Introspectively Envy-Free and Efficient Allocation of Indivisible Mixed Manna

The existence of allocations that are fair and efficient, simultaneously, is a central inquiry in fair division literature. A prominent result in discrete fair division shows that the complementary desiderata of fairness and efficiency can be achieved together when allocating indivisible items with nonnegative values; specifically, for indivisible goods and among agents with additive valuations, there always exists an allocation that is both envy-free up to one item (EF1) and Pareto efficient (PO). While a recent breakthrough extends the EF1 and PO guarantee to indivisible chores (items with negative values), the question remains open for indivisible mixed manna, i.e., for indivisible items whose values can be positive, negative, or zero. The current work makes notable progress in resolving this central question. For indivisible mixed manna and additive valuations, we establish the existence of allocations that are PO and introspectively envy-free up to one item (IEF1). 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 result generalizes the aforementioned existence guarantee for chores. Our techniques can be adopted to obtain an alternative proof for the existence of EF1 and PO allocations of indivisible goods. Hence, along with the result for mixed manna, we provide a unified approach for establishing the EF1 and PO guarantee for indivisible goods and indivisible chores. We also utilize our result for indivisible items to develop a distinct proof of the noted EF and PO guarantee for divisible mixed manna. Our work highlights an interesting application of the Knaster-Kuratowski-Mazurkiewicz (KKM) Theorem in discrete fair division and develops multiple, novel structural insights and algorithmic ideas.