Update:

Thanks bravejia for pointing out one case which was missing in my original post.

the winning strategy is that after flip, two strings are still both can-win-or-lose strings. Basically, starting player can choose not to make decision in his first turn.

Take "++++-++++++" as an example again:

A: ++++---++++

This will leave B with two both can-win-or-lose strings. And there will be no further subset can-win-or-lose strings.

Orignal Post:

I think "++++-++++++" is a lose. And here is why.

Both ++++ and ++++++ are can-win-or-lose strings. A can-win-or-lose string is a case where first player can choose if he wants to win or lose.

For example, in case of ++++, first player can choose flip middle two to win, or to flip first two to lose.

So in case of two can-win-or-lose strings, or even number of can-win-or-lose strings, second player can always adjust his strategy based on first player's move.

Take "++++-++++++" as an example,

If A choose to win "++++", B can win by winning"++++++".

A: +--+-++++++

B: +--+-++--++

If A choose to lose "++++", B can win by losing"++++++".

A: --++-++++++

B: --++-+--+++

B can have same strategy If A choose to flip second can-win-or-lose strings first.

So "++++-++++++" is a lose. Feel free to correct me if I am wrong.