This assumption doesn't make sense
If you prefer, you can play with binomial distributions to calculate P(side X wins with N votes && side X wins with N-1 votes) [1]. Using the normal approximation and ignoring the normalization prefactor (an overestimate) exp[(n(p-0.5)^2 / 1(np(1-p))] odds of having your vote matter. For p=0.501 (i.e., voters are split 50.1%, 49.9%), that works out to about 10^{-87}. For p=0.51, double floats are incapable of representing a number that small.
I stand by my statement. Your vote doesn't matter.
[1] Conceptually, this is slightly different from computing the odds that you cast the deciding vote, but I realize it is more useful from a decision theoretic perspective.
[0] http://redbluerichpoor.com/blog/2008/10/what-is-the-probabil...
[1] http://www.stat.columbia.edu/~gelman/research/published/prob...
(By "significantly", I mean "increases the odds of your vote mattering to a number greater than, say, 10^{-86}. )
The odds that the election would be decided by a single vote have not been considered.
If I actually did have $50k riding on the election, I'd spend more than $0.0005 cents determining which candidate is best, because assessing risk is necessary to plan for the future.
...unless you are alturistic, and you value the impact on the other 300 million citizens (and the rest of the world). Then it can make sense to vote for who you think is best.
Altruistic value is usually very high in the issues that people seem to vote based on: gay marriage, government health care, military operations, etc. Many people do value these things much higher than their tax rate.
The problem arises when we treat dollars as the unit of value. Value is not possible to practically quantify, and using money as the common denominator pushes things that actually are money to the fore, and fails to expose the things that do matter to people.
$50,000? If so, repeat my calculation. $100,000? If so, then voting is worth $0.001 to you. $100 million? Then voting is actually worth $1 to you.
A much simpler explanation of voting behavior is the expressive value of voting. People devote great effort to cheering for their team and signalling allegiance - just look at sales of $50 Yankees pinstripes or $80 Paquiao jackets. Most people don't wait in line for 10 minutes just to donate $1 to a good cause.
Money has value, and events like law changes have value. However, both money and events have wildly varying value to different people. What a dollar is worth to me is different than what a dollar is worth to you, and indeed, what the millionth dollar is worth to me is far different than what the first one is.
The problem is this assumption that money is an appropriate expression of value. This is what I reject. It is an abstraction parallel to the bundles of mortgages that American banks were brought down by - putting it into a unit the MBAs and politicians can understand meant losing essentially all the meaning of what was bought and sold, and ignored fundamental fallacies in assumptions made.
The future that costs you $50,000 (or helps/hurts you via any of the mentioned surrogate currencies) is distasteful. Your irrational brain won't let you do nothing, and makes you go out and cast your vote, even though you may understand that it's a waste of time.
Alternative theory: Risk.
Highly publicized close elections create a scenario that looms large in your mind: The Bad Guy wins by one vote. (Or just a few). You would never forgive yourself. Even though you know it to be fantastically unlikely, you aren't willing to risk it.
/me doesn't vote.
For example, I consider time I spend reading on the toilet basically worthless. So when the voter guide comes in the mail, with the candidates' statements, text of propositions and submitted arguments for or against, it is less than $0.25 to multitask there.
Expected gain, though, is higher than just the tax cuts and beer-having. It can be fun to study that stuff. And, looking at the big picture, the collapse of our democracy by having an uninformed electorate would mean everyone loses everything that is based on the USD.
Taleb's talked about chasing activities where the black swans are positive: http://akkartik.name/blog/9400292
Why is there no net neutrality bill? The rich are overrepresented in our system for a reason. Sad but true.
Suppose your personal stake in a presidential election is $50,000 (e.g., you make $500k/year, and Candidate 1 will raise or lower your taxes by 2.5%/year.) Your odds of swinging the election with your vote are 1 in 100 million (assuming 100 million voters).
Your expected gain from voting is $0.0005. Therefore, you will not rationally be willing to spend more than $0.0005 determining which candidate is best. Further, you might deliberately vote for the wrong candidate for expressive reasons. I.e., if you assign $0.25 worth of utility to supporting the guy you want to share a beer with [1], or to proving you aren't racist, you will vote for the worse candidate.
[1] Because the stakes are so low, most people didn't even bother learning that this candidate doesn't drink. That would take more than $0.25 worth of effort.