The two-parameter Ewens distribution: a discrete approach

The new approaches to macroeconomic modelling that describe macroscopic variables in terms of the behaviour of a large collection of microeconomic entities, has often dealing with the problem of clustering of agents in the market. Aoki and Yoshikawa define “cluster” any group of economic agents (a sector, an industry…) with the same choice or same set of attributes. (f.i. bulls and bears in financial markets). What dynamics emerges in the processes of formation and dissolution of clusters comprising interacting agents? Clustering has often been described by Ewens Sampling Formula (ESF) which has been generalized recently by Pitman to two parameter, where the “weight” for the mutation probability depends on the number of existing clusters. In contrast with the usual nonelementary derivations, we suggest a characterization of the two-parameter ESF pointing to real economic processes. We derive some essential feature of the model without introducing notions like notions like frequency spectrum, structure distribution or complex distributions like Mittag-Leffler which seem difficult to apply to concrete finite populations. We consider a system of n individuals changing attributes over time. The choice is probabilistic, the resulting accommodation probability being a generalization of the famous Ehrenfest-urn-scheme, with the great difference that the creation term is influenced by the results of all previous choices (Ehrenfest-Pitman scheme). The equilibrium probability is understood as the fraction of time the system spends in the considered partition. A finite model of economic interacting agents whose equilibrium aggregation state is described by the two-parameter Ewens distribution is presented. The exact marginal description of a site is derived, wherefrom birth, life and death of clusters is easy to extract.