The Inherent Inaccuracy of Voting

In a New York Times op-ed, New York University sociology professor Dalton Conley points out that vote counting is inherently inaccurate:

The rub in these cases is that we could count and recount, we could examine every ballot four times over and we’d get—you guessed it—four different results. That’s the nature of large numbers—there is inherent measurement error. We’d like to think that there is a “true” answer out there, even if that answer is decided by a single vote. We so desire the certainty of thinking that there is an objective truth in elections and that a fair process will reveal it.

But even in an absolutely clean recount, there is not always a sure answer. Ever count out a large jar of pennies? And then do it again? And then have a friend do it? Do you always converge on a single number? Or do you usually just average the various results you come to? If you are like me, you probably settle on an average. The underlying notion is that each election, like those recounts of the penny jar, is more like a poll of some underlying voting population.

He’s right, but it’s more complicated than that.

There are two basic types of voting errors: random errors and systemic errors. Random errors are just that, random. Votes intended for A that mistakenly go to B are just as likely as votes intended for B that mistakenly go to A. This is why, traditionally, recounts in close elections are unlikely to change things. The recount will find the few percent of the errors in each direction, and they’ll cancel each other out. But in a very close election, a careful recount will yield a more accurate—but almost certainly not perfectly accurate—result.

Systemic errors are more important, because they will cause votes intended for A to go to B at a different rate than the reverse. Those can make a dramatic difference in an election, because they can easily shift thousands of votes from A to B without any counterbalancing shift from B to A. These errors can either be a particular problem in the system—a badly designed ballot, for example—or a random error that only occurs in precincts where A has more supporters than B.

Here’s where the problems of electronic voting machines become critical: they’re more likely to be systemic problems. Vote flipping, for example, seems to generally affect one candidate more than another. Even individual machine failures are going to affect supporters of one candidate more than another, depending on where the particular machine is. And if there are no paper ballots to fall back on, no recount can undo these problems.

Conley proposes to nullify any election where the margin of victory is less than 1%, and have everyone vote again. I agree, but I think his margin is too large. In the Virginia Senate race, Allen was right not to demand a recount. Even though his 7,800-vote loss was only 0.33%, in the absence of systemic flaws it is unlikely that a recount would change things. I think an automatic revote if the margin of victory is less than 0.1% makes more sense.

Conley again:

Yes, it costs more to run an election twice, but keep in mind that many places already use runoffs when the leading candidate fails to cross a particular threshold. If we are willing to go through all that trouble, why not do the same for certainty in an election that teeters on a razor’s edge? One counter-argument is that such a plan merely shifts the realm of debate and uncertainty to a new threshold—the 99 percent threshold. However, candidates who lose by the margin of error have a lot less rhetorical power to argue for redress than those for whom an actual majority is only a few votes away.

It may make us existentially uncomfortable to admit that random chance and sampling error play a role in our governance decisions. But in reality, by requiring a margin of victory greater than one, seemingly arbitrary vote, we would build in a buffer to democracy, one that offers us a more bedrock sense of security that the “winner” really did win.

This is a good idea, but it doesn’t address the systemic problems with voting. If there are systemic problems, there should be another election day limited to only those precincts that had the problem and only those people who can prove they voted—or tried to vote and failed—during the first election day. (Although I could be persuaded that another re-voting protocol would make more sense.)

But most importantly, we need better voting machines and better voting procedures.

EDITED TO ADD (11/17): I mistakenly mischaracterized Conley’s position. He says that there should be a revote when the margin of error is greater than 1 per cent, not a 1 per cent margin of victory.

In terms of a two-candidate race in which each has attained around 50 percent of the vote, a 1 percent margin of error would be represented by 1.29 divided by the square root of the number of votes

That’s a really good system, although it will be impossible to explain to the general public.

Posted on November 13, 2006 at 12:03 PM56 Comments


Robert the Red November 13, 2006 12:24 PM

How about a revote if the margin is less than b*sqrt(N) where N=# votes cast and b is around 3? This would be proportional to the standard deviation of the vote if it were a Bernoulli coin flip. Allen was out at b=4.7 (Web=1172671, Allen=1165440, Parker=26106), kind of past the edge of where a bizarre concatenation of random events could flip things. But try to explain the statistics to a legislator?!

Alan November 13, 2006 12:25 PM

The idea of a revote is rediculous. I would rather spend $$$ on a recount than waste everyone’s time on a revote, which I’m sure is going to cost way more. Furthermore, if its that close, it could be said that one result is just as good as another, so a revote would only waste time and money.

Rich November 13, 2006 12:38 PM

I did the recount of the 1980 presidential primary in New Hampshire (USA). I was a teacher on winter break and volunteered. Each counter had a table and a stack of ballots. Observing the recount at each table were representatives from each of the interested parties. There were four pairs of eyes verifying the count (counter, Republican, Democrat, Independent). The observers for the two major parties were volunteers — the independent candidate had hired guns.

One or two cities had mechanical voting machines whose total was checked, but individual votes were not.

dlg November 13, 2006 12:47 PM

I agree with Alan. Revotes are a waste of time, unless, magically, the problems with the voting procedure have been resolved in the meantime (which hasn’t happened in 8 years, why then in 4 weeks or so?).

Sure, it makes sense to have individual precincts revote. But that would likely create more problems than it solves. Those in the revoting areas will be responsible for the outcome of the election, and they will know it, dramatically changing voting behaviour. They will be the targets of unprecedented influence. Since few voters decide the election, they are worth more money. Hell will break loose for them considering what happens to everyone (even non-voters) before an election.

Also, buying one hundred votes in a state doesn’t change a lot. But buying 100 votes when I already know the election depepnds on them might be worth it.

Roy November 13, 2006 12:49 PM

Even if the system is well-designed, and well-operated, there is still inherent inaccuracy, even with electronic scanners. We tend to imagine electronic counters as deterministic mechanisms, although they are not: they are stochastic.

Counters can fail to detect a pulse, just as they can detect a pulse that never happened. And as counts are carried, the probability that a column — binary or decadal, it doesn’t matter — will carry successfully is not quite unity.

Thus for large counts, ‘the exact true count’ is essentially a useful fiction.

Perhaps the old saying is good advice here: When it’s too close to call, don’t call it at all.

Ted November 13, 2006 12:53 PM

A revote throws in too many additional considerations.

If the vote’s so close, it’s probably a safe bet that more voters may be able to find their way to the polls, perhaps changing the demographic and swaying the result in a direction different than that of the initial set of voters (who, because they bothered in the first place, deserve more weight IMO).

Again, in a close vote, absentee ballots matter a lot. But of course many of those people may not be aware of the revote, or may not be able to get or send a ballot in time.

There are a host of other issues, too – in a given populace, it may be the bloc supporting one candidate may be more inclined to be indisposed at any particular point in time for the revote (the affluent are more likely to be traveling, for instance).

Overall, a revote is a lot different than a runoff.

I see no way

Rob Shein November 13, 2006 12:58 PM

There’s another kind of error that can be uncovered by a recount, however, which is one based on human error in a position that can affect a large number of votes. Years ago, I did some volunteer work for a campaign, and there was a close race which ended up with a recount. Well, in the course of that recount, it turned out that there were some extra digits in the number of votes tallied in a specific district. Through human error, thousands of extra votes were added to the total for one candidate; these weren’t cases of votes flipped from A to B, but more like a case of 512 votes for A and 697697 votes for B. Because of the recount, the error was spotted and addressed.

Valentin November 13, 2006 1:37 PM

You say: “Conley proposes to nullify any election where the margin of victory is less than 1%” – but thats just not what is proposed in the NYTimes article. It is saying that the margin of error (!) should be smaller than 1% – thats something quite different.

Michael T November 13, 2006 2:02 PM

“Random errors are just that, random. Votes intended for A that mistakenly go to B are just as likely as votes intended for B that mistakenly go to A.”

Not if votes are unevenly distributed.

If you have 1000 votes, 600 for A and 400 for B, and 100 votes get flipped at random, 60 votes for A and 40 votes for B will get flipped.

Result: A loses 20 votes, B gains 20 votes, A’s margin reduces from 200 votes to 160 votes.

Obviously, as the distribution of votes approaches even the distribution of error approaches even, suggesting random errors shouldn’t be a problem.

A few moments analysis reveals that:

lead reduction (%) = 2 * flip rate (%)

for instance a 10% flip rate will reduce a candidate’s lead by 20%, a 20% flip rate will reduce a candidate’s lead by 40%, and so on.

It won’t affect the race outcome (assuming a statistically significant number of voters) but random errors do affect the result.

Ray November 13, 2006 2:08 PM

I don’t think that Dalton Conley has ever counted a large jar of pennies.

If your count doesn’t converge on a single number, then you aren’t doing it correctly. Clear a large flat surface. Create stacks of 10 pennies. Group the stacks of 10 by 10s. Collect the groups of 100 by 10. Check the stacks, check the groups, check the collections. Check again. Have your friend repeat the checks. Resolve any discrepancies.

This is accounting, not quantum physics. The pennies are all on the table in front of you and if you can’t get an accurate accounting, you aren’t trying hard enough.

Dalton makes a mistake when talks about measurement error and large numbers. Measurement error on binary values is zero. Either you have one penny, or you don’t. You don’t have one penny with probability 0.9999 or even 1-10^-100. The measurement error on a summation of 0 and 1 values is also zero.

Of course there are practical problems with counting large numbers and even more problems when you are counting things like ballots that can have stray marks, hanging chads, etc. but his original example is flawed.

I agree that recounts are problematic, especially when e-voting machines that don’t give you any “thing” to recount, but if there is a “thing” then the tally can converge with sufficient time and effort. A revote is a counting of entirely different sample. It’s convergence properties are worse than a recount, not better. What are you going to do when the revote is close? Call a re-revote?

Mike Schiraldi November 13, 2006 2:11 PM

It’s worse than the article says — running an election is even harder than counting a jar full of pennies, because with the jar, it’s at least conceptually possible to get an accurate tally.

However, even with a perfect election system, and 0% invalid ballots, you -still- can’t get a perfectly accurate measurement because of things like weather.

Let’s say it happens to rain — or snow — in a county that leans strongly toward one political party. That will suppress voter turnout there, and could swing an election.

So there will always be some error. The goal should be to keep the error due to mistakes smaller than the error due to factors like the weather.

Anonymous November 13, 2006 2:32 PM

If the vote’s so close, it’s probably a safe bet that more voters may be able to find their way to the polls, perhaps changing the demographic and swaying the result in a direction different than that of the initial set of voters (who, because they bothered in the first place, deserve more weight IMO).

Actually this is exactly the reason why a new election is a good thing. People will be going back with the historic knowledge and may indeed exercise a kind of game theory and review their vote in light of the last time; a Bayesian kind of vote, I guess.

jayh November 13, 2006 2:35 PM

… Clear a large flat surface. Create stacks of 10 pennies. Group the stacks of 10 by 10s. Collect the groups of 100 by 10. Check the stacks, check the groups, check the collections. Check again. Have your friend repeat the checks. Resolve any discrepancies

Ballots don’t work that way. Tedious hours of counting ballots (usually with multiple candidates and questions) introduce simple human fatigue whether using chits, chips, hash marks or tabulators. Things get missed, hit twice, distractions occur.

Patience Dowell November 13, 2006 2:45 PM

I disagree with Mr. Conley. Despite the difficulties in counting the votes, which are legitimate, there is most certainly a universal “truth” to the number of ballots cast for each candidate. Whether those ballots are a correct representation of what voters intended is extraneous to his original thesis, as I understand it.

If 1,000,000 people cast votes, regardless of who they are for, 1,000,000 ballots must exist, or there has been an error of some variety. The question is not one of whether the votes are even countable, as Mr. Conley seems to allege, but rather, how we can go about eliminating systemic and random errors. There is no excuse, to me, for votes not to be accurately counted. If the current system doesn’t achieve that outcome, then one must be devised that will.

Of course, that’s just my opinion, I could be wrong…

Dwight November 13, 2006 2:48 PM

The approach suggested by “Robert the Red” (first comment) sounds good, but wouldn’t this have a result that a single error resulting in a change of many votes (as in Rob Shein’s comment) would go unremarked? In general, any algorithm that says that we will ignore errors with certain characteristics will become the foundation for the next generation of fraud.

Bunny November 13, 2006 2:54 PM

I don’t think it’s true that random errors cannot be avoided. Here in Germany, all voting and vote-counting is done by hand, for each individual voting district (“Wahlbezirk”; the rule is that there should be at most 2500 eligible voters in each district), and when votes are counted (always by several people at once, and the process is open to everyone who wants to come in and watch, which many people do), they are always counted in two different ways (due to the fact that you have two votes: one for the candidate you want to see elected in your constituency, and one for the “general” votes where you vote for a party as opposed to a specific candidate). If the numbers don’t match, everything is recounted from scratch.

Mistakes may still happen, of course, but I think that in general, it’s a pretty fool-proof and reliable system (certainly not one where you’d expect random error margins of even 0.1%). And of course, the actual ballots are saved, too, so that if someone demands a recount later on, it can always be done.

Saying things like “We’d like to think that there is a “true” answer out there, even if that answer is decided by a single vote” really makes no sense to me – there is an objective, absolute truth out there when it comes to elections. Conducting elections means trying to figure out what that truth is, and I’m really rather surprised – and sad – to see that (some) people apparently give up so easily and say “we can’t eliminate errors, anyway, so we’ll have to live with them”.

Of course it’s important to guard against errors, and automatic recounts or even revotes if the margin is under a certain threshold makes sense. But this is a kludge – not an excuse for conducting elections in a sloppy way from the start and then claim that problems that might arise (specifically, random errors) are unavoidable. They’re not.

Gerg November 13, 2006 3:08 PM

It’s worse than all that. Systematic errors can introduce errors out of all proportionality to any actual randomness in the process.

In other words the closeness of the election is a red herring and the focus on the closeness is an anachronism from when fraud could only be executed retail.

Now the threat is wholesale fraud. It doesn’t matter how large the winning margin is, if there’s no paper ballot you can’t know that the result was correct.

A reasonable system should have some random sample of precincts being checked against the paper ballots after the results are tabulated to ensure the electronic counting matches the papers the voters actually saw.

Pat November 13, 2006 3:41 PM

I am not like Conley and I know that you can count pennies accurately – there are a determinate, whole-number number of pennies in that jar. Break the large group into small ones, count and check the small groups, then add them up. IOW, Ray’s suggestion works. It just takes time and patience. So if he’s alleging that an election vote can never be exact, then pennies are the wrong analogy.

Further, I don’t see that the proposition is correct. Number of votes cast is a determinate number. There’s no inherent fuzziness about a vote, no quantum issues. It’s a yes/no proposition: either you put the proverbial pebble in Candidate A’s bin or not. And voters make it all the more convenient by going to polling places; it’s not like the Census that has to find everyone, whether they want to be found or not. If you don’t show up at a polling place, you’re not counted. The counting can be broken down in the same way as counting pennies.

There are two problems. One, poorly designed ballots make it hard to determine how the voter voted. (When I voted in 2000, I remember noticing different ways my vote could have come out unclearly because of the poor ballot design.)

Two, counting as we would for pennies would take a long time so we look for faster methods. It is these methods that are inaccurate. If we were willing to wait a few days, we’d get better answers and fewer nagging questions.

gilberto November 13, 2006 3:42 PM

Of course we can talk about the “true count” of ballots which,however complicated, are discrete objects which can be counted and categorized, as opposed to continous variables which we can only measure with so much accuracy. For many years banks have had money counting machines for both paper currency and coins. If banks can do it so can election officials. It may be a little more tricky reading a paper ballot accurately than a dollar bill, but it can be done.

DaveC November 13, 2006 3:51 PM

As previous posters observe, this is an integer measurement problem, not a floating point one, and there is such a thing as the right total. If there isn’t, then the physical world interface needs a redesign (no chads) to make it so.

Random sample audits are an excellent guard against fraud, as are independent tallies …. e.g. have (i) a touchscreen machine produce (ii) a paper ballot which you can then (iii) optically scan, and buy the ballots, touchscreens and scanners from 3 different vendors. Of course, that drives cost.

A key element of the above, however, is random – witness the Illinois election in the Bev Harris documentary where the random recount was also rigged.

Another missing element is accountability – the above county officials are being prosecuted, but they are a rare exception. If you’re a partisan elections official, and there are no consequences to you from an unfair election, why wouldn’t you perform vote rigging and intimidation, with any process available, including hacking, in service of the greater public “good” of putting your party into power?

Llywelyn November 13, 2006 4:15 PM

“””Votes intended for A that mistakenly go to B are just as likely as votes intended for B that mistakenly go to A. “””

This isn’t quite true, because the options are ordered and thus there is a potential for biasing in the “random” selection. A randomized ballot would make this issue go away, however.

Ted November 13, 2006 4:19 PM

… If you’re a partisan elections official, and there are no consequences to you from an unfair election, why wouldn’t you perform vote rigging and intimidation, with any process available, including hacking …

I’d like to think you wouldn’t because you believe that fair elections are the bedrock of a democracy.

I know there are people who actually do believe that I’ll be better off if they make my decisions for me, but I have to believe that they are few and the vast majority of poll workers believe in the process more than any they do in any candidate.

As far as I’m concerned, election fraud should be punishable by public stoning, since I can’t think of anything worse off the top of my head.

Brandioch Conner November 13, 2006 6:24 PM

What the … ?

Counting pennies is incredibly easy as others have noted.

Counting ballots would be almost as easy if people would just look at the problem correctly.

Step #1. Have a Democrat pull out a ballot and evaluate if it can be read or not.

Step #1a. If it cannot be read, clearly, then it goes in the “future count” bin.

Step #1b. If the ballot can be read, it is handed to a Republican who verifies that the Democrat identified the vote correctly.

Step #2. The vote is then put in a pile for that candidate.

Every 10 votes are bundled and marked as 10 votes.

10 bundles of 10 votes are bundled and marked as 100 votes.

And so forth.

If at the end of the count, the margin between the candidates is less than the number of ballots in the “future count” bin, those are sorted and both sides try to agree on what the vote was (if possible).

You can add Independents, Libertarians, Communists, or whatever other representatives you want from whatever parties you want.

If there’s any question AFTER the re-count, the bundles can quickly be verified. Are there 10 votes in the 10 vote bundles? Are there 10 10’s in the 100? Are there 10 100’s in the 1,000?

And the ONLY thing you have to work on is getting a paper ballot that can easily be marked and be read just as easily.

No high-tech solution needed. Just basic common sense and a little design work.

Moz November 13, 2006 7:00 PM

So why not introduce physical token-based counting? Provide tokens for each ballot and the voter simply selects one per vote and deposits them in the receptacle provided. Building a machine to do this would be simple enough and it’s definitely voter verifiable even if the voter never touches them. Making the tokens recyclable after the election shouldn’t be too hard, and making them mechanically softable likewise.

Most voters are cabable of going “ok, the presidential candidates are all triangular tokens, I want the blue one with ‘bob the democrat’ written on it”. The voter drops their chosen tokens in the ballot box, then the unused tokens into the bin. Said bin gets most of the tokens removed periodically and re-used.

Counting consists of sorting tokens by election, then by candidate. The biggest pile wins 🙂

This would be quite expensive to do the first time, but with many metals or plastics re-imprinting tokens would be fairly easy and they could be used for quite a long time.

Much easier would be an electronic paper ballot printing system but that’s complex and technologically tricky to explain to people. Counting tokens is something most people understand.

BruceG November 13, 2006 7:49 PM

Getting the count right is possible. As others have pointed out, it’s an integer arithmetic problem, and easily solved.

Doing the count multiple ways (or multiple folks counting at once – marking off checkboxes for each candidate or a “no vote” or “spoiled/questionable” in a race) is one way to cross-check. Having competing interests present (one from each candidate/party) doing the counts minimizes the issue. The core problem with the electronic machines is not that they’re electronic – it’s that there’s only one thing doing the counting. That allows bias and prevents checking. Having multiple independent systems (tough to do with machines… easy with humans) means you can have a cross check and nothing is reported until there’s agreement between the counts.

Yes, this may take longer to report out. So what? The only issue there is the length of work for the election workers (and it is an issue – 13+ hours of voting is a LONG shift, and counting by hand after that is not going to be a fun job – been there, done that in a different country and many years ago) – we don’t need to know the results immediately – really we don’t. Valid, secure votes are worth a heck of a lot more than results by 10:00 Eastern… (which doesn’t happen anyways)

Jamie November 13, 2006 9:45 PM

I live in New Zealand, and we have a very similar voting system to Germany. You have two votes, one for your local representative, and one for a party. The ballot is a single piece of paper with two columns, one for each vote. You place one mark in each column, then put your ballot into a sealed box. (An example ballot paper can be found at

Once voting has closed, volunteers count the votes in each box. Anyone can watch, and so more eyes means less chance of error. Each electorate counts their own votes, then sends the total in to the central tabulator.

This system has been in place for many years, and so far there have been no controversies. By the end of the night, we generally know who our next government is going to be.

Sure, paper voting takes more manpower, and involves a lot of dead trees, but it works. All of the ballots are kept, so a recount is simple. Anyone can understand the process, and there’s definitive proof if anything does go wrong.

Harry Crumb November 14, 2006 1:00 AM

@Brandioch Conner “…..the bundles can quickly be verified”. Indeed so, I am so well trained and skillful, I can count the number of notes/ballot papers in a bundle simply by listening as I riffle the bundle close to my ear.


Bonzo November 14, 2006 1:20 AM

How about if we use machines for the recounts (e.g. optical scan), but each candidate provides their own counting machine. This way it becomes the incentive and responsibility of each candidate to make sure their counting apparatus does not undercount their votes. The ballots are counted on each machine, and if there are discrepancies, recounted by hand. Ideally, if there are discrepancies hinting that a certain candidate rigged their counting machine, there will be a political fallout.

Wylie November 14, 2006 1:51 AM

America’s voting system is broken – that much is clear. It has been clear since the 2000 election made world headlines.

What is not clear is WHY you people spend millions of dollars on voting machines which (obviously) cannot be demonstrated as being 100% accurate. Just reading about only the theoretical problems (lack of audit, perception issues etc) let alone the actual problems, tells me that dog wont hunt, so why does your govenment continue to spend money on them?

We have a paper system in Australia. We mark the paper with a pencil, its counted by hand (no bar codes, no scanners – just eyeballs) and then tallied by computer. Every polling place and every district can recount if they need to. Its not that hard.

Sure, its not perfect (We got John Howard) but it works!

Im a geek, and I wouldnt vote electronically – especially on what you’re supposed to vote on.

Steve November 14, 2006 5:07 AM

Counting moderate numbers of pennies is pretty easy to an acceptable degree of accuracy – weigh them. The variation in mass of a penny is quite high. Let’s say it’s 1% – so don’t weigh them in groups of more than 50 (or whatever the statistics says is necessary for your desired confidence). Add up the results.

It should be quite straightforward to manufacture ballot papers (or “cards”, since heavier is easier) with mass variations of less than 1%. Unlike pennies they don’t get rolled in mud, stored in people’s pockets, or used as impromptu screwdrivers. So they’ll remain pretty much as constructed.

Counters can divide the ballots into groups of about 50 and weigh them. If the total result is within the expected margin of error (which will still only be a small number of votes, since the chance of an error on each group of 50 is small), then start worrying about counting them. Where it isn’t close, you’ve saved a load of money which you can use to fund multiple counts where they’re needed.

Of course, to do this you need a separate ballot for each race, so it may not be practicable to use it for everything. And you have to prevent voters from sticking gum to their ballots – this can be done by weighing their ballot at the point where the post it. If the card is overweight reject it, destroy it, and force them to fill out a new one.

Steve November 14, 2006 5:12 AM

I forgot to point out that I personally think optical scanning/counting is more accurate than weighing, and doesn’t require extra security surrounding the required weight of the ballot, or humans to separate the ballots in the first place.

I propose the above as a suggestion for folks who for whatever reason don’t like/trust the scanners, but might believe a scales.

Richard November 14, 2006 6:08 AM

In the UK, the Returning Officer in charge of the count can order a recount if it’s close. But the decision is entirely at his discretion. He can order more than one recount, but can also refuse to do a recount. Any candidate can call for a judicial review of the refusal to do a recount and ask for the election to be held again. But that can be a perilous request.

In the one case a few years ago, the sitting Conservative MP lost by a mere handful of votes after the third recount. He went to court to get the vote held again. Whereupon he lost by several thousand votes. We don’t like a sore loser round here. 🙂

Election officials are paid a modest fee and are not allowed to be linked to a particular party. They are usually officers of the local council.

The election is done by electors marking their choice on a paper ballot. These ballots are then counted by hand. No computers are involved. It seems to work.

bob November 14, 2006 6:24 AM

OK, enough with the pennies. The analogy doesnt work. Try counting pennies, not in the hundreds of pounds, but in the hundreds of thousands of pounds.

And they are scattered across the entire country, no single person can see, count or control them all.

And each of the 50 states each have their own techniques, so some states instead of pennies would have dimes, some nickels or quarters (remember – 50 different backs on quarters now – keep track of those, too) with the odd holdout like VT or DC or San Francisco using SBA dollars or Mercury dimes or something (Buffalo NY would obviously use their namesake nickel).

And we not only need to count them per se (votes for president, for example) but also count the breakdown of years they were issued (congressperson), which mint they were created from (senator) and additionally categorize condition, ie are they uncirculated, mint, worn, etc (lightly marked paper, hanging chad, etc) all from the same original token.

And I cant even think of an analogy for state and local issues, school board members, judges, etc all from the initial single unit deposited, all counted at the same time.

Coins dont make a valid comparison.

Adam November 14, 2006 7:04 AM

Bob > Yes, they do count, because although the pennies are distributed across the whole country, so are the counters.

The entire analogy of “break them down into smaller groups, count the groups, add the totals” works at the precinct/state level too.

All the votes in the nation are broken down into states and precincts. Every single precinct should be able to come up with an exact, 100% accurate count for the number of votes cast in it. Those numbers are then added up at the state level, for a 100% accurate count of votes by state. At that point you either count electoral college votes, or just add them all together to get the “popular vote” tally.

If you do the “definitely A”, “definitely B” and “maybes” pile (as suggested by Brandioch) then instead of waiting for the precinct tally to come in before deciding whether to count the maybes, you need to wait for the state count to be finalised. Yes, there might be a bit more hanging around for small precincts in a big state, but given that the large precincts will be the ones who take longest to count their “maybe”s, the total for the state won’t take any longer to come to.

There is nothing quantum or statistical about votes. They’re definitely cast or not, and the number of votes for “candidate A”, “candidate B”, … “candidate n” and “spoiled” should always add up to this value.

Yes, there might be some judgement to figuring out where the difference between a vote and a spoiled ballot are, and individual ballots may need to be reviewed in a close race (although the actual criteria for a spoiled ballot should be well-specified beforehand). But the total number of votes cast should never vary. Any system that allows it to vary is Broken And Wrong(tm) and needs to be replaced with something that works.

This is counting for goodness sake. All the people involved in overseeing an election should have got this well figured out before they reached high school.

Nancy Lebovitz November 14, 2006 7:06 AM

Here’s another notion for counting, though the multiple humans from different parties may well be good enough.

Have computer-printed ballots. Post them all on walls with video cameras feeding the images to the net. Let everyone who’s interested do the counting.

Jerome Lacoste November 14, 2006 8:53 AM

What about using a redundant voting system ?

Each voter votes twice on 2 different machines provided by 2 different companies. The results are counted by 2 separate entities.

If the 2 results are too different, the vote is cancelled. If they are close enough, the election result is the average.

More expensive, but from a user point of view, if you can queue to vote once, you can vote twice…

Of course you have those that will vote differently in the 2 votes (willingly or by accident). But that should end up in the random errors.

Bruce Schneier November 14, 2006 9:12 AM

“You say: ‘Conley proposes to nullify any election where the margin of victory is less than 1%’ – but thats just not what is proposed in the NYTimes article. It is saying that the margin of error (!) should be smaller than 1% – thats something quite different.”

Interesting. How does he propose to determine the margin of error? That seems fraught with its own problems.

Bruce Schneier November 14, 2006 9:18 AM

“The idea of a revote is rediculous. I would rather spend $$$ on a recount than waste everyone’s time on a revote, which I’m sure is going to cost way more. Furthermore, if its that close, it could be said that one result is just as good as another, so a revote would only waste time and money.”

This is certainly true if the error is random, but not if the error is systemic. (I’m going to write my Wired column this week on this.)

derf November 14, 2006 9:46 AM

On top of the inaccuracy of the counting and the process itself, there is also widespread fraud. I often joke about death turning people to democrat because almost everyone that dies near an election ends up somehow voting for a democratic candidate. There are also the illegal “get out the vote” campaigns where “volunteers” in vans round up poor registered voters, hand them food, and send them into their precinct to vote for the whichever candidate’s party paid for the vans. That’s just two examples of what happens in almost every city in every election.

Neither voting machines nor better counting schemes can deal with fraud.

Adam November 14, 2006 10:29 AM

derf > No, problems with verifying the count and dealing with fraud are pretty orthogonal and have very little to do with each other.

So, what’s your point? That we should not fix the counting problem until we’ve fixed the fraud problem? If so, why? Fixing the fraud problem won’t solve the counting problem any more than solving the counting problem will fix the fraud problem.

Therefore we might as well attack both problems at the same time. With different groups of people if necessary.

But this post/thread is specifically about the counting problem. Feel free to attack the fraud problem in response to a different post. 🙂

derf November 14, 2006 2:29 PM


The point is that solving the count problem doesn’t actually solve the one identified, authenticated, authorized voter = one real vote problem.

The overall system needs to be addressed, not just the technology used for vote counting.

Moz November 14, 2006 3:19 PM

I think a lot of the thought on counting accurately is because that’s the easiest problem to solve. Once you have counting the votes worked out hopefully there will be more confidence in fixing other problems, like fraud, corruption voter suppression and only allowing two parties to compete[1]. But right now most people seem convinced that the system is broken and that it’s not possible to fix it.

Without wanting to sound offensive, can I point out that even very poor countries without significant infrastructure manage to run better elections than the US does. East Timor, for instance, or Sri Lanka. About the only thing you can claim is that you’re not having a civil war over the results (but then you’re not occupied by enemy nations either).

[1] yes, any number of parties can enter, but the threshold to win is artificially high.

Adam November 14, 2006 6:10 PM


OK, I guess it’s just matter of how you approach it.

Looks like I’m a “fix each bug separately, one at a time” kind of guy, and you’re a “rewrite from scratch”er.


Mithrandir November 14, 2006 6:24 PM

A recount is good for correcting systemic error. That’s all well and good.

But I don’t think a recount makes any sense in the case of a close election. If the election is so close that the result of a given count constitutes a random variable, then from a preservation of democracy stand point, it doesn’t matter who enters office.

It doesn’t matter.

Both have an equally strong mandate. Half the electorate is going to lose. I suggest that in such situations, we either re-vote (hoping for additional voter turnout, and thus a better sample), or have the candidates Rock-Paper-Scissors for office.

bob November 15, 2006 6:46 AM

I think revotes are a bad idea. A party that was doing poorly (especially if the trend was upward) would be motivated to disqualify the election in order to do better in the revote.

revoter November 15, 2006 10:18 AM

IMHO, revoting if the results from the first time are too close does much more than fixing random errors.

Sometimes many people did not vote the first time because they were too sure that their favourite candidate would win anyway. Or voted for extremist parties, or not at all, just to demonstrate their aggravation about the system or the big parties (this is typical for france).

In any case, during revoting everybody knows their vote will indeed decide between the two who were so close the first time. And more often than not, it turns out the population has a clear preference for one over the other.

Anonymous November 15, 2006 10:31 AM

I believe Conley is wrong because votes can and should be counted correctly.

Even if his point were valid, the error margin should be much smaller than stated, more around 1×10^-6 (one in a million, or 0.0001%).

But he just distracts from the real problems.

The outrageous thing is not inaccuracy, but obvious defectiveness or forgery.
In the last few years we observed:

  • zero votes for one candidate,
  • a negative number of votes for one candidate,
  • more votes for one candidate than there were registered voters

I can’t imagine larger errors than these.
Can you?

derf November 15, 2006 12:59 PM


Hehe – “Format and reinstall”.

IMHO, the voting machine issue isn’t a big problem since there are other accepted methods of voting that don’t have these problems.

Unfortunately, the fraud issue gets lip-service, but no real action most of the time. When the fraud issue is at least attempted to be tackled, the courts kill it because requiring a photo ID (in some states) infringes on voting rights:,,-5947120,00.html

This actually brings up an interesting point – without an ammendment to the US constitution, a national photo ID card (regardless of its supposed security) will be unconstitutional. Apparently, photo ID presents to big a burden on minorities, the elderly, and the poor.

Dan November 15, 2006 3:51 PM

I definitely oppose any revote schemes. It could fundamentally alter the intended outcome. (Much like live exit polling, IMO, but I digress.)

The problem with a revote is that there is no guarantee that a person will vote the same way the second time. Those that revote have an unfair advantage over other voters by knowing the results of the first one, even if all that is disclosed is “the first vote was too close to call”. That is in addition to any other information that might be discovered about a candidate between election day and re-election day. How would a less -popular 3rd party candidate be affected by such a scheme?

And aside from that, what if the results is exactly the same, ad infinum? When does it stop?

Brandioch Conner November 15, 2006 4:50 PM


The easiest solution is to, once again, sort the “problem” into categories.

#1. Identified, confirmed voters.

#2. Partially identified but 100% confirmed voters.

#3. 100% identified but only partially confirmed voters.

#4. Partially identified and partially confirmed voters.

#5. Unidentified and unconfirmed voters.

Then, prior to the election, determine what the criteria will be for each category and notify the people of their categorization and how to “upgrade” their status.

“Identification” means that your identity is verified. This is easiest with photo ID. But not everyone has photo ID.

“Confirmed” means that the fact that you have not voted yet has been verified. In Iraq they used ink to mark the fingers of the voters.

Then decide on whether you want to refuse ballots from anyone below a certain category or allow them to cast “provisional” ballots and under what circumstances they will be counted.

Ed Gerck November 17, 2006 1:36 AM

This recent NYT op-ed piece on voting was, to say the least, very old. Two centuries ago, perhaps, the following paragraph by Conley might be acceptable:

“The rub in these cases is that we could count and recount, we could examine every ballot four times over and we’d get — you guessed it — four different results. That’s the nature of large numbers — there is inherent measurement error.”

There’s no such “inherent measurement error” for small or large numbers. We use very very long numbers daily. As digital signatures with 2048-bits attest to, people see every day that Conley’s assertion is not true.

Further still, if Conley’s assertion would be the case, chemistry, banking and cooking would also not be possible.

The author confused the issues and the reader, asking for an arbitrary “1%” limit to redo an election. With some election technologies and some machines, the measurement error might be even well above 1%. Every time you put punch cards in to tally, you get a different result.

Of course, there is outcome uncertainty in voting, which can be seen as accuracy and reliability problems in both registering and counting votes. They start with the voter registration, that might err in allowing or blocking voters, and include the voter herself, who might cast a vote that was no intended.

In voting, the significant problem is that of providing an accurate and reliable result (two different requirements) while preventing anyone (including the voter) to prove how a voter voted. Errors in the interface that provides such functionality cannot be corrected, unless an anonymized audit system is used (see [1] for an example).

There are also social errors. If the election is important enough, some countries use run-off voting where the candidate must have the majority +1 vote to win. In the US, we also talk about “instant run-off voting” for the same reason, to assure that the election is not only measurement-wise correct but also socially-wise correct.

However, one error we do not have is in counting large numbers.

Ed Gerck, Ph.D.


Yorick November 17, 2006 4:41 AM

“There’s no such “inherent measurement error” for small or large numbers. We use very very long numbers daily. As digital signatures with 2048-bits attest to, people see every day that Conley’s assertion is not true.”

Yah, but you have to take your shoes off to do it.

Nick November 19, 2006 1:28 PM

Any system to be implemented in politics will have to meet political tests or won’t be adopted:
— Margin sizes ignored by public.
— Complex systems rejected by public.
— Revotes unpopular; run-offs more acceptable.
— Revote/run-off timing.
— Revote/run-off voter restriction.
— Counts now done in an hour or so.
— Randomizing paper ballots is costly. Very.
— Who gets bad machines; & paper going uncounted.
— Recounts may defy odds.
— Fake errors.
— Large-number counts: Errors are human, not inherent.
— Accountants stick a penny on one side, unexplained.
— Apolitical counters unlikely.
— Weather misapplied.
— GOTV usually legal.
— Open-source dream.

Margins of victory have political value to winners only for a short time. After that, all that matters is who won. The rest is drama and color, and they fade. Political insiders consider it when an incumbent runs again, but as just one factor among many in the next campaign. A loser may have an easier time raising money next time if they lost by only a little, but about 96% of the public doesn’t give money anyway. In general, except for the Presidency sometimes, in a few months most people forget margins and just remember who won.

Complex systems are hard to explain, so the public rejects them. Not because they’re dumb, but because they’re smart: a complex system unobserved is easier to wring into falsehood, like at the board that said the FBI called a terrorist alert and kicked out the observers (the FBI didn’t call). Someone described a recount of warehoused machines as rushed: the one calling out numbers walked quickly past machines, forcing observers to keep pace and not read too closely. Former Pres. Carter, worldwide observer of elections, had a hard time finding a US pollsite willing to let foreigners observe a US election. Observation is fraught with problems; here’s a complex system that’s not very complex but still failed: Preferential voting (enter first choice as “1”, second choice as “2”, etc.) was in existence in NYC not many years ago, but I suspect almost no one can say which elections used it. Hint: It was widely criticized for low turnout despite voter eligibility being wider than usual. Second hint: I found other NYC elections with even lower turnout that weren’t criticized, and didn’t use preferential voting. Consequence of the system: Whomever had their votes counted before anyone else and won got to give away their extras and affect others’ outcomes, favoring slates rather than minorities. And recall the debate earlier in Bill Clinton’s first term about a Presidential appointment nominee who’d written about several uncommon but established methods of voting as fair: she was pilloried and he dropped her. Likewise, use margin-of-vote, not margin-of-error. Don’t require square roots, or someone unobserved will decide the square root of 800 is 400.

Automatic revotes won’t fly with the public. Better is the run-off: a similar concept. In some races, the run-off is required when no one gets a majority; only the top two candidates face each other in a run-off. To legislate this, come up with a rationale that doesn’t make the counting mechanism look defective, because people will ask why no one can count. Banks count money. They’ve got trillions. The busiest elections board is probably responsible for no more than a few millions. Grand-totaling subtotals is even easier. The public knows they ought to be able to count them. You won’t convince the public that if boards can’t count votes the first time they’ll magically do it the second time. You didn’t say that but that’s how the argument will devolve in public political debate, because people have other things to do than deep analysis of a system when anyone will say people ought to be able to count one plus one plus one plus one until they’re done. If the counters can’t do it, hire more of them or replace them, they’ll say.

Timing is another issue for revotes and run-offs, besides the angle of recontacting travelers and others who need absentee ballots. If it’s too soon after the prior vote, it’s hard to persuade committed people to change their minds; tempers are too raw, depressing turnout. Candidates try to get the candidates who are no longer competing to endorse them or at least shut up; then the endorsers are invited to public events, media events, etc. to persuade voters. This can take a while, like a few weeks. If litigation starts, courts’ political neutrality (if any) can result in slowness (they don’t want political calendars rushing their trials and decisions), which can seriously gum up campaigning and change outcomes; if courts are not neutral, you know what that means. Judges who are not elected are usually appointed by political leaders. And if there needs to be time before an inauguration for job prep, enough of that time has to be left intact.

If eligibility to vote in a rerun or revote is limited to those who tried the first time, turnout can’t be higher and almost certainly will be lower. That won’t be evenly distributed; true believers are likelier to vote. If two candidates got their bases out and also got swing votes and ended up almost equal, that doesn’t mean their bases alone were almost equal. Thus, whether run-off/revote eligibility depends on first-time attempts or is the same as for the first time, or even adds people who registered too late for the first time but now are eligible (as when registration must be n days before any given election), is a political question affecting outcomes. It also adds administrative issues for people who voted the first time by affidavit or absentee ballot whose votes weren’t counted: now those votes will have to be recorded (not counted), the recording difficult when an affidavit is often issued because someone is claiming the wrong district by mistake, and their registration couldn’t be found.

How long to count: We know an outcome from machines in about an hour. That’s in a candidate’s office, not the board of elections; boards may need another couple of hours plus travel for documents. Machine totals are available within about 1-15 minutes after a pollsite closes; candidates’ representatives call in their numbers for both sides to campaign HQ; HQ adds them up for all the pollsites they hear about, probably the ones where they sent watchers, which are probably the ones where they expected to be stronger and had more at stake to protect. A campaign manager may analyze those numbers to see how they distribute across strong and weak districts to predict the outcome, which is why some candidates concede surprisingly early.

Randomizing ballots: Machines can do that, but we may not trust the machines. Paper can’t be random without much more expense. With minor party candidates, randomly permuting 6 names means 6!=720 different ballots for 1 district. There may not be that number of voters needing paper, or even not needing paper. And when you permute for office A and permute for office B, the two permutations must be unrelated to each other to maintain randomness (lest voters be disproportionately influenced by patterns of proximity of names). If so, and if we have 6 offices open at once, a 6×6 grid means (6!)^6=139,314,069,504,000,000 blank forms in 1 district. Can I have the printing contract? I think NYC uses rotation within 1 office to simulate randomness and ignores office adjacency; a 6×6 grid then only needs 6 blanks. The public probably accepts that.

Old-style voting machines (mechanical, etc.) were susceptible to uneven quality and wear: somewhere I heard or read of a manufacturer decades ago offering “nigger machines”, which were more likely to break down and which could be assigned to unfavored districts. It’s not necessary for a manufacturer to be in on that scheme; a board can send whichever old machines are worse to unfavored districts. In NYC, it is no longer necessary for a voter to wait hours until a machine is fixed, because a paper ballot can be issued immediately and entered into the system, but their drawback is that they get counted only if machine totals in a race are close enough for paper, affidavit, and absentee ballots to matter and in a NYC machine booth a voter can change their mind before reopening the curtain but on a paper etc. ballot that’s not allowed (ballots with crossouts are disqualified).

Recounts in New York City usually result in a wider gap than the original final counts, according to a rumor. I have not confirmed this. (An original final count is final unless the top two vote-getters came so close that the government or a candidate invokes a recount.) If this rumor is true, it’s contrary to mathematical probability; it may also be contrary to political probability, because while one party vastly overwhelms the other in numbers of voters most of the important races are intraparty races in primary elections. That leaves another possibility: institutional reputational protection for the Board of Elections (which, although governmental, is run by the 2 parties), because if recounts usually result in widening of gaps then there isn’t likely to be a demand for a recount unless the original final result was very close indeed, and avoiding recounts allows a belief that the original final numbers are good, thus enhancing the reputations of elections officials and the system.

Intentional errors are probably plentiful. We’re talking about politics. People who want to help candidates win sometimes get jobs at polls and counting votes. I don’t know what screening goes on, but I suspect most of it is about which politically-connected person referred an applicant. They usually know better than to proclaim that they’re biasing the counts. The parties watch each other, which helps, but that also pushes some manipulations beneath the surface where both parties can practice their craft, unseen. Often counters likely believe they’re counting honestly, but nonetheless are slipping subconsciously.

Large-number counting has no inherent error, but humans do, and they do all the counting. Machines that count are proxies for humans: built and read by humans. Paper money counting machines in ATMs doubtless make mistakes, too; they only have to make much fewer than people do. (The notion of inherency of error was apparently posited by a sociologist, who studies people, not by a mathematician.) The voting public wants people to certify whatever the machines say, as a check, so humans are going to be in on the counting, and even the best-intentioned humans have psychological, physical, and budgetary limits.

On accounting and physics: The double-entry accounting system that requires that every amount appear twice has a special type of account that allows amounts to differ for the instance where, say, you have about a thousand dollars but balancing your books shows a one-cent error. That error is too small to be considered “material” (usually 3% or more for 1 account), and you don’t have to spend $10,000 on management and bookkeepers’ labor trying to track the penny down. Accountants have a way to enter that missing penny on one side only. It’ll balance the books and survive an audit.

“Election officials . . . are not allowed to be linked to a particular party. They are usually officers of the local council.” (Richard.) Who constitutes the council? Apolitical people? How’d you accomplish that?

The weather problem is misstated (by Mike Schiraldi). As stated, it doesn’t affect counting accuracy, but it really does, because some staff might not come in, burdening others, who get tired and miscount. (And Mike is perhaps describing voters’ choices as not reflecting an entire community’s preference, but that’s more than a weather problem.)

Get-out-the-vote (GOTV) efforts, driving voters to polls (per derf’s comment): Where is that illegal? Some things are, but not that, to my knowledge.

I wish the new machines had been made with well-understood firmware and open-source software only. I think lots of programmers would have lept at the chance. But black boxes are easier to lie about or force to lie (e.g., by versions of the GIGO method). No point in writing the source code if no board will install it.


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