I didn't think anyone would be that interested in my little hypothetical. :)

Well, I may live in a different country to you, so any department I describe may not be familiar to you. But without getting too specific, my first starting observation is that the hiring practices of most public organisations are terrible. They're not stringent enough and they don't encourage self motivated workers. So that would be the first thing I'd want to try and fix. Self motivation within the workforce would seem to be one of - if not the most - important ingredients to get the Valve/GitHub style culture working.

Size of the organisation also would seem to be important. I have seen it discussed here before as to just how large an organisation this kind of flat strucutre would scale out to. I obviously don't know the answer. But again I think it would largely come back to how well the hiring process is managed.

And how such a essentially anarchistic department would interface with the rest of the government and/or the public would also be a interesting experiment.