Question There's a logical scenario which often comes up in discussions around the question of voting. We all know the conversation... Person 1: I don't vote because my vote has no impact on the outcome of the election. Person 2: Not on it's OWN it doesn't, but if everyone thought that, no one would vote, and THEN what would happen?! Person 1: But I don't decide whether all those other people vote, I only have control of my 1 vote! My question here relates not to whether or not one should or shouldn't vote, or to the voting example alone, but rather to the logic of this situation. For this example let us assume (for the sake of the point I am interested in) that it is universally agreed that all people (including Person 1 and 2) agree that nobody voting is an outcome that everyone wishes to avoid. And also assume (despite the conversation above!) that everyone decides privately whether to vote or not, such that their decision cannot influence others decisions) Finally assume that the election involved has never been decided by a margin of less than 1000 votes. To me Person 2's argument is something like the following: There would be negative consequences of a number of people doing X, therefore no one should do X, even if any one of them taken in isolation has no impact. So I guess my question is: Is this a logical fallacy? I've tried to search for discussion of this scenario before on the web but never found anything. There's probably even a name for it! Help... Cheers Pip

Accepted:
November 3, 2010

Comments

I don't think there's a named fallacy here, but I do think the principle proposed by Person 2 is unsound. If this principle were sound, then it would be impermissible to remain childless even in a world as overpopulated as ours.

The principle can be revised to be more plausible. When many people in some group are making a morally motivated effort to achieve a certain good that would not exist (or to avert a certain harm that would not be averted) without their effort, then one has moral reason to do one's fair share if one is a member of this group. This sort of principle against free-riding on the moral efforts of others can explain why one should generally vote and do so conscientiously -- at least unless one has conclusive reason to judge that enough others are already acting and that one's own effort will therefore add nothing to the outcome.

But there is also a more direct explanation of why one ought to vote. As philosopher Derek Parfit has argued, the extremely low probability of one vote affecting the outcome is compensated by the extremely large moral importance of the outcome. Thus Person 1 would likely concede that one ought to vote in "small" elections where one's vote may very well affect the outcome, e.g. in the mayoral election of one's tiny home village -- with 62 other voters, say. In this case, the probability that one will cast a deciding vote is a whopping 10 percent.

I derive this percentage as follows. With 63 people voting, there are 63!/(32!*31!) voting patterns where one side wins by one vote and an equal number of voting patterns where the other side wins by one vote. I divide this sum by the total number of voting patterns -- 2^63 -- to estimate the percentage or probability of "extremely close" outcomes. I then multiply by 32/63 to reflect the fact that only 32 out of the 63 people voting have actually cast decisive votes. More generally, for any odd-numbered electorate of 2n+1 voters, the probability that any given vote will be deciding is (2n)!/(n!^2*2^2n).

Now suppose you live in a town with four times as many other voters: 248. Do you now have less reason to vote? You'll be less likely to affect the outcome. But the outcome also matters more from a moral point of view -- nearly four times as much, I would think, because the new mayor will be seriously affecting the lives of nearly four (249/63 times) time as many people as in the first scenario.

While Parfit concluded that a morally motivated person has as much reason to vote in a larger election as in a smaller one, I can strengthen his conclusion by showing that the moral reason for voting actually becomes ever more weighty as the size of the electorate increases. This is so on the assumption that the importance of voting is proportional to the expected impact of the vote, that is, to the magnitude of the impact of the outcome multiplied by the probability of casting a deciding vote. The former of these factors is proportional to the size of the electorate. The latter of these factors is roughly proportional to squareroot (1/n). Thus, in the village of 249 voters, the probability of casting the deciding vote is still 5.06 percent, about half of 10.09 percent rather than merely a quarter. The expected impact as defined then goes up roughly with the square root of the size of the electorate. A public-spirited voter, then, who considers equally the impact of the vote upon all citizens, arguably has, other things being equal, more reason to vote the larger the election in which s/he is eligible to vote.

Here are the numbers from a quick spreadsheet calculation, showing the number of voters (yourself included), the probability of casting a deciding vote, and the expected impact (probability of casting a deciding vote multiplied by the overall number of voters):

____63_____10.092%______6.36

___249______5.061%_____12.60

___631______3.178%_____20.05

__2491______1.599%_____39.83

__6301______1.005%_____63.34

_24901______0.506%____125.91

_63001______0.318%____200.27

249001______0.160%____398.14

We see here that the probability of casting a deciding vote declines roughly with the square root of the size of the electorate. Assuming that the moral importance of the outcome increases roughly with the size of the electorate, we can conclude that the expected moral impact (and hence the strength of the moral reason in favor) of voting increases roughly with the square root of the size of the electorate.

In a US election, you might have about 120 million votes cast, so here your probability of casting a deciding vote is about 0.0073% (keeping the electorate constant, we can expect one in 6850 US elections to be decided by one vote) and the expected impact of your vote (continuous with the above calculation) is about 8740. Note that, if your vote is deciding in such an election, then a lot of other votes are also, like yours, deciding votes: in the simple case I have been discussing (assuming only two choices and leaving aside complications such as the US Electoral College), you would be one of 60,000,001 people each of whom was needed to outvote 60,000,000 others.