Locally greedy characterizations of interim allocations

The received characterizations of feasible interim allocations are mostly in the spirit of Border (1991): Fix a family of sets, each containing some player-types, and test the interim allocation under consideration against all these sets. In the published literature, such Border-like characterizations are known to be valid only in the paramodularity framework, which rules out combinatorial complexities such as matchings. This paper presents a necessity and suciency test for Border-like characterizations with or without paramodularity. It implies that the validity of the characterizations requires that any interim allocation about to become infeasible be locally greedy: that its domain be covered by a family of subsets within each of which the underlying ex post allocation follows some greedy algorithm. I prove impossibility of Border-like characterizations in the knapsack model of sharing economies and prove Border-like characterizations in a matching model that allows each player to have arbitrary numbers of types, and in a ranked-item auction model with a group-specific quota constraint.