A LOWER BOUND FOR THE DIMENSION OF THE MESSAGE SPACE OF THE DECENTRALIZED MECHANISMS REALIZING A GIVEN GOAL

The dimension of the message space is an important indicator of the informational efficiency of the decentralized economic mechanism. The minimum dimension of the message space of the decentralized mechanisms that realize a given goal reflects how well the goal can be decentralized in its realization. The knowledge of the minimum dimension also helps to identify those mechanisms that are informationally efficient in realizing the goal. Towards determining this minimum dimension, a general method is developed in this paper which leads to obtaining a lower bound for the dimension of the message space of the decentralized mechanisms that realize a given goal. The lower bound is expressed in terms of certain differential property of the goal function. In particular, for economies consisting of two agents the lower bound can be further expressed by the rank of the 'bordered Hessian' of the goal function. It is demonstrated that the method is constructive and algorithmic; the lower bound is computable by computer programs. Our approach is analytic and requires differentiability assumption.