My co-blogger and former (visiting) colleague Ilya Somin has an interesting article on the rationality of voter ignorance forthcoming in Critical Review, which he blogged about yesterday. On empirical matters, Ilya and I usually agree, often using some of the same data sources to make our points. But here, we seem to differ on a theoretical matter based on a difference in modeling assumptions.

In his post, Ilya argues that, because of altruism, voting is rational in order to increase the probability that your preferred candidate will win. As I will explain below, I think that one can come up with a story consistent with rational voting, but probably not the story that Ilya tells.

Ilya summarizes his argument in his post:

The key insight is that the traditional rational choice theory of voting implicitly assumes that the voter cares only about their own self-interest, narrowly defined. But if you care even slightly about the potential benefits to fellow citizens of ensuring that the "right" candidate wins, then the sum total of those benefits might well outweigh the (generally low) costs of voting even after discounting for the fact that there is only a minute chance that your vote will make a difference. . . .

This theory is not entirely original to me; it was in fact proposed by philosopher Derek Parfit in his 1984 book Reasons and Persons (pp. 73-75); what I have done is explore its implications for voting theory more generally. Until now, Parfit's theory has been largely ignored by academic scholars of voting, perhaps because Parfit is not an economist or political scientist, and his book is primarily devoted to other subjects.

Of course, perhaps the other reason that Parfit’s theory has been largely ignored is that it is appears to be wrong.

Ilya’s Equation

In doing his modeling and making what he says are “plausible assumptions,” Ilya assumes that the odds of casting the deciding vote are 100 million to 1 and that people value their own utility 1000 times more than each other individual’s utility.

The equation on the rationality of voting that Ilya refers to is this (Somin manuscript, page 4):

(((Expected Difference in Welfare per person if the preferred candidate wins) x (300 million people) / 1000) / 100 million)) - Cost of Voting = Expected Utility of Voting

Ilya assumes that the cost of voting is $10, and the expected benefit per person of the preferred candidate winning is $5,000. If you plug in these numbers, the result is:

($5000 x 300,000,000 / 1000) / 100,000,000) – $10 = $5 (net expected value of voting).

Accordingly, Ilya concludes that voting would be rational because the expected benefit of $15 is larger than the expected cost of $10, leaving a net expected benefit of $5 for voting.

A Serious Problem With One Assumption

One of the nice things about equations is that their assumptions are usually explicit. Look more closely at Ilya’s equation and what it says about voter preferences BEFORE the 1 in a 100 million discount is applied.

Ilya's equation assumes that, if a voter could guarantee a victory for his preferred candidate, a typical voter would be willing to pay only $5,000 for one person's benefit (presumably his own), but that the same voter would be willing to pay about $1.5 billion dollars to benefit others ($5,000 x 300 million people / 1000). In other words, Ilya assumes that a rational voter when voting values the total utility of other Americans 300,000 times more than he values his own total non-altruistic utility ($1.5 billion to $5,000). Moreover, even leaving aside the comparative valuation, it can’t be that (because of altruism) the utility to each person voting of having one’s preferred candidate certain to win would be $1.5 billion dollars. To say that these are extraordinarily implausible assumptions is an understatement.

I think, for example, that if most individuals were given a choice between half a billion dollars for one's own bank account and certain victory for one’s preferred presidential candidate, most individuals would be willing to switch allegiances and take the half billion dollars. Indeed, I suspect that a majority of individual voter’s allegiances could be purchased for a secret political consulting job paying much less than $200,000 per person (assuming certain victory for the candidate doing the hiring).

So let’s make a much more rational assumption, but still one that is unrealistically generous to Ilya’s position. Let’s assume that a typical voter has a family gross income of $50,000 and a take-home pay of $35,000, and like Ilya, let’s assume that the voter would be willing to pay $5,000 just for one person's benefit (his own) to guarantee a win for his preferred presidential candidate. Now let’s be extraordinarily generous to Ilya’s position and assume that the typical voter is highly altruistic and would be willing to pay another $35,000 (an entire year of his family’s take-home pay) just to benefit others, not himself, by having his preferred candidate win.

Plugging this into Ilya’s analysis, the expected benefit to each person of voting would be $40,000, divided by 100 million (the probability of casting the deciding vote), which is only 4 one-hundredths of a penny. Obviously, voting would be highly irrational, since the expected benefit to even a highly altruistic voter of just 0.04 cents is much less than Ilya’s assumed cost of $10 for voting. Thus, even for an extraordinarily altruistic voter, voting is highly irrational if its purpose is to influence the outcome of a presidential election.