Probably at least some MathOverflow users have been in a situation like this.
I have placed a bounty on some question in hope to get an answer (or a better answer or a specific type of answer, for example, combinatorial proof instead of inductive argument). I did not get an answer I wanted. If there was some answer at all, maybe I awarded the bounty to the closest match, but I still would like to see an answer fulfilling my original requirements. What can I do next?
- One possibility is adding new bounties on the same question repeatedly. This is probably not very efficient and I could exhaust non-negligible part of my reputation after a few iterations. (This would be difficult especially for low rep users.)
- I could mention (in a comment or - if it is my question - directly in a question), that if somebody posts an answer fulfilling some criteria, I will reward such answer with a bounty. A clear disadvantage of this is that this will not be very prominently visible.
- There is also this feature request: Request new function Standing Bounty. But considering that it is two years old and currently at score $-12$, it is not very likely that it gets implemented.
Would creating a list of such bounties be potentially useful. I would imagine each entry to contain description of the question and what the user offering the potential bounty looks for. And the entry would be considered as a commitment to award a bounty if the answer they are satisfied with is posted.
And if something like that could be useful, what would be the best way to implement it? Would it be a dedicated meta post? (I am afraid it might get too long.) Would perhaps a dedicated chat room be more suitable? Or would a better solution be some wiki-like page not directly on MathOverflow website?
To prevent misuse of the list, maybe some rules would be needed? Probably only person who already gave at least one bounty on that question should post there? And maybe there could be some age limit - say questions as least one or two months old?
I will add that there are possibilities to find list of all past bounties. However, not all past bounties are equal. The fact that a user offered bounty for some question in the past does not automatically mean that they are still interested in (new answers for) that question. Somebody might, for example, just add a bounty simply to reward an answer which is already posted. However, the proposal here is mainly for the question where the user offering a bounty is still interested in the question and is willing to reward a new answer by additional bounty, if a satisfactory answer is posted.
Several posts here on meta suggests that some users are not quite satisfied with visibility of the featured questions. For example: Is bounties period sufficient? and Why is it not possible to offer a bounty for longer than 7 days? (In other words, some users feel they did not got their
money reputation points' worth after they offering a bounty.) In fact, I have mentioned rather similar proposal in my answer to Is bounties period sufficient? I am posting this here so that it is slightly more visible and also so that various possible implementations of this can be discussed here. (And also whether something like this could be potentially useful.)
I will also add that I have proposed something similar on Mathematics Meta: Could list of unresolved past bounties be useful? The question was quite well received - the score of the question is currently +37 and answer with a proposed solution is at +8 - but the proposal was not successful. I have posted an answer to this question, giving more details about the "chat-solution". (In the answer I have also tried to make a comparison between MO and MSE concerning the aspects which might influence such undertaking.)
I will also point out that there already is some post with list of interesting questions on this meta. However, it is supposed to be used for questions from the Mathematics site: Interesting (and not sufficiently answered) questions on math.SE. (Do you think something similar could work for this purpose?)
I am also aware that there are users who dislike the whole gamification and reputation aspect of the site. For them, possibility of getting additional points might in fact decrease their motivation to provide an answer. But this is true already for the bounty system. (See, for example, the following for some older related discussions: Opting out of reputation or Hiding reputation.)