On the Preference Relations with Negatively Transitive Asymmetric Part. I (original) (raw)
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On Bubbling of Linearly Ordered Sets. Part I
Given a loset I, every surjective map p: A ---> I endows the set A with a structure of preordered set by "replacing" the elements of I with their inverse images via p considered as "bubbles" (sets endowed with an equivalence relation), lifting the structure of loset on A, and "agglutinating" this structure with the bubbles. Every bubbling A of a structure of loset I is a structure of preordered set A (not necessarily complete) whose preorder has negatively transitive asymmetric part and every such structure on a given set A can be obtained by bubbling up of certain structure of a loset I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of bubbles. As a consequence of this characterization, under certain natural topological conditions on the preordered set A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.
From preference to utility: a problem of descriptive set theory
Notre Dame Journal of Formal Logic, 1985
Some years ago J. H. Silver proved that a co-analytic equivalence relation on a Polish space has either countably many or continuum many equivalence classes. Later L. Harrington greatly simplified the complicated original proof. The present paper is a sort of footnote to Harrington's lectures on these matters. It will be shown that information developed in his proof settles a problem of (hyper-)theoretical mathematical economics first investigated by Wesley [13] and Mauldin [8]. Namely, it will be shown that any family of closed preference orders that is parametrized in a Borel fashion can be represented by a family of continuous utility functions parametrized in an absolutely measurable fashion. Though the author is greatly indebted to Mauldin's work [8], the treatment of the problem here will be self-contained. Background and motivation for problems of this kind can be found in [6], Section 2.1. Terminology and notation pertaining to descriptive set theory will be as in [9]. 2 Definitions Throughout let ψ be a topological space. A preference order on 'ψ is any transitive, connected binary relation <*. Associated are the strict preference and indifference relations given by: x <* y <-> x <* y & ~y <* x X =* y +-> x <* y & y <* χ m Note that Ξ* i s an equivalence relation, and that <* induces a linear order on its equivalence classes, [x]* will denote the equivalence class of x. <* will be *Research in part supported by USA National Science Foundation Grant MCS 8003254.
Mathematical Topics on Representations of Ordered Structures and Utility Theory
Studies in Systems, Decision and Control, 2020
A continuous multi-utility fully represents a not necessarily total preorder on a topological space by means of a family of continuous increasing functions. While it is very attractive for obvious reasons, and therefore it has been applied in different contexts, such as expected utility for example, it is nevertheless very restrictive. In this paper we first present some general characterizations of the existence of a continuous order-preserving function, and respectively a continuous multi-utility representation, for a preorder on a topological space. We then illustrate the restrictiveness associated to the existence of a continuous multi-utility representation, by referring both to appropriate continuity conditions which must be satisfied by a preorder admitting this kind of representation, and to the Hausdorff property of the quotient order topology corresponding to the equivalence relation induced by the preorder. We prove a very restrictive result, which may concisely described as follows: the continuous multi-utility representability of all closed (or equivalently weakly continuous) preorders on a topological space is equivalent to the requirement according to which the quotient topology with respect to the equivalence corresponding to the coincidence of all continuous functions is discrete.
Some algebraic characterizations of preference structures
Journal of Interdisciplinary Mathematics, 2004
In decision theory several preference structures are used for modeling coherence and rational behavior. In this paper we establish, from an algebraic approach, characterizations of some general properties involving preference and indifference relations, as well as the more common preference structures used in the literature.
2018
Throughout the proof, we will denote RC by R to simplify the notation. (That is, for any x and y in X, we have {x} = C{x, y} iff x R y, and {x, y} = C{x, y} iff x R y.) Consequently, C ⊆ R and ∼C ⊆ R. We will use these facts below as a matter of routine. Also, when % ∈ P(C), we write M(S) = MAX(S,%) for any S ∈ X. First, we borrow the following results, each corresponds to Lemma A.5, Claim 1, and Claim 3, from the proof of Theorem 5.3. We note that the proofs of these results do not rely on the assumption that the choice correspondence C is nonempty-valued.
Continuous representation of a preference relation on a connected topological space
Journal of Mathematical Economics, 1987
Necessary and sufficient conditions are given for the existence of two continuous real valued functions u and o on a connected topological space X endowed with a preference relation < (i.e., an asymmetric binary relation) such that y is preferred to x if and only if u(y)>u(x). It is shown that these conditions-a slight generalization of the usual ones encountered in classical utility theory-entail the existence of such a continuous representation a, u with a and u continuous utility functions for two complete preorders intimately connected with the preference relation <.
A note on the representation of preferences
Mathematical Social Sciences, 1995
We consider a class of relations which includes irreflexive preference relations and interdependent preferences. For this class, we obtain necessary and sufficient conditions for representation of the relation by two numerical functions in the sense of a ~ x if and only if u(a) < vex).
Representing preferences with nontransitive indifference by a single real-valued function
Journal of Mathematical Economics, 1995
Let + be an interval order on a topological space (X, r), and let x < L y if and only if [y<z~x+z], and x+2' y if and only if [z <x * z + y]. Then <t and <t" are complete preorders. In the particular case when 4 is a semiorder, let x <t!. y if and only if x.<* _ ~1 and x<"* y. Then 4% is a complete preorder, too. We present sufficient conditions for the existence of continuous utility functions representing < t , -X 2 * and <'1 , by using the notion of strong separability of a preference relation, which was introduced by Chateauneuf (Journal of Mathemarical Economics, 1987, 16, 139-146). Finally, we discuss the existence of a pair of continuous functions u, u representing a strongly separable interval order + on a measurable topological space (X, T, CL, A).