For one, the people typically participating in a question like ship dates are not just from the project team, but from other roles as well, i.e. people who will be involved in marketing the product, selling the product, engineers who work on similar products, etc. These people have less of a direct interest in the outcome vs. simply expressing what they think is going to happen. So diversity of participation by role is pretty key.
Your point about self-fulfilling prophecy is also something we get asked about regularly. In practice, the market is available to trade 24/7 until the outcome is known. So going through your scenario, let's say the market is showing a really high likelihood something is going to ship late. Management decides to mitigate that risk by putting more people on the project. Well the market is still open, so if no one actually thinks putting more people on the project is going to do a damn thing, then the likelihood of being late will remain high in the market. Or maybe those people just got pulled from another project so now that project is in jeopardy. Well the market would hopefully begin to reflect that and when the answer is known everyone will be judged on whether they were right or not.
So ideally if your marketplace is setup correctly and people are actively participating, you're getting all this realtime feedback about what people think is going to happen AND you can even see what people think will happen in reaction to the management decisions you're making. Usually an employee wants management to know something is screwed up without having to stand up and say it to their face in a status meeting.
Also in your scenario you're assuming management is making decisions based on the prediction market alone. I've never seen this - instead it's one factor they consider among others. Despite my belief in the value of prediction markets, I also believe in management being able to make independent decisions. Prediction markets are just a simple and systematic way to get human input from those who have different perspectives.
Without a market, information is disjointed; everybody has some imperfect, incomplete part of the whole picture. Any planner will make irrational plans, he or she is dealing with a minute subset of information.
When the market is for information itself, I'm not sure if Hayek would have much to say. Markets for such things seem like they should be far more fragile. I think stock market volatility in the recent crisis is actually evidence of that.
Incidentally, and ironically, all this technology at our disposal giving us more "freedom", has made the feasibility of central planning more possible. Dys/U-toptian novelists take note.
I am not an economist of Hayek expert, but my understanding is that he address "markets as information" as being markets. Even gambling markets and futures markets, which are both classes of prediction markets generally, are actually information markets even though they are ostensibly about goods or events.
I agree and disagree about your characterisation of technology. While yes, computers make authoritarianism more tractable, I don't think they can completely solve the calculation problem.
Suppose you use Leontiev input-output matrices to do your central planning. Matrix multiplication is an O(n^3) problem (I've been told by a mathematician that it can be done in n^2.75 with some tricks). If we take GDP growth as a proxy for the number of matrices required, it would need to be kept below a few percent or it would outstrip Moore's law in the long run. And that's ignoring the problem of populating the matrices in the first place.
FWIW, I host an economics blogger in Australia, Dr Nicholas Gruen, who has proposed to reduce market volatility by requiring blind clearances on a once-per-minute basis.
For instance, one of the new trends in prediction markets has been corporate elicitation (e.g., Inkling Markets YC W06). For instance, a market might ask a pool of employees "When will product X ship?". Now consider the employee's problem. He knows the product will be delayed until November at the current pace, but that if managers see high November probabilities they'll cut back on the project and re-focus development efforts (or, perhaps instead, shift some staffers over to get the project done faster). How should the employee trade in this market?
If your answer is "it's unclear" then you're exactly right. The self-referentiality of prices (they cause actions which then define prices) makes the problem much harder. It's actually possible to design markets which are so self-referential that any trade will be the right one (the market prices are self-predicting prophecies) or that any trade will be the wrong one (e.g., if there's a high price on terrorist attack Z then the government will always spend enough money to prevent Z from happening).
In fact, the only way around the paradox is to have the prices in a prediction market not matter at all - for the managers to take the same course of action regardless of the prices they see in the market. But then why run a market in the first place?
The whole thing is an interesting and seemingly fundamental perversion of the way we normally think about prices. And again, this is without any malicious intent on the part of participants, which will certainly only further confuse things.