The informational efficiency of finite price mechanisms

This paper obtains finite counterparts of previous results that showed the informational efficiency of the Walrasian mechanism among all mechanisms yielding Pareto-optimal individually rational trades in exchange economies while using a continuum of possible messages. Such “continuum” mechanisms lack realism, even when the space of environments (characteristics) is a continuum, since it is not possible to transmit or announce all elements of a message continuum, and it generally takes infinite time to find an exact equilibrium message among all messages in such a continuum. Accordingly the paper studies finite approximations (having a finite number of messages) of the continuum Walrasian mechanism and of other continuum allocation mechanisms. An approximation’s overall error for a class of exchange economies, is the largest distance, over all economies in the class, between the continuum mechanism’s final allocation and the approximation’s final allocation. In particular, we develop finite counterpart s of the superiority, with respect to message-space dimension, of the Walrasian mechanism over Direct Revelation (DR). We measure a finite mechanism’s cost by the number of its (equilibrium) messages.Our two main results are as follows: (1) For exchange economies we find that the overall error of a (sufficiently fine) approximate Walrasian mechanism is less than the overall error of a not-more-costly approximation of a continuum DR mechanism whose equilibrium outcomes are trades that are Pareto optimal and individually rational. More generally, approximate Walrasian mechanisms are superior, in the same sense, to approximations of any continuum mechanism whose equilibrium outcomes (like those of the continuum Walrasian mechanism) are Pareto optimal individually rational trades and whose message space has higher dimension than that of the Walrasian mechanism. (2) As we increase without limit the dimension of the set of environments (characteristics) defining our class of exchange economies, we find that the extra cost of such DR approximations relative to Walrasian approximations, when both achieve the same overall error, also grows without limit.Thus the informational superiority of the Walrasian mechanism emerges again when we approximate it and take the finite number of messages in the approximation as our cost measure.