Up or Down?

It’s an issue that has plagued humankind ever since men and women began living together - with indoor plumbing.
And while most of us come to some kind of “resolution’ with our better halves (one way or another), the issue of whether the toilet seat should be left up or down is nonetheless one of those things that generates a lot of “passion’ among the parties concerned (particularly in the wee hours of the morning). Scientific inquiries into the matter are almost non-existent. What, you say? ALMOST non-existent?
Indeed, we have unearthed an analysis, titled “The Social Norm of Leaving the Toilet Seat Down: A Game Theoretic Analysis,’ by Hammad Siddiqi that purports to apply scientific analysis to this societal challenge.
Other Studies
Nor is he the first to take on the topic. Siddiqi cites two relevant studies on the matter, specifically (Choi, P., “Up or down? A male economist’s manifesto on the toilet seat etiquette,’ Michigan State Working Paper (2002); and Harter, R., (2005), “A game theoretic approach to the toilet seat problem,’ Science Creative Quarterly. However, he writes that the former “argues that the rule of leaving the toilet seat down after use is inefficient in the sense that there is at least one other rule that outperform this rule. The unit of analysis in Choi (2002) is the household and the efficient rule is defined as one that minimizes the total cost of toilet seat operations per household.’ However, he notes that Choi does not model the issue as a situation of conflict (I’m betting Choi has never been on the wrong side of a toilet seat placement), and hence, from the perspective of Hammad Siddiqi, the author of the newer paper, “ignores the game theoretic aspects of the problem.’
On the other hand, Siddiqi says that Harter’s 2005 analysis models the situation as a cooperative game and “proposes a contract that splits the costs of toilet seat operations evenly among the parties.’ Siddiqi notes, however, that both papers agree that the social norm of leaving the toilet seat down is “inefficient,’ in the sense that it does not minimize the total cost of toilet seat operations per household. He goes on to find fault with the two papers for failing to address an important concern: If a female finds the toilet seat in a wrong position, then she will most probably yell at the male involved. “This yelling inflicts a cost on the male,’ Siddiqi writes, and “based on this omission, women may argue that the analysis in these papers is suspect.’
The Costs of Yelling
In what purports to be an effort to redress this shortcoming, Siddiqi’s paper internalizes the cost of yelling and models the resulting conflict as a non-cooperative game between two species, males and females. The paper proceeds to outline the situation as a non-cooperative game of conflict between two people: one male (John) and one female (Marsha), who initially live alone, and who each have access to a separate restroom with one toilet. The two “use this toilet for two operations” (which he aptly refers to as #1 and #2), he goes on to say – and then assigns an inconvenience cost alongside a probability factor.
Having established that as a baseline, he turns his attention to the situation, “popularly known as marriage,’ when these different approaches can come into conflict.
Ultimately, he finds that, while the social norm of leaving the toilet seat down is inefficient, he notes that, “to our dismay, we also find that the social norm of always leaving the toilet seat down after use is not only a Nash equilibrium in pure strategies, but is also trembling-hand perfect. So, we can complain all we like, but this norm is not likely to go away.’
Not that I understand either a Nash equilibrium (in pure or impure strategies), and I’m almost afraid to ask what “trembling hand perfect’ is supposed to suggest – but the essence seems to be that it may be “inefficient’ to leave the seat down, but that doesn’t mean society won’t expect certain behaviors.