Proximately Envy-Free and Efficient Allocation of Mixed Manna

The existence of fair and efficient allocations of indivisible items is a central problem in fair division. For indivisible goods, the existence of Pareto efficient (PO) and envy free up to one item (EF1) allocations was established by Caragiannis et al. In a recent breakthrough, Mahara established the existence of PO and EF1 allocations for indivisible chores. However, the existence of PO and EF1 allocations of mixed manna remains an intriguing open problem. In this paper, we make significant progress in this direction. We establish the existence of allocations that are PO and introspective envy free up to one item (IEF1) for mixed manna. In an IEF1 allocation, each agent can eliminate its envy towards all the other agents by either adding an item or removing an item from its own bundle. The notion of IEF1 coincides with EF1 for indivisible chores, and hence, our existence result generalizes the aforementioned result of Mahara.