DOI: 10.1007/s00199-004-0512-6Economic Theory 26, 227–235 (2005)

The separability principle in bargaining�

Youngsub Chun

School of Economics, Seoul National University, Seoul 151-742, KOREA(e-mail: ychun@plaza.snu.ac.kr)

Received: August 12, 2002; revised version: March 22, 2004

Summary. We investigate the implications of the separability principle in the con-text of bargaining. For two bargaining problems with the same population, supposethat there is a subgroup of agents who receive the same payoffs in both bargain-ing problems. Moreover, if we imagine the departure of this subgroup with theirpayoffs, then the remaining agents face the same opportunities in both bargainingproblems. The separability principle requires that under these hypotheses, the re-maining agents should receive the same payoffs in both bargaining problems. Webegin with investigating the logical relations between separability and two otheraxioms, contraction independence and consistency. Then, we establish characteri-zations of the Nash and egalitarian solutions on the basis of separability.

Keywords and Phrases: Bargaining problem, Axiomatic approach, Separabilityprinciple, Nash solution, Egalitarian solution.

JEL Classification Numbers: C71, C78.

1 Introduction

The bargaining problem, as formulated by Nash (1950), consists of a feasible setand a disagreement point, represented in utility space. The agents can achieve anypoint in the feasible set if they unanimously agree on it. If they do not agree on anypoint of the feasible set, they end up at the disagreement point. Nash investigates theexistence of solutions to such problems that would satisfy a certain list of appealingproperties or axioms, and lays the foundations of the axiomatic approach.

In this paper, we formulate and investigate the implications of the separabilityprinciple in the context of bargaining. For two bargaining problems with the same

� This work was supported by the Brain Korea 21 Project in 2003. I am grateful to William Thomson,a referee, and an associate editor for their valuable comments.

228 Y. Chun

population, suppose that there is a subgroup of agents who receive the same pay-offs in both bargaining problems. Moreover, if we imagine the departure of thissubgroup with their payoffs, then the remaining agents face the same opportunitiesin both bargaining problems. The separability principle requires that under thesehypotheses, the remaining agents should receive the same payoffs in both bargain-ing problems. It was introduced and studied by Moulin (1987) in the context ofsurplus sharing, and its implications in other contexts have been studied recentlyby Chun (1999a,b, 2000) and Ehlers and Klaus (2003). The current version, whichis reformulated to be suitable for problems with non-transferable utilities such asbargaining, is new to this paper.

We begin with investigating the logical relations between separability and twoother axioms widely discussed in the literature: contraction independence [Nash’s(1950) independence of irrelevant alternatives] and consistency (Harsanyi, 1959;Lensberg, 1987, 1988). As it turns out, separability is implied by contraction inde-pendence alone and Pareto optimality and consistency together. Then, we presentcharacterizations of the Nash and egalitarian solutions on the basis of separability.

2 Preliminaries

Let I ≡ {1, 2, . . . } be the (infinite) universe of “potential” agents, and N be thefamily of finite subsets of I, with at least two agents. Let N and M be the genericelements of N . Given N ∈ N , a bargaining problem, or simply a problem, is apair (S, d), where S is a subset of RN and d is a point in S, such that

(1) d = 0, S ⊂ RN+ , and there exists x ∈ S with x > d,1

(2) S is convex, bounded, closed, and comprehensive, i.e., for all x ∈ S and ally ∈ RN

+ , if y ≤ x, then y ∈ S.

The set S is the feasible set. Each point x of S is a feasible alternative. The pointd is the disagreement point. The intended interpretation of (S, d) is as follows: thecoordinates of each x ∈ S are the von Neumann-Morgenstern utility levels attainedby the agents through the choice of some joint action. The agents can achieve x ifthey unanimously agree on it. If they do not agree on any point of S, they end upat d. Since we assume that d = 0, we use the notation S instead of (S, d). Let ΣN

be the class of all problems for N, and Σ ≡ ∪ΣN .A solution is a function F : Σ → ∪RN such that for all N ∈ N and all

S ∈ ΣN , F (S) ∈ S. The value taken by the solution F when applied to theproblem S, F (S), is the solution outcome of S.

The following notation and terminology will be used frequently. Givenx1, . . . , xk ∈ RN

+ , comp{x1, . . . , xk} is the comprehensive hull of these points(the smallest comprehensive set of RN

+ containing them); con◦comp{x1, . . . , xk}is their convex and comprehensive hull (the smallest convex and comprehensiveset of RN

+ containing them); given x1, . . . , xk ∈ RN+ and S1, . . . , Sk ⊂ RN

+ ,con ◦ comp{x1, . . . , xk, S1, . . . , Sk} is the convex and comprehensive hull of

1 Vector inequalities : given x, y ∈ RN , x >= y, x ≥ y, x > y.

The separability principle in bargaining 229

these points and sets. Let M, N ∈ N be such that M ⊂ N. For any x ∈ RN , xM

is the projection of x onto RM , and similarly for any T ⊂ RN , TM is the projectionof T onto RM ; rx

M (T ) is the projection of the section of T passing through x onto

RM ; also, given S ⊂ RM+ and t ∈ RN\M

++ , cyl{S, t} is the cylinder with base Sand height t, that is, cyl{S, t} = comp{x ∈ RN

+ |xM ∈ S, xN\M = t}. Finally,ei ∈ RN is the unit vector whose ith coordinate is equal to 1, and eN ∈ RN is thevector whose coordinates are all equal to 1.

We impose the following axioms, which are standard in the literature. WeakPareto optimality requires that there should be no feasible alternative at which allagents are strictly better off than at the solution outcome. Pareto optimality requiresthat there should be no feasible alternative at which all agents are weakly better offand at least one agent is strictly better off.

Weak Pareto optimality: For all N ∈ N , all S ∈ ΣN , and all x ∈ RN , ifx > F (S), then x /∈ S.

Pareto optimality: For all N ∈ N , all S ∈ ΣN , and all x ∈ RN , if x ≥ F (S),then x /∈ S.

Let WPO(S) ≡ {x ∈ S| for all x′ ∈ RN , x′ > x implies x′ /∈ S} be the setof weakly Pareto optimal points of S. Similarly, let PO(S) ≡ {x ∈ S| for allx′ ∈ RN , x′ ≥ x implies x′ /∈ S} be the set of Pareto optimal points of S.

An agent who receives a payoff smaller than his/her disagreement payoff wouldnot agree to the solution outcome. The next axiom provides agents an incentive tocooperate.

Strong individual rationality: For all N ∈ N and all S ∈ ΣN , F (S) > 0.

Symmetry requires that if a problem is invariant under all exchanges of agents,then all agents should receive the same payoffs. Anonymity, which is a strengtheningof symmetry, requires that relabellings of agents should not affect the solutionoutcome.

Symmetry: For all N ∈ N and all S ∈ ΣN , if for all one-to-one functionsγ : N → N, S = {x′ ∈ RN

+ | there exists x ∈ S such that for all i ∈ N,x′

γ(i) = xi}, then for all i, j ∈ N, Fi(S) = Fj(S).

Anonymity: For all N, N ′ ∈ N such that |N | = |N ′|, all one-to-one functionsγ : N → N ′, all S ∈ ΣN , and all S′ ∈ ΣN ′

, if S′ = {x′ ∈ RN ′+ | there exists

x ∈ S such that for all i ∈ N, x′γ(i) = xi}, then for all i ∈ N, Fγ(i)(S′) = Fi(S).

Scale invariance requires that the solution should be invariant with respect topositive linear rescalings, independent person by person, of their utilities. Note thatthis requirement precludes establishing compromises by means of interpersonalcomparisons of utility as in the case of the egalitarian solution. Given N ∈ N ,an independent person by person, positive linear transformation is a function λ :RN → RN , given by a ∈ RN

++, such that for all x ∈ RN , λ(x) ≡ (aixi)i∈N .Let Λ be the class of all such transformations.

Scale invariance: For all N ∈ N , all S ∈ ΣN , and all λ ∈ Λ, F (λ(S)) =λ(F (S)).

230 Y. Chun

Next is continuity, which requires that a small change in the feasible set shouldnot cause a large change in the solution outcome. Here, convergence of a sequenceof sets is evaluated in the Hausdorff topology.

Continuity: For all N ∈ N , all sequences {Sν} ⊂ ΣN , and all S ∈ ΣN , ifSν → S, then F (Sν) → F (S).

Now we define solutions for bargaining problems. The Nash (1950) solutionchooses the maximizer of the product of utility gains from the disagreement point.The egalitarian solution, introduced by Kalai (1977), chooses the maximal feasiblepoint at which the utilities of all agents are equal.

Nash solution, N : For all N ∈ N and all S ∈ ΣN , N(S) is the maximizer of theproduct

∏xi in x ∈ S.

Egalitarian solution, E: For all N ∈ N and all S ∈ ΣN , E(S) is the maximalpoint of the set {x ∈ S|xi = xj for all i, j ∈ N}.

3 The separability principle

For two bargaining problems with the same population, suppose that there is asubgroup of agents who receive the same payoffs in both problems. Furthermore,if we imagine the departure of this subgroup with their payoffs, then the remainingagents have the same opportunities in both problems. The separability principlerequires that under these hypotheses, the remaining agents should receive the samepayoffs in both problems.

Separability: For allM,N ∈ N withM ⊂ N, and allS,S′ ∈ ΣN , ifFN\M (S) =

FN\M (S′) and rF (S)M (S) = r

F (S′)M (S′), then FM (S) = FM (S′).

Among the three well-known bargaining solutions, the Nash and egalitarian solu-tions satisfy separability, but the Kalai-Smorodinsky (1975) solution does not.

We begin with investigating important logical relations between separabilityand two other axioms widely discussed in the literature, contraction independenceand consistency. Contraction independence, introduced by Nash (1950) under thename of independence of irrelevant alternatives, requires that if an alternative hasbeen chosen for some problem, then it should be chosen for any problem obtainedby a contraction of the feasible set, as long as the alternative remains feasible (seeThomson [forthcoming] for related literature).

Contraction independence: For all M ∈ N and all S, S′ ∈ ΣM , if S′ ⊂ S andF (S) ∈ S′, then F (S′) = F (S).

Next is the axiom of consistency: if a solution chooses a certain alternative forsome problem, then for any “reduced” problem obtained by imagining the departureof some of the agents with their payoffs and reassessing the situation from theviewpoint of the remaining agents, the solution should assign to the remainingagents the same payoffs as before.

The separability principle in bargaining 231

Consistency: For all M, N ∈ N with M ⊂ N, all S ∈ ΣM , and all T ∈ ΣN , ifS = r

F (T )M (T ), then F (S) = FM (T ).

Consistency in the context of bargaining has been introduced by Harsanyi (1959)under the name of bilateral equilibrium, and extensively studied by Lensberg (1987,1988) (see Thomson and Lensberg [1989] for a survey).

To investigate the logical relations between the three axioms, we will imposethe following axiom of dummy, introduced by Chun (2002). It requires that addingplayers who does not disturb the alternatives feasible to the original agents shouldnot affect their payoffs.

Dummy: For all M, N ∈ N such that M ⊂ N, all S ∈ ΣM , and all T ∈ ΣN , iffor some t ∈ RN\M

++ , T = cyl{S, t}, then FM (T ) = F (S).

It is a weak requirement satisfied by many well-known solutions, but not by theegalitarian solution.2 It is easy to check that consistency implies dummy.

First, we show that separability is implied by either contraction independencealone or Pareto optimality and consistency. Note that if |N | = 2, then separabilityand consistency are vacuously satisfied. Consequently, in our discussion of thelogical relations between axioms, we assume that |N | ≥ 3.

Proposition 1. Contraction independence implies separability.

Proof. Let F be a solution satisfying contraction independence. Let M, N ∈ Nbe such that M ⊂ N. Let T, T ′ ∈ ΣN be such that FN\M (T ) = FN\M (T ′)

and rF (T )M (T ) = r

F (T ′)M (T ′). Let T ′′ ≡ T ∩ T ′. Since T and T ′ are well-defined

problems in ΣN , T ′′ is a well-defined problem in ΣN . Note that T ′′ ⊆ T, T ′′ ⊆T ′, F (T ) ∈ T ′′, and F (T ′) ∈ T ′′. By contraction independence applied to Tand T ′′, F (T ′′) = F (T ). Also, by contraction independence applied to T ′ andT ′′, F (T ′′) = F (T ′). Altogether, F (T ) = F (T ′), and in particular, FM (T ) =FM (T ′). Therefore, separability holds. ��Proposition 2. Pareto optimality and consistency together imply separability.

Proof. Let F be a solution satisfying Pareto optimality and consistency. Let M,N ∈ N with M ⊂ N. Let T, T ′ ∈ ΣN be such that FN\M (T ) = FN\M (T ′)

and rF (T )M (T ) = r

F (T ′)M (T ′). Let L ≡ {i ∈ M | there exists x ∈ r

F (T )M (T )

such that xi > 0}. Note that FM\L(T ) = FM\L(T ′) = 0. If |L| = 0,then we obtain the desired conclusion FM (T ) = FM (T ′) from FM\L(T ) =FM\L(T ′) = 0. If |L| = 1, we obtain the desired conclusion from Paretooptimality and FM\L(T ) = FM\L(T ′) = 0. If |L| ≥ 2, let S, S′ be such

that S ≡ rF (T )L (T ) and S′ ≡ r

F (T ′)L (T ′). It is clear that S and S′ are well-

defined problems in ΣL. By consistency, F (S) = FL(T ) and F (S′) = FL(T ′).

2 Let M = {1, 2}, N = {1, 2, 3}, S = comp{(1, 1)}, and T = comp{(1, 1, 1/2)}. SinceE(S) = (1, 1) and E(T ) = (1/2, 1/2, 1/2), the egalitarian solution does not satisfy dummy. Now

let T ′ = comp{(1, 1, 1)}. Then, E(T ′) = (1, 1, 1). Even though rF (T )M (T ) = r

F (T ′)M (T ′),

EM (T ′) �= EM (T ). Therefore, this example shows that the egalitarian solution does not satisfystrong separability discussed later.

232 Y. Chun

Since S = rF (T )L (T ) = r

F (T ′)L (T ′) = S′, F (S) = F (S′). Together with

FM\L(T ) = FM\L(T ′) = 0, FM (T ) = FM (T ′), the desired conclusion. ��It is also interesting to ask whether the converse of Proposition 2 is true. Since

the egalitarian solution satisfies separability, but not consistency, separability alonedoes not imply consistency. However, by imposing Pareto optimality, continuity,and dummy additionally, we can establish the converse relation.

Proposition 3. Pareto optimality, continuity, dummy, and separability togetherimply consistency.

Proof. Let F be a solution satisfying Pareto optimality, continuity, dummy, andseparability. Let N ∈ N , T ∈ ΣN , and x ≡ F (T ). We need to show that for allM ⊂ N, if rx

M (T ) is a well-defined problem in ΣM , then F (rxM (T )) = xM . From

now on, we assume that rxM (T ) is a well-defined problem. First, suppose that x > 0.

For anyM ⊂ N, letT ′ ∈ ΣN be such thatT ′ ≡ cyl{rxM (T ), xN\M}.Since rx

M (T )is a well-defined problem in ΣM and x > 0, T ′ is a well-defined problem in ΣN . By

Pareto optimality, FN\M (T ′) = xN\M , and moreover, rF (T ′)M (T ′) = rx

M (T ). Byseparability applied to T and T ′, FM (T ′) = xM . By dummy, F (rx

M (T )) = xM .Therefore, consistency holds.

If x > 0, then we approximate T by a sequence {T ν} of problems of ΣN suchthat T ν → T and that for all ν and all y ∈ PO(T ν), y > 0. By Pareto optimality,for all ν, F (T ν) > 0, and we obtain the desired conclusion by continuity. ��Remark 1. Pareto optimality in Proposition 2 and continuity in Proposition 3 canbe replaced by strong individual rationality.

Next we establish the logical relations between contraction independence andconsistency.

Proposition 4. Strong individual rationality, dummy, and contraction independencetogether imply consistency.

Proof. Let F be a solution satisfying strong individual rationality, dummy, andcontraction independence. Let N ∈ N , T ∈ ΣN , and x ≡ F (T ). We needto show that for all M ⊂ N, if rx

M (T ) is a well-defined problem in ΣM , thenF (rx

M (T )) = xM . From now on, we assume that rxM (T ) is a well-defined problem.

For anyM ⊂ N, letT ′ be such thatT ′ ≡ cyl{rxM (T ), xN\M}.By strong individual

rationality, x > 0, so that T ′ is a well-defined problem in ΣN . Since T ′ ⊂ Tand x ∈ T ′, by contraction independence, F (T ′) = F (T ) = x. By dummy,F (rx

M (T )) = xM . Therefore, consistency holds. ��Remark 2. The converse relation that consistency together with Pareto optimalityand continuity implies contraction independence has been established by Lensberg(1988, Lemma 1) (see, also, Thomson and Lensberg [1989, Lemma 11.1]).

We now introduce two variants of separability. First is a stronger form of sep-arability, called strong separability, obtained from separability by dropping thehypothesis that FN\M (T ) and FN\M (T ′) are equal. The Nash solution satisfiesstrong separability. It is clear that strong separability implies separability. How-

The separability principle in bargaining 233

ever, separability is weaker than strong separability. Indeed, the egalitarian solutionsatisfies separability, but not strong separability.3

Our second modification, which we call domination-separability, requires thatunder the hypotheses of strong separability, all the remaining agents should beaffected in the same direction: all gain or all lose together. Of course, strong sepa-rability implies domination-separability. On the other hand, Pareto optimality anddomination-separability together imply strong separability.Also, it is easy to checkthat domination-separability is implied by consistency.

Domination-separability: For all M, N ∈ N with M ⊂ N, and all S, S′ ∈ ΣN ,

if rF (S)M (S) = r

F (S′)M (S′), then FM (S) >= FM (S′) or FM (S) <= FM (S′).

4 Characterizations

Here, we ask whether it is possible to characterize the Nash and egalitarian solutionson the basis of separability.

One possible approach is to use the logical relations established in the previoussection. In Propositions 2 and 3, we show that Pareto optimality and consistencytogether imply separability, and Pareto optimality, continuity, dummy, and separa-bility together imply consistency. Moreover, Lensberg (1988) shows that the Nashsolution is the only solution satisfying Pareto optimality, anonymity, scale invari-ance, continuity, and consistency. Since consistency implies dummy as mentionedearlier, we have: A solution defined on Σ satisfies Pareto optimality, anonymity,scale invariance, continuity, dummy, and separability if and only if it is the Nashsolution.

On the other hand, if the society consists of more than two agents, we canestablish another characterization of the Nash solution without dummy. Moreover,this characterization is proven while keeping the society unchanged.

Lemma 1. Let N ∈ N be such that |N | ≥ 3. Let F be a solution defined onΣN , which satisfies Pareto optimality, anonymity, scale invariance, continuity, andseparability. Let S ∈ ΣN and x ≡ F (S). If for all M ⊂ N such that |M | = 2,rxM (S) ∈ ΣM , then FM (S) = N(rx

M (S)).

Proof. 4 Let F be a solution satisfying the five axioms. Let N ∈ N be such that|N | ≥ 3. Let S ∈ ΣN and x ≡ F (S). First, we assume that x > 0. Then, for allM ⊂ N, rx

M (S) ∈ ΣM . By scale invariance, we may assume that xN\M = eNN\M

and N(rxM (S)) = eM .

Suppose that S contains a nondegenerate segment normal to eM centered atN(rx

M (S)), that is, for some δ > 0, [(1+δ, 1−δ), (1−δ, 1+δ)] lies in the Pareto op-timal boundary of rx

M (S). Let σ be the nondegenerate segment, i and j be two mem-bers of M, and T ≡ cyl{con ◦ comp{2ei, 2ej}, eN

N\M}. By Pareto optimality and

anonymity, F (T ) = eN . Let k ∈ N\M be fixed, and T ′ ≡ cyl{rxM (S), eN

N\M}.

3 See footnote 2 for an example.4 Even though we use separability while Lensberg (1988) uses consistency, our proof of Lemma 1

is related to the proof of Lemma 2 in Lensberg.

234 Y. Chun

For each ν ∈ N+, let Cν be the cone with vertex (1 + 1ν )ek spanned by T ′ and

T ν ≡ Cν ∩ T. As ν → ∞, T ν → T. Let σ′ = σ + eNN\M . Note that for all ν,

σ′ ⊂ PO(T ν) and T ν ∈ ΣN .Consider the sequence {zν}, where zν ≡ F (T ν) for all ν, as ν → ∞. Note that

F (T ) = eN ≡ z, and if zν moves away from z at all, it has to be along the segmentσ′ because of Pareto optimality. Let ν̄ ∈ N+ be such that for all ν > ν̄, zν ∈ σ′.For all ν > ν̄, rzν

M (T ν) = rxM (S) and zν

N\M = eNN\M . Therefore, by separability,

zν is constant for all ν > ν̄, and by continuity, zνM = eN

M . By separability again,FM (S) = eN

M , as desired.If x > 0 or S does not contain the nondegenerate segment, we obtain the desired

conclusion by continuity. ��The announced characterization of the Nash solution follows. The problem

S ∈ ΣN is smooth at x ∈ S if there exists a unique hyperplane supporting S at x.Also, the problem is smooth if it is smooth for all x ∈ PO(S) such that x > 0.

Theorem 1. Let N ∈ N be such that |N | ≥ 3. A solution defined on ΣN satisfiesPareto optimality, anonymity, scale invariance, continuity, and separability if andonly if it is the Nash solution.

Proof. From our earlier discussion, it is obvious that the Nash solution satisfies thefive axioms. Conversely, let F be a solution satisfying the five axioms. Let N ∈ Nbe such that |N | ≥ 3 and S ∈ ΣN be given. If S is smooth, then N(S) is the onlypoint in S satisfying the conclusion of Lemma 1. That is, it is the only point in Ssuch that for all M ⊂ N with |M | = 2, and all x ∈ S, if rx

M (S) ∈ ΣM , thenxM = N(rx

M (S)). Therefore, F (S) = N(S). If S is not smooth, then we obtainthe desired conclusion by continuity. ��

Now we turn our attention to the egalitarian solution and ask whether its char-acterization can be obtained on the basis of separability. As noted earlier, the egal-itarian solution does not satisfy dummy, so that the first approach, which uses thelogical relations established in the previous section, cannot be taken.

Our main result here involves weak Pareto optimality, symmetry, continuity,domination-separability, and the following axiom of converse consistency, assum-ing the society consists of more than two agents. Suppose that there is a feasiblealternative for some problem. Furthermore, the alternative has the property that forall proper subgroups of the agents it involves, the solution chooses the restrictionof the alternative to the subgroup for the associated reduced problem obtained byimagining the departure of some agents with their payoffs and reassessing the situa-tion from the viewpoint of the remaining agents. Converse consistency requires thatunder these hypotheses, the alternative should be chosen as the solution outcomefor the problem. In the context of bargaining, its implications have been studied byChun (2002).

Converse consistency: For all N ∈ N , all T ∈ ΣN , and all x ∈ T, if for allM ⊂ N, rx

M (T ) ∈ ΣM and F (rxM (T )) = xM , then F (T ) = x.

Our characterization of the egalitarian solution follows.

The separability principle in bargaining 235

Theorem 2. Let N ∈ N be such that |N | ≥ 3. A solution defined on ∪M⊆NΣM

satisfies weak Pareto optimality, symmetry, continuity, domination-separability, andconverse consistency if and only if it is the egalitarian solution.

Proof. It is obvious that the egalitarian solution satisfies the five axioms. Conversely,let F be a solution satisfying the five axioms. Also, let N ∈ N such that |N | ≥ 3and S ∈ ΣN be given. First, we consider the case PO(S) = WPO(S). Suppose,by way of contradiction, that F (S) = E(S). Then, there exists M ⊂ N such that|M | = 2, r

F (S)M (S) ∈ ΣM , and that FM (S) = E(rF (S)

M (S)), which together with

PO(S) = WPO(S), implies that for some i ∈ M, Fi(S) > Ei(rF (S)M (S)) ≡ α

and for some j ∈ M, Fj(S) < Ej(rF (S)M (S)) = α.

Let x ∈ RN+ be such that xi ≡ α for all i ∈ N, and T ∈ ΣN be such that

T ≡ cyl{rF (S)M (S), xN\M}. For each ν ∈ N+, let T ν ≡ con ◦ comp{cyl{S, (1 −

1/ν)xN\M}, x}. Note that T ν → T. By weak Pareto optimality and symmetry,for all M ⊂ N and all ν, F (rx

M (T ν)) = xM . By converse consistency, for all ν,

F (T ν) = x. By continuity, F (T ) = x. Since rxM (T ) = r

F (S)M (S), by domination-

separability, either FM (S)>=FM (T ) = xM or FM (S)<=FM (T ) = xM . However,Fi(S) > α = xi and Fj(S) < α = xj , a contradiction.

If WPO(S) = PO(S), we obtain the desired conclusion by continuity. ��

References

Chun, Y.: Equivalence of axioms for bankruptcy problems. International Journal of Game Theory 28,511–520 (1999a)

Chun, Y.: The separability principle in economies with single-peaked preferences. Social Choice andWelfare (forthcoming) (1999b)

Chun, Y.: Agreement, separability, and other axioms for quasi-linear social choice problems. SocialChoice and Welfare 17, 507–521 (2000)

Chun, Y.: The converse consistency principle in bargaining. Games and Economic Behavior 40, 25–43(2002)

Ehlers, L., Klaus, B.: Coalitional strategy-proofness, resource-monotonicity, and separability for mul-tiple assignment problems. Social Choice and Welfare 21, 265–280 (2003)

Harsanyi, J.C.: A bargaining model for the cooperative n-person game. In: Kuhn, H.W., Tucker, A.W.(eds.) Contributions to the theory of games, vol. IV. Annals of Mathematical Studies No. 40.Princeton: Princeton University Press 1959

Kalai, E.: Proportional solutions to bargaining situations: interpersonal utility comparisons. Economet-rica 45, 1623–1630 (1977)

Kalai, E., Smorodinsky, M.: Other solutions to Nash’s bargaining problem. Econometrica 43, 513–518(1975)

Lensberg, T.: Stability and collective rationality. Econometrica 55, 935–961 (1987)Lensberg, T.: Stability and the Nash solution. Journal of Economic Theory 45, 330–341 (1988)Moulin, H.: Equal or proportional division of a surplus, and other methods. International Journal of

Game Theory 16, 161–186 (1987)Nash, J.F.: The bargaining problem. Econometrica 18, 155–162 (1950)Thomson, W.: Bargaining theory: the axiomatic approach. New York: Academic Press (forthcoming)Thomson, W., Lensberg, T.: Axiomatic theory of bargaining with a variable number of agents. Cam-

bridge: Cambridge University Press 1989