Economic Theory 21, 241–261 (2003)

Claims problems and weighted generalizationsof the Talmud rule�

Toru Hokari1 and William Thomson2

1 Institute of Social Sciences, University of Tsukuba, 1-1-1 Ten’no-dai,Tsukuba, Ibaraki 306-8571, JAPAN (e-mail: hokari@social.tsukuba.ac.jp)

2 Department of Economics, University of Rochester, Rochester, NY 14627, USA(e-mail: wth2@troi.cc.rochester.edu)

Received: April 8, 2002; revised version: June 26, 2002

Summary. We investigate the existence of consistent rules for the resolution ofconflicting claims that generalize the Talmud rule but do not necessarily satisfyequal treatment of equal. The first approach we follow starts from the descriptionof the Talmud rule in the two-claimant case as “concede-and-divide”, and an ax-iomatic characterization for the rule. When equal treatment of equals is dropped,we obtain a one-parameter family, “weighted concede-and-divide rules”. The sec-ond approach starts from the description of the Talmud rule as a hybrid of theconstrained equal awards and constrained equal losses rules, and weighted gener-alizations of these rules. We characterize the class of consistent rules that coincidewith weighted concede-and-divide rules in the two-claimant case or with weightedhybrid rules. They are defined by partitioning the set of potential claimants into“priority classes” or “half-priority classes” respectively, and selecting referenceweights for all potential claimants. For the first approach however, and in eachclass with more than two claimants, equal treatment is actually required

Keywords and Phrases: Claims problems, Weighted generalizations of Talmudrule, Consistency, Converse consistency.

JEL Classification Numbers: C71, D71.

1 Introduction

When a firm goes bankrupt, how should its liquidation value be divided amongits creditors? This problem is an example of a general class that we call “claims

� Thomson acknowledges support from NSF under grant SBR-9731431. We thank Jean-Pierre Benoı̂tfor his comments, and the referee for several useful suggestions.Correspondence to: W. Thomson

242 T. Hokari and W. Thomson

problems”. In general, the question is how to allocate a resource among agentshaving incompatible claims on it, the goal being to identify well-behaved methodsof making a recommendation for each problem. We call such methods division“rules”.1 In the standard specification of a claims problem, claimants only differin their claims, and the requirement is imposed on rules that two agents with equalclaims should receive equal amounts. This is the property of “equal treatment ofequals”. However, in applications, in addition to their claims, agents may haverights, needs, obligations and so on, that could, or should, be taken into accountwhen performing the division. Then, two agents with equal claims need not receiveequal amounts.2 In fact, fairness requires that this be the case.

The possibility of treating differently two agents with equal claims can beformally accommodated in two ways. First, claimants can be sorted into priorityclasses, precedence being given to each class before any consideration is given toany lower class. This is actual practice. An extreme case is when a strict order isspecified on the set of claimants and they are compensated one after the other untilmoney runs out. We call such a rule a “priority rule”. Alternatively, and somewhatless radically, we can choose “weights” reflecting the relative importance to be givento each claimant. Using weights so as to gear the social choice towards agents whoare perceived as more deserving is standard in cooperative game theory and invarious branches of the theory of fair allocation. (How to use these weights in thepresent context is made explicit later on.) Finally, rules can be defined by mixingboth procedures, partitioning the set of claimants into priority classes and withineach class, assigning them weights.

Most of the rules that are central to the literature can be redefined in this way.This is in particular the case for two important rules that appear in Medieval writings,the constrained equal awards and constrained equal losses rules. The former assignsequal amounts to all claimants subject to no one receiving more than his claim; thelatter assigns amounts so that all claimants experience equal losses subject to noone receiving a negative amount. However, a rule whose introduction has been veryimportantly responsible for the considerable development of the axiomatic literatureon the subject has not: it is the rule defined by Aumann and Maschler (1985) torationalize the resolutions proposed in the Talmud for certain numerical examplesdescribed there, the “Talmud rule”. Our objective is to investigate the existence ofgeneralizations of the rule that would allow recognizing ways in which claimantsmay differ besides their claims.

We follow two strategies. To explain them, we recall two important facts con-cerning the Talmud rule. First, in the two-claimant case, the rule assigns to eachclaimant the sum of (i) the difference between the amount to divide and the claimof the other claimant (or 0 if this difference is negative), and (ii) half of the remain-der. We call this method “concede-and-divide” because one of its justifications isthe following natural and simple scenario: the difference in (i) is interpreted asthe amount conceded to each agent by the other, and in a “first-round” it certainlymakes sense to award him that amount; the dispute is really about the remainder,

1 For a survey of the literature devoted to this subject, see Thomson (1995).2 For another study of a model of claims resolution that dispenses with the equal treatment assumption,

see Moulin (2000). We discuss this contribution in the concluding section.

Claims problems and weighted generalizations of the Talmud rule 243

and (ii) says that in a “second-round”, this remainder should be divided equally(Aumann and Maschler, 1985, give arguments in favor of equal division in thesecond round). Seen from a different perspective, the difference in (i) can be de-scribed as a “minimal” amount to which the agent is entitled, his “minimal right”.A second important fact concerning the Talmud rule is that it is “consistent”: whensome agents leave with their awards and the situation is re-evaluated, the rule makesthe same recommendation for the remaining agents as before. This property hasrecently played an important role in the axiomatic study of a variety of classes ofproblems. A central justification for the Talmud rule is that, although many waysof looking at the problem coincide with concede-and-divide in the two-claimantcase, the rule is the unique consistent one to do so (Aumann and Maschler, 1985;Benoı̂t, 1997).

Our first strategy starts from the observation just made concerning the coinci-dence of many rules with concede-and-divide in the two-claimant case. Here areimportant examples. For the random arrival rule, imagine claimants arriving oneat a time, and fully compensate them until money runs out; then, take the averageof the awards vectors so obtained under the assumption that all orders of arrivalare equally likely.3 For the minimal overlap rule, imagine that each agent claimsa specific part of the amount to divide; position these claims so as to maximize inthe lexicographic maximin order the part claimed by exactly one claimant, then thepart claimed by exactly two claimants, and so on; finally, divide each part equallyamong all agents claiming it.4 Two other rules that coincide with concede-and-divide in the two-claimant case are the version of the constrained equal awards ruleobtained by first assigning to each claimant his minimal right and then applyingthe rule to divide the remainder, and the version of the constrained equal lossesrule obtained by first truncating claims by the amount available. For each of therules just enumerated, a natural asymmetric version of the scenario that underliesit can be defined. It turns out that all the rules so obtained coincide. We call them“weighted” concede-and-divide rules.

For the passage from two claimants to larger populations, we appeal to consis-tency. Our first main result is a characterization of the consistent rules extending theweighted concede-and-divide rules. They are defined as follows: an ordered parti-tion of the set of potential claimants into “priority classes” is defined, a “referencepartition”, and a positive weight is chosen for each potential claimant belongingto a two-claimant class. For each specific problem, the reference partition inducesan ordered partition of the set of claimants who are present; for a component ofthis partition that coincides with a two-claimant reference priority class, apply theweighted concede-and-divide rule relative to the weights assigned to its two mem-bers; otherwise, apply the Talmud rule itself.

For example, take as reference partition {. . . , 11, 9, 7, 5, 3, 1, 0}, {2, 4, 6},{8, 10}, {12, 14, 16}, {18, . . .}, with the left-to-right direction corresponding tolower and lower priorities. For a problem with claimant set {7, 5, 3, 1, 0, 2, 6, 8, 10,12, 18, 20}, the induced partition is {7, 5, 3, 1, 0}, {2, 6}, {8, 10}, {12}, {18, 20}.

3 This rule is based on the scenario underlying the Shapley value (Shapley, 1953).4 This rule is defined by O’Neill (1982).

244 T. Hokari and W. Thomson

For the component {7, 5, 3, 1, 0}, apply the Talmud rule because this componentis induced from a reference class that has more than two claimants. The next com-ponent, {2, 6}, has two members, but apply the Talmud rule to it as well becausethis component is induced from a reference class that has more than two claimants.The next component, {8, 10}, coincides with a two-claimant reference class, so aweighted concede-and-divide rule can now be used. The next component consistsof a single agent, and all rules coincide in that trivial case. The final componentis {18, 20}, and since it is induced from a reference class that has more than twoclaimants, return to the Talmud rule itself.

Our second strategy takes as point of departure an alternative description ofthe Talmud rule as a hybrid of the constrained equal awards and constrained equallosses rules. Since both have straightforward “weighted” generalizations to the sortof situations we have in mind here, we use those as ingredients in constructinggeneralizations of the Talmud rule. Interestingly, in the two-claimant case, theresulting “weighted hybrids” rules are not the same as the weighted concede-and-divide rules. We then ask about their consistency and show that when the weightsare all positive, and if they are chosen in a “consistent” manner across populations,consistency holds. This conclusion parallels the aforementioned characterizationof Aumann and Maschler’s. Moreover, the limit case where some of the weightsare 0 is also allowed. The choice of weighted hybrid rules or of their limits in thetwo-claimant case, and consistency, give us weighted hybrid rules in general: a“reference ordered partition” of the set of potential claimants is chosen and eachpotential claimant is assigned a positive “reference weight”. This time and for areason to be explained shortly, we refer to its components as “half-priority classes”.For each specific problem, the reference partition induces an ordered partition onthe set of claimants actually present. The awards vector chosen by the rule canthen be described as a function of the amount available as follows: divide thefirst units among the members of the first component of the induced partition byapplying the weighted hybrid rule with weights proportional to the weights of itsmembers, and do so until the amount to divide is equal to the half-sum of theirclaims. Then, turn to the second component of the induced partition and divide thenext units among its members by applying the weighted hybrid rule with weightsproportional to their weights. Proceed in this way until each claimant has receivedhis half-claim. At that point, return to the first component of the induced partitionand divide additional units among its members by picking up the application ofthe first weighted hybrid rule where it was interrupted, and do so until all of itsmembers are fully compensated. Then, return to the second component of theinduced partition and to the second weighted hybrid rule, and divide additionalunits until its members are fully compensated, and so on.

The need to select the half-claims vectors when the amount to divide is equalto the half-sum of the claims limits the extent to which some claimants can begiven precedence over others. Indeed, the weighted hybrid rules have the followingfeature: focusing on the two-claimant case, as it allows for a simple statement, itis that, as we increase the parameter describing the extent to which a claimant isfavored at the expense of the other, the weighted hybrid rule does not get closer tothe priority rule in which the favored agent is first. Our first strategy provides this

Claims problems and weighted generalizations of the Talmud rule 245

limit behavior, which one may consider desirable, but at the price of restricting thechoice of weights, that is, the flexibility with which one can favor certain claimantsat the expense of others. This flexibility is present only for two-claimant priorityclasses.

We will leave it to the reader to decide which of the two classes of rules weidentify is more suitable. In applications, it certainly will be a matter of circum-stances how much of an asymmetric treatment of claimants is needed. It is clearhowever that in generalizing the Talmud rule so as to accommodate the need tofavor particular claimants, consistency places constraints that do not exist for otherrules, and that rarely exist in other contexts. We say rarely because we know ofat least one other model, for which a similar phenomenon occurs (Section 3). An-other revealing study in this regard is discussed in the concluding section (Moulin,2000). There, we also show how to incorporate exogenously given informationabout priority classes and weights, and we present an alternative to consistency thataccommodates an asymmetric treatment of claimants (Dagan and Volij, 1997).

2 The problem of resolving conflicting claims

There is an infinite set of “potential” claimants, indexed by the natural numbersN. Each given problem, however, only involves a finite number of them. Let Ndenote the class of non-empty finite subsets of N. A claims problem is a pair(c, E) ∈ R

N+ × R+, where N ∈ N , such that

∑N ci ≥ E. Let CN denote the

class of all problems with claimant set N . A division rule is a function definedon ∪N∈N CN that associates with each N ∈ N and each (c, E) ∈ CN an awardsvector of (c, E), namely a vector x ∈ R

N such that 0 � x � c and satisfying theefficiency condition

∑N xi = E.5

Let R be a rule. For each N ∈ N and each c ∈ RN+ , the path of awards of

R for c is the locus of the awards vector R selects as E ranges from 0 to∑

N ci:x ∈ PR(N, c) ⇐⇒ there exists E ∈ [0,

∑N ci] such that x = R(c, E).

Here are important rules. For the constrained equal awards rule, awards areequal subject to no-one receiving more than his claim. For the constrained equallosses rule, at the chosen awards vector, the losses agents experience are equalsubject to no-one receiving a negative amount.

Constrained equal awards rule, CEA. For each N ∈ N , each (c, E) ∈ CN ,and each i ∈ N , CEAi(c, E) ≡ min{ci, λ}, where λ ∈ R+ is chosen so as toachieve efficiency.

Constrained equal losses rule, CEL. For each N ∈ N , each (c, E) ∈ CN , andeach i ∈ N , CELi(c, E) ≡ max{ci − λ, 0}, where λ ∈ R+ is chosen so as toachieve efficiency.

The Talmud rule (Aumann and Maschler, 1985) can be described in severalways. Its description as a hybrid of the constrained equal awards and constrainedequal losses rules is our point of departure in our second attempt at defining gener-alizations of it that accommodate a desired preferential treatment of some agents.

5 The notation x � y means that for each i ∈ N , xi ≤ yi.

246 T. Hokari and W. Thomson

Although this description has almost always been adopted in the literature, an im-portant lesson of our work is that which description is used considerably influencesthe way in which one goes about searching for such generalizations.

Talmud rule, T. For each N ∈ N , each (c, E) ∈ CN , and each i ∈ N , Ti(c, E) ≡min{ ci

2 , λ} if∑ cj

2 ≥ E, and Ti(c, E) ≡ max{ ci

2 , ci − λ} otherwise, where ineach case, λ ∈ R+ is chosen so as to achieve efficiency.

The following requirement on a rule is central to our analysis: starting fromsome problem, and having applied the rule, we imagine some of the claimantsleaving with their awards, and we re-evaluate the situation from the viewpoint ofthe remaining ones. We ask that for this “reduced problem”, the rule should assignto them the same awards as it did initially.6

Consistency. For each N ∈ N , each (c, E) ∈ CN , and each N ′ ⊂ N , if x ≡R(c, E), then xN ′ = R(cN ′ , E − ∑

N\N ′ xi).7

Bilateral consistency is obtained by adding the restriction |N ′| = 2. In deriv-ing necessary conditions for our main characterizations, we only use this weakerproperty, but the rules we end up with are consistent.

3 A first approach

We begin with a discussion of the case of two claimants. In order to generalizea given rule so as to favor one of them, we introduce a parameter representingthe extent of this desired bias. The parameter is specified by applying the ruleto a problem in which claims are equal. The choice made then reflects society’spreferences in treating the two claimants. In principle, the parameter could dependon the common value of their claims and on the amount to divide. We limit ourselvesto rules for which asymmetry is “uniform”, as they can be described by means ofa single parameter in the one-dimensional simplex.

We observed earlier that in the two-claimant case, the Talmud rule coincideswith a number of other important rules. We listed concede-and-divide, the randomarrival and minimal overlap rules, the rule obtained from the constrained equalawards rule by first assigning to each agent his minimal right, and the rule obtainedfrom the constrained equal losses rule by first truncating any claim greater thanthe amount to divide by that amount. Our first approach starts from asymmetricversions of the scenarios underlying these rules. In spite of their great diversity,they all result in the same family of rules. We then give an axiomatic justificationfor this family. Finally, we extend its members to arbitrary populations by meansof consistency.

The first part of the program just outlined simply consists in reconsidering thevarious scenarios that we have seen lead to rules that all coincide in the two-claimantcase, but this time allowing for an asymmetric treatment of the two claimants.

6 For a survey, see Thomson (1996).7 Since rules are such that for each i ∈ N , xi ∈ [0, ci], then the sum of the claims of the remaining

claimants is still greater than the amount that the remainder, so that the reduced problem is well-defined.

Claims problems and weighted generalizations of the Talmud rule 247

Figure 1a–d. Weighted versions of concede-and-divide. Starting from concede-and-divide, we giveprogressively more weight to agent 1 than to agent 2. The kinks in the paths of awards still occur whenE = c1 or E = c2. a Concede-and-divide. b and c: Weighted versions of weights α1 ≡ ( 2

3 , 13 ) and

α2 ≡ ( 56 , 1

6 ). d At the limit, we obtain the priority rule relative to the ordering 1 ≺ 2, D1≺2

For “concede-and-divide”, the concession step is unchanged but the remainderof the second step is divided proportionally to the weights assigned to the twoclaimants. For the random arrival rule, we place a greater weight on the order forwhich one of them is first. For the minimal overlap rule, when two agents claimthe same part of the amount available, we perform the division proportionally totheir weights. The constrained equal awards rule operated from minimal rights andthe constrained equal losses rule operated from truncated claims can be similarlyredefined. Remarkably, all of these generalizations coincide with the rules definedas follows. Given pair {i, j} ∈ N , let ∆{i,j} denote the unit simplex in R

{i,j}+ and

int∆{i,j} its relative interior. Let α ≡ (αij , αji) ∈ ∆{i,j}:

Weighted concede-and-divide rule of weights α ∈ ∆{i,j}, Cα. For each(c, E) ∈ C{i,j},

{Cα

i (c, E)≡ max{E − cj , 0}+αij

[E − max{E − cj , 0}− max{E − ci, 0}]

,

Cαj (c, E)≡ max{E − ci, 0}+αji

[E − max{E − cj , 0}− max{E − ci, 0}]

.

Next, we enquire about a possible axiomatic justification for these rules. Anatural starting point is the characterization of concede-and-divide itself offeredby Dagan (1996). It involves three requirements. First, two claimants with equalclaims should receive equal amounts. Second, one should be able to ignore anypart of a claim that is over the amount to divide. Third, one should be able to solveeach problem in either one of the following two ways: (i) directly, or (ii) by firstassigning to each claimant his minimal right, decreasing claims by the amountsreceived in this first round, and then applying the rule to divide what is left.

Equal treatment of equals. For each (c, E) ∈ CN and each pair {i, j} ⊆ N , ifci = cj , then Ri(c, E) = Rj(c, E).

Invariance under claims truncation. For each (c, E) ∈ CN , R(c, E) =R(t(c, E), E), where for each i ∈ N , ti(c, E) ≡ min{ci, E}.

248 T. Hokari and W. Thomson

Minimal rights first. For each (c, E) ∈ CN , R(c, E) = m(c, E) + R(c −

m(c, E), E − ∑mi(c, E)

), where for each i ∈ N , mi(c, E) ≡ max

{E −∑

j∈N\{i} cj , 0}

.8

Concede-and-divide is the only two-claimant rule satisfying these three re-quirements (Dagan, 1996). If equal treatment of equals is dropped, a large class ofadditional rules become admissible, but a small subclass of rules also satisfy therequirement that if claims and amount to divide are multiplied by the same positivenumber, so should all awards. This requirement is met by most rules that have beendiscussed in the literature.

Homogeneity. For each (c, E) ∈ CN and each t > 0, R(tc, tE) = tR(c, E).

Our characterization, whose proof does not depart much from Dagan’s ownargument, is the following:

Proposition 1. For |N | = 2, the weighted concede-and-divide rules are the onlyrules on CN satisfying homogeneity, invariance under claims truncation, and min-imal rights first.

Proof. We omit the proof that the weighted concede-and-divide satisfy the threeproperties. Conversely, let N ≡ {i, j}, R be a rule satisfying these properties, and(c, E) ∈ CN . Without loss of generality, suppose ci ≤ cj .

Case 1: E ≤ ci. Then, min{ci, E} = min{cj , E} = E. By in-variance under claims truncation, R(c, E) = R(E, E; E). Let α ≡R(E, E; E)/‖R(E, E; E)‖ = R(E, E; E)/E. Let E′ ≤ ci. By invariance un-der claims truncation, R(c, E′) = R(E′, E′; E′) and by homogeneity, R(c, E′) =E′α. Altogether, we conclude that for each E ≤ ci, R(c, E) = Cα(c, E).

Case 2: ci < E ≤ cj . Then, minimal rights are max{E − cj , 0} = 0 andmax{E − ci, 0} = E − ci. Claims revised down by the minimal rights are ci andcj−(E−ci). There remains E−(E−ci) = ci to divide. After truncation, claims aremin{ci, ci} = ci and min{cj −(E−ci), ci} = ci. Since, by Case 1, R(ci, ci; ci) =ciα, we obtain, by minimal rights first, R(c, E) = (0, E − ci) + ciα = Cα(c, E).

Case 3: cj < E. Then, minimal rights are E − cj and E − ci. Claims reviseddown by the minimal rights are ci − (E − cj) and cj − (E − ci), which are equal.There remains E − (E − cj) − (E − ci) = ci + cj − E to divide. We have alreadycalculated that R(ci + cj − E, ci + cj − E, ci + cj − E) = (ci + cj − E)α. Byminimal rights first, R(c, E) = (E − ci, E − cj) + (ci + cj − E)α = Cα(c, E).

��Equipped with both a definition of a family of rules that constitute attractive

weighted generalizations of the two-claimant version of the Talmud rule as wellas with an axiomatic justification for them, we now turn to general populations.We appeal to consistency. Our next result is a characterization of the class ofrules satisfying all our requirements. First, define an ordered partition of the set

8 Note that(c − m(c, E), E − ∑

mi(c, E))

is a well-defined problem.

Claims problems and weighted generalizations of the Talmud rule 249

of potential claimants into priority classes. Also, assign a positive weight to eachpotential claimant, with the weights assigned to all members of a class that containsmore than two agents being equal. Without loss of generality, they can be chosento be 1. Then, for each specific group of claimants and each specific problem thisgroup may face, identify the ordered partition of this group induced by the referencepartition. Do not give anything to a class until all classes with higher priorities havebeen fully satisfied. For a component of the partition induced by a two-claimantreference class, divide between them what remains available when its turn comesby applying the weighted concede-and-divide rule with weights proportional totheir reference weights. For a component induced by any larger reference class,(for which all weights are equal,) apply the Talmud rule itself.

Formally, let � be a complete and transitive binary relation on N, with ≺ and ∼denoting its asymmetric and symmetric parts. Let w ∈ R

N++ be such that for each

i ∈ N, if |{j ∈ N |i ∼ j }| �= 2, then wi = 1.

First definition of the sequential Talmud rule relative to � and w, T C,�,w. LetN ∈ N and (c, E) ∈ CN . Let (S1, S2, . . . , SK) be the ordered partition of N suchthat for each pair {k, �} ⊆ {1, 2, . . . , K}, each i ∈ Sk, and each j ∈ S�, (i) ifk = �, then i ∼ j, and (ii) if k < �, then i ≺ j. Let E1 ≡ min{E,

∑i∈S1

ci},E2 ≡ min{E − E1,

∑i∈S2

ci}, E3 ≡ min{E − E1 − E2,∑

i∈S3ci}, and so on

For each pair {i, j} ∈ N , let αij ≡ wi

wi+wj. Then, let TC,�,w(c, E) be the

awards vector x of (c, E) such that for each k ∈ {1, 2, . . . , K},

(i) if Sk = {i}, then xi ≡ Ek;(ii) if Sk = {i, j}, then (xi, xj) ≡ C(αij ,αji)(ci, cj , Ek);(iii) if |Sk| ≥ 3, then xSk

≡ T (cSk, Ek).

Our first main result is the following characterization:

Theorem 1. A rule R on C is such that

(∗) for each pair {i, j} ∈ N , there exists a vector of weights (αij , αji) ∈ ∆{i,j}

such that on C{i,j}, R = C(αij ,αji),

and satisfies consistency only if there exists a complete and transitive binary relation� on the set of potential claimants such that for each pair {i, j} ∈ N ,

(i) (αij , αji) = (1, 0) if and only if i ≺ j;(ii) (αij , αji) ∈ int∆{i,j} if and only if i ∼ j;(iii) if there exists k ∈ N\{i, j} such that i ∼ j ∼ k, then αij = αji.

Then, R coincides with the sequential Talmud rule TC,�,w, where w ∈ RN++ is

defined by setting for each i ∈ N,

wi ≡{

αij if there exists j ∈ N\{i} such that{k ∈ N

∣∣ k ∼ i

}= {i, j},

1 otherwise.

Conversely, each sequential Talmud rule TC,�,w satisfies condition (∗) andconsistency. Here, for each pair {i, j} of claimants, the weights (αij , αji) are

250 T. Hokari and W. Thomson

constructed from � and w by

(αij , αji) ≡{

(1, 0) if i ≺ j,(

wi

wi+wj,

wj

wi+wj

)if i ∼ j.

Clearly the rules TC,�,w satisfy condition (∗). Also:

Lemma 1. The sequential Talmud rules TC,�,w are consistent.

Proof. Let N ∈ N , (c, E) ∈ CN , and x ≡ TC,�,w(c, E). We show that for eachN ′ ⊂ N , xN ′ = TC,�,w

(cN ′ , E − ∑

N\N ′ xi

). By the definition of consistency,

we can suppose that N ′ ≡ N\{j} for some j ∈ N .Let (S1, S2, . . . , SK) be the ordered partition of N induced by �. Let � ∈

{1, 2, . . . , K}be such that j ∈ S�. For eachk ∈ {1, 2, . . . , K}, letEk be the amountto divide among the members of Sk in (c, E). For each k ∈ {1, 2, . . . , K} with k �=�, let E′

k be the amount to divide among the members of Sk in (cN ′ , E −xj). Also,if S�\{j} �= ∅, let E′

� be the corresponding amount for S�\{j} in (cN ′ , E − xj).Clearly, for each k < �, E′

k = Ek. Thus, for each k < �, xSk=

TC,�,wSk

(cN ′ , E − xj).First, suppose that S� = {j}. Then xj = E� and

E′�+1 = min

{E − xj −

( �∑

k=1

Ek − E�

),

∑

i∈S�+1

ci

}= E�+1.

Similarly, for each k ≥ � + 1, E′k = Ek. Thus, for each k > �, xSk

=TC,�,w

Sk(cN ′ , E − xj).

Next, suppose that |S�| ≥ 2. By definition,

E′� =

{min{E − xj ,

∑i∈S�\{j} ci} if � = 1,

min{E − xj − ∑�−1k=1 Ek,

∑i∈S�\{j} ci} if � > 1.

Case 1: xj = cj . Then E′� = E� − xj and E′

�+1 = min{E − xj − (∑�

k=1 Ek −xj),

∑i∈S�+1

ci} = E�+1. Similarly, for each k ≥ � + 1, E′k = Ek.

Case 2: xj < cj . Then, E� = E if � = 1, and E� = E − ∑�−1k=1 Ek if � > 1.

Also, for each k > �, Ek = 0. Recall that for each i ∈ N , xi ≤ ci. Thus,

∑

i∈S�\{j}ci ≥

∑

i∈S�\{j}xi = E� − xj =

{E − xj if � = 1,

E − xj − ∑�−1k=1 Ek if � > 1.

This implies that E′� = E� − xj ≤ ∑

i∈S�\{j} ci, and hence, for each k > �,E′

k = Ek = 0.

In both cases, E′� = E� − xj and for each k > �, E′

k = Ek. If |S�| ≥ 3,then by consistency of the Talmud rule, for each i ∈ S�\{j}, xi = Ti(cS�

, Ek) =Ti(cS�\{j}, E� − xj) = TC,�,w

i (cN ′ , E − xj). If S� = {i, j}, then xi = E� −xj = TC,�,w

i (cN ′ , E − xj). Finally, for each k > �, since E′k = Ek, we have

xSk= TC,�,w

Sk(cN ′ , E − xj). ��

Claims problems and weighted generalizations of the Talmud rule 251

Clearly, the weighted concede-and-divide rules satisfy the following property:

Resource-monotonicity. For each (c, E) ∈ CN and each E′ ∈ R+, if E < E′ ≤∑ci, then for each i ∈ N , Ri(c, E′) ≥ Ri(c, E).

Moreover, if a rule is resource-monotonic in the two-claimant case and consis-tent, it is resource-monotonic in general (Dagan, Serrano, and Volij, 1996; Hokariand Thomson, 2000). Thus, if a rule satisfies condition (∗) and consistency, it isresource-monotonic.

Lemma 2. Let R be a rule on C satisfying condition (∗) and consistency. Let N ≡{i, j, k}. Then the α’s appearing in condition (∗) satisfy the following relations:

(i) If αji, αjk �∈ {0, 1}, then αji = αjk.(ii) If αji < 1 and αjk = 1, then αik = 1.

Proof. (i) Suppose that αji, αjk �∈ {0, 1}. First, we show that αji ≥ αjk.Suppose, by contradiction, that αji < αjk. Let (ci, cj , ck) ≡ (1, 2, 1). Since Ris resource-monotonic, R(c, E) is non-decreasing and continuous in E. Thus,for some E1 > 0, Ri(c, E1) + Rj(c, E1) = 1. By condition (∗) and con-sistency,

(Ri(c, E1), Rj(c, E1)

)= R(ci, cj ; 1) = (αij , αji). By consistency,(

Rj(c, E1), Rk(c, E1)) ∈ PR({j, k}, cj , ck). Thus, as indicated in Figure 2a,

Rk(c, E1) = αji·αkj

αjk. Similarly, since R(c, E) is non-decreasing and continuous

in E, then for some E2 > 0, Rj(c, E2) + Rk(c, E2) = 1. By consistency and con-dition (∗),

(Rj(c, E2), Rk(c, E2)

)= R(cj , ck; 1) = (αjk, αkj). By consistency,(

Ri(c, E2), Rj(c, E2)) ∈ PR({i, j}, ci, cj). Since αji < αjk < 1, it can be seen

from Figure 2(b) that Ri(c, E2) = αij .

By resource-monotonicity, the line segment in R{i,k}+ connecting (αij ,

αji·αkj

αjk)

and (αij , αkj) is a subset of PR({i, k}, ci, ck). Since αji < αjk, then αji·αkj

αjk<

αkj . Thus, PR({i, k}, ci, ck) contains a non-degenerate line segment along whichclaimant i’s award is constant and strictly between 0 and c1 (Fig. 2b). Since ci =ck = 1, this contradicts condition (∗). Thus, αji ≥ αjk.

By a similar argument, it can be shown that αji ≤ αjk. Thus, αji = αjk.(ii) Suppose that αji < 1 and αjk = 1. Let (ci, cj , ck) ≡

(1, 1, 1). Since R(c, E) is non-decreasing and continuous in E, then forsome E1 > 0, Ri(c, E1) + Rj(c, E1) = 1. By consistency and condi-tion (∗),

(Ri(c, E1), Rj(c, E1)

)= R(ci, cj ; 1) = (αij , αji). By consis-

tency,(Rj(c, E1), Rk(c, E1)

) ∈ PR({j, k}, cj , ck). Thus, as indicated in Fig-ure 3, Rk(c, E1) = 0. By consistency, (αij , 0) = (Ri(c, E1), Rk(c, E1)) ∈PR({i, k}, ci, ck). By αij > 0 and condition (∗), αik = 1. ��

Lemma 2 essentially implies the existence of a complete and transitive binaryrelation on N that is used in defining each sequential Talmud rule. The next lemmashows how to construct this relation.

Lemma 3. Let R be a rule on C satisfying condition (∗) and consistency. Definea binary relation � on N as follows: for each pair {i, j} ∈ N , i � j if and only ifαij > 0. Let ∼ denote the symmetric part of �. Then the following two conditionshold:

252 T. Hokari and W. Thomson

�

�

xj

xk

�cj = 2

�

ck = 1

�1

�αjk

�αji

�

αkj

�

�

�

αji·αkj

αjk

a

�

�

��

��

��

��

��

��

xi

xj

xk

�

cj = 2

�

ci = 1

�

ck = 1

�

�

αjk

�

αji

�

αij

�

αkj

�

�

1

�

�

b

Figure 2a,b. Proof of Lemma 2(i). Panel a represents a side-view of the three-claimant problem inpanel b

(i) � is complete and transitive.(ii) If i, j, k ∈ N are distinct and i ∼ j ∼ k, then

αij = αji = αik = αki = αjk = αkj =12.

Proof. Clearly, � is complete. We show that � is transitive. Let i, j, k ∈ N besuch that i � j and j � k. Then αij > 0 and αjk > 0. We want to show thatαik > 0. Suppose, by contradiction, that αik = 0. Then αki = 1. By condition (ii)of Lemma 2, αkj < 1 and αki = 1 imply αji = 1, which contradicts αij > 0.Thus, αik > 0.

Next, let i, j, k ∈ N be distinct and i ∼ j ∼ k. Then, by the definition of �,αij , αji, αik, αki, αjk, αkj �∈ {0, 1}.

By condition (i) of Lemma 2, αij = αji = αik = αki = αjk = αkj = 12 . ��

So far, we have shown that if a rule satisfies condition (∗) and consistency, thenit coincides with a sequential Talmud rule for the two-claimant case. The followingproperty is useful to show that this coincidence holds in general.

Claims problems and weighted generalizations of the Talmud rule 253

xi

�

�xj

��

��

��

��

��

��xk

αjk = cj = 1

�

ci = 1

�

ck = 1

�

�

αji�

αij�

Figure 3. Proof of Lemma 2(ii). Relating the claimants’ weights

Converse consistency. For each N ∈ N with |N | ≥ 3, each (c, E) ∈ CN , andeach x ∈ R

N+ with

∑N xi = E, if for each N ′ ⊂ N with |N ′| = 2, xN ′ =

R(cN ′ , E − ∑

N\N ′ xi

), then x = R(c, E).

Lemma 4. The sequential Talmud rules TC,�,w are conversely consistent.

Proof. Clearly, the rules TC,�,w satisfy condition (∗). By Lemma 1, they areconsistent. As mentioned earlier, condition (∗) and consistency imply resource-monotonicity. Resource-monotonicity and consistency imply converse consistency(Chun, 1999). Thus, they are conversely consistent. ��Proof of Theorem 1. As mentioned before, each rule TC,�,w satisfies condition (∗)and by Lemma 1, consistency. Conversely, let R be a rule on C satisfying condi-tion (∗) and consistency. Let � denote the complete and transitive binary relationdefined as in Lemma 3. For each i ∈ N, let

wi ≡{

αij if there exists j ∈ N\{i} such that{k ∈ N

∣∣ k ∼ i

}= {i, j},

1 otherwise.

Then, by Lemma 3, R coincide with TC,�,w in the two-claimant case. Let N ∈ Nwith |N | > 2, (c, E) ∈ CN , and x ≡ R(c, E). Since R is consistent, for eachN ′ ⊂ N with |N ′| = 2,

xN ′ = R(cN ′ , E − ∑

N\N ′ xi

)= TC,�,w

(cN ′ , E − ∑

N\N ′ xi

).

Finally, by converse consistency of TC,�,w, x = TC,�,w(c, E).9 ��9 This proof is an instance of a very general lemma, “Elevator Lemma” (Thomson, 1996), which

asserts that if a solution correspondence is consistent (bilateral consistency would suffice), and coincideswith a conversely consistent solution correspondence in the two-agent case, then coincidence holds foran arbitrary number of agents.

254 T. Hokari and W. Thomson

Since each rule TC,�,w satisfies homogeneity, invariance under claims trun-cation, and minimal rights first, we obtain the following result as a corollary ofProposition 1 and Theorem 1.

Corollary 1. The sequential Talmud rules TC,�,w are the only rules satisfyinghomogeneity, invariance under claims truncation, minimal rights first, and consis-tency.

In the theory of coalitional games, a similar question to the one we addressedhere has arisen concerning the possibility of defining and justifying asymmetricversions of the “standard solution”, the two-player solution that divides equallybetween the players the surplus above their individual rationality utilities. Suchoperations are in general possible, and rich classes of solutions have emerged thathave been very useful in applications, the primary example being the weightedversions of the Shapley value. One solution however has not been so generalized,the nucleolus (Schmeidler, 1969). Indeed, if the counterpart of the condition thatwe referred to as consistency is imposed on a solution to coalitional games, noweighted generalization exists (Orshan, 1994; Hokari, 2000). Given the correspon-dence between the nucleolus and the Talmud rule (Aumann and Maschler, 1985),10

one may think that our result could be obtained by somehow adapting these authors’proofs. This is indeed the case, although there is no logical relation between theseearlier results and ours because we work on a smaller domain on which the axiomslose force. At the same time, and precisely because claims problems constitute aconsiderably simpler class of problems, a much more direct proof is available. Thatis the proof we have presented.

4 A second approach

If we think of the Talmud rule as a hybrid of the constrained equal awards andconstrained equal losses rules, our search for generalizations of the rule naturallypasses by first defining weighted generalizations of these rules. This is easily done,even for an arbitrary number of agents. Given N ∈ N , let ∆N denote the unitsimplex in R

N+ and int∆N its relative interior.

Weighted constrained equal awards rule of weights α ∈ int∆N , CEAα. Foreach (c, E) ∈ CN and each i ∈ N , CEAα

i (c, E) ≡ min{ci, αiλ}, where λ ∈ R+is chosen so as to achieve efficiency.

Weighted constrained equal losses rule of weights α ∈ int∆N , CELα. For

each (c, E) ∈ CN and each i ∈ N , CELαi (c, E) ≡ max

{ci − λ

αi, 0

}, where

λ ∈ R+ is chosen so as to achieve efficiency.

10 If, as suggested by O’Neill (1983), one associates to a claims problem a coalitional game byassigning to each coalition a “worth” equal to the difference between the amount to divide and the sumof the claims of the members of the complementary coalition (or 0 if this difference is negative), andthen calculate the nucleolus of the game, one obtains the awards vector produced by the Talmud rulefor the problem.

Claims problems and weighted generalizations of the Talmud rule 255

�

�

x1

x1α1

= x2α2

�c

CEAα

� α

a

�

�

x1

x2

α1(c1 − x1) =α2(c2 − x2)

�c

CELα

‘

�α

b

Figure 4a,b. Weighted versions of the constrained equal awards and constrained equal losses rules.a Weighted constrained equal awards rule of weights α ≡ ( 2

3 , 13 ). b Weighted constrained equal losses

rule of weights α

Note that the limit case, where α has coordinates equal to 0, is excluded, asthe formulae would then not produce a well-defined rule. Choosing weights equalto 0 amounts to giving priority to some agents over the others, a possibility that wediscuss a little later. In order to get an idea of the behavior of the rules just introduced,we plot representative paths of awards in the two-claimant case (Fig. 4).

We now combine these rules so as to obtain a rule in the spirit of the Talmud rulebut that incorporates society’s bias towards one or the other of the two claimants.We give the definition for a fixed N , and again, for positive weights.

Weighted hybrid rule of weights α ∈ int∆N , Hα. For each (c, E) ∈ CN

and each i ∈ N , Hαi (c, E) ≡ min{ ci

2 , αiλ} if∑ cj

2 ≥ E, and max{ ci

2 , ci − λαi

}otherwise, where in each case, λ is chosen so as to achieve efficiency.

Figure 5 shows for the two-claimant case how the weighted hybrid rule behavesas the relative weight placed on the claim of agent 1 increases. Also represented isthe limit case, which is naturally associated with the limit weight vector (1, 0). Itis of interest that for each α ∈ ∆N and each claims vector, the path of awards ofHα goes through the half-claims vector.11

We just wrote the three definitions above for a fixed population but when pop-ulation varies, the weights assigned to a given claimant could in principle varydepending upon the identity of his fellow claimants. However, in order to obtainconsistency, we will see that weight vectors assigned to different populations shouldsatisfy certain relations: given two claimants and two groups related by inclusionand containing both, the relative weights assigned to these two claimants should beequal in both groups. One way to achieve this property is to assign a “reference”weight to each potential claimant, and for each actual problem, to use the weightsthat have been assigned to whoever is present. If all reference weights are positive,

11 They satisfy the property called “midpoint property”, formulated by Chun, Schummer, and Thom-son (1999).

256 T. Hokari and W. Thomson

Figure 5a–d. Weighted versions of the Talmud rule, seen as a hybrid of the constrained equal awardsand constrained equal losses rules. Here, N ≡ {1, 2}. In each panel, the slope of the segment emanatingfrom the claims point is the inverse of that of the segment emanating from the origin. (a) The symmetriccase. In the following panels, we give progressively more weight to claimant 1 relative to claimant 2.(b) Here, α1 ≡ ( 2

3 , 13 ), (c) Here, α2 ≡ ( 4

5 , 15 ). (d) At the limit, we obtain a rule that is composed of

“twice” the priority rule in which claimant 1 has priority over claimant 2, the half-claims vector beingused instead of the claims vector itself. We denote this “half-priority” rule H(1,0)

the weighted generalizations of the constrained equal awards and constrained equallosses rules so obtained are consistent. So are the weighted generalizations of thehybrid rules.12

The limit cases can be described as follows. First, define an ordered partitionof the set of potential claimants. For each problem, identify the partition of the setof claimants actually present induced by the reference partition. We naturally referto its components as “priority classes”. Handle each component of the inducedpartition in succession. If the starting point is the constrained equal awards rule,apply the weighted constrained equal awards rule of weights proportional to thereference weights assigned to its members until each of them is fully compensated.If the starting point is the constrained equal losses rule, follow a similar procedure.If the starting point is the hybrid of the two rules, let us refer to the components ofthe reference partition of the set of potential claimants as “half-priority classes”.For each problem, once again, identify the partition of the set of claimants actuallypresent induced by the reference partition; then, apply the weighted hybrid rule ofweights proportional to the reference weights assigned to its members until eachof them has received his half-claim; next, turn to the component of the inducedpartition with the next highest half-priority and apply the weighted hybrid rule ofweights proportional to the reference weights assigned to these claimants until eachof them has received his half-claim, and so on. When each claimant has receivedhis half-claim, revisit each component of the partition, in the same order, and returnto the corresponding hybrid rule (the rule relative to weights proportional to thereference weights assigned to its members), and do so until each of them is fullycompensated.

Formally, let � be a complete and transitive binary relation on the set of poten-tial claimants N. Let ≺ and ∼ denote the asymmetric and symmetric parts of �,respectively. Let w ∈ R

N++ be a list of positive weights.

12 They also all admit parametric representations (Young, 1987).

Claims problems and weighted generalizations of the Talmud rule 257

Second definition of the sequential Talmud rule relative to � and w,� and w, T H,�,w. Let N ∈ N and (c, E) ∈ CN . Let (S1, S2, . . . , SK)be the ordered partition of N such that for each pair {k, �} ⊆ {1, 2, . . . , K},each i ∈ Sk, and each j ∈ S�, (i) if k = �, then i ∼ j, and (ii) if k < �,then i ≺ j. Let E1 ≡ min{E, 1

2

∑i∈S1

ci}, E2 ≡ min{E − E1,12

∑i∈S2

ci},E3 ≡ min{E−E2−E3,

12

∑i∈S1

ci}, and so on. For each E ∈ ] 12

∑N ci,

∑N ci

[

and each k ∈ {1, 2, . . . , K}, let F k be defined as Ek was by replacing E byE − 1

2

∑Sk

ci, and each E� by F�.Now, for each N ∈ N and each i ∈ N , let αi ≡ wi∑

N wjand α ≡ (αi)i∈N .

Then let TH,�,w(c, E) be the awards vector x obtained as follows: for each k ∈{1, 2, . . . , K},

(i) if E ≤ 12

∑N ci, then xSk

≡ CEAα( 1

2cSk, Ek

);

(ii) if E ∈ ] 12

∑N ci,≤

∑N ci

], then xSk

≡ 12cSk

+ CELα( 1

2cSk, Fk

).

We have the following theorem: (The proof is omitted because of the overlapwith that of Theorem 1.)

Theorem 2. A rule R on C is such that

(∗∗) for each pair {i, j} ∈ N , there exists a vector of weights (αij , αji) ∈ ∆{i,j}

such that R = H(αij ,αji) on C{i,j}.

and satisfy consistency only if there exist a complete and transitive binary relation� on the set of potential claimants and a list of positive weights w ∈ R

N++ such

that for each pair {i, j} ∈ N , the following relations hold:

(i) (αij , αji) = (1, 0) if and only if i ≺ j;(ii) (αij , αji) ∈ int∆{i,j} if and only if i ∼ j and (αij , αji) is proportional to

(wi, wj).

Then, R coincides with the sequential hybrid rule TH,�,w.Conversely, each sequential hybrid rule TH,�,w satisfies condition (∗∗) and

consistency. Here, for each pair {i, j} of claimants, the weights (αij , αji) areconstructed from � and w by invoking conditions (i) and (ii).

The condition relating the weights chosen for the two-claimant populations isnecessary for consistency. Another way to express the condition is as follows: ifagent i belongs to a higher half-priority class than agent j, then (αij , αji) = (1, 0),and if agents i, j, and k belong to the same half-priority class, then the weights(αij , αji) and (αjk, αkj) determine the weights (αik, αki).

Note that in our definition, the bias in favor of an agent is independent of theamount to divide. One could argue that a distinction should be made about whoshould be favored depending upon how much is available. This two-regime ideais part of the Talmud rule, which trades-off a particularly favorable treatment ofthe agents with the smaller claims over half the range of the amount to dividefor a particularly favorable treatment of the agents with the larger claims over thecomplementary range. However, it does so in a way that respects equal treatment

258 T. Hokari and W. Thomson

of equals. If this property is dropped, we have the option of choosing weightsthat depend on the regime. However, for each regime, the weights should satisfyconsistency conditions of the kind expressed in the theorem.

A special case is when a reversal of the weights occurs between the two regimes.Let N ≡ {i, j}. The suggestion here is to apply the weighted constrained equalawards rule of weights (αi, αj) for an amount to divide at most as large as the half-sum of the claims (of course, using the half-claims instead of the claims themselves)and the weighted constrained equal losses rule of weights (αj , αi) otherwise (onceagain, using the half-claims).13

If some weights are equal to 0, consistency and this reversal have an interestingimplication. Let N ≡ {1, 2, 3}. Suppose that when the amount to divide is less thanthe half-sum of the claims, the weights are (α12, α21) = (1, 0), (α13, α31) = (1, 0),and (α23, α23) = (1, 0). Suppose also that the weights are reversed when theamount to divide is greater than the half-sum of the claims. Then, a consistentrule that coincides with these hybrid rules for two-claimant problems behaves asfollows. As the amount to divide first increases from 0, agent 1 gets everything untilhis award is his half-claim; agent 2 receives any additional amount available untilhis award is his half-claim; then agent 3 receives any additional amount availableuntil his award is his half-claim. At that point, consistency forces us to reversethe half-priority relation: agent 3 receives any additional amount until he is fullycompensated; agent 2 comes next until he too is fully compensated; and it is onlythen that agent 1 starts receiving more.

5 Concluding comments

1. Several rules have been discussed in the literature that are closely related to theTalmud rule, and it is natural to enquire about the possibility of defining weightedgeneralizations of them. One such rule is introduced by Piniles (1861) in an attemptto explain the numerical examples in the Talmud. This rule can be described asconsisting of a “double application” of the constrained equal awards rule, the half-claims being used instead of the claims themselves. Also, a “reverse” of Piniles’rule and a “reverse” of the Talmud rule can be defined by exchanging the order inwhich the constrained equal awards and constrained equal losses rules are applied.It is easy to construct consistent extensions of the weighted versions of Piniles’rule and of its reverse. On the other hand, for the “reverse Talmud rule”, twoapproaches can be taken to define weighted generalizations. These approaches leadto counterparts of the rules we identified in Theorems 1 and 2 . Like the ruleswe describe in Theorem 1, the “sequential reverse Talmud rules” allow a flexibledifferential treatment of claimants only in the two-claimant case.

2. Instead of letting priority classes and weights emerge from the axioms, one couldenrich the model by adding information of this kind. This possibility is discussedby Thomson (1995), who defines a claims problem with priority classes as a list(c,≺, E) where (c, E) ∈ CN and ≺ is a complete and transitive binary relation

13 We thank Jean-Pierre Benoı̂t for the suggestion.

Claims problems and weighted generalizations of the Talmud rule 259

on N whose equivalence classes constitute a partition PN of N . Also, a claimsproblem with weights is a list (c, w, E), where (c, E) ∈ CN and α ∈ int∆N is apoint in the interior of the (|N | − 1)-dimensional simplex indicating the relativeimportance that should be given to claimants. A richer formulation would includepriority classes and weights.14 Let us then consider rules defined over a domainof such problems when N runs over N . A rule should reflect the value judgmentsincorporated in the choice of classes and weights: for each problem, it should assignnothing to a class until all classes with higher priority are fully compensated; withineach class, it should gear the awards vector towards the members of the class towhom higher weights have been assigned: if they have equal claims, the ratio oftheir awards should be equal to the ratio of their weights unless the agent with thegreater weight is fully compensated.

In the variable population version of the model, one would have to specifyclasses and weights for each population of claimants, and require the rule to behaveaccordingly for each population. However, consistency will imply that the exoge-nous data be “consistent” across populations: if in some population, an agent isassigned to a higher priority class than some other agent, so should he in any otherpopulation to which they both belong. Also, if in some population, two agents areassigned to the same priority class, so should they in any other population to whichthey both belong; also, the ratios of the weights assigned to them in the two pop-ulations should be equal. These requirements will imply two basic restrictions onthe class and weight assignments exogenously chosen for the various populations:these classes should be induced from a reference partition of the set of potentialagents into priority classes, and for each induced class, the exogenous weights as-signed to its members should be proportional to a reference weight vector chosenfor the reference class of which this induced class is a subset. If our first approach istaken, Theorem 1 says more. It implies that in any reference priority class contain-ing more than two agents, the reference weights should be equal. On the other hand,if our second approach is taken, Theorem 2 would imply no additional restrictionson classes and weights.

3. In a recent paper, Moulin (2000) has considered a similar issue to the one weaddress and the similarities between his conclusions and the ones we reached byfollowing our first approach should help shed light on both studies. His startingpoint is a rich family of two-claimant rules, to which he had arrived by imposing acertain list of axioms (homogeneity, and two “composition” properties, expressingthe invariance of the choice with respect to two alternative ways of dealing withpossible increases, or decreases respectively, in the amount to divide). He askedhow such two-claimant rules could be extended to general populations, and showedthat any rule passing all of his tests can be described as follows: the population ofpotential claimants is partitioned into reference priority classes; for each problem,this reference partition induces an ordered partition of the set of claimants actuallypresent, and these classes are handled in succession (this is as we discovered forour problem); within each two-claimant induced class, a rule of the kind he hadobtained for the two-claimant case can be used (again, this conclusion parallels

14 Alternatively, one could have a lexicographic system of weights.

260 T. Hokari and W. Thomson

ours), but within each induced class containing three or more claimants, the choice isnarrowed down to one in the small subclass consisting of the proportional, weightedconstrained equal awards, and weighted constrained equal losses rules; finally, theweights used to modify the constrained equal awards and constrained equal lossesrules have to be proportional to weights assigned to the members of the referenceclass of which this class is a subset (we too obtain an important narrowing, but ofcourse to a different rule, the Talmud rule).

Incidentally, we note that a simple proof of Moulin’s characterization alongthe lines of the proof we develop here can be devised (Thomson, 2001b), therebyshowing the usefulness of our approach beyond our specific application. For anotherpresentation of the approach, see Thomson (2001a).

4. Given the somewhat limiting conclusions uncovered in Theorem 1, one maywonder whether a weaker version of consistency would be available to help guidethe passage from two-claimant populations to general populations. The answer isyes. Dagan and Volij (1997) define a rule to be “average consistent” if for eachproblem, and for each claimant involved in this problem, this agent’s award isequal to the average of his awards in the associated reduced problems relative toall the subgroups to which he belongs. They also propose a version in which theaverage is limited to subgroups of two claimants. They show that each resourcemonotonic rule has an average consistent extension, and this extension is unique.Since the weighted concede-and-divide rules are resource monotonic, they are cov-ered. Therefore average consistency offers a reasonable alternative to consistencythat allows significantly broader opportunities for extending to general populationsrules that have been found desirable in the two-claimant case.

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