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.
There are 100 million votes, one of which decided the election (the N+1'th vote on the winning side, assuming N votes on the losing side).
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.
I can't help you clean up the calculations because the entire premise that the probability of "your vote mattering" times your potential financial gain from one candidate over another gives any reasonable estimate of utility of a vote is absurd.
It doesn't have to be financial gain - it can be altruistic gain from policy choices. As I said in another post, if you personally would be willing to spend $50k to legalize gay marriage (for example), you can replace $50k of financial gain by $50k worth of utility derived from helping others.
The rational voter never votes. At best he spends his effort convincing other people to vote a certain way, persuading others that their vote does indeed count, all the while knowing his own vote doesn't. If he participates at all, he does so as demagogue and charlatan.
Maybe I'm looking at this analogy too hard, but ...
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.
Fine, but this doesn't change the main point. Even if you figured it was worth your time to determine the effect of each candidate on your future, it still wouldn't compel you to vote for the candidate which helps you most; the expected value would still only be a tiny fraction of a penny, and you'd be better off using your vote to signal allegiance, assuage your conscience, etc.
...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.
I think assuming that the only value you derive from voting being tax rate is the major flaw here.
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.
I discussed a tax increase simply because it made things simple, but you can assign a dollar value to any of the items you've mentioned. How much would you, personally, pay to legalize gay marriage?
$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.
What I'm saying, though, is putting a dollar figure on things like gay marriage is disingenuous. We have this conception that "value" automatically means that it can be put in terms of money.
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.
Or, like most people, you're not a rational machine.
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.
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.
Aren't you assuming that all you can do to protect your 50k is vote? And that all votes are independent? What if you could spend 25k mobilizing your side in the election? Or running an issue ad? You gain it all back and then some.
Still seems like a low return on investment, especially when you multiply by the risk of failure. Yes you can increase your investment and reduce the risk, but the ceiling on gains make this unappetizing.
On the contrary, you currently get far more leverage out of your money by participating in the (US) political system and flashing cash than you do by apathy. And the more money you spend, the less risk there is.
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.