"This paper develops an economic analysis of the toilet seat etiquette, that is, whether
the toilet seat should be left up or down. I investigate whether there is any efficiency
justification for the presumption that men should leave the toilet seat down after use. I find
that the “down rule” is inefficient unless there is a large degree of asymmetry in the
inconvenience costs of shifting the position of the toilet seat across genders. I show that the
“selfish” or the “status quo” rule that leaves the toilet seat in the position used dominates the
down rule in a wide range of parameter spaces including the case where the inconvenience
costs are the same. The analysis can be applied to other shared facilities that can becustomized to each user’s preference."
If you're a too long didn't read kind of person, the answer is that - unless the women in a household significantly outnumber the men, it's better to leave the seat as used in terms of time and effort for both parties.
Now, Choi goes into a lot of detail about this throwing about some very complex looking math to prove his point. The paper is very funny, and worth a read and you can check it out at https://www.msu.edu/~choijay/etiquette.pdf
So, in the interests of engendering discussion here, what do you do? Up or down?