Contested garment rule

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. Therefore, we first give to each claimant, the amount conceded to him/her by 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. ^[1] They call the rule, that selects this unique division, the CG-consistent rule (it is also called the Talmud rule). ^[2]

Problem description

There is a divisible resource, denoted by ${\displaystyle E}$ (=Estate or Endowment). 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 ${\displaystyle c_{i}}$. We denote ${\displaystyle c_{N}:=\sum _{i=1}^{n}c_{i}=}$the total claim. ${\displaystyle c_{N}>E}$, that is, the estate is insufficient to satisfy all the claims. The goal is to allocate to each claimant an amount ${\displaystyle x_{i}}$ such that ${\displaystyle \sum _{i=1}^{n}x_{i}=E}$.

Two claimants

With two claimants, the CG rule works in the following way.

• Truncate each claim to the estate (since one is not allowed to claim more than the entire estate). That is, set ${\displaystyle c_{i}':=\min(c_{i},E)}$ for each claimant i.
• Allocate to claimant 1 an amount ${\displaystyle E-c_{2}'}$ that is, the amount not claimed by 2.
• Allocate to claimant 2 an amount ${\displaystyle E-c_{1}'}$ that is, the amount not claimed by 1.
• The remainder is ${\displaystyle E-(E-c_{2}')-(E-c_{1}')=c_{1}'+c_{2}'-E}$; divide it equally among the claimants.

Summing the amounts given to each claimant, we can write the following formula:

${\displaystyle CG(c_{1},c_{2};E)=\left({\frac {E+c_{1}'-c_{2}'}{2}}~,~{\frac {E+c_{2}'-c_{1}'}{2}}\right)}$

For example:

• If ${\displaystyle E=1}$ and ${\displaystyle c_{1}=c_{2}=1}$, then both claimants get 1/2, that is, ${\displaystyle CG(1,1;1)=(1/2,1/2)}$.
• If ${\displaystyle E=1}$ and ${\displaystyle c_{1}=1}$ and ${\displaystyle c_{2}=1/2}$. then claimant 1 gets 3/4 and claimant 2 gets 1/4, that is, ${\displaystyle CG(1,1/2;1)=(3/4,1/4)}$.

These two examples are first mentioned in the first Mishnah of Bava Metzia: ^[3]

"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."

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. ^[4] In particular, we seek an allocation that respects the CG rule for each pair of claimants. That is, for every claimants i and j:

${\displaystyle (x_{i},x_{j})=CG(c_{i},c_{j};x_{i}+x_{j})}$.

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:

• If ${\displaystyle E\leq c_{N}/2}$ (that is, the total estate is less than half the total claims), then apply the rule of constrained equal awards to half the claims, that is, return ${\displaystyle CEA(c_{1}/2,\ldots ,c_{n}/2;E)}$.
• iF ${\displaystyle E\geq c_{N}/2}$, then give each claimant half of his/her claim, and then apply the rule of constrained equal losses to the remainder, that is, return ${\displaystyle (c_{1}/2,\ldots ,c_{n}/2)+CEL(c_{1}/2,\ldots ,c_{n}/2;E-c_{N}/2)}$.

Note that, with two claimants, once the claims are truncated to be at most the estate, the condition ${\displaystyle E\geq c_{N}/2}$ always holds. For example:

• ${\displaystyle CG(1,1/2;1)=(1/2,1/4)+CEL(1/2,1/4;1/4)=(1/2,1/4)+(1/4,0)=(3/4,1/4)}$.

Here are some three-claimant examples:

• ${\displaystyle CG(100,200,300;100)=(33.333,33.333,33.333)}$; here CEA is used.
• ${\displaystyle CG(100,200,300;200)=(50,75,75)}$; here CEA is used.
• ${\displaystyle CG(100,200,300;300)=(50,100,150)}$; here either CEA or CEL can be used (the result is the same); when the sum of claims is exactly half the estate, each claimant gets exactly half his/her claim.
• ${\displaystyle CG(100,200,300;400)=(50,125,225)}$; here CEL is used.
• ${\displaystyle CG(100,200,300;500)=(66.667,166.667,266.667)}$; here CEL is used.
• ${\displaystyle CG(100,200,300;600)=(100,200,300)}$; here CEL is used.

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.

• 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."

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 ${\displaystyle \min _{i}(c_{i}/2)}$. Then, the claimant with the smallest ${\displaystyle c_{i}}$ is put on hold, and the next units are divided equally among the remaining claimants until each of them up to the next-smallest ${\displaystyle c_{i}}$. Then, the claimant with the second-smallest ${\displaystyle c_{i}}$ is put on hold too. This goes on until either the estate is fully divided, or each claimant gets exactly ${\displaystyle c_{i}/2}$. 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 ^[6] 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: ${\displaystyle CG(c,E)=CG(\min(c,E),E)}$. ^[7]

CG is self-dual. This means that it treats gains and losses symmetrically: it divides gains in the same way that it divides losses. Formally: ${\displaystyle CG(c,E)=c-CG(c,C-E)}$, ^{[1] [7]} 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 seperability: define ${\displaystyle v_{i}=\max(0,E-\sum _{j}c_{j})}$ = the sum conceded to i by all other agents. Then, CG can be separated to two phases as follows: first, each agent i gets v_i; 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. ^[8]

CG satisfies securement. This means that each agent with a feasible claim (c_i ≤ E) is guaranteed at least 1/n of his claim: ${\displaystyle CG(c,E)_{i}\geq \min(c_{i},E)/n}$ (this property is similar to proportionality). In fact, CG satisfies a stronger property: ${\displaystyle CG(c,E)_{i}\geq \min(c_{i}/2,E/n)}$. ^[9]

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 his claim: ${\displaystyle c_{i}-CG(c,E)_{i}\geq \min(c_{i},C-E)/n}$. ^[9]

Characterizations

Nir Dagan ^[7] proved two characterizations of CG:

• For two claimants, CG is the unique rule that satisfies both self-duality and independence of irrelevant claims.
• For two claimants, CG is the unique rule that satisfies both self-duality and separability.
• For two claimants, CG is the unique rule that satisfies independence of irrelevant claims, equal treatment of equals, and separability.

Moreno-Ternero and Villar ^[9] proved that CG is characterized by each of the following combinations:

• CG is the unique rule that satisfies self-duality, securement and consistency.
• CG is the unique rule that satisfies securement, dual securement, and consistency.

They show that these characterizations are tight:

• The Proportional rule satisfies self-duality and consistency, but not securement (e.g. if the claims are E and 2E, then the first claimant gets only E/3).
• The Truncated proportional rule (applying the proportional rule after truncating all claims to E) satisfies securement, but it is not self-dual and does not satisfiy dual-securement (e.g. if the claims are E/2 and E, then the first claimant gets E/3. This is more than 1/2 of his claim, so securement is satisfied. However, his loss is only E/6 which is 1/3 of his claim, rather than 1/n).
• The Adjusted proportional rule ^[10] 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. ^{[11] [12]}

Equality

Ly, Zakharevich, Kosheleva and Kreinovich ^[13] 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 he cannot get more than his 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 r*max+(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. ^[13]

Game-theoretic analysis

Nash equilibrium of competitive game

O'Neill ^[14] 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 ${\displaystyle c_{i}}$ 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. ^[14]

Nucleolus of cooperative game

The CG rule can be derived independently, as the nucleolus of a certain cooperative game defined based on the claims. ^[15]

Manipulation by pre-donation

Sertel ^[16] 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 his 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. ^[17] 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: ^[2]

• If the sum of claims is larger than 2E, then it applies the CEA rule on half the claims, that is, it returns ${\displaystyle CEA(c_{1}/2,\ldots ,c_{n}/2;E)}$ .
• Otherwise, it gives each agent half its claim and then applies CEA on the remainder, that is, it returns ${\displaystyle (c_{1}/2,\ldots ,c_{n}/2)+CEA(c_{1}/2,\ldots ,c_{n}/2;E-\sum _{j=1}^{n}c_{j}/2)}$ .

Examples with two claimants:

• ${\displaystyle PINI(60,90;100)=(42.5,57.5)}$. Initially the claimants get (30,45). The remaining claims are (30,45) and the remaining estate is 25, so it is divided equally.
• ${\displaystyle PINI(50,100;100)=(37.5,62.5)}$. Initially the claimants get (25,50). The remaining claims are (25,50) and the remaining estate is 25, so it is divided equally.
• ${\displaystyle PINI(50,100;100)=(37.5,62.5)}$. Initially the claimants get (25,50). The remaining claims are (25,50) and the remaining estate is 25, so it is divided equally.

Examples with three claimants:

• ${\displaystyle PINI(100,200,300;100)=(33.333,33.333,33.333)}$. Here the sum of claims is more than twice the estate, so the outcome is ${\displaystyle CEA(50,100,150;100)=(33.333,33.333,33.333)}$.
• ${\displaystyle PINI(100,200,300;200)=(50,75,75)}$. Again the sum of claims is more than twice the estate, so the outcome is ${\displaystyle CEA(50,100,150;200)=(50,75,75)}$.
• ${\displaystyle PINI(100,200,300;300)=(50,100,150)}$. Again the sum of claims is more than twice the estate, so the outcome is ${\displaystyle CEA(50,100,150;300)=(50,100,150)}$.

Generalization

Moreno-Ternero and Villar ^[18] 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 ${\displaystyle E\leq t\cdot c_{N}}$, then the outcome is ${\displaystyle CEA(tc_{1},\ldots ,tc_{n};E)}$; so each claimant i receives at most ${\displaystyle t\cdot c_{i}}$.
• If ${\displaystyle E=t\cdot c_{N}}$, then each claimant i receives exactly ${\displaystyle t\cdot c_{i}}$.
• If ${\displaystyle E\geq t\cdot c_{N}}$, then the outcome is ${\displaystyle (tc_{1},\ldots ,tc_{n})+CEL((1-t)c_{1},\ldots ,(1-t)c_{n};E-tc_{N})}$; so each claimant i receives at least ${\displaystyle t\cdot c_{i}}$.

An equivalent description is: the claimants receive money in an equal rate, until the lowest claimant (1) has received t*c₁. Then the lowest claimant exits, and the others continue until the second-lowest (2) claimant has received t*c₂. This goes on until all claimants have received ${\displaystyle t\cdot c_{i}}$. 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 c_i 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)

Further reading

• Steven Landsburg, Let the rabbi split the pie: Talmudic wisdom applied to bankruptcy

References

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