Voting and the Heap Paradox

One feels foolish basing a post around a comment made on another blog since it is just this sort of ingrown quality which often gives online discourse a bad name. And yet, it's hard to resist at times, so I'm about to indulge myself. Last week I ran across a comment made by this fellow, on this post, which ran as follows:

What I say is that a person’s individual vote doesn’t matter, in the sense where “matter” refers to having any effect whatsoever on the outcome of the election. It does of course matter in terms of what it does to the person voting.

The interesting thing about this is not, I think, the author's wider claim that voting is merely a training ground for moral relativism (which I find both uncompelling and uninteresting), but rather that it seems like an interesting variation on what is called the "heap paradox".

The heap paradox is as follows. Say you have one grain of sand. You add another grain of sand to it. Do you now have a heap? No. You add yet another grain of sand. Do you have a heap of sand now? No. And yet, if you continue adding grains of sand you will eventually find that you have a heap of sand. Similarly, if you have a heap of sand and you remove one grain, you still have a whole heap; you don't have heap-1.

A heap is, thus, an entity with a definition which is not strictly quantitative, and yet relies on having at least some minimum (though unspecified) number of constituents. There is no specific number of grains of sand which clearly marks the borderline of "heap" and yet it is nonetheless clear that some numbers do constitute a heap (126,452) and others do not (6).

Some similar things seem to be at play in regards to voting. Clearly, in most elections the margin of victory is sufficiently large that if any one person, or any dozen or hundred or thousand had not voted, the outcome would have been the same. But imagine we have an election in which the results are one million votes for candidate A and one million and one votes for candidate B. Whose vote decided the election?

Well, no one vote did. If I voted for candidate B, one could say that without my vote he would not have won, but it is not my vote in specific that decided the issue, because if any one of the million other voters who cast a vote for B had not voted, he not have won. Thus, A was defeated by one vote, and yet no one voter cast that vote, A was defeated by the sum of the actions of all two million and one voters who voted.

Thus, it is in a certain sense true that no person's individual vote has "any effect whatsoever on the outcome of the election", in that no specific vote is "the deciding vote" in an election. Except in the sense of being chronologically the last vote cast, not single vote ever is a deciding vote.

And yet, clearly, if some given number (varying depending on the closeness of the election results) of voters had not voted, or had voted differently, the outcome of the election would have been different. It seems non-sensical to say that the casting of any one vote has no effect, when the sum of several of these "no effect" events could be "effect". It we take "no effect" to be 0 and "effect" to be 1, then if we say that individual votes have no effect we are suggesting that 0+0+0+0+0+0=1

Thus, while it is in a certain sense true that an individual person's vote does not matter, it is true in a sense which arguably doesn't matter. Because while it's true that the vote itself is not by itself decisive, the cumulative (and at times even the individual) effect of not making votes is in fact quite decisive. To suggest that it doesn't matter if you vote because your vote will not "matter" in the above sense makes no more sense than to claim that one can have a heap of sand without actually having any sand grains.