This is a continual issue, that is, the optimality of 'being a gentleman' and leaving the seat down (see here). The latest in The Science Creative Quarterly offers an advance (?) to our previous knowledge:

In this paper, we internalize the cost of yelling and model the conflict as a non-cooperative game between two species, males and females.We find that the social norm of leaving the toilet seat down is inefficient. However, 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.

When dealing with toilet seat equilibrium, a trembling hand equilibrium refinement is compelling.