The proportional solution for rights problems

Xlathematical Social Sciences 15 (1988) 23 l-246 North-Holland

231

THE PROPORTIONAL SOLUTION FOR RIGHTS PROBLEMS

Youngsub CHUN*

Department of Economics, Southern Illinois University, Carbondale, IL 62901-4515, U.S.A.

Communicated by H. Moulin Received 13 June 1987 Revised 15 July 1987

Recently there have been several studies to provide axiomatic characterizations of solutions to rights problems. However, these studies do not give a satisfactory answer to the question why the proportional solution is the most widely used. This is the question addressed in this paper. To that purpose, we adopt the axiomatic approach; we suggest a set of axioms which a desirable solution should satisfy and we show that the proportional solution is the only solution to satisfy these axioms. Our main axioms are no advantageous reallocation and additivity. A solution satisfies no advantageous reallocation if no subgroup of claimants ever benefits by transferring parts of their claims between themselves. A solution satisfies additivity if it yields the same allocation whether the total estate is divided at once or in several steps.

Key words: Rights problem; proportional solution; no advantageous reallocation; additivity.

1. Introduction

How should the net worth of a bankrupt firm be allocated between its creditors? How should the fairness of a tax schedule be evaluated? How should the estate of a man be divided among his heirs? These problems are referred to the bankruptcy, the taxation and the estate problem respectively and can be expressed as one com- mon problem. So we call all these problems rights problems.

Rights problems have been studied for more than two thousand years. For example, the Talmud suggests an interesting solution, discussed in O’Neill (1982), to the estate problem. Recently there have been several studies to provide justifications of solutions to rights problems. In their beautiful paper, Aumann and Maschler (1985) show that the Talmudic solution coincides with the nucleolus of a coalitional game associated in a natural way with the estate problem, and that the solution has ap- pealing properties, especially consistency. Roughly speaking, consistency requires that what a solution recommends for a group should- be compatible with what it

* This paper is based on Chapter 4 of my Ph.D. dissertation at the University of Rochester. I am ex- tremely grateful to Professor William Thomson for his numerous comments. The comments from Pro- fessor HervC Moulin and a referee of this journal are also gratefully acknowledged. I retain, however, full responsibility for any shortcomings.

0165-4896/88/%3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland)

232 Y. Chun / Proportional solution

recommends for subgroups. Young (1987a, b) characterizes all solutions satisfying consistency as well as a few elementary axioms. These solutions include the Talmudic solution and the proportional solution - which divides the estate in proportion to the claims - as special cases.

However, these studies do not give a satisfactory answer to the question why the proportional solution is the solution most widely used to solve rights problems. This is the question addressed in this paper. To that purpose, we adopt the axiomatic approach; we suggest a set of axioms which a desirable solution should satisfy and we show that the proportional solution is the only solution to satisfy these axioms.

Our main axioms are no advantageous reallocation and additivity. A solution satisfies no advantageous reallocation if no subgroup of claimants ever benefits by transferring parts of their claims between themselves. A solution satisfies additivity if it yields the same allocation whether the total estate is divided at once or in several steps.

The paper is organized as follows. Section 2 contains some preliminaries and in- troduces the concept of a solution and basic axioms. In Section 3, we study the implication of imposing the axiom of no advantageous reallocation. In doing so, we derive axiomatic characterizations of the proportional solution. In Section 4, we study the implication of imposing the axiom of additivity. Once again, we derive axiomatic characterizations of the proportional solution.

2. The problem

Let N={l,..., n}, where n E trJ, ’ be the set of claimants, with generic element i.

Let ci?O be claimant i’s claim (or right) against an estate of value E.

Definition. An n-person rights problem is a pair (c,E) E @!+l such that c~(Ci)i~~ and CipNci >O. We denote by B” the class of n-person rights problems.

Note that we do not impose the condition ES CiarJCi. Although this condition is used by all four authors quoted earlier, we allow for the case E> CieNCie The imposition of the condition, however, would not affect our conclusions.

Definition. An n-person solution is a function F” : B”+ R” which associates with any problem (c,E) E B” a vector F”(c, E)=(F/‘(c,E))~~~. A solution is a list F=(F”),,..

Notation. For any coalition Tc N, we define c~= Ci, TCi, F;(c, E)- Ci, TFF(~, E) and so on.

’ N designates the set of natural numbers.

Y. Chun / Proportional solution 233

Let F be a solution. We impose the following axioms on f,

Pareto Optimality (PO). For all n and for all problems (c, E) E B”,

F;(c, E) = E.

Anonymity (AN). For all n, for all permutations Q : N-N, for all i = 1,. . . , n and for all problems (GE) E B”,

F,“(c,E) = F&(c’, E) where c'= (C~(i))ieN.

Continuity (CONT). For all n, for all problems (c, E) E B” and for all sequences {(ck,Ek)} of elements of B”, if (ck,Ek)-+(c,E), then F”(ck,Ek)+Fn(c,E).

The axiom PO requires that the whole value of the estate be distributed (even if cn;< E). The axiom AN says that the only relevant characteristics of the agents are their claims. The axiom CONT requires that a small change in the parameters of the problem cause only a small change in the outcome.

In all of our characterization theorems, solutions will be required to satisfy these three axioms.

Definition. The proportional solution is defined by

Fi”(c,E)=c’E cN

for all n, for all i= 1, . . . , n and for all problems (c, E) E Bn. (Recall that we assumed

that c,> 0.)

Definition. The egalitarian solution is defined by

F:(c,E)=E/n

for all n, for all i = 1, . . . , n and for all problems (c, E) E B”.

3. Transfer of claims

In this section, we discuss the transfer of claims. First, we introduce the main axiom.

No advantageous reallocation (NAR). For all n, for all problems (c, E), (c’, E’) E B” and for all coalitions TG N, if Ci = c,! for all i E N \ T, CT= c; and E = E’, then F;(c, E) = F;(c’, E’).

Let n, F” and (c,E) E B” be given and imagine the following scenario. First, all

234 Y. Chun / Proporrional solurion

claimants calculate their shares according to the solution F”. Then they consider possible transfers of claims between themselves and recalculate their shares. If there is some group T such that the total share of its members is greater than their total original share, then they have an incentive to transfer their claims (and maybe re- distribute their shares later). However, if a solution satisfies NAR, claimants never have such an incentive.

The phenomenon of advantageous reallocation has been analyzed in traditional economic setups by Gale (1974) and Aumann and Peleg (1974) and may threaten the competitive equilibrium arbitration method. The axiom of NAR was introduced by Moulin (1985a) to characterize the egalitarian and utilitarian solutions in quasi- linear social choice problems.

Now we characterize all solutions satisfying NAR in addition to PO, AN and CONT. Note that, for n = 2, NAR is just a restatement of PO and any solution satisfying PO also satisfies NAR.

Theorem 1. Let n be given. An n-person solution F” satisfies PO, AN, CONT and NAR if there exists a continuous function g” : IF?*+ IR such that

Fi”(c, E) s 2 E- ’ cN cN

(n - l)Ci- C Cj g”(cN,E) j+i I

(1)

for all i= I,..., n and for all problems (c, E) E B”. Also, for n 13, a solution satisfying PO, AN, CONT and NAR must have the form given in (1).

Proof. It can easily be checked that any solution of the form given in (1) satisfies the four axioms. To prove the second statement, let NE { 1, . . . , n}, where n r 3, and cm@,,..., c,,) be given. Also let F” be an n-person solution satisfying the four axioms. Now let c’= (c, + c2, 0, c3, . . . , c,). By NAR we have

F;(c,E)+F,“(c,E)=F,“(c’,E)+F,“(c’,E). (2)

Let c”= (c~,cN\~,O, . . . . 0) and T= (2, . . . . n}. By NAR applied to c, C” and T, we obtain

F;(c, E) = F;(c”, E).

This equation and PO imply

E-F;(c,E)=E-Ff(c”,E), i.e.,

Fi’(c, . ..-.cn,E)=F~(c,,c,\,,O,...,4E). (3)

Similarly, we have

Fi”(C,E)=F;(Ci,CN\i,O,..., O,E), for all i, (by (3) and AN),

F~(c;E)=F~(c,+Cz,cN\{,,2},0 ,... ,QE} (by (3)) and

>

(4) F;(c’,E)=F;(O,CN,O ,..., 0,E) (by (3) and AN).

k: Chun / Proportional solution 235

Using equations (4), we have from (2)

F;(C&V\t,O,..., O,E)+FI”(Cz,C,v,z,O, . . ..O.E)

=~;(C,+C~,C,-\(,,~jr4 . . . . O,E)+F;(O,C,,O ,... ,O,E). (5)

Let f” : lR3 -+ R and g” : R2 + !R be defined by

fn(Ci,CN,E)‘F;(CirCN\i,0,...,O,E)-F;(O,CNIO,...,0,E) (6)

and

gR(C/%j,E)=F;(O,cN,O ,..., 0,E). (7)

Taking (6) into account, we obtain from (5)

f”(C,, C,, E) +f”(C2, C,,,E) =f”(C, + C2, CN, I!?) for all Cl, C, and E.

Since nL 3, f” is additive with respect to its first argument for each CN and E.

Moreover, by CONT, it is continuous. Therefore, the theorem on Cauchy’s equation 2 applied to f n implies that there exists a continuous function h” : R2 --t R such that

~“(C~,CN,E)=C~~*(CN,E)* (8)

Substituting (8) into (6) and using (4) and (7). we obtain

F~(c,E)=cihR(c,v,E)+gn(cN,E) for all i=l,..., n. (9)

Summing up these equations and using PO, we obtain

CNhn(Cpj, E) + ng”(cN, E) = E,

or equivalently,

hn(CN,E)={E-ng”(CN,E)}/CN.

This equation and (9) yield

F,“(C,E)=(C~/CN){E-~S”(CN,E)} +g”(CN,E)

=(C~/CN)E-(I/CN) (n- I)Ci- C Cj gn(CN,E), j+i 1

as desired. 0

The proportional and the egalitarian solutions are members of the family characterized in Theorem 1. They are obtained by setting gn(cN, E) 50 and g”(CN, E) = E/n, respectively, for all n and for all problems (c, E) E B”.

Next we characterize the proportional solution by imposing the natural require- ment that an individual with zero claim should not affect the decision.

’ For a discussion of Cauchy’s equation, see Eichhorn (1978, pp. 4-7).

236 Y. Chun / Proporrional solution

Dummy (DU). For all nr2, for all i= 1, . . . , n - 1 and for all problems (c, E) E B”, if c,, = 0, then Fy(c, E) = F,?- ‘(c’, E) where c’ is obtained from c by deleting the last component c, .

Among the solutions characterized by Theorem 1, only the proportional solution satisfies the dummy axiom. The following result has been independently proven by Moulin (1985c, Theorem 5).3

Theorem 2. A solution satisfies PO, AN, CONT, NAR and DU if and only if it is the proportional solution.

Proof. It is clear that the proportional solution satisfies the five axioms. To prove the ‘only if’ part of theorem, let n r 3 and (c, E) E B” with c,, = 0 be given. By DU, we have

Fi”(c,E)=F;-‘(c 1 ,..., c,_,,E) for i=l,..., n-l.

Summung up these equations and adding Fi(c, E) to both sides, we obtain by PO,

F;(c,E)=O.

Recalling the definition of g”, given by (7), this result implies that

g”(+, E) = 0 for all n and for all (c, E) E B”.

So we have, as desired,

(10)

Fi”(C,E)=(Ci/C,)E for nr3, for all i= l,...,n and for all (c, E) E B”. (11)

For n = 2, let (c, E) E B2 be given. Also let (c’, E) E B3 such that c,! = Ci for i = 1,2 and c< = 0 be given. Now we derive F3(c’, E) from (11). Applying DU, we obtain the desired result.

Finally, we note that for n = 1, equation (9) is just a restatement of PO. 0

Remark 1. In Theorem 2, CONT can be replaced by the weaker condition that F” be continuous at at least one point of its domain. This change affects only the properties of g” in the statement of Theorem 1; g” is no longer required to be continuous. However, this change does not affect equation (8), so the conclusion of Theorem 2 still holds.

No Transfer Paradox (NTP). For all n and for all problems (c, E), (c’, E’) E B”, if C,IC;, c;lci for aN i#l, cN=ch and E=E’, then FF(c,E)IF,“(c‘,E’).

3 Moulin (198%) uses slightly weaker conditions, Individual Rationality and No Free Lunch, while I use DU and CONT. As a result, he can prove the only if part of the theorem for n~3.

’ An analogous condition has been introduced by Thomson (1987) under the name of disagreement point monotonicity in his study of solutions for bargaining problems.

Y. Chun / Proporiional solution 237

No Transfer Paradox (NTP). For aN n and for all problems (c, E), (c’, E ‘) E B”, if C, I c;, ~(5 Ci for all i # 1, c,~ = ch and E = E ‘, then F,“(c, E) 4 F,“(c’, E ‘).

The imposition of NTP instead of CONT changes only the properties of g” in the statement of Theorem 1; g” is no longer required to be continuous. As earlier, this change does not affect equation (8), so the conclusion of Theorem 2 still holds.

O’Neill (1982, p. 368) gave an axiomatic characterization of the proportional solution. He used PO, AN, CONT and DU together with the following axiom.

Strategy-proofness (SP). For all n, for all T, T’such that TU T’= N, Tfl T’= C#J and for all problems (c, E) E B”,

F;e(c’, c”, E) = F;,+;(c), c;,, E)

where c’= (ci)i, ry C"E (Ci)i, T' and 1 TI = t.

While NAR requires that transfers of claims are not beneficial to all claimants in- volved, SP requires that merging or splitting groups of claimants (an operation which changes the number of claimants) is never globally beneficial to the members of that group. As we will show in Lemma 1 below, SP implies NAR. Since the egalitarian solution satisfies NAR, but not SP, NAR does not imply SP. However, NAR together with PO and DU implies SP, as proved in Lemma 2 below.

The dummy axiom is redundant in the ‘only if’ part of O’Neill’s characterization theorem since PO, AN and SP together imply DU, as proved in Lemma 3 below.

In the following lemmas, we discuss the logical relations between the axioms used by O’Neill and those used by us.

Lemma 1. SP implies NAR.

Proof. Let n, TG (1, . . . , n} where T= @J, (c, E) E B”, (c’, E’) E B” such that ci= ci for all ieN\ T, c~=c$ and E=E’ be given. Let ITI =t. We have

F~(~,E)=F,“_:,+:((C~)~.N\*,CT,E) (by SP),

= F,“If,f:((C;)ie,\,, Cf, E) (by assumption on c and c'),

= F;(c’, E) (by SW.

Therefore,

F;(c, E) = F;(c’, E).

as desired. 0

Lemma 2. PO, NAR and DU together imply SP.


Proof. Let N={l,..., n}, T={r+l,..., n> where Orr<n--1, c=(c ,,..., c,,) and c’= (c,, . . . , c,, cT, 0, . . . ,O) be given. We have

F?(c, E) = I+(c’, E) (by NAN,

4”+,(c’,E)=&%?c I,... ,c,,cr,E) (by DU),

and

F,“(c’,E)=O for i=t+2,...,n (by PO and DU).

So we obtain

F;(c,E)=F:,f:(c~ ,..., c,,c~,E)r

as desired. q

Lemma 3. PO, AN and SP together imply DU.

Proof. Let n L 3 and (c, E) E B” with c, =0 be given. Suppose now that the claims of agents n and n - 1 are merged. We have

F,“_,(c,E)+F,“(c,E)=F,“_:(c, ,..., c,_,,E) (by SW.

Similarly, we obtain

F,“(c,E)+F,“(c,E)=F/‘-‘(c,, . . . . c,_,,E) for i= l,..., n- 1 (by AN). (12)

Summing up these (n- 1) equations, we have

n-1 n-l

C Fi”(c,E)+(n-~)F,“(c,E)=~~~~~-‘(c,,...,c,-,.E), i=I

which can be written as

f F;(c,E)+(~-~)F,“(c,E)=~&+(c,,...,c~_~.E). i=l i=l

Since PO implies that I:=‘=, F,f’(c,E)= Cy:; Fr-‘(cl, . . ..c._,,E)=E, we obtain

F,“(c,E)=O. (13)

Using (13), we have from (12)

Ff(c,E)=F/‘-‘(cl ,..., c,,_,,E) for i=l,..., n-l.

For (c, E) E BZ with c, = 0, we can obtain the desired conclusion by adding one

more dummy agent. q

The next lemma is a direct consequence of Lemma 2 and Theorem 2.

Lemma 4. The proportional solution satisfies SP.

Y. Chun / Proporrional soiurion 239

Now Lemmas 1, 3, 4 and Theorem 2 together imply

Theorem 3. A solution satisfies PO, AN, CONT and SP if and only if it is the proportional solution.’

Remark 3. The replacements of axioms suggested in Remarks 1 and 2 can also be carried out in Theorem 3.

4. Division of estate

In this section, we discuss the problem of estate division. First, we introduce the main axiom.

Additivity (AD). For all n, for all i = 1, . . . , n andfor aNproblems (c, E), (c’, E’) E B”, if c=c’ then

F,“(c,E+E’)=Fi”(c,E)+F,“(c,E’).

Let n, F” and (c, E), (c’, E’) E B” such that c = c’ be given. Since the claims are the same against E and E’, the claimants have two options: one is to solve the two separate problems (c,E) and (c, E’) and the other to solve the augmented problem (c, E + E’).6 If a solution satisfies additivity, then the claimants are indifferent between the two options and, thereby, the choice between two options has no impor- tance. A related axiom was introduced by Young (1987a, b) under the name of path independence.’ Path independence requires that, if the value of the estate is divided into two parts E and E’, then F”(c,E+E’) be equal to the sum of F”(c,E) and F”(c- F”(c, E), E’). The difference between additivity and path independence is whether claims are adjusted or not when the second problem is defined. The joint implications of additivity, path independence and no advantageous reallocation are explored in Moulin (1987).

Now we characterize all solutions satisfying AD in addition to PO, AN and CONT.

Theorem 4. Let n be given. An n-person solution F” satisfies PO, AN, CONT and

5 A direct proof of this theorem is given in Banker (1981) for cost sharing problems.

6 If we set the augmented problem to be (2c,E+E’), then the axiom is equivalent to weak linearity discussed later, in Remark 7.

’ An axiom in the same spirit was introduced by Kalai (1977) for bargaining problems under the name of step-by-step negotiations.


AD if and only if there exists a continuous function h” : R”+ IR such that, for all i= 1 , . . . , n and for all problems (c, E) E B”,

FT(c,E)=E+E n n t

(n-l)h”(cj,c_i)- c h”(cj,ci) I

(14) jti

where c__i is obtained from c by deleting the component Ci.

Proof. It is clear that any solution of the form given in (14) satisfies the four axioms. So we prove the ‘only if’ part of the theorem. Since F” satisfies AD and CONT, the theorem on Cauchy’s equation applied to Fr(c,E) implies that there exist continuous functions h,f’ : lR”-t IR such that

FF(c,E)=Eh,!‘(c) for all i= 1, . . . . n. (15)

By AN, it is meaningful to define

hl(C)~~‘(Ci,C_i) for all i=l,...,n.

Using this definition and summing up the equations in (15), we have

F,$(c, E) = E C h”(Ci, C-i). ieN

By PO, we have

l= C h”(Ci,C_j)e ieN

Using (16), we obtain from (15)

F:(c,E)=Eh”(c,,c_i)+~ l- C h”(Cj,c-j) jeN 1

=E+E n n I

nh’(Ci,C_i)- C h"(Cj,C_j) jeN 1

=E+F ~-l)h”(ci,C-i)-j~ih”(Cj,C-j)l, J

as desired. 0

The proportional and the egalitarian solutions characterized in Theorem 4. They are obtained by

(16)

are members of the family setting h”(ci, C_i)‘Ci/CN and

h”(ci, c_i) = 0, respectively, for all n, for all i= 1, . . . , n and for all problems (c,E)EB”.

Next we characterize the proportional solution by imposing an additional axiom. Suppose that the estate E is equal to the sum of the claims cN. Under this circum- stance, it is natural to require that each agent gets exactly the amount that he claims, as formalized below.


Exact Clearance (EC). For all n, for aN i = 1, . . . , n and for all problems (c, E) E B”, if c,~=E, then F,“(c,E)=c,.

Among the solutions characterized by Theorem 4, only the proportional solution satisfies the exact clearance axiom. This result still holds even if PO and AN are dropped from the list.

Theorem 5. A solution satisfies CONT. AD and EC if and only if it is the proportional solution.

Proof. It is clear that the proportional solution satisfies the three axioms. So we prove the ‘only if’ part of the theorem. Let F” be an n-person solution satisfying the three axioms. By AD and CONT, as explained earlier before deriving equations (15), there exist continuous functions hi”: IT?“- IR such that

FF(c,E) = Eh,“(c) for all i= 1, . . . ,n.

By setting E = cN and applying EC, we have

(17)

C;= CNh,!(C),

or equivalently,

h,!'(C)=Ci/CN for all c and for all i= 1, . . ..n. (18)

By substituting (18) into (16), we obtain

F;(c, E) = SE for all n, and for all i= 1, . . . . n, for all (c, E)E B”, cN

as desired. 0

Remark 4. In Theorem 5, CONT can be weakened as in Remark 1.

Remark 5. In Theorem 5, CONT can be replaced by the following axiom, also used by Moulin (1985b) in his study of quasi-linear social choice problems.

Cost Monotonicity (CM). For all n, for all i = 1, . . . , n and for all problems (c, E), (c‘,E’)eB”, ifc=c’and ErE’, then F,?(c,E)cF/‘(c,E’).

The imposition of CM instead of CONT changes only the properties of h” in the statement of Theorem 4; h” is no longer required to be continuous. However, this change does not affect equation (17), so the conclusion of Theorem 5 still holds.

Remark 6. In Theorem 5, EC can be replaced by the following axiom, introduced by Aumann and Maschler (1985).


Self-duality (SD). For all n, for all i= 1, . . . , n and for all problems (c, E) E B” such that c&E,

ci-F”(c,C,C.-E)=F~(c,E).

In fact, it can easily be proved that (i) AD and SD together imply EC, and

(ii) AD and EC together imply SD. However, EC and SD together do not imply AD. The Talmudic solution in Aumann and Maschler satisfies EC, SD, but not AD. That the solution violates AD can easily be shown from the numerical example in Aumann and Maschler (1985, p. 196).

In traditional bargaining theory, the additivity axiom is closely related to the following axiom.

Linearity (LI). For all n. for all i= 1, . . . , n, for all problems (c, E), (6, E’) E B” and

for aflIER such that 011~~1, ifc=c’, then

F,“(c,%E+(l -QE’)=AFi”(c,E)+(l -QFi”(c,E’).

However, LI is not sufficient to characterize the proportional solution, even in con- junction with PO, AN, CONT and EC.

Theorem 6. Let n be given. An n-person solution F” satisfies PO, AN, CONT and LI if and only if there exist continuous functions h” : I?” + IT? and g” : R” + iI? such that

E E Fi”(c,E)=- + -

n n i (n - l)h”(citC_i)- C h”(cj, c-i)

j+i

+l n

(n-l)g”(Ci,C_i)- C g"(Cj,C_j)

jti (1%

for all i= 1, . . . . n and for all (c, E) E B”.

Proof. It is clear that any solution of the form given in (19) satisfies the four axioms. So we prove the ‘only if’ part of the theorem. Since F” satisfies LI and CONT, the theorem on Jensen’s equation’ applied to F/‘(c, E) implies that, for each i= 1, . . . , n, there exist continuous functions hi” : lR”-+ IR and g: : IR” + R such that

F-i”@, E) = Eh;(c) +gi”(c). (20)

By AN, it is meaningful to define

hl(C)~h’(Ci,C_i) for all i=l,...,n

and

gl(C)~gg”(Cj,C_i) for all i= l,...,n.

* For a discussion of Jensen’s equation, see Dhombres (1979, pp. 2.18-2.22).


Summing up the equations in (20), we have

F#,E)=E c h”(c;,c_j)+ c g”(c;,c-j). iaN iE.V

By PO, this gives

E=E C h”(Ci,C-i)+ C g”(CivC_i). itN i6N

Using (21), we obtain from (20)

(21)

F’f(C,E)=Eh”(Ci,C_i)+g”(Ci,C_i)+ i L

E-E C h”(Cj$C-j)- C g”(Cj*C-j) jeN jeN

=t+ E

I

tlh”(Ci*C_i)- 1 h”(CjSc_j) jGN 1

+ i I

ng”(Ci*C_i)- C g”(Cj,C_j) jsN I

=- f+f[(n-l)h”(Ci,C-i)- C h”(Cj,C_j)) j+i

(tl- l)g”(Ci, C- j)- C g”(Cj, C-j) 9 j+i 1

as desired. Cl

Remark 7. In Theorem 4, LI can be weakened to the following condition.

Weak Linearity (W.LI). For all n, for at1 i = 1,. . . , n and for all problems (c, E), (c’, E’) E B”, if c = c’, then

F;(c,(~/~)(E+E’))=(~/~)F~“(c,E)+(~/~)F,”(c,E’).

Note that W.LI and CONT are sufficient to derive equation (20).

Next we impose EC to get a further characterization result.

Theorem 7. Let n be given. An n-person solution F” satisfies PO, AN, CONT, L1 and EC if and only if there exists a continuous function g” : IR” + R such that

F,“(c,+C’E+: (n- l)g”(C;,c-i)- C g”(Cj,C-j) (22) cN n j+i

for all i= 1, . . . . n and for all (c, E) E B”.

Proof. It is clear that any solution of the form given in (22) satisfies the five axioms. So we prove the ‘only if’ part of the theorem. Let n, (c, E) E B”, (c’, E’) E B” with c=c’ and E’=cN be given. Also let F” be an n-person solution satisfying the five

244 Y. Chun / Proportional solulion

axioms. By LI, we have

F~(c,~E+(~-~)c~)=~F~(c,E)+(~-~)F’~(c,c~) for all i=l,...,n.

By EC, this gives

Fp(c,AE+(l-A)cN)=AF~(c,E)+(l-A)ci for all i=l,...,n.

By Theorem 6, this equation implies that, for all i= 1, . . . , n.

${~E+(1-i.)cN}+~{rlE+(l-r\)cN}~-l)h”(c~~C_~)- C hn(Cj,C-j)] j+i

+ ’ n I

(n - l)g”(Cj, C-j)- C g”(cj9 c-j) jti 1

=A -+

[ t

f i (n-l)h”(Ci,C-i)- C h”(cj,c-j)] jti

+’ (n-l)g”(Ci,C_i)- C g”(Cj*C_j) +(l-A)ci* n I jti 11

or equivalently,

~(1_1)CN+~(l_A)CN I

(n-l)h”(ci,c-i)- C h”(cjtc-j) j+i I

+i(l-l) (n-l)g”(Ci~C_i)- C g”(Cj,C_j) =(l-/.)Ci* n j+i

This yields

(n- l)h”(Ci,C_i)- C h”(Cj,C-j) j+i

(n-l)g”(Ci,C_i)- C g”(Cj,C-j) . (23) j+i

Using (23), we have from (19)

F;(c,E)=-+ f ~(~)[~ci-cN-~~-l)g~(ci,c-i)-~ign(cj,c-j)~~

+’ n

(fZ- l)g”(Ci,C_i)- C g”(cj*c-j) j+i 1

,C’E+i (n-l)g”(Ci,C-i)- C g”(Cj,C_j) c.v n j*i

as desired. Cl

As we have shown in Theorem 7, PO, AN, CONT, LI and EC are not sufficient

Y. Chun / Proporrional solution 24s

to characterize the proportional solution. However, adding a mild axiom to this list will yield such a characterization.

Suppose that the estate E is equal to zero. It seems very natural to require that every claimant gets nothing. Therefore, we impose the following axiom.

Trivial Clearance (TO For al/ n, for a/l i = 1, . . . , n and for all problems (c, E) E B”, if E = 0, then

F;(c,E)=O.

Among the solutions characterized by Theorem 7, only the proportional solution satisfies TC. This result still holds even if PO and AN are dropped from the list.

Theorem 8. A solution satisfies CONT, LI, EC and TC if and only if it is the proportional solution.

Proof. It is immediate that the proportional solution satisfies the four axioms. So we prove the ‘only if’ part of the theorem. Let F” be an n-person solution satisfying the four axioms. By LI and CONT, as explained earlier before deriving equations (20), there exist continuous functions hl : IT?” -+ R and g: : IR” + IR such that

F,“(c,E)=Eh/(c)+g,!‘(c) for all i=l,...,n. (24)

First, we apply TC to equations (24) by setting E=O and obtain

g:(c)=0 for all c and for all i= 1, . . ..n. (25)

Next, we apply EC to equations (24) by setting E =c, and obtain

Ci=CNh/(C)+gr(C), (26)

which, together with equations (29, imply

h/‘(c)=3 for all c and for all i=l,...,n. (27) cN

By substituting (25) and (27) into (24), we have

F,“(c,E)=‘E for all n, for all i=l,...,n, and for all (c,E)EB”, cN

as desired. Cl

Remark 8. Remarks 1, 5, 6 and 7 apply to Theorem 8.

Remark 9. From the proofs of Theorems 5, 6, 7 and 8, we can easily show the following:

(i) CONT and AD together imply LI, and (ii) CONT, LI and TC together imply AD.


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The proportional solution for rights problems

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