Hammad Siddiqi gives us THE SOCIAL NORM OF LEAVING THE TOILET SEAT DOWN: A GAME THEORETIC ANALYSIS in the The Science Creative Quarterly. He looks at the question of leave-it-in-the-last-position vs. leave-it-down via game theory. Here is the conclusion:
For “mankind”, the analysis in this paper has the following appeal: Once again, it has been found that the social norm of leaving the toilet seat down is inefficient; hence, “mankind” may feel vindicated.
For “womankind”, the analysis in this paper is appealing for the following reason: It has been shown that the social norm of leaving the seat down is a trembling-hand perfect equilibrium. Hence, this norm is not likely to go away, at least in the near future.
In our household this battle has been settled, not by yelling or being "right," but by adding other variables. Namely, we don't want the cat(s) drinking out of the toilet. And now that there are children about: we don't want them diving in. (Granted, they are big enough to open the toilet themselves now.)
[found via Twitter]