This paper is concerned with the amount of communication that must be provided to implement a performance standard by a mechanism whose stationary messages have the Nash property. We study the question whether a given message space is large enough to implement a given performance standard. In general, an implementing mechanism with the Nash property in messages requires a larger message space than sufficies for decentralized realization without regard to individual incentives. In particular, we study implementation of Walrasian allocations in exchange environments. We show that the smallest message space that implements Walrasian allocations is one of dimension, roughly, n. (l - 1) + l/(n - 1), where l is the number of commodities and n the number of agents We exhibit an implementing mechanism whose message space has that dimension.