Pair formation.

A multitype pair formation model for a one-sex population, without separation, with given type distribution of singles, produces a distribution of pairs with the given type distribution as a marginal distribution. The pair distribution can be seen as a nonnegative symmetric matrix. For this matrix representation formulas have been given years ago and have been widely used. The goal of the paper is to understand these formulas in probabilistic terms and give a meaning to their coefficients. Our approach connects the formulas to the problem of completing a substochastic matrix to a stochastic matrix. In this way the coefficients in the representation formula can be interpreted as preferences and insight can be gained into the set of distributions respecting given preferences. In order to put these questions into a wider perspective, the classical two-sex pair formation models are reviewed and embedded into the class of one-sex models, and dynamic models are designed that yield pair distributions as limit elements.