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Fair Division*

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Book cover Computational Complexity

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Article Outline

Glossary

Definition of the Subject

Introduction

Single Heterogeneous Good

Several Divisible Goods

Indivisible Goods

Conclusions

Future Directions

Bibliography

Adapted from Barry R. Weingast and DonaldWittman (eds) Oxford Handbook of Political Economy (Oxford University Press, 2006) by permission of Oxford UniversityPress.

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Notes

  1. 1.

    [28], pp. 28–29; additional information on the minimum numbers of cuts required to give envy‐freeness is given in [19], and [29].

  2. 2.

    See [10,20,26] for other approaches, based on bidding, to the housemates problem discussed in [31]. On approximate solutions to envy‐freeness, see [33]. For recent results on pie‐cutting, in which radial cuts are made from the center of a pie to divide it into wedge‐shaped pieces, see [3,8].

  3. 3.

    Dividing several homogeneous goods is very different from cake‐cutting. Cake‐cutting is most applicable to a problem like land division, in which hills, dales, ponds, and trees form an incongruous mix, making it impossible to give all or one thing (e. g., trees) to one player.By contrast, in property division it is possible to give all of one good to one player. Under certain conditions, 2‑player cake division, and the procedure to be discussed next (adjusted winner), are equivalent [22].

  4. 4.

    A website for AW can be found at http://www.nyu.edu/projects/adjustedwinner.Procedures applicable to more than two players are discussed in [13,15,23,32].

  5. 5.

    AW may require the transfer of more than one issue, but at most one issue must be divided in the end.

  6. 6.

    A procedure called proportional allocation (PA) awards issues to the players in proportion to the points they allocate to them. While inefficient, PA is less vulnerable to strategic manipulation than AW, with which it can be combined ([13], pp. 75–80).

  7. 7.

    For a more systematic treatment of conflicts in fairness criteria and trade-offs that are possible, see [5,6,9,11,18,21].

  8. 8.

    Recall that an envy-free division of indivisible items is one in which, no matter how the players value subsets of items consistent with their rankings, no player prefers any other player's allocation to its own. If a division is not envy-free, it is envy‐possible if a player's allocation may make it envious of another player, depending on how it values subsets of items, as illustrated for player C by division (12, 34, 56). It is envy‐ensuring if it causes envy, independent of how the players value subsets of items. In effect, a division that is envy‐possible has the potential to cause envy. By comparison, an envy‐ensuring division always causes envy, and an envy-free division never causes envy.

  9. 9.

    This is somewhat different from Rawls's (1971) proposal to maximize the utility of the player with minimum utility, so it might be considered a modified Rawlsian criterion. I introduce a rough measure of utility next with a modified Borda count.

  10. 10.

    This might be considered a second‐order application of the maximin criterion: If, for two divisions, players rank the worst item any player receives the same, consider the player that receives a next-worst item in each, and choose the division in which this item is ranked higher. This is an example of a lexicographic decision rule, whereby alternatives are ordered on the basis of a most important criterion; if that is not determinative, a next-most important criterion is invoked, and so on, to narrow down the set of feasible alternatives.

  11. 11.

    The standard scoring rules for the Borda count in this 6-item example would give 5 points to a best item, 4 points to a 2nd-best item, …, 0 points to a worst item. I depart slightly from this standard scoring rule to ensure that each player obtains some positive value for all items, including its worst choice, as assumed earlier.

Abbreviations

Efficiency :

An allocation is efficient if there is no other allocation that is better for one player and at least as good for all the other players.

Envy‐freeness :

An allocation is envy-free if each player thinks it receives at least a tied-for‐largest portion and so does not envy the portion of any other player.

Equitability :

An allocation is equitable if each player values the portion that it receives the same as every other player values its portion.

Bibliography

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Author information

Authors and Affiliations

  1. Department of Politics, New York University, New York, USA

    Steven J. Brams

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  1. Steven J. Brams
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Brams, S.J. (2012). Fair Division* . In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_68

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