Knowing whether it's rational to vote depends upon knowing the odds. My previous post pointed out how our bad polling data makes it difficult to assess this. But there is an even more worrisome problem.
Imagine I tell you I have a lottery ticket for sale. Suppose it costs you, at bare minimum, whatever income you would earn in half an hour. I tell you the ticket has unknown odds of winning an unknown prize. Should you buy the ticket?
Answer: No. To know whether's rational to buy the ticket, you need to know roughly what the expected utility of the ticket is. Here, that would require you to know the odds and the potential payoffs, or, at least, have someone you can reliably defer to who can tell you whether the ticket is a good deal or not.
Imagine you said, "Well, I've found a solution to the lottery ticket problem. I'm just going to convince myself to believe that the expected utility of the ticket is $5,000. So it's a good deal and I'm buying it." Not very convincing, is it? There is a puzzle in decision/game theory about dealing with false beliefs, but we recognize that in some way or other something has gone wrong here. You now believe the ticket has a high expected payoff and so is worth the cost, but you have no real reason to do so.
Now consider the issue of voting for the purpose of changing the outcome of the election. Everyone in the literature models a vote as being like a lottery ticket. There is some chance it will break a tie and decide the election. So, we take the expected difference in value between the two (or more) candidates and discount that by the chance of being decisive.
What is the expected difference between the candidates? Well, that's usually really hard to say. A recent paper in APSR says that Republican vs Democratic leadership makes no measurable difference in outcomes in a bunch of issues after a few years. What about, say, presidents? Well, again, hard to say. No one could have predicted that Bush II would have to deal with 9/11, or that he would have invaded Iraq and Afghanistan. Very few people, other than some libertarian dudes who are always predicting a crash, predicted the housing crash under Bush II. (Even the libertarians dudes are questionable because they constantly say there's going to be a crash. They might be little more reliable than the Marxists who say that every recession proves capitalism is unsustainable.) After Trump won in 2016, a bunch of my academic friends made apocalyptic predictions about what he would do, which they were not willing to bet on (too bad for me), and which have not yet come true. Hmmm.
Ask yourself: Imagine you wanted to publish a paper in APSR arguing that Dukakis would have done a better job governing the US than Bush I, and further, you have to estimate how much better. Think of how difficult that would be, even for an expert. Most political scientists, economists, and so on, would not be able to successfully model the counterfactuals and make any real assessment. Any such paper would be provisional, and even experts wouldn't really know.
Ask yourself: What is the probability of a vote being decisive? Should you use the same kinds of estimates Anthony Downs did? Lomasky and Brennan? Edlin, Gelman, and Kaplan? Should you use Barnett's new method? Understanding this requires tremendous mathematical aptitude. Alternatively, knowing whom to defer to requires having some reliable method of determining whose judgment you should accept.
Now consider what all this means for the typical voter. Even if some experts know the odds and know the expected value difference between candidates, they don't. My mom has no idea. Your uncle Joe has no clue. Your postal delivery worker has no idea. Further, many of them wouldn't be able to form a reliable judgment if they tried--at least not without extraordinary effort--because they don't and probably can't understand the math, the empirics, or the counterfactuals needed to assess the expected value of their votes. Further, they are not in a good position to judge whom to defer to. Should they listen to Barnett? to Gelman? To Downs? To Caplan? How would they know? Should they use the odds the Economist published a week before the election, which now look to have been very wrong?
For them, casting a vote remains--and will always remain--an unknown unknown. They don't know the difference in the value between the candidates and they don't know they probably of being decisive. If it's not rational to buy a lottery ticket in such situations, why would it be rational to vote?