Well, that depends on how you set up the bounty program. What if I find a solution, share it to a friend so that both of us can claim the prize?

bug bounty programs have never paid out independent disclosure for the same bug though; they might split or even pay-out larger coordinated efforts. It's largely a first place award only.

assume there exists 2+ different bugs

after the 1st bug is found, no payout for any other of the bugs