Do Stackoverflow have certain policy to follow?
No, I don't think so. But here is a recent blog post of Jeff's on the issue which has quite a relaxed stance and I think can be viewed as the somewhat official policy. Final paragraph:
"So, as always, use your good judgment and please continue to close and merge duplicates as you see fit. However, bear in mind that cultivating and supporting a moderate amount of natural duplication actively helps the community. I wasn’t kidding when I said learn to stop worrying and love (some) duplication. Use the above guidelines and try to find a happy, reasonable medium somewhere in the middle there."
Is there any solution not considering participants being some sort of boy-scouts crazy after some virtual badges, signs, bells and whistles?
There are no bells and whistles for pointing out duplicates. It takes active effort and thinking to find a good original question that really answers the new OP's problem. Sometimes, the OP will even hate you because they don't want to read through some other question, but would like to have custom-tailored answers.
It's still a good thing to do, like picking up litter in the park. The only reward is a good feeling inside. I don't see a way to make this work without using "Boy Scout" tactics.
If not - why not to get rid of this button at all?
Getting rid of the "close as duplicate" button would be getting rid of the last barrier against an endless stream of the same questions, every time getting the same copy+pasted answers. If anything, the process needs to be further improved, maybe even gain reputation so more people do it. I don't think getting rid of it would do any good.
How to make “exact duplicate” close reason to work?
Give more incentive to find good duplicates - either through reputation points, badges, a list of the users with the most successful duplicate votes. Take away reputation earned from duplicate questions. That's all I can think of.
