Friday, July 22, 2016

The Instrumental Value of One Vote

Over in this Leiter thread, some philosophers seem to be dismissing the instrumental value of voting (for Clinton over Trump) for misguided reasons:

(1) That a marginal vote is "astronomically unlikely to change the outcome."

This is not true,* at least for those who are able to vote in a swing state. According to Gelman, Silver and Edlin (p.325), the chance of a marginal vote altering the election outcome is as high as 1 in 10 million, depending on the state.  Given that the outcome will in turn affect hundreds of millions (or even billions) of people, voting for Clinton in a swing state arguably has significant expected value.

(2) That the system is not sensitive to a single vote, and anything close to even will be decided by the courts or the like.

The claim that insensitivity undermines marginal impact is generally fallacious.

Given that a large collection of votes together makes a difference, it is logically impossible for each individual addition to the collection to make no difference.  While it may be true that an objectively tied vote and an objective 1-vote victory would not be distinguished by the system, there must be some smallest and largest numbers of votes that would in fact trigger a recount or a court case (or whatever), in which case one of those numbers [specifically, whichever one is the difference between a straight victory and a court-delivered loss] provides the new threshold that matters for a marginal vote to make a decisive difference.  (See also the final page of this paper by Gelman et al.)

* = I've previously been led astray by Jason Brennan's model from p.19 of The Ethics of Voting, which really does yield astronomically small chances -- on the order of 10 ^ -2650.  I thank Toby Ord and Carl Shulman for their corrections in this public Facebook thread.

In short, Brennan's mistake (and that of the past researchers he draws on) is to model voters as having a fixed non-50/50 propensity to favour a particular candidate over the other.  Even if the fixed propensity is just 50.5, repeating the odds over 100+ million voters makes the result an astronomically certain victory for the favoured candidate (with a vanishing small standard deviation from the expected result of their securing 50.5% of the total votes).  This is obviously not an accurate reflection of either our epistemic position prior to an election, or of any kind of objective probability distribution over the possible outcomes.  It's a bad model.  A better model would either model different voters as having different propensities [as per section 5 of this Gelman et al paper] or at least take on board our credences over a range of possible propensities (including 50/50) rather than stipulating that a particular non-50/50 propensity holds.

As Gelman once wrote in a comment on Brennan's blog:
[T]he claim of "10 to the −2,650th power" is indeed innumerate. This can be seen in many ways. For example, several presidential elections have been decided by less than a million votes, so a number of the order 1 in a million can't be too far off, for a voter in a swing state in a close national election. For another data point, in a sample of 20,000 congressional elections, a few were within 10 votes of being tied and about 500 were within 1000 votes of being tied. This suggests an average probability of a tie vote of about 1/80,000 in any randomly selected congressional election. It's hard to see how the probability of a tie could be of order 10^-5 for congressional elections and then become 10^-2650 for presidential elections. 

Finally, even if you accept the fixed-propensity model (despite its being demonstrably wrong), my old post on the Best Case for Voting makes the case for co-operating as part of a group that (collectively) achieves its maximum marginal impact by having each (or most) of its members vote.


  1. FWIW, if the Gelman-Kaplan model is right, that makes it considerably easier for me to get to my conclusions in The Ethics of Voting!


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