On Proportionally Consistent Solutions to the Divorced-Parents Problem.

When Dutch parents with children divorce, a mediator compiles a matrix with the financial needs of the children and the financial capacities of parents to meet these needs. Moreover, in case parents have children from previous marriages or are prepared to contribute to stepchildren, a bipartite graph shows which parent is financially responsible for which child. The Dutch high court ruled that the final contributions should be proportionally consistent, implying that shortages for children should be prevented if possible and any remaining parental capacities should be proportionally divided among parents responsible for the same child. Finding by hand this proportional solution is difficult for realistic court cases, as these can include several (step)parents and children. The paper on the divorced-parents problem shows that the final unique solution can be found when parents start court cases iteratively and provides efficient algorithms that can deal with large (country-size) networks. When Dutch parents divorce, Dutch law dictates that the parental contributions to cover the financial needs of the children have to be proportionally consistent. 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 modeled as a bipartite rationing problem for which a unique global proportionally proportional solution exists. Moreover, we develop two efficient algorithms for obtaining this proportionally proportional 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 (FPA), 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 because, 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 proportional parental contributions before bringing their case to court. [ABSTRACT FROM AUTHOR]