Hmmm. The "twice as likely" one seems wrong in the middle somehow, if twice p=0.5 is "just a bit more than that", the function would sort of accord, but restricting the usage seems as good an answer.

Of course, I need to go and find some suitable functions Fs anyway for my own satisfaction.

That was my eventual conclusion, yeah. I found functions which had all the properties I'd aimed for, but they behaved so unhelpfully in the mid-range that I despaired of getting anyone to even begin to agree that they were a good definition of the phrase "n times as likely" in the general case :-)

Which is always very unsatisfying to write up, isn't it?

I think I will leave this for more productive persuits ;)