Note: This is a final rewrite and combination of two essays that were written before and after the presidential election of 2000, the closest major election in history.
We have just experienced the closest election in history. Al Gore is ahead in the popular vote by approximately 0.35%. Bush's "certified" victory in Florida by 537 votes represents a margin of only 0.008%. It is the mathematical equivalent of flipping a coin and having it land on its edge. From a math junkie perspective, a famous mathematical paradox became reality. And, it ain't pretty.
What follows are some facts of this election, and some interesting mathematical perspective of elections in general.
A few years ago, we in Arizona were electing a Governor. Just before election time, three major polls were released to predict a potential winner. Two of them showed the Democrat candidate winning by a landslide, the third showed the Republican candidate winning by a landslide. Between the three polls there was a swing of close to 20% difference.
The Republican won. Why? Because even though the state of Arizona is about 50:50 party registration (actually we are 43% Rep:42% Dem:15% Lib/Ref/Ind/Other). Voter turnout was weak, Republicans showed up, Democrats did not. Only one of the political polls correctly predicted the outcome, because it was the only poll based on most likely voters, the rest were just a sampling of adults. Since then I have decided that the majority of political polls are of little value.
Those that vote in an election determine the outcome, this poll is the only one that will count.
The US Constitution prescribes a way to elect the President of the US:
The executive power shall be vested in a President of the United States of America. He shall hold his office during the term of four years, and, together with the Vice President, chosen for the same term, be elected, as follows:
Each state shall appoint, in such manner as the Legislature thereof may direct, a number of electors, equal to the whole number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or person holding an office of trust or profit under the United States, shall be appointed an elector. [...]The electors shall meet in their respective states and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves; they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President, ... and transmit sealed to the ... President of the Senate [who] shall, ... open all the certificates and the votes shall then be counted;--the person having the greatest number of votes for President, shall be the President, if such number be a majority of the whole number of electors appointed... (Article 1, Section 1 and Ammendment 12, italics are mine)
Basically, what this says is that the voters do not decide who will be President, this body of electors, chosen from each state, decides who the next President will be. We merely vote on the electors themselves. This system was designed so that elections would be run by the states themselves, and not by the federal government.
The problem is that this system makes it mathematically possible for one to get the most votes and lose the election. How? If a candidate wins a state with 51% of the vote, he gets 100% of the electors for that state not just some. George W Bush will win in Texas by a significant margin. Texas is the second largest state by population, and thus has lots of electoral votes. But, if Bush loses enough smaller states by close margins, he could lose enough electoral votes to lose the election, but the close margins could be made up by his significant win in Texas. Al Gore could face the same fate winning New York (the third largest state) significantly, but losing the small states. The winner will be the candidate with the most electoral votes, not the one with the most actual votes.
This is exactly what happened in the election of 2000. Gore actually won more popular votes than Bush, but Bush took more states than Gore. Since each state no matter how small is guaranteed 3 electoral votes, the difference favored Bush.
This is the fourth time it has happened in American history. (1824 - Andrew Jackson wins popular vote, but lost electoral vote to John Quincy Adams; 1876 - Samuel Tilden wins popular vote but lost electoral vote to Rutherford B. Hayes; 1888 - Grover Cleveland wins popular vote but lost electoral vote to Benjamin Harrison.)
From a mathematical perspective, weird outcomes like the electoral college scenario happen all the time. An election can be said to be fair as long as the manner in which the vote will be taken and outcome decided is agreed upon ahead of time. Both Bush and Gore know that the electoral vote is more important than the popular vote, otherwise they would be campaigning everywhere instead of just in a few close states.
If there is no agreed upon manner for a decisive outcome, it is possible for there to be multiple outcomes, especially with more than two candidates. This is called the "election paradox".
For example, lets look at the 1992 Presidential race. There were three major candidates George Bush Sr., Bill Clinton, and Ross Perot. Clinton easily won the electoral college, and he also won the popular vote (but not by a 50% majority). What if we picked the President like Yugoslavia, and the President HAD to have 50% or more of the popular vote? Bush and Clinton would have been forced into a runoff. Perot got 11% of the vote. Exit polls that year show that given a choice between Clinton or Bush, most Perot voters would have picked Bush. A runoff election probably would have gone to Bush, and Clinton would have been sent back to Arkansas.
Don't feel sorry for Bush Sr. just yet, it is possible to imagine scenarios in which even Perot would have won. What if we handled three way elections like a "round robin" sports competition. Perot being the centrist candidate would have won in a two way against either Bush or Clinton. If there were three seperate votes (Bush vs. Clinton, Bush vs. Perot, and Clinton vs. Perot), the final outcome would have been Perot (2-0), Bush (1-1), and Clinton (0-2). Perot would have gone to Washington simply by being everyone's second favorite candidate.
The most important life lesson of the "election paradox" has to do with statistical interpretation. Given raw scientific data, it may be possible to manipulate the data to say anything you want it to say. The multiple scenario outcomes of a three-way Presidential election proves that. You just have to present the data in the right way. Just like elections, the only valid statistical interpretation is one in which the way the data is interpreted is agreed upon prior to collecting the data.
Update (Post Election November 2000): Then there is the whole Palm Beach voting fiasco which may mean a difference of as many as 19,000 (currently uncounted) votes. Such fuzziness of election results exist during every election, rarely does it make a difference. In this, the closest election in American history, it just may make a difference.
Mathematically, it should not make a difference at all. A replacement election in one county to decide the matter would mathematically invalidate the whole presidential election by virtue of the "election paradox" principle. The "butterfly" ballot that Palm Beach County was going to use was published ahead of time, it was approved by election officials in both parties, there were no pre-election objections to the format, and there are no signs of election fraud. Read my definition of the election paradox again. The contrapositive states: If there is an agreed upon manner for a decisive outcome, then multiple outcomes cannot happen. The use of these ballots was agreed upon before hand, thus whatever the outcome of these ballots, including legally throwing out the votes of people who made mistakes, must be accepted. To throw out the whole election because of confusion over the ballot would be just as bad as holding a runoff election back in 1992. Heck, we may as well just have the candidates flip a coin to decide a winner.
My opinion: I would rather trust the people who were smart enough to apply for absentee ballots and dedicated enough go through the trouble of mailing them in to decide the next President, than leaving in the hands of those people too stupid to figure out how to fill out a "butterfly" ballot and not dedicated enough to ask for help when they got confused.
A Mathematically Tied Election
Before the election we were inundated by polls. Everyday one of the candidates had a very small lead over the other, and most polls showed that the race was basically even when we took into account margins of error. A random sample of say 1000 Americans has approximately a 4 to 5% margin of error.
What the recount in Florida taught us, is that even an election has a margin of error. Using real numbers there were 5,941,639 legitimate ballots cast in the presidential race according to the first accounting. The sum of all changed votes, both positive and negative, during the recount was 4,297. Dividing the low number by the high number and multiplying by 100 equals a margin of error of 0.07%. The difference between Bush and Gore in the original count was 1784 votes which is a gap of 0.03%. Thus the difference is within this margin of error. Mathematically, we have a tie. On pointing this out in the update of the last essay, I got this message from a reader:
If I follow your line of argument regarding the statistics of the election in Florida correctly, this means there is no conclusive outcome. Because the margin of error in the counting process is greater than the margin of difference between the candidates, from a mathematical perspective, out of 50 recounts some would come out in favor of Bush, some in favor of Gore. Then which do we choose retrospectively - back to the election paradox.
In statistical theory, when the margin of difference is less than the margin of error, are the results simply declared inconclusive? That doesn't seem to be a viable option at the moment for this situation, because neither the media nor the candidates view the election process in this way, as a collection of statistical data. - Philip von Maltzahn
Mathematically, the results are inconclusive. There are enough problems with the Florida vote count that anyone could have ended up the winner. Obviously, calling Florida a "tie" would not be politically acceptable to anyone. Hence, the endless parade of lawsuits.
The Florida county with the biggest change from first count to recount is, to no one's surprise, Palm Beach County with a change of 892 votes, a net gain of 682 in favor of Gore. With 422,683 votes cast for either Bush or Gore (we will dismiss the disputed 3,412 votes for Buchanan), that is a margin of error of 0.21%. Right now Gore leads Bush in the nationwide popular vote by a mere 0.22%, so a margin of error of 0.21% is rather significant.
On December 8th, the Florida Supreme Court once again made the mathematical mistake of changing the way the state of Florida counts ballots after the election. They deemed necessary the inclusion of partial hand recounts in the election returns. This has never been done before, and is contrary to pre-election state policies, which is why they were never included in the first place.
On December 12th, The U.S. Supreme Court overturned the math mistake of the Florida Supreme Court. They overturned 7-2 the decision to order more recounts on the grounds that there were no uniform standards for the counting of those questionable ballots. Without an exact standard for counting votes, the margin of error will increase, making the platonic ideal outcome even less likely. While legal scholars debate the decision, mathematically it was the right thing to do.
The Florida Supreme Court may have had the disenfranchised voters in mind when they made their solomonic decision. But, they basically made the election scientifically invalid. The Supreme Court was understandably blinded by the fact that this election will decide the President of the United States. A just and mathematically valid decision would have treated this election the way they would treat a close election to the state water board. By changing the rules of this election, and nobody denies that is what they did, the election in Florida is now a victim of the election paradox. The Supreme Court basically tried to rule in a way that would eliminate uncertainty. They have ended up guaranteeing uncertainty.
In another victory for uncertain outcome is the decision in Palm Beach to consider "dimpled chad's" From Fox News:
In a major victory for Al Gore's presidential campaign, a Florida circuit judge ruled Wednesday afternoon that Palm Beach County must consider dimpled chad in its manual recount
Palm Beach has worked since 1990 under a rule saying at least one corner of a chad — the bit of paper that gets pushed through on a punchcard ballot — must be dislodged for a ballot to count.
This is a clear violation of the Election Paradox. If Palm Beach County hand counts the ballots differently this time than in the past, this post election change in policy mathematically invalidates the election the same way a "revote" mentioned above would.
By the way, two different recounts were done on the so called "undercount" ballots in Palm Beach County. In one done by the Palm Beach newspaper, Bush actually gained more votes and widened his lead. In one done by the AP which took into consideration dimples far more liberally, Gore gained 57 votes, not enough to overtake Bush.
Bush would have won Florida no matter what.
Tied elections are uncommon, but not unheard of. What happens in a tied election varies from state to state. In Arizona and New Mexico, tied elections are decided by a game of chance. In the past two decades, tied elections have only happened a combined three times in both states. Traditionally, the game of choice has been one hand of five card stud poker, a reminder of our "old west" heritage. Two of the ties were decided this way, a judicial election in New Mexico and a state legislature election in Arizona. The third was a city council seat in the town of Showlow, Arizona. The town is actually named after a card game, so they decided the city council race by cutting a deck of cards until someone showed the deuce of clubs (the "low" card).
The Election Map and the "Two Cultures"
Another outcome of interest in this election is the rural vs. urban vote. While this makes no difference to the outcome of the election, it has some interest from a mathematical perspective. Here is the now infamous county map from USA Today:
Gore won the popular vote, by winning just those counties marked in blue. Bush not only won the most states (30), but also the most counties. How then did Gore win the popular vote? If you note where all the major cities in America are located, they are all in blue. Of the 40 largest metropolitan areas, only Phoenix, Dallas, San Diego, Tampa Bay, Salt Lake City, and Indianapolis went for Bush.
From a math mistake perspective, this is a bad way to look at things. Maps that break down things by states or counties are always misleading, because of the large but low populated states that cover most of the western half of the country. This is another graph error I should add to the Glossary.
This graph does have interest from a political perspective. There is strong evidence that the country is fractionalizing into two camps: the rural and the urban. These "two cultures" are growing more and more separate. Rural America is generally conservative, not because they are "old fashioned", but because liberal agendas have harmed rural communities. Liberal environmental laws ruin rural economies, liberal estate taxes ruin family farms, while federal dollars for roads, housing, and modernization all go to the cities. Urban America is generally Liberal for exactly the opposite reasons. The environment needs to be preserved and pristine so that urban people can go four wheeling on the weekends.
Ultimately, this may be most harmful to the Democrats. The most telling fact is the bar chart in the bottom right corner. The growth rate in Democratic strongholds is shrinking while in Republican strongholds it is growing. The 2000 Census was released on December 28th and had the adjusted electoral college been put into effect before the election, Bush would have won 278 to 260 instead of 271 to 267. If the electoral college would have been adjusted by the 2000 census before the election, Bush's electoral victory would have been easier. Many of the Gore states will be losing electoral votes, while many of the Bush states will be gaining.
Mathematics and Election Reform
No doubt, one of the outcomes of this election will be some major reforms in how we elect politicians and how election tallies are made. Some of these issues will be the following:
This quote has been getting a lot of attention lately in that it
essentially means that the Florida Legislature can indeed vote for electors
regardless of the outcome of the lawsuits. But, it also puts a good argument
forward in keeping the Electoral College. I would like to see the Electoral
College stay but with some changes. Two states, Nebraska and Maine, have a
system in place in awarding electors more fairly. The winner of the whole
state would get two electoral votes, then the winner in each congressional
district would get an additional electoral vote. This reform would not
require a Constitutional Amendment to implement, just a law passed in each
of the states with more than 3 electoral votes available. And it would be to
the States advantage to do so. Candidates do not go to states they have no
chance of winning. All 54 electoral votes in California went to Gore, but if
this program were in place, 15 or 16 of those votes would have gone to Bush.
Gore could have made up the difference in other states like Arizona with 8
votes that went to Bush, but at least two votes would have gone to Gore
under this system. Florida would be split 11 for Gore and 14 for Bush. An
analysis of this plan using the election returns since 1960 would have sent
Nixon instead of Kennedy to the White House in 1960, and would have resulted
in a tied electoral count (269 to 269) in the Ford and Carter race in 1976.
The biggest advantage would be the nearly complete elimination of post
election lawsuits. Questionable outcomes would be limited for the most part
to individual congressional districts, not statewide like we have now or
nationwide like we would have under a popular vote scenario. Yes, it would
still be mathematically possible to win the popular vote, but lose the
electoral vote, but such a reform would ultimately be more decisive.
I thought about that when I heard all of the controversy about double punched butterfly ballots in Palm Beach. If they used the same system we do here, the people would have been given a chance to correct their mistake before leaving the polling place. While the Federal Government has no say so in how elections are run, they could offer to pay for any county that wants to throw out their inaccurate punch card systems for optical scan systems. Such systems have an error rate as low as 0.01% and would have reported accurate enough to decisively decide the Florida outcome. And if that is not enough, determining voter intent in hand counts is blatantly obvious and does not require magnifying glasses and back lighting.
Here is how it worked: Weeks before the election, voters had to apply for an absentee ballot application, this could be done electronically. The application was sent via snail mail where it had to be signed by the voter to be legally binding, then snail mailed back to the county election board which upon receipt compares the signature to the one on their original voter registration, then sends the voter a secure password, again via regular mail. The voter could then log on to a secure server at election.com where they had to enter their name, voter ID number, and the secure password. If these did not match, the vote would not be legal. This process is complicated with many hacker proof safeguards built in. Details about this election can be found at election.com.
Since the votes are dated on the Internet, there is no question about post marks. And since all you need is a computer connected to the internet, absentee votes could be done from anywhere in the world. As troubling as it is to some, this will be the wave of the future. I believe at least one state will offer Internet absentee voting by the 2004 presidential election.
A tie election means neither candidate can assume the Presidency with a legitimate mandate of the people. The historical trend is that such candidates only stay one term. We cannot stop elections from being this close again, but we can stop the election process from being so difficult in future virtual tied elections.