Falling under the broad category of "papers I never thought I'd see written" comes this article by Hammad Siddiqi about the social norm of leaving the toilet seat down and whether or not it represents a Nash Equilibrium. He models the situation in terms of a two-player, non-cooperative game with payoffs given in terms of C "cost of changing seat position" and D "cost of husband getting yelled at by irate wife". Needless to say, D>>>C. Not exactly gender-neutral, but read on if you're curious...
Structure of the Game
There are two people, one is a representative of the male species, call him John and the other is
a representative of the female species, call her Marsha. They use
the toilet for two operations; #1 and #2. Marsha performs both with the seat in
the down position whereas John performs #1 with the seat in the up position and
#2 with the seat in the down position. That means, he must change the seat
position appropriately before performing the corresponding operation.1 Assume
that the inconvenience cost of changing the seat position is C. Further assume
that the need for #1 arises with a probability p....Consider the situation in which John and Marsha decide to cohabit and both use the same toilet. This situation is popularly known as marriage. That
changes things for the worse for both parties as far as the toilet operations are
concerned. John argues 'Why does it matter if the seat is up or down? Let's leave
the seat in the position used." Let's call that strategy J. Marsha fights back, "It
must be down or else." Let's call this strategy M.
Too see the payoff matrix, click below:
Siddiqi concludes that while the social norm of leaving the toilet seat down is inefficient, it represents a trembling-hand Nash Equilibrium. You can see in the payoff matrix that both JJ (leaving the seat in the position in which it was last used) and MM (always leaving it down) are Nash Equilibrium strategies, but if Marsha "trembles" to M, the formulation of D>>>C means that JJ cannot survive. The seat will remain down.
Sorry boys, we've got Game Theory on our side.
Hat tip: Crooked Timber
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The equation could use a few more points. I have a tendency to leave toilet seat up because of an experience with a roommate who left the toilet seat as he found it and who also had bad aim. A couple of times of coming in, finding the toilet seat down, and sitting in liquid convinced me to always leave with the toilet seat up no matter what I'd just finished doing. The couple of women I've had as roommates since that time agree that the ick factor is enough to excuse my trouble in leaving the toilet seat down.
Since many men are sufficiently deaf and deficient in long term memory such that D=C, maybe the real-world Nash equilibrium is JM...
What if Marsha has bad aim? And if the cost to John of pointing this out is also >>C? Or is this just my own small hell?
"trembling hand", indeed! I learned long ago to just sit when I pee. Problem solved. Besides, I can use that moment to do some reading.
Yes, I am a total loser.
That, plus my wife keeps my nuts in her purse.
What might I ask would be the perturbation in this Nash Equilibrium if the "trembler" in question was a transexual??