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A Selection of Maximal Elements Under Non-Transitive Indifferences

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A Selection of Maximal Elements Under Non-Transitive Indifferences Journal of Mathematical Psychology 54 (2010) 481–484 Contents lists available at ScienceDirect Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp A selection of maximal elements under non-transitive indifferences José Carlos R. Alcantud a,∗, Gianni Bosi b, Magalì Zuanon c a Facultad de Economía y Empresa, Universidad de Salamanca, E 37008 Salamanca, Spain b Facoltà di Economia, Università degli Studi di Trieste, Piazzale Europa 1, 34127 Trieste, Italy c Facoltà di Economia, Università degli Studi di Brescia, Contrada Santa Chiara 50, 25122 Brescia, Italy article info a b s t r a c t Article history: In this work we are concerned with maximality issues under intransitivity of the indifference. Our Received 19 August 2009 approach relies on the analysis of ``undominated maximals'' (cf., Peris & Subiza, 2002). Provided that an Received in revised form agent's binary relation is acyclic, this is a selection of its maximal elements that can always be done when 4 August 2010 the set of alternatives is finite. In the case of semiorders, proceeding in this way is the same as using Luce's Available online 13 October 2010 selected maximals. We present a sufficient condition for the existence of undominated maximals for interval orders Keywords: without any cardinality restriction. Its application to certain types of continuous semiorders is very Maximal element Selection of maximals intuitive and accommodates the well-known ``sugar example'' by Luce. Acyclicity ' 2010 Elsevier Inc. All rights reserved. Interval order Semiorder 1. Introduction In light of these two facts, it seems interesting to provide conditions for the existence of undominated maximals without any Even though there are arguments to ensure the existence of cardinality restriction. We present a condition that is sufficient for maximal elements for binary relations in very general settings, this the existence of undominated maximal elements in interval orders. concept does not always explain the choice under non-transitive Its specialization to certain types of continuous semiorders is very indifference well. Luce (1956) argued that in order to account for intuitive and accommodates the well-known ``sugar example'' by certain procedural aspects better, some ``selection of maximals'' Luce. helps the researcher. From his Introduction: ``... a maximization This work is organized as follows. In Section 2, we establish our principle is almost always employed which states in effect that notation and preliminary definitions. Then in Section 3, we analyse a rational being will respond to any finite difference in utility, the existence of undominated maximals in the case of unrestricted however small. It is, of course, false that people behave in this cardinality of the set of alternatives. As an application, a concrete manner''. This author proposed the concept of a semiorder as specification leading to Luce's analysis of the ``sugar example'' is a way to deal with intransitive indifferences without giving up provided. Finally, we investigate the role of different assumptions transitivity of the strict preference. in our results. Section 4 contains some conclusions and remarks. In this work we are concerned with maximality considerations under intransitivity of the indifference. It is based on the analysis 2. Notation and preliminaries of ``undominated maximals'', a concept introduced and explored by Peris and Subiza (2002). They established two particularly remarkable facts. For one thing, such selection of maximals can be Let us fix a set X of alternatives. Unless otherwise stated, ≻ ≻ ≻ · · · ≻ done when the set of alternatives is finite provided that the binary henceforth denotes an acyclic relation, i.e., x1 x2 xn 6D 2 relation is acyclic. Then, proceeding in this way is the same as using implies x1 xn for all x1;:::; xn X. Its lower (resp., upper) 2 f 2 V ≻ g Luce's selected maximals in the case of semiorders.1 contour set associated with x X is z X x z (resp., fz 2 X V z ≻ xg). A subset A ⊆ X is a lower (resp., upper) set of ≻ when a 2 A; x 2 X, and a ≻ x (resp., x ≻ a) implies x 2 A. ∗ Corresponding author. E-mail address: [email protected] (J.C.R. Alcantud). URL: http://web.usal.es/jcr (J.C.R. Alcantud). weak order then the induced weak ordering is identical to the given one. The term 1 Luce (1956, Section 3) states that in terms of a semiorder on a set it is possible Luce's selected maximals of a semiorder refers to the maximal elements of such weak to define a natural weak ordering on the same set. Further, if the semiorder is a ordering, provided that they exist. Example 2 formalizes this construction. 0022-2496/$ – see front matter ' 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmp.2010.08.001 482 J.C.R. Alcantud et al. / Journal of Mathematical Psychology 54 (2010) 481–484 Denote by % the complement of the dual of ≻ (i.e., x % y if and various approaches to that issue but the most celebrated one only if y ≻ x is false), and by ∼ the indifference relation associated probably is the Bergstrom–Walker theorem and its variations with ≻ (i.e., x ∼ y if and only if both x % y and y % x). (cf., Bergstrom, 1975; Walker, 1977). Its basic form states that With every acyclic relation ≻ on X we associate the traces ≻∗ upper semicontinuous acyclic relations on compact topological and ≻∗∗ defined as follows: for all x; y 2 X, spaces have maximal elements. Example 1 shows that even in the ∗ case of semiorders on countable sets, this specification does not x ≻ y , 9ξ 2 X V x ≻ ξ % y; suffice to ensure that undominated maximals exist. ∗∗ x ≻ y , 9η 2 X V x % η ≻ y: Example 1. Let us fix A D N D f1; 2; 3;:::g. The next expression ∗ ∗∗ Therefore, if we denote by % and % the respective complements produces an upper semicontinuous semiorder with respect to the of the duals of ≻∗ and ≻∗∗, we have excluded point topology associated with f1g on A, which is always 2 ∗ compact : x % y , .y ≻ z ) x ≻ z/; ∗∗ m ≻ n if and only if m is odd; n is even, and m C 1 > n: x % y , .z ≻ x ) z ≻ y/: Although A has an infinite number of maximal elements (namely, ≻ We recall that a binary relation on X is an interval order if it is the odd numbers) there are no undominated maximals because irreflexive and the following condition is verified for all x; y; z; w 2 .m C 2/ ≻D m when m is odd. X: If we adhere to the topological approach in our quest for condi- .x ≻ z/ and .y ≻ w/ ) tions that guarantee that undominated maximals do exist, then we .x ≻ w/ or .y ≻ z/: need to consider other suitable assumptions. That is the purpose of Section 3.2. Further, a binary relation ≻ on X is a semiorder if ≻ is an interval order and the following condition is verified for all x; y; z; w 2 X: 3.2. Existence of undominated maximals for unrestricted domains ≻ ≻ ) .x y/ and .y z/ The next lemma shows that an alternative expression for the set .x ≻ w/ or .w ≻ z/: of undominated maximals can be given only under acyclicity of ≻. ∗ ∗∗ If ≻ is an interval order then ≻ and ≻ are weak orders (i.e., Lemma 1. Suppose that ≻ is an acyclic relation on X. Then asymmetric and negatively transitive binary relations). The traces D ∗ ∗∗ UM X ≻ D M M X ≻ ≻ ≻ and ≻ extend the interval order ≻ (i.e., x ≻ y implies both . ; / . ; /; /: x ≻∗ y and x ≻∗∗ y for all x; y 2 X). If ≻ is a semiorder then the 0 ∗ ∗∗ Proof. Along the proof of Peris and Subiza (2002, Theorem 2) the binary relation ≻ D ≻ [ ≻ is a weak order (cf., Fishburn, D ∗ inclusion M.M.X; ≻/; ≻ / ⊆ UM.X; ≻/ is proved. The fact that 1985, Theorem 2 of Section 2) and therefore we have that x ≻ y 2 ≻ ≻ ≻D ∗∗ every x UM.X; / belongs to M.M.X; /; / is immediate. implies that x % y for all x; y 2 X. Using the terminology of Peris and Subiza (2002), the weak Next we present some technical and useful properties that hold D D in our setting. dominance relation % and the strict dominance relation ≻ associated with an interval order ≻ on a set X can be defined as Lemma 2. Suppose that ≻ is an acyclic relation on X. Then, follows: for each x; y 2 X, ∗ (1) ≻D ⊆ ≻ on M.X; ≻/, ∗ D ∗ ∗∗ (2) M.X; ≻ / ⊆ M.X; ≻/. x % y , x % y and x % y; D D D Proof. In order to check (1) we notice that the original expression x ≻ y , x % y and not .y % x/: for ≻D, namely We denote by M.X; ≻/ the set of maximal elements relative to ≻ on ( ≻∗ ∗∗ X, i.e., M.X; ≻/ D fx 2 X V 8z 2 X; z ≻ x is falseg. x y and x % y .a/ ≻D , If τ is a topology on X; ≻ is upper semicontinuous if its lower x y or ≻∗∗ ∗ contour sets are open. From Alcantud (2002), we say that .X; τ/ x y and x % y .b/ is ≻-upper compact if for each collection of lower open sets that can be simplified because now x ≻∗∗ y is impossible since y is covers X there exists a finite subcollection that also covers X. maximal for ≻. This fact rules out (b) and yields the conclusion. ∗ Part (2) is direct because ≻ extends ≻: if x ≻ y then x ≻ y % y 3. Selection of maximal elements for acyclic relations due to irreflexivity of ≻. The set of undominated maximal elements of X is defined as Remark 1. Besides Lemma 2(1), we can further note that if ≻ is a semiorder then ≻D D ≻∗ on M.X; ≻/.
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