Generally an idea can be any kind of know how. You could've invented a faster algorithm that gives you competitive advantage for some time. If you share the algorithm the advantage is lost.

I don't believe that sharing all your business know hows is going to be beneficial to you.

Feedback. The benefit of sharing your ideas is that other people get to spot holes in them before you find those holes the expensive way.

In your analogy: your faster algorithm is probably not optimal. Sharing it with some people who can give feedback will make it better. Those people are probably not your competition, and your competition can talk to those people even if you don't. If the algorithm is going to make that much difference, then if your competition aren't working on it they're incompetent, and if they are working on it then who's to say they haven't got the better version already?

Your losses from sharing the algorithm are negligible (because the competition are either incompetent or working on it themselves) but your benefits from sharing and getting feedback are large.

You completely dismissed the possibility that you come up with something better than anyone has and by sharing it you just give it to them for free.

Yes, because the chances of you coming up with something better than anyone else has, on your own with no consultation with anyone else, no feedback and no collaboration, is unlikely in the extreme.

I wouldn't consider an innovative algorithm to be "an idea." It's as much an invention or a trade secret as anything else.

Ideas are cheap (approaching worthless but not quite) - trade secrets are incredibly valuable.