The idle ramblings of a Jack of some trades, Master of none

In 2002, Jay Pil Choi wrote a neat little paper1 on the economics of the position of a toilet seat. It is a remarkable tour-de-force, very tongue-in cheek but also filled with insight. He investigates whether there is any economic rationale for women insisting that men leave the toilet seat down after use. His use of the word 'economic', of course, is rather general - the cost of inconvenience to women is not merely financial, but also, very likely, hygienic and psychological.

It is fairly clear that the perpetually 'down' position is not efficient over the entire population of toilet users. Men will have to raise the seat and lower it after use, whereas women don't have to touch the seat at all. (We ignore the horrible consequences of flushing without closing the lid.) An alternative formulation is the 'selfish' one (Choi's usage), where men leave the seat up after use, and for women to leave it down. The inconvenience cost applies only when the previous user was of the opposite gender. The worst case scenario is for the toilet to be used alternately by men and women, with each incurring the operational cost of moving the seat up or down. If we assume that men and women use the toilet at random, the overall inconvenience cost is less. The selfish strategy is better than the perpetually 'down' strategy.

Choi goes on to show that the 'selfish' strategy is in general the best possible strategy for toilet users - if the inconvenience cost of changing the toilet seat position is the same for both genders.

This assumption may appear to be a rather big one. There are some hidden inconvenience costs involved, which Choi helpfully lists: "unwittingly placing one's bottom directly on the bowl"; "risk of falling in by sitting down without looking when the seat is up"; "leaving sprinkles on the seat when it is down." Women may accuse men of that last inconvenience, but as Ellen Degeneres has pointed out, women are not exactly saints in this matter either. (I wish I could find the video where she announces this finding.) Men might claim they are better aimers of the stream, but let's face it - the aim is not always great and anyway we do it from a greater height, and the spray is not always deterministic in direction. And inebriation does not help.

Choi shows that if the proportion of male users in any given period is α, and the inconvenience cost of lowering the seat for women is cf, and the inconvenience cost of raising the seat for men is cm, then the women-friendly perpetually 'down' rule is better than the men-friendly perpetually 'up' rule if and only if
Further, he shows that the perpetually 'down' rule is better than the 'selfish' rule if and only if
So if men and women visit the toilet with the same frequency (α = ½), the inconvenience to women should be three times higher than that to men to justify the 'perpetually' up policy.

When you consider that men are afflicted with nocturia more commonly than women, it makes sense that - at night, at least - the 'down' rule is patently unfair.

Choi goes on to try variations on the policy governing the toilet position, and shows that the selfish rule outperforms all of them.

In all this analysis, Choi misses out one glaring fact: the reason for the toilet visit is the chief determinant for the seat position. After all, men do lower the seat to poop. So we have need for a finer discrimination in our analysis. This is provided by Richard Harter2.

If we assume that men and women perform toilet actions at the same frequency, but perform one operation (which he euphemistically calls #1) with probability p, then the cost to a bachelorette is 0, because she'll always leave the seat down, whereas for a bachelor, the cost is
where C is the inconvenience cost to the bachelor.

If the man and woman are living together, both of them will find their inconvenience cost jump. Harter studies the 'why does it matter if the seat is up or down' strategy proposed by the man, and the perpetually 'down' strategy, and determines the cost (M, for men, W for women) under each:

The marginal costs (difference between bachelorhood and cohabitation) then are

Clearly, the woman's cost is greater than the man's for any p less than 1. She therefore objects.

Under the perpetually 'down' strategy, the woman incurs no incremental cost at all, whereas the man faces
In the interests of marital harmony, therefore, neither strategy is fully satisfactory. The only via media is for both parties to equalise the respective incremental costs. Harter shows that this can happen only if the man leaves the seat up after performing operation #1 with a frequency
He goes on to say that since p is not readily measurable, we can assume p = 2/3;, in which case f = ½, and so the man should follow the simple strategy: in the morning, after performing operation #1, leave the seat up; in the evening, put it down.

This is not the end of the story, of course. As Hammad Siddiqi points out3, if the woman finds the seat up, she will yell at the man, leading to an incremental cost to him of D over and above the inconvenience cost C he incurs in moving the seat down. The yelling is somewhat shrill, so we can assume that D is much, much greater than C.

In this case, the Harter approach of a cooperative game is flawed, and Siddiqi proposes a non-cooperative game, and determines its Nash equilibrium. He finds, to his dismay, that the social norm of always leaving the toilet seat down after use is not only a Nash equilibrium in pure strategies but is also trembling-hand perfect. So, we can complain all we like, but this norm is not likely to go away.

Fortunately, though, all hope is not lost. Siddiqi notes that an important part of social norms is to improve social welfare. He is able to show that the perpetually 'down' strategy decreases social welfare, and makes a strong plea that this particular social norm be quickly amended.

This is why I think the Eastern-style squat toilet is the answer to all these ills.


1. Jay Pil Choi, "Up or Down? A Male Economist's Manifesto on the Toilet Seat Etiquette", Michigan State University, November 2002.

2. Richard Harter, A Game Theoretic Analysis of the Toilet Seat Problem, Science Creative Quarterly, Number 4, January 2006.

3. Hammad Siddiqi, "The social norm of leaving the toilet seat down: A game theoretic analysis," MPRA Paper 856, University Library of Munich, Germany.


km said...

Bravo. I need to read this to the wife.

//just kidding. I *dare* not question the Edict.

Maddy said...

hey man - in much of the east e.g india the toilet lid has no real purpose - it is never used and lies as a flap for some unforseen future...

Fëanor said...

KM: I assume from your ensuing silence that you've been chewed out and hung to dry.

Maddy: I shudder to think of the backsplash spreading what not all over the environs when the toilet is flushed with the lid up.

Sue VanHattum said...

Sexist, the part about the yelling being shrill. Also, if you think about why women have gotten to choose preferred seat position, you might realize it has something to do with who cleans all the misdirected pee (more of which comes from men). I skimmed and did not see mention of this.

I know one man who does pee sitting down. Bravo for him. My son peed sitting down, until various people took it upon themselves to 'teach' him to pee standing. No one has managed to teach him how to get it all in the potty, and I've been cleaning pee from the floor for 5 years now. (He's 8 and very squeamish about touching icky stuff; eventually, I hope to turn this job over to him.)

If he starts doing the cleaning, I'll be ok with the seat staying up (although it does look tacky).

Anonymous said...

Actually, there is a very good reason why the toilet lids of flush toilets should always be lowered before flushing, whether users are male or female.

"It is well known that toilet flushing produces aerosols that can spread infectious diseases. In the paper cited below, the authors deliberately contaminated a toilet with enteric bacteria (Serratia marcescens) and a virus (bacteriophage MS2). Serratia and MS2 were detected in the air after the first flush. Subsequent flushing also released microorganisms into the air, although the numbers declined after each flush. The authors conclude:

Many individuals may be unaware of the risk of air-borne dissemination of microbes when flushing the toilet and the consequent surface contamination that may spread infection within the household, via direct surface-to-hand-to mouth contact. Some enteric viruses could persist in the air after toilet flushing and infection may be acquired after inhalation and swallowing.

There is evidence that plumbing was responsible for spread of SARS coronavirus during the 2003 epidemic. In contrast to influenza virus, it is clear that the SARS coronavirus can replicate in the human gastrointestinal tract, and that fecal shedding is a significant mechanism of transmission."


Anonymous said...

Flushing with the lid up can pose health risks, certain viruses can be shed in urine and feces, and will spread in the air and walls when the toilet is flushed. Rotavirus are frequently found in feces, so you can get gastroenteritis just by inhaling the air in a toilet that was flushed in the past few hours. Nastier viruses like SARS, hepatitis A can be also be spread that way. So put the lid down.

Post a Comment