Jim Lindgren's most recent post is a rejoinder to my original reply. To recap, I argued that Jim's initial argument against my theory of rational voting doesn't refute it because Jim assumes that, for my theory to be correct, there must be a linear relationship between "the amount of money you are willing to give up to benefit others and the amount of benefit they receive from the sacrifice... That is, if you are willing to give up $1 so that your neighbor will get $1000, you are also willing to give up $1.5 billion in order to give your fellow Americans $1.5 trillion." To my mind, there is a big difference between the willingness to make relatively small, even trivial sacrifices and very large ones. The person who is willing to sacrifice $10 to give his fellow citizens an expected benefit of $15,000 (the tradeoff in the equation from my paper) won't necessarily be willing to sacrifice $1.5 million to give them $1.5 trillion.

Jim replies:

But Ilya, surely you understand that it is YOUR EQUATION that is linear. If you run the numbers through your own equation, you get EXACTLY the results I reported. You assume that each and every voter on average would value a guarantee of his own preference for president at $5,000 for one person's benefit and altruistically internalizes 1.5 billion dollars of the benefit of others.

Dividing the $1.5 billion altruistic utility that your equation and example say that each voter on average acts on in choosing to vote by 100 million to generate a benefit of $15 does not make your equation any less linear, a fact that you surely know. If the relationship is not "purely linear" (or even remotely linear), then your equation is dead wrong because your equation is purely linear.

Yes, the equation is purely linear. That is because I did not consider, in setting up the model, cases where the decisionmaker must decide whether to make a very large sacrifice rather than a small one. Quite likely, I was wrong to ignore that scenario, although I don't think that it is relevant to an analysis of the utility of voting (almost always an act that requires only small sacrifices). But building it into the equation would not significantly alter the result. For example, one could change the cost term in the equation (cost of voting) to include an additional variable that accounts for the fact that the tradeoff will change as C (the dollar cost of voting) becomes an increasingly high percentage of the potential voter's total wealth. Thus, the Total Cost (TC) of voting=C + (CxP), with P representing the percentage of the voter's wealth that $C is. For most people, however the cost of voting (which I assumed to be $10) will be a tiny percentage of their total wealth, and thus CxP will be utterly insignificant in real world terms (at least when dealing with voting). For example, if $10 is an unrealistically high 0.1% of the person's total wealth, then CxP will be $1.

One could even posit a geometric rather than arithmetic increase in the value of CxP as C rises, since sacrificing 50% of your wealth may well pinch a lot more than fifty times as much as the suffering you feel from sacrificing %1.In technical economic terms, the marginal utility of income may go down as your wealth increases, but it may actually go up as your wealth decreases. Losing $100 is a small sacrifice for me, but perhaps a big one for a very poor person. An implication of this point is that very poor people will have lower turnout rates than others (because they are the only ones for whom CxP might be significant), and the evidence of numerous turnout studies does indeed bear this out.

In sum, the unwillingness of people to make very large sacrifices to vote, in my view, says little about their willingness to make very small ones. However, Jim's analysis does highlight the need to deal with this point explicitly in future versions of the model (sadly, it is too late to incorporate in this particular publication, which will soon be going to press).

Lastly, Jim, I think, continues to make the mistake of conflating dollar cost with utility, that I pointed out in my previous post. In the paper, I use hypothetical dollars as a convenient symbol for total utility, but I also emphasized that, in the real world, monetary cost is just one part of the total utility of voting.

UPDATE: Here's a simpler way of making my point that occurred to me after I'd already written the body of this post:

Many people routinely contribute small portions of their income to charity, say $100, in order to produce benefits for others. Let's say that a $100 donation produces 100 units of utility for those who benefit from the charity. Most of those who are willing to donate $100 to charity in order to produce 100 utiles would not be willing to donate $50,000 to produce 50,000 utiles. Yet this is not irrational; it simply represents the fact there is a nonlinear relationship between people's willingness to make small sacrifices for others and their willingness to make large ones. Many people are willing to be moderately generous, but very few are willing to be martyrs for a cause.

The same point applies to voting. The person who is willing to sacrifice the equivalent of $10 in order to give his fellow citizens an expected value of $15,000 is not necessarily irrational if she is simultaneously unwilling to sacrifice $1.5 million in order to give them $1.5 trillion, or even to sacrifice $35,000 in order to do so (as in one of the examples Jim analyzes).