Bargaining problems with claims

Mathematical Social Sciences 24 (1992) 19-33

North-Holland

Bargaining problems with claims

Youngsub Chun

Department of Economics, Seoul National University, Seoul 151-742, Korea

William Thomson

Department of Economics, University of Rochester, Rochester, NY 14627, USA

Communicated by J. Harsanyi

Received 26 June 1989

Revised 8 October 1991

We enrich the traditional model of bargaining by adding to the disagreement point and the

feasible set a point representing claims (or expectations) that agents may have when they come to the

bargaining table. We assume this point to be outside of the feasible set: the claims are incompatible.

We look for solutions to the class of problems so defined, using the axiomatic method. We follow the

various approaches that have been found useful in the classical axiomatic theory of bargaining. We

successively consider how solutions respond to certain changes in (i) the feasible set, (ii) the disagreement

point and the claims point, and (iii) the number of agents. In each case we formulate appropriate axioms

and study their implications. Surprisingly, each of these approaches leads to the same solution:

it is the solution that associates with each problem the maximal point of the feasible set on the line seg-

ment connecting the disagreement point to the claims point. We name this solution the proportional solution.

Key words: Bargaining problems; claims point; proportional solution.

1. Introduction

A firm goes bankrupt and its net worth is not sufficient to pay back all of

its creditors. How should it be divided among them? In this paper we con-

sider such situations and propose ways in which the creditors’ claims should be

taken into consideration in the computation of the amount attributed to each of

them. Our objective is to identify desirable ways of performing this division.

We follow the axiomatic approach: we formulate requirements on how the

division should be carried out and we search for rules that satisfy all the re-

quirements.

Cqrrespondence to: William Thomson, Department of Economics, University of Rochester, Rochester,

NY 14627, USA.

01654896/92/$05.00 0 1992-Elsevier Science Publishers B.V. All rights reserved

20 Y. Chun, W. Thomson / Bargaining problems with claims

Such bankruptcy problems (and taxation problems,’ which are mathematically

equivalent) have been studied by O’Neill (1982), Aumann and Maschler (1985),

Moulin (1987), Young (1987, 1988), Chun (1988), and Curiel, Maschler and Tijs

(1988). These authors have assumed each agent’s utility to be a linear function of

the monetary amount he is awarded. In the language of game theory, this is a

‘transferable utility’ assumption, i.e. the set of utility vectors available to the

creditors is a hyperplane normal to a vector of equal coordinates. Here, we

generalize the model to cover the ‘non-transferable utility case’: we allow the feasi-

ble set to have a more general shape. We also specify in our model a ‘reference

point’ from which utility gains can be measured. This point is implicitly taken to

be the origin in the above-mentioned papers.

Our domain can alternatively be obtained by enriching the domain defined by

Nash (1950) in his study of bargaining through the addition of the claims point and

for that reason we refer to the problems we analyze as ‘bargaining problems with

claims’. Consider a labor-management conflict: there is a set of feasible alternatives

consisting of all the utility vectors attainable by some division of the profits of the

firm. The two parties have to agree on one of them. If they do not agree, a strike

or a lockout may occur. This is the ‘disagreement point’. Here, it will play the role

of what we referred to earlier by the deliberately neutral term of ‘reference point’.

In addition, labor and management come to the negotiation table with certain ex-

pectations, or with certain claims. These expectations may have been formed by ob-

serving the resolution of similar conflicts in related industries but may not be

mutually compatible in the actual situation; the claims may represent commitments

made to the agents in earlier negotiations which, because of changes in the industry

that may have adversely affected the feasible set of the firm, cannot all be honored

any more.

The theory of bargaining will serve as a very convenient template for the theory

of bargaining with claims that we would like to initiate here (see Thomson, forth-

coming, for a review of this literature). Axiomatic studies of the bargaining problem

can be divided into three groups. Starting from Nash (1950) himself, the traditional

literature has been mainly based on axioms pertaining to changes in the feasible set.

More recently, however, the focus has shifted to the dependence of solutions on the

disagreement point (Thomson, 1987; Chun and Thomson, 1990a), and on the

number of agents (Thomson, 1983a, 1983b). We will show that each of these ap-

proaches can be fruitfully adapted to develop a theory of bargaining with claims.

Indeed, the mathematical structure of the class of problems under study here re-

mains sufficiently close to that of bargaining theory, and the various points of view

that have proved useful there will prove equally useful here. The overall picture will,

however, be quite different. Indeed, the various developments that have occurred

in bargaining theory in the last decade have not permitted the identification of a

’ In a taxation problem, the issue is how to raise a given total amount by assessing taxpayers as a function of their incomes.

Y. Chun, W. Thomson / Bargaining problems with claims 21

d l

Fig. 1.

single solution as being best,2 whereas here our various angles of attack all lead to

the same solution. This is the solution that selects the maximal feasible point on the

line connecting the disagreement point with the claims point. Therefore, one can

much more confidently than would perhaps have been expected, advocate one solu-

tion as being the most desirable.

2. Preliminaries

An n-person bargaining problem ivith claims, or simply a problem, is a triple

(S, d, c), where S is a subset of !I?‘, d and c are points in W, such that (see Fig. 1)

(1) S is convex and closed,

(2) there exist PEST+ and r E $2 such that for all XE S, C pixis r,3 (3) S is comprehensive, i.e. for all XES and for all ye ‘W, if y~x, then YES,

(4) there exists XE S with x> d, and

(5) c$S, crd and csa(S), where a,(S)=max{xiIxES} for all i if this max-

imum exists and a;(S) = 03 otherwise.

’ Three solutions (and variants of these solutions), however, reappear in study after study: they are

the Nash (1950) solution, the Kalai-Smorodinsky (1975) solution, and the egalitarian (Kalai, 1977)

solution.

3 Vector inequalities: given x,y~%“, xgy, xry, x>y, !J?~+=(~E%“) x>O}.


Fig. 2.

Let Z” be the class of all n-person problems.

The notation S designates the feasible set. Each point x of S is a feasible alternative. The point d is the disagreement point4 and c is the claims point. The in-

tended interpretation of (S, d, c) is as follows: the agents can achieve any point x of

S if they unanimously agree on it; the coordinates of x are the utility levels,

measured in some von Neumann-Morgenstern scales, attained by the n agents

through the choice of some joint action; the point d is an alternative at which the

agents end up in the case of no agreement, or more generally a reference point from

which they find it natural to measure their utility gains in order to evaluate a pro-

posed compromise; each coordinate of the claims point may represent a promise

made to the corresponding agent, or his understanding of what the others had

agreed that he would get; or an agreement, perhaps a contract, made before some

unfavorable circumstances led to changes in the feasible set that make it impossible

to execute. It is assumed that claims are made in good faith, and that although they

cannot all be honored, they should be taken into account in the determination of

the final compromise. What will be a reasonable compromise?

A solution is a function F: Z” + !Xn that associates with each (S, d, c) EZ” a

unique point of S, F(S, d, c), called the solution outcome of (S, d, c). The following solution, illustrated in Fig. 2, will play the central role in our analysis.

Definition. The proportional solution, P: for all (S, d, c) E .Z”, P(S, d, c) is the max-

imal point of S on the segment connecting d and c.

4 Even though d could be interpreted in a variety of ways, we keep its naming from standard bargain-

ing theory.


The term ‘proportional solution’ may partially obscure the fact that the solution

appears less closely related to the solutions already known in bargaining theory as

proportional (the utility gains at the compromise are proportional to some fixed vec-

tor) than to the Kalai-Smorodinsky solution (for which the utility gains at the com-

promise are proportional to what they would be at the ideal point, the point whose

ith coordinate is the maximal utility agent i could obtain in the part of S dominating

d). However, in addition to the fact that the Kalai-Smorodinsky solution can be

seen as a ‘normalized’ version of the egalitarian solution, which is the symmetric

member of the proportional family, we will propose several characterizations of our

solution that are patterned after characterizations of the egalitarian solution.

Perhaps the most important property satisfied by both our solution and the

Kalai-Smorodinsky solution, but not by the egalitarian solution, is scale invariance

(see below for the statement of this condition). Therefore we prefer thinking of it

as a hybrid between the egalitarian and Kalai-Smorodinsky solutions appropriate

for the domain at hand.

We will offer several characterizations of our solution. They will involve a

number of axioms adapted in a straightforward way from the standard theory of

bargaining. These axioms are presented first. In addition, a variety of specific addi-

tional axioms will be introduced in each of the subsequent sections.

Weak Pareto Optimality (WPO). For all (S, d, c) EC” and for all XE 91fl, if

x>F(S, d, c), then xe S.

This axiom requires that there be no feasible alternative that all agents strictly prefer

to the solution outcome.

Let WPO(S)={xES 1 Vx’E!I?‘“, x’>x= x’$S} be the set of weakly Pareto- optimal points of S. Similarly, let PO(S)= {XE S / VX’E Ytn, x’>x * x’$ S} be the

set of Pareto-optimal points of S.

Symmetry (SY). For all (S, d, c) EC”, if for all permutations rc : { 1, . . . , n) +

II, . . ..n). S=TC(S), d=n(d), and c=n(c),’ then <(S, d, c) =Fj(S, d, c) for all

i,jE(l,..., n].

This axiom requires that if the problem is invariant under all exchanges of agents,

then all agents should be treated identically.

Boundedness (BDD). For all (S, d, c) E .Z’“, ds F(S, d, c) 5 c.

This axiom requires that no agent be worse off at the solution outcome than at the

disagreement point (a condition which is often considered in bargaining theory

under the name of ‘individual rationality’), and that no agent be better off at the

solution outcome than at the claims point.

5 ~(x)=(x,~,,),=, ,..., n and n(S)-{x’EW 1 3x~‘W’such that x’=n(x)}.


In what follows convergence of a sequence of sets is evaluated in the Hausdorff

topology.

Continuity (CONT). For all sequences {(S”, d”, c”)> of elements of 2” and for all

(S, d, c) ~27, if S” --t S, d” -+ d and c” --t c, then F(S”, d”, c”) -+ F(S, d, c).

This axiom requires that a small change in the problem never cause a large change

in the solution outcome.

Let A” be the class of transformations I : W --f (31” defined as follows: for each

ie N, there exist USE%++ and bi E ‘$3 such that for all XE Sn, Ai = U;X; + b;. Given

SC%“, A(S)={~E!I?I 3x~S with y=l(x)}. Note that if (S,d,c)eF, then

(A(S), A(d), A(c)) E 2’ as well.

Scale Invariance (SC.INV). For all (S, d, c) E Z” and for all A E A”, F@(S), A(d), A(c)) =

WV, d, c)).

This axiom says that subjecting a problem to a positive linear transformation

acting independently on each coordinate leads to a new problem that should be

solved at the image under this transformation of the solution outcome of the

original problem. It is justified by the fact that agents’ utilities are of the von

Neumann-Morgenstern type. But note that it precludes basing the recommendation

for a compromise on interpersonal comparisons of utility.

It is easily verified that all of the above conditions are satisfied by the propor-

tional solution. In the two-person case, the proportional solution satisfies Pareto

optimality:

Pareto Optimality (PO). For all (S, d, c) E 27’ and for all x E %‘I, if x 2 F(S, d, c),

then xe S.

The proportional solution also satisfies Pareto optimality on the domain of n-

person ‘strictly comprehensive’ problems (problems (S, d, c) such that for all x, y E S

with xry there exists z ES with z>y). To obtain Pareto optimality over the whole

of Z‘“, we could follow standard procedure@ and define a lexicographic extension

of the solution. This extension would not, however, satisfy continuity or bounded-

ness. Other appealing properties discussed later would also fail to be satisfied.

The following notation and terminology will be used frequently. Given SC%‘,

int(S) is the interior of S, and r.int(PO(S)) is the relative interior of PO(S) with

respect to WPO(S). Given x, y E !R2” such that x#y, l(x, y) is the line passing through

x and y. Given x1, . . . ,xk E ‘W, cch{x’, . . . , xk} is the smallest convex and compre-

hensive set containing these k points, and given S’, . . . , SkCS2”, cch{ S ‘, . . . , Sk} is

the smallest convex and comprehensive set containing these k sets.

6 See, for example, Imai (1983).


3. Axioms concerning changes in the feasible set

In this section we consider certain changes in the feasible sets and formulate

axioms specifying how the solution outcome should respond to these changes. These

requirements, together with a few of the standard conditions, lead to our first char-

acterization of the proportional solution. The axiom and the proof presented first

are adapted from Kalai (1977).

Strong Monotonicity (ST.MON). For all (S, d, c), (S’, d’, c’) with (d, c) = (d’, c’) and

ScS’, F(S’,d’,c’)zF(S,d,c).

This axiom says that an expansion of opportunities, other things being equal,

benefits all agents.

Theorem 1. The proportional solution is the unique solution satisfying WPO, SY, SC.INV, and STMON.

Proof. It is obvious that the proportional solution satisfies the four axioms. Con-

versely, let F be a solution satisfying the four axioms. Let (S, d, c) ~2’” be given.

First, suppose that P(S, d, c) E PO(S). By SC.INV, we can assume that d= 0 and

c=(l,..., 1). Then P(S,d,c)=(a ,..., (x)=x for some (Y. Let S’s r)n(S), where the

intersection is taken over all permutations rc : { 1, . . . . n} + { 1, . . . ,n>. Note that

(S: d, c) EZ’“. Since (S’, d,c) is invariant under all exchanges of agents and

XE PO(S’), we have by WPO and SY, F(S’, d, c) =x. Now S’CS and by ST.MON,

F(S’, d, c) 5 F(S, d, c). Since XE PO(S), we obtain F(S, d, c) =x = P(S, d, c), as desired.

To conclude in the case P(S, d, c) E WPO(S) \ PO(S), we apply a continuity argu-

ment involving the earlier conclusion and ST.MON. Q.E.D.

Next we analyze situations where the feasible set may be uncertain. The require-

ment that we will consider for that case is adapted to our model from an axiom of

bargaining theory proposed by Perles and Maschler (1981) and further studied by

Peters (1986).

Concavity with Respect to the Feasible Set (S.CAV). For all (S,d, c), (S’,d’,c’) with (d,c) = (d’,c’) and for all a~ [O,l], F(aS+ (1 -a)S’,d,c)zaF(S,d,c)+ (1 - a)F(S’, d’, c’). (Note that (as+ (I- a)S’, d, c) is a well-defined element of 2’“).

This axiom can be motivated on the basis of timing of decisions. Consider agents

today, who tomorrow will face one of two equally likely problems (S,d,c) and

(S’, d, c) that differ only in their feasible sets. They have two options: they can wait

until tomorrow for the uncertainty to be lifted and solve then whatever problem has

come up, or they can make contingent contracts. Since the set of feasible utility vec-

tors obtained by such contracts is simply the ‘average’ of S and S’, the expected


payoff vector associated with the second option isF((S+ S/)/2, d,c). On the other hand,

the expected payoff vector associated with the first option is [F(S, d, c) + F(S’, d, c)]/2.

This point is typically Pareto-dominated by a point of (S + S’)/2 so that it would

be desirable that the agents avail themselves of the possibility of writing contingent

contracts. Imposing S.CAV on the solutions guarantees that all agents will benefit

from such contracts.

We are now ready to state a characterization of the proportional solution based

on S.CAV. Its proof is a simple adaptation of proofs in Aumann (1985) and Peters

(1986).

Theorem 2. The proportional solution is the unique solution satisfying WPO, SY, CONT, SC.ZNV, and S. CA V.

Proof. It is obvious that the proportional solution satisfies the five axioms. Con-

versely, let F be a solution satisfying all of the axioms. Let (S, d, c) E 2’” be given.

By SC.INV, we can assume that d = 0 and c = (1, . . . , 1). Then P(S, d, c) = (a, . . . , a) =x*

for some a>O.

(i) There exist p E ‘!I?: + and rE 33 such that, for all XE WPO(S), C pixj=r. Let

S’= nrc(S), where the intersection is taken over all permutations of { 1, . . . , n}. Note

that (S’, d, c) E 27’ and x* E PO(S’). Since (S’, d, c) is invariant under all exchanges

of agents, we have by WPO and SY, F(S’, d, c) =x*. Since pS’+ (1 - /?>S = S for

O<p< 1, by S.CAV, F(S,d,c)zPF(S’,d,c)+(l -P)F(S,d,c), which implies that

F(S, d, c) z F(S’, d, c). Since F(S’, d, c) =x* E PO(S), we conclude that F(S, d, c) = x* = P(S, d, c), as desired.

(ii) P(S, d, c) =x* E r.int(PO(S)). Since x* E r.int(PO(S)), there exists a supporting

hyperplane at x* with a normal vector p* E 32: + . Let T= (X E Sn / C PjFXiz C p,*x:>.

From the previous step, F(T,d,c) =x*. Since pS+(l-p)T=T for O<p< 1, by

S.CAV, F(T,d,c)zPF(S,d,c)+(l-P)F(T,d,c), which implies that F(T,d,c)s F(S, d, c). Since F(T, d, c) =x* E r.int(PO(S)), we conclude by WPO that F(S, d, c) = x* = P(S, d, c), as desired.

(iii) For the case when P(S, d, c) $ r.int(PO(S)), we apply CONT and use the con-

clusion of case (ii). Q.E.D.

4. Axioms concerning changes in the disagreement point and the claims point

In this section we consider changes in the disagreement point and claims points.

Disagreement Point Quasi-Concavity (d.Q-CAV). For all (S, d, c), (S’, d’, c’) E 2”, for all i and for all (r E [O, 11, if (S, c) = (S’, c’), then &(S, ad + (1 - a)d’, c)~ min{F;(S, d, c),F,(S’, d’,c’)). (Note that (S, ad+ (1- a)d’, c) is a well-defined ele-

ment of Z” .)


This axiom, adapted from a condition introduced by Chun and Thomson (1990b) for

bargaining problems, can be motivated on the basis of timing of bargaining, as in

the case of S.CAV, but this time the uncertainty pertains to the disagreement point.

Another difference is that instead of requiring that agents be unanimous in preferring

contingent contracts (this would result in the payoff vector F(S, (d+ d’)/2, c)), we

impose the weaker condition that no agent be worse off under contingent contracts

than he would be at the solution outcome of the worse for him of the two consti-

tuent problems. The reason for using the weaker condition is that the exact counter-

part of S.CAV is actually incompatible with WPO and BDD, as shown later. Yet,

the condition we will use does provide agents some incentive to reach an agreement

early.

Mid-Point Domination (MPD). For all (S, d, c) E _Z”‘, either F(S, d, c) 2 (c + d)/2 or

F(S, d, c) 5 (c + d)/2.

Instead of taking all the features of the problem they face into account, agents may

find it natural to evaluate proposed compromises with reference to some summary

information. An appealing way of summarizing the important features of a problem

is to take the average of the disagreement point and the claims point. We will require

that all agents end up above this average or that they all end up below. This is the

content of MPD, an axiom which is adapted from conditions used in bargaining

theory (Moulin, 1983; Thomson, forthcoming).

We now turn to the results. The proof of Lemma 1 is similar to the proof of Lem-

ma 2 in Chun and Thomson (1990a).

Lemma 1. Let F be a solution satisfying WPO, BDD, and d.Q-CA V. Also let (S, d, c) E Z” such that F(S, d, c) E PO(S) be given. Then for all d’ E [d, F(S, d, c)[, F(S, d: c) = F(S, d, c).

Proof. Since F satisfies BDD, (S,d’,c) ~2:” for all d’E [d,F(S,d,c)[. Let d’E ]d,F(S,d,c)[ be given. Let a~]9 11 be such that d’=ad+(l -d)F(S,d,c), and {cr”}

be a sequence of elements of IO, l[ be such that cz” < d for all v and (Y” --f d. Also,

let d”=(d’-cx”d)/(l -cx”) for all v. Note that (S,d”,c)~z” for all v. By d.Q-CAV,

fi(S,d’,c)zmin{I;;:(S,d”,c), e(S,d,c)} for all i and for all v. As v+ce, d”--+ F(S, d, c), and since F(S,d, c) E PO(S), it follows from BDD that F(S, d”, c) -+ F(S, d, c). Therefore we obtain F(S, d’, c) 2 F(S, d, c). Since F(S, d, c) E PO(S), we

conclude by WPO that F(S, d’, c) =F(S, d, c). Q.E.D.

Theorem 3. The proportional solution is the unique solution satisfying WPO, BDD, CONT, MPD, and d.Q-CA V.

Proof. It is obvious that the proportional solution satisfies the five axioms.


Conversely, let F be a solution satisfying the five axioms. Let (S, 4 c) EZ” be

such that F(S, d,c)=x~ r.int(PO(S)). From Lemma 1, for all d’~ [d, F(S,d, c)[, F(S, d’, c) = F(S, d, c). Suppose, by way of contradiction, that x#y=p(S, d, c). Pick

d’ E [d,F(S,d,c)[ and z~r.int(PO(S)) such that z~l(d’,c) and I/c-z(\ z jjz-d’/.

Since x~r.int(PO(S)), and xsc by BDD, such d’ and z exist. Moreover, since

x#y, it follows that x#z. Now let d2=2z-c. Then d*E[(d’,c) and IIc-z/=

([z-d*/. Since d’zd*, (S,d*,c)EZ”. By WPO, MPD, and the fact that

zEI.int(PO(S)), F(S,d*,c)=z. Since d’EI(d*,z), from Lemma 1, F(S,d’,c)=z, a

contradiction (see Fig. 3).

Since an arbitrary problem (S, d, c) E Z” can be approximated by a sequence of

problems ((S”,d”,c”)} of elements of Z” such that, for all v, d” =d, c”=c and

F(S”, d”, c”) E r.int(PO(S”)), we conclude by CONT that F(S, d, c) = P(S, d, c) for all

(S,d,c)EZ”, as desired. Q.E.D.

In the above theorem, WPO cannot be strengthened to PO, as shown in the

following:

Theorem 4. There is no solution satisfying PO and BDD if n 2 3.

Proof. Let n = 3, S = cch{(l, l,O), (0, 1, l)}, d= (O,O, 0), and c = (2/3,2/3,2/3). It is

easily verified that (S, d, c) E Z” and that PO(S) = [( 1, 1, 0), (0, 1, l)]. However, for all

XEPO(S), x2>c2, in contradiction with BDD. Q.E.D.

Also, we should note that in Theorem 3, d.Q-CAV cannot be strengthened to the

following condition which is the counterpart for our domain of a condition, in-

troduced by Chun and, Thomson (1990a) for bargaining problems.

Fig. 3.

Y. Chun, W. Thomson / Bargaining problems wirh claims 29

Disagreement Point Concavity (d.CAV). For all (S, d, c), (S’, d’, c’) EL’” and for all

a~[O,l], if (S,c)=(S’,c’), thenF(S,crd+(l-cr)d’,c)~a~(S,d,c)+(l-a)F(S’,d’,c’).

Another impossibility result is obtained by imposing the following condition, which

can be motivated on the basis of timing of bargaining, as in the case of S.CAV and

d.CAV.

Claims Point Concavity (c.CAV). For all (S, d, c), (S’, d’, c’) EL’” and for all a E [O, 11,

if (S, d) = (S’, d’), then F(S, d, ac+ (1 - a)~‘)2 crF(S, d, c) + (1 - a)F(S’, d’, c’).

The proofs of both parts of the next theorem are obtained by simple adaptations

of an argument found in Chun and Thomson (1990a).

Theorem 5. There is no solution satisfying either (i) WPO, BDD, and d. CA V, or (ii) WPO, BDD, and c.CA V.

5. Axioms concerning changes in the population

In this section we focus on the behavior of solutions across cardinalities. To ac-

commodate changes in the number of agents, we generalize the model along the lines

of a similar generalization of the bargaining problem (Thomson 1983a, 1983b).

There is an infinite universe I= { 1,2, . . . } of potential agents, only a finite number

of whom are present at a given time. Let JZ be the class of finite subsets of I. Given

ME&, !RM is the Cartesian product of IMI copies of $I indexed by the members

of M. Let EM be the class of subsets of !RM satisfying all the assumptions previous-

ly imposed on the members of Z”. Let _Z’=UMEcx EM. A solution is a function

defined on _Z that associates with every ME&~ and every (S, d, c) ~2~ a unique

member of S interpreted as the compromise recommended for that problem.

We will present an alternative characterization of the proportional solution using

axioms concerning changes in the set of agents. All of the axioms introduced earlier

for the fixed population case can of course be extended to the variable population

case by requiring that they hold for all ME JGk. For instance, WPO can be written as:

Weak Pareto Optimality (WPO). For all ME& and for all (S, d, c) EL’~, F(S, d, c) E WPO(S).

We omit the straightforward reformulation of those of our earlier conditions that

we will also need. We will also use the following.

Anonymity (AN). For all M, M’E & with ) M( = / M'j , for all SE .Z”, S’E .Z”’ and for all one-to-one functions y : M-t M’, if S’= {X’E 5_R”’ / 3x~ S with xl(;, =xj for

all i E M}, d;,,, = d, and cicij = c, for all i E M, then Q)(S: d’, c’) = F;-(S, d, c) for all iEM.

30 Y, Chun, W. Thomson / Bargaining problems with claims

This axiom says that the solution should be invariant under exchanges of the names

of agents. It is a straightforward generalization of SY.

Fixed Claims Contraction Independence (FCCI). For all ME&, for all (S,d,c), (S ‘, d ‘, c’) E Z”, if S’CS, (d’,c’)=(d,c), and F(S,d,c)eS’, then F(S’,d’,c’)=

F(S, d, c) .

This axiom, which is adapted from Nash’s well-known axiom of contraction independence, requires that if an alternative is thought to be the best compromise for

some feasible set, the disagreement point and the claims point being fixed, then it

should still be thought the best for any subset that contains it.

To introduce the next axiom we need additional pieces of notation. For ME&,

let eM be the (Ml-dimensional vector with all coordinates equal to 1. Given

M,NE&! such that MCN, and xe!R N, let x, be the projection of x onto !X”, and

given (S,d,c)EZN, let ~&S)={XE~~) (x,d,,,,M)ES}.

Population Monotonicity (POP.MON). For all M, NE A, for all (S, d, c) E ZM and

for all (S’,d’, c’) l _ZN, if MC N, (dh, CL) =(d,c), and tg(S’)=S, thenF,(S’,d’,c’)s

F(S, d, 4.

This axiom, which is adapted from a condition used by Thomson (1983a, 1983b),

says that the arrival of new claimants, resources remaining fixed, penalizes all agents

initially present.

The proof of the next result is closely related to Thomson’s (1983b) characteriza-

tion of the egalitarian solution.

Theorem 6. The proportional solution is the unique solution satisfying WPO, BDD, CONT, AN, SC.INV, FCCI, and POP.MON.

Proof. It is obvious that the proportional solution satisfies all of the axioms. Con-

versely, let F be a solution satisfying all of the axioms. Let ME ._4 and (S, d, c) E ZM be given. By SC.INV we may assume that d =0 and c= eM, so that P(S,d, c) = x=aeM for some O<a<l. Let sksn{Xm”IXg. Of course, (S’,d,c)EZM and by BDD and FCCI, F(S’, d, c) = F(S, d, c) =x. Now let n 2 IMI + 1 be the smallest

integer such that for all XE S’, CjEM ,= x.-=na. Let NE& with McNand INI =n be

given. Let Tc!??~ be defined by T={xE%~\ CjE:Nxi(=INja,xseN}. Further-

more, let d’,c’EBN be such that d’=O and c’=eN. Since (T,d’,c’)~x~, by WPO

and AN, we have F( T, d’, c’) = aeN. Let T’={zE%~) 3y~S and te$l with zM=y

and q = t} and T”= Tn T’. Note that (T”, d’, c’) E ZN. Since T”C T, by FCCI, we

have F(T”,d’,c’)=F(T,d’,c’). Since t’&(T”)=S’, by POP.MON, we have

F(S’, d, c) 2 FM(T”, d’, c’) =x. If x E PO(S), we are done. Otherwise, we conclude by

a continuity argument involving an approximation of S by a sequence of problems

(SV,dY,cV)EZM for which F(S”,d”,c”)EPO(S”) for all v. Q.E.D.


Our final characterization of the proportional solution involves the following

axioms.

Independence of Non-Individually Rational Alternatives (INIR). For all ME&

and for all (S, d, c), (S’, d’, c’) E Z”, if (d,c)=(d’,c’), and S’=comp{xES/xzd},

then F(S’, d’, c’) = F(S, d, c).

This axiom, which is adapted from a condition used by Peters (1986), says that the

non-individually rational alternatives are irrelevant to the determination of the solu-

tion outcome. It is a natural requirement since all agents can guarantee themselves

their utilities of the disagreement point.

Independence of Unclaimed Alternatives (IUA). For all ME& and for all (S, d, c), (S’,d‘,c’)~z~, if (d,c)=(d’,c’), and S’= {XCZS / xsc}, then F(S’,d’,c’)=F(S,d,c).

This axiom requires that the part of the feasible set that is not dominated by the

claims point be ignored. Since each coordinate of the claims point can be regarded

as the maximal utility level to which each agent is entitled, feasible alternatives that

are not dominated by the claims point can be thought of as being irrelevant.

Sequential Invariance (SEQ.INV). For all M, NE&, for all (S, d, c) E_Z’~ and for

all (S’, d’, c’) E ZN, if Mc_N, (dh,ch)=(d,c), tg(S’)=S, and (S,F,(S’,d’,c’),c)E Z”,’ then F(S, d, c) = F(S, FM@‘, d’, c’), c).

The main axiom of this section, SEQ.INV, is motivated by considering the

possibility of a varying population. Let N be a population of agents facing the prob-

lem (S’, d’, c’). Assuming the solution F is being used, the compromise F(S’, d’, c’) is chosen. Then, it is determined that the agents in some subgroup N \Mare, in fact,

not entitled to anything. Is the original agreement moot, or should the remaining

agents take it as a starting point? An agent who is hurt if it is ignored might feel

that the community is reneging on its promise, whereas agents who benefit might

argue that any earlier agreement is irrelevant. Imposing SEQ.INV permits a recon-

ciliation of these two viewpoints. A related axiom, introduced for bargaining theory

by Thomson (forthcoming), was itself inspired by a condition proposed by Kalai

(1977) for bargaining problems with a fixed population under the name step-by-step negotiation .

For the next result, we need to introduce a smaller domain.

(5’) CBS, crd, and csa(S,d), where a,(S,d)=max(x;jxES, xzd} for all i. Let 2” be the class of all n-person problems satisfying (l)-(4) and (5’).

The proof of the next result, which is closely related to Thomson’s (1983a)

characterization of the Kalai-Smorodinsky solution, is omitted.

’ As we will see later, if F satisfies BDD, then ds&(S’, d’, c’) 5 c, and for (S, FM(S’, d ‘, c’), c) to be in

Z”, we only need &(S’, d’, c’) E int(S).


Theorem 1. The proportional solution is the unique solution on 2” satisfying WPO, CONT, AN, SC. INV, INIR, IUA, and SEQ.INV.

6. Concluding comment

In some recent work, Gupta and Livne (1988) considered bargaining problems

enriched by specifying a reference point inside the feasible set, which they inter-

preted in a variety of ways, one of which being a starting point for the negotiations.

Their structure is ‘dual’ to the claims problems analyzed here.

We should also note that, when specialized to the class of bargaining problems

with claims that exhibit the transferable utility property, our solution reduces to one

of the solutions discussed in the literature quoted in the Introduction. Axiomatic

characterizations of that solution in that context can be found in Chun (1988). The

solution is also a special case of a larger class characterized by Young (1987).

The present study is not intended to constitute an exhaustive theory of how

bargaining problems with claims should be resolved. Instead, our purpose was to

offer a sample of results that can be obtained relatively easily by drawing upon the

conceptual apparatus and the proof techniques of the theory of bargaining.

A direction of particular interest in which this investigation could be extended

concerns the way solutions may depend on the cIaims point. We have used several

conditions in which the claims point is central (BDD, MPD, c.CAV, and IUA). Other

conditions easily come to mind: monotonicity properties with respect to changes in

the claims point (as pointed out by a referee) that would parallel the monotonicity

properties with respect to the disagreement point considered by Thomson (1987)

(these are of course still relevant here), and properties pertaining to uncertainty in

the claims point, that would parallel similar properties involving uncertainty in the

disagreement point (two of which were examined in Section 4). Finally, we note once

again that a lexicographic extension of the proportional solution can easily be defin-

ed by adapting the techniques used by Imai (1983), Chun (1989), and Chun and

Peters (1988, 1989). This extension satisfies Pareto optimality, but at the price of

several other properties, among which the most obvious one is continuity.

Acknowledgments

W. Thomson gratefully acknowledges support from NSF grant No. 8511136. We

thank a referee for his/her helpful comments.

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Bargaining problems with claims