"Obama Won and I Helped!"

I've always been intrigued by the voter's paradox: the difficulty of explaining why you should vote, if there's just about no chance you will cast the deciding vote.  One way to go is to give up the idea that we vote to have an impact.  We can just capitulate and say the only reason to vote is for the enjoyment of being with our neighbors and participating in democracy.  Or some such. But that just doesn't seem to be the whole story.  The fact that voting determines the outcome of the election is a large part of the reason why it's enjoyable.  So I want to hang on to some sort of an impact rationale for voting.

Maybe we can, if we're not too greedy about it.  As a voter, you are a little like A, B, C, D, or E in the above diagram. Suppose I am A.  I get a little bit of the credit for holding up F even though it's true that if A were removed, F would still be aloft. A is not "the deciding vote" for F remaining aloft, but still contributes to F's elevation. Surely you have to say A is helping keep up F. If you say A isn't contributing, you'll have to say the same about each of the other blocks.  Then you'll have to generalize and say none of the blocks are holding up F--which is ludicrous.

In the block situation, the weight of F is resting on all five lower blocks, so A gets some credit, even though A could be removed without F falling down. I'm suggesting voting is like that.  My vote helped Obama get re-elected, even though it's true that he won by a couple of million votes, so if I hadn't voted, he still would have been elected. The analogy isn't perfect, but I think there's something to it.  If you say my vote didn't help, you're going to have to say the same thing about any other vote for Obama.  You'll again have to generalize and say none of the voters made him win--which is ludicrous. The blocks all help, even if there are more than needed.  The voters all help the winning candidate, even if there are more than needed.

Let's have one more analogy.  Suppose I'm trying to make a styrofoam bottle sink, so put a thousand equal-sized marbles in it.  It would have sunk if I'd put in just 800. Any one of the marbles could have an existential crisis. "What's the point?  If I hadn't been in the bottle, it still would have sunk!" That's true, but if it made any one marble superfluous it would make each and every marble superfluous. No, each marble contributes, even though there are more than enough.

So I say each Obama-voter can take pride in causing Obama to win yesterday, even if it's true that none of us cast the deciding vote.* We did have good reason to vote: in order to be causes of his winning. And if you voted for Romney? You get to be proud too. There was a reasonable chance your vote would be part of the cause of Romney winning, even if there was virtually no chance you would cast the deciding vote.

* Thanks to the first comment, I need to clarify. I'm ignoring the US electoral college system here. Given the electoral college system, it really is true that a voter in a very red state like mine--Texas--couldn't help Obama win. But on the other hand, we all think it was important for Obama to win the popular vote, nationally, and we in Texas did help him do that (despite not casting the deciding vote)--or so I am arguing.


Aeolus said...

But you voted in Texas under a winner-take-the-whole-state system. So your vote didn't support Obama's victory, after all. If every Democrat in every red state had stayed home, Obama still would have won with exactly the same electoral-college margin of victory. And every Republican presidential vote in the whole country went for naught.

And it doesn't stop there. Consider 1000 people supporting a load where only 800 are required. Each person is bearing a little less of the load than they would otherwise. So even in those states Obama won, the contribution of individual Democrat voters is reduced in proportion to the excess -- e.g., each Democrat voter in Massachusetts contributed, but not as much as a Democrat voter in Ohio.

Still, a good result.

Jean Kazez said...

OK, yeah, I wasn't thinking about the specifics of this political system, just about the general puzzle of why we should vote, when we're very unlikely to cast the deciding vote. Thanks for the point--I need to clarify.

I think it was at least politically important for Obama to win a majority of the votes, nationally, and we in Texas did contribute to that happening. On the two analogies, it's true that the size of a causal contribution depends on what/who else is contributing. Is it really so implausible that a voter would contribute more to the national outcome if a tiny number of people were voting? That doesn't seem like a terribly strange thing to say.

ʯɲʑɩʛʯɖʋɪʉ ɕɑʒʝɪɪʧʠʘɶ said...

Suppose you need X votes to win. X-1 votes have the same value as 0 votes. By voting you have a low 1/X chance of being that decisive vote, with a high impact of X. So X/X = 1, and your vote, surprise surprise, is worth just one vote.

The same reasoning can be applied to the usual argument when buying meat in a supermarket that buys from suppliers only big quantities. It's very unlikely that your buying one portion of meat will result in the supermarket buying X portions from their suppliers, but the low probability and the high impact factor each other out, and when buying one portion you are still responsible for the impact of one portion.

Deepak Shetty said...

which straw breaks the camels back?

Craig Urias said...

Reminds me of Parfit's harmless torturers.

I don't buy that the electoral college has any fundamental impact on Jean's argument. If votes are "chunkified" rather than tallied directly, it's still the case that Jean contributed to a chunk, and from there the same argument applies. The logistics of the chunkification process don't really matter. That some votes effectively count more than others is just a bit of trivia about this particular chunkification.

Craig Urias said...

@Aeolus "If every Democrat in every red state had stayed home..."

But this kind of post hoc reasoning does not work. The fallacy we are up against is that small probabilities do not matter. You are using hindsight to turn small probabilities into certainties, and yes from that perspective you are right. But that's not something that could apply before the election results were known.

Adam Merberg said...

I once went to a talk by a mathematician friend who argued that while voting doesn't make sense from a selfish point of view, it actually constitutes an effective form of charity. He said data suggested that your chance of casting a deciding vote in an election with N voters is on the order of 10/N. Then, if you think the difference in outcomes in a national election is worth, say, a trillion dollars, then you can look at voting as being like giving thousands of dollars to charity.

Of course, the electoral college raises some complications. I also have not entirely convinced myself that this view of voting makes philosophical sense.