RATIONAL CHOICE PROCEDURES
Taradas Bandyopadhyay
Department of Economics
University of California
Riverside, CA 92521 U.S.A.
E-mail: [email protected]
and
Kunal Sengupta
School of Economics, University of Sydney
Sydney, NSW 2006, Australia
E-mail: [email protected]
ABSTRACT
This paper introduces a notion of a choice procedure that is a mapping which, for every ordered subsets of
alternatives (so that their collection is a cover of the entire set), specifies a subset of the set of presented
alternatives. A choice procedure is sequential if choice proceeds in a sequence of the order in which
subsets of alternatives are considered. This paper shows that although a path independent quasi-transitive
rationalizable choice procedure is necessarily sequential, a path independent sequential choice procedure
is not necessarily rational. Furthermore this paper characterizes a class of choice procedures that are
respectively transitive, quasi-transitive and acyclic rationalizable. This paper axiomatizes the path
independence property of the various rationalizable sequential choice procedures.
September 2012
RATIONAL CHOICE PROCEDURES
1. INTRODUCTION
The vast literature on consistency and rationality is about the relation between the axioms that relate
choice from a subset to the superset and rationalization of a choice function. However, the study of the
process which describes how the choice from smaller sets relate to the larger set has been largely
ignored, except in works of Plott (1973), Sertel and Van der Bellen (1980), Bandyopadhyay (1988) and
Bandyopadhyay and Sengupta (1998). Being curious about Arrow' s (1963) justification of introducing
transitive rationalization in social choice is that "the final choice should be independent from the path to
it," Plott (1973) described a choice process which is a mechanism of "divide and conquer", where a set
of alternatives is divided into subsets, a choice is made over each of these sets and then a final choice is
made over the chosen alternatives in the first round. Since the final choice depends on the alternatives
chosen from the subsets, the final outcome of a choice process here crucially depends on the way the
alternatives were initially divided up for consideration. The sequence in which subsets are taken up for
consideration is irrelevant here. The path independence of Plolt' s two-stage procedure makes the way
the entire set is divided up into subsets irrelevant for the final choice. Sertel and Van der Bellen (1980)
essentially described a choice procedure in which choice proceeds in a sequence of a finite set of
alternatives by considering the first two alternatives over which a choice is made, then choosing from
the union of the chosen elements) with the third alternative, then choosing from the union of the chosen
elements) with die next alternative, and so on, until all the alternatives have been considered Clearly, in
this choice process the final choice depends solely on the sequence in which alternatives are taken up for
consideration.
The path independence of either of the choice procedures does not guarantee that a choice is
necessarily rational. The requirement of path independence of a choice procedure comes largely from
the desirability of having rational choice. Bandyopadhyay (1988) proposed a modification of Afriat-SartelVan der Bellen procedure by making elementary choice always to binary. In his description, choice
proceeds in a sequence of a finite set of alternatives by considering the first two elements of the sequence
to make a choice, men for every chosen element, compare with the third element of the sequence and
take the union. Then for every chosen element of the earlier round, compare with the next element of
the sequence and take the union, and so on, until all the alternatives have been considered. Clearly, like
Arriat-Sertel-Van der Bellen, in this choice procedure the final choice depends on the sequence in which
alternatives are taken up for consideration. Bandyopadhyay (1988) has shown that the path
independence of such a choice procedure is equivalent to transitive rationalization. This result in turn
resolves Arrow' s observation about the relation between a weak preference ordering and a path
independent choice procedure.
Some observations on the present state of the literature are in order. First, the outcome of the
requirement of a path independent choice function crucially depends on the description of a choice
procedure. Second, all the choice procedures that are discussed hi the literature are based either on the
way the entire set is divided into the subsets or on the sequence in which the successive alternatives are
taken up for consideration.
This paper introduces a general structure in which primitive is not merely from a set the choice is
made, but also the procedure one is adopted to make the choice. In contrast to a choice function
framework, a choice procedure depends both on the way a set of alternatives is broken up into subsets
and the sequence in which each of these subsets is taken up for consideration. Specifically, for any
given set of alternatives A, a choice procedure is a rule which, for every collection of the subsets of A
that constitutes the entire set and for every sequence on the collection of these subsets, specifies a subset
of the set A. Recall that the choice procedure introduced by Bandyopadhyay (1988) is an element based
sequential choice procedure, its path independence is a necessary and sufficient condition for transitive
rationalization. However, the path independence of an element based sequential choice procedure which is
introduced by Sertel-Van der Bellen does not guarantee any rationalization, transitive or otherwise.
Now in a generalized framework it is of interest in characterizing the class of sequential choice
procedures whose path independence ensures various transitive rationalization. Then we investigate the
importance of sequential character of a path independent choice procedure in establishing the transitive
rationalization. It is shown that a path independent choice procedure that ensures quasi-transitive
rationalization is necessarily sequential. Since every path independent sequential choice procedure is not
necessarily rational, we characterize the path independent sequential choice procedures that are quasitransitive rational. The path independence requires that for every sequence over all possible divisions of
the set of alternatives, a choice procedure gives the same final outcome. Under certain situation it is
certainly a restrictive condition. In the absence of path independence, we propose to characterize the
elements that are chosen for every sequence over all possible divisions of the set of alternatives, i.e.,
the intersection of the sets of chosen elements where each set is an outcome of a choice process
corresponds to a given sequence. We characterize the class of sequential choice procedures under which
the intersection of the sets turns out to be the set of best elements. We conclude the paper by
axiomatizing the path independence property of the rationalizable sequential choice procedures.
2. PRELIMINARIES
Let X be a finite set of alternatives. For every A
of A. For A
n
X, [A] denotes the set of all nonempty subsets
[X], let A1,A2,...,An be a sequence of non-empty subsets of A such that it is a cover of the
set A, i.e.,
Ai = A. Note that except for n = 1 or | A| , there are many collections of n elements of
i= 1
[A] such that each collection is a cover of the set A. Let < A1,A2,...,An> denote an ordered set of
subsets of A, and let
(A) be the set of all such ordered sets of subset of A. An element
k
(A) is the
sequence k over the elements of a particular collection {A1(k), A2(k),....,An(k)}. We write,
k
= <
A1(k), A2(k),....,An(k) > . When specification of a sequence is not crucial, we will write
< A1,A2,...,An>
as an element of
(A). Under the restriction that n = | A| ,
(A) is a set of all one-
to-one functions from (1, 2,....| A| } to A where | A| denotes the number of elements in A.
DEFINITION 1. Let A
[X]. A choice procedure is a rule C(.) which, for every integer n
{1,
2,...| A| }, and for every admissible ordered n-tuple of subsets of A, < A1(k), A2(k),....,An(k) >
(A), specifies a subset of A, i.e., C(< A1(k), A2(k),....,An(k) > )
A For n = 1, a choice procedure
is to be called a choice function, and we write C(.)
Throughout this paper, C(.) is to be any choice procedure.
In words, a choice function is a rule which specifies a subset from a set of alternatives presented
for choice. Note that as an outcome of a choice function, more than one element may be chosen. Both
demand functions and preference orderings can be regarded as special cases of choice functions. For a
given set of alternatives presented for choice, a choice procedure is a rule which also specifies its
subset when the alternatives are considered in a sequence of subsets. Thus, in our general framework,
the way a given set of alternatives A is divided into smaller sets and the sequence in which these subsets
are to be considered are both crucial.
Now, we introduce a procedure in which only the division of a set is crucial. Plott (1973)
proposed a choice procedure in which in the first round, the choice is made from each of the subsets
and then the final choice is made from the collection of all the alternatives that are chosen in the subset
comparisons. This "divide and conquer" rule is defined below.
DEFINITION 2. Let A
[X]. For every integer n
subsets of A, < A1,A2,...,An>
(1, 2,....| A| } and for every ordered n-tuple of
(A), C(.) is said to be the multi-stage choice procedure iff
C(< A1,A2,...,An> )= C(∪ [ C(Bn-1)]) such that B1 = ∪i C(Ai) and Bt = C(Bt-1) for t = 2, 3,...m.
Note, in the multi-stage choice procedure, the division of A into the subsets is crucial and the
sequence in which the subsets are considered is irrelevant. For t = 2, the multi-stage choice procedure
reduces to a two-stage choice procedure.
Next we introduce a choice procedure where both the division of A and the sequence in which
subsets are to be considered are crucial.
DEFINITION 3. Let A
[X]. For every integer n
subsets of A, < A1,A2,...,An>
{1, 2,....| A| } and for every ordered n-tuple of
(A), C(.) is said to be a sequential choice procedure iff
C(< A1,A2,...,An> ) = H(n) such that C(< A1> ) = H(1) and C(< H(t-l), A> ) = H(t), t = 2, 3, ..., n.
H(l) is chosen when the first element of the sequence A, is presented for choice. In the next
round, keeping the order, the set H(l) and the next element of the sequence A2 are considered, and the
chosen elements are H(2). The process continues until the last element of the sequence is taken into
account. Note that a sequential choice procedure depends, in general, both on the collection of subsets
of the set A and the sequence or order in which each element of the collection of subsets of A is taken up
for consideration. Under the restriction that the elements of the sequence must be singleton, the
outcome of a sequential choice procedure depends only on the sequence or order in which each element
of A is taken up for consideration. This special class of sequential choice procedures is in the opposite
end of the multi-stage choice procedure which depends only on the collection of subsets of the set A,
and the sequence in which each element of the particular collection of A is to be considered has no effect
on the outcome.
We now introduce three specific sequential choice procedures by restricting that the every subset
of A to be singleton.
DEFINITION 4. Let A
[X]. For every integer n
subsets of A, < A1,A2,...,An>
{1, 2,....| A| } and for every ordered n-tuple of
(A), C(.) is said to be the SV sequential choice procedure iff
C(< A1,A2,...,An> ) = J(n) such that C(<A1>) = J(1) and C(<J(t-l) ∪ At>) = J(t), t = 2, 3, ..., n.
In words, J(i) is the set of alternatives chosen from a set containing the ith element of the sequence
and the alternatives that are chosen in the earlier round. Note that J(n) is the terminal choice. For example,
for a sequence < x, y, z > suppose both x and y are chosen in the second round, then J(3) is the set of
elements chosen from {x, y, z}. In a published literature this procedure was first introduced by Sertel and
Van der Bellen (1980) where each element of the sequence, < A1,A2,...,An> is restricted to be singleton.
DEFINTTION 5. Let A
[X]. For every integer n
subsets of A, < A1,A2,...,An>
{1, 2,.... |A|} and for every ordered n-tuple of
(A), C(.) is said to be the generalized binary sequential choice
procedure iff C(< A1,A2,...,An> ) =T(n) such that C(<A1>) = T(l) and for t
({a} ∪ Ai). Note that T(n) = ∪a
1)
{2,3,...,n}, T(t) = ∪a
T(t-
C({a} ∪ An) is the terminal choice.
T(n-1)
In words, T(t) is the collection of alternatives that are chosen by comparing the tth element of the
sequence with each of the alternatives that are chosen in the earlier round. This procedure was introduced
by Bandyopadhyay (1988) where each element of the sequence, < A1,A2,...,An> , is restricted to be
singleton; and this condition is known as the binary sequential choice procedure. For example, for a
sequence < {x}, {y}, {z}> , suppose both x and y are chosen in the second round, then T(3) is the set of
elements that are chosen pairwise between x and z, and y and z. Note that from the definition of a choice
procedure, it is possible in this case to have J(3) = {x}, but T(3)
| A| , T(t)
{x}. Thus, in general, even for n =
J(t) for t = 3, 4, ..., | A| and J(2) = T(2). However, if a choice procedure is single-valued
over every pair of alternatives, T(t) = J(i) for all t
{1, 2,....| A| }.
Note that the binary sequential choice procedure allows some inefficiency, since at any stage of
comparison, the new element may be the winner in some pairwise comparison and at the same time may
be the loser in some other comparison that in turn carries the rejected element to be considered in the
next round of comparison; however, a rational choice set does not contain any element which is
rejected in a subset comparison (Chemoff, 1954). To take care of this inefficiency problem, we introduce
the following sequential choice procedure.
DEFINITION 6. Let A
[X]. For every ordered | A| - tuple (< {x1}, {x2},...,{x| A| }> )
(A), C(.) is
said to be the refined binary sequential choice procedure iff C(< {x1}, {x2},...,{x| A| }> ) = T*(n) such
that C(<{x1}>) = T*(l) and for t
xt}> ) for all a
{2,3,...,| A| }, T*(t) = ∪a
({a} ∪ {xt}), whenever xt
T*(t-1)
C(< {a,
T*(t-l), otherwise T*(t) = T*(t-1).
3. PATH INDEPENDENT SEQUENTIAL CHOICE PROCEDURES
DEFINITION 7. Let A [X]. A choice procedure C(.) is said to be path independent iff for all
< A1, A2,...,An,>
(A), C(< A> ) = C(< A1, A2,...,An,> ).
DEFINITION 8. A binary relation Q defined over X, and also let Q* is a subrelation of Q such that for
all x1, x2
X, [x1Q*x2 ~ (x1Qx2 and ~ x2Qx1)]. Q is said to be (i) reflexive iff, for all x
connected iff, of all distinct x1, x2
and x2Qx3)
X, x1Qx2 or x2Qx1; (iii) transitive iff, for all x1, x2, x3
X, [(x1Qx2
x1Qx3];
(iv) quasi-transitive iff, for all x1, x2, x3
..., xn
X, xQx; (ii)
X, [x1Q*x2 and x2Q*x3)
X, [(x1Q*x2 and x2Q*x3 and ... and xn-1Q*xn,)
DEFINITION 9. For all A
[X] and all n
rational iff for < A1, A2> ...,An>
X such that for any integer t
and xQy for all y
x1Q*x3]; (v) acyclic iff, for all x1, x2,
x1Qxn.
{1, 2,....| A| }, a choice procedure C(.) is said to be
(A), there exists a reflexive, connected and acyclic relation Q on
(1, 2,...,n}, C(< ∪tAt> ) = M(∪tAt, Q) where M(∪tAt, Q) = {x| x
∪tAt
∪tAt }. In addition, if Q is quasi-transitive (resp. transitive), then C(.) is said to be
quasi-transitive rational (resp. transitive rational).
In other words, a suborder Q on X is said to rationalize a choice procedure C(.) if the set of
chosen elements is always the set of Q-best elements of any set of alternatives that are considered for
comparison at any stage of the choice process. Q is then called the rationalization of C(.). Similarly for
quasi-transitivity of Q, Q is called the quasi-transitive rationalization of C(.), and for transitivity of Q,
Q is called the transitive rationalization of C(.).
Note that under the assumption of minimal path independence, for any A
[X], a choice
procedure is rational if there is a reflexive, connected and acyclic Q such that C(A) = M(A, Q).
Next, following Uzawa (1956) and Arrow (1959), we introduce a binary relation of preference
which could be generated from any choice procedure.
DEFINITION 10. Let C(.) be a choice procedure. For all x, y
iff x
X, a binary relation R is defined: xRy
C({x, y}). Then define xPy iff (xRy and ~ yRx), and define xly iff (xRy and yRx). The relation
R is called the base relation (see Herzberger (1973)).
THEOREM A (Bandyopadhyay (1988)). Let C(.) be a choice procedure. Then C(.) is transitive rational
if and only if the binary sequential choice procedure is path independent
Theorem A establishes that when an agent makes a decision according to the description of the
binary sequential choice procedure, then the chosen elements are transitive rationalizable if and only if
the chosen elements are not vulnerable to the sequence in which the alternatives are considered to make
the decision. This result reinforces the argument made by Arrow (1951) for pain independence in
introducing the requirement of transitive rationalization in making a social decision. In Theorem A, the
choice process is a specific description of a sequential choice procedure, and then its path independence turns
out to be the necessary and sufficient condition for transitive rationalization. The question is whether path
independence of every sequential choice procedure ensures transitive rationalization. The following example
shows that there is a sequential choice procedure whose path independence is not sufficient to have transitive
rationalization.
EXAMPLE 1: Let C(.) be a choice procedure. For A = {x, y, z}, consider the SV sequential choice
procedure. Suppose C(< {x, y}> ) = {x, y}, C(< {y, z}> ) = {y, z}, C(< {x, z}> )= {x}, and C(< {x, y,
z}> ) = {x, y}. Note that the path independence of SV sequential choice procedure is satisfied, however, we
have xly, ylz and xPz.
It is now interesting to examine the property of a class of path independent sequential choice procedures
which is necessarily transitive rational. In other words, we are interested in characterizing the transitive
rational choice procedures.
DEFINITION 11. Axiom of Non-dominance (ND). Let C(.) be a choice procedure and A
set. If for every x
A, there exists y
...,| A| }, < A1, A2,...,An>
(A), x
[X] be any
A such that C(< {x, y}> ) = {x, y}, then for any integer n = {1,2,
C(< A1, A2... An> ) and z
A\ C(< A1, A2... An> ) implies C(< {x,
z}> ) = {x}.
THEOREM 1. Let C(.) be a path independent sequential choice procedure. Then C(.) is transitive rational if
and only if it satisfies the Axiom of Non-dominance.
It is left to the reader to check that the refined binary sequential choice procedure does not satisfy the
condition ND, however, its path independence ensures quasi-transitive rationalization of a choice procedure.
Now we explore the relation between quasi-transitive rationalization and the path independence of sequential
choice procedures.
THEOREM 2. Let C(.) be any path independent sequential choice procedure. Then binary relation R is
quasi-transitive.
The example below shows that every path independent sequential choice procedure is not quasitransitive rational.
EXAMPLE 2. Let C(.) be a choice procedure. For A = {x, y, z} consider the SV sequential choice
procedure.
Suppose C(< {x, y}> ) = {x, y}, C(< {y, z}> ) = {y, z}, C(< {x, z}> ) = {x, z} and C(< {x, y, z}> )
= {x}.
The path independence of SV sequential choice procedure is satisfied, however, C(.) is not rational.
THEOREM 3. Let C(.) be any path independent choice procedure. If C(.) is quasi-transitive rational,
then it must be a sequential choice procedure.
In the presence of Example 2, Theorems 2 and 3 together show that every quasi-transitive
rationalizable path independent choice procedure is sequential whereas every sequential path
independent choice procedure is not rationalizable, although it ensures quasi-transitivity in a binary
relation R. We now characterize the path independent sequential choice procedures that are necessarily
quasi-transitive rational.
DEFINITION 12. Axiom of Weak Non-dominance (WND). Let C(.) be a choice procedure
and A
[X] be any set. For every x
A and for every y
for any integer n = {1,2, ...,| A| }, any < A1, A2,...,An>
A if C(< {x, y}> ) = {x, y}, then
(A), C(< A1, A2> ...,An> ) = A.
A related consistency condition used extensively in the literature on choice is stated below.
Minimum Rationality (MR). Let C(.) be a choice procedure and A
y}> ) for all y
A, then x
| X| be any set. If x
C(< {x,
C(< A> ).
Clearly, MR implies WND, however, the converse is not true.
THEOREM 4. Let C(.) be any path independent sequential choice procedure. Then C(.) is quasitransitive rational if and only if it satisfies the Weak Axiom of Non-Dominance.
4. RATIONALIZABLE TWO-STAGE CHOICE PROCEDURE AND SEQUENTIAL CHOICE
THEOREM 5. Let C(.) be the two-stage procedure. Then C(.) is quasi-transitive rational if and only if it
satisfies path independence and the Weak Axiom of Non-Dominance.
COROLLARY OF THEOREM 5. (Plott (1973)). Let C(.) be the two-stage procedure. Then C(.) is
quasi-transitive rational if and only if it satisfies path independence and the Minimum Rationality condition.
Consider a path independent choice procedure C(.) which satisfies WAND. By Theorem 4, C(.) is
quasi-transitive rational if and only if it is sequential; by Theorem 5, C(.) is quasi-transitive rational if and
only if it is the two-stage choice procedure. In other words, C(.) being a path independent choice
procedure which satisfies WAND, the sequential procedure and the two-stage procedure are equivalent.
We do not consider now at the outset that the choice procedure is path independent. To begin with,
suppose that the two-stage procedure and a sequential procedure yields identical outcome for any set A.
Then one can conclude:
THEOREM 6.
Let C(.) be the two-stage procedure which satisfies the Weak Axiom of Non-
Dominance. Then C(.) is sequential if and only if it is quasi-transitive rational and path independent
We have shown that if the two-stage choice procedure satisfying WAND is sequential, then it
must be quasi-transitive rationalizable. The rest of the proof follows from theorem in Plott (1973), which
states that if the two-stage procedure is quasi-transitive rationalizable, then it is path independent
5. RATIONAL CHOICE PROCEDURES
Let Vc and Wc respectively be the set of all quasi-transitive and acyclic rationalizable choice
procedures. Theorem 3 shows that any path independent choice procedure C(.) which is an element of
Vc is necessarily a sequential choice procedure; while Theorem 4 shows that every path independent
sequential choice procedure satisfying WAND must be an element of Vc. It is obvious that an element
of We\Vc cannot be a path independent sequential choice procedure. In the absence of path
independence, it is of interest to examine the regularity condition of a sequential choice procedure that
ensures rational choice.
DEFINITION 13. Let A
[X]. For any < A1,A2,...,An>
(A), let
"(A) be a set of all sequences
over the elements of < A1,A2,...,An> . A choice procedure C(.) is said to satisfy the core property (CP)
iff for all < A1(k), A2(k)> ...,An(k)> , < A' 1(h). A' 2(h);...,A' n(h)>
A2(k)> ...,An(k)> ) ≠
(A),
C(< A1(k),
.
THEOREM 7. Let C(.) be a sequential choice procedure which satisfies the core property. Then
C(.) is rational if and only if it satisfies the Weak Axiom of Non-Dominance.
6. CHARACTERIZATION OF PATH INDEPENDENCE
Blair (1975) axiomatized the path independence of two-stage choice procedure. We now propose to
characterize the path independence of a choice procedure in our generalized framework. First, we
introduce the following two conditions.
DEFINITION 15. Contraction Consistency (CC). Let C(.) be a choice procedure. For all A, B
such that B
A, if x
B then [x
C(A) implies x
C( k) for all
k
(B)].
DEFINITION 16. Expansion Consistency (EC). Let C(.) be a choice procedure. For all A, B
B
A then C(A)
C( k) for all
k
[X]
[X], if
(B).
In words, the CC condition requires that if an element is chosen in a superset A, then for any
sequence over the elements of [B], where B
A, it must be chosen. For
(B) = < B> , CC is
equivalent to the Chemoff condition, which says that an element being rejected in a smaller set must be
rejected in a larger set. For
(B) to be the sequences of all ordered pairs of [B], we refer this condition
as pair-wise CC. The EC condition says that the chosen elements of a superset A cannot be a subset of
chosen elements of its subset B irrespective of the sequence over the elements of [B]. For
(B) =
< B> , EC is equivalent to the Superset Property. Once again, for
(B) to be the sequences of all
ordered pairs of [B], we refer this condition as pairwise EC. Blair (1975) showed that the Chernoff
condition and the superset property together are equivalent to the path independence of two-stage
procedure. Now, one can easily check the following result.
THEOREM 8. Let C(.) be a choice procedure. Then C(.) is path independent if and only if it satisfies CC and
EC.
Theorem 8 shows that strengthening the Chemoff condition and the superset property, the path independence of
the two-stage procedures are extended to path independence of any choice procedure. Next, we would like to
investigate the path independence of the sequential choice procedures.
DEFINITION 17. Axiom of Pairwise Path Independence (PPI). Let C(.) be a choice procedure and let A
For all ordered pairs <A1 A2>, <A2, A1>
[X].
(A), C(<A1, A2>) = C(<A2, A1>) = C(A).
DEFFNITION 18. Weak Axiom of Pairwise Path Independence (WPPI). Let C(.) be a choice procedure and let
A
[X]. For all ordered pairs <A1 A2>, <A2, A1>
(A), C(<A1, A2>) = C(<A2, A1>).
DEFINITION 19. Axiom of Pairwise Optimally (PO). Let C(.) be a choice procedure and let A
ordered pair <A1 A2>
(A), for x C(A) and for all y
A2, if x C({x, y}) then x
C(<A1 A2>).
DEFINTTTON 20 . Axiom of Irreducibility (IR). Let C(.) be a choice procedure and let A
ordered n-tuple < A1, A2> ...,An>
[X]. For an
[X]. For all
( A), C(C(< A1, A2> ...,An> ) = C(< A1, A2> ...,An> ).
It is obvious that the pairwise CC and the pairwise EC together are equivalent to PPI. Now, weakening CC
and EC we characterize the path independence of sequential choice procedure.
THEOREM 9. Let C(.) be a sequential choice procedure. Then C(.) is quasi-transitive rational and path
independent if and only it satisfies the Axioms PPI and PO.
It would be obvious from the proof that a two-stage choice procedure satisfying PPI and PO is quasitransitive rational and path independent as well. In this context, readers should recall Theorem 6. Next,
weakening PPI, we characterize quasi-transitive rationalizable choice procedures.
THEOREM 10. Let C(.) be a quasi-transitive rationalizable sequential choice procedure. Then C(.) is path
independent if and only if it satisfies Axioms WPPI, PO and IR.
It is obvious that for the element based sequential choice procedures, the conditions WPPI and PO are
innocuous. Therefore the above result shows that the condition IR is globalizing the path independence of choice
over any two elements to a larger set of alternatives.
7. PROOFS
In the process of converting the documents in Microsoft Word, the notations are all messed up and therefore
proofs are not included at this version of the paper at this point.
REFERENCES
1. Afriat, S.N. (1967), "Principles of Choice and Preference," Paper No. 160, Institute for
Research in the Behavioral, Economic and Management Sciences, Herman C. Krannert
Graduate School of Industrial Administration, Purdue University.
2. Arrow, K.J. (1959), "Rational Choice Functions and Orderings," Economica. N.S. 26, 121-127.
3. Arrow, K.J. (1963), Social Choice and Individual Values. Second Edition, New York: Wiley.
4. Bandyopadhyay, T. (1988), "Revealed Preference Theory, Ordering and the Axiom of
Sequential PathIndependence," Review of Economic Studies 55, 343-351.
5. Bandyopadhyay, T. (1990), "Revealed Preference and the Axiomatic Foundations of Intransitive
Indifference: The Case of Asymmetric Subrelations," Journal of Mathematical Psychology. 34,419434.
6. Bandyopadhyay, T. and K. Sengupta (1991), "Revealed Preference Axioms for Rational
Choice",
Economic Journal 101,202-213.
7. Bandyopadhyay, T. and K. Sengupta (2006), "Rational Choice and von NeumannMorgenstern’s Stable Set: The Case of Path Dependent Procedures," Social Choice and
Welfare, 611-619.
8. Blair, D.H.(1975), "Path Independent Social Choice Functions: A Further Result,"
Econometrica 43, 174-175.
9. ChernofF, H. (1954), "Rational Selections of Decision Functions," Econometrica 22, 423-443.
10. Fishbum, P.C. (1970), "Intransitive Indifference in Preference Theory: A survey," Operations
Research. 18, 207-228.
11. Georgescu-Roegen, N. (1936), "The Price Theory of Consumer Behavior," Quarterly Journal of
Economics. 50, 545-593.
12. Georgescu-Roegen, N. (1958), ' Threshold in Choice and the Theory of Demand," Econometrica 26,
157-168.
13. Hansson, B. (1968), "Choice Structures and Preference Relations," Synthese. 18, 443-458.
14. Herzberger, H. (1973), "Ordinal Preference and Rational Choice," Econometrica 41, 187-237.
15. Hicks, J.R. (1946), Value and Capital 2nd ed., Oxford: Oxford University Press.
16. Hicks, J.R. (1956), A Revision of Demand Theory. Oxford: Clarendon Press.
17. Kalai, E. and N. Megiddo (1980), "Path Independent Choices," Econometrica 48, 1980.
18. Plott, C. (1973), "Path Independence, Rationality and Social Choice," Econometrica 41, 1075-91.
19. Richter, M. (1966), "Revealed Preference Theory," Econometrica 34, 635-645.
20. Sen, A.K. (1970), Collective Choice and Social Welfare. San Francisco: Holden-Day.
21. Sen, A.K. (1971), "Choice Functions and Revealed Preference," Review of Economic Studies. 38,
307-317.
22. Sertel, M. R. and A Van der Bellen (1979), "Synopses in the Theory of Choice," Econometrica 47,
1367-1387.
23. Sertel, M. R. and A Van der Bellen (1980), "On the Routewise Application of Choice," Journal
of Economic Iheory, 12.