As a new user user experiencing my first rush of SO addiction, it was interesting to see that the random rewards of finding questions to answer and getting notifications on my own questions didn't extend to badges. I complete a well defined task (edit 500 6+ month old posts), I get a pat on the head/badge. Could some of the behaviours be better encouraged by setting a general aim (edit old posts) with unpredictable and variable rewards, or is there an element of this system that I am missing?
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Intermittent rewards would work for what you want (encourage users to do X), but not in the way that we at SO want to encourage users. Stack Overflow isn't, and never will be, a RPG. The badges are there so that users can see what's encouraged (answering your own question, for instance) not so that people can collect and compare. The addictive nature of intermittent rewards would put the focus on the badges, rather than the work.
(Plus, adding dumb luck to badges would decrease their value. There are too many people firing the SO question shotgun already)