A Representation of the Maximal Set in Choice Problems Where Information Is Incomplete

Banerjee and Pattanaik [1] proved that the maximal set generated by a qua-si-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.


Introduction
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 ( ) In general (for any binary relation R and any non-empty feasible set A) we have that , where the equality holds in case that R is complete. If a binary relation R is transitive, Banerjee and Pattanaik ([1], Proposition 3.2) showed that the maximal set generated by R is the union of the optimal choice sets generated by all possible orderings extending R. That is, where  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 [2] extended Banerjee and Pattanaik's result by relaxing the axiom of transitivity to the axiom that Suzumura calls consistency [6]. Arló   , 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.
Thef non-comparable part 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 The set of all best elements in A with respect to R is the greatest set of A, to be denoted by ( ) 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 ( ) A binary relation R is an extension of a binary relation R if and only if 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 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  is a class such that , 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 ([10], page 64), Hansson [11] and Fishburn [12] prove that any quasi-ordering has an ordering extension. Banerjee  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

Theorem 1. A quasi-ordering R is choice-functionally recoverable if and only if
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 [7] show by counter-example that the same is not necessarily true for a reflexive and consistent binary relation.
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 With Assumption (  ), Suzumura and Xu generalize the result of Banerjee and Pattanaik as follows: The following definition is of use in the next results. Lemma. Let R be a binary relation on X and let  be a closed upward and arc-receptive collection of binary relations on X such that R ∈  . Suppose that x X ∈ and A is an arbitrary subset of X. Then, there exists a binary relation R * ∈ such that for each y A Proof. Fix an x X ∈ . Given y A ∈ , let Well-Ordering Principle asserts that every set X can be well-ordered; that is, if K is any set, then there exists a well-ordered set Λ which serves as an index set for the elements of K, so we may write By definition, K has a first element, a second element, a third element and so on. Let y K λ ∈ . Then, (  be a chain in   and let ˆi Since  is closed upward, we conclude that R ∈ . But then, Therefore, any chain in   has an upper bound in   (with respect to set inclusion). By Zorn's lemma, there is a a contradiction to maximality of R * . It follows that M * = Λ . Therefore, for each y A ∈ with ( ) ( ) , , x y N x y ∈ , we have y y λ = for some λ ∈ Λ . Hence, For a given binary relation R on X, let be the set of all strict total extensions of R. Theorem 3. Let R be a binary relation on X and let  be a be a closed upward and arc-receptive collection of binary relations on X such that R ∈  .
We first show the ⊇ inclusion. Take any , and thus, It suffices to show the ⊆ inclusion. Take holds. On the other hand, By Lemma 3, there exists a binary relation R R * ⊇ in  such that for each y A Therefore, for each y A Consider now the case where R * is non-total. Let We have that R * ∈ , so this class is non-empty. Let It follows that Ĉ ∈   . Since  is closed upward, we conclude that Ĉ ∈ which implies that ˆĈ ∈ . Therefore, any chain in  has an upper bound in  (with respect to set inclusion). By Zorn's lemma, there is a maximal element R in  . If there existed distinct , z w X ∈ not comparable with respect to R , then the fact that  is arc-receptive would imply the existence of an extension R * of R ∈ with ( ), z w R * ∈ . But then, R * ∈ , which contradicts the maximality of R in  . The last contradiction shows that R is total. Therefore, for each y A ∈ , we have ( ), 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 [1] show that the maximal set generated by a quasi-ordering R is the union of the optimal sets generated by all possible ordering extensions of R. This result was extended by Suzumura and Xu [2] by relaxing the axiom of transitivity to the axiom of consistency [6]. The choice rule, which relates the maximal set generated by a binary relation R and the union of the sets of best elements generated by all possible orderings extending R, is called by the name of "the choice functional recoverability" (see [2], Definition 3.1). In the case of choice-functional recoverability achieved by the above mentioned authors, we focus on the recoverability of the choice set defined in terms of R by means of the set of the best elements defined in terms of each and every transitive and complete extension of R.
However, as Arló Costa pointed out in [3]