If you're a fellow American, you're probably cleaning off your barbeque and putting out your flag in preparation for the July 4th holiday. This is the day set aside once per year to celebrate the independence of the United States from England, and to reflect upon the mathematical theorems that led to the founding of our nation. What? Are you thinking that July 4th isn't about math? Clearly something has been missing from all the specials you've been watching on TV. Actually, there was at least one mathematical result that had a direct influence on our Founding Fathers: Condorcet's Jury Theorem. This theorem states that if you are trying to decide on a topic by voting, and the average voter has at least a 50-50 chance of getting it right, increasing the number of voters gives a more accurate result.
How did Condorcet's Jury Theorem come about? Back around the time of America's founding in the late 1700s, there was a colorful French mathematician, the Marquis de Condorcet, who spent a lot of time thinking about mathematical aspects of democracy, voting, and probability theory. Condorcet is known to have collaborated with colleagues in the US and worked personally with Benjamin Franklin, and thus his theorem is thought by many to have directly influenced the United States Consitution of 1787. The Jury Theorem was part of a larger 1785 work, Essay on the Application of Analysis to the Probability of Majority Decisions, considered a major contribution in the development of probability theory. Condorcet was concerned both with the basic concepts of probability, and how they could be used to help develop rational public policies-- one of the earliest mathematicians to attempt such direct applicaton to the social sciences.
To get an idea of how we prove that increasing the number of voters leads to a more accurate total, think about a situation where we have some odd number <n> of votes, the probability of an individual vote being correct is <p>, and we add two more votes that might change the result. In this case, you can think of the nth vote as having been a deciding vote: our oddness assumption prevents ties, and otherwise the two new votes would not have been able to reverse the answer. So we just need to calculate probabilities for the last 3 votes, in 2 cases: did they convert an incorrect majority to a correct majority, or a correct majority to an incorrect majority? Remember, each individiual vote has a probability <p> of being correct and <1-p> of being wrong. Also remember that you can compute the overall probability of a set of independent events by multiplying their probabilities together. So for the previous deciding vote to be incorrect, and the two new votes reversed the tide, the probability would be (1-p)*p^2. For the previous vote to have been right, and the two new ones to reverse the total, it would be p * (1-p)^2. The first one of these is larger than the second if and only if p is greater than 1/2.
We should be careful, though, to look at the limitations of this theorem before relying on it too much in our political system. At its heart is the notion of some kind of objective correctness, the ability to state a right answer that most people will come to with a definite probability. Except when I'm one of the candidates, there are very rarely such clearly correct answers in politics. It also depends on the voters being independent, and ignores the influence of neighbors, popular delusions, people too lazy to gather the information needed to make an intelligent decision, or people held in sway to demagogues. I'm pretty sure that, depending on your political leanings, you consider the election of one or more of our past two U.S. presidents a result of such factors. And the theorem only works when choosing between precisely two alternatives: it does not account for voting on multiple choices.
The last limitation leads to another of Condorcet's important insights, his "voting paradox". Assume we have 3 opponents A, B, and C, and the public is divided into 3 segments with cyclic preferences: some prefer A over B over C, some want B over C over A, and some want C over A over B. Let's say they vote and get basically a 3-way tie, with just a very tiny margin putting C over the 1/3 number needed to win. You could argue that if C won, then 2/3 of the population would have wanted B to win over C: since the ones that wanted A preferred B over C, and the ones that wanted B also preferred B over C. Yet if B won, you could say precisely the same thing about the people preferring A, or if A won, could make a simlar complaint about C! Thus, somehow a popular vote will always result in an outcome that 2/3 disagree with. This can be solved by voting with a "Condorcet method", basically a more complex voting method that guarantees the winner will be someone who would have won in a pairwise competition with every other candidate. This can be done by having voters list candidates in order of preference instead of just choosing one result, or by having multiple rounds of voting with runoffs. Personally, I think these kinds of alternative methods make a lot of sense, though they suffer from the disadvantage of being much more complex to administer and understand.
So, with all these great insights into how government should work, you would infer that Condorcet had a brilliant political career, right? Unfortunately, he suffered the results known since Socrates to those who embrace their philosophical insights too enthusiastically without sufficiently understanding the flawed human psychology all around them. Initially a supporter and active participant in the 1789 French Revolution, he failed to support the radical faction that took control, instead trying to convince them to embrace his ideas. While this got him a position on the Assembly for a while and a decent amount of popular support, it eventually led to his arrest and death. Technically he killed himself in jail, but many believe this was permitted or staged to avoid the embarrasment of publicly executing such a well-known revolutionary figure. Either way, in the most important applied mathematics experiment of his life, the basic assumptions of Condorcet's theorems just didn't hold. Let's hope my term on the Hillsboro School Board leads to a better fate.
And this has been your math mutation for today.