The contested garment (CG) rule,[1] also called concede-and-divide,[2] is a division rule for solving problems of conflicting claims (also called "bankruptcy problems").
The idea is that, if one claimant's claim is less than 100% of the estate to divide, then he effectively concedes the unclaimed estate to the other claimant.
The remaining amount is then divided equally among the two claimants.
The CG rule first appeared in the Mishnah, exemplified by a case of conflict over a garment, hence the name.
In the Mishnah, it was described only for two-people problems.
But in 1985, Robert Aumann and Michael Maschler have proved that, in every bankruptcy problem, there is a unique division that is consistent with the CG rule for each pair of claimants.
[2] There is a divisible resource, denoted by
There are n people who claim this resource or parts of it; they are called claimants.
The amount claimed by each claimant i is denoted by
, that is, the estate is insufficient to satisfy all the claims.
The goal is to allocate to each claimant an amount
With two claimants, the CG rule works in the following way.
Summing the amounts given to each claimant, we can write the following formula:
These two examples are first mentioned in the first Mishnah of Bava Metzia:[3]"Two are holding a garment.
To extend the CG rule to problems with three or more claimants, we apply the general principle of consistency (also called coherence), which says that every part of a fair division should be fair.
[4] In particular, we seek an allocation that respects the CG rule for each pair of claimants.
.Apriori, it is not clear that such an allocation always exists, or that it is unique.
However, it can be proved that a unique CG-consistent allocation always exists.
[1] It can be described by the following algorithm: Note that, with two claimants, once the claims are truncated to be at most the estate, the condition
The first three examples appear in another Mishnah, in Ketubot:[5]"Suppose a man, who was married to three women, died; the marriage contract of one wife was for 100 dinars, and the marriage contract of the second wife was for 200 dinars, and the marriage contract of the third wife was for 300, and all three contracts were issued on the same date so that none of the wives has precedence over any of the others.
The CG rule can be described in a constructive way.
Suppose E increases from 0 to the half-sum of the claims: the first units are divided equally, until each claimant receives
is put on hold, and the next units are divided equally among the remaining claimants until each of them up to the next-smallest
This goes on until either the estate is fully divided, or each claimant gets exactly
If some estate remains, then the losses are divided in a symmetric way, starting with an estate equal to the sum of all claims, and decreasing down to half this sum.
The CG rule is self-dual.
This means that it treats gains and losses symmetrically: it divides gains in the same way that it divides losses.
[1][6] The CG rule can be derived independently, as the nucleolus of a certain cooperative game defined based on the claims.
[7] Zvi Menahem Piniles, a 19th-century Jewish scholar, presented a different rule to explain the cases in Ketubot.
The rule works as follows:[2] Examples with two claimants: Examples with three claimants: