A continuum model as a limit of large finite matching markets

The continuum matching model has been instrumental in the studies of finite economies. Yet, there is limited theoretical justification on whether and when a continuum model approximates large finite problems. In this paper, I study the following question: if we randomly sample finite economies from some distribution, will the stable assignments of these finite economies converge to a stable assignment of the continuum economy as we increase the sample size? I provide a simple condition, which I call rich preferences, that guarantees the convergence. I also provide approximate convergence results under weaker conditions.