Idea averaging is not an example of "the wisdom of crowds". It's a manifestation of groupthink.
More specifically, it's an example of when a group of people will tend to agree on something that is the least objectionable on average. This means that it's less likely that an unconventional idea will be accepted.
I was watching the election results on my computer in 2012. By 8 or 9pm most of the rural area votes had been counted in Florida and the urban votes were still being counted. Most of the votes left to count would have been for Obama who was just over 50% at that point. However, on the prediction market, Intrade, at the moment the contract on an Obama win was $0.22 paying a dollar if he does. Holy cow!. I tried so hard to get as many thousands of dollars in because I figured that very hopeful conservatives had been betting on Romney and probably watching FoxNews so they didn't know he lost Florida for a long time after. Unfortunately, I couldn't at that time get money into the market because of laws and it had to be held in escrow or something for 24 hours before a person could bet. I just did a search and there is a discussion about how lopsided the bets on Romney were.[1]
Since most unconventional ideas tend to fail (because people have thought for a long time about what would be good, and over time those things have often become conventional), it is a useful heuristic to go with conservative, conventional ideas. I think this totally fits in with "wisdom of crowds".
Of course, this also means you won't generate that once-in-a-million super great unconventional idea. But then again, I think that's the point. If you try to generate the great unconventional idea, the base rate of failure is so high that you should not have so much hubris as to believe you actually can do it, and most often you should admit that what you think is a great unconventional idea really probably can't do any better than whatever conventional wisdom would choose.
I think my view on this is not popular because everyone wants to believe in the idea underdog -- the idea that was spurned by conventional wisdom but proved itself regardless and made someone famous for executing upon it.
That's great and all, but in the same way that putting all of your money on a single stock is a bad strategy, failing to average across ideas embraced by conventional wisdom is also (often, but not always) a bad strategy.
On a different note though, there is a lot of value in your comment even connecting it back to the "wisdom of crowds" stuff. One of the issues with "wisdom of crowds" is that the aggregated average tends only to be better whenever there is a sufficient variance of opinions in the crowd. This way, there is not systematic bias in everyone's thinking, meaning that the average too will have that bias. The more variation, the more that each participant's particular way of being wrong will contribute to "canceling out" the errors, leaving the average in a good position.
To me it suggests that when unconventional ideas do hit it big, it is because they are highly related to some source of demonstrable bias in conventional wisdom, and by removing that bias, such ideas can actually improve upon whatever is the current state of conventional wisdom.
But most of the time, it seems we only hit upon these bias-based improvements by accident because tons and tons of people keep trying (and failing, often to their personal detriment) to commit to unconventional ideas that aren't articulated in terms of any demonstrable bias, and it ends up basically being an idea lottery to see who happens to hit the "jackpot" idea. That's what makes conventional wisdom appealing as an alternative, steady expected return with low risk, but it's also what means it will never lead you to the next big thing.
Ironic, then, conventional wisdom is a useful heuristic because of all these unconventional ideas individuals do attempt, and so the answer to keep moving humanity forward is, yes, go ahead and try that unconventional idea, or you will contribute to systemic bias.
I would expect the "wisdom of the crowds" to have a different result if, when asking for each person how many peas are in a bottle, each person is first told of the current running average or median. I think it would achieve a better result if everyone made their own guess independent of the "crowd". Otherwise, the random bias that occurred from the first 20 people's guesses is going to percolate because everyone who is guessing after will have their guess anchored when they're told of the group opinion.
More specifically, it's an example of when a group of people will tend to agree on something that is the least objectionable on average. This means that it's less likely that an unconventional idea will be accepted.
This concept has been studied for a long time in the social psychology literature. For a survey of related topics: http://chicagounbound.uchicago.edu/cgi/viewcontent.cgi?artic... (see specifically "Hidden Profiles and Common Knowledge").