On the Lattice Structure of Stable Allocations in a Two-Sided Discrete-Concave Market

The stable allocation model is a many-to-many matching model in which each pair’s partnership is represented by a nonnegative integer. This paper establishes a link between two different formulations of this model: the choice function model studied thoroughly by Alkan and Gale and the discrete-concave (M♮-concave) value function model introduced by Eguchi, Fujishige, and Tamura. We show that the choice functions induced from M♮-concave value functions are endowed with consistency, persistence, and size monotonicity. This implies, by the result of Alkan and Gale, that the stable allocations for M♮-concave value functions form a distributive lattice with several significant properties such as polarity, complementarity, and uni-size property. Furthermore, we point out that these results can be extended for quasi M♮-concave value functions.