The help center makes it very clear how Community awards bounties:
If you do not award your bounty within 7 days (plus the grace period), the highest voted answer created after the bounty started with a minimum score of 2 will be awarded half the bounty amount (or the full amount, if the answer is also accepted). If two or more eligible answers have the same score (their scores are tied), the oldest answer is chosen. If there's no answer meeting those criteria, no bounty is awarded to anyone.
The bold emphasis there is mine. And yes, an ulterior motive prompted this question, but I'm genuinely curious what other people think about the process in general.
I understand that the point of a bounty is often to attract new attention to a question, and I assume that this is the logic behind awarding the bounty to whatever answer received the most votes after the bounty is posted. But at the same time, if a user doesn't award their own bounty, that often suggests that they didn't get an answer that satisfied their expectations from anyone. In essence, their attempt to earn a certain kind of attention for a question failed.
My suggestion is to award the bounty to whatever answer gets the most upvotes since the bounty was posted, but regardless of when the answer was posted. This would still prevent old answers from easily earning abandoned bounties because of their high vote counts unless those old answers earned the most upvotes even after the bounty was posted.
This is essentially a discussion question, but since it also involves a suggestion, feel free to downvote if you disagree with the suggested approach and prefer the way bounties are awarded now.