Does your vote make a difference?

Tomorrow is election day here in the US, so I'm going to have a post about elections.  It's strictly nonpartisan—people seeking bile can look elsewhere.

People often say things like "I'm not voting because it's unlikely that one vote could make a difference".  Sometimes they sound rather cynical about it, which I find a little odd, seeing as the whole point of democracy is that no individual gets to dictatorially make all the decisions for everyone.  But let's suppose they are just trying to be practical.  Is this a rational attitude to take?

There are many possible reasons to vote, even when your vote won't make a difference.   For example, it could be urged that voting fulfils a civic duty, that casting an informed vote forces you to become educated about the issues of the day, and to come to a decision about them.  It could also be urged that there is some political responsibility being exercised by the members of the majority even when the majority is by greater than one, or a value of protest in voting for the minority position.  Presumably people who "waste" their vote on third parties, or other hopeless causes, are primarily thinking about the communicative effects of voting.

But let's leave all this aside, and just think about the sort of power that comes from exercising a vote that changes the outcome.  This only happens when 1) the votes for two options are tied, so that you cast the deciding vote, or else 2) the votes for two options differ by one, so that you can accept that result or else cause a tie (the resolution of ties depends on the election system.  If ties are resolved randomly, the final result will change half the time).  For simplicity, let's assume the election has just two options: Yes and No.

How unlikely this is depends on the number of voters V, and the odds that the election will end up tied (within one vote).  Voters don't vote randomly, instead the more popular candidate will usually receive more votes. A certain fraction $x$ of the voters will choose to vote Yes, while $1-x$ will vote No.  Since we aren't certain what $x$ will be in advance, we have to model it by a probability distribution over possible values of $x$.  This is a function $p(x)$ describing the probability density of getting any particular outcome $x$.  (Here I'm assuming that $V$ is large so that $x$ can be approximated as a continous variable.  Note that because $p(.5)$ is a probability density rather than a probability, it can be greater than 1).

The odds of your vote making a difference can now be calculated to be:

$$\frac{p(.5)}{V}$$

This is a small number.  However, how important would it be if you got lucky and your vote did make a difference?  The results of the election presumably affect a large number of people, roughly comparable to the number $V$ of voters.  So the factors cancel out, and the expected effect of your vote does not depend on the size of V!  Instead it is proportional to $p(.5)$, which is equal to 100 times the odds that the vote lies within the critical percentile between 49.5% and 50.5%.

Another way of putting this, is if it would be worthwhile for everyone to miss their lunch to vote, so that everyone can get (what you believe is) a political benefit, then it is worthwhile for you to vote so that maybe everyone can get the political benefit.  Here I'm assuming you are voting for altruisic reasons, to benefit everyone.  The expected selfish benefits to you personally are indeed tiny.

It might still be rational to abstain from voting if 1) the odds of the election being close are small, or 2) the expected difference in benefit from the options is small.  These factors obviously depend on the specific election, but in general one expects that important issues are at stake.  So unless the outcome is a foregone conclusion, it is indeed rational to vote because of its deciding effects.

Does that mean that, if you're a resident of a non-Swing-State like California, you shouldn't vote, because (regardless of whether you support or oppose him) we all know that the state will go to Barack Obama?  If you said that, shame on you!  Go look at the local races instead of just thinking about the Presidency.  Although the number $V$ of voters affected by local politics is much less, the odds of you making a difference are proportionally greater.  The size cancels out, so you should regard local, state, and national races as being approximately equally important.  (Although I suppose one could argue that the President is twice as important, because he determines foreign policy as well as domestic policy.)