The existence of allocations that are simultaneously fair and efficient is a central inquiry in the 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, allocations that are both envy-free up to one item (EF1) and Pareto efficient (PO) always exist for indivisible goods and additive valuations. 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, where values can be positive, negative, or zero.
The talk will describe our recent work that 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 (in fact a generalization of) the existence of EF1 and PO allocations of indivisible goods, as well as recover a distinct proof for the existence of PO and envy-free allocations of divisible mixed manna.
Hence, our results provide a unified approach that subsumes several state-of-the-art existence results concerning fair and efficient allocation of both indivisible and divisible items.
Based on joint work with Dr. Siddharth Barman (IISc)
