By toilet seat problem I refer to the problem of a couple living together, one man and one woman, sharing one toilet. To be more mathematically specific:

For Marsha the seat position transfer cost is 0 since all operations are performed with the seat in the down position. For John the cost is greater than 0 since seat position transfers must be performed.

Let p be the probability that John will perform a #1 operation vs a #2 operation. Assume that John optimizes his seat position transfer cost (see remark 3 below.) Then it is easy to determine that John’s average cost of seat position transfer per toilet opeation is

B = 2p(1-p)C

where B is the bachelor cost of toilet seat position transfers per toilet operation.

Now let us consider the scenario where John and Marsha cohabit and both use the same toilet. In our analysis we shall assume that John and Marsha perform toilet operations with the same frequency (see remark 4 below) and that the order in which they perform them is random. They discover to their mutual displeasure that their cohabitation adversely alters the toilet seat position transfer cost function for each of them. What is more there is an inherent conflict of interest.

This is one of the more rigorous game theory considerations of the toilet seat problem I've read. The solution proposed at the end seems sensible enough.

Let's not allow our current technological constraints and limited imagination confine our solution set, however. I propose a different, even more ideal solution.

We develop a toilet seat that is in communication with the Apple Watch worn by both the man and the woman. When the woman walks into the bathroom, her Apple Watch authenticates itself to the toilet seat which then automatically lowers itself. Meanwhile, when the man walks in, the toilet seat remains in whatever position it's in, per the widely accepted bachelor toilet seat strategy. One could try to further optimize for the man by learning, Nest-style, the general pattern of #1 and #2 operations and caching the last 24 to 48 hours worth of such operations, but the added complexity may only capture a slight marginal decrease in cost to him.

There is yet another solution, brought to mind by episode 4 of season 4 of Curb Your Enthusiasm, in which Larry David admits to peeing sitting down. Optimal for her, and, David claims, good for him as well.

“If I pee twenty times in a day I can get through the whole New York Times, for god's sake!”

That's two posts today that mention bathroom operations. My mind is really in the toilet.