## Costs and Benefits of Voting

Economics has lots of odd results. Many of them are deeply insightful, like the principle of comparative advantage, with the potential to improve lots of people’s lives when understood and acted upon. On the other hand, some of them are just bizarre, which for my money includes the claim that voting is irrational. Basically, the argument goes that your individual vote is unlikely to make any difference (it’s pretty rare that there’s going to a final margin of a single vote after all), and since you have to spend time and energy to vote, it’s just not worth the effort.

Now you can recover that with things like the categorical imperative (if everyone did it, things wouldn’t be so rosy, would they? therefore you shouldn’t do it), but that’s not terribly persuasive — and besides, even when everyone else votes, it still seems intuitively good that you vote too.

There are some other easy ways to avoid the conundrum too: compulsory voting means you’re forced to pay the costs of voting either way (by voting, or by paying a fine), so even a trivial benefit makes it worthwhile. And you might argue that getting an “I voted” button, or the feeling of participating in democracy is worthwhile. Or you might argue that your vote isn’t just about who wins, but also helps finance their future campaigns (to the tune of about $2 per vote in Australia), or you just think the final margin of victory will affect the mandate the winner can claim, and be useful on that basis.

But again, intuitively, I’d like to believe it’s still a good idea to vote even without any of those factors, and even if most everyone else votes. And personally, I prefer it when my reason either matches my intuition, or doesn’t just suggest a different approach, but can show why the approach my intuition suggests is actually harmful.

With that in mind, I found Robin Hanson’s post on Noble Abstention pretty fascinating. It basically assumes something like the worst case for voting — everyone who’s already voted is smarter than you, and everyone who votes after you isn’t. It also assumes that nobody is perfect at picking the best candidate, and that people get exponentially worse at doing so. By varying the degree of the exponent, Robin finds that for exponents above a half, the best candidate is most likely to win when only one person or a small handful of people vote, and for exponents below a half, the best candidate is most likely to win when everyone votes.

The analysis is somewhat more interesting when you look at individual incentives: whether your vote will make the best candidate (who may or may not be who you vote for) more or less likely to win. In the cases he examines, with exponents above about 0.7, assuming both that each voter is (very slightly) more likely to vote for the better candidate, and that everyone else is voting, including everyone smarter and stupider than them, *half* of the ten thousand voters are actually in the position where their vote will actually *decrease* the liklihood of the better candidate winning.

To be clear: you’re probably going to vote for the better candidate. The better candidate is already likely to win. Of the 9,999 other people who’ve voted, you’re smarter than 4,999 of them. Yet your vote will decrease the odds of the better candidate winning compared to you not voting.

That seems pretty unintuitive, but it actually falls out from a very particular arrangement of probabilities. Namely, your vote will make a difference in two circumstances: you vote for Alice, when she has just under half the votes, or you vote for Bob, when he has just under half the votes. If *P(A)* are the odds you vote for Alice, and *P(n for B)* are the odds Bob has n votes currently, then assuming there are *2n* other voters, your probabilities are:

*P(n for A) * P(A)*— you win the election for Alice*P(n for B) * P(B)*— you win the election for Bob*1 – P(n for B)*— your vote doesn’t end up mattering

(Note that there are *n* voters for Bob precisely when there are also *n* voters for Alice)

If there are *2n+1* other voters you can’t force a win, but you can force a draw, and your probabilities are:

*P(n for B) * P(B)*— you stop Alice from winning the election*P(n for A) * P(A)*— you stop Bob from winning the election*1 – P(n for A) – P(n for B)*— your vote doesn’t end up mattering

And the problem is that you’re making things worse when *P(n for X)*P(X)* is lower for the better candidate than for the worse one — which isn’t only true when you’re more likely to vote for the worse candidate (ie, *P(X)* is lower for the better candidate), but also when *P(n for X)* is lower.

But that in turn depends closely on the distribution of everyone else’s likelihood for voting for the better candidate. Robin’s assumptions naturally imply a particular range of distributions that cover both possibilities — where the vote distributions of prior voters sets future voters up to prevent their votes from being useful; and where the vote distributions of later voters keep improving the odds of the better candidate winning. There are plenty of other possible outcomes too; what assumptions might be valid depend a lot on how much information you actually think voters can obtain about the candidates and how much pre-existing biasses might prevent that information from making it through to the election, and how much you think people differ in their odds of picking the better candidate.

But beyond that, it gives an interesting way of analysing whether or not it’s worth voting, even under the assumption that your individual vote might not make a difference, and that you might end up voting for the worse candidate. Taking Robin’s distribution, for example, with an exponent of 0.5, gives the 10,001st voter a 50.05% chance of picking the better candidate, and overall odds of 57.869279% of the population selecting the best candidate. Based on Robin’s assumptions, the first voter had a 55% chance of doing that alone, and the following 9998 voters improved that to 57.869274% (ie, by 2.869274%), leaving the last two voters (#10,000 who could only force the election into a draw, and #10,001 who could actually provide a winner), to improve on that by 0.000005%, or 1 in 20,000,000 times, or 1 in 40,000,000 times each.

So supposing you’re the dumbest person in an electorate of 10,000 people, and your odds of picking the better candidate are only better than tossing a coin one time in 2000, then it’s worth spending half an hour voting, if you think the difference between the two candidates is enough that improving the better candidates chances by 1 in 40,000,000 is worth the effort.

In general, if there are *n* voters, and an overall chance of electing the best candidate of *p*, the average voter can claim a contribution of *(p-0.5)/n*. If you want to be pretty generous, and say the best candidate wins 90% of the time, the average voter in an electorate of 50,000 people might be able to claim to make a difference 1 time out of 125,000. If you think it’s only 75% of the time, you’re down to 1 time out of 200,000. If you think the 58% that resulted from the previous calculations sounded pretty good, you’re down to 1 time out of 625,000.

For comparison, the chance of winning Oz Lotto is reportedly about one in four million, and the size of Australian Federal Electorates tends to be around 90,000 people.

So the question is, do you think you’re above average in picking good candidates? Do you think the difference between the candidates is noticable? And if both of those are true, do you think that a one in 125,000, or one in 625,000, or one in 40,000,000 chance of ending up with the better candidate is worth the effort of voting?

To give a concrete example: if you think Robin’s model for your electorate is vaguely plausible with an exponent of 0.5, and you’re in the top 10,000 voters, and you think choosing the better candidate might save the country a billion dollars (ie, thousand million), then the one in forty million difference your vote makes is worth $25 to the country.

There’s a *lot* of variables there, but I think it’s a plausible economic analysis: Robin’s structure seems somewhat pessimistic to me, a billion dollars’ difference between candidates seems within the realm of plausibility on a national level and possibly even a state a level, and saying your vote is worth $25 to the country matches the amount you’ll be fined in Australia if you don’t vote ($20).

I think that also ends up as a potential justification for “compulsory” voting: given a sufficiently precise model of how much additional information each voter brings to the election, and how much variance there is between candidates’ performance, you can calculate a monetary value for how much each person’s vote is worth to the country, and assuming that has a low variance (eg, if it turns out to be between $15 and $25 for everyone in the country), it would then be reasonable for the country to pay everyone who votes the average amount, or if the country’s funded by taxation and fines anyway, to fine everyone who doesn’t vote that same amount.

And if you have economically justified compulsory voting, the individual economic analysis for whether it’s rational to vote becomes simple: is the nuisance of voting worth the cash reward, or worse than the nuisance of the fine as the case may be?

But again, it depends heavily on how the population’s electoral nous is distributed — the value of people’s votes might range from $19.99 to $20.01 or from $0.02 to $100.00, or from -$50.00 to $10,000. As an egalitarian, I have an a priori assumption it doesn’t vary much. Your mileage may vary.