Abstract
Suppose that two players, P1 and P2, must divide a set of indivisible items that each strictly ranks from best to worst. Assuming that the number of items is even, suppose also 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 so that it becomes a complete allocation (all items are allocated) that is EF for one player (Pi) and almost-EF for the other player (Pj) in the sense that it is EF for Pj except for one item — it would be EF for Pj if a specific item assigned to Pi were removed. Moreover, we show how low-ranked (for Pj) that exceptional item may be, 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.
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Notes
Throughout, we follow the convention that when one player is named Pi, the other player is Pj. Thus, when i = 1, j = 2, and when i = 2, j = 1.
The matching need not be unique. Assume P1 ranks four items 1 \(\succ\) 2 \(\succ\) 3 \(\succ\) 4, and P2 ranks them in reverse order: 4 \(\succ\) 3 \(\succ\) 2 \(\succ\) 1. Then assigning {1, 2} to P1 and {3, 4} to P2 yields two matchings in which P1 pairwise prefers each of its items to the matched item of P2: (1 \(\succ\) 3, 2 \(\succ\) 4) and (1 \(\succ\) 4, 2 \(\succ\) 3). The same two matches, exactly reversed, show that P2 pairwise prefers its allocation to P1’s. Thus, if an allocation is EF, P1’s and P2’s matchings may be inverses—as illustrated by this example—but need not be, as illustrated by Example 1 in Sect. 2.
Pruhs and Woeginger [17] provide a different, but equivalent, condition.
We use the same symbol, \(\succ\) , for a player’s ranking of items and for the player’s preference on subsets of items, as the latter is an extension of the former.
Statement (iii) in Lemma 5 of Pruhs and Woeginger [17] is equivalent to D(S).
The definition to follow can be compared to the definition of “EF up to 1 item, or EF1” in the literature [11]. In our context, where the only available information about preferences over subsets is what can be inferred from the rankings of items, it is a natural analog and extension. For a different approach to almost-EF, see Bilò et al. [3].
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D. Marc Kilgour’s research is supported by an NSERC Discovery Grant.
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Brams, S.J., Kilgour, D.M. & Klamler, C. Two-Person Fair Division of Indivisible Items when Envy-Freeness is Impossible. Oper. Res. Forum 3, 24 (2022). https://doi.org/10.1007/s43069-021-00115-7
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DOI: https://doi.org/10.1007/s43069-021-00115-7