The non-existence of a utility function and the structure of non-representable preference relations

Beardon, Alan F.; Candeal, Juan Carlos; Herden, Gerhard; Induráin, Esteban; Mehta, Ghanshyam B.

doi:10.1016/s0304-4068(02)00003-4

Cited by 61 publications

(32 citation statements)

References 18 publications

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“…Furthermore, the analytical form of these functions can be explicitly given in a remarkably simple way. This sheds further light on the significance of [1] and [2], since the lexicographically decomposable chains isolated and examined in these papers do not only constitute an important class of linear orders lacking real-valued utility functions, but also turn out to be a class of linear orders admitting uncountably large families of analytically specifiable utility functions on non-Archimedean extensions of the reals. In view of the last fact, it is also possible to introduce uncountable families of utility functions for certain lexicographic products of linear orders, in particular R n , for every n > 2.…”

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confidence: 79%

“…This suggests that a utility representation for L 2 should assign to every vector of the form 0, n a value that is infinitely close to that of 0, 0 . The same argument naturally extends to encompass the class of lexicographically decomposable chains described in [2] and studied in [1]. Since these chains are isomorphic to certain lexicographic orders, and one may regard lexicographic orders as a linear arrangement of clusters of infinitely close points, it is natural to associate them with a utility function on a non-Archimedean structure.…”

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confidence: 94%

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Nonstandard utilities for lexicographically decomposable orderings

Rizza

2015

Journal of Mathematical Economics

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Using a basic theorem from mathematical logic, I show that there are field-extensions of R on which a class of orderings that do not admit any real-valued utility functions can be represented by uncountably large families of utility functions. These are the lexicographically decomposable orderings studied in [1]. A corollary to this result yields an uncountably large family of very simple utility functions for the lexicographic ordering * Email: d.rizza@uea.ac.uk 1 of the real Cartesian plane. I generalise these results to the lexicographic ordering of R n , for every n > 2, and to lexicographic products of lexicographically decomposable chains. I conclude by showing how almost all of these results may be obtained without any appeal to the Axiom of Choice. Non-Archimedean utility functionsUtility functions may be seen as strong homomorphisms from a complete preorder Z = Z, into a numerical ordering. Although a customary choice for the codomain of a utility function is the set R of real numbers, alternatives that violate the Archimedean property have been studied since at least the 1950's (see e.g. [6] 2 preference to be represented. In this paper I adopt this perspective on a class of linear orders that lack real-valued utility functions and show that each of them has an uncountable family of utility functions on an arbitrary, elementary extension of the real field containing positive infinitesimals. As a consequence of this result, I also establish a connection between lexicographically ordered, real vector spaces and elementary extensions of the reals, the two more prominent choices of non-Archimedean codomain for utility functions. The best known linear order from the class I consider was introduced by Debreu in [4]: it is the lexicographic ordering of the real Cartesian plane, i.e., the chain L 2 = R 2 , 2 , where the binary relation 2 is defined by the condition: r, s 2 r , s iff r < r or r = r and s ≤ s .Seen as a vector space, L 2 is non-Archimedean, since there is no positive, integer multiple of 0, 1 that is greater than 1, 0 . This suggests that a utility representation for L 2 should assign to every vector of the form 0, n a value that is infinitely close to that of 0, 0 . The same argument naturally extends to encompass the class of lexicographically decomposable chains described in [2] and studied in [1]. Since these chains are isomorphic to certain lexicographic orders, and one may regard lexicographic orders as a linear arrangement of clusters of infinitely close points, it is natural to associate them with a utility function on a non-Archimedean structure. Theorem 3.2 shows that uncountable families of such functions always exist 2 . This result locally improves in several ways the general theorem, proved by Skala in [14] and more concisely presented by Narens in [12], to the effect that every transitive and complete relation has a utility function on a particular ultrapower extension of the reals.I show in section 3 that, with regard to lexicographically decomposable chains,2 To be preci...

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Nonstandard utilities for lexicographically decomposable orderings

Rizza

2015

Journal of Mathematical Economics

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“…Intuitively, we may notice that, if there were a utility function representing L each vertical line should be mapped into an open interval of the real line R. However, since in the real line any family of pairwise disjoint open intervals is countable, we would not have room enough to represent the whole plane R 2 , because the set of real lines is uncountable. (For further details, see [37,44,59,60], as well as p. 73 in [61], among many other possible sources.) Remark 10.…”

Section: Remarkmentioning

confidence: 99%

A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings

Campión

Gómez‐Polo

Induráin

et al. 2018

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Different abstract versions of entropy, encountered in science, are interpreted in the light of numerical representations of several ordered structures, as total-preorders, interval-orders and semiorders. Intransitivities, other aspects of entropy as competitive systems, additivity, etc., are also viewed in terms of representability of algebraic structures endowed with some compatible ordering. A particular attention is paid to the problem of the construction of an entropy function or their mathematical equivalents. Multidisciplinary comparisons to other similar frameworks are also discussed, pointing out the mathematical foundations.

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“…As analyzed in Beardon et al [5] (see also Giarlotta [23,24]), there are three different classes of totally ordered structures that cannot be represented in the real line through an order-isomorphism. These classes are: (i) long chains, i.e., chains containing a subchain orderisomorphic to the first uncountable ordinal, (ii) Aronszajn chains, i.e., uncountable chains that neither are long, nor contain any uncountable representable subchain, (iii) planar chains, i.e., chains containing a subchain order-isomorphic to a non-representable subset of the lexicographic plane.…”

Section: Introductionmentioning

confidence: 99%