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 of
the 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 children
from previous marriages. We show that, mathematically, this settlement problem can be modelled as a bipartite
rationing problem for which a unique global proportionally fair solution exists. Moreover, we develop two efficient
algorithms for obtaining this proportionally fair solution, and we show numerically that both algorithms are
considerably faster than standard convex optimization techniques. The first algorithm is a novel tailor-made
fixed-point iteration algorithm, whereas the second algorithm only iteratively applies simple lawsuits involving
a single child and its parents. The inspiration for this latter algorithm comes from our main convergence proof
in which we show that iteratively applying settlements on smaller subnetworks eventually leads to the same
settlement on the network as a whole. This has significant societal importance since in practice lawsuits are
often only held between two or a few parents. Moreover, our iterative algorithm is easy to understand, also
by parents, legal counselors, and judges, which is crucial for its acceptance in practice. Finally, as the method
provides a unique solution to any dispute, it removes the legal inequality perceived by parents. Consequently, it
may considerably reduce the workload of courts because parents and lawyers can compute the proportionally fair
parental contributions before bringing their case to court.