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Maximin Envy-Free Division of Indivisible Items

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Abstract

Assume that two players have strict rankings over an even number of indivisible items. We propose two algorithms to find balanced allocations of these items that are maximin—maximize the minimum rank of the items that the players receive—and are envy-free and Pareto-optimal, if such allocations exist. To determine whether an envy-free allocation exists, we introduce a simple condition on preference profiles; in fact, our condition guarantees the existence of a maximin, envy-free, and Pareto-optimal allocation. Although not strategy-proof, our algorithms would be difficult to manipulate unless a player has complete information about its opponent’s ranking. We assess the applicability of the algorithms to real-world problems, such as allocating marital property in a divorce or assigning people to committees or projects.

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Notes

  1. Procedures, based on cardinal utility, to divide an odd number of items are of course available, but the properties that we focus on would have to be redefined to allow cardinal comparisons of the players’ subsets.

  2. Note that we consider only balanced allocations, i.e., the two players’ subsets have equal cardinality.

  3. For a detailed study of how to rank sets of items, see Barbera et al. (2004).

  4. See Bouveret et al. (2016) for a discussion.

  5. Note that several distinct injective mappings may guarantee EF. Moreover, the injective mappings \(f_A\) and \(f_B\) may or may not be inverses. If EF is guaranteed by some bijection, then it is guaranteed by a canonical bijection defined as follows: Order A’s assigned items according to A’s preference and then order B’s items to A’s preference. The bijection maps the \(k{\text {th}}\) item on the first list to the \(k{\text {th}}\) item on the second list.

  6. It could as well be called a minimax allocation, because minimizing the maximum depth, and maximizing the minimum depth, are equivalent processes, though in game theory maximin and minimax outcomes may not be the same.

  7. Note that the maximin depth cannot be less than m, even for unbalanced allocations.

  8. Step 4 is taken from the AL algorithm of Brams et al. (2014).

  9. Repetitions of step 1 ensure the envy-freeness of the items it allocates, because a player is assigned its own singles, which it ranks at or above the current maximin rank, m, to the opponent’s singles, which it ranks below rank m. In other words, each player item-wise prefers its singles to the other player’s singles in each repetition.

  10. However, if the least-preferred unassigned items are identical after the first assignment of singles, the procedure followed under ISD is exactly the same as under SD.

  11. An allocation is PO if and only if it is the product of a sequence of sincere item choices by the players, i.e., choices that are consistent with the players’ preference rankings (Brams and King 2005). Thus in Example 1, if A and B sincerely choose items in the alternating order ABABABAB, they obtain the allocation (1234, 8765); if the alternating order is BABABABA, they obtain the allocation (1235, 8764). The first allocation is PO but neither EF nor MX, whereas the second is PO and EF but not MX.

  12. If an SD allocation were not PO, there would be a preferred allocation of singles or doubles to each player. But at every stage at which singles are allocated, neither player would prefer any of its opponent’s singles to its own, so neither player can—by trading any of its singles for the other player’s singles—improve its allocation at that stage. As for the doubles, they are allocated according to AL, which guarantees that at least one allocation will be PO (Brams et al. 2014).

  13. In fact, any “bottom-up” procedure involving two players who submit rankings is vulnerable to misrepresentation (Brams and Taylor 1999, pp. 21–24). Under such a procedure, one player chooses first—by giving its worst item to the other player—and the players anticipate a sequence of later choices (in which, for example, their choices alternate). Optimal choices in such a game constitute a Nash equilibrium (Kohler and Chandrasekaran 1971; Brams and Straffin 1979; Levine and Stange 2012), but they do not necessarily yield an EF allocation when one exists.

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Correspondence to Steven J. Brams.

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We acknowledge the valuable comments of the audience of the Dagstuhl Seminar 15241 “Computational Social Choice: Theory and Applications”. Klamler was supported by the University of Graz project, “Voting-Selection-Decision”.

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Brams, S.J., Kilgour, D.M. & Klamler, C. Maximin Envy-Free Division of Indivisible Items. Group Decis Negot 26, 115–131 (2017). https://doi.org/10.1007/s10726-016-9502-x

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