Continuous-Time Random Matching

Abstract

Continuous-time random matching with a large (continuum) population is widely exploited in the literature, but has not had a rigorous formulation, nor a demonstration of its key assumed properties. This paper provides the first probabilistic foundation for this approach by presenting a mathematical model of continuous-time random matching and showing its existence and properties. The agents' types, which can change due to random matching and random mutation, form a continuum of independent continuous-time Markov chains. Using the exact law of large numbers, we show how the cross-sectional distribution of agent types evolves deterministically according to an explicit ordinary differential equation. Nonstandard analysis is used in proving the main theorem.

Keywords: Independent dynamic random matching, Markov chain, exact law of large numbers, continuous-time, random mutation.

JEL Classification: C78, D50, D53, D83