We model the toilet seat problem as a 2 player non-cooperative game. We find that the social norm of leaving the toilet seat down is inefficient. However, to the dismay of “mankind”, we also find that the social norm of leaving the seat down after use is a trembling-hand perfect equilibrium. Hence, sadly, this norm is not likely to go away.
It appears there is a literature on the subject: Jay Pil Choi argues that the selfish rule of leaving the toilet seat how one prefers it is the most efficient one. Any other rule (always up, or always down) inconveniences each member of one gender twice on each bathroom trip. With the selfish rule, one may not need to alter the seat, depending who came before, and never alters it after doing one's business. The always down rule is only efficient if the inconvenience to females is much larger than to males.
Hammad Siddiqi argues that a crucial element is missing from this analysis: conflict, or in other words game theory. One could do it like in a cooperative game, like Richard Harter. This assumes that members of a household minimize the joint cost of toilet seat moving. But what if they act selfishly, and possibly strategically?
Within a non-cooperative game, player can inflict penalties on others. In this case it would be females yelling at males for leaving the seat up. While leaving the seat always down remains inefficient, as in the two previous analyses, it is now a Nash equilibrium. There are often many such equilibria in a game, so this does not say much. But it is also trembling-hand robust, which means it is still the best strategy if the man (or the woman) sometimes forgets. Hammad Siddiqi concludes by asserting that if females realized that the always-down rule is inefficient, they would stop yelling and allow for a better strategy. There is hope.