Deterministic approximation of stochastic evolutionary dynamics

Abstract

Deterministic dynamical systems are often used in economic models to approximate stochastic systems with large numbers of agents. A number of papers have provided conditions that guarantee that the deterministic models are in fact good approximations to the stochastic models. Much of this work has concentrated on the continuous time case with systems of differential equations approximating discrete time stochastic systems, although some important early work in this field considered discrete time approximations. A crucial aspect of the existing work is the assumption that the stochastic models involve agents of finitely many types. However, many existing economic models assume agent types may take on infinitely many distinct values. For example, some models assume agents hold divisible amounts of money or goods, and therefore agent types form a continuum. In this paper we examine discrete time deterministic approximations of stochastic systems, and we allow agent attributes to be described by infinitely many types. If the set of types that describe agents is a continuum, then individuals, in some sense, are unique, and it is not obvious that the stochastic models that rely on random matching of agents as the source of uncertainty can be approximated in a deterministic manner. Indeed, we give two examples that show why a law of large numbers may not lead to deterministic approximations. In the positive direction, we provide conditions that allow for good deterministic approximations even in the case of a continuum of types.