Stable matching with dynamic preference lists


I have a set $ F$ of $ n_1$ families, a set $ C$ of $ n_2$ children ($ n_1<n_2$ ) and a set $ M$ of feasible one-to-one matchings of the families with the children. All the children have the same preference list over the families, but the families have different preference lists.

Our objective is to select from the set $ M$ a stable matching. It is clear that the family with rank 1 will be matched with the child she ranks the best among those that is matched with them in $ M$ , the family with rank 2 will be matched with the child she ranks the best among those that is matched with them in the matchings selected for family 1 and so on.

When the preference lists are fixed, the constraint can be modeled as follows:

$ 1 \leq \sum\limits_{j’\in J^{\leq}_{C_j^i}(i)}x_{ij’} + \sum\limits_{i’\in F^{\leq}_{i}}x_{i’C_j^i} $ ; $ i \in F$ , $ j=1,\ldots,i$

with

  • $ x_{ij}$ equals one if family $ i$ is matched with rank $ j$ .
  • $ C_j^i$ is the child with rank $ j$ for family $ i$ .
  • $ J^{\leq}_{s}(i)$ is the set of children that family $ i$ ranks at the same level or better than child $ s$ .
  • $ C^{\leq}_{i}$ is the set of families that are at the same level or better than family $ i$ .

How to modify this constraint to take into account the changes in the preference lists?