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.
- 2.
- 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.
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.
AW may require the transfer of more than one issue, but at most one issue must be divided in the end.
- 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.
- 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.
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.
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.
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.
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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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