The goal of the great toilet seat debate must be to decide what set of actions for a given household will result in the least amount of effort expended as a whole.
First, lets assume that 1 out of every 4 trips to the toilet for a man is a #2. Thus the seat is needed down for 1/4 trips for men and 4/4 for women, equaling 5/8 of all trips. This means that for any given trip to the bathroom where men and women use it with equal frequency, 62.5% of the time the seat will be used in a down position and 37.5% of the time the seat will be used in an up position.
Scenario #1: To make our math easy, lets say a household is one married couple. Each person uses the toilet 4 times per day, so 8 total uses. If the couple agrees to leave the seat as they found it, and the default position is down, then 5 of the trips (62.5%) will require no action, and 3 of the trips will require two movements each, a total of 6 individual up-or-down actions taken. So for 8 toilet trips, 6 actions were taken, all by the man. Under this scenario, it doesn't matter what order the people use the toilet in, as it has no bearing on how much effort is expended.
Scenario #2: Now it gets more complicated. Sticking with the single couple household, lets change the rules so that each person only flips the lid to his or her own requirements, and doesn't change it when he/she is done. 8 trips during the course of a day, alternating, means the seat is needed Down, Up, D, U, D, U, D, D. Assuming this pattern continues, the seat is down at the end of the day and therefore it's down at the beginning of the next day. Here the seat is moved 6 times per day, 3 for each person. This scenario results in more effort sharing between the two people than scenario #1, but it does not increase or decrease overall efficiency.
Scenario #2b: Unlike scenario #1, #2 will vary according to what order the people use the toilet. If the man uses it all four times, and then the woman uses her four turns afterward, the pattern changes to this: UUUDDDDD. This results in 2 actions, both by the man, resulting in an increase in efficiency. If the pattern is more randomized, say, DDUUDUDUDD, there are again 6 seat moves, 3 by each person.
Scenario #3: Family of 5; Mom and Dad, a son, two daughters. Policy is to leave the seat down when finished. If each person uses the toilet 4 times per day, that means the seat is needed up 30% and down 70%. This would be 12 seat moves per day, 6 per male, and none for the females. If policy changes to leaving the seat how you use it, then given a pseudo-random pattern of DDDUUDDUDDUDUDDDUDDD results in 10 or fewer seat moves per day, 5 for men, 5 for women. On a per-person basis,
Scenario #4: Family of 5; Mom and Dad, two sons and a daughter. Policy is to leave the seat down when finished. If each person uses the toilet 4 times per day, that means the seat is needed up 45% and down 55%. This would be 18 seat moves per day, 6 per male, and none for the females. If policy changes to leaving the seat how you use it, then given a pseudo-random pattern of DUDUDUDDUUDUDDUDUDUD results in 16 seat moves per day, 9 for the men, and 6 for women.
There are many more scenarios possible and lots of tweaking that could be done. What if men only go #2 20% of the time? What if women average 5 trips per day and men only 3?
I think that what these scenarios show clearly is that a policy of leaving the seat down when finished has the potential to waste effort. Men can argue that total effort can be reduced if each person simply puts the seat up or down according to his/her own requirements and then leave it that way.
Actually, the most efficient system would be to have a rigid schedule for bathroom breaks; not only
when each person can go, but
what they can do during his or her time. :devil:
that went on when they found out what I had done. After a while I could take no more and reluctantly agreed to dismantle it. 
