What sort of mathematical claim is made by theorems in Welfare Econ? Essentially, it is that Walrasian Equilibria- which can be shown to have a representation as computably enumerable reals- aren't dominated by non-enumerable solutions.
What happens when we introduce information asymmetry into a model? Essentially, we are saying that there is a missing market for that information. If its being missing makes a material difference there will be both demand and supply and stuff on sale. The fact that the stuff on sale doesn't do the job doesn't matter. Indeed, strictly speaking, all markets are missing since all demand is derived demand, all supply is joint supply. Simply by redefining markets as lotteries (when I buy a beer I'm taking a gamble that it will get me drunk, when I sell a beer, I'm taking a gamble that whatever action I take is in joint supply with the shipping of what the market judges to be beer) we get back to an Arrow Debreu Platonism albeit with things like negative probabilities for incompossible states of the world and weird stuff like that..
In other words, information asymmetry- though of historical interest as a 'paradigm shifting' heuristic- isn't worth mathematical treatment- as in Greenwald-Stiglitz- unless it incorporates mathematical objects of fundamentally different orders of complexity.
Stiglitz's 'Whither Socialism?' drew plenty of criticism in its time but what if we think of markets as a canalisation upon capacitance diversity? Essentially, the real 'Socialist calculation' debate isn't about existing preferences or production functions. It's about Muth rational preferences. The interesting question is how to target and get epigenetic stabilization around something in capacitance diversity space which is almost definitely not computably enumerable.