Implementing a generic smooth function

Abstract

We define a realization of a smooth, real-valued function in a manner which is based upon Hurwicz's model of privacy-preserving economic mechanisms. Realizations are distinguished by ‘type’ — a measure of the kind of information that the realization requires from each agent in order to operate. Necessary conditions for the existence of a realization of a particular type are given, and the types of realizations which exist for a generic smooth function are classified. Most of the mathematical techniques in this paper are commonly used in differential geometry and in the study of smooth mappings. Some results from the theory of algebraic equations are also employed.