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Banerjee and Pattanaik [1] proved that the maximal set generated by a quasi-ordering is equal to the union of the sets of best elements of its ordering extensions. Suzumura and Xu [2] extended Banerjee and Pattanaik’s result by relaxing the axiom of transitivity to the axiom that Suzumura calls consistency. Arló Costa in [3] pointed out that in general, an optimizing model cannot require the transitivity of the binary relation used in an optimizing model. In this paper, by using two important ideas of John Duggan [4] , I extend the above mentioned results to arbitrary binary relations whose extensions are complete and not necessarily transitive.

The economic approach to rational behaviour assumes that each individual makes choices by selecting, from each feasible set of alternatives, those which maximize his own preference relation. The classical framework of optimization used in standard choice theory recommends choosing, among the feasible options, a best alternative. According to this modeling of a choice process, the optimal choice set consists of the best alternatives according to a binary relation R. So, if A is the feasible set of alternatives and R is a binary relation over A, a formalization of this idea requires the following definition of the optimal choice set G ( A , R ) :

G ( A , R ) = { x ∈ A | forall y ∈ A , x R y } .

Many economists have pointed out that this stringent form of maximization might not be the kind of optimization that one can apply in problems where information is incomplete. For example, the Nobel-Prize winner Amartya Sen ( [

M ( A , R ) = { x ∈ A | forno y ∈ A , y P ( R ) x } .

In general (for any binary relation R and any non-empty feasible set A) we have that G ( A , R ) ⊆ M ( A , R ) , where the equality holds in case that R is complete. If a binary relation R is transitive, Banerjee and Pattanaik ( [

M ( A , R ) = ∪ R ′ ∈ R G ( A , R ′ ) ,

where R is the set of ordering extensions of R. In other words, Banerjee and Pattanaik’s result starts from a transitive binary relation R in that there are some ordered pairs, say ( x , y ) ∈ X × X , over which R does not convey any information, and answers whether all the information originally conveyed by R can be recovered in terms of the set of all ordering extensions of R. Suzumura and Xu [

R ∗ = { ( x , x ) , ( y , y ) , ( z , z ) , ( y , x ) , ( x , z ) , ( z , x ) , ( z , y ) , ( y , z ) }

is an extension of R satisfying G ( A , R ) = M ( A , R ∗ ) . Obviously, R ∗ is complete but not transitive. This example shows that the extended binary relations used in order to replicate maximizing process tend not to be transitive. So, we must expect that an optimizing model cannot require the transitivity of the binary relation used in the optimizing model.

In this paper, I extend the Banerjee-Pattanaik’s and Suzumura-Xu’s results to arbitrary binary relations whose extensions are complete and not necessarily transitive.

We recall some definitions from Suzumura [

Let X be a non-empty universal set of alternatives and R ⊆ X × X be a binary relation on X. Let P ( X ) be the set of all subsets of X. We sometimes abbreviate ( x , y ) ∈ R as x R y . The asymmetric part of R is defined by

P ( R ) = { ( x , y ) ∈ X × X | ( x , y ) ∈ R and ( y , x ) ∉ R }

and the symmetric part of R is defined by

I ( R ) = { ( x , y ) ∈ X × X | ( x , y ) ∈ R and ( y , x ) ∈ R } .

Thef non-comparable part N ( R ) of R is defined by letting, for all x , y ∈ X , ( x , y ) ∈ N ( R ) if and only if ( x , y ) ∉ R and ( y , x ) ∉ R . For any subset A of X, an element x ∈ A is a maximal element in A with respect to R if for all y ∈ A , ( y , x ) ∉ P ( R ) . Dually is defined the notion of minimal element. The set of all maximal elements in A with respect to R is the maximal set of A, to be denoted by M ( A , R ) . Likewise, an element x ∈ A is a best element in A with respect to R if ( x , y ) ∈ R holds for all y ∈ A . The set of all best elements in A with respect to R is the greatest set of A, to be denoted by G ( A , R ) . We say that R on X is 1) reflexive if for each x ∈ X ( x , x ) ∈ R ; 2) asymmetric if for each x ∈ X ( x , y ) ∈ R implies ( y , x ) ∉ R ; 3) transitive if for all x , y , z ∈ X , [ ( x , z ) ∈ R and ( z , y ) ∈ R ] ⇒ ( x , y ) ∈ R ; 4) anti-symmetric if for each x , y ∈ X , [ ( x , y ) ∈ R and ( y , x ) ∈ R ] ⇒ x = y ; 5) total if for each x , y ∈ X , x ≠ y we have x R y or y R x ; 6) complete if for each x , y ∈ X , we have x R y or y R x . It follows that R is complete if and only if it is reflexive and total. The following combination of properties are considered in the next theorems. A binary relation R on X is 1) a quasi-ordering if R is reflexive and transitive; 2) an ordering if R is a total quasi-ordering; 4) a partial order if R is an anti-symmetric quasi-ordering; 5) a linear ordering if R is an anti-symmetric ordering; 6) tournament if R is asymmetric and total. For any binary relation R, let R ¯ be the transitive closure of R, which is defined by ( x , y ) ∈ R ¯ if and only if there exist m ∈ ℕ and z 0 , ⋯ , z m ∈ X such that x = z 0 , ( z k , z k + 1 ) ∈ R for all k ∈ { 0, ⋯ , m − 1 } and z m = y . The relation R is consistent, if for all x , y ∈ X , ( x , y ) ∈ R ¯ implies ( y , x ) ∉ P ( R ) (see [

A binary relation R ^ is an extension of a binary relation R if and only if R ⊆ R ^ and P ( R ) ⊆ P ( R ^ ) . If an extension R ^ of R is an ordering, we call it an ordering extension of R. We call a binary relation R ^ on X satisfying R ⊂ R ^ , P ( R ) ⊂ P ( R ^ ) and R ≠ R ^ a strict extension of R. For a given binary relation R on X, let R ( R ) be the set of all ordering extensions of R and R Σ ( R ) be the set of all strict ordering extensions of R.

A set X is well-ordered if there is a binary relation ≤ on X which is a linear order and for which every non-empty subset of X has a minimal element. A chain C is a class such that B , B ′ ∈ C implies B ⊆ B ′ or B ′ ⊆ B . A partially ordered set is a set X together with a partial ordering ≤ . Zorn’s lemma states that if X is a partially ordered set such that every chain in X has an upper bound, then X has a maximal element.

Arrow ( [

M ( A , R ) = ∪ R ′ ∈ R ( R ) G ( A , R ′ )

is equivalent to the non-emptiness of R Σ ( R ) . Suzumura and Xu [

Definition 3.1. ( [

M ( A , R ) = ∪ R ′ ∈ R Σ ( R ) G ( A , R ′ )

holds for all non-empty subsets A ∈ X .

According to what we have said above, the choice-functional recoverability of a quasi-ordering is equivalent to the non-emptiness of the set of its strict ordering extensions. The following theorem, which is due to Banerjee and Pattanaik [

Theorem 1. A quasi-ordering R is choice-functionally recoverable if and only if R Σ ( R ) ≠ ∅ .

While the choice-functional recoverability of a quasi-ordering R is equivalent to the non-emptiness of the set of its strict ordering extensions, Suzumura and Xu [

In order to generalize Theorem 1, Suzumura and Xu consider the following assumption ( ⋆ ):

( ⋆ ) Let R be a binary relation on X. For all x , y ∈ X , ( x , y ) ∈ P ( R ¯ ) implies ( x , y ) ∈ R .

With Assumption ( ⋆ ), Suzumura and Xu generalize the result of Banerjee and Pattanaik as follows:

Theorem 2. A reflexive and consistent binary relation R on X is choice-functionally recoverable if and only if R Σ ( R ) ≠ ∅ and Assumption ( ⋆ ) hold.

Now, I give a more general theorem of recoverability of choice functions for binary relations whose extensions are total and not necessarily reflexive and transitive.

The following definition is of use in the next results.

Definition 3.2. (see [

Lemma. Let R be a binary relation on X and let

Proof. Fix an

By definition, K has a first element, a second element, a third element and so on. Let

Let

with

a contradiction to maximality of

For a given binary relation R on X, let

Theorem 3. Let R be a binary relation on X and let

Proof. Let R,

We first show the

It suffices to show the

By Lemma 3, there exists a binary relation

Consider now the case where

We have that

we conclude that

It follows that

Therefore, R is choice-functionally recoverable.

To prove the converse, suppose that R is choice-functionally recoverable, so that we have

for all non-empty subsets A of X. Clearly,

Theorem 1 of Banerjee and Pattanaik as well as Theorem 2 of Suzumura and Xu are corollaries of Theorem 3.

Proof of Theorem 1. Let R be a quasi-ordering on X and let

Proof of Theorem 2. Let R be a reflexive and consistent binary relation on X satisfying assumption (

Then, there exists

Therefore,

Clearly,

For much of the economic analysis, the characterization of maximizing the utility of individuals as a criterion of rationality (optimization) can pose serious problems, especially in the case where no alternative can be identified as the best choice. This kind of optimization is often ineffective for finding a choice set, which only requires choosing an alternative that is not judged to be worse than any other. A nice regularity condition to the above procedure for the construction of non-empty choice sets is acyclicity. This is because acyclicity is sufficient for the existence of maximal elements when the set of alternatives is finite, and it is also necessary for the existence of maximal elements in all subsets of alternatives. In the special case in which R is transitive, Banerjee and Pattanaik [

The authors declare no conflicts of interest regarding the publication of this paper.

Athanasios, A. (2018) A Representation of the Maximal Set in Choice Problems Where Information Is Incomplete. Theoretical Economics Letters, 8, 2631-2639. https://doi.org/10.4236/tel.2018.811167