Is there really an answer to this age old question?
A toilet looks best and was esthetically designed, when not in use, to have the seat AND lid down. However . . .
Are you looking for a more scientific answer? There have been mathematical attempts to create a formula that will work in a shared household or office setting to determine correct protocol. These formulas take into account number of male users VS. female users. Taking the average number of trips per day and breaking down the ratio of those trips into #1 and #2 (each requiring a different seat position for the men!). The easiest way would be to place a log (no pun intended) in the bathroom for a week and have people mark down whether they used the seat up or down. Then you will know what works best for your situation.
Are you looking for a more humanistic answer? This can be looked at a few ways. Does one person care more than the other? If so, Just go with that person wants. Is there a practical reason to leave the seat up (in house full of girls with one guy)? What if that guy is 7 years old and can’t aim properly? Maybe its best to leave the seat up so the ladies don’t have to sit in puddles.
I do have one compelling argument for keeping the seat and lid down all the time. Hygiene! If you have a toothbrush within 10 feet of the toilet, every time you flush you are stirring up bacteria and possible making some airborne. Making it possible some bacteria gets transferred to your toothbrush.
