Alice and Bob share a house which has a single lavatory. Given the various possible strategies that each housemate could use in their toilet seat interactions, which combination of strategies is least inconvenient overall?
There are multiple possible definitions of "inconvenient". We gather data for the eight most obvious metrics, but it's up to the housemates to decide which metric is most important.
This simulation (which is really one simulation for each pair of strategies) runs continuously as you leave the browser window open, gathering long-term averages.
The default strategy for lazy humans, highlighted in yellow, is to always leave the seat as it was used. This is the benchmark upon which we are attempting to improve.
The simulated toilet seat has two positions: up and down (see Further Work).
The simulated toilet was installed with the seat down.
Bob always lifts the seat when possible.
Some strategies yield identical results. E.g. for Alice, "leave the seat down" and "leave the seat as it was used" are identical strategies.
I tried to go into this piece of work with an open mind and few preconceptions, although I had a few hunches. The results were extremely interesting.
There is an answer!
The first and most shocking result is that there is a solution to the problem which is provably perfectly fair - for some metrics. If we consider the metrics "absolute difference in mean flips per day" or "absolute difference in mean flips per visit", then the optimal strategy is for both Alice and Bob to leave the toilet seat in the opposite state from how it was found. Using this strategy, all toilet users always flip the toilet seat exactly once per visit.
It is difficult to imagine a more even-handed approach. Alice and Bob are able to use the same strategy, and the strategy extends to additional housemates, additional lavatories and arbitrary lavatory usage patterns. (Alice and Bob are assumed to have identical lavatory usage patterns. If they do not, the result still holds for absolute difference in mean flips per visit, but possibly not for absolute difference in mean flips per day.)
This was a very surprising result. Although the logic is obvious in retrospect, I had no expectation that this piece of work would find a strategy that was provably fair and perfectly fair. The only way a "better" result could be obtained is if we found a "better" metric of fairness/inconvenience...
What about other metrics?
The other metrics are "mean flips per day" and "mean flips per visit". Here I found another surprise. The default strategy for lazy humans, whereby Alice and Bob both leave the seat as used, is apparently the optimal one.
This is not what I expected to find at all! However, this, too, is quite logical and obvious in retrospect. Minimising mean flips per day/visit is the same as minimising total flips, since we are not able to modify the number of elapsed days/visits at will. This in turn indicates that the best strategy is the laziest one, which is the default one.
Separate data is gathered for Alice and Bob, so that we can compare their individual experiences.
In the default scenario, whereby Alice and Bob leave the seat as used, Alice must flip the seat ~1.63 times per day / ~0.36 times per visit, whereas Bob must flip the seat ~2.7 times per day / ~0.60 times per visit. In either case, Bob is far more accustomed to inconveniently configured toilet seats than Alice. This may explain why the Bobs of the world consider the entire matter to be far less significant than the Alices.
Furthermore, while Alice may feel that ~1.63 flips per day / ~0.36 flips per visit is too many, note that even under ideal circumstances for Bob (Alice leaves the seat up, Bob leaves the seat as used), Bob must flip the seat more than this: ~1.83 times per day / ~0.41 times per visit.
More realistic toilet usage distributions
More metrics. Are there factors other than "how many times I have to flip the seat" that should be considered?
What about closing the toilet seat lid entirely?