Let me briefly revisit my random variables class....
Define an (admittedly minor) element of work to be changing the position of the toilet seat. Assume a random normal distribution of bathroom habits -- men and women visit the bathroom with equal regularity and have either reason for visiting with equal frequency. We neglect vomiting as a rare occurrence outside of pregnancy.
Under the assumption that the seat should always be left down, there are no elements of work involved in a woman going to the bathroom -- it starts down, ends down, doesn't move. There is an expected one element of work done each time a man goes to the bathroom -- either zero or two, depending on where he needs it. The average expected work per person per visit to the bathroom is one-half unit.
If the seat is left where it was last used, then moved to where it is needed, things become a little more complicated. Assuming equal distribution again, there is a 75% chance that the seat was already down. A woman would never need to move it, a man might need to move it. In the remaining 25% of the time, a woman would always need to move it, a man might need to move it. Average expected work per person per visit is three-eights unit.
Moving the seat where you need it requires 25% less overall work.
Consider gender allocation: If the seat is always down, the man's expected work per bathroom visit is one; the woman's expected work is zero. If the seat is moved as needed, the woman's expected work is one-quarter; the man's is one-half. Guys are basically asking women to take a share of the effort.
Of course, the assumption of all probabilities being equal is not necessarily accurate. If the number of males and females in the house is not equal, if one (or more) of the women are pregnant, if people in the house have markedly different bladder capacities, etc. then adjust probabilities accordingly.
Less tongue-in-cheek, I can understand wanting the
lid down for aesthetic, pet-drinking, or accidental-object-dropping reasons. Lid down implies seat down. Too, that way work is shared -- everyone has to lift the lid (and possibly seat) and put it back down. Work per person per visit is
always two. Higher work for the aesthetic you want, but equality is achieved.
Pets and people falling in? Really an issue of what you're used to. My cats never had an issue of falling in, and actually they didn't drink from the toilet as long as their water bowl was full. But if you don't care about the
lid then trying to specify a seat position just seems pointless to me.
