(An aside: While sitting in a lecture today – Al Roth was giving a talk on his organ donation chains – I was working through a result from Myerson’s famous optimal auction paper, and rather stumped on a technical point, I was rudely surprised to see Myerson himself sit down behind me. This inspired me to look back through a few of the famous early 80s papers on mechanisms and information transfer so as to avoid any embarrassment should one of these guys catch me fumbling through their proofs!)
This Milgrom/Stokey paper is the source of the famous “no-trade theorem”. Before Aumann and his followers showed just how strong the assumption of common knowledge is – and therefore how strong the informational requirements of a rational expectations model with public prices are – there was an assumption that traders could profit on their inside information if they were “small” relative to a market. This turns out to be false. Assume that beliefs are concordant; that is, if we agreed on the state of the world, we would agree on what outcome will occur. If traders are (even slightly) risk averse and we are at an equilibrium with known prices, then none of us will trade (we will infer that everyone knows there must be a “sucker” who is willing to accept the bet, since all trades for insurance reasons have already occurred by the assumption that we are in equilibrium). If there are markets before and after private information is revealed, then there exists a fully revealing ex-post equilibrium; that is, we all learn everyone’s private signal by Aumann’s common knowledge result applied to the change in prices, and therefore update our belief about the true state identically, leading to a new price vector that reveals all the private information. Indeed, with some technical assumptions, any ex-post equilibrium, even one that is not fully revealing, will have prices that incorporate all of the private information available to the agent; he could forget his private signal and will nonetheless be willing to trade in exactly the same way.
The no-trade theorem is a bit worrying, since we do in fact see people trying to trade on private information all the time, even in markets (like a stock market) where the prices are surely common knowledge. A great amount of work has tried to escape this conclusion – a particularly successful argument involves a small subset of ignorant traders (“noise traders”) whose existence suffices to break common knowledge of rationality and allow for trade to resume even among the non-noise traders.
A Fine Theorem 
Kevin. It sounds like the M&S’s no-trade equilibrium is founded upon a very restrictive set of assumptions. The market is so efficient/rational/transparent with nearly no disagreement about what it will turn out to be like in the next period (or just generally the near future) that it can make a trader with private information a sucker, which is why it is regarded unrealistic as well. Having read this, I’m starting to wonder if there is a paper out there on the effect of uncertainty (or discrepancy in opinion) on the trading behavior of informed and uninformed investors?
Not too restrictive – the point is simply that with common priors and an object with common value, there can be no discrepancy in opinion. That is, from the choice on whether to accept the trade, both you and I can infer what private information the other person must have. This result holds even under uncertainty, as both of us may only have signals about the value of the good.
From Aumann, basically, we know discrepancy of opinion must rely on differences in the prior of each agent, which is difficult to philosophically interpret. But of course, Milgrom-Stokey has thousands of citations, and many papers have tried to do exactly that.