In the budget-feasible allocation problem, a set of items with varied sizes and values are to be allocated to a group of agents. Each agent has a budget constraint on the total size of items she can receive. The goal is to compute a feasible allocati
on that is envy-free (EF), in which the agents do not envy each other for the items they receive, nor do they envy a charity, who is endowed with all the unallocated items. Since EF allocations barely exist even without budget constraints, we are interested in the relaxed notion of envy-freeness up to one item (EF1). The computation of both exact and approximate EF1 allocations remains largely open, despite a recent effort by Wu et al. (IJCAI 2021) in showing that any budget-feasible allocation that maximizes the Nash Social Welfare (NSW) is 1/4-approximate EF1. In this paper, we move one step forward by showing that for agents with identical additive valuations, a 1/2-approximate EF1 allocation can be computed in polynomial time. For the uniform-budget and two-agent cases, we propose efficient algorithms for computing an exact EF1 allocation. We also consider the large budget setting, i.e., when the item sizes are infinitesimal compared with the agents budgets, and show that both the NSW maximizing allocation and the allocation our polynomial-time algorithm computes have an approximation close to 1 regarding EF1.
We study a fair resource sharing problem, where a set of resources are to be shared among a set of agents. Each agent demands one resource and each resource can serve a limited number of agents. An agent cares about what resource they get as well as
the externalities imposed by their mates, whom they share the same resource with. Apparently, the strong notion of envy-freeness, where no agent envies another for their resource or mates, cannot always be achieved and we show that even to decide the existence of such a strongly envy-free assignment is an intractable problem. Thus, a more interesting question is whether (and in what situations) a relaxed notion of envy-freeness, the Pareto envy-freeness, can be achieved: an agent i envies another agent j only when i envies both the resource and the mates of j. In particular, we are interested in a dorm assignment problem, where students are to be assigned to dorms with the same capacity and they have dichotomous preference over their dorm-mates. We show that when the capacity of the dorms is 2, a Pareto envy-free assignment always exists and we present a polynomial-time algorithm to compute such an assignment; nevertheless, the result fails to hold immediately when the capacities increase to 3, in which case even Pareto envy-freeness cannot be guaranteed. In addition to the existential results, we also investigate the implications of envy-freeness on proportionality in our model and show that envy-freeness in general implies approximations of proportionality.
The Nash social welfare (NSW) is a well-known social welfare measurement that balances individual utilities and the overall efficiency. In the context of fair allocation of indivisible goods, it has been shown by Caragiannis et al. (EC 2016 and TEAC
2019) that an allocation maximizing the NSW is envy-free up to one good (EF1). In this paper, we are interested in the fairness of the NSW in a budget-feasible allocation problem, in which each item has a cost that will be incurred to the agent it is allocated to, and each agent has a budget constraint on the total cost of items she receives. We show that a budget-feasible allocation that maximizes the NSW achieves a 1/4-approximation of EF1 and the approximation ratio is tight. The approximation ratio improves gracefully when the items have small costs compared with the agents budgets; it converges to 1/2 when the budget-cost ratio approaches infinity.
