Arrow’s Impossibility Theorem

Sean Carroll at Preposterous Universe:

Here is the setup. You have a set of voters {1, 2, 3, …} and a set of choices {A, B, C, …}. The choices may be candidates for office, but they may equally well be where a group of friends is going to meet for dinner; it doesn’t matter. Each voter has a ranking of the choices, from most favorite to least, so that for example voter 1 might rank D first, A second, C third, and so on. We will ignore the possibility of ties or indifference concerning certain choices, but they’re not hard to include. What we don’t include is any measure of intensity of feeling: we know that a certain voter prefers A to B and B to C, but we don’t know whether (for example) they could live with B but hate C with a burning passion. As Kenneth Arrow observed in his original 1950 paper, it’s hard to objectively compare intensity of feeling between different people.

The question is: how best to aggregate these individual preferences into a single group preference? Maybe there is one bully who just always gets their way. But alternatively, we could try to be democratic about it and have a vote. When there is more than one choice, however, voting becomes tricky.

More here.