I set a bounty to a question (How fat is a triangle?). After about 6.5 days I got two answers. Both look good anad relevant, but in order to fully understand them, I need more time, since the question and answers are full of details. Currently I have only 1.5 day to award the bounty (including the grace period), which is insufficient for reading, asking clarification questions and waiting for clarifying answers.
Previously, when a similar thing happened to me (in The square cover number and the number of horizontal sides ), I just gave the bounty to the (single) answer, although it was imperfect, because I believed it could be improved and there was no time to wait. I think this time-pressure is bad for the site because it encourages the OP to give the bounty to imperfect answers, instead of encouraging the answerer to improve it.
I understand that the limitation on the grace perion comes to encourage the OP to make a choice rather than think forever. However, the time might be too short. So here are several suggestions:
A. The grace period should automatically grow in 1 day for every new answer.
B. (anorton's suggestion): The grace period should grow for each new user posting an answer.
C. The grace period should be 7 days from the posting of the first answer.