Weird but interesting question, it kinda makes me curious as well. Would've been a cool thing for some people I guess. The only problem with a competition, as I see it, is the difference between this and just pitching your idea to a publisher the standard way. Either it's the big guy (i.e. publisher) who decides the winner, in which case its just like pitching your idea normally (and thus redundant). Or, it's decided by vote, in which case the publisher must potentially forego his own Brand and preference to cater to a public opinion determined by random people. Not only do they need to have actually heard of this competition, but decided to participate as well (which in sum may not be an adequately large or accurate focus group, unless the competition is thoroughly adverticed).
Both are unlikely, IMO. Some publishers might perhaps try this, though. It could be interesting to make games based on public opinion, as long as that opinion comes from relatively educated people. For anything factual, meritocracy is usually better than democracy. But I imagine that one could advertice this competition on Gamasutra, Gamedev and various zines (IGN, PC Gamer, GameSpot, and so of) as a major event. It could be an annual or similar event. That would at least attract the attention of a good number of fellow game developers, hardcore gamers and design enthusiasts (which might be good and relevant enough for a publisher to consider it).