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I posted a question on math.SE a few weeks back, which received a fair bit of attention and some promising leads in the comments but no clear answers. I posted a +100 bounty on the question, and during the bounty period an answer was posted which provided some mildly interesting discussion of why $6!\cdot 7!=10!$ but did not provide anything resembling a bijection, which was the explicit aim of the question and associated bounty. The bounty has now expired with no additional answers.

Because the question has been fairly popular, the answer has been upvoted by 4 of the 684 viewers of the question (and downvoted by one), even though it doesn't seem like the community's overall opinion is particularly favorable - the question itself was upvoted 16 times over the same period since the answer was posted. As such, it will receive a +100 bounty if I manually award it, and a +50 bounty if I let the bounty expire, per my understanding of the bounty system.

I am unsure of the proper response here. On the one hand, I appreciate the answer, and found it somewhat interesting - I would rather have read it than not. (I haven't voted on it either way.) I also don't want to let the 100 reputation "go to waste", as it were. This would suggest directly awarding the bounty.

On the other hand, the answer does not even attempt to provide the bijection my question was looking for, and it seems like it would be bad for the site to set the expectation that posting incomplete answers on popular-but-hard bountied questions is a way to earn easy reputation. It feels sort of unfair to me to award the same bounty to an incomplete answer that I would have liked to award to a full response to the question. This would suggest not awarding the bounty, but as the voting patterns make this impossible, letting it expire for half-credit as a fallback plan.

Right now, I am thinking of letting the bounty expire and giving +50 reputation to the existing answer, and editing my question or adding a comment to make clear that I will create and award a new +100 bounty to any future answer that resolves this question. Is this a reasonable choice to make? Are there factors I'm neglecting, or any other thoughts on what the right choice is in this situation? I'm relatively new to the site and don't want to violate any math.SE norms or slight the user who posted an answer, if possible.

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    $\begingroup$ I think you've considered this matter in a very rational manner, both options.So to your second to last question in your last paragraph, I think "this" is without question a reasonable choice to make: "Right now, I am thinking of letting the bounty expire and giving +50 karma to the existing answer, and editing my question or adding a comment to make clear that I will create and award a new +100 bounty to any future answer that resolves this question." Thanks for the very well written question you've posted here ... $\endgroup$
    – amWhy
    Nov 27 '20 at 21:58
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    $\begingroup$ ...In doing what you are thinking in the last paragraph, in no way does it violate any math.SE norms, nor does it slight the user who posted an answer. $\endgroup$
    – amWhy
    Nov 27 '20 at 21:59
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    $\begingroup$ Just so you know, you can't put another +100 bounty on the question. Any subsequent bounties you place on the same question must double each time. $\endgroup$ Nov 28 '20 at 2:15
  • $\begingroup$ At least you got a partial answer. It is better than getting no answer (in one case after my bounty expired I had to post the same on mathoverflow and even offer a bounty on mathoverflow to get the answer). So it is OK if half the award goes to that answer. $\endgroup$ Nov 29 '20 at 2:47
  • $\begingroup$ I'm well aware that bounties are no guarantee of an answer; thus far, I have spent 250 of my reputation across four bounties, all of which have proved fruitless or yielded very subpar responses. My question is about the ethics of deliberately letting a bounty expire and auto-award to an answer, not a complaint about the uncertainty inherent in staking a bounty. $\endgroup$ Nov 29 '20 at 3:06
  • $\begingroup$ Bounties often don't result in better answers, but rather in answers that take more time and effort. $\endgroup$ Nov 29 '20 at 4:08
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    $\begingroup$ @MichaelHardy there's nothing really stopping a user from waiting to post an answer after some time has passed. Of course, fastest-gun is a problem on this site but much less so on medium-high quality questions. $\endgroup$
    – Integrand
    Dec 1 '20 at 1:47
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Welcome to the meta site!

You are entirely within your rights to let the bounty expire. Personally, I think of bounties as a sort of 'custom order'. When you post a question on the site, you're de facto opening the door to any answers that appear, which you can accept or vote up/down as you please. When you post a bounty, however, because you're carving your own rep you are allowed to be more discerning, and in my opinion sometimes that means letting the bounty pass for another day.

I've developed a personal habit of leaving comments on answers to questions I've posted a bounty that explain why or why not the answer is worthy of the bounty. Many times, it's something to the effect of, 'this is just what I wanted; if nothing better comes along, the bounty is yours'. Other times, I'm more critical or explain why I feel my request isn't being met.

Somewhere on the main site I read something to the effect of 'part of what you are paying for is increased exposure to your question.' This view in particular supports the idea that bounties are not a guaranteed inheritance to those who answer the question but are a reward to be disbursed at the whim of whoever put the bounty up.

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  • $\begingroup$ Thanks! In a case like this (where the bounty will be awarded no matter what), do you think leaving such a comment is reasonable? It feels a little rude to say in effect "I don't think your answer deserves this bounty, but I am forced to give it to you", but doing nothing seems to give the false impression that the question was merely forgotten about. $\endgroup$ Nov 28 '20 at 3:35
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    $\begingroup$ I don't think it's wrong to not give them the bounty or to leave a comment saying so because I don't have the mindset that someone deserves the bounty just because they answer. If anything, if you post it with a few days left, there's always a chance someone can change their answer to better suit what you were looking for. $\endgroup$
    – Integrand
    Nov 28 '20 at 4:55
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    $\begingroup$ The first bounty I awarded was, in hindsight, to a very subpar answer. While I don't lose sleep over it (it's just numbers on a website at the end of the day), it has certainly made me a little more exacting on subsequent ones. $\endgroup$
    – Integrand
    Nov 28 '20 at 4:57
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I add some related thoughts.

be bad for the site to set the expectation that posting incomplete answers on popular-but-hard bountied questions is a way to earn easy reputation.

I think it is natural to reputation to come easy, because it is provided by a poll, so it measures rather popularity than value, which sometimes can be understood and evaluated by only a few experts. I can illustrate this with my answers. The highest upvoted among them is a joke, whereas a gem of my habilitation thesis, a solution of a more than fifty-year-old open problem, received only three upvotes.

I rarely ask questions, but when I do this I usually upvote an answer, to express my gratitude to a man who devoted (for free!) his time and efforts to care about my problem. Sadly, such attitude is often neglected by askers at MSE. By similar reasons, offering a bounty I would award the best relevant answer, also keeping in mind not to waste bounty points.

But your case is specific and I think it should be your decision to which extent support such answers.

I think that a bountied question usually is rather hard, otherwise it would be answered before a bounty was proposed. So such a question can miss a perfect answer even after bountied. Also it can happen that a question has no affirmative answer. In your case it is unclear for me why there exists a required natural bijection corresponding to an equality, which can be accidental and hold only for specific numbers, but do not be an instance of a pattern holding for a sequence.

Finally I note that in a bounty comment usually is specified a purpose of the bounty. In your comment is stated that you want to draw more attention to this question. Looking at the number of its upvotes, I guess that this aim was reached.

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    $\begingroup$ I like your answer, but I'm concerned at the assumption that anyone answering your questions is a man: "I rarely ask questions, but when I do this I usually upvote an answer, to express my gratitude to a man who devoted (for free!) his time and efforts to care about my problem." $\endgroup$
    – amWhy
    Nov 29 '20 at 1:26
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    $\begingroup$ @amWhy In that sentence "man" means "human", this is the first meaning of this word in my dictionary. $\endgroup$ Nov 29 '20 at 4:50

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