Wasting votes
Now this isn't super high. Definitely low enough that the symbolic value of voting may predominate. But what about voting for a third party candidate or a write-in?
The decisiveness here is awkward to analyze. If you have a preference between the two main candidates, you're probably going to be really sad if it turns out the other one won by one vote. That is, the chances you'll actively regret your vote (that is, wish that your vote for your more preferred main candidate was there to be counted) are again something like 10-5. Not that high. But the chances that your vote will be decisive for your candidate are really so near zero as to make numerical calculation difficult.
Think about what such a result would mean. Let's say the polls show the race at say 51-44-5, meaning someone is leading with 51% of the poll, the second place is at 44%, and 5% are undecided or going for another candidate. Margin of error might be 4% or so. So the most naive calculation says that the two leading candidates poll numbers must be off by 8-10 standard deviations, and the 'other' number also must be off by that much, AND everyone must really be thinking that your candidate is the best in the 'other' category.
The probability that the real population dislikes someone at 6 standard deviations from the measured value is about 10-9. With sampling bias and so on, let's take that as the value. So then the probability of the polls being off by so much in all these cases are somewhere in the 10-30 range. At any rate, we're so far out that the shape of what we're computing is in real doubt. Perhaps it is 10 orders of magnitude better. Perhaps it is a hundred orders of magnitude worse. At this point, does it really matter?
The point is, you are accepting a dramatically higher (but measurable) chance that you will regret your vote for a probability of your vote meaning something that is too small to measure. Not a good trade, unless the value of the vote is purely symbolic. Now given the fairly low probability of a vote being decisive, perhaps that's mostly what this is about: the vote is mostly a conversational tool for later: "Well, I voted for XYZ!"
Another interpretation is that on any ballot there are probably races with much higher probabilities of any vote being decisive (or, more loosely, regretted if not cast). The highest of these may be a couple or three orders of magnitude higher -- they may be for local races with much lower total vote counts, or for tighter races. There's good reason to think that on any ballot, the estimate of the value of filling out some part of it may be as high as 10-2 or even 10-1. In that case, given the small marginal cost of filling in the rest, it may be rational to do so.
But whatever the case, it seems to me that voters who vote for third-party candidates or write-in candidates haven't understood that their neighbors are allowed to vote strategically and that they are too. That is, at some level, they think it would be better if everyone just wrote down the name of the person who they really, truly thought was the most qualified, and that the best person would then win by some sort of acclamation, or something. Real-life democracy can't work that way without a mind-reading machine, though. And given the necessary possibility of voting strategically (that is, voting not for the person you think is best, but for a weighted estimate of the person you think is most likely to win who is the most preferable to you), voting non-strategically will always be a waste of your ballot.
Unless you happen to be one of the super-partisans who really does believe that one of the main candidates really truly is the best possible candidate for office. ¶ 11:13 PM
Saturday, November 8, 2008
Why do people vote? Interesting question. There's an analysis [via Yglesias] that estimates that the probability your vote is decisive as about 10-8. Some states are an order of magnitude better (in 2008, topped it off at about 2*10-7).
But presumably the actual value is somewhat blurrier, and you might be really glad you voted if the win happened by only a few votes, up to a more "symbolic value only" value if there's no recount. Assuming recounts are at about half a percent, which translates to several thousand, this gives us perhaps three orders of magnitude, or somewhere around 10-5 chance that you'll be really glad (based on the outcome) that your vote for your candidate was there to be counted.
Now this isn't super high. Definitely low enough that the symbolic value of voting may predominate. But what about voting for a third party candidate or a write-in?
The decisiveness here is awkward to analyze. If you have a preference between the two main candidates, you're probably going to be really sad if it turns out the other one won by one vote. That is, the chances you'll actively regret your vote (that is, wish that your vote for your more preferred main candidate was there to be counted) are again something like 10-5. Not that high. But the chances that your vote will be decisive for your candidate are really so near zero as to make numerical calculation difficult.
Think about what such a result would mean. Let's say the polls show the race at say 51-44-5, meaning someone is leading with 51% of the poll, the second place is at 44%, and 5% are undecided or going for another candidate. Margin of error might be 4% or so. So the most naive calculation says that the two leading candidates poll numbers must be off by 8-10 standard deviations, and the 'other' number also must be off by that much, AND everyone must really be thinking that your candidate is the best in the 'other' category.
The probability that the real population dislikes someone at 6 standard deviations from the measured value is about 10-9. With sampling bias and so on, let's take that as the value. So then the probability of the polls being off by so much in all these cases are somewhere in the 10-30 range. At any rate, we're so far out that the shape of what we're computing is in real doubt. Perhaps it is 10 orders of magnitude better. Perhaps it is a hundred orders of magnitude worse. At this point, does it really matter?
The point is, you are accepting a dramatically higher (but measurable) chance that you will regret your vote for a probability of your vote meaning something that is too small to measure. Not a good trade, unless the value of the vote is purely symbolic. Now given the fairly low probability of a vote being decisive, perhaps that's mostly what this is about: the vote is mostly a conversational tool for later: "Well, I voted for XYZ!"
Another interpretation is that on any ballot there are probably races with much higher probabilities of any vote being decisive (or, more loosely, regretted if not cast). The highest of these may be a couple or three orders of magnitude higher -- they may be for local races with much lower total vote counts, or for tighter races. There's good reason to think that on any ballot, the estimate of the value of filling out some part of it may be as high as 10-2 or even 10-1. In that case, given the small marginal cost of filling in the rest, it may be rational to do so.
But whatever the case, it seems to me that voters who vote for third-party candidates or write-in candidates haven't understood that their neighbors are allowed to vote strategically and that they are too. That is, at some level, they think it would be better if everyone just wrote down the name of the person who they really, truly thought was the most qualified, and that the best person would then win by some sort of acclamation, or something. Real-life democracy can't work that way without a mind-reading machine, though. And given the necessary possibility of voting strategically (that is, voting not for the person you think is best, but for a weighted estimate of the person you think is most likely to win who is the most preferable to you), voting non-strategically will always be a waste of your ballot.
Unless you happen to be one of the super-partisans who really does believe that one of the main candidates really truly is the best possible candidate for office. ¶ 11:13 PM
