On Proportionally Fair Solutions for the Divorced-Parents Problem

When Dutch parents divorce, Dutch law dictates that the parental contributions to cover the financial needs ofthe children have to be proportionally fair. This rule is clear when parents only have common children. However,cases can be considerably more complicated, for example when parents have financial responsibilities to childrenfrom previous marriages. We show that, mathematically, this settlement problem can be modelled as a bipartiterationing problem for which a unique global proportionally fair solution exists. Moreover, we develop two efficientalgorithms for obtaining this proportionally fair solution, and we show numerically that both algorithms areconsiderably faster than standard convex optimization techniques. The first algorithm is a novel tailor-madefixed-point iteration algorithm, whereas the second algorithm only iteratively applies simple lawsuits involvinga single child and its parents. The inspiration for this latter algorithm comes from our main convergence proofin which we show that iteratively applying settlements on smaller subnetworks eventually leads to the samesettlement on the network as a whole. This has significant societal importance since in practice lawsuits areoften only held between two or a few parents. Moreover, our iterative algorithm is easy to understand, alsoby parents, legal counselors, and judges, which is crucial for its acceptance in practice. Finally, as the methodprovides a unique solution to any dispute, it removes the legal inequality perceived by parents. Consequently, itmay considerably reduce the workload of courts because parents and lawyers can compute the proportionally fairparental contributions before bringing their case to court.