Contested garment rule
Contested garment rule
The contested garment (CG) rule, also called concede-and-divide, 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 they effectively concede the unclaimed estate to the other claimant. Therefore, the amount conceded by the claimant is first given 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.<blockquote>"Two are holding a garment. One says, "I found it," and the other says, "I found it":
If one says "all of it is mine" and the other says "all of it is mine", then this one shall swear that he owns no less than half of it, and this one shall swear that he owns no less than half of it, and they shall divide it between them. If one says, "all of it is mine" and the other says "half of it is mine", then the one who says "all of it is mine" shall swear that he owns no less than three quarters of it; and the one who says "half of it is mine" shall swear that he owns no less than one quarter of it; the former takes three quarters and the latter takes one quarter." </blockquote>
Many claimants 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. In particular, we seek an allocation that respects the CG rule for each pair of claimants. That is, for every claimants i and j:<blockquote><math>(x_i, x_j) = CG(c_i, c_j; x_i+x_j)</math>.</blockquote>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.<blockquote>"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.
If the total value of the estate is only 100 dinars, the wives divide the estate equally. If there were 200 dinars in the estate, the first wife takes 50 dinars, while the other two wives take three dinars of gold each, which are the equivalent of 75 silver dinars. If there were 300 dinars in the estate, the first wife takes 50 dinars, the second takes 100 dinars, and the third takes six dinars of gold, the equivalent of 150 silver dinars."* </blockquote>
Constructive description 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 <math>\min_i(c_i/2)</math>. Then, the claimant with the smallest <math>c_i</math> is put on hold, and the next units are divided equally among the remaining claimants until each of them up to the next-smallest <math>c_i</math>. Then, the claimant with the second-smallest <math>c_i</math> is put on hold too. This goes on until either the estate is fully divided, or each claimant gets exactly <math>c_i/2</math>. 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.
Explicit formula Elishakoff and Dancygier present an explicit formula for the CG rule for n claimants.
Properties CG satisfies independence of irrelevant claims. This means that increasing the claim above the total estate does not change the allocation. Formally: <math>CG(c,E) = CG(\min(c,E), E)</math>. where C is the sum of all claims.
CG satisfies [[equal-treatment-of-equals]]: agents with the same claim will get exactly the same allocation.
CG satisfies separability: define <math>v_i = \max(0, E- \sum_j c_j)</math> = the sum conceded to i by all other agents. Then, CG can be separated to two phases as follows: first, each agent i gets vi; then, the same rule is activated on the remaining claims and the remaining estate.
- Note that separability is the dual of independence-of-irrelevant-claims.
CG satisfies securement. This means that each agent with a feasible claim (ci ≤ E) is guaranteed at least 1/n of their claim: <math>CG(c,E)_i \geq \min(c_i,E)/n</math> (this property is similar to proportionality). In fact, CG satisfies a stronger property: <math>CG(c,E)_i \geq \min(c_i/2,E/n)</math>.
CG also satisfies the dual property to securement: the loss of each agent i with claim at most the total loss C-E, is at least 1/n of their claim: <math>c_i - CG(c,E)_i \geq \min(c_i,C-E)/n</math>. is self-dual and satisfies securement and its dual, but it is not consistent. The [[constrained-equal-awards]] rule satisfies securement and consistency, but it is not self-dual and does not satisfiy dual-securement. The constrained-equal-losses rule rule satisfies dual-securement and consistency, but it is not self-dual and does not satisfiy securement.
See also: More characterization of the Talmud rule.
Equality Ly, Zakharevich, Kosheleva and Kreinovich prove that CG for two agents satisfies a fairness notion based on equal distance from a status quo point. Several other rules are based on this fairness notion, e.g.:
- constrained-equal-awards aims to equalize the distance from the status-quo point (0,0);
- constrained-equal-losses aims to equalize the distance from the status-quo point (c1,c2).
This raises the question of what status-quo points are reasonable. For each claimant, there can be a whole interval of possible status-quo points, for example:
- If E ≤ c1 ≤ c2, then for both agents the range of possible outcomes is [0,*E*], so the status quo point can be any point in [0,*E*]x[0,*E*].
- If c1 ≤ E ≤ c2, then for agent 1 the range of possible outcomes is [0, c1] as the agent cannot get more than their claim. For agent 2, the worst outcome is that agent 1 gets c1; hence the range of possible outcomes is [E-c1, E].
- If c1 ≤ c2 ≤ E, then for agent 1 the range of possible outcomes is [E-c2, c1], and similarly for agent 2 the range is [E-c1, c2].
The agents can be optimistic and look at the highest values in their interval, or be pessimistic and look at the lowest values in their interval, or in general look at any intermediate point rmax+(1-r)min, where r is the "optimism coefficient". For any optimism coefficient r, we get a different status-quo point.
The CG rule selects, for any optimism coefficient r, an outcome in which both claimants are equally distant from their status-quo point corresponding to r. describes the following game.
- The estate is divided to small units (for example, if all claims are integers, then the estate can be divided into E units of size 1).
- Each claimant i chooses some <math>c_i</math> units.
- Each unit is divided equally among all agents who claim it.
Naturally, the agents would try to choose units such that the overlap between different agents is minimal. This game has a nash-equilibrium. In any Nash equilibrium, there is some integer k such that each unit is claimed by either k or k+1 claimants. When there are two claimants, there is a unique equilibrium payoff vector, and it is identical to the one returned by CG.
Manipulation by pre-donation Sertel considers a special case of a two-claimant setting, in which the endowment E is equal to the larger claim (E = c2 ≥ c1). This special case corresponds to a cooperative-bargaining problem in which the feasible set is a triangle with vertices (0,0), (c1,0), (0,c2), and the disagreement point is (0,0). The payoff is calculated using the Nash Bargaining Solution. A claimant may manipulate by pre-donating some of their claims to the other claimant. In equilibrium, both claimants receive the payoffs prescribed by CG.
Piniles' rule Zvi Menahem Piniles, a 19th-century Jewish scholar, presented a different rule to explain the cases in Ketubot. His rule is similar to the CG rule, but it is not consistent with the CG rule when there are two claimants. The rule works as follows: define a family of rules, which they call the TAL family, which generalizes the Talmud rule, as well as constrained-equal-awards and constrained-equal-losses. Each rule in the TAL family is parameterized by a parameter t in [0,1]. The TAL_t rule divides the estate as follows:
- If <math>E \leq t\cdot c_N</math>, then the outcome is <math>CEA(t c_1,\ldots,t c_n; E)</math>; so each claimant i receives at most <math>t\cdot c_i</math>.
- If <math>E = t\cdot c_N</math>, then each claimant i receives exactly <math>t\cdot c_i</math>.
- If <math>E \geq t\cdot c_N</math>, then the outcome is <math>(t c_1,\ldots,t c_n) + CEL((1-t) c_1,\ldots,(1-t) c_n; E-t c_N)</math>; so each claimant i receives at least <math>t\cdot c_i</math>.
An equivalent description is: the claimants receive money in an equal rate, until the lowest claimant (1) has received tc1. Then the lowest claimant exits, and the others continue until the second-lowest (2) claimant has received tc2. This goes on until all claimants have received <math>t\cdot c_i</math>. If there is remaining amount, then the claimants enter again, from the highest to the lowest, and get money until their losses are equal.
In this family, TAL-0 is CEL; TAL-1/2 is CG; and TAL-1 is CEA. The dual of TAL_t is TAL_(1-t). All rules in this family have the following properties:
They are parametric: agent i's award depends only on ci and on some parameter that depends on E. They satisfy equal-treatment-of-equals. They are continuous. They are consistent. They are order-preserving: agents with higher claims get higher rewards and suffer higher losses. They satisfy claims-monotonicity: increasing a claim weakly increases the award. They are homogeneous: multiplying the claims and the endowment by the same positive number yields multiplication of the outcome by the same number. They satisfy resource-monotonicity. Some properties are satisfied only by subsets of the TAL family:
- All and only TAL-t rules with t in [1/2,1] are independent of irrelevant claims (claims larger than E can be truncated to E with no effect on the outcome), and satisfy securement.
- All and only TAL-t rules with t in [0,1/2] are separable (satisfy composition from minimal independent), and satisfy the dual of securement.
- Only CEA and CEL satisfy composition up and composition down.
- If t1 ≥ t2, then the outcome of TAL-t1 always dominates the outcome TAL-t2 in the Lorenz ordering (which means, intuitively, that it is more egalitarian)