Because most philosophies that frown on reproduction don't survive.

Tuesday, November 27, 2007

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.


Patrick said...

Very interesting. However I think there is a sense in which the decision to vote or not vote can matter very much. Apply a cost-benefit analysis.

First, what are the odds that MY vote will be decisive in a national election? This can be measured statistically. It is very low.

Second, what are the odds I will be run over by a truck, mugged, or otherwise harmed while I am on the way to the polling place? This is also measurable. It is also very low - but is it lower than the chance of my vote being decisive?

If the chance of my vote being decisive is lower than the chance of my being harmed while in the act of voting, then voting is an irrational thing to do. The cost is greater than the benefit.

If fewer people vote then the calculation changes, of course.

IlĂ­on said...

It seems to me (and Patrick's comment is in line with this idea) is that the whole "my vote doesn't count" whine/excuse is but a variation on the infantile demand that the universe revolve around oneself.

"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"

Perhaps, someday, you'll look at "Darwinism" in light of this idea.

Darwin said...

"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"

Perhaps, someday, you'll look at "Darwinism" in light of this idea.

I would tend to dispute the contention that "Darwinism" involves a sum greater than the parts, except to the extent that physical science in general limits itself to a subset of reality -- examining only those things that admit to direct and repeatable physical causes.

LogEyed Roman said...

Darwin, I like your analysis. I also have been going by an idea something like Patrick's since my early 20's. I felt discouraged by voting because it seemed like my one vote could not possibly make any difference. But I came up with a less elegent version of Patrick's cost-analysis. You never know.

I also have since realized that we have no idea just how much our actions affect the world around us. I have had prodigious effect on my life due to apparently minor actions, even simply acts of speech, of others. In fact one single vote can have repercussions we cannot imagine. Moreover, every single vote does make a change in statistics. Over time a single voter voting consistently in a certain way can have more effect than thousands who either do not vote or who change their approach a lot.

Grains of sand? Yes. How may grains does it take to start an avalanche?

LogEyed Roman

Vitae Scrutator said...

It seems to me that you're falling into a fallacy here, the fallacy of decomposition, where you attribute some property of a whole to its various parts.

I usually hear this argument from my wife who, whenever I suggest that a single vote "doesn't count" always responds by saying "well, if everybody thought that way, then nobody would vote."

But, of course, it is verifiably never the case that nobody votes, so in practical terms a single vote really doesn't count in the sense that an election is going to be determined by a majority of one (of course such an outcome is logically possible, but I don't know that it has ever actually happened in non-rigged elections). It seems irrelevant to say that the election is determined by a whole bunch of "single votes" all added together.

On the other hand, I think that I do agree that voting is important to do, and I have argued as much here. I vote even knowing that my vote isn't going to have any effect on the outcome, which just goes to show you what a Platonist I really am.

Patrick said...

I must clarify that my example above was just an illustration. I vote every time and have so far not been injured in the process.

The strict analysis I described leaves out some intangible benefits of voting. For example, we may want our children to see us vote in order to teach them the importance of being good citizens. We get that benefit even if we lose the election.

My example also overstates the risk. Yes, I could get hurt while driving to the polling place - but I could also get hurt just by staying at home. It's the marginal change in risk that would have to be considered.

Anonymous said...

I agree generally with Scott Carson's understanding (including his linked post). I would merely add that it isn't just important to vote against the Nazis; it is often important to vote against the ballot itself. After all, by the time the ballot is printed most of the important decisions have already been made, the ballot and its context being a product of those decisions.

In addition to what Scott said I would continue by observing that voting does have a significant impact, though not on the outer world, but rather on the person who votes. Since that is where in point of fact all of the significant impact of an act of voting occurs, that is where the evaluation of that act should be assessed, even if on utilitarian terms. Impact on the outside world is of negligible magnitude in comparison, in Patrick's cost/benefit framework.

Finally, I also appreciate Ilion's preemptive verdict of "infantile". Nothing presents a rational argument quite like gratuitous insult.