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:

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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