Counting Ballots

One often hears that the one-person-one-vote principle must prevail. Presumably, the reason is that it is supposed to assure a final tally consistent with voters' choices.

The assumption implicit in the one-person-one-vote principle is that including more ballots in the count increases the accuracy of the final proportion of votes assigned to each candidate. This is true for one unbiased counting process. However, actual counting processes usually are biased even if they are impartial. Combining the results of two impartial counting processes having significantly different error rates can reduce accuracy relative to using only the results from the process with the lowest error rate. For example, reasonable assumptions imply that adding the results of a hand count of machine rejected ballots to the results of a machine count reduces accuracy.

In all likelihood, applying the one-person-one-vote rule rigidly will work against what it was designed to achieve.

What could the one-person-one-vote principle have been intended to achieve? Probably, a result that is consistent with voter's choices. If so, then the appropriate goal should be to count ballots only if doing so increases the accuracy of the final proportion of votes assigned to each candidate. Counting all ballots is not usually consistent with this goal.

Here is an example (albeit extreme) that proves the point. Suppose an election is held to choose one of two candidates and:

¨ 6,000,000 people vote.

¨ Machines count 5,999,000 of the ballots.

¨ People count 1,000 of the ballots.

¨ The proportion of the votes for each candidate is the same for the machine and people counted ballots.

¨ Machines make errors at the rate of one per trillion ballots.

¨ People make errors at the rate of one per two ballots.

For all practical purposes, the machine count will be completely accurate. The error in the counted proportion of the votes for each candidate is far less for the machine count taken alone than for a combined machine-people count. The people count just adds noise. It is best not to include the people count in the results.

The previous example overstates the accuracy of machines relative to people. The next example probably understates it. If so, the truth is more discouraging than portrayed.

¨ 450,000 people vote.

¨ Machines count 405,000 of the ballots.

¨ People count 45,000 of the ballots.

¨ The proportion of the votes for each candidate is the same for the machine and people counted ballots.

¨ Machines make errors at the rate of one in a thousand.

¨ People make errors at the rate of one in a hundred.

¨ The true vote is 65/35 for candidate A over candidate B.

The machine determined proportion of votes for each candidate has a typical error (root mean square error, for the techies) of 0.000608. The corresponding figure for the people count is 0.00607 (898% greater than for the machine count). The figure for the combined count is 0.00115 (89% greater than for the machine count alone).

An unbiased counting process is one where votes assigned in error are assigned to each candidate with a probability equal to his true proportion of the vote. For example, suppose Candidates A and B received 65% and 35% of the votes, respectively. Then when an unbiased counting process makes an error, the probabilities of Candidates A and B being assigned the vote are 65% and 35%, respectively.

Unbiased counting processes are desirable. However, for all practical purposes, they do not exist. For example, as shown below, an honest count generally is biased.

An impartial ballot counting process is one that is blind to the candidates. Counting errors occur at the same rate for each candidate and each candidate has an equal chance that the vote will be assigned to him.

According to this definition, a typical machine or honest person can be considered an impartial counter.

Impartial counting processes are feasible, at least approximately. However, as shown below, they are virtually always biased.

Any counting process assigns a proportion of the ballots to each candidate. Counting errors cause the assigned proportion to differ from the true proportions (the will of the voters). A counting process is unbiased if the expected error in the assigned proportions is zero. It is biased if the expected error in the assigned proportions is not zero. Bias is defined as the expected error in the assigned proportions.

Consider an impartial machine that is so poorly functioning that it errs on every ballot. Then it assigns about 50% of the ballots to candidate A and the other 50% to candidate B. There are four possible outcomes.

¨ A ballot marked for candidate A is assigned to candidate A.

¨ A ballot marked for candidate B is assigned to candidate B.

¨ A ballot marked for candidate A is assigned to candidate B.

¨ A ballot marked for candidate B is assigned to candidate A.

If each candidate truly has 50% of the votes, then these outcomes are equally likely. Consequently, the expected number of net votes received by candidate A due to machine error is 0, and the counting process is unbiased. Nevertheless, the likelihood that the true vote in any election will be perfectly equally divided among the candidates is essentially zero.

If candidate A truly has 100% of the votes, then these outcomes are not equally likely. The second and fourth outcomes are impossible. The first and third outcomes each occur about 50% of the time. Each candidate will be assigned about 50% of the votes. Candidate A, who received 100% of the true votes will tally about 50% of them. Candidate B, who received 0% of the true votes also will tally about 50% of them. The bias is 50% in favor of candidate B.

The second example illustrates a fundamental fact. To the extent an impartial machine makes mistakes, the expected result is bias in favor of the underdog. A person can be thought of as an unusually error prone machine. Thus, impartial hand counts are particularly biased when the true vote favors one candidate.

Other things equal, an impartial counting process's bias is large to the extent one candidate is favored over another. Thus, a recount of ballots already counted should, on average, show a net gain for the underdog if the recounting process has a higher error rate than the original counting process and both counting processes are impartial. Arguably, hand recounts of ballots already counted by machine that increase the winner's margin are evidence against impartiality.
To the extent that voters truly favor one candidate over another, it is likely that this kind of bias will overwhelm other sources of impartial counting error. In the second example, the bias in the machine count proportion is 0.000600 in favor of candidate B and that in the people count proportion is 0.00600 in favor of candidate B. The bias in the machine-person count is 0.001140, about 90% greater than for the machine count alone. All three bias figures are almost 100% of the corresponding total typical errors.

Impartial counting process errors tend to cluster around the bias. The error standard deviation is a measure of the typical difference between the proportionate error and the proportionate bias. In the second example, the error standard deviations for the machine count, person count, and machine-person count are 0.0000993, 0.000938, and 0.000130, respectively. The machine-person count figure is about 31% greater than for the machine count.

The total typical error due to both the bias and the clustering of errors around the bias is measured by the root-mean-square error, or RMS for short. This is the measure of counting process error used in the second example.

All impartial recounts simply replace one error series with another. This makes it possible to convert a loss to a win. Thus, the underdog benefits from any impartial recount, complete or partial. Impartial counting of representative ballots not previously counted also helps the underdog, for the same reason.

The higher the recounting process's error rate, the more the underdog is helped. Suppose the recounting process's error rate is higher than that for the original counting process. Then the bias in favor of the underdog is increased. The typical cumulative error also is increased. A recounting process with the highest feasible error rate maximizes the probability of a net gain for the underdog large enough to win. Replacing a machine count with a hand count is an example of this.

Think of the proportion of the ballots assigned to the underdog after a count as a stock price. Then the underdog's situation is akin to holding an out of the money call option that has just expired. This call option is worthless. An impartial recount will result in a new proportion for the underdog, either higher or lower. This change is like a stock price change. Allowing the recount is like extending the call option's life. It adds value. Allowing recounts is equivalent to forcing the candidate who is ahead to give a free call option to the underdog.

The proportion of all uncounted ballots representing votes for the underdog should be about the same as for the counted ballots. However, it is easy to find localities where uncounted ballots particularly favor the underdog. Counting only or predominantly these ballots introduces unusually large bias; hence is particularly likely to convert a loss to a win. This practice is equivalent to an error process that is not blind to the candidates, hence does not satisfy the criteria for impartiality.