On continuous multi-utility representations of semi-closed and closed preorders

Bosi, Gianni; Herden, Gerhard

doi:10.1016/j.mathsocsci.2015.10.006

Cited by 19 publications

(16 citation statements)

References 13 publications

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“…Furthermore, these concepts are strongly related to similar ideas on economics (e.g. utilities, Richter-Peleg multi-utility representations, ... 1,5,16,31,33 ). Besides, partial functions are quite common in computing so, dealing with partial functions in order to represent orderings could be a good technique.…”

Section: Partial Representability Of Preordersmentioning

confidence: 99%

“…It is trivial too that this semiorder fails to be regular and so, it cannot be represented through a SSrepresentation. 5 Nevertheless, it is easily partial SS-representable just by means of the following two partial functions:…”

Section: Accepted Manuscriptmentioning

confidence: 99%

“…Several recent papers have studied the problem of representability 1,2,13,7,16,20,26,29 , as well as the corresponding continuity 4,5,14,12 . Although new results and some characterizations have been achieved, some questions are still open.…”

Section: Further Commentsmentioning

confidence: 99%

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Partial Representations of Orderings

Bosi

Estevan

Zuanon

2018

Int. J. Unc. Fuzz. Knowl. Based Syst.

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In the present paper a new concept of representability is introduced, which can be applied to not total and also to intransitive relations (semiorders in particular). This idea tries to represent the orderings in the simplest manner, avoiding any unnecessary information. For this purpose, the new concept of representability is developed by means of partial functions, so that other common definitions of representability (i.e. (Richter-Peleg) multi-utility, Scott-Suppes representability, … ) are now particular cases in which the partial functions are actually functions. The paper also presents a collection of examples and propositions showing the advantages of this kind of representations, particularly in the case of partial orders and semiorders, as well as some results showing the connections between distinct kinds of representations.

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Section: Partial Representability Of Preordersmentioning

confidence: 99%

Section: Accepted Manuscriptmentioning

confidence: 99%

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Partial Representations of Orderings

Bosi

Estevan

Zuanon

2018

Int. J. Unc. Fuzz. Knowl. Based Syst.

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“…We hope that in the future we shall be able to characterize the preordered spaces in which every closed preorder has a countable continuous multi-utility representation. We note that Theorem 3.4 in [5] represents a result of this type with respect to the existence of continuous multi-utility representations of every closed preorder.…”

Section: Discussionmentioning

confidence: 94%

Normally preordered spaces and continuous multi-utilities

2016

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“…One source of counterexamples is a topological vector space (and hence mixture space): L p [0, 1], with the usual norm, with 0 < p < 1, which has no non-zero continuous linear functionals (Rudin, 1991, §1.47). As far as we know, the strongest necessary condition for CMR to hold is given by Bosi and Herden (2016), under the assumption that the topology is first countable. Bosi and Herden remark that they do not see any possibility for satisfactorily avoiding that assumption.…”

Section: Related Literaturementioning

confidence: 99%

Expected utility theory on mixture spaces without the completeness axiom

McCarthy¹,

Mikkola²,

Thomas³

2021

Preprint

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A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a mixture preorder has a multirepresentation that consists of real-valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the mixture space is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result.

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On continuous multi-utility representations of semi-closed and closed preorders

Cited by 19 publications

References 13 publications

Partial Representations of Orderings

Partial Representations of Orderings

Normally preordered spaces and continuous multi-utilities

Expected utility theory on mixture spaces without the completeness axiom

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