Guaranteeing Envy-Freeness under Generalized Assignment Constraints

Abstract

We study fair division of goods under the broad class of generalized assignment constraints. In this constraint framework, the sizes and values of the goods are agent-specific, and one needs to allocate the goods among the agents fairly while further ensuring that each agent receives a bundle of total size at most the corresponding budget of the agent. Since, in such a constraint setting, it may not always be feasible to partition all the goods among the agents, we conform---as in recent works---to the construct of charity to designate the set of unassigned goods. For this allocation framework, we obtain existential and computational guarantees for envy-free (appropriately defined) allocation of divisible and indivisible goods, respectively, among agents with individual, additive valuations for the goods. We deem allocations to be fair by evaluating envy only with respect to feasible subsets. In particular, an allocation is said to be feasibly envy-free (FEF) iff each agent prefers its bundle over every (budget) feasible subset within any other agent's bundle (and within the charity). The current work establishes that, for divisible goods, FEF allocations are guaranteed to exist and can be computed efficiently under generalized assignment constraints. Note that, in the presence of generalized assignment constraints, even the existence of such fair allocations of divisible goods is nonobvious, a priori. Our existential and computational guarantee for FEF allocations is built upon an incongruity property satisfied across a family of linear programs. This novel proof template is interesting in its own right. In the context of indivisible goods, FEF allocations do not necessarily exist, and hence, we consider the fairness notion of feasible envy-freeness up to any good (FEFx). Under this notion, an allocation of indivisible goods is declared to be fair iff for each pair of agents, a and b, envy-freeness holds for agent a against every feasible and strict subset of b's bundle; a similar guarantee is required with respect to the charity. We show that, under generalized assignment constraints, an FEFx allocation of indivisible goods always exists. In fact, our FEFx result resolves open problems posed in prior works, which provide existence guarantees under weaker fairness notions and more specialized constraints. Further, for indivisible goods and under generalized assignment constraints, we provide a pseudo-polynomial time algorithm for computing FEFx allocations, and a fully polynomial-time approximation scheme (FPTAS) for computing approximate FEFx allocations.