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From the Reputation & Moderation page from the Help Center, we read that:

A bounty is a special reputation award given to answers. It is funded by the personal reputation of the user who offers it, and is non-refundable.

And in particular:

  • All bounties are paid for up front and non-refundable under any circumstances.

Imagine you have a very big, complex, and time-consuming task to perform that you are not willing to do by yourself. Being an experimented StackExchange user (with 300 reputation), you know that you cannot straightforwardly ask for someone to do it for you, because your question would be quickly red flagged.

So you decide to split your hard task into (say) 3 easy ones. But now you want these problems to be solved as fast as possible, so you decide to place attractive bounties on top of your question. In order for your 3 sub-questions to receive the maximum (that you can afford) attention, you decide to invest your whole reputation, and place 3 bounties of 100 each on these sub-questions.

This is how things should work, right? (actually, you should not make others do your job, but that's out of the scope of this question)

I was wondering how exactly setting bounties work, since apparently, with reputation R, it is possible to place n bounties of b_i reputation so that \sum_{i=0}^n b_i > R. In other words, the sum of the bounties that you are placing can be strictly greater than your reputation.

This is currently happening in TeX.SX, see these 3 questions:

These 3 questions are all asked by a person having currently 125 reputation. So my question is the following: did this guy had 125+3*50 = 275 reputation before setting the bounties or is there something going wrong is setting bounties. By that I mean that somehow your reputation is not escrowed when you offer a bounty on some question.

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    As far as I know the reputation for a bounty is deducted immediately when you click OK on the dialogue to set the bounty. So the user would have had 275 points before they set the bounties. And that checks out: people start with 1 point and the reputation changes in tex.stackexchange.com/users/119955/hyperplane?tab=reputation (excluding the -150 for the bounty) add up to exactly +274. – moewe Jan 5 '20 at 16:23
  • Of course they did. Do you have evidence to the contrary showing they didn't have 275 reputation? Why would you think there's something wrong with the bounty system? – Werner Jan 6 '20 at 7:03
  • I didn't know that it was possible to check the history of one's reputation, I was simply asking myself if this was a possible bug, thanks. – jchd Jan 6 '20 at 16:13
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The person you are giving as an example has more than 200 reputation on others SE sites, so she or he had a +100 association bonus when coming to TEX S.E. If you add his 17*10 reputations gained for Q&A on this site, +4 for accepting answers and 1 initial point, you have a total of 275 reputations before he offered those three bounties.

From TeX S.E help center

If you have a registered account on any other site in our network with at least 200 reputations, you will receive a +100 points reputation bonus when you first log in

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