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. In p articular, the authors study implementation of Walrasian allocations in exchange environments. They show that the smallest message space t hat implements Walrasian allo-cations is one of dimension, roughly, n E (. 1 1)& ./(n 1 1), where . is the number of commodities and n the number of agents. The authors exhibit an implementing mechanism whos e message space has that dimension.