The Efficiency of Competitive Mechanisms under Private Information, Journal of Economic Theory
We consider the efficiency properties of exchange economies where privately informed traders behave strategically. Specifically, a competitive mechanism is any mapping of traders' reports about their types to an equilibrium price vector and allocation of the reported economy. In our model, some traders may have non-vanishing impact on prices and allocations regardless of the size of the economy. Although truthful reporting by all traders cannot be achieved, we show that, given any desired level of approximation, there is N such that any Bayesian-Nash equilibrium of any competitive mechanism of any private information economy with N or more traders leads, with high probability, to prices and allocations that are close to a competitive equilibrium of the true economy. In particular, allocations are approximately efficient. A key assumption is that there is small probability that traders behave non-strategically.
Nabil Al-Najjar, Rann Smorodinsky
Al-Najjar, Nabil, and Rann Smorodinsky. 2005. The Efficiency of Competitive Mechanisms under Private Information. Journal of Economic Theory. 137(1): 383-403.