Summary: People often say that voting is irrational, because the probability of affecting the outcome is so small. But the outcome itself is extremely large when you consider its impact on other people. I estimate that for most people, voting is worth a charitable donation of somewhere between $100 and $1.5 million. For me, the value came out to around $56,000.
Moreover, in swing states the value is much higher, so taking a 10% chance at convincing a friend in a swing state to vote similarly to you is probably worth thousands of expected donation dollars, too.
I find this much more compelling than the typical attempts to justify voting purely in terms of signal value or the resulting sense of pride in fulfilling a civic duty. And voting for selfish reasons is still almost completely worthless, in terms of direct effect. If you're on the way to the polls only to vote for the party that will benefit you the most, you're better off using that time to earn $5 mowing someone's lawn. But if you're even a little altruistic... vote away!
Time for a Fermi estimate
Below is an example Fermi calculation for the value of voting in the USA. Of course, the estimates are all rough and fuzzy, so I'll be conservative, and we can adjust upward based on your opinion.
I'll be estimating the value of voting in marginal expected altruistic dollars, the expected number of dollars being spent in a way that is in line with your altruistic preferences.1 If you don't like measuring the altruistic value of the outcome in dollars, please consider making up your own measure, and keep reading. Perhaps use the number of smiles per year, or number of lives saved. Your measure doesn't have to be total or average utilitarian, either; as long as it's roughly commensurate with the size of the country, it will lead you to a similar conclusion in terms of orders of magnitude.
At least 1/(100 million) = probability estimate that my vote would affect the outcome. This is the most interesting thing to estimate. There are approximately 100 million voters in the USA, and if you assume a naive fair coin-flip model of other voters, and a naive majority-rule voting system (i.e. not the electoral college), with a fair coin deciding ties, then the probability of a vote being decisive is around √(2/(pi*100 million)) = 8/10,000.
But this is too big, considering the way voters cluster: we are not independent coin flips. As well, the USA uses the electoral college system, not majority rule. So I found this paper by Gelman, King, and Boscodarin (1998), where they simulate the electoral college using models fit to previous US elections, and find that the probability of a decisive vote came out between 1/(3 million) and 1/(100 million) for voters in most states in most elections, with most states lying very close to 1/(10 million).
At least 55% = my subjective credence that I know which candidate is "better", where I'm using the word "better" subjectively to mean which candidate would turn out to do the most good for others, in my view, if elected. If you don't like this, please make up your own definition of better and keep reading :) In any case, 55% is pretty conservative; it means I consider myself to have almost no information.
At least $100 billion = the approximate marginal altruistic value of the "better" candidate. I think this is also very conservative. The annual federal budget is around $3 trillion right now, making $12 trillion over a 4-year term, and Barack Obama and Mitt Romney differ on trillions of dollars in their proposed budgets. It would be pretty strange to me if, given a perfect understanding of what they'd both do, I would only care altruistically about 100 billion of those dollars, marginally speaking.
I don't know which candidate would turn out "better for the world" in my estimation, but I'd consider myself as having at least a 55%*1/(100 million) chance of affecting the outcome in the better-for-the-world direction, and a 45%*1/(100 million) chance of affecting it in the worse-for-the-world direction, so in expectation I'm donating at least around
(55%-45%)*1/(100 million)*($100 billion) = $100
Again, this was pretty conservative:
- I'm more like 70% sure,
- Being in California, Gelman et al. put my probability of a decisive vote around 1/(5 million).
- To me, the outcome matters more on the order of a $700 billion donation, given that Obama and Romney's budgets differ on around $7 trillion, and I figure at least 10% of that is stuff that I'd care about relative to other shifts in money I could imagine.
That makes (70%-30%)*1/(5 million)*($700 billion) = $56,000. Going further, if you're
- 90% sure,
- voting in Virginia -- 1/(3.5 million), and
- care about the whole $7 trillion dollar difference in budgets,
you get (90%-30%)*1/(3.5 million)*($7 trillion) = $1.2 million. This is so large, it becomes a valuable use of my time to take 1% chances at convincing other people to vote... which I'm hopefully doing by writing this post.
Now, I'm sure all these values are quite wrong in the sense that taking account everything we know about the current election would give very different answers. If anyone has a more nuanced model of the electoral college than Gelman et al, or a way of helping me better estimate how much the outcome matters to me, please post it! My $700 billion outcome value still feels a bit out-of-a-hat-ish.
But the intuition to take away here is that a country is a very large operation, much larger than the number of people in it, and that's what makes voting worth it... if you care about other people. If you don't care about others, voting is probably not worth it to you. That expected $100 - $1,500,000 is going to get spread around to 300 million people... you're not expecting much of it yourself! That's a nice conclusion, isn't it? Nice people should vote, and selfish people shouldn't?
Of course, politics is the mind killer, and there are debates to be had about whether voting in the current system is immoral because the right thing to do is abstain in silent protest that we aren't using approval voting, which has better properties than the current system... but I don't think that's how to get a new voting system. I think while we're making whatever efforts we can to build a better global community, it's no sacrifice to vote in the current system if it's really worth that much in expected donations.
So if you weren't going to vote already, give some thought to this expected donation angle, and maybe you'll start. Maybe you'll start telling your swing state friends to vote, too. And if you do vote to experience a sense of pride in doing your civic duty, I say go ahead and keep feeling it!
I've found a couple of papers by authors with similar thoughts to these:
- Jankowski (2002), "Buying a Lottery Ticket to Help the Poor: Altruism, Civic Duty, and Self-interest in the Decision to Vote", and
- Edlin, Gelman and Kaplan (2007), "Voting as a Rational Choice: Why and How People Vote To Improve the Well-Being of Others.
Also, just today I found this this interesting Overcoming Bias post, by Andrew Gelman as well.
1 A nitpick, for people like me who are very particular about what they mean by utility: in this post, I'm calculating expected altruistic dollars, not expected utility. However, while our personal utility functions are (or would be, if we managed to have them!) certainly non-linear in the amount of money we spend on ourselves, there is a compelling argument for having the altruistic part of your utility function be approximately linear in altruistic dollars: there are just so many dollars in the world, and it's reasonable to assume utility is approximately differentiable in commodities. So on the scale of the world, your affect on how altruistic dollars are spent is small enough that you should value them approximately linearly.