The seven hats problem

This is a game played cooperatively by seven people.
They are each given at random either a black or a white
hat; they can each see the color of each other's hat, but
not his/her own. They can not communicate in any way.
After a fixed time, they each have to write on a piece
of paper either "white", "black" or nothing. They win if
at least one of them has written something, and if all
those who have written something have written the
color of their own hat.

Question: what strategy gives them the highest probability
of winning ?

You can find some comments here, in particular what the winning probability of the best strategy is.
If you really want to, you can also check the solution.
If you don't find it clear enough send me an email.