A complementary approach to transitive rationalizability

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Munich Personal RePEc Archive

A complementary approach to transitive

rationalizability

Magyarkuti, Gyula

Corvinus University of Budapest

7 December 1999

Online at

https://mpra.ub.uni-muenchen.de/20164/

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A complementary approach to transitive rationalizability

Gyula Magyarkuti

∗

Department of Mathematics

Corvinus University of Budapest

February 1, 2010

Abstract

In this article, we study the axiomatic foundations of revealed preference theory. We define two revealed relations from the weak and strong revealed preference. The alternativex is preferred toy

with respect toUifx, being available in an admissible set implies, the rejecting ofy; andxis preferred toy with respect toQif the rejecting ofximplies the rejecting ofy. The purpose of the paper is to show that the strong axiom of revealed preference and Hansson’s axiom of revealed preference can be given with the help ofUandQand their extension properties.

JEL: C60

1 Introduction

A choice correspondence leads to at least two relations. On the one hand,x is weakly preferred toy ifx is chosen, whiley could have been selected under some set of alternatives. On the other hand,x is strictly preferred toy ifxis chosen, whiley is available and rejected. The traditional formulation of the weak (WARP) and the strong (SARP) axioms of revealed preference have been given in terms of the relationship between the weak and the strict revealed preference relations. The strong congruence ax-iom (SCA) is equivalent to the rationality of the choice correspondence with transitive (and complete) underlying preference (see Richter (1966)). Suzumura (1976) introduced an auxiliary concept – con-nected chain, that provides a uniform framework of WARP, SARP and SCA. Hansson’s axiom of revealed preference (HARP) (see Hansson (1968)) can be regarded as an equivalent formulation of SCA in terms of connected chain.

The extension theorem of Szpilrajn was the transfinite tool of Richter’s proof; this is why the relation-ship of revealed preference axioms and extension theorems arose. Recently, Duggan (1999) summarized and gave a unified treatment of Szpilrajn–type extension theorems. He pointed out the possibility of the application of an “intension” theorem in the theory of revealed preferences.

It will be shown that HARP can also be written as a special relationship between the weak and the strict revealed relations, eliminating the distinction between the introduction of Hansson’s axiom and the introduction of the weak and strong axioms of revealed preference. It is known that there are equiv-alent formulations of SCA and SARP (Theorems 1 and 2). This note gives an alternative characterization (Theorem 3) with the help of extensions properties ofUandQ.

2 Preliminaries

For a binary relationS ⊆ X×X, letcS

g(B)denote theS-greatest elements of the setB, i.e.,cSg(B)

.

=

x∈B: x,y

∈S∀y∈B , and letcS

m(B)denote theS-maximal elementsof the setB, that is,cSm(B)

.

=

{x∈B:Sx_∩_B₌∅}, whereSx _{is the upper level set of}_S _at_x_{. Borrowing the terminology from Clark}

∗_{email: [email protected], web: http://www.uni-corvinus.hu/magyarkuti}

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(1985), we introduce thecomplementarity operator, defined upon relations as follows:Γ (S)= (. Sc₎−1

, whereSc _{= (}. _X_×_X₎r_S_{. It is easy to see that the complementarity operator establishes a relationship}

between the two concepts of optimality, that is,cS g(B) =c

Γ(S)

m (B)for everyB⊆X. Lett(S)andn t(S)

denote the transitive closure and the negative transitive interior of a relationS, respectively.

The triple(X,B_,_c₎_{is called a}_{decision structure}_{, where}B_{is a subset of the power set of}_X _and_c _: B_→P₍_X₎_{is a}_{choice correspondence}_{such that}∅6=c(B)⊆B. There are two kinds of definition for the revealed preference relation. The first one is theweak: x,y

∈Rif and only if there existsB∈B_for

whichx∈c(B)andy ∈B; the second one is thestrict: x,y

∈Pif and only if there existsB∈B_for

whichx∈c(B)andy ∈Brc(B).

The traditional formulation of the WARP is the inclusionP ⊆Γ (R), and the SARP is the inclusion t(P)⊆Γ (R), wherePis the strict andRis the weak revealed relations. The SCA introduced by Richter says: x,y

∈t(R) andy ∈c(B),x ∈ Bimpliesx ∈c(B). The choice correspondence is said to be transitive g -rationalwhenever there exists a transitive relationSwithc=cS

g.

Duggan (1999) defined the notion of compatible extension and compatible “intension” of a binary relation. Given relationsD andD′_,_D′ _{is a}_{compatible extension}_of_D _if_D _⊆_D′_and_A₍_D₎ _⊆_A₍_D′₎_;

hereA(D) =. D∩Γ (D) is the asymmetric part of a relation D. We denote by E₍_D₎ _{the set of}

com-patible extensions of D. SimilarlyD′ _{is a}_{compatible “intension”} _of_D _if _D′ _⊆ _D _and_A₍_D₎ _⊆_A₍_D′₎_.

We denote byI₍_D₎_{the set of compatible “intensions” of a relation}_D_{. It is not too hard to see that} I₍_D_{) =}_{_{Γ (}_D′₎_:_D′_∈_E_{(Γ (}_D₎₎_}_{, that is, the operation}_Γ_{is a one-to-one correspondence between}_I₍_D₎

andE_{(Γ (}_D₎₎_.

Suzumura (1976a) showed that the relationD has complete, transitive, compatible extensions if and only ifA(D)◦t(D)⊆Γ (D), which is equivalent tot(D)∈E(D)as Duggan clarified. Of course the statement thatD has asymmetric, negative transitive compatible “intension” if and only ifn t(D) ∈

I(D), is an easy consequence of the previous one.

3 Results

PutU = Γ (. R), whereRis the weak revealed relation of a decision structure. The x,y

∈U can be interpreted as follows. By the definition ofΓit is equivalent to y,x

/

∈R, which is equivalent to the fact thaty ∈c(B)andx∈Bcan not both hold for anyB∈B_{, that is, for every}_B∈B_,_x∈Bimpliesy∈/c(B). This can be phrased asthe availability of x implies the rejecting of y. Since SARP is formulated by t(P)⊆U, we can give a characterization of SARP as a congruence axiom as follows:if x,y

∈t(P), then the selection of y is prohibited if x is available.

SetQ = Γ (. P), whereP is the strong revealed relation. As before, we investigate the meaning of x,y

∈Q. The inclusion x,y

∈Qis equivalent to y,x

/

∈P, which is equivalent toy ∈/c(B)orx∈/B orx ∈c(B)for anyB∈B_{. Therefore} _x_,_y

∈Q if and only ify ∈c(B)andx ∈Btogether imply that x∈c(B)for allB∈B_{. Comparing this with the original notion of SCA we get the equivalence of SCA}

and the inclusiont(R)⊆Q. This equivalence illuminates the distinction between the introduction of Hansson’s axiom and the weak and strong axioms of revealed preferences.

The inclusion x,y

∈Q can also be expressed in the following way: x∈Bandx∈/c(B)together imply thaty ∈/c(B)for anyB∈B_{, or}_x∈Brc(B)implies thaty ∈/c(B). Thusthe rejecting of x implies the rejecting of y. Since SCA is equivalent to the assumptiont(R)⊆Q, the strong congruence axiom can be expressed as follows:if x,y

∈t(R),then the selection of y is prohibited if x is rejected. The equivalence oft(R)⊆Γ (P)and SCA leads to the summarization of the transitive rationalizabil-ity in terms of revealed preferencesR,P,U, and operations of transitive hull and negative transitive interior.

Theorem 3.1 _Let₍_X_,B_,_c)be a decision structure. All of the following assumptions are equivalent to SCA: 1. t(R)⊆Γ (P);

2. P⊆n t(U);

3. c=ctg(R), that is, c is transitive g -rational;

4. c=cmn t(U);

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y,x

∈n t(U);

6. for any B∈B_{and for all x}_∈_Br_c(B)and for all y ∈c(B), the inclusion y,x

∈n t(U)holds. Here R is the weak, and P the strict revealed, relation of c and U= Γ (R).

Proof. _{The equivalence of assumption}(1)and SCA has just been proved. SinceΓ (t(S)) =n t(Γ (S))for any relationS, condition(2)is equivalent to condition(1). The equivalence of condition(3)and SCA is easy and well-known. The identitiesctg(R)=c

Γ(t(R))

m =cn t

(U)

m hold becausecSg =c Γ(S)

m for any relationS,

which shows the equivalence of(3)and(4). The identityc=ctg(R)is equivalent to the inclusionct (R)

g (B)⊆

c(B)for anyB∈B_{. This inclusion implies that, when}_x∈Bandx∈/c(B), thenx ∈/ctg(R)(B), that is, if

x∈Brc(B), then there existsy ∈Bsuch that x,y

∈t(R)does not hold. Hence for allx∈Brc(B)

there existsy ∈Bsuch that y,x

∈n t(U), which is assumption(5). The equivalence of assumption(2)

and(6)is an easy consequence of the definition of the strict revealed relationP. ❏

The original definition of SARP is the inclusiont(P)⊆Γ (R); hence a parallel theorem can be stated in terms of revealed preferencesR,P,Q, and operations of negative transitive interior and transitive hull. The equivalence of(1)and(3)is due to Clark (1985) and the equivalence of the other conditions are obvious.

Theorem 3.2 _Let₍_X_,B_,_c)is a decision structure. All of following assumptions are equivalent to SARP: 1. t(P)⊆Γ (R);

2. R⊆n t(Q); 3. c=cmt(P);

4. c=cn tg (Q);

5. for any B∈B_{and for every x}∈c(B)and for all y ∈B , the inclusion x,y

∈n t(Q)holds,

where R is the weak, and P the strict revealed, relation and Q= Γ (P).

The difference between HARP and SARP is one of the most interesting problems in revealed prefer-ence theory. We are going to show that this differprefer-ence can be restated in terms of compatible extension and “intension”.

It can be easily proved thatD′_{is a compatible extension of}_D_{if and only if one of the following}

con-ditions holds:

·S(D)⊆S(D′₎_and_A₍_D₎_⊆_A₍_D′₎_; ·D⊆D′_⊆_W₍_D₎_,

whereS(D)=. D∩D−1_{is the symmetric part of relation}_D _and_W₍_D_{) =}_D_∪_{Γ (}_D₎_{is the weak extension}

of relationD. It is also easy to verify that either of the following conditions:

·Sc(D)⊆Sc(D′)andA(D)⊆A(D′);

·A(D)⊆D′_⊆_D

is equivalent toD′_{being a compatible “intension” of}_D_{, where}_S

c(D)=.S(Dc)is the symmetric

comple-ment of the relationD.

Theorem 3.3 _{Suppose that, the decision structure}(X,B_,_c)has the property WARP. Let R and P denote the weak and the strict revealed relations respectively, and let U= Γ (R), Q= Γ (P).

The following assumptions are equivalent to SCA (and therefore to HARP as well): 1. R⊆t(R)⊆Q;

2. t(R)∈E(R); 3. n t(U)∈I(U).

The following assumptions are equivalent to SARP: 1. R⊆n t(Q)⊆Q;

2. n t(Q)∈E(R); 3. t(P)∈I₍_U₎_.

Proof. _{We have seen that SCA is equivalent to}_t₍_R₎_⊆_Q_{. It is well-known that WARP is equivalent to}

the equationA(R) =P, whenceW(R) = Γ (A(R)) = Γ (P) =Q. ThusE(R) ={D′_:_R_⊆_D′_⊆_Q_}_{on the}

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The definition of SARP ist(P)⊆Γ (R), that is,R⊆Γ (t(P)) =n t(Q). This shows the equivalence of SARP and the inclusionsR⊆n t(Q)⊆Q. Therefore, as we have seen,n t(Q)∈E₍_R₎_{, and the involution}

property ofΓimplies thatn t(Q)∈E₍_R₎_and_t₍_P₎_∈I₍_U₎_{are equivalent conditions.} _❏

References

[1] CLARK, S. A. (1985):A complementary approach to the strong and weak axioms of revealed prefer-ence.Econometrica53_{, 1459–1463.}

[2] DUGGAN, J. (1999):A general extension theorem for binary relations.Journal of Economic Theory

86_{, 1–16.}

[3] HANSSON, B. (1968):Choice structures and preference relations.Synthese18, 443–458.

[4] RICHTER, M. (1966):Revealed preference theory.Econometrica34, 635–645.

[5] SUZUMURA, K. (1976):Rational choice and revealed preference.The Review of Economic Studies43,

149–158.