Gelman’s argument is essentially that pB – c > 0 will hold for altruists even though it cannot plausibly hold for narrowly self-interested voters. (If you haven’t seen this formulation before: p is the probability that your individual vote determines the outcome of the election, B is the benefit of having your preferred candidate win, and c is the cost of voting.) He first stipulates, quite reasonably, that p is approximately 1/n, where n is the size of the electorate. A narrowly self-interested voter is very unlikely to find the cost of voting to be worth incurring because p is so small that B provides almost no off-set to c. For an altruistic voter, B is assumed to increase in direct proportion to N, where N is the total population. As N goes up, n likely goes up, and p goes down, but if B goes up too, then that’s no problem, and it’s very likely that the cost of voting is worth bearing.
My critique is essentially this: B cannot go up anywhere near as rapidly in N as Gelman says for a true altruist, because an altruist would also care about the fact that the number of people made worse off by the election of their preferred candidate is also strictly increasing in N. If B increases in N, but at a slower rate than p decreases with N, then the instrumental argument falls apart. My claim is that one must either hold implausible beliefs about the extent to which politics creates both winners and losers or must be frighteningly insensitive to the well-being of the losers in order for B to increase in N rapidly enough to offset the decrease in p. As I said, that doesn’t necessarily make one stupid, as Levitt said that voting to affect the outcome does. But it doesn’t say anything particularly flattering about you either.