Here are a few views:
1. Economizing hand motions is the key, so just leave it "as is" when done. It might be needed in that same position again.
2. Such matters should be arranged to please your wife. It is signaling and a symbolic recognition of her value. The only question is what you get in return, but if you get anything at all it is worth it.
3. Avoiding midnight surprises is the key, which means always leave it down.
4. Many women don’t like the idea that guests could show up and see the insides of their toilet bowls.
#2-4 all point in the same direction, and I don’t give a damn about #1. But somehow I, like many other men, fail to optimize on this question. The more interesting question is why this remains a issue. Here goes:
1. Women keep it an issue, rather than delivering decisive argumentation, to test their men and their sense of commitment.
2. Men cannot help but rebel against the female ethic of caring, especially when it concerns something so infantile as a toilet seat.
3. Existential freedom. I once had a European roommate, and it drove me crazy that he closed all the doors around the apartment. Perhaps an occasional open seat is a quixotic demand that our universe show true randomness and openness.
4. Men prefer to focus more intensely on a smaller number of issues and this isn’t one of them. But obviously that explanation can no longer apply to me.
Addendum: Mikhail directs my attention to this paper.