Last week, in my Intro Machine Learning class, we were asked to vote on whether raising the threshold of a K-Nearest Neighbors regression model would overfit or underfit the model. It had already been a long day, and my group was not feeling particularly confident about either answer. Instead of searching through our notes and Google to find the answer to this question, we decided to be rationally ignorant and wait until the rest of the room had placed their tally under one of the two options and just put our tally, or cast our vote, on whichever option was already winning. The marginal cost of looking finding the answer on our own was far higher than the marginal benefit of being the first group to place their tally under the right answer.
The rest of the groups took turns marking their tallies on the board and, low and behold, the votes were perfectly even. I thought to myself, “Oh no. Our p is 1”. My group all looked at each other with sheepish stares. Not only did we not know the answer to the question on the board, but we also were suddenly the deciding factor of the entire class’s consensus. In fear of placing our tally under the wrong answer in front of the entire class, we sunk into our seats and waited for further conversation from the professor. This made me wonder how often voting abstention in the US is in fact a result of rational ignorance. If we were confident in our answer, our p being 1 would've been an easy opportunity to look super smart in front of the whole class. Does the fear of making the wrong choice prevent people from voting in the same way that the prediction that your p is 0 does?
1 comment:
"Does the fear of making the wrong choice prevent people from voting in the same way that the prediction that your p is 0 does?"
I love this question. In order to try an answer it, we may consider the following minimax-regret voter:
Imagine an election between two fictional candidates. We'll call them Hillary and Donald. Let the election be over just one issue, building a border wall, and let it be certain that Hillary would not build the wall while Trump would do the opposite. In order for a voter to have a "fear of making the wrong choice," we require two more conditions: first, that there is a "right choice," and second, that the voter is not sure what it is, but instead assigns probabilities to wall/no wall being correct. The voter will learn what the correct choice was sometime after the election - hence the regret. Assume the cost of voting is low.
Now we assign values. Say the voter assigns a 60% chance to "build the wall" being the right choice (and thus a 40% chance to its converse being correct).
If the wall being built is the right choice and the wall is actually built, there is a U1=1 change in utility.
If the wall being built not the the right choice and the wall is actually built, there is a U2=-2 change in utility.
If the wall being built is the right choice and the wall is not actually built, there is a U3=-3 change in utility.
If the wall being built not the the right choice and the wall is not actually built, there is a U4=6 change in utility.
So the voter reasons that the expected utility from Trump winning is (0.6*1)+(0.4*-2)=-0.2 and from Hillary winning is (0.6*-3)+(0.4*6)=0.6. If they choose to vote, they will vote for Hillary.
But will they vote? No! A true minimax-regret voter will vote or not vote according to the option which minimizes their maximum possible regret. Because we assumed that the cost of voting was low, the voter's regret when their choice was not decisive can both be taken to be near-zero. But if they were decisive and chose Hillary incorrectly, they would feel greater regret than if they would have been decisive and failed to vote! To see this, note that the voter would regret not building the wall when it ought to have been built (U3=-3) more than they would would regret building the wall when it ought not have been built (U2=-2). To avoid the pain of voting for the wrong candidate and regretting it later, this voter will choose to stay home on election day.
The math, in general form.
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