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Case study of the escalation rule
for the
2009 New Mexico post-election audit law
Paul Stokes
United Voters of New Mexico
23 July 2009
Introduction
The 2009 New Mexico post-election audit bill includes an escalation rule that was developed as a compromise to meet objections of lawmakers and other stakeholders, and yet provide some measure of protection in the event that errors that might overturn an election were observed in the audit. In brief, escalation to a second, equal-sized sample would occur if the normalized errors in the initial audit sample exceeded 90 percent of the unofficial winning margin expressed as a fraction of the vote total. If the sum of the normalized errors in the two samples exceeded 90% of the unofficial winning margin, then a full recount would be required. This escalation rule was intended to provide, at the least, a high probability of a full recount when the number of uniformly-distributed random errors was large enough to overturn the election, and yet not result in full recounts very often when the number of errors would not result in overturning the election. It was clear that if just enough errors to overturn an election were concentrated in a small number of precincts, the probability of escalation would not be very high, but there wasnt time in the rush of the legislative process to analyze the severity of this problem.
The New Mexico post-election audit uses probability-proportional-to-size sampling to account for larger potential voting shifts in larger precincts. The law assumes a maximum within-precinct-miscount of 15% (30% margin shift), and the sample size is chosen to assure that the probability is at least 90% that errors that would overturn an election would result in one or more precincts in the sample having errors.
This paper is intended to provide bounds on the probability of escalating to a full recount in the event that there were enough errors to overturn an election, subject to the assumption that the within-precinct-miscount was no greater than 15%. The effects of varying the number of errors present and of unequal precinct sizes are also discussed.
The numerical results in this paper are primarily for a specific case, intended to approximate the situation in New Mexico, as far as that can be accomplished when making the assumption that all precincts are of equal size - an assumption that was necessary to carry out the calculations with the tools at hand. It would be desirable to consider other cases, such as a range of precinct sizes, unequal precinct sizes, and other escalation rules.
Summary
For a race won by an unofficial winning margin of 1%, this paper shows that with an error threshold in an audit sample of 90% of the unofficial winning margin, escalation to a full recount in the following situations would occur with the probabilities given:
If votes were switched randomly and uniformly in favor of the putative winner by exactly the amount of the unofficial margin, the probability of escalation to a full recount would be 0.9762.
If 15% of the votes were switched in favor of the putative winner in a small number of precincts to obtain exactly the amount of switching in the total population that matched the unofficial margin (i.e., 1%), the probability of escalation to a full recount would only be 0.5183.
However, the presumption that errors or fraud would exactly match the unofficial winning margin is not the only case that should be considered. For example, it seems useful to know that
If 15% of the votes were switched in favor of the putative winner in a small number of precincts to shift the margin by 2% (i.e., the putative winner actually lost by 1%) in the total population, the probability of escalation to a full recount would be 0.9325.
If 15% of the votes were switched in favor of the putative winner in a small number of precincts to shift the margin by 1.5% in the total population, the probability of escalation to a full recount would be 0.8092.
Calculations show that these probabilities do not vary much for unofficial winning margins other than 1% when the number of votes switched equals the unofficial winning margin. I speculate that the probabilities are also similar for the cases where the number of votes switched exceeds the unofficial winning margin.
Analytical Model
Two extreme situations are analyzed to provide numerical results for the probabilities of escalation to a full recount given that a number of votes are switched to the unofficial winner of the contest from the unofficial runner-up, resulting in the actual runner-up being credited with a winning number of votes. The two extremes are 1) uniformly-distributed random vote switching, and 2) vote switching of a number of votes equal to a defined within precinct miscount in a small number of precincts that would overturn the election.
Analysis
The example chosen for the analysis is the case where the unofficial winning margin is 1%. The same calculations could be used for any other unofficial winning margin, but the case using a 1% margin should be illustrative of other cases as well.
In order to perform the calculations in this paper, it was necessary to use the simplifying assumption that all precincts are of equal size. The qualitative effects of this assumption on the numerical results are discussed.
The numerical values used for the various parameters that influence the probability of escalation represent those that apply to the New Mexico situation, except as otherwise noted. These values are:
Approximate number of precincts in a congressional district = 533
Approximate number of voters in a congressional district = 300,000
Within precinct miscount = 15%
Probability of election-overturning errors in the audit sample = 0.9
Uniformly-distributed random vote switching
A contest with a 1% unofficial winning margin requires a sample size of 62 precincts to have a 0.9 probability of having errors in the sample if the number of errors in the population would overturn the election and they were concentrated in as few precincts as possible.
Margin shift in the population to overturn the election = 0.01(300,000) = 3000
The number of votes per precinct = 300,000/533 = 563
The number of votes in a sample = 62(563) = 34906
The threshold for escalation is 0.9(0.01)(34906) = 314
The hypergeometric probability of having more than 314 errors in the sample and thus escalating to the second sample if the number of errors in the population is 1% = 0.9773
The hypergeometric probability of having more than 628 errors in the first and second samples
and thus escalating to a full recount = 0.9989
The probability of escalating to a full recount = probability of escalating to a second sample times the probability that the sum of the errors in the first and second sample exceeds the threshold = 0.9773(0.9989) = 0.9762.
Vote switching of 15% in a small number of precincts, with a total margin shift of 1% in the population
As in the above example, the sample would consist of 62 precincts.
The number of votes in each precinct is 300,000/533 = 563.
The margin shift in each precinct with errors = 2(0.15)(563) = 169 votes.
The threshold for escalating to the second sample is 0.9(0.01)(62)(563) = 314 votes.
Thus, at least two of these precincts would have to be in the sample to escalate to the second sample, and a total of four or more would have to be in both samples to escalate to a full recount.
All of the events that would not result in escalating to a full recount are:
No precincts with errors in the first sample, probability = 0.1 (by definition)
One precinct with errors in the first sample, probability = 0.2553
Two precincts with errors in the first sample and no precincts with errors in the second sample, probability = 0.2909(0.1004) = 0.0291
Two precincts with errors in the first sample and one precinct with errors in the second sample, probability = 0.2909(0.2529) = 0.0736
Three precincts with errors in first sample, no precincts with errors in second sample, probability = 0.2042(0.1163) = 0.0237
Thus, the probability of escalating to a full recount = 1 - (0.1 + 0.2553 + 0.0291 + 0.0736 + 0.0237) = 0.5183.
So the chance of escalating in this case is barely better than even -- not very satisfying.
However, on further thought, the situation may not be as bad as that, depending on how conservative one wishes to be. In particular, it does not seem very likely that errors or fraud would exactly match the unofficial winning margin. If there were fewer errors, the election wouldnt be overturned, so that occurrence would not be a problem in terms of electing the wrong person. But suppose the unofficial margin were 1%, and votes were switched to shift the margin by 2%; i.e., the putative winner actually lost by 1%.
Then, the probability of escalating to a full recount = 1 - (0.0099 + 0.0506 + 0.0008 + 0.0047 + 0.0015) = 0.9325.
If votes were switched to shift the margin by 1.5%, the probability of escalating to a full recount = 1 - (0.0324 + 0.1220 + 0.0057 + 0.0232 + 0.0075) = 0.8092.
It has been suggested that vote switching in all precincts by an amount just less than the 90% threshold of the unofficial winning margin at which escalation would occur, except in a few precincts in which vote switching would occur in large amounts, could overcome the true winning margin while significantly reducing the probability of escalating to a full recount. A problem with this technique, if it were to be used to conduct fraud, is that the perpetrator would not know in advance what the unofficial winning margin will be. If the perpetrator guesses too high, escalation would be almost certain.
Consider, for example, a race that polls say is 50-50, which would have error bounds of roughly 52-48 either way. Such a poll would then provide little help in knowing what the unofficial margin would be within those error bounds.
It also seems unlikely that anyone intent on conducting fraud would want to expose such an attempt to almost certain discovery by election officials or the public observing the unlikely regularity of errors always being about the same magnitude in most or all of the audit sample. Indeed, the premise of the auditing concept discussed in this paper is that detecting at least one precinct with errors would indicate the possibility that enough errors might be present to overturn the election and that further action is warranted.
Thus, I have not considered this technique any further in evaluating the probability of escalation.
A way to significantly increase the probability of escalating to a full recount would be to reduce the error threshold so that a single precinct in a sample would cause escalation. For the example above, this would require the error threshold to be less than 169 votes. To accomplish that, the threshold of the error rate in the sample for escalation would need to be reduced from 90% of the unofficial winning margin to about 45% of the unofficial winning margin. Then, the probability of escalating to a full recount would be one minus the probability of having no precincts with errors in the first sample, or having one precinct with errors in the first sample and no precincts with errors in the second sample. The result would be 1 - [0.1 + (0.2553)(0.0868)] = 0.8778.
The effect of unequal sized precincts
The above results are for cases when all precincts are equal in size. When they are unequal in size (essentially always, of course, although if batches of equal size were used instead of precincts, they might be equal in size), the probabilities of escalation are increased. The amount of increase would be difficult to calculate for the general case, but a specific extreme example illustrates why the probabilities of escalation would be increased.
Suppose a jurisdiction had one precinct with 10,000 votes, and the remaining 380 precincts all had 500 votes for a total of 200,000 votes. The unofficial winning margin is 1% or 2000 votes. But 1000 votes were switched in the precinct with 10,000 votes, to account for all of the 2000 vote unofficial winning margin. The 1% margin calls for 62 precincts to be sampled. Using probability-proportional-to-size sampling, the precinct with 10,000 votes is selected with probability approximately equal to 1 - (1 - 10,000/200,000)**62 = 0.96, and the other 61 precincts in the sample have 500 votes. Thus, there is a 0.96 probability that the number of votes in the sample is 10,000 + 61(500) = 40,500 votes. In that event, the 2000 vote margin shift in the large precinct divided by 40,500 votes is 0.049, much higher than the unofficial winning margin of 1%. Thus, according to the New Mexico rule, another sample would be taken (silly in this case), adding no more errors. The sum of the errors in the two samples would be 2000 out of a total number of votes in the sample of 71,500. 2000/71,500 = 0.028, well in excess of the unofficial winning margin, so a full recount would be required. The probability of that full recount is 0.96, the probability that the large precinct was selected in the initial sample. So in this extreme example, the probability of escalation to a full recount is much higher than for the cases for equal sized precincts.
In general, the reason that the probability of escalation is increased when precinct sizes are unequal is that the number of votes that can credibly be switched in a larger than average precinct using the within precinct miscount rationale is proportionately larger, and the probability that the larger precincts are in the sample is similarly proportionately larger when using probability-proportional-to-size sampling.
Other non-quantifiable effects of the post-election audit
An important reason for conducting post-election audits is to establish some measure of the reliability of the vote tabulators, whether or not errors made by the machines would have overturned elections. Errors revealed by the audit are expected to lead to follow-up by election officials to determine the cause of the errors and to make corrections.
Thus, just because there were not enough errors to lead to a full recount does not mean that the audit results would not lead to remedial action. For example, one or more precincts with any switched votes from one candidate to another would be expected to result in an investigation, and 15% switched votes could be expected to result in in a public outcry for action, not to mention candidate concern.
The post-election audit is only conducted for a few key races, and the kinds of errors revealed in an audit might also affect other races. Therefore, candidates, the media, or the public may press for further investigation, depending on the nature of the errors. Moreover, if the vote tabulators were to be used in subsequent elections, there would be pressure from candidates, the media, and the public to correct any problems that had been revealed.
Some lawmakers objected to an earlier provision allowing the election auditor to target suspicious precincts, because this would not be objective, and could be used to cause mischief. Some stakeholders believe that the audit is to identify problems, but correcting the problems should be left to recount provisions in the election code wherein candidates may request a recount.
15% was chosen based on voting results in Bernalillo County (Albuquerque) using nearest neighbor analysis.
The New Mexico post-election audit requires 73 precincts in the sample because of the granularity of the table of unofficial margins and corresponding sample sizes required in the law and because that sample size was required for the statewide case. For simplicity, the same sample size is used for congressional districts.
For comparison, the probability of escalating to a full recount when the unofficial winning margin is 0.5% is 0.4384. For an unofficial winning margin of 2%, the probability is 0.5347.
Private conversation with Mark Lindeman
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