Actually, the default position should not be down for two important reasons: there will be more lid-raising/lowering (which we are assuming to be distasteful) and this distasteful task will be distributed in an inequitable manner. And we can prove it with math! (I've seen a similar argument somewhere, so the idea for this is not my own.)
We will need two assumptions regarding relative probabilities which are, admittedly, simplifications. If we knew the actual numbers it would be fun to break it down.
Assumption 1: a co-ed bathroom is used equally by men and women.
Assumption 2: the probability of urination/defecation is 50% for both genders. (If this assumption is changed to take into account the greater need to eliminate liquid wastes, the argument becomes stronger. So this is a very conservative assumption.)
Consider two policies: a) the seat must always be set down, and b) the seat should be left in the position in which it was just used.
Now we can calculate the estimated number of lid raises for these two strategies. We'll let X represent the number of seat raises expected on each visit, so the term of interest is E(X). Letting P(Y) be the probability of event Y and Q(Y) be the expected number of adjustments necessary in event Y.
E(X) = P(female, liquid) * Q(female, liquid) + P(female, solid) * Q(female, solid) + P(male, liquid) * Q(male, liquid) + P(male, solid) * Q(male, solid)
a) The seat always starts and ends down, so Q(Y) = 0 when Y is not (male, liquid) but Q(Y) = 2 otherwise (a raise followed by a lowering).
E(X) = 1/4 * 0 + 1/4 * 0 + 1/4 * 2 + 1/4 * 0 = 1/2 = 0.5. So, on average we expect a trip to a restroom by a random individual to involve half a seat adjustment.
b) The seat always starts in the position it was last used. Since there is a 0.25 probability the seat is up, Q(Y) = 0.25 (always a lowering) when Y is not (male, liquid) and Q(Y) = 0.75 (always a raising) otherwise.
E(X) = 1/4 * 1/4 + 1/4 * 1/4 + 1/4 * 3/4 + 1/4 * 1/4 = 6/16 = 0.375. So, on
average we expect a trip to a restroom by a random individual to
involve 3/8 < 1/2 a seat adjustment.
From a strict utilitarian standpoint, then, we should favor a scheme in which the seat is merely left in the same position as its last use. But now, consider the inequity. In scheme (a), all raising and lowering is done by men, none by women. But in scheme (b), 2/3 of the raising is done by men, 1/3 by women. (In fact, men on average double their number of seat adjustments under the seat-down scheme.)
The leave-it distribution of labor is still inequitable, but it is less inequitable than the alternative. The only way to argue against this inequity is to claim that, by virtue of their gender, an accident of birth, one sex bears additional responsibilities, and that men are the gender that should shoulder this responsibility. For those, like myself, who believe in the equality of the genders, this is untenable.
In conclusion, it's easy to see why women like having the seat down- they are able to eliminate all of their seat adjustments relative to a leave-it-as-you-use-it scheme. The cost is then entirely borne by men, who have to perform an extra 2 seat adjustments for every 1 saved by a woman. The system with greater social utility and equity is obvious. Also, dogs and cats will live together in harmony.