On the equivalence of the two existing extensions of the leximax criterion to the infinite case

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Abstract

Using a common framework, we consider the two existing extensions of the leximax criterion to infinite environments [Arlegi, R., Besada, M., Nieto, J., Vázquez, C., 2005. Freedom of choice: the leximax criterion in the infinite case. Mathematical Social Sciences 49, 1–15; Ballester, M., De Miguel, J.R., 2003. Extending an order to the power set: the leximax criterion. Social Choice and Welfare 21, 63–71], and show that, though the respective definitions of the rules and their axiomatic characterizations appear to differ considerably, they actually propose the same extension of the leximax criterion to the infinite case.

Introduction

Consider a given set of alternatives X over which there is defined a preference ordering R. Consider also the problem of ranking all the possible finite subsets of X. This formal problem can be understood in many different decisional contexts. Specifically, if we interpret the subsets as opportunity sets, and take the ranking of these to reflect the degree of freedom of choice with which they provide the decision maker, the mentioned problem is a natural way to formally describe people's valuation of freedom of choice. This is the focus of many authors in the so-called freedom of choice literature.1

Among the many different plausible rules one could imagine, one attractive proposal is the leximax ranking, which is defined, and axiomatically characterized by Bossert et al. (1994). This rule compares any two sets of alternatives by first looking at the best alternative in each set (with respect to R); if the comparison is not decisive (the elements are indifferent), then the rule considers the second-best alternative in each set, and so the procedure continues, if necessary, until there are no more alternatives to be compared in either or both of the sets. In the first case, the set with the greater number of alternatives is declared to be better; in the second, the two sets are indifferent.

The interest of this rule lies in the fact that it incorporates the intrinsic value of freedom of choice (any enlargement of an opportunity set leads to a strict improvement, unlike the standard indirect utility criterion), and the instrumental value of freedom of choice (preferences over the individual alternatives in the set matter, unlike the purely cardinalist rule proposed by Pattanaik and Xu (1990)).

A major drawback of the leximax rule as established in Bossert et al. (1994), however, is that it is not well defined for the case of infinite opportunity sets, and is therefore not applicable in meaningful economic contexts, such as the ranking of standard budget sets, which are typically compact (and infinite).

In an attempt to address this shortcoming, Ballester and De Miguel (2003) and Arlegi et al. (2005) have separately proposed and axiomatically characterized an extension of the leximax rule to the infinite case. The two proposals look very different both in their formulation and their axiomatic structure. To the best of our knowledge, moreover, these are, to date, the only two existing extensions of the leximax rule to the infinite case.

The aim of this study is to prove that, despite their apparent differences, the two extensions of the leximax are equivalent. In Section 2, we introduce the basic notation and definitions. Section 3 presents the definitions of the two extensions of the leximax as originally proposed by their respective authors. Section 4 contains the equivalence result and its proof. In Section 5 we make some comments on the logical relationship between the axioms used for the characterization in each article, and finish with some concluding remarks in Section 6.

Section snippets

Notation and definitions2

N and denote the set of all positive integers and the set of all real numbers, respectively, and n is the n-fold cartesian product of . Let X ⊂ n be a nonempty set of alternatives. In order to ensure that the axioms used are independent, X is assumed to contain at least three elements.

Let R be a complete, reflexive, transitive ordering on X that can be represented by a utility function. The asymmetric and symmetric parts of R are denoted, respectively, by P and I. The set of all subsets of

The two extensions of the leximax criterion to the infinite case

First, a preference relation on F(X) is considered. We denote by   l the finite leximax criterion. Let {a1,…, ak} and {b1,…, br} be two subsets of X, whose elements are labelled from best to worst (with respect to R). We consider A  l B when

  • i)

    there exists i  k such that aiPbi and ajIbj, for all j < i, or

  • ii)

    aiIbi, for all i  r and k  r.

Next, two preference relations on 2X are considered. We denote by   L the leximax criterion defined in Ballester and De Miguel (2003): For any pair of sets A, B  X, A  L B if

The main result

Theorem 1

The two preference relations defined on 2X,  L and  L, are the same.

Proof of Theorem 1

Let A, B  X such that A L B. We prove that A L B implies A  L B and that A L B implies A L B.

First, note that, whenever the empty set is involved in the comparison, A L ∅∼L ∅. Thus, we will concentrate on the remaining comparisons. For notational convenience throughout this proof, we assume that for the comparison between the sets A and B, the first step of the procedure (whether a1 and b1 exist or not) is the decisive one. Note

The axioms

As pointed out earlier, the axiomatic characterizations of the leximax in Ballester and De Miguel (2003), and in Arlegi et al. (2005) are different. Next, we present the axioms used in each article and briefly comment on the logical relationships among them. For the sake of fluency, we assume throughout that R is a linear ordering. While this affects the wording of some of the axioms, the logical relationships between them remain unchanged up to the corresponding restatements of them.3

Conclusions

In this work we have proved that the two so far existing extensions of the leximax rule to infinite environments are equivalent. Before this result was proved we had two possible ways of applying the leximax in infinite environments, each with advantages that were lacking the other: The leximax extension by Ballester and De Miguel (2003) is more synthetic and “compact”. We think it is mathematically more elegant and useful, for example, as a tool for the investigation of further results. On the

Acknowledgements

We thank an anonymous referee for his/her comments on a previous version of this work. This research has been supported by Spanish Ministry of Education grants SEC2003-08105 (a), BEC2002-3780 (c) from DGICYT, and SEJ2005-01481/ECON, FEDER and CREA (b).

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