Another thing that I think is relevant is the "revelation principle", which says that any "social choice function" that can be implemented by some mechanism, ca

Another thing that I think is relevant is the "revelation principle", which says that any "social choice function" that can be implemented by some mechanism, can be implemented by a truthful social choice mechanism. But it seems clear that the thing that I proposed as being an improvement on random ballot can't count as a mechanism which implements a "social choice function", because there can't be an equivalent mechanism which is truthful (... well, I assume that no probability mix of pairwise votes and random ballot stuff is equivalent to the thing I described. Can't see how it could be.). And I think the reason why is that the thing that I described (or, any fully specified version of it) would have to have multiple Nash equilibria. So, for it to make sense to call it a Pareto improvement, I think would need to have that _all_ of the Nash equilibria would be Pareto improvements on the thing. I am now less confident than I was that such an improvement exists.