Lessons on Electoral Fraud from Benford and Brandolini

The election is over, and as expected, Joe Biden has emerged as the clear winner. As was also expected, the President said that this was because of ‘fraud’, of which neither he nor his supporters had an initial theory.

But the point of that first press conference wasn’t to present evidence, but to demand it, and in this space, demand creates its own supply. What has surprised me, though, is the sheer scale of the response. I’d expected the usual suspects to coalesce around a single, central story, or maybe two, but right now there are literally dozens floating around.

On election night, the key theme was the ‘pause’ in the counting process. “We were winning everything and all of a sudden it was just called off.” Trump mused. “A fraud is taking place… why have they stopped counting? Why? Why?’ pleaded a host on Ben Shapiro’s livestream.

By the following afternoon, this fraud-by-delay theory had all but disappeared, as its adherents awoke and remembered that an unusually large number of slow-to-process mail-in ballots was always going to take an unusually long time to process.

Next up was the claim that the number of votes cast in Wisconsin was greater than the number of registered voters, as of November 1st. Of course, the election wasn’t until November 3rd, and in a state that allows for same-day registration, a lot of late registration goes on, especially in a year with record voter turnout. Variations on this theory of phantom voters were briefly shared far and wide — pundits from Infowars’ Paul Joseph Watson to the WSJ editorial board’s Kimberly Strassel were sounding the alarm— but before long, it too had disappeared.

These early confusions, it turned out, were just a prelude to the main event — an avalanche of unconnected claims, from poll workers-turned Democratic operatives brandishing Sharpies, to electoral necromancers summoning armies of deceased citizens to vote and Fox News camera crews sneaking fake Democratic ballots into Phoenix’s counting rooms in broad daylight.

Brandolini’s Law, or the ‘bullshit asymmetry principle’ as it's sometimes called, says that the effort and knowledge necessary to refute a false claim is usually an order of magnitude greater than that necessary to produce it. As much as I'm a fan of Hitchens’ razor, in practice this is almost always the case.

Just one delusional or cynical Twitter user can hatch a rumour that convinces swathes of the electorate that their electoral system has been hijacked by dark and secretive forces. At a minimum, it takes time and effort from friends and relatives to bring those convinced back from the brink, and in the short term it’s often impossible.

Brandolini’s Law might be the most important competitive advantage that false and misleading claims have in the so-called marketplace of ideas. Fortunately, there are a lot of people around, myself included, who enjoy the arduous task of sifting through and refuting them.

Now, amid the blizzard of lies and confirmation bias, one claim in particular stood out. It doesn’t presuppose any particular method of fraud, but rather infers that it probably took place, using a statistical concept known as Benford’s Law. It’s been frequently used to analyse electoral manipulation in Eastern Europe and Russia, and — according to Gateway Pundit and a host of popular MAGA accounts — can be used to show that fraud likely occurred in Michigan, as well as in a number of important counties in other states, including Milwaukee, Wisconsin and Fulton County, Georgia, with greater-than-99% confidence.

Benford’s Law is the observation that smaller leading digits occur more frequently than larger digits, and in a predictable pattern. It emerges from all sorts of data, from street numbers and investment returns to macroeconomic time-series and Covid-19 mortality figures. Exactly why Benford’s Law holds depends on the realm in question.

Take the case of investment: imagine you have a $1000 share in a mutual fund that gives you a 10% fixed return each year for 25 years. After 25 years, if you look back over a list of each year's total, you’ll see $1100; $1210; $1331, and so on. As the sum compounds, that 10% return will grow larger exponentially — in the 25th year alone, you earned an extra $984. Only one of your annual totals will start with a 9, because you leapt past $9000 in a single bound, but a lot will start with a 1, because it took several years of one-thousand-and-something totals before you got past the $2000 mark, and fewer two-thousand-and-something totals before you got past $3000, and so on.

Or take street numbers. Every street has a building number 1, and the vast majority have a building number 2, and 3. As you go higher, fewer and fewer streets include a house of that number, because they‘re too short. Long streets might contain 150 houses, in which every house between 10 and 19, as well as 100 to 150, starts with a 1.

There’s a closely related concept in linguistics known as Zipf’s Law (For those who aren’t familiar, Vsauce made a video on it a few years back), the observation that the most commonly used word in any large collection of text — which is always “the” — appears roughly twice as many times as the second most common one, “of”, and roughly three times as many as the third, and so on.

The key thing to understand about Benford’s Law is that the distribution it predicts arises naturally, so when we see something radically diverge from it, it's often a sign that the data have been artificially produced.

So for vote totals, deviations can signal large-scale manipulation, either because it’s hard for manipulators to simulate randomness, or because they've never heard of Benford's Law and wouldn't think to try. Introducing batches of illegitimate votes into the mix that strongly favour one candidate will push the distribution away from its expected shape. As Gateway Pundit put it,

“Benford’s Law has been used to prove election fraud in the past — Joe Biden’s numbers in Michigan are 99% flawed.”

Now, had Gateway Pundit et al bothered to read any of the literature on election fraud analysis, they’d know there’s no reason to expect first-digit vote totals to follow this law.

Imagine a county made up of precincts whose vote sizes ranged between 1000 and 10,000. In a close race between two candidates, the most frequent first-digit of each of their tallies will probably be 4, 5 or 6, not 1, as each secures around half of the votes in every precinct. The key factor that determines the modal first-digits is simply the sizes of the precincts, and so they don’t tell us anything about whether the election was free and fair.

What analysts look at are the second-digits, which have their own expected frequencies, and these haven’t shown any major deviations this year.

. . .

I’ve been noticing this kind of thing more and more from the type of conservative that liberal social scientists describe as an ‘anti-intellectual’. I think they mean the sort of person who, confronted with a conceptually dense or statistical argument against one of their cherished beliefs, defaults to epistemic relativism - ‘you can use statistics to prove anything!’

The pattern seems to go like this: a technically-inclined member of the Red Tribe presents an argument, in this case ‘Biden cheated, it’s a mathematical certainty’ (other common examples include 'global temperatures are only rising because of increased solar radiation’, or ‘the American government is facing a public debt crisis because the debt/GDP ratio is X%’, etc).

Average members of the Tribe see it and start shouting it from the digital rooftops. They haven't even tried to understand the logic behind it, much less interrogate it — as long as it forwards their overall cause, it's a winner.

Why? Because politics is warfare, and so arguments are soldiers. In Desperate Times like these, the last thing we ought to be doing is second-guessing our own side. After all, the Blue Tribe is quite literally Wrong About Everything, and its unwavering certainty in the face of such wrongness means that it will stop at nothing short of the complete destruction of everything our Tribe holds dear.

A brief response to two obvious objections before we move on: a) yes, of course people on the left behave in similar (albeit subtly different) ways, and b) it's easy to imagine extreme political scenarios in which behaving this way would be a rational strategy. As a general operating mode, though, it's going to make those scenarios a reality about as fast as anything could.

I don’t think we can pin this on illiteracy, either scientific, economic or statistical. If people could stop and take off their motivated reasoning hat, even for a moment, that could break the circuit. The forest just wouldn’t be dry enough for these sorts of ideas to spread like wildfire. Brandolini isn’t always right — it takes less than a minute to find a graph of recent solar radiation patterns, or the US's historic debt/GDP trend. Even without Google, common sense is very often enough.

Maybe the fact that there's a sliver of truth, one that’s apparent to the headline-reader, is what gives them legs. Sunspots effect global temperatures, even a vast economy like the United States could face a debt rollover crisis, and Benford’s Law is indeed used to assess electoral fraud.

I remember when a lot of the standard political wedge issues didn’t hinge on facts — most of which enjoyed bipartisan support — and were instead about different ways of weighting the shared set of facts, based on different ethical world-views.

Somewhere along the line, ‘I believe that an unborn foetus, whatever it might be biologically-speaking, has an intrinsic right to life’ was crowded-out by claims like ‘It’s proven that three-month-old foetuses are conscious persons like you and me.’ ‘Columbine was terrible but I believe that gun possession is an inalienable right’ morphed into ‘Reduced gun deaths that result from tighter gun restrictions will simply be offset by increased knife homicides.’

If it's the case that a lot of people on the Right have started throwing around opaque statistical arguments that support their priors, and that typically they themselves don’t understand these arguments, why might we be seeing more of this now? I can think of several possible reasons.

The charitable one is that they’ve gained a newfound respect for the tenets of empiricism, as opposed to a priori moral argument, having taken heed of the ‘facts over feels’ mantra of millennial conservative luminaries like Charlie Kirk or Candace Owens. But doesn't a blind trust in apparent factoids constitute a disrespect for empiricism?

Maybe it comes from their devotion to a president who almost exclusively makes factual (i.e. falsifiable) rather than moral arguments. I suspect that this is part of it.

Or perhaps I’m being unfair, and they really have understood these arguments, and really did come away convinced. I suggest reading any of the comment threads below any of these posts on Twitter or sites like Gateway Pundit if you think this is the case.

In the end, I think that a lot of what’s going on is simply fire with fire. That in the midst of the endless fact-checking of their president (Kessler's WaPo total stands at 22,510 as of Sept. 3), a lot of Republicans decided that for a long time, their liberal media opponents have been doing the same thing to them — cynically tossing liberal-conclusion-reaching studies in their face, or ‘Eulering’ them as it’s sometimes called online — and deserve to choke on their own medicine.

In any case, watch this space: the vote delivered a decisive result, but one that will remain the subject of intense discussion for years to come— both as a modern Stab In The Back myth and as a temperature gauge of America's culture war.