You're about to brush your teeth but you're all out of toothpaste, so you walk over to the drugstore. They're out of your favorite toothpaste too, but there's a shampoo available for the same price. On the efficient market hypothesis, you expect that the market prices already contain all relevant information about the products, so you have no reason to think the shampoo is less valuable to you. So you buy it, go home, and wash your hair.

What's wrong with this story?

The problem is that you're using a marginalist heuristic. The efficient market hypothesis applies to markets in which people have roughly the same preferences. Everyone wants roughly the same thing from their financial investments - to make more money. So at any given level of risk, you should expect to evaluate tradeoffs the same as anyone else does.

In the case of the drugstore, you have a lot of information about whether you prefer shampoo or toothpaste, that is unlikely to be reflected in the price. The efficient market hypothesis suggests that you shouldn't expect to get a much better deal in a nearby store, but not that you should be indifferent between all similarly priced goods. You value toothpaste over shampoo a lot more than any price difference is likely to reflect, because you have what is called an inframarginal preference: you need toothpaste, and you've already got enough shampoo.

Critch just reposted an old argument in favor of voting, by doing a back of the envelope calculation of its expected impact. The model is perfectly fine, but to estimate the value, he uses a related cost. I don't think this seems like a reasonable thing to do if you're not making the shampoo-for-toothpaste error.

Critch lays out a simple back of the envelope calculation to assess the value of a single vote in a presidential election. His estimate of the value of his vote in the Obama-Romney election of 2012 is the product of three factors:

The probability that his vote swings the election.

Conditioned on his vote swinging the election, the increased probability that the better candidate won.

The value of the better candidate winning, expressed in dollars.

He starts with a simple calculation. There are about 100 million voters, so he conservatively estimates a 1 in 100 million chance of swinging the election. He estimates a 55% chance that he has correctly identified the better candidate, which translates into a 55-45=10 percentage point improvement in the chance that the better candidate wins. He guesses that a very conservative lower bound for the improvement due to the better candidate is $100 billion, since that's a small fraction of the expected $12 trillion in federal expenditures over the president's 4-year term. Finally, he multiplies together these numbers for an expected value of $100 billion * 10% / 100 million = $100 given to charity per vote cast.

He then gives a couple of less conservative estimates:

Say you’re more like 70% sure,

Say you’re a randomly chosen american, so your probability of a decisive vote is around 1/10 million;

Say the outcome matters more on the order of a $700 billion in charitable donations, given that Obama and Romney’s budgets differ on around $7 trillion, and say 10% of that is stuff that money is being used as well as moving charitable donations about things you care about. That makes (70%-30%)*1/(10 million)*($700 billion) = $28,000. 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.

What's wrong with these numbers?

A comparatively minor issue is that it looks like the probability you cast a deciding vote is somewhat too optimistic - based on a more recent paper, the probability of a Californian casting a deciding vote is about 1 in a billion, and even in the swing state of Virginia, it's more like 1 in 10 million. But that's a nitpick, especially since the new paper wasn't even published until 2012.

The big problem is that he's treating dollars of government expenditure as roughly equal in value, and equating them with charitable dollars.

Critch does mention that his metric isn't an universal one:

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. 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.

But, I claim that using the Federal budget as a benchmark isn't just wrong - it's unreasonable. You don't want to look at resources consumed to compare the value of two things. You want to look at what they give you, minus what they cost. If you're out of toothpaste, the fact that shampoo costs the same doesn't matter - you still want the toothpaste and not the shampoo. Similarly, if two government programs cost about the same, there's no particular reason to think they do similar things, or provide similar amounts of value, unless you think government expenditures are the product of an efficient market in positive impact maximization.

Government expenditures are not in fact generated by an efficient market process. Some of them buy bombs to drop on people and kill them. Some of them buy medicine. Some of them go to helping people. There's no strong reason to think that the difference in value between a billion dollars' worth of bombs, and a billion dollars' worth of school lunch funding, is within an order of magnitude of a billion dollars. It could easily be much smaller - or much larger.

The comparison to giving to charity also seems unreasonable - there's no particular reason to think that the cost-effectiveness of your favored charity is similar to the difference in impact per dollar between federal expenditures under two potential presidents.

"Dollars given to charity" is also a cost, not a benefit, at least when you're not specifying the charity's impact per dollar given. There's no particular reason to anchor on the size of the Federal budget, when measuring the impact of the presidency in terms of dollars given to your favored charity (which is I think how people reading the Critch's post will generally interpret the claim). It's not that the federal budget is a biased estimate of effect size - it's that it's irrelevant.

It's as if you tried to estimate the importance of picking the right president, by noting that since there about 4 million federal employees, and the president's term is 4 years, that means that the president affects 16 million person-years, so if the president makes a 10% difference, that's 1.6 million DALYs. You have a number, and it's related in some way to the bigness of a relevant thing - but the number doesn't represent the thing you care about. It's as if you decided that shampoo was a good substitute for toothpaste, on the basis of price.

To be clear, I think Critch's basic framework is a good one, and makes an important point. Arguments that your vote doesn't make a difference because it's so diluted by others' votes really do ignore the scope and power of the US government. There are lots of measures you might want to use for this. For instance, you could look at variation from president to president in the number of foreigners killed by the US military, directly or indirectly. You could look for estimates of how much variation in economic growth is attributable to who's president. You might want to add these measures together. But it's important to choose based on estimates that have something to do with the outcomes you care about, and dollars influenced is not that.

I think it's very important to vote this year, but that's because I believe that a Trump presidency would increase the chance of a nuclear exchange by at least a tenth of a percentage point, and the chance of a major military conflict between great powers by at least a percentage point, which adds up to a lot of expected lives, especially if you think a nuclear exchange is likely to be bad for our prospects of surviving into the far future. More on my reasoning for this later. I don't have any numbers to back up my probability estimates, just my intuition - but it's an intuition about a relevant quantity.