In a large election, an individual vote is almost certain to make no difference to the outcome. Given that voting is inconvenient and time-consuming, this raises the question whether rational citizens should bother to vote.
It obviously depends on the citizen's values. For a completely selfish person, the answer may well be 'no'. Different election outcomes typically don't matter too much for an ordinary citizen's selfish interests; and a miniscule chance of a medium-sized gain does not offset the cost in time and inconvenience.
But most people aren't completely selfish. (And if the few completely selfish voters stay at home, that's no reason for concern.)
What if a potential voter cares about the outcome for others? To simplify, what if she is a pure act-utilitarian who wants to maximize the total expected happiness (or whatever) in the electorate? It is often said that this would still not make voting rational, so that we need to postulate some intrinsic value to voting, or a non-consequentialist obligation to vote.
Along the same lines, in his 1980 paper "Rule utilitarianism, rights, obligations and the theory of rational behavior", John Harsanyi uses the voting case to argue that rule utilitarianism leads to better outcomes than act utilitarianism: rule utilitarians would vote, act utilitarians would abstain.
But let's think this through. Here is one of Harsanyi's examples.
EXAMPLE 1. 1000 voters have to decide the fate of a socially very desirable policy measure M. All of them favor the measure. Yet it will pass only if all 1000 voters actually come to the polls and vote for it. But voting entails some minor costs in terms of time and inconvenience. The voters cannot communicate and cannot find out how many other voters actually voted or will vote.
Under these assumptions, if the voters are act utilitarians then each voter will vote only if he is reasonably sure that all other 999 voters will vote. Therefore, if even one voter doubts that all other voters will vote then he will stay home and the measure will fail. Thus, defeat of the measure will be a fairly likely outcome.
Is this correct? Let's figure out the decision matrix for an arbitrary member of the group.
We'll assume that everyone loses 1 util by voting. If everyone votes, this means that the group has lost 1000 utils in total. To get an interesting social dilemma (or an argument for rule utilitarianism), we want the state in which everyone votes to be better than the state in which everyone stays at home. So the net utility of the "very desirable measure" M must exceed 1000 utils. Let's say it is 2000 utils. The decision matrix for an arbitrary act-utilitarian voter now looks like this.
... 998 others vote 999 others vote vote ... -999 1000 don't vote ... -998 -999
Here, voting has highest expected utility iff the probability of 999 others voting is at least 1/2000. Our voter does not have to be "reasonably sure", as Harsanyi claims, that all the others will vote. Only if she is very confident that some of the others will stay at home is it rational for her to abstain.
Admittedly, in real life it may be reasonable to assign a probability of less than 1/2000 to the assumption that everyone else in a large group will do their share. But in real life we also rarely need absolutely everyone to do their share in order to reach a desirable outcome. Moreover, if we may be confident that at least one person will not show up, this is typically because it is reasonable to expect that at least one person isn't motivated or has forgotten or is unable to come. But if the chance of such a disturbance is in fact greater than 1/2, then groups of (act-utilitarian) non-voters perform better, in the long run, than groups of (rule-utiliarian) voters who often waste almost everyone's efforts.
So Harsanyi's example doesn't work. Nor do his other examples.
I think this illustrates a general fact: people aren't very good at calculating expected utilities -- not even experts in decision theory. We often use heuristics, such as only looking at the most probable state. Of course the (by far) most probable state is that your vote will make no difference. Similarly: reducing your carbon-offprint won't affect global warming, donating to cancer research won't affect whether new cures will be found, going vegetarian will not prevent the rise of antibiotic resistant bacteria. But for rational decision-makers, this is irrelevant. What matters is the expectation of the difference. It's worth sitting down and doing the math.