We have recently been subjected to a barrage of messages from corporations, Bollywood stars and page three familiars exhorting us to go out and vote. The subtext is that if you do not vote, you do not "care". After the poor turnout in South Mumbai, there was much anguish and soul-searching by perpetually anguished, professional soul-searchers such as Barkha Dutt on exactly this matter.
I wish to point out that it is perfectly rational for voters who "deeply care" (in a sense which I will make precise below) to abstain from voting. The argument is very simple and runs as follows. It is extremely improbable for a voter to be able to influence the outcome in a large election. A voter can be influential only if the other voters are exactly divided in their votes for the best candidate. One does not need a Ph.D in probability theory to realize that (i) this is an extremely unlikely event and (ii) this probability will decline rapidly as the number of voters increases. For instance if there are only two candidates and voter preferences over the candidates are equally likely, then the chances of being influential is 0.5 if there are three voters, about 0.03 if there are 1000 voters and very close to zero if there are 10,000 voters. Therefore, even though you care deeply about the outcome of the election, your expected payoff from voting is likely to be very small; if you have to offset these gains against the cost of voting (these costs are not necessarily monetary; they may represent the discomfort of standing in queues and so on) you may decide quite rationally not to vote even if these costs are very low (as they are in places like Delhi and Mumbai).
The argument above for not voting involves a curious inconsistency. Suppose all voters argued in the same way and concluded that they should not vote. Then every voter would be influential and would gain by voting! A formal game theoretic way of saying this is to say that for all voters not to vote, is not a Nash equilibrium of the game (Nash, here, is John Nash of ``A Beautiful Mind''). So what is the Nash equilibrium here? Suppose that there are N eligible voters (N large) with different voting costs denoted by c. Assume that the proportion of voters with voting costs less than c is given by F(c). Clearly F(c) increases as c increases. Assume that each voter benefits an amount α (let us not quibble at this moment about how these things are measured) if her preferred candidate wins. A "caring" voter has a large positive α and an apathetic one, presumably a small positive one. Let p(n) denote the probability of a voter influencing the outcome when n voters actually vote. It is clear that p(n) declines as n increases. Let c* be a solution to the equation p(NF(c*))α =c*. Some harmless assumptions regarding the functions p and F (continuity, etc) will guarantee the existence of a solution. The Nash equilibrium of the game is that voters with costs below c* will vote while those with costs above c*, will not. The point here is that the turnout on which voters' decisions to vote are based, is exactly the one generated by those decisions.
Is the discussion above "useful" in any sense? I think it is, if you are interested in motivating voters to vote. If your message is "Vote because you can choose a better Government", you are trying to get voters to increase their α. This is not likely to have a large effect because the p(n) term is already very close to zero. A better strategy is to emphasize that voting is duty just like paying taxes and not throwing garbage into your street. The effect of this is to add a positive constant K on the left hand side of the equilibrium equation. Voters get this benefit independently of the outcome of the vote - you can think of this as the "warm glow" you get when they put that ink on your index finger. It is quite easy to verify that if K is large enough, you get a corner solution where all voters irrespective of their voting costs, vote.