I don't think we should promote not awarding a bounty too much.
I'm not saying they should just be awarded to any old answer, but the OP loses the rep regardless, so why not someone who spent their time trying to help?
An answer might not be perfect or even answer the question, but it might have given the OP food for thought.
The current text seems pretty clear, and isn't exactly pushy:
Please review the answers. If you feel one of them deserves your
bounty, award it by clicking the bounty icon to the left of the
answer. Otherwise, you might consider providing feedback in the form
of comments or edits.
It's up to the user if they want to award it, including if an answer didn't really answer their question.
And messages from the site shouldn't try to sway their decision one way or another. So you have to be careful with what you say.
The site should only inform of the options, not be too suggestive towards how and when a bounty should applied.
I think the current message "if one of them deserves.." is enough really.
Bounty is a chance. You pay rep for attention on your question, but doesn't guarantee good answers.
Look at it this way:
- If user gets no answers, OP loses bounty rep
- If user gets answer(s) but is/are rubbish, OP loses bounty rep
- If user gets answer(s) but don't really answer the question, OP
loses bounty rep
1 has no choices with the bounty, and 2 I would likely not award (no control of potential auto award). If 3, then someone has tried to help, and while might not be a perfect answer to the question, the choice is let rep go to waste, or award someone for their time and efforts.