Originally Posted by Dianeva
Or at least that it's so unlikely to make a difference (for there to be a point off from a tie) that it isn't worth the effort of voting. Just as lottery tickets aren't even worth spending $1 to buy no matter how big the prize is if the probability of winning is like 1/10^7.
This seems to be the crux of the voting question. As in the case of lotteries, the decision to participate should ideally be based not just on the probability of there being some kind of payoff that follows from your participation, but also on the size of that payoff. If the lottery prize is large enough relative to the probability of that payoff being realized, it might make good sense to enter the lottery. This notion is traditionally formalized in terms of the expected value (EV) of choosing to participate. Letting u and p be the size of the payoff and its probability of attainment, respectively, EV is defined as u*p. In other words, if a lottery offers a 1 in a million chance of winning a million dollars, and the price of a ticket is greater than $1, then entering the lottery is probably not a smart idea; but if the price of a ticket is less than $1, then I might want to buy a ticket (or two).
We can use these basic concepts to do a (very) rough calculation of a person's EV of voting, restricting ourselves to the particular case of an American voting in a presidential election.
As above, let u = the amount of money such that you are indifferent between being given u and getting to choose who wins the election. So u is an index of how much you would value being able to choose the candidate of your choice to win, in American dollars.
Let p = the probability that your vote will be decisive. In a recent paper (LINK), some statisticians (including Nate Silver of FiveThirtyEight fame) estimated this to be between 1 in 10 million and 1 in 10 billion, depending on the state you live in, with most Americans being around the 1 in 60 million range. So let's let p = 1/60000000.
At this point I should perhaps mention that using the probability of a decisive vote as an index of individual vote influence is not considered controversial by academics and scholars in this area. While there is substantive disagreement on whether it makes good sense or not to vote, nearly all of the disagreement stems from these broader issues of the expected value of voting, not from disagreements about whether it is appropriate to consider probability of a decisive vote as a good indication of individual influence. There is no way of deciding which individual vote was decisive (in the event that a single vote was decisive), but it is fairly straightforward to determine which groups of people had more influence on the outcome than others.
With that aside, let's say that in order for us to consider voting to be worth it for us, we need EV = u*p > $5 (for the nonAmericans, this is roughly the price of an average McDonald's lunch ticket).
For this to be true, we need u > $300 million. In other words, we need the relative value of our preferred election outcome, compared to the other possible election outcomes, to be worth more than $300 milllion to us.
The first impression of this number (at least for me) is that it is awfully high. Our estimate is subject to some variation based on, e.g., what you consider to be a sensible minimum EV to make it worth voting, where you live, etc. But within any reasonable tweaking of these parameters, you're always going to end up with a Big Number.
There are two basic insights we can begin to draw from this. First, if you are a purely selfinterested voter, there is virtually no conceivable way that voting in an American presidential election makes sense for you.
However (and this is the second point), if you start to view voting as an altruistic act, akin to donating to a charity, it starts to make more sense. Andrew Gelman has a nice way of framing this HERE (I adopt his framing although he illustrates using different figures; see the point above about variation). As Gelman points out, if your vote is decisive, it will make a difference for 300 million people. So, using our figures from above, if you believe that the difference between outcomes of your preferred candidate winning and the other candidate winning is equivalent to, on average, every American's quality of life being improved by > $1, and this is what you care about rather than how much it will benefit you personally, then voting could make sense. Of course, reasonable people could disagree on whether this belief about the anticipated outcome is realistic. (The "on average" clause from above is important because, realistically, the difference in outcomes will mean huge losses for some people, huge gains for other people, and relatively little for many others in between  the consideration here is whether, after averaging across all those gains and losses, we have a net gain of > $1 for all, or > $300 million in total.)
Whether a similar line of reasoning would lead you to quitting your job and becoming a fulltime lobbyist is left as an exercise to the reader...


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