For n-person bargaining problems the family of proportional solutions (introduced and characterized by Kalai) is generalized to bargaining problems with non-convex payoff sets. The so-called "efficient proportional solutions" are characterized axiomatically using natural extensions of the original axioms provided by Kalai.n-person non-convex bargaining; proportional solutions; egalitarian solution

Jens Leth Hougaard

Mich Tvede

English

Research Papers in Economics

Discussion Papers 
Department of Economics 
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ISSN: 1601-2461 (E) 
 
No. 10-21 
 
 
 
 
n-Person Nonconvex Bargaining:  
Efficient Proportional Solutions 
 
 
 
Jens Leth Hougaard, Mich Tvede 
 
 
 
 
 
 
 
 
 
 
  
  
 
  
 n-Person Nonconvex Bargaining:
Ecient Proportional Solutions
Jens Leth Hougaard Mich Tvedey
Abstract
For n-person bargaining problems the family of proportional solutions
(introduced and characterized by Kalai) is generalized to bargaining
problems with non-convex payo sets. The so-called "ecient propor-
tional solutions" are characterized axiomatically using natural exten-
sions of the original axioms provided by Kalai.
Keywords: n-person non-convex bargaining, proportional solutions, the
egalitarian solution.
JEL-classication: C78.
Corresponding author. Department of Food and Resource Economics, University of
Copenhagen, Rolighedsvej 25, DK-1958 Frederiksberg C, Denmark. Email: jlh@foi.dk
yDepartment of Economics, University of Copenhagen, Oester Farimagsgade 5 Build-
ning 26, DK-1353 Copenhagen K, Denmark. Email mich.tvede@econ.ku.dk.
11 Introduction
In axiomatic bargaining theory it has traditionally been assumed that the set
of payos is convex since if two dierent payos (considered as von Neumann-
Morgenstern utilities) are feasible so is any lottery between them. However,
there are bargaining problems where randomization seems unreasonable, for
example, in moral hazard problems where random contracts may not be
allowed. Moreover, even though randomization may be reasonable, agents
may still violate the von Neumann-Morgenstern axioms.
Consequently a series of recent papers have examined well known bargain-
ing solutions when the set of payos is non-convex: The Nash solution ([12])
has been considered in e.g. [8], [5], [14] and [13]. The Kalai-Smorodinsky
solution ([9]) has been considered in e.g. [1], [4] and [7].
The family of proportional solutions (by Kalai [8]), comprising the egali-
tarian solution, has received less attention in this respect. [4] consider a char-
acterization of the egalitarian solution when the set of payos is non-convex
by relaxing Pareto optimality but it seems dicult to justify bargaining so-
lutions which are not Pareto optimal.
In the present paper we dene a generalization of the family of pro-
portional solutions to bargaining problems when the set of payos is non-
convex insisting on Pareto optimality - hence called ecient proportional
solutions. We demonstrate that a natural extension of Kalai's original ax-
ioms (Pareto optimality, Scale invariance and Monotonicity) for bargaining
problems where the set of payos is convex may be used to characterize e-
cient proportional solutions for bargaining problems where the set of payos
is non-convex.
By insisting on Pareto-optimality, the ecient proportional solution need
not be unique for some bargaining problems but it can be shown that typically
the ecient proportional solution will in fact be unique by adapting the proof
of Theorem 2 in [7] to the present set-up.
22 The model
An n-person bargaining problem is described by a threat point a 2 Rn, that
is the result in case of disagreement, and a set of feasible payos S  Rn. Let
U be the set of n-person bargaining problems (a;S) where S \(fag+Rn
+) is
compact and S \ (fag + Rn
++) is non-empty. A solution is map f from the
set of bargaining problems U to the set of payos Rn such that f(a;S) 2 S
for all (a;S) 2 U.
Let E(S)  Rn be the set of Pareto optimal payos in S and let D(a;S) 
Rn be the set of individually rational payos so D(a;S) = fx 2 Sjx  ag.
Let the reference point sv : U ! Rn be the ecient point in the intersection
of the comprehensive hull of the set of feasible payo and the line through a
in direction v so
sv(a;S) = E((S   R
n
+) \ fx 2 R
njx = a + tv for some t 2 Rg):
Let Uc  U be the set of bargaining problems where the set of feasible
payos is comprehensive and convex, then for all v 2 Rn
+ n f0g the reference
point sv : Uc ! Rn is a proportional solution. The family of proportional
solutions is analyzed in [8] where it is shown that in Uc the proportional
solution is characterized by the following properties:
 Pareto optimality so f(a;S) 2 E(S):
 Scale Invariance so (f(a;S)) = f((a);(S)) for all strictly increasing
maps,  : Rn ! Rn with j(x) = xj + j.
 Monotonicity so f(a;S)  f(a;T) for S  T.
Note, that the `proportional solution' in [3] diers from the above deni-
tion and is closer related to the Kalai-Smorodinsky solution ([9]).
2.1 Ecient proportional solutions
For the set of n-person bargaining problems where the threat point is zero
and the set of feasible payos is compact and comprehensive, [4] consider an
3extension of the egalitarian solution sv for v = (1;:::;1). However in their
characterization of the egalitarian solution they use weak Pareto optimality
and we nd it dicult to justify bargaining solutions that are not Pareto
optimal.
Therefore insisting on Pareto optimal solutions we must accept that pro-
portions (as given by the direction v) may only remain xed up to a certain
point simply because the boundary of S may not be Pareto optimal for all
directions. Intuitively, a straightforward ecient generalization of the family
of proportinal solutions is to `move' from the threat point a in direction v
until the reference point sv(a;S) is reached and if it is not Pareto optimal,
then jump to an Pareto optimal point x 2 E(S) dominating the reference
point, that is to some x  sv(a;S). Formally: a bargaining solution f is
called ecient proportional if and only if there exists a vector v 2 Rn
+ n f0g
such that f(a;S) 2 fx 2 E(S)jx  sv(a;S)g for all (a;S) 2 U. Let fv be an
ecient proportional solution with direction v.
Clearly, on the set of bargaining problems where the set of feasible pay-
os is comprehensive and convex Uc, the family of ecient proportional so-
lutions and family of the proportional solutions of [8] coincide. Moreover
for a bargaining problem (a;S) and a xed direction v the ecient propor-
tional solution need not be unique. Indeed the set of Pareto optimal payos,
that dominate sv(a;S), fx 2 E(S)jx  sv(a;S)g may contain many points.
However the approach and proof of Theorem 2 in [7] may be adapted to the
present framework to demonstrate that typically the ecient proportional
solution is in fact unique.
3 Characterization of ecient proportional so-
lutions
Let Uh  Uc be the set of bargaining problems where the sets of feasible
payos is the intersection of a nite number of half spaces containing the
threat point. Therefore, (a;S) 2 Uh if and only if there exist a nite number
4of strictly positive vectors and numbers (pk;bk)k where pk 2 Rm
++ and bk > 0
for all k such that
S =
\
k
fx 2 R
njpi  x  pi  a + bi and x  ag:
For all bargaining problems (a;S) 2 Uh and directions v 2 R++, the
reference point is Pareto optimal because Uh  Uc so sv(a;S) 2 E(S).
Ecient proportional solutions can be characterized by four axioms: Pareto
Optimality, Restricted Scale Invariance (that is scale invariance restricted to
problems in Uh), Positive Directions (that is the solution must improve upon
the threat point in all directions) and Restricted Monotonicity (that is mono-
tonicity restricted to comparing pairs of problems where one problem is in
Uh and the other problem is in U).
Axiom 1 (Pareto Optimality) f(a;S) 2 E(S) for all (a;S) 2 U.
Axiom 2 (Restricted Scale Invariance) (f(a;S)) = f((a);(S)) for all
strictly increasing maps  : Rn ! Rn where j(x) = xj +j for all (a;S) 2
Uh.
Axiom 3 (Positive Directions) f(a;S)   a 2 Rn
++ for all (a;S) 2 U.
Axiom 4 (Restricted Monotonicity) f(a;S)  f(a;S0) for all (a;S) 2 Uh
and (a;S0) 2 U where S  S0   Rn
+.
The main result of the present paper is the following characterization of
the family of ecient proportional solutions.
Theorem 1 A bargaining solution is an ecient proportional bargaining so-
lution if and only if it satises Pareto Optimality, Restricted Scale Invariance,
Positive Directions and Restricted Monotonicity.
Proof: Clearly, for all v 2 R++ the solution fv : U ! Rn satises Pareto
Optimality, Restricted Scale Invariance, Positive Directions and Restricted
Monotonicity. Next, we prove the converse.
5Firstly, we show that if a solution satises Pareto Optimality, Restricted
Scale Invariance, Positive Directions, and Restricted Monotonicity, then it is
ecient proportional on the class Uh. Secondly, we extend the result to the
class U.
Suppose that there exist two problems (a;S) and (a0;S0) (both in Uh) and
two directions v and v0, where v;v0 2 Rn
++ and v0 6= v, such that f(a;S) =
sv(a;S) and f(a0;S0) = sv0(a0;S0).
Let an increasing ane map  : Rn ! Rn where j(x) = xj + j for
 > 0 be dened by (a) = a0 and (sv(a;S)) 2 E(S0). Then sv(a0;S0) =
(sv(a;S)) and f((a);(S)) = (f(a;S)) according to Restricted Scale
Invariance so f((a);(S)) = sv(a0;S0). Next, let S00 = S0 \ (S), then
(a0;S00) 2 Uh so f(a0;S00) = sv(a0;S0) according to Pareto Optimality and Re-
stricted Monotonicity applied to the problems (a0;(S)) and (a0;S00). How-
ever f(a0;S0)  f(a0;S00) according to Restricted Monotonicity applied to the
problems (a0;S0) and (a0;S00), but this contradicts that f(a0;S0) = sv0(a0;S0)
and f(a0;S00) = sv(a0;S0), because sv0(a0;S0);sv(a0;S0) 2 E(S0). Therefore,
we conclude that there exists a unique direction v 2 Rn
++.
Suppose that there exists (a;S) 2 U n Uh such that f(a;S) = 2 fx 2
E(S)jx  sv(a;S)g. Then there exist t 2]0;1[ and j 2 f1;:::;ng such that
(1   t)aj + tsj
v(a;S) > fj(a;S). Let ej 2 Rn be the vector, where the j'th
coordinate is one and all other coordinates are zero, and let 1 2 Rn be the
vector, where all coordinates are one. For " > 0 such that
max
k
fs
k
v(a;S)   a
kg  "
t
1   t
X
j
(s
j
v(a;S)   a
j)
let Sj dened by
Sj = fxj(ej + "1)  (x   (1   t)a   tsv(a;S))  0 and x  ag:
Then xj  sj
v(a;S) for all x 2 Sj. Let S0 = \jSj, then (a;S0) 2 Uh so
f(a;S0) = (1   t)a + tsv(a;S). Moreover S0  S because x  sv(a;S) for
all x 2 S0. Therefore Restricted Monotonicity is violated for the problems
(a;S) and (a;S0).
6Q.E.D
The following four examples will demonstrate logical independence of the
above axioms.
Example 1 For t 2]0;1[ solution f(a;S) = (1   t)a + tsv(a;S) satises Re-
stricted Scale Invariance, Positive Directions and Restricted Monotonicity,
but not Pareto Optimality.
Example 2 Let (t) be a path in t starting in 0 where j(t0) > j(t) for all
t0 > t such that k(1)k = 1. Moreover, let t(a;S) = supftja + (t) 2 Sg.
Then for (a;S) 2 Uh the solution f(a;S) = (t(a;S)) and otherwise f(a;S) 2
fy 2 E(S)jy  (t(a;S))g. This solution satises Pareto Optimality, Positive
Directions and Restricted Monotonicity, but not Restricted Scale Invariance.
Example 3 The solution f(a;S) 2 argmaxf
Qn
j=1(xj   aj)jx 2 D(a;S)g sat-
ises Pareto Optimality, Restricted Scale Invariance, Positive Directions but
not Restricted Monotonicity.
Example 4 The solution fv(a;S) where v = ej satises Pareto Optimality,
Restricted Scale Invariance and Restricted Monotonicity. but not Positive
Directions.
Remark An appropriate generalization of the egalitarian solution is singled
out by replacing the axiom of positive directions with an axiom of restricted
symmetry: (Restricted Symmetry) if a = 0 and S = fxj1  x  1g, then
fi(a;S) = 1=n for all i.
4 Final remarks
Axiomatic characterization of bargaining solutions may alternatively be in-
terpreted as characterizations of benchmark selections within a production
economic framework, see e.g. [6]. The family of Kalai solutions characterized
in the present paper resembles selections by the so-called directional distance
7functions introduced in [11], see e.g. also [2]. Our axioms all have natural
interpretations in this respect. Pareto Optimality is equivalent to a require-
ment of technical eciency. Scale Invariance is obviously also relevant in
production space. Positive Direction implies that the benchmark should be
strictly better in all production factors and Restricted Monotonicity may re-
fer to monotonicity in production sets. The present characterization therefore
also characterizes benchmark selection by the directional distance function
on non-convex production sets with the obvious changes in the modeling
framework.
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9

An extension of the Nash bargaining problem and the Nash social welfare function, Theory and Decision,

Bargaining problems with claims,

Bargaining without convexity,

Benchmark selection: an axiomatic approach,

Bene and distance functions,

Bene functions and duality,

Beyond Nash bargaining theory: the Nash set,

Nash bargaining theory with nonconvex problems,

Nonconvex n-person bargaining: ecient maxmin solutions,

Proportional solutions to bargaining situations: Interpersonal utility comparisons,

n-Person Nonconvex Bargaining: Efficient Proportional Solution

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