up or down redux

Here's the actual toilet paper, uh, paper.

It's a different approach than I would have done. He's gone the pure theory route whereas I would have probably set up a Monte Carlo with a few more parameters.

I said in the previous post it was a fairly intuitive answer that people should just leave the seat where they left it (selfish mode) rather than the guy putting the seat down (down mode).  One can think of it this way:

1. In down mode the male will more often than not have to raise the seat and then lower the seat. Presumably he pees more than he poops. This incurs 2 seat movements for most visits to the toilet. Plus unnecessary movements when he returns to pee again before the female pees.
2. In selfish mode the guy will always at most have to move the seat once while the female will always at most have to move the seat once.

Two seems more equitable and intuitively seems like it should involve less seat movements.

The paper's summary is as follows

I show that the selfish rule always dominates the other two [down mode or up mode] if the inconvenience costs of changing the toilet seat position are the same across genders. In section III, I characterize the optimal rule for the toilet seat position. It turns out that the selfish rule is the most efficient rule in a wide range of parameter spaces. I also derive the condition that the down rule can be the most efficient one when the inconvenience costs are asymmetric.

By aysmmetric inconvenience costs the author means that one person finds it a bigger pain in the ass to lift the seat up or down compared to the other person. I'll leave it up to individual couples to determine whether this asymmetry exists and to what degree.

Analysis:

• Alpha denotes the share of pee usage of the toilet by the male gender.
• The inconvenience costs is given by Cw and Cm for woman and man
• As far as I can tell the author assumes males and females only pee. This is a problem as males need the seat down to poop. It's possible one could incorporate this parameter into Alpha.
• The author then goes on to calculate the overall Value to the couple based on their inconvenience costs and a discount factor

If the inconvenience cost is the same for male and female then the Selfish Mode is always the best Value as seen in the plot below (the vertical access is costs or sort of a negative value; lower is more value).

The only way that the Selfish Mode does not win out and the Up or Down Mode wins is when the inconvenience cost is significantly more for one gender than the other.

In particular that transition is when the ratio of the convenience cost of the female to male is greater than (1+Alpha)/(1-Alpha). For example when Alpha = 50% (the male pees as much as the female) then the convenience cost of the female must be THREE times that of the male for the Down mode to make sense. Again I will leave it up to the reader to determine their and their partner's convenience costs.

Please don't call me. I'm just the messenger.