The Palm Beach County Vote

Reallocation and statistical counting methods

IMPORTANT NOTE

Due to new information (i.e., we found the correct statute that prohibits the counting double voted ballots), this paper is NOT yet self consistent. 

Conclusion: this is recent and self-consistent and has some useful arguments and links to statistical papers. 
Appendix 3: is useful for case law.

Hopefully, this web page will be useful for the case in front of Judge Sauls, and for Jesse Jackson's efforts.

I have pages of names of people who were disenfranchised by the Palm Beach Ballot.

A note about this page from Arnold Barnett, George Eastman Professor of Management Science, Sloan School of Management, MIT

Thanks for sending the piece, which I've read and found interesting. Let me comment quickly on the statistical framework you laid out.

If one believes that:

(i) the double votes in Palm Beach County should be counted

(ii) each double-vote was intended to be for one candidate (as opposed to expressing equal preference for the two candidates chosen)

(iii) it is permissible to use statistical approximations to estimate how the double votes would have been distributed across candidates

then I think you present a reasonable and lucid discussion of how one might proceed.

No method, of course, is perfect. It could be, for example, that those who double-voted are different in some salient respect (e.g. age) from those who voted only once. Thus, one cannot be sure that the intended voting pattern of double-voters actually mirrors that of other voters. Moreover, inferring a voter's party preference based on votes other than for President is not always a straightforward business.

However, so long as the starting premises of a calculation are clearly stated at the outset (and it is done correctly), there are no logical problems with performing it. If the calculation is completed under different assumptions, one gets a sense of the sensitivity of the outcome to the starting premises. Such an approach to making approximations is used in statistics all the time: I recently used three different starting assumptions to estimate the death toll in an airport runway collision.

Good luck.

Executive summary

• Voting machine ballots with multiple punches are legal ballots under Florida law. There is no statute that disqualifies such ballots. Section 101.011(4) does not apply to voting machine ballots (it applies only to manual paper ballots)
• Yet, despite being legal ballots, none of the 19,120 multi-punched ballots were counted in Palm Beach County
• This is a sufficient number of ballots in question as to impact the results of the election and opens the door for a court to determine a proper remedy, if any.
• Whether or not the ballot was "illegal" as some have argued is debatable. Whether the ballot was confusing is certain. But most importantly, the ballot was strongly biased against one candidate. Due to a poorly designed ballot, voters were 5 times more confused than "normal" (i.e., on other parts of the same ballot). In fact, two different statistical analyses done below (and by others) show definitively that that the additional confusion  above the normal level was primarily from voters who tried to vote for Gore (see Appendix 2).
• Ballots should be fair. They should not be biased for or against one or more candidates. If there is a situation that can affect the outcome of an election, there is a wrong and it deserves a remedy. There were actually at least three wrongs here, and possibly four:
  • The ballot was abnormally confusing
  • The ballot was biased heavily against one candidate (Gore)
  • Ballot with multiple punches are legal, but have not been counted
  • The ballot design itself may have been illegal.
• There is a right to vote and we have a duty to count all votes and try to capture voter intent whenever possible. A court should choose an allocation scheme that best reflects voter intent under the circumstances.
• Using statistical techniques, we can very accurately infer the intent of these "multiply punched" ballots in aggregate, which is, in the end, the only measure that matters in an election. Statistical allocation techniques are proven in many court cases and have not been successfully challenged (cases cited below). While there is a margin of error for statistical techniques, there is also a margin of error for machine counting and, many Republicans would argue, for hand counting. Properly applied, statistical estimation techniques are as accurate as other counting techniques, and, in some cases, such as severe ballot mutilation, statistical techniques may be the only accurate counting technique available.
• Ten different allocation techniques are explained in a table further down this page, including when the technique is applicable, and the advantages and disadvantages of the technique.
• A court should pick a remedy which best fits the observed facts, statistics, and constraints (e.g., time constraints or lack of data constraints). Here are a few examples:

• If the Florida courts decline to hear a challenge to the ballot counting, they are, defacto, choosing the "ignore" remedy. This would be the worst possible allocation scheme as it, in effect, disenfranchises voters. Ignoring the ballot entirely is ONLY an appropriate remedy for illegally cast ballots or ballots where no voter intent whatsoever is indicated. For any ballot from a legitimate voter, where there is any indication of an attempt to make a choice (one or more dimples, chads, etc), statistical estimation techniques are always superior.  For example, if voted ballots are so mutilated that voter intent cannot be ascertained at all (e.g., all holes are punched), it is always better to allocate such ballots on the same ratio as recorded for normally tabulated (single vote) ballots (the "simple ratio" technique), than it is to throw out such ballots since the former better captures voter intent. 

  For example, suppose only County PB uses chad ballots and suppose that 1 out of every 10 ballots is defective due to a manufacturing glitch. When the ballots with the defective chad were counted by a voting machine, all the defective chads come out of the ballot. The appropriate remedy is to allocate such ballots proportionally, rather than not count these ballots at all since this would best reflect the will of the voters. Statistically, even though we cannot determine such a fully punched ballot on an individual basis, we can extremely accurately determine the will of the voters in aggregate. In fact, our statistical count would, on average, exactly match a machine or hand count of those same ballots before the chads fell out (within normal margins of error for machine counts and hand counts, i.e., a fraction of 1%). 

• The results of using the best allocation techniques are consistent. Using completely different approaches to estimating the vote, one based on a visual analysis of the graphical design of the ballot ("biased") and a second approach looking at actual overpunch statistics ("punch ratio"), you arrive at results which are very similar (we didn't expect these results to be identical since the "punch ratio" technique is a much better estimator for unknown ballots than a qualitative inspection).
• Using the most accurate statistical allocation techniques, it can be accurately estimated with a high degree of confidence that Gore would pick up well over 10,000 votes relative to Bush (see Appendix 2D), enough votes to create a convincing margin of victory so large as to render all other legal contests in Florida moot. 
• Other independent statistical analyses, using entirely different methods (both here and in other papers), predict similar results.
• Therefore, if the Florida Supreme court were to choose a statistical allocation of these ballots, which we believe is the fairest way to infer voter intent, then immediately upon rendering that decision (and subject only to challenge by the US Supreme Court), Gore will have convincingly won the popular vote in Florida. This leaves no ambiguity as to who should receive the electors. Furthermore, the law does not say that the court must choose a remedy that has no margin of error. The important matter immediately at hand is to determine which candidate won the popular vote in Florida, not work forever to determine the most accurate possible count.
• Finally, once again we emphasize that no matter which allocation method is finally chosen, it is important that we choose a method other than throwing out the ballots. For example, suppose that half the voting machines in Palm Beach County had a "defect" that caused a Buchanan chad to fall out whenever a Gore chad was punched. Which remedy would be most fair?
  • Discard all those double punched ballots
  • Use a statistical allocation technique (such as the "excess confusion ratio"  technique described below) that can recover the originally expressed voter intent (in aggregate) with a margin of error of less than 1%, while not taking unfair advantage of other counties that have not counted overvoted ballots

Summary of the argument

According to numbers reported in the media and confirmed through phone conversations with election officials in Palm Beach, 19,120 ballots were not counted in Palm Beach because, due to a confusing design and a variety other reasons, they were punched for more than one candidate. These ballots were set aside and never counted even once.

Because this election is so close, these ballots, if properly counted, may change the results of this election. Many independent scholarly analyses such as:

• http://www.ssc.wisc.edu/~bhansen/vote/vote.html
• "A Simple Analysis of Discarded Votes by Precinct in Palm Beach," by Peter Orszag and Jonathan Orszag, November 10, 2000 www.sbgo.com/Papers/Election/ANALYSIS OF DISCARDED VOTES BY PRECINCT IN PALM BEACH.pdf
• http://pages.about.com/bgspence/index.html (has a list of more links)

all argue that these ballots did not affect all voters equally. One paper (the first one cited in the bullet list above) estimates are that Gore should pick up at least 4,270 votes relative to Bush if these votes are properly counted, and perhaps as many as 17,710 votes. Our own calculations show that Gore should get at least 4,800 votes relative to Bush (if the ballot were completely unbiased). However, since all the data indicates that the ballots was strongly biased, our lowest estimate of the number of net votes Gore would gain is10,682 votes (this number assumes that we only want to count ballots over the normal confusion number (see Appendix 2D)), 13,957 votes (estimating the "punch ratio" value and counting all the overvoted ballots). And he might get as many as 16,380 votes relative to Bush (using the "biased" method which tends to overestimate and includes all the ballots). We also confirmed that the ballots that were misvoted were mostly voters trying to vote for Gore. This is because the "punch ratio" calculation yielded a result that is numerically very similar to the "biased" calculation in which we assumed that only Gore voters were confused and also because the ratio of misvotes involving Gore were 5 times the misvotes that involve Bush. Other papers using different techniques confirm our results that the ballots that were set aside were primarily Gore votes. 

In most contests, overvoting is randomly distributed among all voters and among all counties at a very small level (approximately 1% according to the ACLU brief at www.aclu.org/court/fladell.pdf). That means that counting these overvotes is likely to yield votes in the same proportion to the votes actually cast, and thus unlikely to impact the outcome of the election. In other words, the total number of counted ballots will increase by 1% in every county, but the ratios between all the candidates is unlikely to change. This explains why, in virtually all counties and contests, these votes may be set aside and not counted; because they are unlikely to change the outcome. Everyone is treated fairly under the normal circumstances even though those votes are not counted.

But the Palm Beach County ballot was different. It was not equally confusing to all voters (see Appendix 1). The analyzed overvoted ballots indicate that Gore voters were far more likely to overvote than Bush voters. Furthermore, the magnitude of the overvoting in the President section was 4 times higher than in a normal election (as noted in the ACLU brief www.aclu.org/court/fladell.pdf), and 5 times higher than other parts (the Senate portion) of the same ballot, according to statistics provided to the public from Palm Beach.

Thus, this is unlike the normal overvoting situation because a widely acknowledged design problem with this particular ballot led to a situation in which:

• voters for Gore were 5 times more likely to overvote than voters for Bush ("the wrong")
• the magnitude of the effect was significantly higher than normal, i.e., 4.15% of the ballots instead of less than 1% ("the second wrong")

To not count these ballots would be disenfranchising voters, it would be unfair relative to other voters, and it would be contrary to the directive of the Florida Supreme Court. 

While some people might argue that these votes don't deserve to be counted because the voter did not comply with the instructions ("vote for group"), this is not what the law says. We cannot change the laws in the middle of the game just because some people do not like the outcome.

The law says if there is a wrong (an unfortunate design error that was not caught in advance by either party), the court may provide "any relief appropriate under such circumstances." 

These ballots are not illegal. The law does not say that these ballots must be set aside. There were a number of ways they could have become overvoted ranging from handling, tampering, voter confusion, etc. However, the source of the confusion does not matter at this point. Nor does it matter who approved that ballot, or from what party that person was. What matters is how we can best capture the intent of these ballots because these ballots may likely change the outcome.

There are a number of ways to accurately determine the intent of these ballots in aggregate. We list the methods these votes can be tabulated and expect that a court would choose the method which yields the most accurate capture of voter intent. 

We believe that one of the statistically fairest, and completely non-partisan method, is to allocate multiply punched ballots relative to the proportion of votes for those punched candidates in the county where the ballot came from ("punch ratio" technique). An even better way is to use the "excess confusion ratio" technique in the table below.

Let's look at the "punch ratio" technique. Assuming we have 100 ballots double punched for candidate A and B, and we know that the ratio of votes of candidate A to candidate B in that county is X, then we should allocate those ballots in the same ratio X. In fact, a simple experiment proves that this is the most accurate technique if we are trying to count every overvoted ballot.  For example, if we are given a randomly selected set of 1,000 ballots that we single punched for either A or B and we were not allowed to look at each ballot, and we had to allocated them with only the knowledge that they were either voted for A or B, then using the county % method, we would find that we would yield a statistically computed count that, on average, exactly matches the actual machine and hand count of those same ballots!

Under Florida law, a court may consider the list of available remedies and choose the remedy which best captures the intent of the voters. We show that ignoring these ballots is actually the worst possible remedy because it captures no voter intent whatsoever. Yet this is the remedy that a court would, in effect, choose should that court choose to not hear a case that contains the arguments presented here.

What the law says

Where there is a right, there must be a remedy.

We should focus on the right to vote and strive to count all votes (Florida Supreme Court/Harris)

We should not cast a ballot for a candidate that the voter has not indicated. This argument has been used in deciding other cases.

Equally important is what the law does NOT say. The law does NOT say that:

• Overvoted ballots are illegal. 
• Overvoted ballots must not be counted
• You must divine voter intent on a "ballot by ballot basis" (as opposed to "in the aggregate")

Because we have a contested election, and because these uncounted ballots (set-aside overvoted ballots) in Palm Beach can make a difference, it is reasonable to bring them in and try to ascertain voter intent as is allowed under florida law 102.168: http://www.leg.state.fl.us/statutes/index.cfm?App_mode=Display_Statute&Search_String=&URL=Ch0102/SEC168.HTM&Title=->2000->Ch0102->Section%20168

Grounds for contesting an election include (among others)

"(c) Receipt of a number of illegal votes or rejection of a number of legal votes sufficient to change or place in doubt the result of the election." or

"(e) Any other cause or allegation which, if sustained, would show that a person other than the successful candidate was the person duly nominated or elected to the office in question or that the outcome of the election on a question submitted by referendum was contrary to the result declared by the canvassing board or election board."

The relief which may be granted in a contest is as follows:

"8) The circuit judge to whom the contest is presented may fashion such orders as he or she deems necessary to ensure that each allegation in the complaint is investigated, examined, or checked, to prevent or correct any alleged wrong, and to provide any relief appropriate under such circumstances."

Indisputable fact 

In Palm Beach County, there were at least 19,120 ballots with overvotes which were not counted even once. More importantly, those ballots, if we can determine voter intent, may make a difference in the outcome of the election. Using the "excess confusion ratio" approach in Appendix 2D, we estimate that Gore will pick up 10,682 votes relative to Bush.

The "wrong": Why these ballots were overvoted
It really doesn't matter how these ballots got this way. The point is that they exist. And they exist in numbers 5 times higher than normal (e.g., relative to the senate portion of the same ballot) and Gore overvotes are 5 times more likely than Bush overvotes. There could be a number of errors that happened after the fact: ballot handling, machine counting, loose chads, ballot tampering. However, those effects would likely have equally affected voters for either candidate. It is instead almost certain that most of these overpunches were due to a confusing ballot layout. See below for a more complete list of causes. (see Appendix 1 below).

The argument

We can't know how these ballots got an extra hole, but we know that these holes were not punched in at random because the punches are not evenly distributed among all the holes. There is clearly "voter intent" in these ballots that we can very accurately capture using a statistical remedy. It would be a travesty of our system of democracy if we did not at least try to capture the will of the people. I have statements from hundreds of Palm Beach voters who called in to say they were confused by the ballot.

We owe each individual the "right" to vote and to have that vote counted.

We know:

• overvoted ballots exist. It does not matter that 5th graders, after the fact, could figure it out. That does not change the facts. It is universally accepted by multiple independent sources that the ballot was abnormally confusing (see Appendix 1)
• it isn't illegal to count these. In fact, these are not illegal ballots under the law! Under 101.011(4), it says that "If the elector marks more names than there are persons to be elected to an office, or if it is impossible to determine the elector's choice, his or her ballot shall not be counted for the office." However, 101.011(4) is only applicable to paper ballots (where a voter marks an "X" on the ballot), so that it does not apply in this case. There is no comparable statute governing the discarding of "voting machine ballots" which is the type of ballot used in Palm Beach County. This may well have been deliberate in not having such a provision since, as we have noted, it is possible to infer, with a great deal of accuracy, the intent of such voters using statistical allocations. So the general principle of count all ballots should apply and we should count to the best of our ability. It was improper to have not counted these double punched ballots in an election that is this close. While some may argue, “If you double punched a ballot you blew it.” That's nice, but that's not what the law says. We cannot change the laws in the middle of the game just because some people do not like the outcome. So it's not illegal to count these ballots. But since the level of confusion is roughly the same everywhere, it doesn't matter whether we count or don't count these since  it doesn't change the outcome or the ratios between candidates (only the absolute count). It's a different story though when we have an abnormal level of confusion and/or biased confusion. Here we have both.
• discerning what was intended by these excess overvoted ballots can change the outcome of the election 
• the courts are allowed to determine voter intent
• the courts can pick among many ways to discern voter intent and would most logically pick the remedy that is the best way to capture voter intent most accurately.
• if a court chooses to ignore a case based on these arguments, that court would de facto be picking one of the remedies listed. So that court would be saying, in effect, that the court believes that the "ignore" remedy is superior to the other remedies. We would find such a ruling astonishing in light of the many clearly and unambiguously superior methods for determining voter intent for such ballots.

Remedies that a court could pick from

Note: In the descriptions below, we only include supplemental information. The main description is included in the table above.

Re-vote
We believe that while some argue that there is sufficient law to indicate that a court could call a re-vote, that this is unlikely to happen. There are over punches and errors in all elections. No election is perfect. There is no end to a re-vote cycle if this were allowed since there would be no standard for when an election is "good enough," especially in a close race between two candidates. And there is not time enough in this particular instance for a re-vote and the challenges that might be pursued under a re-vote. More importantly, statistical techniques allow us to very accurately recover information which appears to be "obscured." So a re-vote is a wasteful exercise and may be an endless cycle. And it may be illegal.

Simple ratio
If we are sure that the ballot is unbiased with respect to any candidate, this technique is simple, fast, and easy.

Punch ratio
This is the statistically fairest, statistically most accurate, and completely non-partisan method. We allocate multiply punched ballots relative to the proportion of votes for those punched candidates based on the ratio of the normal ballots for those candidates. 

So if we have a stack of 100 ballots which are all double punched for candidate A and B, and we know that the ratio of votes of candidate A to candidate B in that county (or precinct if we are using this on a precinct level) is X, then we should allocate those ballots in the same ratio X. In fact, a simple experiment proves that this is the most accurate technique.  

For example, if we are handed a randomly selected set of 1,000 ballots that we single punched for either A or B and we were not allowed to look at each ballot, and we had to allocate them with only the knowledge that they were either voted for A or B, then using the punch ratio method, we would find that we would yield a statistically computed count that, on average, EXACTLY matches the actual machine and hand count of those same ballots!

Similarly, if we hand a court 100 ballots that were punched for Gore/Buchanan and tell the court that all we know about these ballots is just two things: (a) the voter intended that each ballot is either a vote for Gore or a vote for Buchanan, (b) the ratio of Gore to Buchanan voters in the county these ballots came from is 99 to 1, and then ask that court to allocate these ballots among the two possible candidates in the fairest possible manner, we believe that a court would allocate 99 ballots to Gore and 1 ballot to Buchanan since statistically this is the most logical outcome. Of course, the court could not possibly know which single ballot should be cast for Buchanan. That is not important. It is only important the final ratio reflect the ratio in the county (precinct) of origin as this is most likely to reflect the will of the people within that county (precinct).

Another way to view this is as follows. Suppose we have a deck of playing cards and we draw 20 cards at random from the deck, face down. We can allocate the 20 cards into 4 piles of 5 cards each and claim that the first pile accurately counts the number of Spades, the second pile accurately counts the number of Clubs, the third pile counts the number of Hearts, and the fourth pile the number of Diamonds (these might be our 4 candidates for office assuming that our voters split evenly among all 4 candidates). When we actually turn over the cards, we'll find that we misallocated the cards within the various piles. But when we add up the total vote count, we'd find that, on average, the numbers we deduced from statistics exact;u match the observed numbers. And the larger our deck of cards, and the more cards we draw in our sample, or the more times we repeat the experiment, the closer our statistical guess will come to our actual count (usually within an error of a card or two). So this simple example shows that our aggregate counts are accurate using this technique, though on an individual ballot basis, it is not accurate. In an election, only the aggregate amount is of interest.

Another advantage of this method, beyond statistical accuracy, is that it is very easy to implement. You can program a voting machine to count all the double punched ballots and tally how many had each type of double punch. Then it is a simple matter to take each type of double punch, apply the ratio for those two candidates, to determine the final allocation.

It is also relatively easy to do this process by hand. You sort ballots into stacks based on the type of double punch, e.g., a stack for 4/5 punches, a stack for 3/5 punches, etc. Then you take each stack and divide up the stack for the two candidates that are punched based on the county wide ratios observed for single punched ballots, and mark each stack. Then you combine all stacks marked for each candidate and count the stacks.

A further advantage of this approach is that it appears, based on analysis of existing lawsuits filed in Florida courts, that the winner in Florida can be determined as soon as the decision is made to select this technique by using the data we know about the double punched ballots (although in this case, we may wish to draw a larger sample size, and draw ballots at random from all precincts).

See Appendix 2A for a calculation of the Gore margin of victory if this technique is chosen.

Also, note that while this technique is accurate across a broad range of ballot problems, it is not necessarily the most perfect technique available in every case. For example, see the "biased" technique below which may be more accurate in our case.

Modified punch ratio
This is the same as the "punch ratio" technique, but adds in some common sense so that we don't count nonsense ballots and so that we can greatly simplify the amount of work we have to do while minimizing the impact on accuracy. So we do things like only consider double punched, adjacent-hole ballots.

Excess confusion ratio
This technique will work fine as long as our main candidates are uncorrelated. In other words, in this case, as long as there are relatively few multi-punched ballots with both Bush and Gore, and as long as the other candidates are mostly immaterial (1% or less), this technique works well. This is the case with Palm Beach ballots. In the correllated case, it's much more complicated. For example, suppose our voting machines are rigged so that every time we knock out a Gore chad, the machine knocks out a Buchanan chad. A suppose our voters are perfect. Then the only double punches are for Gore-Buchanan, and using our original technique, we'd allocate half to Gore and half to Buchanan. This would be a big mistake. However, this is not the case here since we are only looking the Bush/Gore numbers and these are not correlated since the number of Bush/Gore punched ballots is relatively small as a percentage of the total number of ballots.

Biased
Suppose our ballot was designed in such a way that only Gore voters would be likely to double punch or multi-punch. In that case, every single double punched ballot above the "normal" noise level of confusion should be allocated to Gore. In fact, this is probably the closest we have to the truth in this particular case based on all the evidence we cite in Appendix 1 below. 

See the results of such an approach in Appendix 2B

Dominant (allocate to the leading candidate)
In this technique, if we have a race between two leading candidates, we allocate ballots to the primary candidate when a ballot is marked for only one primary candidate and discard the ballot if it is marked for both primary candidates. 

For example, in our Palm Beach case, we'd use the following simple rules to count multi-punched ballots to help break a tie:

The advantages of this technique include:

• non-partisan
• easy to understand
• easy to explain
• easy to implement

This technique is most applicable under the following conditions:

• there is tremendous time pressure to make a decision as to which technique to use
• there is little time to perform the actual count
• there are a relatively small number of ballots cast for the next leading candidate (e.g., in Palm Beach it is 1.29%)
• the number of ballots in question is are relatively few (under 20,000) compared to the total number of ballots cast in the state
• there are only two principal candidates that matter
• our primary interest is to determine which candidate had the most votes
• we are trying to maximize our chance of capturing voter intent
• we have little faith in statistics
• we like simple rules

Random
If  a set of ballots is punched for A and B, we allocate half the ballots to candidate A and half to candidate B. This is much worse than the methods above because it is ignoring all the data we have about where the ballot came from. So we are discarding information that would help us determine voter intent. A simple test with single punched ballots (say ballots that were cast for Gore and ballots that were cast for Buchanan that were mistakenly mixed in together), shows this is not very accurate at all. However this technique is still superior to other techniques, such as ignoring the ballot entirely, since using this technique, we have exactly a 50% chance of voting for the intended candidate, whereas if we ignore the ballot, we have 0% chance of capturing the intent of the voter.

However, there is a risk of using this technique when there are more than two principal candidates that can accumulate votes. For example, suppose our candidates are A, B, and C. Suppose our machine punches A/B when we hit A, and B/C when we hit C. Then if all voters vote evenly for A and C, and we count ballots using the 50/50 technique, then B will win the election by a convincing margin even though no voter intended to cast a vote for B!

Party
We can allocate double punched ballots by choosing the punch corresponding to the party of the voter. So if we have a Gore/Buchanan double punch on a Democratic ballot (as inferred from the other votes on the ballot), we'd count that as a vote for Gore. This technique does not work well if the party of the voter doesn't correspond to any of the punches, so we'd have to select a secondary technique (such as the "allocate based on % in county" technique) to be used when the primary "allocate by party" technique fails.

Analysis
If we extensively statistically analyze the voting pattern of each individual ballot compared to all other ballots, we may develop statistical patterns that give us a clue as to the most likely intent on a given ballot for the Presidential vote, given the votes in the rest of that ballot. This is computationally difficult and is very subjective. There is not time to program or test such a remedy.

Even 
This is not equivalent to ignoring the ballot. So for example, suppose we have only 100 double punched ballot for candidate A and B only. And suppose we have 10 candidates on the ballot. We'd allocate 10 ballots to A, 10 to B, and so on. In other words, we'd assign ballots to candidates C, D, E, etc. even though these ballots were only punched for A and B. This is clearly very unfair. And it clearly violates voter intent since we are allocating ballots to a candidate that the voter definitely did not pick. 

Ignore the ballot
Ignoring the ballot is an appropriate remedy when it can be determined that a ballot cast in an election should not have been cast. For example, if counterfeit ballots were stuffed into a ballot box, those ballots should be identified an then not counted. Or if the law explicitly states that in order to be counted, a ballot must be filled out in a certain way. For a ballot from a legal voter, ignoring the ballot is the most extreme remedy available and should only be used if no holes are punched and no intent is indicated (e.g., no dimples). If the intent of a legal voter cannot be determined on an individual basis, such as multi-punch ballots, statistical techniques described above better capture voter intent than ignoring the ballot. 

More on the legal validity of statistical allocation

A court should pick the remedy that best captures voter intent, since to ignore the arguments is, DE FACTO, picking the "ignore" method which we argue is the WORST OF ALL BY FAR since it captures no voter intent whatsoever. Such a remedy (or lack of a chosen remedy) disenfranchises voters whose intent can still be accurately captured (in aggregate) even if their ballots are randomly mutilated beyond all recognition.

See Appendix 3 for more information. 

Conclusion

In light of the dismissal of the butterfly ballot cases by the Supreme Court, is there a case here?
Yes, because the in Fladell/Katz, the court ruled only that the ballot complied with the law to a sufficient degree as to not warrant voiding the election. The court did not rule as to whether the magnitude of the minor problems that had been pointed out were sufficient to enable a court to provide a remedy to ensure that the result of the election properly reflected the will of the people. 

In other words, the court ruled that the ballot was not so bad that we can't figure out who won. 

We totally agree with the court's finding and it is in fact required that this be the case for our case to have any merit whatsoever. Because if the ballot was found to be invalid and the election results nonsensical, it would be difficult to draw statistical inferences from random or extremely noisy data. However, if the data is mostly clean, as the courts have ruled, then statistical techniques are very applicable and can give us great insight as to how to correct errors so that the true will of the people is reflected in the election.

What does Florida law say?
Florida law (IX. 102.168) says:

Grounds for contesting an election include (among others)

"(3)(c) ... rejection of a number of legal votes sufficient to change or place in doubt the result of the election." or

"(3)(e) Any other cause or allegation which, if sustained, would show that a person other than the successful candidate was the person duly nominated or elected to the office in question or that the outcome of the election on a question submitted by referendum was contrary to the result declared by the canvassing board or election board."

The relief which may be granted in a contest is as follows:

"8) The circuit judge to whom the contest is presented may fashion such orders as he or she deems necessary to ensure that each allegation in the complaint is investigated, examined, or checked, to prevent or correct any alleged wrong, and to provide any relief appropriate under such circumstances."

IX 101.5614(5) says:

"No vote shall be declared invalid or void if there is a clear indication of the intent of the voter as determined by the canvassing board."

IX 101.5614(6), says:

"If an elector marks more names than there are persons to be elected to an office or if it is impossible to determine the elector's choice, the elector's ballot shall not be counted for that office, but the ballot shall not be invalidated as to those names which are properly marked."

The Florida Supreme Court in Fladell/Katz ruled  (on 12/1/2000) that "the Palm Beach County ballot does not constitute substantial noncompliance with the statutory requirements mandating the voiding of the election."

Twenty-five years ago, this Court commented that the will of the people, not a hyper-technical reliance upon statutory provisions, should be our guiding principle in election cases: 

[T]he real parties in interest here, not in the legal sense but in realistic terms, are the voters. They are possessed of the ultimate interest and it is they whom we must give primary consideration. The contestants have direct interests certainly, but the office they seek is one of high public service and of upmost importance to the people, thus subordinating their interest to that of the people. Ours is a government of, by and for the people. Our federal and state constitutions guarantee the right of the people to take an active part in the process of that government, which for most of our citizens means participation via the election process. The right to vote is the right to participate; it is also the right to speak, but more importantly the right to be heard. We must tread carefully on that right or we risk the unnecessary and unjustified muting of the public voice. By refusing to recognize an otherwise valid exercise of the right of a citizen to vote for the sake of sacred, unyielding adherence to statutory scripture, we would in effect nullify that right. 

Boardman v. Esteva, 323 So. 2d 259, 263 (Fla. 1975) (emphasis added). We consistently have adhered to the principle that the will of the people is the paramount consideration.9 Our goal today remains the same as it was a quarter of a century ago, i.e., to reach the result that reflects the will of the voters, whatever that might be.

The US Supreme Court (Thornburg v. Gingles, 478 US 30) has held that you can use statistical evidence to prove an argument. Many other courts have used statistical techniques to allocate votes including the Illinois Supreme Court and the Alaska Supreme Court (Appendix 3). Of course statistics are only estimates. We can calculate the answer only with a level of confidence (such as 99% confidence). But all systems have confidence levels including our voting systems. There were missing ballots, mutilated ballots, voters turned away from the polls, voters who were indimidated, voters who punched the wrong hole, miscounted ballots, missing chads, confusing ballots, vote counting machines that have an accuracy of only 99.9%, and so on. In short, due to these flaws, every election is only an estimate of the true will of the people. 

Was there a wrong? Or a right that could not be exercised?
Our claim is that there is abundant statistical evidence that suggests the Palm Beach ballot had some subtle design defects that, while not great enough to mandate the voiding of an election, were of a sufficient magnitude (because of the closeness of this election) as to meet the requirements of 3(e). Evidence of this includes:

• Statistical evidence by leading experts (over 20 independently authored expert papers such as Brett) have shown that it is statistically certain (99.9999% sure) that the Buchanan vote total is wrong. Another study says that we can say with 99 percent confidence that at least 1966 Buchanan ballots were erroneous. This analysis is further confirmed by Henry Brady's study which also looked at the statistics of the absentee ballots from the same county. We cannot expect the court to analyze each of these studies but it is reasonable for the court to pick a conservative estimate from the many credible papers written on the subject. Since Brady's study showed that Gore is entitled to approximately 2,000 of the miscast Buchanan votes (relative to Bush), and because this election is so close that a 2,000 vote difference can cause a different person to be elected, 102.168 3(e) is satisfied. 
• Statistical evidence has also showed that there is a high probability that because of a design problem in the Palm Beach ballot, at least 15,377 voters who voice would normally have been heard, were disenfranchised due to a hyper-technical reliance on 101.5614(6) which caused their votes not to be counted even though it can be argued that such voters attempted to follow the instructions and, from a statistical point of view, with a high degree of certainty, we can determine the will of those voters so that their voice will be heard. Based on ballot data from Palm Beach, we  statistically estimate that counting such ballots in an absolutely fair manner (as detailed below) will result in a net shift of over 10,000 votes to one of the candidates. This is more than enough votes to give this candidate a clear and convincing victory in this contest. Again, 102.168 3(e) is satisfied. 

How could this have happened?
The instructions on the voting machine say in big red letters "The maximum legal voting time is 5 minutes." With such a long ballot is it possible that people, especially first time voters, or voters who hadn't voted in a while, didn't read, or didn't feel they had time to read, the instructions to remind them of what to do and quickly tried to follow the instructions? These voters thought that they complied with the instructions and should have their vote counted if we can infer their intent.

The instructions printed on the voting machine are to "vote for the candidate(s) of your choice" and the instructions printed on the ballot for President say "vote for group." 

How a Gore voter could have mistakenly voted for Buchanan
This interpretation is supported by visual inspection of the ballot. A voter for the Democratic group could follow the "Democratic line" see an arrow to the right of that line, then punch the hole indicated by the arrow. Statistically analysis of the data indicates that this is exactly what happened.

How a Gore voter could have mistakenly punched holes for Buchanan and Gore
This interpretation is supported by visual inspection of the ballot. The instructions said "vote for group" and Electors for President and Vice President. Is it unreasonable then for us to count ballots of voters who interpreted those instructions as voting for the presidential candidate by punching the hole indicated by the arrow, and voting for the democratic party by following the democratic line and punching the hole indicated by the arrow on the opposite that line? Another interpretation is that some voters read the "Vote for group" and "Electors" language as meaning to punch once for each candidate and punched two holes, one indicated by the arrow to the right of the presidential candidate, and another by following the line under the vice-presidential candidate to the hole (with an arrow on the other side of the ballot indicating you can punch there. Inspection of the actual double punch data shows this is exactly what happened! Using the above interpretation which we can get from visual inspection, and knowing that virtually all voters voted for either Bush or Gore, and knowing we'd expect more Gore voters than Bush voters in this county, we'd expect that the three most common double punch combinations would be 4/5, 5/6, and 3/4. When we view the actual data, we find that this is exactly the case. These are, by far, the three most prevalent double punches, and occur in practice in exactly in the order of frequency we'd expect from our visual analysis.

It is certain, from a statistical analysis, that a subtle design defect, while insufficient to void the election, nevertheless caused a subtle bias which, due to the large size of Palm Beach County and the closeness of the current margin, has with great statistical certainty has resulted in errors of such magnitude as to cause the candidate that has won the contest by a convincing margin to be declared to have lost the election.

Using statistical techniques, or a simple counting technique, we can determine who the proper winner of the election should be by a definitive margin. This design defect resulted in voters who voted for the wrong candidate or punched two holes instead of one. Both of these problems, while insufficient to cause the election results to be nullified, resulted in disenfranchisement of a small, but for this election extremely significant, percentage of voters (approximately 4%) who were trying to comply with the instructions.

Should a remedy be ordered?
Since 102.168 (3)(e) is satisfied, under 102.168 (8), the court may choose a remedy or ignore this issue. And the court has broad discretion as to the remedy.

These ballots were cast by real registered voters who were trying to express their intent and trying to follow the instructions. They have a right to have their vote count (that is the "wrong" in this case that deserves a remedy). These votes have not been counted, even once. Furthermore, people who intended to vote for Gore have had their votes miscounted for Buchanan.

The key question for the Florida Supreme court is this: 

If you do not count these double punched ballots, are you 99% certain that we are electing the candidate that the voters intended to elect?

Our analysis indicates the opposite: if you ignore these ballots (and other similar valid contests that can produce a shift in the balance of power of comparable magnitude as this case), it is 99% certain that the wrong candidate will win. Even worse, we can state confidently based on statistical analysis, that the candidate that should have won by a convincing margin, will have lost the election.

IX 101.5614(6), says:

"If an elector marks more names than there are persons to be elected to an office or if it is impossible to determine the elector's choice, the elector's ballot shall not be counted for that office, but the ballot shall not be invalidated as to those names which are properly marked."

A strict reading of this statute suggests that we are not allowed to count a double punched ballot. But this statute must be interpreted in the context of other statements in the same statute, such as " if it is impossible to determine the elector's choice," and a "clear indication of the intent of the voter," as well as the principles of the Florida Supreme Court, "the will of the people, not a hyper-technical reliance upon statutory provisions, should be our guiding principle in election cases" and "The right to vote is the right to participate; it is also the right to speak, but more importantly the right to be heard." Hence, 101.5614(6) is in conflict with these principles in cases where we can accurately infer the meaning in such overvoted ballots. 

For example, suppose our ballot had a subtle design defect that affected a small portion of the people and caused them to interpret a ballot that indicated to them that in order to vote for candidate G, they must punch out 2 holes. And suppose that the only ballots that had double punches had this combination. And suppose further that we had affadavits from such voters that this is what they saw when they voted and they were confused and they punched twice. Shall we disenfranchise those voters for their confusion if we can accurately infer their intent? We see nowhere in the law that either the legislature or the Courts intend to disenfranchise a voter who is confused. Indeed, it seems the whole purpose of the statute 101.5614(6) is to not count a vote unless we can determine who to cast the vote for. And we have the overriding directive from the Supreme Court that is to be used in election cases:

• "the will of the people, not a hyper-technical reliance upon statutory provisions, should be our guiding principle in election cases" 
• "The right to vote is the right to participate; it is also the right to speak, but more importantly the right to be heard.

Because the Florida Supreme Court has ruled that the Palm Beach ballot is fair, that means that all types of voters were likely to be confused and double punch. Essentially, they said the ballot wasn't so overly biased as to throw out the election. But they didn't say it was unbiased either. So that means that the normal voter statistics should apply to these ballots. In such a case we can very accurately determine voter intent in the aggregate on such double punched ballots. For example, if the Gore/Buchanan ratio of voters in single punched votes is 99 to 1, and we have 100 double-punch Gore-Buchanan ballots, we know with 99% confidence that 99 voters voted for Gore and 1 voted for Buchanan. So is it better to not count such ballots? Or will we better reflect the will of the people if we count them statistically?

Furthermore, we absolutely know from inspection of the data and affidavits of registered voters, that a large number of voters have had their votes tabulated for the candidate which they did not intend to vote for. Yet we can estimate the amount of error with extremely high confidence using statistical techniques. And furthermore, using the same unbiased, non-partisan statistical techniques, we can determine where those votes should have gone so that the over-riding principle of the will of the people is reflected in the outcome of the election.

Virtually every piece of credible statistical analysis of the data indicates that Gore won the popular vote. On December 2, the Miami Herald also published an analysis that they had commissioned that showed that under a fair, full and accurate count, Gore won Florida by 23,000. And even under the most absurd assumption (that 100% of the 185,000 discarded ballots did not mean to vote in the election), Gore still wins. So no matter how you cut the data or what assumptions you make, Gore should win.

So if our job is to reflect the will of the people, and we have a choice to make:

• Choice A (not providing a remedy/ignoring these ballots): Has ZERO chance of reflecting the will of those voter who double punched or had their votes counted for Buchanan and has 99% chance of electing the wrong candidate.
• Choice B (using a technique to accurately count the intent of double punched ballots or using a statistical technique to estimate the proper vote adjustment): Has 99% chance of reflecting the will of the voter.

Which choice would we make?

What should the remedy be?
We are not asking the court to void this election. We are asking the court to adhere to its guiding principle of ensuring that the election results reflect the true will of the people. And right now, the evidence is overwhelming that this is not the case. It is virtually certain that we have errors of such magnitude as to cause the candidate that has won the contest by a convincing margin to be declared to have lost the election.

We are asking that a simple, straightforward, non-partisan remedy be ordered to correct for the design defects and give a more true and accurate reflection of the will of the people who voted in Palm Beach in this election:

These ballots were not randomly punched. These were legal votes. We just have trouble knowing how best to count them. But now that it's down to just two candidates, and that the Florida Supreme Court has ruled that the ballot is fair, it is much more clear how to do this (see the counting technique below). A court's job is to infer intent of these voters and preserve their right to have their vote counted.

Using statistics, we can do an excellent job at inferring intent. We hope that a court would evaluate each of the possible remedies listed above and choose the remedy which best reflects voter intent for these ballots. 

• For the Buchanan votes, an available remedy under 102.168 (8) is to allocate those votes to the statistically proper recipient as determined by analyzing the data. For example, if all credible technical papers indicate that 99% should go to one candidate and 1% to the other, we suggest that the court invoke such a remedy. Or a more conservative approach would be to look at several papers and choose the paper which minimizes the shift in the "balance of the votes" between the candidates. For example, if we have 3 credible papers and the papers suggest the follow allocations most fit the data, 99/1, 90/10, 80/20, then the court should pick the 80/20 allocation since this will be closer to the truth than the current allocation, but it will minimize the shift in the balance of the votes.
• For the 19,120 overvotes, we suggest the following method:
  • Because the ballot has been ruled to be a fair ballot, we can use any "fair" counting technique to count these ballots. 
  • Because only two candidates matter at this point, we need only figure out how to fairly allocate these ballots between these two candidates. But we only allocate the ballot to a candidate if and only if the ballot is clearly punched for the candidate. And because the votes were virtually all for one the two major candidates, the most obvious fair counting technique (based on the instructions presented to the voter) is this:

  Ballot marks	Action
  Bush and Gore only	Do not count
  Bush, but not Gore, and optionally one other adjacent punch	Count for Bush
  Gore, but not Bush, and optionally one other adjacent punch	Count for Gore
  Neither Bush or Gore	Do not count

  • After getting vote totals, multiply each candidates total by .975 to account for the other candidates in the race (so that we have not overcounted for our leading candidates).
  • Because of the lack of time, the fastest way to perform this count is by (a) hand count, (b) partial hand count of a random sample with a statistical extrapolation, or (c) machine count using the above algorithm
  • The argument that you might count a ballot for a candidate that the voter did not intend is specious for two reasons: (1) since it's a fair ballot, voters who intended to vote for a minority candidate will, on average, help each of the two leading candidates equally under this counting technique, (2) not counting that vote for the minority candidate that the voter intended doesn't matter since that candidates has lost. There is no harm to that candidate, there is no wrong (especially at the tiny numbers we are talking about here for minority candidates (less than 200 votes in the election).
  • The argument that this is unfair to other counties which did not count double-punched ballots is specious because we already allow some counties to hand count hanging and pregnant and dimpled chads, while other counties did not.
  • Optionally, you may choose to count either: (a) all multi-punched ballots, (b) only double punched ballots, or (c) only double-punched ballots where the punches are adjacent to each other.

This remedy will produce a greater confidence that the result is correct than not applying the remedy. In other words, if we are really concerned that the voice of the people be heard, and we are told that it is virtually certain that we have made a mistake, shouldn't we fix that mistake?

A more accurate (but perhaps less acceptable politically) solution for the situation in Palm Beach County, is in the table at the top of this web page (row #5, as calculated in Appendix 2D). A statistical best-case/worst-case analysis of these multi-punch ballots would tell us that Al Gore should receive at least 10,000 votes relative to Bush.

Some people have told me that because of the gravity of the situation, the Supreme Court would never make such a bold move. We could argue that the other way. Because of the gravity of the situation, the Supreme Court must make a bold move to ensure that the will of the people is preserved and the right candidate is elected president. Had this been a normal election, no one would contest the "fair counting technique" in the table above. It is simple, logical, non-partisan, and can be done by a machine.

Appendix 1: The ballot was confusing

The ACLU has filed an extensive brief on the 19,120 ballots in Palm Beach that were set aside and not counted..

We argue that the ballot was abnormally (5 times more than the Senate part and at least 4 times more than the normal 1% confusion amount) confusing to people who wanted to vote for Gore for these reasons:

• confusing layout: if you focus on reading the text on the left, the second arrow you see points to the Buchanan hole and the line above democratic points to the Buchanan hole
• confusing order: the second hole is not the hole for gore
• confusing language: it says "vote for Group", not "vote for one" (however, this is as specified in Florida Law)

These confusions are validated by the following observations:

• The double punches are not equally distributed among all combinations, but mostly on two combinations and over 87% involved hole 5 (Gore) while only 16% involved hole 3 (Bush). Does that sound like a fair ballot? In other words, people trying to vote for Gore were 5 times more likely to multi-punch than people trying to vote for Bush. This is the most telling and objective measure: the lack of randomness shows these ballots were not punched at random, which means we can infer voter intent. In fact, you can even precisely infer voter intent on ballots which have been completely mutilated if those ballots were randomly selected from the set of ballots.
• This additional level of confusion (5 times the Bush voter level) is not because Gore voters are more senile or more confused. On the Senate portion of the same ballot, the number of multi-punched ballots was reduced by that same factor of 5. Hence, the Bush and Gore voters tended to make mistakes at the same rate on unconfusing parts of the ballot. The logical conclusion is that the ballot itself caused the extreme confusion.
• The single biggest double punch was 4&5, ie., someone confused by the ballot who voted for Buchanan, then realized their error and voted for Gore. The confusion we most expected to see, the confusion that most people complained about, was in fact, the error that occurred most often
• the number of double punches for President was 5 times the "normal" amount (e.g., the number of double punches in the Senate part of the ballot for the same voters). In Palm Beach, some 19,120 county ballots for the presidential race were tossed out before they were counted because more than one candidate was picked. Only 3,783 voters made that mistake on the U.S. Senate portion of the ballot, a ratio of over 5 to 1.
• When CNN asked people on the street in NYC what they thought, everyone said "it was confusing" (although everyone got the vote correct)
• There was an abnormally high number of votes for Buchanan in Palm Beach that a large number of scholars believe was due to the confusing nature of the ballot
• There are many affidavits of people who were confused by the ballot and had voted for the wrong candidate
• Exit polls indicated that Gore won the state
• Several ballots were found that had a #1 hole punched. That's impossible to do in the voting machine and could indicate ballot tampering or "loose chads".
• numerical analyses of the data in this paper as well as in other papers all confirm that the ballot design made it significantly more difficult to vote for Gore without double punching
• the only county in Florida with a "butterfly" ballot is Palm Beach.

Furthermore, these additional observations are true only for the Palm Beach ballots, and not for any other county in Florida:

• the satires (the squiggly line ballot)
• the Apple Computer full page ads, e.g., USA Today, Nov 17, p 7A: "Good design isn't a luxury"
• appeared on the cover of Time magazine (Nov 20, 2000)
• the extremely large number of academic papers written on the subject 
• the scatter and bar charts (see The Palm Beach balloting)
• the extremely high percentage of "double punched" ballots in palm beach relative to other counties (and relative to other parts of the same ballot)
• there are many affidavits of people who were confused by the ballot and had voted for the wrong candidate

The fact that people, AFTER THE FACT, people in surveys got it right isn't surprising... 96% of the Palm Beach voters got it right. So if you ask a class of 40 5th graders, it's not that surprising that they all got it right...the sample size is too small and the venue hardly approaches a realistic voting booth where you aren't asked to do just a single task that you can really focus on.

The fact it was approved by a democrat doesn't make it a fair ballot. People make mistakes. Republicans do too. We should be blind to which party designed and approved the ballot. It matters only that the ballot was clearly confusing.

Double punching indicates a voter either made a mistake and tried to correct it by punching the proper hole, or the voter was confused by "vote for group" instruction.

Bottom line: The preponderance of the evidence indicates that the ballot was abnormally confusing, 5 times more confusing than the Senate portion of the same ballot, for example.

Appendix 2A: Approximate calculation of Gore gain using "punch ratio" allocation

Suppose we use the multi-punch data reported in: http://pages.about.com/bgspence/index.html and in www.sbgo.com/Papers/Election/Overvotes in Four Palm Beach Precincts.xls and include all multi-punched ballots since the statistical techniques are equally accurate even if every hole is punched on the ballot!

We compute the Bush/Gore vote by knowing that Gore gains 277 votes for every 1000 Bush or Gore ballots counted in Palm Beach ((270-153)/(270+153)*1000)

In our analysis below, we simplify things somewhat and rather than compute a ratio for each candidate, since other candidates only got around 2% of the vote (see official Palm Beach Results), we just focus on Bush/Gore. A more accurate analysis would treat each multi-punch type separately and allocate those punches among the punched candidates based on the percentages observed in the counted ballot. See the "modified punch ratio" technique in the table for more information.

This implies that for every 1,000 multi-punched ballot we count in Palm Beach, Gore picks up 730 votes relative to Bush. So for 19,120 ballots, we'd estimate that if these ballots were statistically analyzed and statistically counted, it would mean a net Gore gain of 13,957 votes. 

We'd get a more accurate estimate if we fully did all the calculations and used data based on all the multi-punched ballots, and not just a 1% sample.

Our result compares quite favorably with the 16,380 net Gore votes we calculated using an entirely different approach in Appendix 2B below in which we didn't use individual multi-punch ballot data at all. In that analysis, we just assumed that all voter over-punching, beyond the normal level, was for people voting for Gore.

The similarity of these two numbers leads us to conclude that the vast majority all of the over-punching of ballots (above the normal level) were voters trying to vote for Gore.

Because the numbers computed using the two techniques ("punch ratio" and "biased") are similar, it further suggests that virtually all the bias was against Gore.

Appendix 2B: Calculation of Gore gain using "biased" allocation

We can make a quick estimate of the result in this case (ignoring other candidates for simplicity). From analysis of the multi-punched ballots analyzed, voters were 5 times more confused votes on the Presidental selection than on the Senate section (19,120/3,783). So subtracting, we get 19,120-3,783=15,337 net Gore votes. If we assume equal confusion among the 3,783 votes (the normal confusion level), then we'd allocate 3,783 among Gore and Bush on the same ratio as the vote in Palm Beach, namely, (269,754/152,964)=1.76. So we'd expect 3783/2.76=1370 Bush votes and 2413 Gore votes for a net gain of 1043 Gore votes on the normal confusion part. Added to the 15,337 abnormal confusion caused by the butterfly ballot design, we'd get 16,380 net gain for Gore, which is somewhat higher than our estimate using the actual ballot statistics using the "punch ratio" approach (Appendix 2A).

Appendix 2C: Calculation of Gore gain using "simple ratio" allocation

If the ballot were just abnormally confusing and unbiased with respect to candidate, we'd expect that the confused ballots should occur in exactly the same proportion as the votes for candidates as a whole. 

For every 1,000 ballots cast in Palm Beach, Gore will gain (269-153)/462*1000 = 251 relative to Bush

So in 19,120 unbiased confused (or tampered or defective) ballots, we'd expect Gore to gain 19,120*.251=4,800

Appendix 2D: Calculation of Gore gain under an "excess confusion ratio" allocation

Since in a normal election, we leave set aside the normally multi-punched ballots as we argued earlier, we can do the same here if we want to be as "fair" as possible.

So we have (overpunch in Presidential portion)-(overpunch in Senate portion)=19,120 - 3,783=15,337 abnormally high overpunches.

Now, since we know that Gore voters were 5.59 times more likely to overpunch than Bush voters (from the table in Appendix 2A we have 123/22=5.59)

So therefore, Gore picks up 4.59/6.59 * 15337=10,682 votes relative to Bush.

We can get a more accurate result if we had used  the overpunch stats for 500 randomly picked ballots (the current stats are from a small number of precincts).

Hence, the court might rule that 10,682 is the number Gore picks up relative to Bush, unless a new number, using better data, can be computed using the same technique as we used here by a certain set date. That way, the resolution is guaranteed by a date.

Appendix 3: Re-allocation legal argument background (case law)

It would be wrong to ignore these ballots since these ballots can decisively tip the election. The US Supreme Court (Thornburg v. Gingles, 478 US 30) has held that you can use statistical evidence to prove an argument (statistical best-case/worst-case arguments from Dr. Grofman that whites and blacks vote cohesively was accepted even though we cannot prove this via direct measurement). We should use the statistics of those ballots to determine what they are telling us based on all the evidence available. The court is allowed to do that under Florida law. 

As the court in Bakken , 329 N.W.2d at 580, eloquently put it, "[t]he very heart of our form of government depends upon the legal and moral principle that each valid vote should be counted."

Voters in Palm Beach maintain that they have been disfranchised because they reasonably misunderstood the ballot, the ballot did not itself comply with law, and election officials acted improperly. 

Assuming that a court agrees that there was disfranchisement in Palm Beach, it is necessary to determine what remedy is appropriate. Guidance in selecting a remedy was provided in at least three respects by the Florida Supreme Court in Palm Beach County Canvassing Board v. Harris. First, the Court made clear that the right to vote is the right to participate, the right to speak, and, most importantly, the right to be heard. The Court repeatedly emphasized that the overarching principles of interpretation to be applied to election law matters included the recognition that all political power rests in the hands of the people and the election laws should be interpreted to ensure that the intent of the voters is ascertained and respected. Second, the Court made clear that in unusual situations such as the present one, it is appropriate to use equitable powers to fashion a remedy that serves to effect the right of the voters to be heard. Third, the Court held that in fashioning an equitable remedy, practical considerations and constraints (such as the date of the meeting of the Electoral College) may be taken into account. 

A court confronting the disfranchisement claims from Palm Beach voters could order a new vote, throw out all ballots in dispute, hold a hearing as to each ballot in dispute and then allocate the disputed votes one-by-one, or allocate the disputed ballots using some general criterion, such as proportion of votes throughout the county, voter affiliation, etc. 

The first option, a new vote, is presently being sought in lawsuits filed by voters in Palm Beach. This option is time-consuming and is highly unlikely given the current time frame. 

The second option would be to throw out disputed ballots in Palm Beach. This option would directly violate the overarching principle that the right to vote is the right to be heard. If the disputed ballots are all discarded, then this right is ignored completely with respect to those voters whose ballots are ignored. This option contradicts the entire premise of Palm Beach County Canvassing Board v. Harris. 

The third option is to allocate the disputed ballots based on a case-by-case determination of voter intent. This option has the advantage of allowing for a determination of individual voter intent, but would be more time consuming than a new vote. 

The fourth option is to allocate the votes using some general formula, such as proportion of votes throughout the county or party affiliation. This option has the advantage of respecting the right of Palm Beach voters to be heard, but has the disadvantage of presenting a risk of imprecision. For example, if party affiliation is used, it is possible that a Democrat who wanted to vote for Buchanan will be counted as voting for Gore. Nevertheless, this option is practical in light of time constraints and some amount of imprecision is surely preferable to the denial of all voter voices (option 2).

Our brief research on the matter revealed no Florida decisions directly on point. There are, however, other courts that have dealt with allocation issues. For example, an Illinois appellate court held in 1998 that illegal votes should be allocated to the candidates in the same proportion that each candidate received votes in the precincts where the illegal votes were cast. See In re The Purported Election of Bill Durkin, 299 Ill. App. 3d 192; 700 N.E.2d 1089 (Ill. App. 2d Dist.). The court recognized that the Illinois Supreme Court had approved of both party affiliation (Talbott v. Thompson, 350 Ill. 86, 182 N.E. 784 (1932)) and the proportion method (Thornton v. Gardner, 30 Ill. 2d 234, 195 N.E.2d 723 (1964)) as ways of allocating illegal votes. The court in Durkin affirmed the trial court's decision not to use the party affiliation method because there was an independent candidate who could lose votes under that method. 

Although there is a difference between the allocation of illegal votes and the allocation of disputed votes, these allocative acts are similar in that they risk imprecision. Nevertheless, the Illinois courts, upon whose election law jurisprudence the Florida Supreme Court rested in its decision in Palm Beach Canvassing Board v . Harris, consider allocation using general criteria to be appropriate in ensuring that the integrity of the voting process is maintained.

In any event, it would be wrong to ignore these ballots since these ballots can decisively tip the election. The US Supreme Court (Thornburg v. Gingles, 478 US 30) has held that you can use statistical evidence to prove an argument (statistical best-case/worst-case arguments from Dr. Grofman that whites and blacks vote cohesively was accepted even though we cannot prove this via direct measurement). We should use the statistics of those ballots to determine what they are telling us based on all the evidence available. The court is allowed to do that under Florida law. 

There is a variety of case law to support statistical techniques and these decisions have survived intact a broad range of challenges by a variety of experts. 

Here are the cases from ANDRE FLADELL, ET AL. vs. PALM BEACH COUNTY CANVASSING BOARD, ETC., ET AL. filed with the Florida Supreme Court on November 28, 2000. Here is a link to that case (see section VI):

Can a court better decide the intent of these voters than casting these votes aside? We believe the answer is an emphatic "yes!" According to the ACLU, there is a variety of case law to support statistical techniques and these decisions have survived intact a broad range of challenges by a variety of experts. And extensive list of such cases can be found in ANDRE FLADELL, ET AL. vs. PALM BEACH COUNTY CANVASSING BOARD, ETC., ET AL. filed with the Florida Supreme Court on November 28, 2000. Here is an excerpt from the Fladell case (see section VI).

Courts have often held that where votes are affected by election officials misconduct or other illegality - - such as the clear noncompliance alleged by plaintiffs - - one remedial option available is to statistically adjust the election totals based on examination of the overall election returns. In Curry v. Baker, 802 F.2d 1302, 1318 (11th Cir. 1986) the Eleventh Circuit held that the district court properly used expert statistical and survey testimony in the voting election contest " . . . as the most reliable evidence available to protect the fairness of the election process . . .”)

Similarly, in Hammond v. Hickel, 588 P.2d 256 (Alaska 1978), the Supreme Court of Alaska endorsed just such a procedure in its discussion of how lower courts should deal with election contests. The Court explained that the determination of whether electoral irregularities could have changed the election result depends on whether the irregularities introduced "bias" into the system. Id. at 260. The Court explained:

If the bias has tended to favor one candidate over another and the number of votes affected by the malconduct can be ascertained with precision, all such votes will be awarded to the disfavored candidate to determine if the results of the election would be changed. If the number of votes affected by the bias cannot be ascertained with precision, a new election may be ordered, depending upon the nature of the bias and the margin of votes separating the candidates. Boucher v. Bomhoff, 495 P. 2d 77 (Alaska 1972). Where the malconduct has not injected any bias into the vote, but instead affects individual votes in a random fashion, those votes should be either counted or disregarded, if they can be identified, and the results tabulated accordingly. Finally, if the malconduct has a random impact on votes and those votes cannot be precisely identified, we hold that the contaminated votes must be deducted from the vote totals of each candidate in proportion to the votes received by each candidate in the precinct or district where the contaminated votes were cast. (citing cases).

Similarly, if a specified number of votes should have been counted but are no longer available for counting, they should be added to the vote totals of each candidate in proportion to the votes received by the candidate in the precinct or district in which the votes would otherwise be counted. Id. Thus, the Court recognized that, depending on the facts of each case, a court may choose between awarding disputed votes to the disfavored candidate altogether, scheduling a special election, or making a proportional adjustment of the vote totals. Regarding the latter "pro rata" approach, the Court added that if the local election returns were such "as to render this method unsuitable based on a statistical approach," the court should adjust the vote totals based on the statewide ratio of the candidates’ votes. Id. The Court added that "[t]he remedy of a re-election may be required under certain circumstances." Id., citing Finkelstein & Robbins, “Mathematical Probability in Election Challenges”, 73 Columbia Law Review, 241 (1973) (describing how statistical formulas can assist courts faced with close election challenges). See also, Krauss, “Analyze This: A Physicist on Applied Politics,” The New York Times, November 21, 2000.

Nor is this flexible approach allowing statistical readjustment of vote totals limited to Eleventh Circuit and Alaska law. See, e.g., Thornton v. Gardner, 195 N.E. 2d 723, 724 (Ill. 1964)(deducting illegal votes cast from each side in proportion to the total vote); Grounds v. Lowe, 193 P.2d 447, 453 (Ariz. 1948)(same); Russell v. McDowell, 23 P. 183, 184 (Cal. 1980)(making findings as to some disputed votes based on testimony, and adjusting the remaining votes on a pro rata basis); see also Singletary v. Kelley, 51 Cal. Rptr. 682, 683 (Cal. 1st DCA 1966)(deducting illegal votes from each candidate in proportion to the total vote).

Courts in other states have used a more intuitive, informal mathematical approach in other election contest cases to determine whether the outcome was reasonably in doubt. See, e.g., Ippolito v. Power, 241 N.E. 2d 232, 294 (N.Y. 1968)(ordering new election where there were 101 suspect votes and declared winner had only 17-vote margin out of 2827 total votes cast); Santucci v. Power, 252 N.E. 2d 128 (N.Y. 169) (same, where suspect votes totaled 650 and margin of victory was only 95 votes). While these cases often involve simple calculations of ratios based on total vote, courts have used more sophisticated statistical techniques to discern what adjustment to the vote would be appropriate. See, e.g., Cellar v. Larkin, 335 N.Y.S. 2d 791 (Sup. Ct.) aff'd mem, 288 N.E. 2d 135 (N.Y. 1972)(in congressional election challenge, court relied on statistical probability analysis to show that the number of suspect votes would have to be over 2.5 times as large to create even a one-in-a-thousand probability of changing the outcome). In LaCaze, supra, even the dissenting justice noted that one alternate remedy (to a re-vote or special election) available to the Court would have been to do a statistical analysis of the results and conclude that a sufficient number of the missing votes would have been cast for the challenger to make him the winner. 310 So.2d at 87-88.

The significance of the above cited authority is that the hands of the judiciary are not tied in fashioning remedies due correct election irregularities - - in fact legal precedent indicates that judicial involvement in rectifying such illegalities was expressly envisioned.

Be sure to read the Conclusion section above for a continuation of the legal argument and a compelling legal argument as to why these ballots must be counted, and a fair and easy way to machine count these ballots without using statistical techniques.

Background reading
official Palm Beach Results

"A Simple Analysis of Discarded Votes by Precinct in Palm Beach," by Peter Orszag and Jonathan Orszag, November 10, 2000 www.sbgo.com/Papers/Election/ANALYSIS OF DISCARDED VOTES BY PRECINCT IN PALM BEACH.pdf

http://pages.about.com/bgspence/index.html (has the multi-punch data and a really good list of more links)

Large number of academic papers written on the subject 

A partisan dissenting analysis comes to a totally different theory. But it doesn't fit the observed multi-punch facts, so it's a cute hypothesis that doesn't fit the facts (he fits the aggregate data, but ignores the statistics of the multi-punch cards that were tabulated). In fact, it never (a) refers to the actual muli-punched cards which were analyzed and (b) explains how his theory could possibly fit the observed facts when the ballots themselves were analyzed. I'd love to see his version of the calculation I did in Appendix 2A above which makes NO assumptions whatsoever and just interprets the ballots as they stand.

Scatter and bar charts (see The Palm Beach balloting)

ACLU brief on the Fladell case (www.aclu.org/court/fladell.pdf)

Fladell case 

Palm Beach County Vote- Reallocation and statistical counting (supplemental arguments)

Palm Beach ballot: Was it legal?

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