Fair Division

Fair-division procedures offer ways to restrain the exercise of power by preventing one player from gaining more than its “fair share.” Such procedures go back to the Hebrew Bible.

For example, Abram (later to become Abraham) and Lot had to decide who would get Canaan and who Jordan. They reached a decision by using a form of divide-and-choose when Abram said to Lot, “Let there be no strife between you and me, between my herdsmen and yours, for we are kinsmen. Is not the whole land before you? Let us separate: if you go north, I will go south; and if you go south, I will go north” (Gen. 14:8–9). In a similar vein, Solomon had to decide which of two women was the mother of a disputed baby and proposed cutting the baby in half, which elicited a protest from the real mother that revealed her identity.

The oldest known procedure for dividing a single divisible good, such as a cake or land, between two players is “I cut, you choose,” or divide-and-choose. The same procedure can also be used if there are multiple goods: the divider partitions the goods into two piles, and the chooser selects one pile.

If the divider has no knowledge of the chooser's preferences, the divider should divide the goods 50–50 for itself. That way, whichever pile the chooser selects, the divider is assured of getting 50%. The chooser, on the other hand, will get more than 50% if it thinks the two piles are unequal and selects the one that it thinks is more valuable. If the divider knows the chooser's preferences, it can exploit this information to make one pile slightly more valuable than 50% for the chooser, so the chooser will select it, keeping for itself the pile it values more (assuming it values different goods from the chooser).

Is divide-and-choose fair? To make an assessment, consider the following criteria for determining what a fair share is:

• 1.
  Proportionality. If there are n players and they are equally entitled to the goods, a division is proportional if each player thinks it received at least 1/n of the total value.
• 2.
  Envy-freeness. If no player is willing to give up its portion in exchange for the portion another received, this player will not envy any other player. In two-player disputes, a division is envy-free if and only if it is proportional.

  In the case of three players, however, envy-freeness is more demanding than proportionality. For example, I may think I'm getting 1/3, but if I think you are getting 1/2 (because the third player, in my eyes, is getting only 1/6), then I will envy you. Whereas an envy-free allocation is always proportional, even when there are more than two players, a proportional allocation is not necessarily envy-free.

• 3.
  Equitability. A division is equitable if all players think they received exactly the same value above 1/n. Thus, no player P envies another's [Page 240]happiness for getting more of the value in its eyes than P got in its eyes. To illustrate the difference between equitability and envy-freeness for two players, an allocation is envy-free if one player thinks it received 51% and the other 90%. But it is not equitable, because they both did not receive the same amount above 50%, making the 51% player less happy than the 90% player.
• 4.
  Efficiency. An allocation is efficient if there is no other allocation that is better for one player without being worse for another. Efficiency by itself—that is, when not linked with properties such as proportionality, envy-freeness, or equitability—is no guarantee that an allocation will be fair. For example, an allocation that gives everything to me and nothing to you is efficient: Any other allocation will make me worse off when it makes you better off. The other properties of fairness, combined with efficiency, ensure that the total value is distributed according to the aforementioned properties.

The modern mathematical theory of fair division has its roots in the 1940s, particularly in the work of Polish mathematicians who proposed different procedures for dividing both divisible and indivisible goods.

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