Bargaining on a non-convex domain

Faculty Working Paper 91-0104

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B385

1991:104 COPY 2

Bargaining on a Non-Convex Domain

John P. Conley

Department of Economics

Simon Wilkie

Bell Communications Research

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Bureau of Economic and Business Research

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

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BEBR

FACULTY WORKING PAPER NO. 91-0104

College of Commerce and Business Administration

University of Illinois at (Jrbana-Champaign

January 1991

Bargaining on a Non-Convex Domain

John P. Conley*

and

Simon Wilkie**

Department of Economics

r

* Department of Economics, University of Illinois, Champaign, IL 61821
** Bell Communications Research, Morristown, New Jersey 07960-1910

Abstract

In this note we show that the characterizations of the Kalai-Smorodinsky solution gr

in Kalai and Smorodinsky (1975). and of the Egalitarian solution given in Kalai I 1977) for
the domain of convex bargaining problems can be extended to a domain of compri :
(free disposal) bargaining problems. We also discuss the literature in this area.

I

I

I

1. Introduction

An n-person bargaining problem consists of a pair (5, d) where S is a non-empty su
of !ft", and d £ 5. The set S is interpreted as the set of utility allocations that are attainable
through joint action on the part of all n agents. If the agents fail to reach an agreement,
then the problem is settled at the point d, which is called the disagreement pond. A
bargaining solution F, defined on a class of problems Sn, is a map that associates with
each problem (S.d) E Hn a unique point in 5. In the axiomatic approach to bargaining
we start by specifying a list of properties (Pareto-optimality, for example) that we would
like a solution to have. If it can be shown that there is a unique solution that satisfies a
given list of axioms, then the solution is said to be characterized this list.

It is common to restrict the domain to problems with convex feasible sets. However,
bargaining problems can arise from a variety of political, social and economic situations.
The requirement that S be convex seems to remove many important cases from consider-
ation. For example, the image in utility space of a finite set of resource allocations will
be a finite set of points, not a convex set. The standard justification for restricting at-
tention to convex problems is an assumption that agents' preference's can be represented
by von Neumann-Morgenstern utility functions. The feasible set may then be convexified
by usimz; lotteries. We find this approach unappealing for two reasons. First, the von
Neumann-Morgenstern hypothesis is often rejected in empirical studies, and there is no
shortage of alternatives in the literature. See Fishburn (19S9) for a systematic exposition.
Second, allowing problems to be settled at lotteries gives rise to serious questions in the
interpretation of the axioms. We discuss this a length in Conley and Wilkie (19S9).

In this paper we require only that the feasible set be comprehensive. This is equiv-
alent to assuming free disposal in tin' underlying economic problem. Our results may be
stated succinctly: (1) on our domain, there does not exit a solution that
Pareto optimality and symmetry; (2) if we replace strong Pareto-optimality with weak
Pareto-optimality, then Kalai and Smorodinsky's characterization of their solution on the
domain of convex problems may be carried over to the domain of comprehensive prob-

lems; and (3) Kalai's characterization of the egalitarian solution on the domain of convex
and comprehensive problems may be extended directly to the domain of comprehensive
problems.

2. Definitions and Axioms

We start with some definitions and formal statements of the axioms used in the char-
acterizations. Given a point d G 3£n, and a set S C 3£n, we say S is d- comprehensive if
d < x < y and y G 5 implies x G S.1

The comprehensive hall of a set S C 3£n, with respect to a point d G 9ftn is the smallest
d-comprehensive set containing 5:

comp(S; d) = {x G 9£n | a; G 5 or 3y G 5 such that d < x < y). (1)

The convex hull of a set S C -ft" is the smallest convex set containing the set S:

n+l n+l

i=l i=l

i{s) = I x G 9£n | x = ^ X'Vi where 51 A|' = 1? A< : - ° V *' and lJi e S V

Define the weak Pareto set of S as:

WP(S) = {x G 5 | y > .r implies t/ £ 5}. (3)

Define the strong Pareto set of S as:

P(5) = {z e S\y> x implies y g 5}. (4)

The domain of bargaining problems considered in this paper is '£.''. This is defined as

the class of pairs (5. d) where S C ft" and </ G R" such that:
Al ) S is compact.

I lie vector inequalities are represented by >, >, unci ^>.

A2) S is d-comprehensive.

A3) There exists x G 5 and x >> d.

We now present the axioms used in this paper.
Weak Pareto-Optimality (W.P.O.): F(S, d) G WP(S).

A permutation operator, 7r, is a bijection from {1,2,. . . ,n} to {1,2, .. . ,?i}. II" is the class
of all such operators. Let it(x) = {x^x\x<2\ . . . ,x<n>>).2 and rr(5) = {y G K" | </ =
7t(2*)j; G S}.

Symmetry (SYM): If for all permutation operators tt G IIn, ir(S) — S and ir{d) = c/, then
Fi(S,d) = FJ(S,d±VilJ.

An affine transformation on 3£n is a map, A : 3£n — * %n, where A(.r) = a -f 6a: for some
a G "ftn,/> G ^+4.- A" is the class of all such transformations. Let \{S) — {y G ft" | y =
X(x).x G 5}.

Sca/e /nvanance (S.INV): V A G A'1, F(A(5), A(</)) = A(F(S,d)).
Translation Invariance (T.INV): V x G 3£n, F(5 + {^},<i + a;) = F(S,d) + .r.
Srfnm*/ Monotonicity (S.MON): If S C 5' and d = d', then F(S',d') > F(S,d).
The iV/ea/ Point of a problem (S,d) is defined as:

a(S,d) = ( max x l , max x , . . . , max .r " ) . ( 5 )

x > d £>d x>d

Restricted Monotonicity (R.MON): If 5 C 5', d = c/', and a(S,d) = a(S\d'), then
F(S',d') > F(5,c/).

Superscripts stand for the components of a vector

3. The Results

First we show the impossibility result.

Theorem 1. fi f : E£ -> $n such that f satisfies SYM and PO.

Proof/

Consider the problem (5,d) where 5 = comp( {(1, 2) (J(2, 1)}; (0, 0)) and d = (0.0).
By PO. /(5,d) = (2.1) or /(5,d) = (1,2). But this contradicts SYM.

Now we consider the Kalai-Smorodinsky solution, K:

K{S,cl) = max [a; £ S \ x £ con(a(S,d),d)] . (6)

The axioms used are those employed by Kalai and Smorodinsky( 1975) to characterize A'
on the convex domain with two agents, except that only weak Pareto-optimality is used.
The generalization to more agents is not immediate since A' docs nor even satisfy \YP()
on E"on for n > 2. No such difficulty arises on the comprehensive domain. For further
discussion see Kalai and Smorodinsky(1975) and Thomson(19SC).

Theorem 2. A solution F on ££ satisfies SYM S.INV, W.P.O, and R.MON if and only
it it is the Kalai-Smorodinsky solution.

Proof/

The proof that K satisfies the axioms is elementary and is omitted.

Conversely let F be a solution satisfying the four axioms. Given any (S.d) G -,".
assume by S.INV that the problem has been normalized such that d — 0 and u[S.d) =
[jj i) = //. Then A'( S,d) = (q or) = x for some a > 0. Let T be defined as:

r = comp(y;0)\ {* + ȣ+} (7)

and consider the problem (T, 0). We distinguish two cases:

Case 1) S C 3£+. Since S is comprehensive and x E WP(S), we have S C T. Also, since T
is symmetric, ci = 0, and x is the only symmetric element WP(T), by W.P.O. and
SYM, F(T,0) = x. However, since 5 C T, and a(5,0) = a(T,0) = y, by R.MON
F(5,0) <F(T,0) = a?
Now let T" be defined by,

T' = comp((^,0,...,0),(0,/?,...,0),...,(0,...,/9),a;;0). (8)

Consider the problem (T", 0). Since T is symmetric, c? = 0, and x is the only symmetric
element in VVP{V), then by W.P.O. and SYM, F(T',0) = x. Also, since V C S and
a(S,d) = a(T',0) = y, by R.MON, F(S,d) > F{T',d) = x. Thus F(S.d) = x =
K(S,d).
Case 2) S tf_ 9ft™. Let V be defined as follows,

v = r(jju*(5)l. (9)

Note that V is symmetric and S C V. If we replace (T, 0) with (V, 0) then the
argument of case 1 goes through as before.

Last we turn to the egalitarian solution. We show that Kalai's (1977) characterization
is true on the comprehensive domain.

E{S,d) = {max [a: E 5 | x{ - dt = Xj - dj ViJ E (1, . . . n)]} . (10)

Theorem 3. A solution F on £" satisfies SYM, T.INV, W.P.O, and S.MON if and only

if it is the egalitarian solution.

O'

Proof/

The proof that E satisfies the four axioms is elementary and is omitted. Conversely let
F be a solution satisfying the four axioms. Given any (S, d) G E™, we can assume by T.INV
that the problem has been normalized such that d = 0. Thus E(S,d) = (a, . . . , a) = x for
some a > 0. Now let T be defined by:

T = comp(x;0), (11)

and consider the problem (T, 0). Since T is symmetric, d = 0, and x is the only symmetric
element of WP(T), by W.P.O. and SYM, F(T,d) = x. Also, since S is comprehensive
rcS. Hence, by S.MON, F{S,d) > x.

By assumption, S is compact. Thus, there exists ft 6 3? such that x £ S implies f — ft,
— 3. . . . , — ft) < (x1,^2, . . . ,xn) < (ft, ft, . . . ,ft). Let Z be the symmetric closed hypercube
defined by:

Z = {y€Rn| \y\ < (ft, ft,..., ft)}. (12)

Also define T' as:

r = z\ {* + ȣ+}. (i3)

Consider the' problem (T';0). Since T' is symmetric, c/ = 0 and .r is the only symmetric
element of WP(V), by W.P.O. and SYM, F(T',d) = x. But since 5 C T'. l>y S.MON,
F(S,<!) < x. Thus, F(S,d) = x = E(S,d).

4. Conclusion

In a recent paper, Anant et al [1990] show that the Kalai-Smorodinsky theorem can
be extended directly on the domain of "NE- Regular" problems. Our first theorem shows
this characterization is not true on the domain of comprehensive problems. However, since
the set of comprehensive problems includes this class of NE- Regular problems, and the
Kalai-Smorodinsky solution is always strongly Pareto-optimal on this class, our axioms
imply strong Pareto-optimality on the domain of NE- Regular problems. Thus our second
theorem implies the Anat et al [1990] theorem. In addition, the comprehensive domain
arises naturally from an assumption of free disposal on the underlying economic problem.
It is not clear what class of economic problems would give rise to NE-Regular feasible sets.

In general, work suggests that the assumption of a convex feasible set is not essential
for any Monotone Path Solution. Since any Monotone Path Solution is well-defined on the
domain of comprehensive problems any characterization found on the domain of convex
problems should be easy to adapt. This class of solutions is discussed and axiomatized
Thomson (19S6), pp 52-57. The solution proposed by Nash (1950) is not well defined on
our domain. We examine an approach to extending the Nash solution in a companion
paper, Conley-Wilkie (19S9).

References

Anant, T.C.A, Badal Mukherji and Kaushik Basu (1990): "Bargaining Without
Convexity, Generalizing the Kalai-Smorodinsky Solution," Economics Letters. 33. pp.
115-119.

Conley, John and Simon Wilkie (1989): "The Bargaining Problem Without Convex-
ity," BBER paper No. 89-1620, .

Fishburn, P. C. (1989): Non-Linear Preference and Utility Theory. Johns Hopkins
University Press.

Kalai, Ehud (1977): "Proportional Solutions to Bargaining Situations: interpersonal
Utility Comparisons," Econometrica, 45, No. 7, pp. 1623-37.

Kalai, Ehud, and Meir Smorodinsky (1975): "Other Solutions to Nash's Bargaining
Problem," Econometrica, 43, No. 3, pp. 513-8.

Nash, John (1950): "The Bargaining Problem," Econometrica, 18, ppl55-62.

Thomson, William (1986): Bargaining Theory: The Axiomatic Approach. Unpublished
Manuscript, Rochester New York.

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