Axiomatic and Game-Theoretic Analysis of Bankruptcy And

Axiomatic and GameTheoretic Analysis of

Bankruptcy and Taxation Problems a Survey

William Thomson

This version August

I am grateful to Bettina Klaus Eiichi Miyagawa Juan MorenoTernero Anne van den

Nouweland James Schummer Oscar Volij and esp ecially Nir Dagan and ChunHsien Yeh

for their very useful comments I also thank a referee for detailed comments and the NSF

for its supp ort under grant SES and SBR

Abstract

When a rm go es bankrupt how should its liquidation value be

divided among its creditors This essayisanintro duction to the lit

erature devoted to the formal analysis of such problems We present

the rules that are commonly used in practice or discussed in theoret

ical work We show howmany can be obtained by applying solution

concepts develop ed in co op erative game theory for bargaining games

and for coalitional games We formulate prop erties of rules rst

when the p opulation of agents is xed then when it may vary com

pare the rules on the basis of these prop erties and search for rules

satisfying the greatest number of the prop erties together We mo del

the resolution of conicting claims as strategic games and extend the

mo del to handle surplus sharing and situations in which the feasible

set is sp ecied in utilityspace

Identifying wellb ehaved taxation rules is formally identical to

identifying rules to reconcile conicting claims and all of the results

we present can b e reinterpreted in that context

Keywords Axiomatic analysis Bankruptcy Claims problems

Prop ortional rule Talmud rule Constrained equal awards rule Con

strained equal losses rule

JEL Classication numb ers CDD

William Thomson Department of Economics University of Ro chester Ro chester

New York Tel fax email

wthtroiccro chesteredu

Contents

Intro duction

Claims problems and division rules

An inventory of rules

Relating division rules and solution concepts of the theory

of co op erative games

Bargaining solutions

Solutions to coalitional games

Prop erties of rules the xedp opulation case

Basic prop erties

Monotonicity requirements

Indep endence additivityand related prop erties

Op erators

Prop erties of rules the variablep opulation case

Population monotonicity

Replication invariance

Consistency

Average consistency

Merging and splitting claims

Op erators

Multiple parameter changes

Strategic mo dels

Extensions of the basic mo del

Surplussharing

Nontransferable utility problems

Mo del with group constraints

Exp erimental testing

Conclusion

References

Intro duction

When a rm go es bankrupt what is the fair way of dividing its liquidation

value among its creditors This essay is an intro duction to the literature

devoted to the formal analysis of problems of this kind whichwe call claims

problems The ob jective of this literature which originates in a fundamental

pap er by ONeill is to identify wellb ehaved rules for asso ciating

with each claims problem a division between the claimants of the amount

available

We rst present several rules that are commonly used in practice or dis

cussed in theoretical work We then formulate a number of app ealing prop

erties that one may want rules to satisfy compare the rules on the basis

of these prop erties and identify rules satisfying various combinations of the

prop erties Indeed the axiomatic metho d underlies most of the developments

on which we rep ort here and they illustrate the increasingly imp ortant role

the metho d has been playing in the design of allo cation rules The rapid

progress witnessed in the literature on the adjudication of conicting claims

is largely due to researchers being able to draw on the conceptual appara

tus and the pro of techniques elab orated in the axiomatic analysis of other

mo dels running the gamut from abstract mo dels of game theory and so cial

choice to concrete mo dels of resource allo cation We do not limit ourselves to

axiomatic studies however We also showhow the to ols of co op erative game

theory from b oth the theory of bargaining and the theory of coalitional

games can be used to dene rules and we discuss a variety of strategic

approaches

The b estknown rule is the prop ortional rule which cho oses awards pro

p ortional to claims Prop ortionality is in fact often taken as the denition

of fairness for claims problems but we will challenge this p osition and start

from more elementary considerations An imp ortant source of inspiration

for the research we describ e is the Talmud in which several numerical ex

amples are discussed and recommendations are made for them that conict

with prop ortionality Can these recommendations be rationalized by means

of wellb ehaved rules Among all existing rules are there grounds for prefer

ring some to the others Are there yet other rules that deserve our attention

Finallywe consider extensions of the mo del in particular some covering

situations where the amount to divide is more than sucient to honor all

the claimsthis is the problem of surplus sharingand mo dels where the

data are sp ecied in utility space and the upp er b oundary of the feasible set

is not restricted to b e contained in a hyp erplane normal to a vector of ones

We close this intro duction by noting that the problem of assessing taxes as

a function of incomes when the total tax to b e collected is xed is formally

identical to the problem of adjudicating conicting claims All of the results

we present can be reinterpreted in that context and more generally in the

context of the assessment of liabilities

An imp ortant question that we will not address is the extent to which

the choice of particular division rules aects agents incentives to make com

mitments that one party may in the end be unable to honor In the context

of bankruptcy these are the incentives to loan and to b orrow In many

of the other applications the parameters of the problems to be solved also

result from decisions that agents have made and whatever rule is used at

the division stage will in general have had an eect on these earlier choices

In order to handle these kinds of issues we would need to embed division

rules in a more complete mo del in which risktaking eort and other vari

ables chosen by agents such as lenders b orrowers tax payers government

agencies and others are explicitly describ ed sto chastic returns to economic

activities are factored in and so on But the theory develop ed here which ig

nores incentives is a necessary comp onent of the comprehensive treatment

it would have to be formulated in a generalequilibrium and gametheoretic

frameworkthat we envision

Claims problems and division rules

An amount E R has to b e divided among a set N of agents with claims

adding up to more than E For each i N let c R denote agent is

claimandc c the vector of claims Initiallywetake N to b e a nite

i iN

subset of the set of natural numb ers N We sometimes designate by n the

cardinalityof N Altogether a claims problem is a pair c E R R

suchthat c E Let C denote the class of all problems In Section

An imp ortant step in that direction is taken by Araujo and Pascoa

We denote by R the cartesian pro duct of jN j copies of R indexed by the memb ers

of N The sup erscript N also indicates an ob ject p ertaining to the set N Whichinter

pretation is the right one should b e clear from the context A summation without explicit

b ound should be understo o d to be carried out over all agents We allow the equality

c E for convenience

Vector inequalities x y x y xy

we consider situations in which the p opulation of claimants may vary and

we generalize the mo del accordingly

In our primary application E is the liquidation value of a bankrupt rm

the members of N are creditors and c is the claim of creditor i against the

rm A closely related application of the mo del is to estate division a man

dies and the debts he leaves b ehind written as the co ordinates of c are found

to add up to more than the worth of his estate E How should the estate b e

divided Alternatively each c could simply b e an upp er b ound on agent is

consumption without a higher bound necessarily giving him greater rights

on the resource

is interpreted as atax assessment problem the When a pair c E C

members of N are taxpayers the co ordinates of c are their incomes and

they must cover the cost E of a pro ject among themselves The inequality

c E indicates that they can jointly aord the pro ject A dierent

interpretation of c is as the b enet that consumer i derives from the pro ject

Although these various situations can be given the same mathematical

description and the principles relevant to their analysis are essentially the

same the app eal of each particular prop erty may of course dep end on the

application In what follows we mainly think of the resolution of conicting

claims Our mo del is indeed a faithful description of the actual situation

faced by bankruptcy courts for instance By contrast the issue of taxation is

not always sp ecied by rst stating an amount to be collected p erhaps due

to the uncertainty p ertaining to the taxpayers incomes Taxation schedules

are usually published rst and the amount collected falls wherever it may

dep ending up on the realized incomes

We will search for ways of calculating for each claims problem a division

between the claimants of the amount available this division being under

sto o d as a recommendation for the problem Formally a division rule is

N N

a function that asso ciates with each problem c E C a vector x R

whose co ordinates add up to E and satisfy the inequalities x c Such

a vector is an awards vector for c E Our generic notation for a rule

is the letter R Given a claims vector the lo cus of the awards vector chosen

However we should note that a numb er of prop erties of rules that we will use later

have b een rst considered in the context of taxation

We limit ourselves to the searchfor singlevalued rules since for this mo del in contrast

withanumb er of other mo dels that are commonly studied a great varietyofinteresting

rules enjoy this prop erty

x x

c c

k k

e e e

E E

a b

Figure The Talmud rule The worth of the estate is measured horizontally

and claims and awards vertically a The rule applied to the contested garment

problem for which claims are c c The Talmud considers an estate

worth and recommends the awards vector g b The rule applied

to the estate division problem for which claims are c c c

For an estate worth the Talmud recommends e if worth

it recommends k and if worth it recommends p

by a rule as the amount to divide varies from to the sum of the claims is

the path of awards of the rule for the claims vector

We will also oer several results concerning a version of the mo del in

which the inequality c E is not imp osed and generalized rules

such a rule may select for some c E C an ecient vector x that do es

not satisfy the inequalities x c

An inventory of rules

We pro ceed with a presentation of two intriguing problems discussed in the

Talmud The Talmud sp ecies only a few numerical examples but the de

sire to understand them has provided much of the imp etus underlying the

theoretical eorts describ ed in these pages

The contested garment problem Figure a two men disagree on

the ownership of a garment worth say The rst man claims half of it

and the other claims it all Assuming b oth claims to be made in

go o d faith how should the worth of the garment be divided among them

The Talmud recommends the awards vector g Baba Metzia a

All references to the relevant passages of the Talmud and medieval literature are taken

from ONeill Aumann and Maschler and Dagan

The estate division problem Figure b a man has three wives whose

marriage contracts sp ecify that up on his death they should receive

and resp ectively The mandiesand his estate is found to b e worth only

The Talmud recommends e If the estate is worth

it recommends p but if it is worth it recommends

k Kethub ot a the author of this Mishna is Rabbi Nathan

To clarify the mystery p osed bythenumb ers given as resolutions of these

problems we should rst identify a general and natural formula that gener

ates them and it is only recently that such a formula was found We give

the formula after intro ducing several more elementary rules

The following is a simple division scenario for the twoclaimantcase

that delivers the numbers prop osed by the Talmud for the contested gar

ment problem Aumann and Maschler when agent i claims c he is

essentially conceding to agent j the amount E c if this dierence is non

negative and otherwise that is maxfE c g Similarly claimant j s

concession to claimant i is maxfE c g Let us then assign to each claimant

the amount conceded to him by the other and divide equally b etween them

what remains the part that is truly contested Equal division in this second

round makes sense since after b eing revised down by the amounts received

in the rst round as is very natural and truncated by the amount that

remains available the truncation idea is formally intro duced and justied

below in many cases revised claims are all smaller than the remainder so

no truncation is needed b oth claims are equal

Concedeanddivide CD For jN j For each c E C and each

E maxfE c g

i N CD c E maxfE c g

i j

We will discover a number of other ways of thinking ab out the issue

that lead to that same formula in the twoclaimant case However it is

not obvious how to generalize the concedeanddivide scenario itself to the

nclaimant case The dierence for each claimant between the amount to

divide and the sum of the claims of the other agents or if this dierence

is negative can certainly still b e understo o d as a concession that together

they make to him and it is still natural to assign to him this amount in

a rst step However after being revised down by these amounts claims

need not be equal anymore even if truncated by the remainder so equal

division at the second step is not as comp elling as in the twoclaimantcase

Besides it may result in an agent receiving in total more than his claim

In the paragraphs to follow we therefore explore other ideas In each case

we invite the reader to verify that not all of the numbers in the Talmud are

accounted for until we reach the Talmud rule of Aumann and Maschler

The rule most commonly used in practice makes awards prop ortional

to claims

Prop ortional rule P For each c E C P c E c where is

chosen so as c E

Two other versions of the prop ortional rule have b een prop osed The rst

one is dened by making awards prop ortional to the claims truncated by the

amount to divide We refer to it as the truncatedclaims proportional

rule The other requires rst assigning to each claimant the amount that

remains if the other claimants have b een fully comp ensated or if that is

not p ossible We used this dierence ab ove in dening concedeanddivide

It can be seen as a minimum to which the claimantis entitled and shifting

perspectives slightly we will refer to it by an expression that b etter reects

that interpretation Formally for each c E C and each i N let

c g b e the minimal right of claimant i m c E maxfE

j i

j N nfig

in c E and mc E m c E Then the rule selects the awards

i iN

vector at which each claimant receives his minimal right these payments

are feasible each agents claim is revised down to the minimum of i the

remainder and ii the dierence between his initial claim and his minimal

right nally the remainder is divided prop ortionately to the revised claims

Adjusted prop ortional rule A Curiel Maschler and Tijs For

each c E C Ac E mc E P minfc m c E E

i i

P P

m c E g E m c E g

j iN j

The idea of equality underlies many theories of economic justice The

question though is what exactly should be equated esp ecially when agents

are not identical Here agents dier in their claims and equating awards im

plies ignoring these dierences In particular some agents mayreceivemore

than their claims which is precluded by our denition of a rule The rule

presented next remains close in spirit but it is dened so as to resp ect these

upp er b ounds on awards It assigns equal amounts to all claimants sub ject to

no one receiving more than his claim Although this departure from equality

do es not seem to b e much of a step towards recognizing dierences in claims

we will nevertheless provide app ealing axiomatic justications for the rule

It is also an ingredient in the denitions of other interesting rules as we

will recognize in the formulae for Piniles and the Talmud rules b elow It has

been advocated by many authors including Maimonides th Century

Constrained equal awards rule CEA For each c E C and

each i N CEA c E minfc g where is chosen so that

i i

minfc g E

Our next rule Piniles can be understo o d as resulting from a

double application of the constrained equal awards rule the halfclaims

b eing used in the formula instead of the claims themselves First the rule

is applied to divide the minimum of i the amount available and ii the

halfsum of the claims If the amount available is less than the halfsum of

the claims we are done Otherwise each agent rst receives half of his claim

then the constrained equal awards rule is reapplied to divide the remainder

still using the halfclaims This rule accounts for all of the numb ers given in

the Talmud for the estate division problem

Piniles rule Pin For each c E C and each i N Pin c E

P P

j c c

c c

i i

CEA Eif E andPin c E CEA E otherwise

i i i

Another way of implementing the idea of equality leads to the following

formula Chun Schummer and Thomson It is inspired by a solution

to the problem of fair division when preferences are singlep eaked known as

the uniform rule Sprumont As in Piniles it gives the halfclaims a

central role and otherwise it makes the minimal adjustment in the formula

for the uniform rule that guarantees that awards are ordered as claims are

Constrained egalitarian rule CE For eachc E C and each i N

c j c

i i

CE c E minf g if E and CE c E maxf minfc gg

i i i

otherwise where in each case is chosen so that CE c E E

An alternative to the constrained equal awards rule is obtained by

fo cusing on the losses claimants incur what they do not receive as opp osed

to what they receive and cho osing the awards vector at which these losses

are equal sub ject to no one receiving a negativeamount It to o is discussed

by Maimonides Aumann and Maschler

Constrained equal losses rule CEL For eachc E C and each i

N CEL c E maxfc g where is chosen so that maxfc

i i i

g E

We now have all that we need to dene a rule that nally generates

all of the numbers app earing in the Talmud Aumann and Maschler

see Figure As in Piniles formula two regimes are dened dep ending

up on the side of the halfsum of the claims on which the amount to divide

falls For an amount to divide equal to the halfsum of the claims everyone

receives his halfclaim If there is less the constrained equal awards formula

is applied if there is more the constrained equal losses formula is in each

case the halfclaims are used in the formula instead of the claims themselves

Alternatively the rule can b e dened by means of an algorithm First imag

ine the amount available increasing from to the halfsum of the claims

the rst units are divided equally until each agent receives an amountequal

to half of the smallest claim then the agent with the smallest claim stops

receiving anything for a while and the next units are divided equally among

all others until each of them receives an amount equal to half of the second

smallest claim Then the agent with the second smallest claim also stops

receiving anything for a while and the next units are divided equally among

the other claimants until each of them receives an amount equal to half of

the third smallest claim The algorithm pro ceeds in this way until the

amount available is at that point each agent receives his halfclaim

For amounts available greater than awards are computed in a symmet

ric way Starting from an amount available equal to the sum of the claims

in which case each agent receives his claim consider shortfalls of increasing

sizes initial shortfalls are divided equally until each agent incurs a loss equal

to half of the smallest claim the loss incurred by the agent with the smallest

claim stops at that point and any additional shortfall is b orn equally by the

others until their common loss is equal to half of the second smallest claim

The algorithm pro ceeds in this manner until the amount available is

It is a simple matter to see that when applied to the two problems in the

In the context of taxation this rule is known as head tax and the constrained equal

awards rule as the leveling tax

Talmud it do es yield the numb ers given there Henceforth we call the rule

it denes the Talmud rule The following is a compact denition

Talmud rule T For each c E C and each i N

c c

i i

If E then T c E minf g where is chosen so that

minf g E

c c

i i

If E then T c E c minf gwhere is chosen so that

i i

c minf g E

We refer to the twoclaimantversion of the Talmud rule as the contested

garment rule It is easy to check that this rule coincides with concedeand

divide but as already noted it is not the only one to do so

Our inventory of rules is indeed far from b eing exhausted and our next

two rules also have this feature To dene the rst one imagine claimants

arriving one at a time to get comp ensated and supp ose that each claim is

fully honored until money runs out The resulting awards vector of course

dep ends on the order in which claimants arrive To remove the unfairness

asso ciated with a particular order take the arithmetic average over all orders

of arrival of the awards vectors calculated in this way ONeill For

a formal denition of the rule let be the class of bijections from N into

itself

Random arrival rule RA For each c E C and each i N

P P

RA c E c gg minfc maxfE

i j i

j Nj i

Another rule is oered by ONeill as a generalization of an ex

ample found in Ibn Ezra th Century and of an incompletely sp ecied

when n Of course the Talmud not oering any example for the case E

we can only sp eculate as to what it would have recommended then However we nd

the sort of considerations that led Aumann and Maschler to the interp olation and

extrap olation they dene very comp elling and this is whywe refer to the rule they prop ose

as the Talmud rule Moreover their formula is in agreement with another numerical

example in the Talmud that they discuss

P P

c c

c c c

i i

We have T c E CEA E if E and T c E CEL E

otherwise

This rule coincides with the contested garmentruleinthetwoclaimant case ONeill

rule due to Rabad th Century The problem discussed by Ibn Ezra is

that of dividing an estate worth among four sons whose claims are

and He recommends for the rst son for the second

son for the third son and for the fourth son

Rabads suggestion which gives Ibn Ezras numbers in his particular

application is dened for problems such that the estate is worth no more

than the greatest claim Aumann and Maschler if the estate is worth

less than the smallest claim it is divided equally as its worth increases from

the smallest to the second smallest claim the agent with the smallest claim

continues to receive of his claim and the remainder is divided equally among

the other claimants In general when the worth of the estate increases from

the k th smallest claim to the k th smallest claim the amounts received

by the agents with the k smallest claims do not change and the remainder

is divided equally among the other claimants

Here is ONeills prop osal for general problems Instead of thinking of

claims abstractly think of the amount to divide as comp osed of individual

and distinct units Then distribute each agents claim over sp ecic units

so as to maximize the fraction of the estate claimed by exactly one claimant

and sub ject to that so as to maximize the fraction claimed by exactly two

claimants and so on nally for each unit separately apply equal division

among all agents claiming it

Minimal overlap rule MO Claims on sp ecic parts of the amountavail

able or units are arranged so that the numb er of units claimed by exactly

k claimants is maximized given that the number of units claimed by

k claimants is maximized for k n Then for each unit equal

division prevails among all agents claiming it Each claimant collects the

partial comp ensations assigned to him for each of the units that he claimed

The arrangement of claims solving this lexicographic maximization is

unique up to inessential relab elling of units A claim greater than the amount

available is equivalent to a claim equal to the amountavailable and if there

is at least one such claim the solution consists in nesting the claims If

not representing the amount available as an interval E the solution is

obtained by nding a number t R such that each agent i N claims

the interval minfc tgthis takes care of the agents whose claims are at

most tand each of the agents whose claim is larger than t claims t plus

a subinterval of t E these subintervals do not overlap and together they

c t E t As easily cover t E therefore t satises

fiN c tg

veried the rule so dened generates Ibn Ezras numb ers in the example he

considers nesting applies Also in the twoclaimant case it coincides with

concedeanddivide

We will close our inventory by presenting several families of rules These

families tie together several of the rules that we have listed but they are

innite families The rst family called the ICI family for Increasing

ConstantIncreasing an expression reecting the evolution of each claimants

award as a function of the amount to divide can be seen as generalizing

the Talmud rule The only dierence is that the points at which agents

temp orarily stop receiving additional units and the p oints at which they are

invited back in are allowed to dep end on the claims vector Otherwise

agents leave and come back in the same order as for the Talmud rule and

any two agents who are present receive equal shares of any increment

Here is the formal denition Thomson Let G be the family of

where n jN j of realvalued functions of the claims lists G fE F g

k k

vector satisfying for each c R the following relations which we call the

ICI relations These relations are imp osed to guarantee that at the end of

the pro cess just describ ed each agent is fully comp ensated Let C c

E c C F c

n n

E cE c F cF c

c c

n n

E cE c F cF c

k k k k

c c

k k

nk nk

F c E c

n n

c c

n n

N G N

ICI rule relative to G fE F g G R For each c R the

k k

ONeill denes another metho d of random claims as follows agents randomly

make claims on sp ecic parts of the estate the total amount claimed byeachagent b eing

equal to his claim for each part of the estate equal division prevails among all agents

claiming it Unfortunately this metho d may not attribute the whole estate it is not

ecient as formally dened b elow Moreover when claims are compatible it need not

award to eachagentanamount equal to his claim In order to recover eciency ONeill

suggests taking the amounts awarded by the metho d as a starting p oint and applying the

metho d again to distribute the remainder Then an agentmay get more than his claim

To remedy this problem adjust down each agents claim by what he receives initially

Rep eat the pro cess and take the limit

awards vector is given as the amount available E varies from to c as

follows As E increases from to E c equal division prevails as it increases

from E ctoE c claimant s award remains constant and equal division

of each additional unit prevails among the other claimants As E increases

from E c to E c claimants and s awards remain constant and equal

division of each additional unit prevails among the other claimants and so

on This pro cess go es on until E reaches E c The next units go to

claimant n until E reaches F c at which point equal division of each

additional unit prevails b etween claimants n and n This go es on until E

reaches F c at which p oint equal division of each additional unit prevails

between claimants n through n The pro cess continues until E reaches

F c at which p oint claimant reenters the scene and equal division of

each additional unit prevails among all claimants

The constrained equal awards constrained equal losses Talmud and min

imal overlap rules b elong to the family A subfamily of the ICI family is

obtained by cho osing and having the k th agent as determined by

the order of claims drop out when his award reaches the fraction of his

claim and having him return when his loss reaches the fraction of

his claim MorenoTernero and Villar The constrained equal awards

rule is obtained for the constrained equal awards rule for and

the Talmud rule for

A reverse family of the ICI family can b e dened in which the order in

whichagents are handled is reversed The pro cess starts with the agent with

the largest claim and the remaining agents arrive in the order of decreasing

claims The agent with the smallest claim stays until he is fully reimbursed

and agents drop out in the order of increasing claims until each of them is

fully comp ensated Thomson As a function of the amounttodivide

each agents award is constant in some initial interval then increases then

is constant again As for the ICI family any two agents who are present

receive equal shares of any increment This CIC family for Constant

IncreasingConstant contains the constrained equal awards and constrained

equal losses rules

Eachmemb er of the third family Young a is indexed by a function

f R a b R where a b that is continuous nowhere

decreasing in its second argument and such that for eachc R f c a

and f cb c Let be the family of these functions They can be

interpreted as measuring how much each agent should receive in order to

max

f c

c c

f c

a b a b c

max

a b

Figure Parametric representations of two rules a Prop ortional rule

the schedules dened over the interval are segments through the origin of

slop es equal to claims b Talmud rule we assume for simplicity that there is

a maximal value that a claim can take c Then the schedule relative to a

max

typical claim c dened over the interval c follows the line up to the

max

c c

continues horizontally until it meets the line of slop e emanating point

from c then again follows a line of slop e

max

exp erience a certain welfare if his claim has a certain value The resource is

then divided so as to ensure that all agents exp erience equal welfares

Parametric rules of representation f R For each N N and

N f

each c E C R c E is the awards vector x such that there exists

a b for which for each i N x f c

i i

Many rules belong to the family the prop ortional constrained equal

awards constrained equal losses Piniles Talmud and constrained egalitarian

rules are memb ers Figures a and b give parametric representations of the

prop ortional and Talmud rules the latter in the case where an upp er b ound

on claims exists c

max

In the context of taxation the following parametric rules have also b een

discussed for Stuarts rule x maxfc c g and for Cassels rule

i i

x for

This assumption restricts somewhat the scop e of the rule but it p ermits a very simple

piecewise linear representation Chun Schummer and Thomson See Young

a for a representation without the upp er b ound

Relating division rules and solution con

cepts of the theory of co op erative games

In this section we exhibit interesting relations b etween certain division rules

and various solution concepts of the theory of co op erative games For this

theory to be applicable we need rst to dene a formal way of asso ciating

with each claims problem a co op erative game Two main classes of such

games have b een studied bargaining games and coalitional games and ac

cordinglywe establish two kinds of relations

Bargaining solutions

A bargaining game is a pair B d where B is a subset of R the n

dimensional Euclidean space and d is a p ointof B The set B thefeasible

set consists of all utility vectors attainable by the group N by unanimous

agreement and d the disagreement point is the utility vector that re

sults if they fail to reach an agreement A bargaining solution is a func

tion dened on a class of bargaining games that asso ciates with each game

in the class a unique point in the feasible set of the game The following

are imp ortant examples The egalitarian solution Kalai selects

the maximal p oint of B at which the utility gains from d are equal The

lexicographic egalitarian solution Imai selects the point of B

at which these gains are maximal in the lexicographic maximin order

The KalaiSmorodinsky solution Kalai and Smoro dinsky selects

the maximal p oint of B on the segment connecting d to the ideal point of

B d the point whose ith co ordinate is the maximal utility agent i can

obtain sub ject to the condition that all other agents receive at least their

utilities at d The Nash solution Nash selects the point maximiz

ing the pro duct of utility gains from d among all p oints of B dominating d

N N

Given a vector of weights int where denotes the unit simplex

in R and int its interior the weighted Nash solution with weights

selects the p oint of B at which the pro duct x d is maximized

i i

among all p oints of B dominating d The extended equal losses solu

Given x y R letx designate the vector obtained from x by rewriting its co ordi

nates in increasing order y b eing similarly dened Wesaythatx is greater than y in

the lexicographic maximin order if either x y or x y and x y or more

generally for some k n x y x y and x y

k k k k

x x

E E

c a

y a

x c

B c E

d d

x x E

b a

y CEAc E N B c E dE B c E d

t c

z P c E K B c E dN B c E d

v CEL c E XELB c E d

Figure Claims problems and their associated bargaining games Key

for bargaining solutions N Nash solution K KalaiSmoro dinsky solution E

lexicographic egalitarian solution N weighted Nash solution with weights c

XEL extended equal losses solution The shaded region B c E represents the

set of vectors x R such that x E and that are dominated by the claims

vector The region is taken as the feasible set of the bargaining game B c E d

asso ciated with c E The egalitarian bargaining solution selects the maximal

point of B c E d of equal co ordinates At such a point x the whole estate

need not be divided as shown in panel a In the twoagent case the extended

equal losses bargaining solution simply selects the maximal p oint of B c E d

at which losses from the ideal point of B c E d the p oint a are equal the

point v

tion Bossert in a contribution building on the equal losses solution

of Chun b selects the maximal p ointatwhich the losses from the ideal

point of all agents whose utility gains are equal except that any agent who

would exp erience a negative gain is assigned his disagreement utility instead

The most natural way to asso ciate a bargaining game with a claims prob

lem is to take as feasible set the set of all nonnegative vectors dominated

by the claims vector and whose co ordinates do not add up to more than

the amount available and to cho ose the origin as disagreement point This

makes sense since we require that a rule should never assign to any agent

more than his claim However a rule that satises this requirement could

be resp onsive to changes in claims that do not aect the asso ciated bar

gaining game as just dened the prop ortional rule is an example and

one could argue that to o much information is lost in the passage from

claims problems to bargaining games Summarizing given a claims problem

its associated bargaining game is the game with feasible set c E C

B c E fx R x E x cg and with disagreement p oint

d Note that d do es not dep end on c E

This denition is illustrated in Figure for two examples For the rst

example the vector of equal awards do es not b elong to the undominated

b oundary of the feasible set but for the second it do es

In bargaining theory the feasible set is allowed to b e an arbitrary compact

and convex set but here we have the sp ecial case of a feasible set whose

ecient b oundary is a subset of a plane normal to a vector of ones An

extension of the mo del accommo dating more general shap es is discussed in

Subsection

If for each claims problem the recommendation made by a given rule

coincides with the recommendation made by a given bargaining solution when

applied to the asso ciated bargaining game the rule corresponds to the

solution Our rst prop osition describ es a number of such corresp ondences

Theorem The fol lowing correspondences between division rules and bar

gaining solutions hold

The constrained equal awards rule and the Nash bargaining solution

Dagan and Volij

The constrained equal awards rule and the lexicographic egalitarian so

lution

The proportional rule and the weighted Nash solution with the weights

chosen equal to the claims Dagan and Volij

The truncatedclaims proportional rule and the KalaiSmorodinsky so

lution Dagan and Volij

The truncatedclaims constrained equal losses rule obtained from the

constrained equal losses rule by truncating claims by the amount to

divide and the extended equal losses bargaining solution

An alternative sp ecication of the disagreement p oint is p ossible that do es namely

the vector of minimal rights Dagan and Volij

Dagan and Volij also show that the adjusted prop ortional rule corresp onds to

the KalaiSmoro dinsky solution applied to the problem in which the disagreementpoint

is set equal to the vector of minimal rights instead of the origin

The recommendations made by various division rules and the bargaining

solutions to which they corresp ond are indicated in Figure for two examples

Although Theorem establishes useful links between the theory of the

resolution of conicting claims and the theory of bargaining one should p er

haps not attachtoomuch imp ortance to any particular one of them Indeed

since the bargaining games asso ciated with claims problems constitute a very

narrow sub class of the class of bargaining games traditionally studied it fol

lows that bargaining solutions that in general give dierent payo vectors

often coincide on this sub class This phenomenon is illustrated by the fact

that the constrained equal awards rule corresp onds to b oth the Nash solution

and the lexicographic egalitarian solution

Another conversion of claims problems into bargaining games is p ossible

however in which the claims point remains as separate data The relevant

concept is then the generalization of the notion of a bargaining game obtained

by adding a claims point Chun and Thomson study these problems

under the name of bargaining problems with claims See Subsection for

further discussion Then the prop ortional rule corresp onds to the solution

also called prop ortional in that theory

Solutions to coalitional games

Wenow turn to the richer class of coalitional games Such games are formal

representations of situations in which all groups or coalitions and not just

the group of the whole can achieve something Formally a transferable

jN j

utility coalitional game is a vector v v S R where for

S N

each coalition S N v S R is the worth of S This number is

interpreted as what the coalition can obtain on its ownorcanguarantee

itself A solution is a mapping that asso ciates with each such game v a

payo vectorapointin R whose co ordinates add up to v N a prop erty

we will also refer to as eciency

In order to b e able to apply the solutions discussed in the theory of coali

tional games we need a pro cedure for asso ciating with each claims problem

The reader maywonder why a solution that is scale invariant invariant with resp ect

to p ositive linear transformations indep endentagentby agent of their utilities suchas

the Nash solution coincides with a solution that involves utility comparisons suchasthe

lexicographic egalitarian solution The answer is simply that the sub class of bargaining

games asso ciated with claims problems is not rich enough for scale transformations in

which the scale co ecients dier b etween agents to ever b e applicable

such a game The most common one is to set the worth of each coalition

S equal to the dierence between the amount available and the sum of the

claims of the memb ers of the complementary coalition N n S if this dif

ference is nonnegative and otherwise Using terminology intro duced ear

lier the dierence can be understo o d as what the complementary coalition

concedes to S It is certainly what S can secure without going to court

Formally given a claims problem c E C its associated coalitional

jN j

game ONeill is the game v c E R dened by setting for

c g each S N v c E S maxfE

N nS

Note that our denition is in agreement with the standard manner in

which TU games are constructed to represent conicts If the worth of a

coalition is interpreted instead as the amount the coalition can exp ect to

receive the denition is somewhat p essimistic However the bias being

systematic across coalitions we might still feel that the resulting game ap

propriately summarizes the situation

The game v c E is convex Aumann and Maschler Therefore

its core is nonempty In fact this set is simply the set of awards vectors

of c E recall that these are the vectors x R such that x E and

x c

If for each claims problem the recommendation made by a given division

rule coincides with the recommendation made by a given solution to coali

tional games when applied to the asso ciated coalitional game once again

we say that the rule corresponds to the solution Just as we saw for

bargaining solutions a numb er of corresp ondences exist b etween the division

rules intro duced in Section and solutions to coalitional games

In the twoclaimant case for each i N v fig is equal to E c where

j i if this dierence is nonnegative and otherwise the worth of the

grand coalition is equal to E Dividing equally the amount that remains when

A coalitional game is a p oint in a space of considerably greater dimension than a

claims problem One could argue that the passage from claims problems to coalitional

games involves a cumb ersome increase in dimensionality

This means that the contribution of a player to any coalition is at least as large as his

contribution to any sub coalition of it

This is the set of ecientpayo vectors such that each coalition receives at least its

jN j

worth more precisely the core of v R is the set of payo vectors x R such

P P

that x v N and for each S N x v S

i i

What we called the minimal right of a claimantissimplytheworth of the coalition

consisting only of that claimant

each claimant i is rst paid v fig is what virtually all solutions to coalitional

games that are commonly discussed recommend It also corresp onds to the

recommendation made by concedeanddivide

x maxfE c g E maxfE c g

i j k

The rst corresp ondence we describ e for the nclaimant case involves

the random arrival rule and the solution to coalitional games intro duced

by Shapley Most convenient here is the random arrival denition

of this solution The Shapley value of player i N in the game

jN j

v R is the exp ected amountby which his arrival changes the worth

of the coalition consisting of all the players who have arrived b efore him

assuming all orders of arrival to be equally likely with the convention that

the worth of the empty set is Recall that is the class of bijections

jN j

from N into itself Then for each v R and each i N Sh v

v fj N j ig i v fj N j ig

jN j

Another imp ortant solution to coalition games is the prenucleolus

Schmeidler First dene the dissatisfaction of a coalition at a pro

p osed payo vector to be the dierence between its worth and the sum of

the payos to its memb ers Then the prenucleolus is obtained by p erforming

the following sequence of minimizations rst identify the ecient vectors

at which the dissatisfaction of the most dissatised coalition is the smallest

among the minimizers identify the vectors at which the dissatisfaction of the

second most dissatised coalition is the smallest and so on

The DuttaRay solution selects for eachconvex game the payo vector

in the core that is Lorenzmaximal Dutta and Ray

Finally consider the value Tijs It is dened by rst calcu

lating a maximal payo and a minimal payo for each player then it

ONeill discusses a metho d of stepbystep adjustments of particular historical

signicance under the name of recursive completion that pro duces the Shapley value

On the sub class of convex games the prenucleolus coincides with the nucleolus and

more interestingly as this solution is usually multivalued with the kernel

Given x we denote by x the vector obtained from x by rewriting its co ordinates in

P P

increasing order Given x and y R with x y wesay that x is greater than

i i

y in the Lorenz ordering ifx y andx x y y andx x x y y y

and with at least one strict inequality

cho oses the ecientpayo vector that lies on the segment connecting the vec

jN j

tor of minima to the vector of maxima For each v R and each i N

M v v N v N nfig and m v max v S M v

i i S NiS j

j S nfig

Then v M v mv where is chosen so as to obtain e

ciency

The next theorem links four of the division rules intro duced earlier to

solutions to coalitional games

Theorem The fol lowing correspondences between division rules and solu

tions to coalitional games hold

The random arrival rule and the Shapley value ONeil l

The Talmud rule and the prenucleolus Aumann and Maschler

The constrained equal awards rule and the DuttaRay solution Dutta

and Ray

The adjusted proportional rule and the value Curiel Maschler and

Tijs

Of course not every division rule corresp onds to some solution to coali

tional games A necessary and sucient condition for such a corresp ondence

to exist is that the rule dep ends only on the truncated claims and the amount

available Curiel Maschler and Tijs

We close this section by noting that other ways of asso ciating a coalitional

game to a claims problem are conceivable Indeed recall that our earlier def

inition reects a rather p essimistic assessment of what a coalition can exp ect

An alternative assessment this time optimistic of the situation leads to the

formula w c E S minf c Eg for each S N The resulting game

w c E is studied by Driessen Here to o in spite of the bias of the

denition the fact that it is systematic across coalitions gives us the hop e

that the game still provides a useful summary of the situation Driessens

results should reassure us with resp ect to the sensitivity of our conclusions

For a convex game v M v dominates mv

R Lee and Benot give alternative pro ofs of this result

This means that R c E can b e written as R min fc Eg E for some function R

i iN

to how the game is sp ecied the core of v c E actually coincides with the

anticore of w c E and their Shapley values and nucleoli coincide

Prop erties of rules the xedp opulation

case

In this section and the next we formulate prop erties of rules and examine

how p ermissive or restrictive they are We start with the ones we consider

the most natural We continue with prop erties that one may or may not

want to imp ose dep ending up on the range of situations to be covered and

up on the legal or informational constraints that have to b e resp ected

Basic prop erties

Feasibility is simply the requirement that for each problem the sum of the

awards should not exceed the amountavailable and eciency the require

ment that the entire amountavailable should b e allo cated For convenience

wehave incorp orated eciency in the denition of a rule It is obvious that

we cannot distribute more than there is but conceivablywe could distribute

less and if this allowed recovering other prop erties of interest be willing to

consider the p ossibility Nevertheless we will insist on equality here as this

entails no loss

This is the set of vectors x R such that x v N and for each S N

x w c E S In fact Driessen shows that the two games v and w are duals

of each other in a formal sense

A third formulation providing a compromise b etween the p essimistic and optimistic

outlo oks leading to v c E andw c E resp ectively simply consists in taking the average

v cE w cE

The analysis of the resulting game remains to b e carried out although it is

easy to see that in the twoclaimant case its core is a singleton and this singleton coincides

with the recommendation made by concedeanddivide Finally ONeill suggests

having each coalition and its complementplayatwoplayer strategic game of p ositioning

of claims as describ ed in Section b elow and denes the worth of a coalition as what it

receives at the essentially unique Nash equilibrium of the game see Theorem

It may seem that we should never allo cate less than is available but in other settings

this option has proved extremely useful In the context of public go o d decision for ex

ample the socalled ClarkeGroves mechanism succeeds in eliciting truthful information

ab out agents preferences only b ecause it fails budget balance Recall the cost allo cation

interpretation of our mo del

Next are two requirements placing bounds on awards Both are very

natural and wehave also incorp orated them in the denition of a rule Non

negativity gives a lower b ound each claimant should receive a nonnegative

amount Claims boundedness gives an upp er bound each agent should

receive at most his claim

Another lower b ound requirement is that each claimant should receiveat

least the dierence b etween the amountavailable and the sum of the claims of

the other claimants if this dierence is nonnegative and otherwise Recall

that this quantity the claimants minimal right app ears in the denition

of the adjusted prop ortional rule

Resp ect of minimal rights For each c E C and each i N

R c E maxfE c g

i j

N nfig

Respect of minimal rights is a consequence of eciency nonnegativity

and claims boundedness together

Fully comp ensating agents with small claims is of course relatively easier

and in some circumstances it is tempting to do so rst Which criterion

one should adopt to decide how small a claim should be to deserve this

preferential treatment is a matter of judgment however One interesting

critical value of a claim is obtained by substituting it for the claim of any

other agent whose claim is higher and checking whether there would then b e

enough to comp ensate everyone A plausible requirement is that if yes the

agent holding this claim should be fully comp ensated Herrero and Villar

b they use the term sustainability Of all of the rules that we have

seen only the constrained equal awards rule satises the requirement

Conditional full comp ensation For each c E C and each i N if

minfc c g E then R c E c

j i i i

j N

An alternativechoice for a critical value of a claim b elowwhich full com

p ensation could b e required is simply

Symmetrically one could adopt the viewp oint that if a claim is to o small

one should not bother assigning anything to its owner In the context of

bankruptcy for example the ob jectivewould b e to give prioritytoagents who

have risked relatively greater amounts In the context of taxation exempting

Using the language of the theory of co op erative games this prop erty could b e called

individual rationality

agents with lower incomes is a feature of almost all realworld tax laws We

refer to this prop erty as conditional nul l compensation Herrero and

Villar b they use the expression indep endence of residual claims

Next we require that the awards to agents whose claims are equal should

be equal

Equal treatment of equals For each c E C and each fi j gN if

c c then R c E R c E

i j i j

The requirement is not always justied and in fact it is often violated in

practice In actual bankruptcy pro ceedings for example some claims often

have higher priority than others Toallow dierential treatment of otherwise

identical agents we can enrich the mo del and explicitly intro duce priority

parameters A claims problem with priorities is a list c E where

c E C and is a complete and transitive binary relation on N Given

i N the equivalence class containing i is his priority class two agents

in the same priority class are treated dierently only to the extent that

their claims dier but agents in dierent priority classes are usually treated

dierently even though their claims are equal Priorities classes are handled

in succession nothing b eing assigned to any class b efore the claims of all

members of the higher classes are fully satised

All rules can be adapted to accommo date priorities To illustrate and

taking the prop ortional rule as p oint of departure we would make awards

prop ortional to claims within each priority class In fact this is commonly

done Aggarwal

Alternatively and somewhat more exibly we could dene a claims

problem with weights to be a list c E where c E C and

N N

int is a point in the interior of the unit simplex of R indicating

what could be called the relative priorities as opp osed to the absolute

priorities of the previous paragraph that should be given to agents Most

rules can easily be adapted to this setting to o For instance to obtain a

weighted version of the prop ortional rule make awards prop ortional to the

vector of weighted claims sub ject to no one receiving more than his claim

For a weighted version of the constrained equal awards rule p erform the

division prop ortionally to the claimants weights sub ject to no one receiving

more than his claim For a weighted version of concedeanddivide calculate

the concessions as in the initial denition but in the second stage p erform

the division prop ortionally to the claimants weights

Priorities and weights can b e combined Divide agents into priority classes

to be handled in succession within each class use weights to deect the

awards vector in the desired direction

A strengthening of equal treatment of equals is the requirement that the

identity of agents should not matter The chosen awards vector should de

pend only on the list of claims not on who holds them Recall that

denotes the class of bijections from N into itself

N N

Anonymity For each c E C each and each i N

R c E R c E

iN i

i i

Another strengthening is the requirement that the rule should resp ect the

ordering of claims if agent is claim is at least as large as agent j s claim

he should receive at least as much as agent j do es moreover the dierences

calculated agentby agent between claims and awards should b e ordered as

the claims are This prop erty app ears in Aumann and Maschler

Order preservation For each c E C and each fi j gN if c c

i j

then R c E R c E Also c R c E c R c E

i j i i j j

All of the rules of Section satisfy this prop erty but few satisfy strict

order preservation the requirement that if in addition agent is claim is

greater than agent j s claim and E he should receive more equality is

not p ermitted any more a parallel statement b eing made ab out losses

Group order preservation go es beyond order preservation by apply

ing the idea to groups if the sum of the claims of the members of some

group is at least as large as the sum of the claims of the members of some

other group then the sums of the awards to the memb ers of the two groups

should b e related in a similar way Thomson a parallel statementbe

ing made ab out losses For each problem the set of awards vectors satisfying

these inequalities is a p olygonal region it is nonempty as the prop ortional

awards vector always b elongs to it Therefore rules satisfying group order

preservation are easily dened it suces to select from the set Equal

treatment of equal groups the counterpart for groups of equal treatment

NC Lee discusses a weighted version of the constrained equal awards rule

Some authors only imp ose the rst part

of equals says that two groups with equal aggregate claims should receive

equal aggregate awards

Next is the requirement that claimants with greater claims should receive

prop ortionately at most as much

Regressivity For each c E C and each fi j g N if c c

i j

R cE

then

c c

i j

In the context of taxation it is mainly the reverse inequality that has

been imp osed Underlying it is the desire to imp ose equal sacrices on all

agents under the assumption that they have concave and identical utility

functions

Progressivity For each c E C and each fi j g N if c c

i j

R cE

then

c c

i j

Monotonicity requirements

We turn to monotonicity requirements Such requirements have played an

imp ortant role in the analysis of other domains and they often have strong

implications Sometimes they are even incompatible with very elementary

requirements of eciency and fairness in distribution In the presentcontext

they are quite weak however

First is the requirement that if an agents claim increases he should

receive at least as much as he did initially

Claims monotonicity For eachc E C each i N and each c c

we have R c c E R c E

i i i

Under the same hyp otheses wemightwant each of the other claimants to

receiveatmostasmuch as he did initially For eachc E C each i N

and each c c we have R c c E R c E Together with

i N nfig i N nfig

i i

eciency this requirement implies the previous one Claims monotonicity

and even this stronger version are met very generally

By the notation c we mean the vector c from which the ith co ordinate has been

removed

In the study of allo cation in classical economies requirements of this typ e fo cusing

on endowments haveplayed an imp ortant role Thomson

A requirement adapted from the international trade literature is that if

an agent transfers part or all of his claim to some other agent he should

receive at most as much as he did initially Chun a

N N

No transfer paradox For eachc E C each i N and each c R

P P

if c c and c c then R c E R c E

i j i i

i j

Alternatively we could fo cus on a claim transfer from one agent to a

sp ecic other agent and require not only that the former should receive at

most as much as he did initially but also that the latter should receive at

least as much as he did initially This requirement is also easily satised

The next requirement is that if the amount to divide increases each

claimant should receive at leastasmuch as he did initially

Resource monotonicity For each c E C and each E R if

c E E then R c E R c E

Most of the rules that have b een considered in the literature and all of the

rules we have formally dened satisfy resource monotonicity However the

stricter condition obtained by requiring that under the same hyp otheses the

inequalities app earing in the conclusion should b e strict when p ossible that

is if E and for each claimant whose claim is p ositive is not satised

by most of them A conditional version of this stricter prop erty obtained

by adding the strict inequality only for each claimant whose initial award

is neither nor equal to his claim this eliminates corner situations is

satised more generally Similar comments apply to the conditions describ ed

earlier

The nal requirement in this section is that if the amount to divide in

creases of twoagents the one with the greater claim should receive a greater

share of the increment than the other Dagan Serrano and Volij

Sup ermo dularity For each c E C each E R and each pair

fi j g N if c E E and c c then R c E R c E

i i j i i

R c E R c E

j j

The prop ertyis considered by several authors including Curiel Maschler and Tijs

and Young

Apart from the constrained egalitarian rule all of the rules that we have

seen satisfy this prop erty Here to o a strict version can be formulated but

it is rarely met A supermodular rule satises the rst part of order preser

vation but may violate the second part as well as resource monotonicity

As their names indicate both the constrained equal awards and con

strained egalitarian rules are intended to achieve an ob jective of equality

sub ject to constraints and in fact the two rules can b e characterized as b est

in that resp ect diering only in the sp ecication of these constraints For

the former the only constraints are those imp osed on rules that they should

select ecient vectors b ounded below by and ab ove by the claims vector

For the latter the additional constraint is imp osed that if the amountavail

able is equal to the halfsum of the claims each agent should receive half

of his claimlet us refer to this prop erty as the midpoint property and

resource monotonicity The imp ortance in the Talmud of the midp ointas a

psychological watershed is do cumented by Aumann and Maschler

Theorem Schummer and Thomson a The constrained equal

awards rule is the only rule such that for each problem the gap between

the smal lest amount any claimant receives and the largest such amount is

the smal lest

b It is also the only rule such that for each problem the varianceofthe

amounts received by al l the claimants is the smal lest

For the constrained egalitarian rule a parallel statement holds for gap

minimization in the twoclaimant case and for variance minimization in the

general case sub ject to the midpoint property and resource monotonicity

Chun Schummer and Thomson

Indep endence additivity and related prop erties

In this section we consider prop erties of indep endence of rules with resp ect

to certain op erations p erformed on the data of the problem

The rst requirement is that if claims and amountavailable are multiplied

by the same p ositive number then so should all awards It is not always

natural however to treat in a similar way situations in which amountavailable

and claims are small and situations in which these variables are large For

instance in situations in which it is felt that each agent should b e guaranteed

a minimal amount if p ossible this requirement is not reasonable Similarly

in the context of taxation one may want to exempt agents whose income is

below some threshold

Homogeneity For each c E C and each R c E

R c E

The next requirement is that the part of a claim that is ab ove the amount

to divide should be ignored Since this part cannot be reimbursed anyway

replacing c by E for each i N such that c E should not aect the

i i

chosen awards vector

Invariance under claims truncation For each c E C R c E

R minfc Eg E

i iN

This prop erty is satised by the constrained equal awards minimal over

lap random arrival and Talmud rules but not by the prop ortional or con

strained equal losses rules If we feel strongly that invariance under claims

truncation should be imp osed we could of course redene the domain and

only consider problems in which no claim is greater than the amount to di

vide Alternatively we could limit attention to lists c E R

where for each i N c is interpreted as the percentage of the amount to

divide claimed by agent i This restriction might b e particularly meaningful

in the context of estate division think of contradictory wills each of which

sp ecies the p ercentage of the estate that some heir should receive

Next we require that the awards vector should equivalently b e obtainable

i directly or ii by rst assigning to each agent his minimal right adjusting

claims down by these amounts and nally applying the rule to divide the

remainder

However as argued byYoung the problem could then b e redened as p ertain

ing to the division of whatever surplus exists after such thresholds have b een reached

Note that the problem app earing in the prop ertyiswelldened The prop ertyisfound

in Dagan and Volij By analogy to the prop erty studied in bargaining theory these

authors call it indep endence of irrelevant claims We prefer the more neutral expression

of invariance under claims truncation since it can b e argued that the part of an agents

claim that is ab ove the amountavailable is not irrelevant

ONeill calls these problems simple claims problems

Minimal rights rst For each c E C R c E mc E R c

mc E E m c E

This prop erty is satised by the constrained equal losses random arrival

and Talmud rules but not by the prop ortional constrained equal awards or

minimal overlap rules these facts are discussed by Thomson and Yeh

Now consider the following situation after having divided the liquidation

value of a rm among its creditors its assets are reevaluated and found to

be worth less than initially thought p erhaps their market value has changed

in the meantime or certain assets are found to b e inaccessible To deal with

the new situation two options are available i we cancel the initial division

and apply the rule to the revised problem or ii we consider the initial

awards as claims on the revised value and apply the rule to the problem so

dened Our next requirement isthatbothways of pro ceeding should result

in the same awards vectors Moulin

Comp osition down For each c E C and each E E we have

R c E R R c E E

Composition down is satised by the prop ortional constrained equal

awards and constrained equal losses rules but not by the random arrival

Talmud or minimal overlap rules

The opp osite p ossibility is just as plausible namely that after having

divided the liquidation value of a bankrupt rm among its creditors its

assets are reevaluated but this time they are found to be worth more than

originally thought Here we have two parallel options i we cancel the

initial division and apply the rule to the revised problem or ii we let agents

keep their initial awards adjust claims down by these amounts and reapply

the rule to divide the incremental worth The requirement formulated next

is that b oth ways of pro ceeding should result in the same awards vectors

Young It is in the spirit of minimal rights rst but there is no

logical relation b etween the two the constrained equal awards rule satises

This problem is welldened since for each c mc E E m c E and

P P

c m c E E m c E The prop ertyisintro duced by Curiel Maschler and

i i i

Tijs under the name of the minimal rights prop erty Dagan refers to it

as v separability

A prop ertyofstep by step negotiation in the same spirit is analyzed in the context of

bargaining by Kalai

composition up but not minimal rights rst whereas the opp osite holds for

the random arrival rule

Comp osition up For each c E C and each E R if c E

E then R c E R c E R c R c E E E

This prop erty is satised or violated by the same examples we gave for

composition down but the two comp osition prop erties are not logically re

lated

The next requirement is that no group of agents should receive more by

transferring claims among themselves Chun a

No advantageous transfer For each c E C each M N

P P P

and each c R if c c then R c E

iM i i

i i

M M M

R c c E

i iM N nM

Obviously in the presence of eciency which is incorp orated in our

denition of a rule this prop erty is vacuously satised for jN j Of our

main rules only the prop ortional rule passes this test

We continue with a somewhat more technical requirement the awards

vector should be a linear function of the amounttodivide

Resource linearity For each c E C each E such that c E

and each wehave R c E E R c E R c E

The prop ortional rule is the only one of our main rules to satisfy the

prop erty

The next requirement p ertains to situations in which the amount to divide

comes in two parts It states that dividing the rst part rst and then dividing

the second part no adjustment in claims b eing made should yield the same

awards vector as consolidating the two parts and dividing the sum at once

The problem app earing in this expression is a welldened problem since we only

consider solutions satisfying nonnegativity and claims boundedness

A prop erty of this typ e is considered by Gale and Aumann and Peleg

in the context of allo cation in classical exchange economies and by Moulin in the

context of quasilinear so cial choice

By the notation c c we mean the vector in which the claim of each i M

N nM

and the claim of each i N nM is c is c

The implications of this prop erty are studied by Chun a It is most

app ealing in situations in whichthevector c is given a broader interpretation

than a vector of claims as understo o d so far but instead represents rights

that are not commensurable with what is to be divided Then the fact

that a rst amount has already been allo cated cannot be very meaningfully

accompanied by an adjustment in claims when the second amount b ecomes

available

Resource additivity For each c E C and each pair fE E g of ele

ments of R if E E E then R c E R c E R c E

The prop ortional rule satises the prop erty A stronger version is ob

tained by dropping the hyp othesis of equal claims vectors No rule satises

it but if nonnegativity is dropp ed then it can be met

The next requirement is sometimes needed for technical reasons but it

makes much intuitive sense and it is satised by all of the rules that have

been considered in the literature simply small changes in the data of the

problem should not lead to large changes in the chosen awards vector

Continuity For each sequence fc E g of elements of C and each

c E C if c E converges to c E then R c E converges to

R c E

Partial notions of continuity with resp ect to the amount to divide or with

resp ect to each agents claim separately can also b e formulated

Our last requirement for the xedp opulation version of the mo del is that

the problem of dividing what is available and the problem of dividing what

is missing should be treated symmetrically The prop erty is formulated by

Aumann and Maschler who note several passages in the Talmud

where the idea is implicit

Selfduality For each c E C R c E c R c c E

Then of course the inequality c E has no meaning and it makes sense to enlarge

the class of problems under consideration by dropping it

The problem app earing in this prop erty is welldened since we only consider rules

satisfying claims boundedness

Many rules are selfdual including the prop ortional Talmud and ad

justed prop ortional rules

An op eration asso ciating to each rule its dual can b e dened as follows

Given R the dual of R selects for each problem c E C the awards

vector c R c c E the righthand side of the formula app earing in

the statement of selfduality It is clear that the constrained equal awards

and constrained equal losses rules are dual Tosay thataruleisselfdual is

to say that it coincides with its dual

Now two properties are dual of each other if whenever a rule sat

ises one of them its dual satises the other Composition down and compo

sition up are dual Moulin So are invariance under claims truncation

and minimal rights rst Herrero and Villar a Dagan proves a

related result and conditional ful l compensation and conditional nul l com

pensation Herrero and Villar b A prop erty is selfdual if whenever

a rule satises the prop erty so do es its dual Many prop erties are selfdual

examples b eing equal treatment of equals resource monotonicity and conti

nuity The duality notion leads us to the formulation of new prop erties For

instance the dual of claims monotonicity says that if an agents claim

and the amount to divide increase by the same amount the agents award

should increase by at most that amount Thomson and Yeh

The duality notion is useful for another reason it allows us to derive from

each characterization of a rule a characterization of the dual rule by simply

replacing each prop erty by its dual Herrero and Villar a and Moulin

exploit this fact The same comment applies to characterizations of a

family of rules Here the dual result is a characterization of the dual family

The theorems stated b elow includeanumber of such pairs

We continue with a list of characterizations based on the prop erties we

have dened Mostly they are group ed according to which solution comes

out of the axioms

Theorem The constrained equal awards rule is the only rule satisfying

Aumann and Maschler observethattheTalmud rule is the only selfdual rule

that coincides with the constrained equal awards rule on the sub domain of problems c E

suchthat E provided the halfclaims are used instead of the claims themselves

Aumann and Maschler More generally consider a rule dened for each claims

vector and for each amount to divide running from to the halfsum of the claims If for

the halfsum of the claims it selects the halfclaims vector then it has a unique selfdual

extension to all values of the resource

a equal treatment of equals invariance under claims truncation and

comp osition up Dagan

b conditional full comp ensation and comp osition down Herrero and

Vil lar b

c conditional full comp ensation and claims monotonicityYeh a

Dual characterizations of the constrained equal losses rules follow

Theorem The constrained equal losses rule is the only rule satisfying

a equal treatment of equals minimal rights rstand comp osition down

Herrero a

b conditional null comp ensation and comp osition up Herrero and Vil

lar b

c conditional null comp ensation and the dual of claims monotonicity

Yeh a

The next two theorems p ertain to the twoclaimant case

Theorem Dagan For jN j Concedeanddivide is the only

rule satisfying

a invariance under claims truncation and selfduality

b minimal rights rst and selfduality

c equal treatment of equals invariance under claims truncation and

minimal rights rst

Our next characterization which also p ertains to the twoclaimantcase

identies a family of twoclaimant rules that connect the prop ortional

constrained equal awards and constrained equal losses rules Moulin

Family D For jN j Awards space is partitioned into cones each non

degenerate cone is spanned by a homothetic family of piecewise linear curves

in two pieces one b eing a segment containing the origin and contained in

one of the b oundary rays of the cone the rst ray and the other a half

line parallel to the other b oundary ray the second ray Cones can be

degenerate that is can b e rays For each claims vector the path of awards of

the rule is obtained byidentifying the cone to which the claims vector b elongs

and the curve in the cone passing through it and taking the restriction of

the curve to the box from the origin to the claims vector

A simpleproofisgiven byYeh b

The family D is largenote in particular that many of its memb ers violate

symmetry requirements To obtain equal treatment of equals require the

line to b e a cone in the partition For anonymity require the partition to b e

symmetric with resp ect to the line and the designation of rays as rst

or second in each pair of symmetric cones to share this symmetry

Theorem Moulin For jN j The members of the family D

are the only rules satisfying homogeneity comp osition down and comp osi

tion up

Returning to the case of an arbitrary number of claimants we have the

following characterizations of the prop ortional rule

Theorem The proportional rule is the only rule satisfying

a selfduality and comp osition up Young

b selfduality and comp osition down

c for jN j no advantageous transfer Moulin Chun a

Ju and Miyagawa

d resource linearity Chun a

Op erators

The prop erties dened in the previous sections suggest the denition of op

erators on the space of rules Let R be a rule The claims truncation

op erator asso ciates with R the rule dened as follows for each problem rst

truncate claims and then apply R The attribution of minimal rights

op erator asso ciates with R the rule dened as follows for each problem

rst assign to each claimant his minimal right revise claims down by these

amounts and the amount to divide by their sum and now apply R to the

resulting problem We have already dened the duality op erator Finally

given a list of rules and a list of nonnegativeweights for them the convex

ity op erator gives the rule that asso ciates with each problem the weighted

average of the awards vectors chosen by these rules for the problem

A systematic analysis of these op erators is carried out by Thomson and

Yeh who establish the following structural prop erties b etween them

Part c involves neither continuity nor equal treatment of equals These prop erties

were included in early versions of the uniqueness part Moulins result p ertains to a