Conditions α and β are two well-known rationality conditions in the theory of rational choice. This paper examines the implication of weaker versions of these two rationality conditions in the context of solutions to non-convex bargaining problems. It is shown that, together with the standard axioms of efficiency, anonymity and scale invariance, they characterize the Nash solution. This result makes a further connection between solutions to non-convex bargaining problems and rationalizability of choice functions in the theory of rational choice.

Xu, Yongsheng

Yoshihara, Naoki

English

Research Papers in Economics

Hitotsubashi University Repository
Title
Rationality and the Nash Solution to Non-convex
Bargaining Problems
Author(s) Xu, Yongsheng; Yoshihara, Naoki
Citation
Issue Date 2010-04
Type Technical Report
Text Version publisher
URL http://hdl.handle.net/10086/18498
Right 
 
Discussion  Paper  Series  A    No.537 
 
 
Rationality  and  the  Nash  Solution  to  Non-convex  Bargaining  Problems 
 
 
Yongsheng  Xu 
(Department  of  Economics,  Andrew  Young  School  of  Policy  Studies, 
Georgia  State  University) 
and 
Naoki  Yoshihara 
(Institute  of  Economic  Research,  Hitotsubashi  University) 
 
 
 
 
 
 
 
 
April,  2010 
 
 
 
 
 
 
 
 
 
 
 
Institute  of  Economic  Research         
Hitotsubashi  University          
Kunitachi,  Tokyo,  186-8603  Japan      
 
   Rationality and the Nash solution to non-convex bargaining
problems
Yongsheng Xu
Department of Economics
Andrew Young School of Policy Studies
Georgia State University, Atlanta, GA 30302
Naoki Yoshihara
Institute of Economic Research
Hitotsubashi University, Kunitachi, Tokyo
This Version: April 2010Abstract. Conditions α and β are two well-known rationality conditions in
the theory of rational choice. This paper examines the implication of weaker
versions of these two rationality conditions in the context of solutions to non-
convex bargaining problems. It is shown that, together with the standard
axioms of eﬃciency, anonymity and scale invariance, they characterize the
Nash solution. This result makes a further connection between solutions to
non-convex bargaining problems and rationalizability of choice functions in
the theory of rational choice.
J.E.L. Classiﬁcation Numbers: C71, C78, D63, D71
21 Introduction
In this paper, we study the Nash solution to non-convex bargaining problem
by examining its connection to two well-known rationality conditions, namely
conditions α and β, in the theory of rational choice (see, for example, Sen
(1971)). Condition α says that, when a set A contracts to another set B,a n d
if an option x chosen from A continues to be available in B,t h e nx must be
chosen from B.C o n d i t i o nβ, on the other hand, says that, when two options
x and y are chosen from a set A and when A expands to another set B, then,
either both x and y are chosen from B or neither x nor y is chosen from B.
In the literature on non-convex bargaining problems, a stronger version of
condition α, often called contraction independence, has been used for charac-
terizing the Nash solution (see, for example, Xu and Yoshihara (2006)). Con-
traction independence requires that, when a bargaining problem A shrinks
to another bargaining problem B and if B contains some options of the solu-
tion to A, then the solution to B coincides with the intersection of B and the
solution to A. This version of contraction independence can also be regarded
as a natural generalization of Nash’s independence of irrelevant alternatives
(Nash (1950)) introduced for convex bargaining problems where a solution
picks a single option from a bargaining problem.
Building on the intuitions of conditions α and β,i nt h i sp a p e r ,w ec o n -
sider weaker versions of conditions α and β,t ob ec a l l e dbinary condition α
and binary condition β, respectively. Binary condition α requires that, for
any two options x and y,i fe i t h e rx or y is part of the solution to a bar-
gaining problem A, then the solution to the bargaining problem given by the
comprehensive hull (see Section 2 for a formal deﬁnition) of x and y must
contain the intersection of {x,y} and the solution to A. Binary condition
β requires that, if two options x and y are both chosen from the problem
of the comprehensive hull of x and y, then when the problem is enlarged,
either both belong to the solution to the enlarged problem or neither does
not belong to the solution to the enlarged problem. Together with the stan-
dard axioms of eﬃciency, scale invariance and anonymity, we then show that
binary condition α and binary condition β characterize the Nash solution to
non-convex bargaining problems. Note that the contraction independence
discussed in non-convex bargaining problems implies binary condition α and
binary condition β, but the converse relation does not hold. Our result
therefore strengthens the existing characterization of the Nash solution to
non-convex bargaining problems (see Xu and Yoshihara (2006)) and makes
3a close connection between solutions to non-convex bargaining problems and
rationality conditions in the theory of rational choice.
The remainder of the paper is organized as follows. In the following sec-
tion, Section 2, we present notation and deﬁnitions. Section 3 introduces our
axioms and studies their implications on non-convex problems. We conclude
in Section 4.
2 Notation and deﬁnitions
Let N = {1,2,...,n} be the set of all individuals in the society. Let R+ be
the set of all non-negative real numbers, and R++ be the set of all positive
numbers. Let Rn
+ (resp. Rn
++)b et h en-fold Cartesian product of R+ (resp.
R++). For any x,y ∈ Rn
+,w ew r i t ex ≥ y to mean [xi ≥ yi for all i ∈ N],
x>yto mean [xi ≥ yi for all i ∈ N and x 6= y], and x À y to mean [xi >y i
for all i ∈ N]. For any x ∈ Rn
+ and any non-negative number q,w ew r i t e
z =( q;x−i) ∈ Rn
+ to mean that zi = q and zj = xj for all j ∈ N \{ i}.
For any subset A ⊆ Rn
+, A is said to be (i) non-trivial if there exists
a ∈ A such that a À 0, and (ii) comprehensive if for all x,y ∈ Rn
+, [x ≥ y
and x ∈ A] implies y ∈ A. For all A ⊆ Rn
+,d e ﬁne the comprehensive hull of
A,t ob ed e n o t e db ycompA, as follows:
compA ≡
©
z ∈ R
n
+ | z ≤ x for some x ∈ A
ª
.
Let Σ be the set of all non-trivial, compact and comprehensive subsets
of Rn
+.E l e m e n t si nΣ are interpreted as (normalized) bargaining problems.
A bargaining solution F assigns a nonempty subset F(A) of A for every
bargaining problem A ∈ Σ.
Let π be a permutation of N. The set of all permutations of N is denoted
by Π. For all x =( xi)i∈N ∈ Rn
+,l e tπ(x)=( xπ(i))i∈N. For all A ∈ Σ and
any permutation π ∈ Π,l e tπ(A)={π(a) | a ∈ A}. For any A ∈ Σ,w es a y
that A is symmetric if A = π(A) for all π ∈ Π.
Deﬁnition 1: A bargaining solution F over Σ is the Nash solution if for all
A ∈ Σ, F(A)={a ∈ A |
Q
i∈N ai ≥
Q
i∈N xi for all x ∈ A}.
Denote the Nash solution by FN. Note that, for non-convex bargaining
problems, the Nash solution is typically multi-valued.
43 Basic axioms and their implications
In this section, we present several standard axioms, introduce weaker versions
of two well-known rationality conditions, and examine their implications for
solutions to bargaining problems. In particular, we show that the Nash so-
lution to non-convex bargaining problems is characterized by the standard
axioms and the weaker versions of two well-known rationality conditions.
We begin by introducing three axioms, Eﬃciency, Anonymity and Scale In-
variance, which are standard in the literature on Nash bargaining problems.
Eﬃciency (E): For any A ∈ Σ and any a ∈ F(A),t h e r ei sn ox ∈ A such
that x>a .
Anonymity (A): For any A ∈ Σ,i fA is symmetric, then [a ∈ F(A) ⇒
π(a) ∈ F(A) for all π ∈ Π].
Scale Invariance (SI): For all A ∈ Σ and all t ∈ Rn
++,i ftA = {(tiai)i∈N |
a ∈ A} then F(tA)={(tiai)i∈N | a ∈ F(A)}.
In the literature on Nash bargaining problems and on rational choice
theory, various contraction independence properties have been proposed. The
idea behind a contraction independence property is the following: given two
bargaining problems, A and B,i nw h i c hA is a subset of B, and suppose that
ap o i n tx chosen from B as a solution to B continues to be available in A,
then x should continue to be a solution to A provided certain restrictions are
satisﬁed. Nash’s independence of irrelevant alternatives represents a natural
property of contraction independence in convex bargaining problems, and
it, in conjunction with some other conditions such as Pareto eﬃciency and
continuity in Peters and Wakker (1991) and Bossert (1994), and Monotonicity
and a weaker continuity in Sánchez (2000), implies the rationalizability of
solutions to two-person convex bargaining problems.
The following axiom is a weaker version of contraction independence
used in non-convex bargaining problems (see, for example, Xu and Yoshi-
hara (2006)). It requires that, for any two points x and y in a bargaining
problem A,i fe i t h e rx or y is part of the solution to A, then the common
points in {x,y} and the solution to A m u s tb ec o n t a i n e di nt h es o l u t i o nt o
the problem given by the comprehensive hull of x and y.C l e a r l y ,t h i sw e a k e r
5version of contraction independence is also a weaker version of condition α.
The axiom is formulated as below.
Binary Condition α (BCα): For all A ∈ Σ and all x,y ∈ A,i f{x,y} ∩
F(A) 6= ∅ then F(A) ∩ {x,y} ⊆ F(comp{x,y}).
It may be noted that (BCα)i ss p e c i ﬁc to non-convex bargaining problems
and is not applicable to convex bargaining problems. We next introduce a
weaker version of condition β (see Sen (1971)).
Binary Condition β (BCβ): For all A ∈ Σ and all x,y ∈ A,i f{x,y} =
F(comp{x,y}),t h e n[ x ∈ F(A) ⇔ y ∈ F(A)].
Thus, (BCβ) requires that, whenever x and y are both chosen as solutions
to the problem comp{x,y}, then, for any problem A containing both x and
y,e i t h e r[ x and y are both chosen as solutions to A] or neither x nor y is
chosen as a solution to A.
We now explore implications of the above axioms to be imposed on a
solution to bargaining problems. Our ﬁr s tr e s u l ts h o w st h a t ,w h e nas o l u t i o n
satisﬁes (E), (A), (SI), (BCα)a n d( B C β), then, for any x,y ∈ Rn
+,t h e
solution to the problem A = comp{x,y} must be such that F(A)={x,y}
if Πi∈Nxi = Πi∈Nyi > 0 and F(A)={x} if Πi∈Nxi > Πi∈Nyi.T h a ti s ,f o ra
speciﬁc, simple problem given by A, these axioms imply that the solution to
A must be given by the Nash solution. After establishing the above result,
our next result, Proposition 2, shows that, when a solution satisﬁes (E), (A),
(SI), (BCα)a n d( B C β), then the solution to any bargaining problem must
be the Nash solution.
Proposition 1. Let a solution F satisfy (E), (A), (SI), (BCα) and (BCβ).
Then, for all x,y ∈ Rn
+,
(1.1)
Q
i∈N xi =
Q
i∈N yi > 0 ⇒ F(comp(x,y)) = {x,y},a n d
(1.2)
Q
i∈N xi >
Q
i∈N yi ≥ 0 ⇒ F(comp(x,y)) = {x}.
Proof. (1.1). Let
Q
i∈N xi =
Q
i∈N yi > 0. Consider an appropriate t ∈ Rn
++
such that tx =( t1x1,···,t ixi,···tnxn) and ty =( t1y1,···,t iyi,···,t nyn) are
permutations of each other (this is always possible due to the fact that x
and y h a v et h es a m eN a s hp r o d u c t ) .L e tS ≡ comp{tx,ty}. Then, let T ≡
∪π∈Ππ(S). By construction, T is symmetric, and {π(tx),π(ty) | π ∈ Π} ⊆
T is the set of all eﬃcient outcomes in T.T h u s ,F (T) ⊆ {π(tx),π(ty) | π ∈ Π},
6and let tx ∈ F (T). Then, by (A), {π(tx) | π ∈ Π} ⊆ F (T).A l s o ,s i n c etx
and ty are permutations of each other, ty ∈ F (T) by (A). Then, again
by (A), {π(ty) | π ∈ Π} ⊆ F (T).T h u s , F (T)={π(αx),π(ty) | π ∈ Π}.
Thus, tx,ty ∈ F (T). Then, by (E) and (BCα), {tx,ty} = F (S). Thus, by
(SI), F(comp{x,y})={x,y}.
(1.2). Let
Q
i∈N xi >
Q
i∈N yi ≥ 0. Then, by choosing an appropri-
ate ε ∈ Rn
+ with εj > 0 for some j,w ec a nh a v e
Q
i∈N xi =
Q
i∈N zi for
z ≡ y + ε. Then, by (1.1), F(comp{x,z})={x,z}.N o t e t h a t , b y t h e
construction, y ∈ comp{x,z},a n dF(comp{x,z}) ∩ {x,y} = {x}.T h e r e -
fore, x ∈ F(comp{x,y}) follows from (BCα). If y ∈ F(comp{x,y}),t h e n
(BCβ) would imply that y ∈ F(comp{x,z}), a contradiction. Therefore,
y/ ∈ F(comp{x,y}).
Proposition 2. Let a solution F satisfy (E), (A), (SI), (BCα) and (BCβ).
Then, for any A ∈ Σ, F(A)=FN (A).
Proof. Take any A ∈ Σ and x ∈ F (A). Suppose x/ ∈ FN (A). Then, there
exists y ∈ FN (A) such that
Q
i∈N yi >
Q
i∈N xi ≥ 0. Then, by Proposition
1.2, F(comp{x,y})={y}. On the other hand, note that x ∈ F(A).B y
(BCα), x ∈ F(comp{x,y}), a contradiction. Therefore, F(A) ⊆ FN(A).B y
(BCβ) and from Proposition 1.1, F(A)=FN(A) then follows immediately.
Note that FN satisﬁes (E), (A), (SI), (BCα)a n d( B C β). The following
result then follows from Proposition 2 immediately.
Proposition 3. A solution F satisﬁes (E), (A), (SI), (BCα) and (BCβ) if
and only if F = FN.
Proposition 3 thus gives an alternative characterization of the Nash solu-
tion to non-convex bargaining problems. From the characterization result of
the Nash solution to non-convex bargaining problems in Xu and Yoshihara
(2006), it is clear that, in the presence of (E), (A) and (SI), Contraction In-
dependence is equivalent to (BCα)a n d( B C β). Moreover, it is easy to check
that Contraction Independence implies (BCα)a n d( B C β), but the converse
does not hold. Thus, Proposition 3 is a strengthening of the characterization
result of the Nash solution to non-convex bargaining problems in Xu and
Yoshihara (2006).
74C o n c l u s i o n
In this paper, we have examined the implications of two weaker versions
of conditions α and β in the context of solutions to non-convex bargaining
problems. In particular, we have shown that, together with other standard
axioms in this context, they restrict a solution to be the Nash solution. Con-
ditions α and β, together, characterize rationalizability of a choice function
deﬁned over the set of all non-empty subsets of a ﬁnite universal set in terms
of an ordering. It is therefore interesting to note that, in non-convex bargain-
ing problems, (BCα)a n d( B C β) are associated with “rationalizability” of a
solution to bargaining problems. Our results thus make a further connection
between two widely used rationality conditions in rational choice theory and
solutions to non-convex bargaining problems.
References
Bossert, W. 1994, Rational choice and two-person bargaining solutions,
Journal of Mathematical Economics, 23: 549-563.
Nash, J. 1950, The bargaining problem, Econometrica, 18: 155-162.
Peters, H. and P. P. Wakker 1991, Independence of irrelevant alternatives
and revealed groups preferences, Econometrica, 59: 1787-1801.
Sánchez, M. C. 2000, Rationality of bargaining solutions, Journal of Math-
ematical Economics, 33: 389-399.
Sen, A.K. 1971, Choice functions and revealed preference, Review of Eco-
nomic Studies, 38, 307-317.
Xu, Y. and N. Yoshihara 2006, Alternative characterizations of three bar-
gaining solutions for nonconvex problems, Games and Economic Be-
havior, 57: 86-92,
8

Alternative characterizations of three bargaining solutions for nonconvex problems,

Choice functions and revealed preference,

Rational choice and two-person bargaining solutions,

Rationality of bargaining solutions,

The bargaining problem,

Wakker 1991, Independence of irrelevant alternatives and revealed groups preferences,

Rationality and the Nash Solution to Non-convex Bargaining Problems

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Rationality and the Nash Solution to Non-convex Bargaining Problems

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