Statistical Description of Market Shares in Emergent Market
This paper discusses distributions of the composition of a large number of agents by their types, their choices or some other characteristics such as the market shares of shops when the buyers are classified according to the stores they shop. Models with a large number of types are considered in this paper. We describe the distribution of the order statistics of the fractions of agents as the average of multinomial distributions conditional on the vector of random fractions by Dirichlet distributions for the random fractions. The result is the same as the Ewens formula known in the field of population genetics. The interaction processes of a large number of agents are modeled as Markov processes on the set of exchangeable random partitions of a set of integers. Patterns of composition or shares evolve as random partitions of customers. By introducing some simple transition rates for the types of agents a master equation or a diffusion equation approximation describes the time evolution of the process. We show that the frequency spectrum or intensity of fractions have the same form for the market shares and for the relative sizes of the basins of attractions associated with random maps. The characterization given in this paper of distributions may be useful in designing simulation studies or interpreting results of such studies.
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