I wish there was a better science of the economics of imperfect information, or if there is that I would know about it.

Mechanism design? BTW, this is a special case of a mechanism design result known as the Myerson-Satterthwaite theorem. It's not a novel result by any means. The Gibbard-Satterthwaite theorem has a similar flavor, though it is in fact a separate result--for one thing, it applies to voting systems with three or more agents.