Two-Person Fair Division of Indivisible Items when Envy-Freeness Is Impossible

Assume two players, A and B, must divide a set of indivisible items that each strictly ranks from best to worst. If the number of items is even, assume that the players desire that the allocations be balanced (each player gets half the items), item-wise envy-free (EF), and Pareto-optimal (PO). Meeting this ideal is frequently impossible. If so, we find a balanced maximal partial allocation of items to the players that is EF, though it may not be PO. Then we show how to augment it in a way that makes it a complete allocation that is EF for one player (say, A) and almost-EF for the other player (B) in the sense that it is EF for B except for one item – it would be EF for B if a specific item assigned to A were removed. Moreover, we show how low-ranked that exceptional item can be for B, thereby finding an almost-EF allocation that is as close as possible to EF – as well as complete, balanced, and PO. We provide algorithms to find such almost-EF allocations, adapted from algorithms that apply when complete balanced EF-PO allocations are possible.