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Characterizations of Proportional

Rules in Claims Problems

Hirofumi YAMAMURA

Discussion Paper No. 2006-04

April 14, 2006

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Characterizations of Proportional

Rules in Claims Problems*

Hirofumi YAMAMURA+

Tokyo Institute of Technology

April 14, 2006

Abstract

When resources are divided among agents, resources are in many cases divided

proportionally to their claims. In this paper, we provide new axiomatizations of

generalized proportional rules based on the axiom decentralizability due to Moulin

[13,14,15]. Decentralizability requires that no reallocation by any coalition can affect

the awards of the agents outside the coalition. Our results can explain the results of

Chun [7] and J u=Miyagawa=Sakai [10] based on non-manipulability requirements as

corollaries.

We moreover introduce a coalitional form game called a claims reallocation game

which describes a claims problem under any established division rule. We require the

core of a claims reallocation game be always nonempty for claimants to make an

agreement on how to divide. We show that under a certain condition, the core of a claims

reallocation game is always nonempty if and only if the division rule is a generalized

proportional rule.

J EL Classification: C71, D63, D70.

Key words: Bankruptcy problem; Proportional Rule; No Advantageous Reallocation;

Decentralizability; Claims Reallocation Game; Core.

* The author thanks Professor Takehiko Yamato for his helpful comments.+Department of Social Engineering, Graduate School of Decision Science and Technology,

2-12-1 Oh-okayama, Meguro-ku, Tokyo, 152-8552, J apan; yamamura-h@soc. titech.ac.jp

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1. Introduction

When a firm goes bankrupt, how should its liquidation value be divided among

creditors? When cooperation in a group yields a surplus, how its surplus be divided

among the members of the group? Or how should a government redistribute its citizens

incomes among them? There are many kinds of division problems that should be solved

according to a rule.

Such problems are usually solved according to each agents property, which we call

an agents claim. For example, in a bankruptcy problem, the liquidation value is

divided among creditors according to their credits. A claims problem we are going to

study in this paper is as follows. There are a set of agents and a divisible resource to be

divided among them. Each agent has a claim to this resource. A claims problem is how

this divisible resource should be divided among agents according to their claims.

We should note that claims are transferable in many cases. Then, a division rule to

solve claims problems should be established in consideration for the transferability of

claims. That is, a division rule should be made so as not to cause troubles by any

reallocation of claims. In this paper, we study the class of division rules which are not

troubled by reallocations of claims by the following two approaches: an axiomatic

approach and a game theoretic approach.

In axiomatic studies on claims problems, non-manipulability condition, which

demands that there be no agents who benefit by any reallocation of their claims, is the

most studied property to be satisfied in consideration for the transferability of claims.

This property is studied by ONeill [16] in bankruptcy problems and by Banker [5] in

cost allocation problems. And then, Moulin [12] defined this requirement as an axiom

called No Advantageous Reallocation (in short NAR) and Chun [7] introduced NAR in

claims problems. After them, de Frutos [8], J u [9] and J u=Miyagawa=Sakai [10] have

studied non-manipulability. (See Thomson [18] for a survey.)

A typical example of a non-manipulable division rule is the proportional rule, which is

widely used in modern laws. Chun [7] showed that the proportional rule is the only rule

satisfying NAR and supplementary axioms. J u=Miyagawa=Sakai [10] defined

generalized proportional rules and characterize them based on NAR.

J u=Miyagawa=Sakai [10] also axiomatize the proportional rule based on a weaker

version of NAR, they call pairwise reallocation proofness, which takes only pairwise

reallocations into consideration.

In the axiomatic part of this paper, we introduce other weaker versions of NAR which

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take only k-person reallocations into consideration and study to what extent we can

weaken axioms required to characterize generalized proportional rules. To study this

problem, we reconsider the axiom Monlin [13,14,15] defined asdecentralizability

and

introduce its descendent axioms. These axioms help us to order the relationships among

axioms and to study the axiomatization of generalized proportional rules. Our results

can explain the results of Chun [7] and J u=Miyagawa=Sakai [10] based on

non-manipulability requirements as corollaries.

In the game theoretic part of this paper, we investigate a new coalitional game

theoretic approach that rationalizes generalized proportional rules. In the previous

game theoretic approaches, like in ONeill [16] and Aumann=Maschler [3], a problem

under the nature state, where there is no rule to solve claims problems, has been

described as a coalitional form game. Game theoretic studies have provided the

rationales for several division rules as solutions of this game, such as the random

arrival rule (ONeill [16]) and the Talmudic rule (Aumann= Maschler [3]). However,

they have not given a satisfactory answer to the question why the proportional rule is

actually used. The reason why they have not rationalized the proportional rule may

have something to do with Atsumi [2]s criticism about previous game theoretic

approaches.

Atsumi [2]s criticism is summarized as follows. Game theoretic studies have

analyzed a coalitional form game where the worth of a coalition is defined as what it can

get without going to court. However, the establishment of a division rule causes a

different situation from what they have analyzed, because claimants act with this

division rule taking into consideration. So, the division rules they have conducted

through the analysis of this games may not be conclusive.

In this paper, following Atsumi [2]s criticism, we investigate a new coalitional form

game which describes a bankruptcy problem after a rule is established as an act. We

assume claims are transferable among creditors as in modern laws. Then, each

subgroup of claimants may manipulate the sum of their awards by reallocating their

claims. So, the game we consider in this paper might be called a claims reallocation

game.

We provide the rationale for proportional division rules through the analysis of claims

reallocation games. We require that the core of a claims reallocation game be always

non-empty in order for claimants to make an agreement on how to divide the liquidation

value. We show only proportional rules satisfy this requirement under a certain

condition.

The remainder of this paper is as follows. In section 2, we define a claims problem and

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generalized proportional rules. In section 3, we introduce axioms and consider the

relationships among axioms. In section 4, we axiomatize generalized proportional rules

by axioms introduced in section 3 to consider to what extent we can weaken the axiomsrequired to characterize generalized proportional rules. In section 5, we investigate a

new coalitional form game called a Claims Reallocation Game and characterize

generalized proportional rules through the analysis ofClaims Reallocation Games. In

section 6, we state some concluding remarks.

2. The Model

There is a finite set nN ,,2,1 L of claimants. For each iN, i has a claim

kKkiki

Rcc where kK ,,2,1 L , characterized by a finite dimensional vector.

A claims profile c is a list of individual claims knNii

Rcc . For each SN, let

Sc Si

ic . RE is a divisible value to be divided among N. Then, an n-person claims

problem is formalized by the pair RREc

kn

),( such that

k

N Rc .n

D denotes the set of feasible n-person claims problems. That is,

kNknn RcRREcD |),( .

Letnn DD be the domain of n-person claims problems. The domain nD is rich if

nDEc ),( , then for any nDEc ),( such that NN cc and EE ,nDEc ),( .

The richness of the domain is necessary for reallocations of claims to be taken into

consideration. In this paper, we consider only rich domains.

Given a domainn

D , a division rule over a domainn

D is a function nnn RDf :

which associates with each problemnDEc ),( an award profile n

Ni

n

i REcf ),( .

In this paper, we focus on proportional division rules and its generalizations. Then, we

define what J u=Miyagawa=Sakai [10] calls a generalized proportional rule.

Definition 2.1. A rule nnn RDf : is a generalized proportional rule if there existmappings

nk

RRRA : ,kk

RRRW : such that,

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Kk

Nk

Nk

ikNi

n

i EcWc

cEcAEcf ),(),(),( ,

for each iN and each

n

DEc ),( .

What is called the proportional rule is a special case of a generalized proportional

rule.

Example 2.2. Let k=1. A rule nnn RDf : is the proportional rule if it is ageneralized proportional rule which satisfies NiEcA Ni ,0),( and EEcW N ),( .

On the other hand, the egalitarian rule is one of a generalized proportional rule.

Example 2.3. Let k=1. Define the egalitarian rule nnn RDf : such thatE

nEcfni

1),( , for each iN andeach nDEc ),( .

The egalitarian rule is one of a generalized proportional rule such that EEcW N ),(

and Nin

EEcA Ni ,),( .

3. Axioms

In this chapter, we define a number of axioms which a rulenf should satisfy. First

of all, we introduce one of our main axioms, which is interpreted as a condition for

non-manipulability, called no advantageous reallocation.

No Advantageous Reallocation (NAR)1.For any ),( Ec , nDEc ),( , if E=Eand thereexists NS such that SS cc and SNjcc jj \, , then

Si Sin

i

n

i EcfEcf ),(),( . In other words, for any NS , Sin

if )( is

dependent only on Nc , E, and SNjjc / .

1 J u=Miyagawa=Sakai[10] calls this axiomReallocation Proofness.

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This axiom NAR is introduced by Moulin [12] in quasi-linear bargaining problems. In

the context of claims problem, Chun [7] introduced NAR. NAR requires that no coalition

can benefit by reallocating their claims. Since claims are in principle transferable inmodern law, we should regard NAR as a necessary axiom to avoid manipulation.

NAR deals with any feasible coalition. Then a way to weaken NAR is to deal only with

particular coalitions. Actually, J u=Miyagawa=Sakai [10] introduced an axiom which

takes only two parsons coalitions into consideration2. In this paper, we consider more

generally weaker versions of NAR which take only k-person coalitions into

consideration.

No Advantageous Reallocation for k-person Coalitions (NAR-k). For any ),( Ec ,nDEc ),( , if E=Eand there exists NS , | S| =k such that SS cc and jj cc ,

SNj \ , then Si Sin

i

n

i EcfEcf ),(),( . In other words, for any NS ,

| S| =k, Sin

if )( is dependent only on Nc , E, and SNjjc / .

Next, we introduce another one of our main axiom Moulin [13,14,15] defined as

Decentralizability.

Decentralizabil ity (DEC).For any ),( Ec , nDEc ),( , if NN cc , E=Eand ii cc ,

then ),(),( EcfEcf nin

i . In other words, for any Ni , )(nif is dependent only on

Nc , E, and ic .

There are some reasons we consider DEC again, which seems similar with NAR. First,

we have to note that DEC itself has a property a rulenf should satisfy. Referred by

Moulin[15], only Nc , E and ic are required to compute i s award. Therefore, we can

interpret DEC as a requirement to simplify calculating each agents award.

Moreover, when we consider the relationship between NAR and DEC, we notice that

DEC contrasts strikingly with NAR. NAR demands that no reallocation by any coalition

can benefit the members of this coalition. On the other hand, DEC demands that no

2

J u=Miyagawa=Sakai[10] calls this axiomPairwise Reallocation Proofness,

which isequivalent to NAR-2 defined as follows.

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reallocation by any coalition can affect the awards of agents outside this coalition. I t is

mainly because no coalitional reallocation can damage agents outside this coalition why

we require NAE. Therefore, we can regard DEC as a more essential axiom.However, these are not all of reasons why we reintroduce DEC. We reconsider DEC

because DEC and its descendant axioms help us to order the relationships among

axioms. Moreover, DEC and its descendant axioms are useful to study the possibility (or

impossibility) of the axiomatizations of generalized proportional rules. To study these

problems, the following axiom, we call constantness, is a key axiom.

Constantness (CON).For any nDEcEc ),(),,( , if NN cc and EE , then

Nin

iNi

n

i EcfEcf ),(),( . In other words,

Ni

n

if )( is dependent only on Nc

and E.

The axiom Pareto Optimality, which is usually requested, is a special case of

constantness.

Pareto Optimality (PO).For anyn

DEc ),( , EEcfNin

i ),( .

The axiom PO is usually considered to be satisfied, but PO may not be satisfied when

it costs agents to solve claims problems. For example, mediation fees or trial costs are

required. Then, we can justify CON as Moulin [15] justifies DEC. By CON, only Nc

and E are required to compute the sum of awards to all agents. So, CON can make the

computation more simple, which may make a cost to solve problems cheaper.

Next, we consider descendant axioms of DEC. One way to introduce descendant

axioms of DEC is to introduce coalitional versions of DEC, which deal only with

particular coalitions, as we introduce descendant axioms of NAR. To introduce

coalitional decentralizability, we can consider the following two axioms.

Decentralizabili ty for k-person coalitions (DEC-k).For any NS , | S| =k, and ),( Ec ,nDEc ),( , if NN cc , E=Eand ii cc , Si , then

Si

n

i

Si

n

i EcfEcf ),(),( .

In other words, for any NS , | S| =k,

Si

n

if )( is dependent only on Nc , E, and

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Sii

c .

Strong Decentralizability for k-person coalitions (SDEC-k).For any NS , | S| =k,and ),( Ec , nDEc ),( if NN cc , E=E, and SS cc , then

Si

n

i

Si

n

i EcfEcf ),(),( . In other words, for any NS , | S| =k,

Si

n

if )( is

dependent only on Nc , E, and Sc .

DEC-krequires that for any k-person coalition the sum of their awards be never

affected by reallocation by agents outside this coalition. I f they make a coalition, their

awards are dependent of any reallocation outside them. SDEC-kadditionally requests

that any reallocation among any k-person coalition can benefit the members of this

coalition.

Another way to introduce a descendant axiom of DEC is to deal only with agents with

a particular property. In this paper, we consider a weaker version of DEC, which deals

only nulls, those whose claims are zero.

Decentralizability for Nulls (DECN). For any ),( Ec andn

DEc ),( , if NN cc ,

E=E, and 0 ii cc , then ),(),( EcfEcfn

i

n

i . In other words, for any Ni ,

)(nif is dependent only on Nc , E i f 0ic .

The axiom No Awards for Nulls, which is usually considered, is a special case of

DENC.

No Awards for Nulls (NAN).For any ),( Ec nD , if 0ic , then 0),( Ecfni .

To solve a huge bankruptcy problem, taxes are sometimes used. Then, those who have

no claims must accept negative awards through taxes to them. Or, in the context of

income redistribution problems, citizens are usually assured of a living standard with a

certain level, regardless of their incomes. Then, even if a citizens income is zero, he

must redistribute a positive award. These examples are showing that a null agent may

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not get a zero award in a certain rule. This axiom DECN requires that awards for nulls

be dependent only on Nc and E even if No Award for Null is not satisfied.

These are all descendant axioms of NAR and DEC we study as follows. Next, we studythe relationships among these axioms. First of all, we consider the relationship between

NAR and DEC. The following claim 3.1. and example 3.2. show that DEC is strictly

weaker than NAR. That is, NAR implies DEC but DEC does not always imply NAR.

Claim 3.1. NAR implies CON and DEC.

Proof. First, we are going to proof NAR implies CON. For any ),( Ec , nDEc ),( , if

NN cc and E=E , then by NAR,

Ni

n

i

Ni

n

i EcfEcf ),(),( .

So,nf satisfies CON. Second let us proof NAR implies DEC. For any ),( Ec ,

nDEc ),( , if NN cc , E=E , ii cc and, then

),(),(),(),(),(),(

\\EcfEcfEcfEcfEcfEcf jiNj

n

jNj

n

jiNj

n

jNj

n

j

n

i

by NAR. So, for any Ni , )(

n

if is dependent only on C, E, and ic .

Example 3.2. Let k=1 and consider a rule nf such that NiEc

cEcf

N

in

i

,),(

2

satisfies DEC, but this solution does not satisfy NAR.

However, assuming constrantness, we can easily show that NAR becomes equivalent

to DEC3. This fact also indicates that constantness has an important role to associate

NAR with DEC.

Claim 3.3. Assume CON. Then, DEC is equivalent to NAR.

Proof. By claim 3.1, it is sufficient to prove that NAR implies DEC under CON. For any

),( Ec , nDEc ),( , if NN cc , E=E and there exists NS such that SS cc and

3 Moulin [13] shows this fact under PO.

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jj cc , SNj / , then Ni Nin

i

n

i EcfEcf ),(),( by DEC and

),(),( EcfEcf nin

i , SNj / by DEC. Therefore,

Si SNjn

jNi

n

i

n

i EcfEcfEcf \ ),(),(),(

Ni Sin

iSNj

n

j

n

i EcfEcfEcf ),(),(),( /

Next we consider the relationships among descendant axioms of DEC and NAR. The

following claim3.4, 3.5, and 3.6 shows the relationships.

Claim 3,4. For k

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Si

n

i

SNi

n

i

Ni

n

i

SNi

n

i

Ni

n

i

Si

n

i EcfEcfEcfEcfEcfEcf ),(),(),(),(),(),(//

since | N/S| =n-k. So,

n

f satisfies DEC-k.

Claim 3.6. Assume CON, then for k

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4. Main Results

In this chapter, we characterize generalized proportional rules based on DEC or NAR

or their descendent axioms.

Theorem 4.1. Let n3. A rule satisfies DEC, CON and OB if and only if it is ageneralized proportional rule.

Proof. I t is clear that a generalized proportional rule satisfies DEC, CON and OB. Then,

we prove the converse. Letnf be a rule satisfying DEC, CON and OB, then define a

mappingnk RRRA : and

nknn RRRW : such that,

)0,,(),( EcfEcA Nn

N , ),()0,,(),( EcAEcfEcW NNnn .

By DEC ofnf , A is well defined, and by DEC and CON, OB of nf , nW also

satisfies DEC, CON and OB. Then we replace ),( EcW ni by ),,( iNn

i cEcW for each i

N.

Step 1. I fnf satisfies DEC and CON, then nW is symmetric. That is, for any

nDEc ),( , if ji cc , then ),,(),,( jNn

jiN

n

i cEcWcEcW .

Proof of Step 1. Take any jiNk ,\ , and consider ),( Ec , ),( Ec nD such that

EEE , ccc ji , cccc Nkk , ikNhch ,\,0 , and 0hc ,

jkNh ,\ . Since )(nW satisfies DEC and CON, and 0)0,,( EcW Nn

i , iN,

),,(),,(),,(),,( hNn

h

Ni

iN

n

iiNN

n

iiN

n

i cEcWcEcWcEcWcEcW

),,(),,(),,(),,( jNn

jjN

n

jhN

n

h

Ni

iN

n

i cEcWcEcWcEcWcEcW

Step 2. I fnf satisfies DEC and CON, for each Ni ),,( EcW N

n

i is additive with

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regard to ic . That is, if iii ccc , then

),,(),,(),,( iNn

iiN

n

iiN

n

i cEcWcEcWcEcW .

Proof of Step 2. Take any iNkj \, and consider ),( Ec and nDEc ),( such

that

.,,/,0

)(

0,

,

kjiNhc

ccccc

ccc

ccccc

EE

h

Nkk

jj

ii

Then, ),(),(),,(),,( EcWEcWcEcWcEcW njn

iNn

jNn

i

))(,,(),( cccEcWEcW NNn

k

Ni

n

i

),,(),( ccEcWEcW Nn

i

n

i .

SincenW is symmetric by step 1, ),,(),,( cEcWcEcW NiNj . Therefore,

),,(),,(),,( ccEcWcEcWcEcW Nn

iN

n

iN

n

i .

For each kK, let us define a mapping RRRRW knk : such that,

)0,,0,,0,,0,,(),,( 1 LL kNn

kN

n

k cEcWcEcW .

Then, by step 2,

Kk

kN

n

kN

n cEcWcEcW ),,(),,( 111 .

and for any 1/Ni , by step 1,

Kk

ikN

n

kiN

n

iN

n

i cEcWcEcWcEcW ),,(),,(),,( 1 .

Then, for each kK, let us define a mapping RRREcW kNn

k :),( such that,

),,(),( NkNn

kN

n

k cEcWEcW .

Since, for each kK , ),,( EcW Nn

k is additive with regards to kc and satisfies OB,

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then

),(),,( EcW

c

ccEcW Nk

Nk

ikikN

n

k

by J ensens equation 4 (See Aczel=Dhombres [1].) Therefore, only generalized

proportional rules satisfy DEC, CON and OB.

We moreover investigate the axiomatizations of generalized proportional rules based

on weaker versions of NAR or DEC. The following theorem shows to what extent axioms

required to characterize generalized proportional can be weaken without any other

constraints. To proof this theorem, the idea of coalitional decentralizability is useful.

Theorem 4.2. Let 2kn-3. A rule satisfies DEC-k ( or NAR-(n-k) ), CON and OB if andonly if it is a generalized proportional rule.

Proof. We shownf satisfies DEC if nf satisfies DEC-k, 2kn-3. Then, by theorem

4.1,nf must be a generalized proportional rule.

FornDEc ),( , consider a problem nDEc ),( such that E=E , 11 cc , 02 c ,

323 ccc , ),,4(, nicc ii L . Assuming DEC-(n-k), 3kn-1,nf must

satisfy DEC-(n-3) and DEC-(n-2) by Claim 3.4. Since, 23,2\ nN and

33,2,1\ nN , then by DCE-(n-3) ,DEC-(n-2),

).,(),(),(),(),(),( 13,2,1\

1

3,2\3,2,1\3,2\

1 EcfEcfEcfEcfEcfEcfn

Ni

n

Ni

n

i

Ni

n

i

Ni

n

i

n

Taking the similar steps, fornDEc ),( such that E=E , 11 cc , 0ic ,

)1,,2( ni L and 1ccc Nn , we can say ),(),( 11 EcfEcfnn

. Therefore, for

any ),( Ec , nDEc ),( , if NN cc , E=E and 11 cc , then ),(),( 11 EcfEcfnn .

For any iN, 1i , we can similarly show )(nif is dependent only on Nc and E and

ic .

4 Let X be an interval of R which contains 0 and consider a mapping RXf : whichsatisfies for any x,yX, if x+yX then f(x)+f(y)=f(x+y) and one sided boundedness.

Then, there exists cR such that f(x)=cx, for any xX.

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However, we cannot characterize generalized proportional rule by NAR-2 (or DEC-

(n-2) ) only. The following example shows a rule which satisfies NAR-2 (or DEC-(n-2) ),

CON and OB but is not a generalized proportional rule.

Example 4.3. Let k=1 and n4. Consider a rule nf such thatNieachforE

c

cccEcf

N

iiin

i

,3

),( 11 ,

where n+1=1. This rulenf satisfies CON and NAR-2 ( or DEC-(n-2) ), because

EEc

c

EcfN

Ni

i

Ni

n

i

)113(

),( and

Ec

ccc

Ec

ccccEcfEcf

N

ji

jiNk

k

N

jjiin

j

n

i

11

).(\11

2)(2

),(),(

, if j=i+1,

Ec

ccccccEcfEcf

N

jjjiiin

j

n

i

1111 33),(),(

E

c

ccccc

N

jiNk

kjjii

,\1111 3

, otherwise.

Needless to say, nf satisfies OB. However, this rule nf is not a generalized

proportional rule.

In order to axiomatize generalized proportional rules based on NAR-2 (or DEC-(n-2) ),

an additional axiom is required. The following theorem 4.4. is an axiomatization of

generalized proportional rules based on NAR-2 (or DEC-(n-2) ) with an additional axiom

DECN.

Theorem 4.4. Let n3. A rule satisfies DEC-(n-2) ( or NAR-2 ), DECN, CON and OB ifand only if it is a generalized proportional rule.5

Proof. For SN, define SNjcDEcSD jnn /,0|),()( and a mapping

),( EcA N as in theorem 4.1. By DECN, ),( EcA N is well-defined.

5 This theorem is a generalized rule version of J u=Miyagawa=Sakai [10 Theorem 4].

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Step 1. For any SN, | S| =3, there exists a mappingkkn

S RRRW : such that,

Kk Nkn

S

Nk

ikNi

ni EcW

ccEcAEcf ),(),(),( , for each iN and each )(),( SDEc n .

Proof of Step 1. We can prove this lemma similarly with the proof of Theorem 4.1.

Step 2. For any S,TN, such that | S| =| T| =3,n

T

n

S WW .

Proof of Step 2. When 2 TS , take any kK, i,jST, ij, and nDEc ),( such

that )0,,,0( LL Nki cc , iNj ccc and 0kc , k N/ i,j . Since

)(),( SDEc n and )(),( TDEc n ,

),(),(),(),( EcWEcWEcAEcf Nkn

TNk

n

SNi

n

i .

Therefore,n

T

n

S WW .

When 2 TS , we can take S,TN, | S| =| T| =3, such that SS TS

2 TT . So, nTm

T

n

S

n

S WWWW .

Step 3. If for any UmSS

n SDEc

:

)(),(

Kk

Nk

Nk

ikNi

n

i EcWc

cEcAEcf ),(),(),( , then

for any U1:

)(),(

mSS

n SDEc

Kk

Nk

Nk

ikNi

n

i EcWc

cEcAEcf ),(),(),( .

Proof of step 3. Let (c,E) )(SD n , | S| =k+1, then for any jN, ),(),( EcAEcf Njn

j ,

and for any i,i S,

Kk

Nk

Nk

kiikii

n

i

n

i EcWc

ccAAEcfEcf ),(),(),( by NAR-2.

By these equations, we can show

Kk

Nk

Nk

iki

n

i EcWc

cAEcf ),(),( for any iS.

By step 2 and step 3, for any (c,E) )(NDD

nn

,

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18

Kk

Nk

Nk

iki

n

i EcWc

cAEcf ),(),( for each iN which accords with a generalized

proportional rule.

Next, we axiomatize generalized proportional rules based on stronger versions of

coalitional decentralizability. Based on SDEC, we can axiomatize generalized

proportional rules without constantness.

Theorem 4.5.(1) Let n5. A rule satisfies SDEC-2 and OB if and only if it is a generalized

proportional rule.

(2) Let n5 and n-2k3. A rule satisfies SDEC-k and OB if and only if it is a

generalized proportional rule.

Proof. We provenf satisfies CON and DEC-k (kn-3) under the condition (1) or (2).

Then,nf must be a generalized proportional rule by theorem 4.2.

Step 1. Let k2 and nk+2. Ifnf satisfies SDCE-k, then nf satisfies CON.

Proof of Step 1. For nDEc ),( , consider a problem nDEc ),( such that E=E ,

01 c , 212 ccc , 2,1\, Nicc ii . Then, by SDEC-k,

n

i

n

i

kSnS Si

n

i

knkS

nS Si

n

i

kn

n

i

n

i EcfEcfk

n

CEcf

k

n

CEcf

3,,32,,323

),(),(21

),(21

),(LL

and,

n

i

n

i

kSS knSi

n

i

nn Ecf

k

n

C

Ecf

kSSNS

EcfEcf32,1 22

21 ),(

2

21),(

,2,1|

1),(),(

).,()(),(

2

21),(

,2,1|

121

32,1 22

EcfEcfEcfk

n

CEcf

kSSNS

nnn

i

n

i

kSS knSi

n

i

Hence,

n

i

n

i

n

i

n

i EcfEcf11

),(),( . Taking the same steps, fornDEc ),( , such that

E =E, )1,,1(,0 nici L and Nn cc , we can show

n

i

n

i

n

i

n

i EcfEcf

11

),(),( .

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19

Therefore,nf must satisfy CON.

Hence, in the case of (1),

n

f satisfies CON and DEC-k. Therefore,n

f must be ageneralized proportional rule by theorem4.2.

Step 2. Let n5 and k=n-2. I fnf satisfies SDCE-k, then nf satisfies SDEC-2.

Proof of Step 2. FornDEc ),( , consider a problem nDEc ),( such that E =E,

01 c , 212 ccc , 1,,3,0 nici L and )( 21 cccc Nn . Sincenf

satisfies SDEC-k and CON by step 1,

).,(),(),(),(),(),(),(),( 213131

21 EcfEcfEcfEcfEcfEcfEcfEcfnn

n

i

n

i

n

i

n

i

n

i

n

i

n

i

n

i

nn

Hence, )()( 21 nn ff is dependent only on Nc , E and )( 21 cc . For any Nji , ,

ji , we can similarly show )()( njn

i ff is dependent only on Nc , E and )( ji cc .

By step 1 and step 2, nf satisfies CON and DEC-2 in the case of (2). Therefore, nf

must be a generalized proportional rule by theorem4.2.

By, theorem 4.5, we have proved that generalized proportional rules can be

characterized based on SDEC-k if kn-2 with an exception of 4-person problems. In the

last of this section, we show an example of a rule of 4-person claims problems that

satisfies SDEC-2 and OB but is not a generalized proportional rule.

Example 4.6. Let k=1 and n=4. Let ipN

i

cc for each iN. Consider a rule nf such

that,

Epppppp

Ecfn

2

1)()()(),(

2

41

2

31

2

211

,

Epppppp

Ecfn

2

1)()()(),(

2

41

2

31

2

212

,

Epppppp

Ecfn

2

1)()()(),(

2

41

2

31

2

213

, and

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20

Epppppp

Ecfn

2

1)()()(),(

2

41

2

31

2

214

.

This rule

n

f satisfies SDEC-2 and OB, but is not a generalized proportional rule.

5. Game Theoretic AnalysisIn this section, we introduce a new coalitional form game which describes a

bankruptcy problem after a division is established as a law. Before defining it, we

review the previous coalitional game theoretic analyses of bankruptcy problems. Given

an n-person bankruptcy problem (c,E), ONeill [16] defined a coalitional form game

),( ),( EcvN as a bankruptcy game corresponding to a problem(c,E)6 such that,

0,max)( /),( SNEc cESv , for each SNS , .

Aumann=Maschler [3] explain this characteristic function )(),( Sv Ec indicates what

S can get without going to court. In the nature state where there is no rule to solve

claims problems, it is natural for the members of S to evaluate the worth of S by what S

can get without going to court. So, this game ),( ),( EcvN might be called a bankruptcy

game under the nature state.

However, once a rule is established as an act, the members of S will evaluate the worth

of S by what S can get by going to court, as is mentioned by Atsumi [2]. So, we should

redefine the worth of S after a rulenf is established as a law. In modern laws, claims

are in principle transferable among agents. Then, the members of each coalition may

manipulate the sum of their awards by reallocating their claims. Since we should take

such manipulations into consideration, we call a coalitional form game, which describes

a bankruptcy problem under an established rule, a claims reallocation game.

Applying Aumann=Peleg [3]s idea of coalitional form representation of strategic

games, we evaluate the worth of S by the following two ways.

6 ONeill [16] considered only bankruptcy problems. A bankruptcy problem is defined by

the pair RREcn),( such that EcN and 0Nc . Let

n denote the set of

n-person bankruptcy problems, thenn 0|),( NNn candEcRREc ,

which has a rich domain.

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21

One way is to evaluate the worth of S by the assurable level. Given a rulenf and a

problem (c,E), an aggregated award RxS is assurable for S , if

Si SNjjSiiniSSNSNSNjjSSSii Eccfxcccccc ),,(,,)(,,)( //// .

Clearlywhen

Si

SNjjSii

n

icccccc

S EccfxSNSNSNjjSSSii

,,minmax)/(|| ///

,

Sx is assurable for S. Then, we define an assurable(or)representation of a claims

reallocation game as follows.

Definiton 5.1.n

f EcvN ),(, is an assurable (or) representation of a claims reallocation

game corresponding to a problems (c,E) under a rulenf , if

Si

SNjjSii

n

icccccc

f

Ec EccfSvSNSNSNjjSSSii

n

,,minmax)()/(||

),(///

, for each SNS , .

Another way is to evaluate the worth of S by the unpreventable level. Given a rule

nf and a problem (c,E), an aggregated award RxS is unpreventable for S , if

SiSNjjSii

n

iSSSSiiSNSNSNjjEccfxcccccc ),,(,,,,)(

////.

Clearlywhen

Si

SNjjSii

n

icccccc

S EccfxSSSiiSNSNSNjj

,,maxmin)/(|| ///

,

Sx is unpreventable for S. Then, we define an unpreventable(or)representation of

a claims reallocation game as follows.

Definiton 5.2. nf EcvN ),(, is an unpreventable (or ) representation of a claimsreallocation game corresponding to a problems (c,E) under a rule

nf , if

Si

SNjjSii

n

icccccc

f

Ec EccfSvSSSiiSNSNSNjj

n

,,maxmin)()/(||

),(///

, for each SNS , .

It is clear that )()( ),(),( SvSvnn f

Ec

f

Ec . This fact indicates that claims reallocation

games are based on the more pessimistic evaluation of the worth of each coalition than

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22

claims reallocation games.

We should note that claims reallocation games are always superadditive but that

additional assumptions are required for claims reallocation games to satisfysuperadditivity.

Claim 5,3.For any nDEc ),( and nf , nf EcvN ),(, is superadditive.

Proof of Claim 5.3. For each ,,, TSNTS take Siic )( and Tiic

)( such that

Si

SNijSii

n

icccccc

Sii EccfArgcSNSNSNjjSSSii

,,minmax)()/(|| ///

Ti

SNijTii

n

icccccc

Tii EccfArgcTNTNTNjjTTTii

,,minmax)()/(|| ///

.

Then, for each TSNTSNTSNjj ccc //\ , ,

SiTSNjjTiiSii

n

i

SiSNjjSii

n

iccc

EcccfEccfSNSNSNjj

,,,,,min/)/(| ///

TiTSNjjTiiSii

n

i

TiTNijTii

n

iccc

EcccfEccfSNSNSNjj

,,,,,min/)/(| ///

.

Therefore,

)(,,min)()( ),(/|)(),(),( /// TSvEccfTvSv

n

TSNTSNTSNjj

nn f

Ec

TSiTSNjjTSii

n

iccc

f

Ec

f

Ec

,

by definition ofnf

Ecv ),( .

Claim 5.4. nf EcvN ),(, is superadditive if

(1) for any iN,n

if is concave on c.

(2) for any SN,Si

n

if is quasi-concave on Sc and for any iNn

if is continuous on c.

Proof of Claim 5.4. For each ,,, TSNTS take TSNjjc

/)( such that

TSi

TSNijTSii

n

icccccc

TSNjj EccfArgcTSTSTSiiTSNTSNTSNjj

,,maxmin)()/(||

////

And define correspondences SSSiiTTTiiS cccccc ||: and

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23

TTTiiSSSiiT cccccc ||: such that,

Si

TSNjjTiiSii

n

iccc

TiiSEcccfArgc

SSSii

),,(max)(/|

and

Ti

TSNjjTiiSii

n

iccc

SiiTEcccfArgc

TTTii

),,(max)(/|

.

In both case (1) and case (2), )(S and )(T are non-empty, convex, and upper

hemi-continuous. SSSii ccc | and TTTii ccc | are compact. Then, by

Kakutanis fixed point theorem (Kakutani [11]), there must exist Siic )( and Tiic

)(

such that )()(TiiSSii

cc

and )()(

SiiTTiicc

. Therefore,

Ti

TNjjTii

n

iccc

SiSNjjSii

n

iccc

f

Ec

f

Ec EccfEccfTvSvTTTiiSSSii

nn

,,max,,max)()(/|)(/|)(

),(),(

)(),( ),( TSvEcfnf

Ec

TSiNii

n

i

.

Next, we analyze claims reallocation games by considering their cores. Given a

coalitional form game (N,v), an award vector nNii

Rx is a feasible allocation if

)(NvxN . A feasible allocation Niix is dominated by S if there exists a feasible

allocation Nii

y such that Siyx ii , and )(),( Svynf

EcS . Since )(),( Svynf

EcS ,

the members of S can get at least Sii

y by themselves under the assurable scenario.

So, if a division plan Nii

x , which is dominated by S, is proposed, the members of S

must have an objection to this proposal.

The core of a claims reallocation game is the set of feasible allocations which is not

dominated by any SN. If an allocation which belongs in core is proposed, then

no coalition can improve their awards by themselves in the assurable level. Therefore,

under the assurable scenario, no coalition has an incentive to reject this allocation. In

this sense, an allocation which belongs in core is a stable consequences of the

bargaining among N. Since an claims reallocation games under a rulenf is super

additive, we can define the core ofnf

EcvN ),(, , which denotesnf

EcvC ),( , as

SNSNvxandNvxRxvC nnn f EcSf EcNnf Ec .),()(| ),(),(),( .

We can similarly define the core of a claims reallocation game.

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24

If there exists a problem (c,E) such that the core ofnf

EcvN ),(, is empty, then for each

feasible allocation, there must exist SN which can improve their awards byreallocating their claims. Then, it becomes difficult to make an agreement on how to

divide a resource under an established rule nf . Therefore, we require the following

axiom to division rules in order for claimants to bargain smoothly how to divide E.

Core NonEmptiness(-CNE). For any nDEc ),( , nf EcvN ),(, has the nonemptycore.

In the context of implementation theory, we can interpret core nonemptiness in

another way. Let us regardnf as a social choice function social planners want to

implement. I f there exists a problem (c,E) such that Nin

i Ecf ),(nf

EcvC ),( , then

there is no guarantee forNi

n

i Ecf ),( to be actually implemented even if this social

choice functionnf is established as a law, since there exists SN the members of

which have a objection to the allocation Nin

i Ecf ),( under a rule

n

f . Then,

Ni

n

i Ecf ),( should be in the core ofnf

EcvN ),(, in order to guarantee that

Ni

n

i Ecf ),( is actually implemented. Core nonemptiness is of course a necessary

condition fornf to be implemented in the core allocation.

Core Implementabil ity (-CI).For any nDEc ),( , Ni

n

i Ecf ),(

nf

EcvC ),( .

I f for each SN the members of S evaluate the worth of S by unpreventable level,

core nonemptiness and implementability can be rewritten as follows.

Core NonEmptiness(-CNE). For any nDEc ),( , nf EcvN ),(, has the nonemptycore.

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Core Implementabil ity (-CI).For any nDEc ),( , Nini Ecf ),(nf

EcvC ),( .

Since core core, we can easily check that -CNE implies -CNE and that

-CI implies -CI.

Next, we characterize generalized proportional rules based on the axioms related to

the core of claims reallocation games. Since -CNE is the weakest among these axioms,

we first study the axiomatization of the proportional rules based on -CNE.

Theorem 5.5. Let n3. A rule satisfies-CNE, CON and OB if and only if it is ageneralized proportional rule.

Proof. To proof this theorem, we make use of theorem 4.1. By theorem 4.1, it is

sufficient to prove that -CNE is equivalent to DEC.

Claim 5.6. Assume CON, then-CNE is equivalent to DEC.Proof. For any SN and (c,E), (c,E )

nD , if NN cc , E= E and SS cc , consider

(c,E

)

n

D such that ii cc

,iS and jj cc

,jN/S. Assuming CON,

n

f

must satisfy NAR. Then,

SNi

n

i

Ni

n

i

Si

n

i

Si

n

i EcfEcfEcfEcf/

),(),(),(),(

Si

n

i

SNi

n

i

Ni

n

i EcfEcfEcf ),()','()','(/

by CON and NAR. Therefore,

Si

n

i

SiSNjjSii

n

i

cccccc

f

EcEcfEccfSv

SNSNSNjjSSSii

n

),(,,minmax)()/(

||

),(///

,

for each SNS , .

So, an allocation Nii

Ecf ),( must be in the core of )(),( Svnf

Ec which implies the core

ofnf

EcvN ),(, is non-empty.

Next we prove the converse. Assumenf does not satisfy DEC. Then there must exist

Ni and nDEcEc ),(),,( satisfying ii cc , NN cc , and EE , such that

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26

),(),( EcfEcf ii . For each Nj , by CON,

EccfEcfjNv jNkkjnj

ccc

Ni

n

i

f

EcjNkNjNkk

n

,)(,min),())/(( )/(|)(

),()/()/()/(

)/( ),(,),( jNkn

k

n

jNk

n

k EcfEcfEcf .

And for Ni ,

EccfEcfiNv iNkkini

cccNi

n

i

f

EciNiNiNkk

n

,)(,min),())/(( )/(|)(

),()/()/()/(

)/( ),(,),(,),( iNkn

i

n

iNk

n

k

n

iNk

n

k EcfEcfEcfEcfEcf .

Therefore,

)(),(),(1

1))/((

1

1),()/(),(

NvEcfEcfn

kNvn

nn f

EcNj

n

jNk kNj

n

jNk

f

Ec

.

This inequity implies thatnf

EcvN ),(, is not balanced. So the core ofnf

EcvN ),(, must be

empty. (Bondereva [6] or Shapley [17].) Thus, -CNE must imply DEC.

As corollaries of theorem, we can conduct easily show the following statements.

Corollary 5.7. Let n3. A rule satisfies-CI, CON and OB if and only if it is ageneralized proportional rule.Corollary 5.8. Assume n3. A rule satisfies-CNE , CON and OB if and only if it is ageneralized proportional rule. -CNE can be replaced by-CI.

In the last of this chapter, we show two examples which show that -CNE is not

necessarily equivalent to DEC.

Example 5.9. Consider a rule nf such that Ec

cEcf

N

i

Ni

n

),( , for each iN. This

rule does not satisfy DEC. But since En

SSv

n

f

Ec

n

1)(),( for each SN and

En

NNv

n

f

Ec

n

1)(),( ,

E

nE

n

nn1

,,1

L must be in the core of nf EcvN ),(, .

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27

Example 5.10. Assume n=3. Consider a problem

E

EEE,

3

.

3

.

3

and a rule3f such

that

otherwise

ccifE

fN

ii

02

3 , for each iN.

Of course, this rule3f satisfies DEC. But, the core of

nf

EcvN ),(, must be empty since

ESvnf

Ec )(),( , for any SN,| S|2.

6. Concluding Remarks

In this paper, we focus on four axioms which a division rule should be satisfied: no

advantageous reallocation, decentralizability, core nonemptiness and core

implementability. Though these four axioms are derived from different requirements,

they are mathematically equivalent to one another if a division rule satisfies

constantness. This fact makes it possible to characterize generalized proportional rules

in different ways.

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