Generalized coarse matching

This paper analyzes the problem of matching two heterogeneous populations, e.g., men and women. If the payo¤ from a match exhibits complementarities, it is well known that absent any friction positive assortative matching is optimal (PAM). Coarse matching refers to a situation in which the populations into a …nite number of classes, then randomly matched within these classes. In a seminal paper, McAfee (2002) showed that when the match payo¤ function is multiplicative and type distributions satisfy certain hazard rate conditions, then coarse matching with exactly two classes can capture at least half of the total surplus generated by PAM. A limitation of McAfee’s result is that he restricts attention only to two classes and also to a …xed way of partitioning the two populations, namely, at the mean. The focus of this paper is the performance of coarse matching schemes with a …nite number of classes. The main results of the paper are the following ones. First, assuming a multiplicative match payo¤ function, we derive a lower bound on the performance (fraction of the total surplus) of n-class coarse matching under mild conditions on the distributions of agents’characteristics, which provides more ‡exibility of partitioning the populations. Second, we prove that this result generalizes to a large class of match payo¤ functions. Third, we show that these results are applicable to a broad class of applications, including a monopoly pricing problem with incomplete information, as well as to a cost-sharing problem with incomplete information. In these problems, standard models predict that optimal contracts sort types completely. For example, a monopolist will o¤er a continuum of product qualities. Our result implies that a monopolist can capture a large fraction of the second-best pro…ts by o¤ering pooling contracts with a small number of qualities.