A Finitary Characterization of the Ewens Sampling Formula

Abstract

The clustering of agents in the market is a typical problem dealt with by the new approaches to macroeconomic modeling, describing macroscopic variables in terms of the behavior of a large collection of microeconomic entities. Clustering has many economic interpretations [3], that are often described by Ewens' Sampling Formula (ESF). This formula can be traced back to Fisher as “species sampling”, and its main use was restricted to genetics for a long time. Contrary to the usual complex derivations [18], we suggest a finitary characterization of the ESF pointing to real economic processes. Our approach is finitary in the sense that we provide a probabilistic characterisation of a system of n individuals considered as a closed system, a population, where individuals can change attributes as time moves on. The intuitive meaning of the probability is the fraction of time the system spends in the considered partition. As ESF represents an equilibrium distribution satisfying detailed balance, some properties which are otherwise difficult to prove are derived in a simple way.