Stable partitions for proportional generalized claims problems

We consider a set of agents, e.g., a group of researchers, who have claims on an endowment, e.g., a research budget from a national science foundation. The research budget is not large enough to cover all claims. Agents can form coalitions and coalitional funding is proportional to the sum of the claims of its members, except for singleton coalitions which receive no funding. We analyze the structure of stable partitions when coalition members use well-behaved rules to allocate coalitional endowments, e.g., the well-known constrained equal awards rule (CEA) or the constrained equal losses rule (CEL).For continuous, (strictly) resource monotonic, and consistent rules, stable partitions with (mostly) pairwise coalitions emerge. For CEA and CEL we provide algorithms to construct such a stable pairwise partition. While for CEL the resulting stable pairwise partition is assortative and sequentially matches up lowest-claims pairs, for CEA the resulting stable pairwise partition is obtained sequentially by matching up in each step either a highest-claims pair or a highest-lowest-claims pair.More generally, we also assume that the minimal coalition size to have a positive endowment is larger or equal to two. We then show how all results described above are extended to this general case.