A guy named Anand Venkataraman at the Speech Technology and Research Lab at SRI International did such an analysis, and he found that the second strategy is optimal in the sense that it leads to less total work performed by all members of the household. Yet his study did not take into account the fact that men do not always need the seat up. Furthermore, his analysis did not take into account the fact that people have competing interests, and do not necessarily seek to minimize the total work of the household. Instead, people usually want to minimize their own total work, and possible that of other people weighted by how much they care about those people over themselves. My analysis draws on but extends Venkataraman's methods. If you check my math and it's wrong, sue me. It's Saturday morning and I'm kind of tipsy.
There are two grown men in my household and two grown women. There is also one three-year-old girl, but we will ignore her potty habits for the moment because whenever she goes potty, someone has to help her do it. Assume that both men and women defecate on average once per day but urinate on average six times per day. This means that men will want the seat to be up six out of seven times per day they go to the toilet. So:
x = 6/7
is the probability that a man would want the seat up on a given trip to the restroom and
k = n x p = 4 x 7 = 28
is the total number of household visits to the toilet by an adult, where n is the number of household members and p is the number of toilet visits per household member.
Each time a male must put the toilet seat up and then down if it is down and he wants it up, he performs two units of work. Each time a female must put the toilet seat down when it is up (and she always wants it down), she performs one unit of work. Assume that people don't like to do work (or perhaps that they like to minimize their risk of infection by minimizing their contact with toilet seats), but that they also don't like others to do work, weighted by how much they care about other people compared to how much they care about themselves. Therefore:
(1) W(g|s) = cw + Σby
where W is the altruism-weighted work that an individual of gender g is trying to minimize subject to the household toilet seat strategy profile s, w is the work done by the focal individual, c is the amount of weight one places on the amount of work they do themselves, and Σby is the altruism-weighted work summed over all other adult members of the household, allowing for a and y to differ among fellow household members. For simplicity, however, let us assume that b is equal across all fellow household members. In this case, (1) simplifies to:
(2) W(g|s) = cw + bΣy
Since all that matters is the relative weight that one places on their own work versus others, let a be the ratio b/c, in which case (2) becomes:
(3) W(g|s) = (1 - a)w + aΣy
Individuals seek to minimize W(g|s) under the constraints placed upon them by the fact that people have to pee and poop, and men prefer to pee standing up (Aside for women: If you wonder why men prefer to pee standing up, imagine having a protuberance between your legs that threatens to drag against the underside of a toilet bowl every time you sit down on it. Then you'll understand.)
I was going to do a detailed analysis in which we do not assume any particular household strategy, but instead calculate the work individuals do under an arbitrary household toilet seat strategy profile. But I don't have all day. Instead, I will follow Venkataraman's strategy and compare the altruism-weighted work that individuals would do under two possible household-level strategies: (1) men always make sure the seat is down (Scheme 1 in Venkataraman's analysis); (2) men leave the seat up and women leave the seat down (Scheme 2 in Venkataraman's analysis).
It makes sense to assume that there is a household level social norm if we assume that toilet seat behavior is actively agreed upon within a household according to preconceived notions about what is proper. Ideally, however, we should endeavor to do a full analysis under arbitrary individual-level strategies. For now, just bear with me.
First, let's calculate the work that male and female household members would do under Scheme 1 (the chivalrous strategy). Under Scheme 1, a female does no work. In contrast, a male does an amount of work that is proportional to the probability that a man's trip to the bathroom is to go piddle, as my Great Auntie Alice would call it, and another man has not gone piddle in a household member's previous trip to the poitty. (I love you, Auntie Alice!)
Before we calculate the amount of work that males do under Scheme 1, we need to calculate the conditional probability that a man's trip to the bathroom is to go piddle and that the previous person to go potty was either a male who went poop or a female. This is equal to:
(4) x[(1 - x)/2 + 1/2] = x(1 - x/2) = z
assuming that there are equal numbers of adult males and females in the household, as in true in my household, and is approximately true in the average human household. Therefore, the work that a female will do under Scheme 1 is
(5) w(f|1) = 0
while the work that a male will do under Scheme 1 is
(6) w(m|1) = 2zp
because putting the seat up and down is two units of work. Thus the altruism-weighted work of a female under Scheme 1 is
(7) W(f|1) = (1 - a)0 + a2zpn/2 = azpn
and the number of altruism-weighted work of a man under Scheme 1 is
(8) W(m|1) =
(1 - a)2zp + a2zp(n/2 - 1) + a0/2 =
(1 - a)2zp + a2zp(n/2 - 1)
where zpn/2 is the total number of potty visits by males in which the male has to put the toilet seat up then down, and n/2 - 1 is the number of other males in the household given that a household member is male. Notice that
(9) W(f|1) < W(m|1)
holds so long as a < 1 (i.e., if individuals care about their own work effort at all; note that this assumes that everyone has equal other-regarding preference a).
What about Scheme 2? Under Scheme 2, a female must put the seat down if the person who went potty previous was a male who went pee, which occurs with probability
(10) (x/2)/2 = x/4 = q
(Recall that females always go potty sitting down, and about half of adult household members are male). A male must put the seat up if he has to go pee with probably z (see equation (4)). Under Scheme 2, the work that a woman does is
(11) w(f|2) = qp
The work that a male does under Scheme 2 is a bit more complicated, because we have to keep track of the both the probability that he is going piss when the last person needed the seat down, and the probability that he is going poop when the last person was a guy taking a piss. The first probability is simply z. The second probability is
(12) (1 - x)x/2 = r
Therefore, the work that a male does in Scheme 2 is
(13) w(m|2) = (z + r)p
Thus the altruism-weighted work that a woman does under Scheme 2 is
(14) W(f|2) =
(1 - a)qp + aqp(n/2 - 1) + a(z + r)pn/2
and the altruism-weighted work that a man does under Scheme 2 is
(15) W(m|2) =
(1 - a)(z + r)p + a(z + r)p(n/2 - 1) +
Note that the condition
(16) W(f|2) < W(m|2)
is true so long as q < z + r, which is always true because 0 < x < 1 < 1.25. Once again, men are stuck doing most of the work, and suffering from the work that others are doing. Why? Because in a household with equal numbers of males and females, it is more likely that a male will have to change the state of the toilet seat than a female will have to change the toilet seat. And why is that? Because all females do is sit down, whereas males change their position to the toilet depending on whether they are going poo or piddle.
While you might say, "Why don't men just sit down when they pee? They should feel so lucky that they can stand up," I've already mentioned that, for most men, sitting down on a toilet runs the risk of having the tip your man bits scrape along the nasty insides of a toilet bowl. We're not lucky that we "get to" stand up while we pee. We do it because sitting down on a toilet is, for a man, an unsanitary thing to do. In fact, heterosexual ladies (and homosexual men), the less often your male sex partner's man bits scrape along the insides of a toilet, the less often you will come into contact with a penis that has been in such a nasty place. Think about it.
Anyway, another question we should answer if it is the case that both males and females are better off under one scheme or another. Mathematically, this addresses the condition
(17) W(f|s(i)) < W(f|s(j) & W(m|s(i)) W(m|s(j))
Women are indifferent to the two Schemes if a = 1/(n/2 + 1). They would prefer Scheme 1 (when women do less work) if a < 1/(n/2 + 1), and they would prefer Scheme 2 if a > 1/(n/2 + 1). So if women care about their fellow household members a sufficient amount, they will prefer to live in a household where they deal with their own toilet seat moving duties. If they don't care enough about their fellow household members, they'd rather free ride on the work of men (Aside: I'm not saying women free ride on the work of men overall house work. Obviously that is not the case. Men are, compared to women, lazy when it comes to housework, even in households where the women have careers. I'm just saying that women free ride on the work of men when it comes to the toilet seat. It is an open question whether men putting down the toilet seat after they piss is just given the amount of work women do in the household compared to men). By the way, in my household, the threshold level of altruistic inclination happens to be 1/3.
What would men prefer? Honestly, men would prefer to sit down and play video games and drink beer all day, but I'm talking specifically about the toilet seat norm. It turns out that men would prefer to Scheme 1 so long as a > 1/(2 - 4/n). In my house, that value is equal to 1. It is impossible for a to be greater than one because it is a weight parameter.
So in my household, males should prefer Scheme 2. Yet females only prefer Scheme 2 if they care more than half as much about other people in the household as they do about themselves (because two thirds divided by one third is one half). That's unrealistic. People usually care more about themselves than they do about other people. So we've got ourselves a dilemma. I have some fieldwork to do, so we'll have to wait until later today to resolve it.
If you find that some of my half-drunk derivation is mathematically incorrect (I don't care if my assumptions are correct, we can quibble about that later), tell me.