remaining 95% had been tampered with?

Even without those boxes, how is it possible to reliably rig an election using false ballots without producing huge statistical anomalies?

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The trouble is that the "known accurate" sample is probably not a random sample from the population of all ballots, as it is made up of 100% (approximately) of the votes from a small collection of specific polling sites, and we don't know how those specific polling sites were selected. If they were randomly selected, and there were enough of them, then you could compare them with the results of other polling places and have some hope of detecting fraud, although the power of whatever tests you might construct might not be high unless you have many hundreds of polling sites in your known accurate sample. Of course, Russia is very big, so I assume they could have thousands of polling places in their known accurate sample.

In many cases, rigging an election sometimes does produce huge statistical anomalies. Often the government has little interest in reporting enough information for people to find that out, and in many countries, the press is largely compliant with government wishes and won't really investigate.

If it's done with some care, though, it can be hard to tell. Imagine Chicago in the 1960s, which was a) very large and b) heavily Democratic. If an extra 4-6% Democratic ballots were added across the city, consistently, year after year, who could tell? (Ignoring for the sake of the example the pointlessness of such an effort.)

Here's a link to an interesting look at the 2009 Iranian election that reviews some techniques (good and bad) that can be used even in situations where you have no clean polling place data: Thomas Lotze

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  • $\begingroup$ Comparing votes at a sample of polling places to votes at other polling places is not a reliable way of detecting fraud. If a modest amount of ballot stuffing has occurred in all of the polling places, or in a significant fraction of them, then you're not going to see any statistical anomalies; they're all going to look identical. $\endgroup$ – D.W. Dec 4 '11 at 21:34
  • $\begingroup$ Right, that was the point of the Chicago example, although I realize on rereading that I didn't make it clear. $\endgroup$ – jbowman Dec 4 '11 at 22:17
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This is a complex topic. Elections are a complex societal mechanism, so you should not expect there to be any simple "silver bullet" solution to election fraud.

There are many kinds of election fraud. Let me draw one distinction between fraud where the attacker inserts extra ballots not associated with any eligible voter (call this "ballot stuffing"), vs. those where the attacker causes correctly-cast votes to be miscounted. It sounds like you are concerned with the former, but random sampling is generally not a very effective way of dealing with ballot stuffing, for a variety of reasons. The best defenses against ballot stuffing tend to be procedural, not statistical.

You were vague about the details of what you were proposing, but I infer that you are proposing the following: at some point after the polls close, randomly select 5% of the ballots from among all the ballots in the ballot box at that time, manually count them, and see if the candidate with the most votes among your sample is the same as the officially declared candidate (the one who allegedly has the largest number of votes among all of the ballots).

This proposal has a number of serious shortcomings, which means that it likely will not be effective at detecting large-scale "ballot stuffing":

  • It does nothing to detect any ballot stuffing that may have occurred before the random sampling takes place. If dishonest poll workers stuff extra ballots into the ballot box before the polls open, or during the day, or after the polls close but before the ballots are sampled, you'll never detect that sort of ballot stuffing.

    It does no good to take a random sample of a population of ballots that is not representative of the will of the populace. Counting 5% of the stuffed ballots will not give you any more accuracy than counting 100% of the stuffed ballots.

  • Your proposal is awfully vague about who will perform the sampling and recounting, and when they will do it. If you had in mind that, at every polling station, the poll workers would be responsible for sampling 5% of their ballots and counting them and reporting their counts, then this does nothing to detect misbehavior by dishonest poll workers; if poll workers are dishonest, they can conduct this stage dishonestly or lie about the results of this stage. On the other hand, if you had in mind that the ballots would all be transferred to some central location where election workers perform the sampling, it introduces a different set of problems; it does nothing to detect ballot stuffing that may occur during the day or during transit (which is probably the most common form of ballot stuffing), and it also doesn't work if those workers are dishonest.

  • Your proposal doesn't say about how to provide transparency to the public. An essential requirement for elections is that they must provide transparency. As Dan Wallach has written, the winners almost never complain about the results of the election; an election has to convince the losers, and their supporters. If random sampling and recounting is done in the polling places, it is too hard for concerned members of the public to observe this. If it is done at a central location, at a fixed time, then observation becomes possible -- but we need to preserve the chain of custody for the physical ballots until then, and we need to make sure no ballots have been stuffed (that every ballot comes from an eligible voter).

  • Finally, the statistical power of this approach is less than state-of-the-art methods for election auditing. With your scheme, you need to sample $O(1/\epsilon^2)$ ballots to detect errors where an $\epsilon$ fraction of the ballots have been miscounted. State-of-the-art schemes only need to sample $O(1/\epsilon)$ ballots.

Of these, the first is probably the most severe for the Russian application you mention.

All of these problems can be solved, given appropriate design of the election mechanism (e.g., selection of poll workers, publicly observable processes, careful design of the audit mechanism), but it takes care. There has been a tremendous amount of work on this problem. If you are interested, I urge you to read some of the following references:

As you can see, the statistics community is making important contributions to this topic.

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