Introduction to Formal Preference Spaces

Eliza Niewiadomska; Adam Grabowski

Formalized Mathematics (2013)

  • Volume: 21, Issue: 3, page 223-233
  • ISSN: 1426-2630

Abstract

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In the article the formal characterization of preference spaces [1] is given. As the preference relation is one of the very basic notions of mathematical economics [9], it prepares some ground for a more thorough formalization of consumer theory (although some work has already been done - see [17]). There was an attempt to formalize similar results in Mizar, but this work seems still unfinished [18]. There are many approaches to preferences in literature. We modelled them in a rather illustrative way (similar structures were considered in [8]): either the consumer (strictly) prefers an alternative, or they are of equal interest; he/she could also have no opinion of the choice. Then our structures are based on three relations on the (arbitrary, not necessarily finite) set of alternatives. The completeness property can however also be modelled, although we rather follow [2] which is more general [12]. Additionally we assume all three relations are disjoint and their set-theoretic union gives a whole universe of alternatives. We constructed some positive and negative examples of preference structures; the main aim of the article however is to give the characterization of consumer preference structures in terms of a binary relation, called characteristic relation [10], and to show the way the corresponding structure can be obtained only using this relation. Finally, we show the connection between tournament and total spaces and usual properties of the ordering relations.

How to cite

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Eliza Niewiadomska, and Adam Grabowski. "Introduction to Formal Preference Spaces." Formalized Mathematics 21.3 (2013): 223-233. <http://eudml.org/doc/267147>.

@article{ElizaNiewiadomska2013,
abstract = {In the article the formal characterization of preference spaces [1] is given. As the preference relation is one of the very basic notions of mathematical economics [9], it prepares some ground for a more thorough formalization of consumer theory (although some work has already been done - see [17]). There was an attempt to formalize similar results in Mizar, but this work seems still unfinished [18]. There are many approaches to preferences in literature. We modelled them in a rather illustrative way (similar structures were considered in [8]): either the consumer (strictly) prefers an alternative, or they are of equal interest; he/she could also have no opinion of the choice. Then our structures are based on three relations on the (arbitrary, not necessarily finite) set of alternatives. The completeness property can however also be modelled, although we rather follow [2] which is more general [12]. Additionally we assume all three relations are disjoint and their set-theoretic union gives a whole universe of alternatives. We constructed some positive and negative examples of preference structures; the main aim of the article however is to give the characterization of consumer preference structures in terms of a binary relation, called characteristic relation [10], and to show the way the corresponding structure can be obtained only using this relation. Finally, we show the connection between tournament and total spaces and usual properties of the ordering relations.},
author = {Eliza Niewiadomska, Adam Grabowski},
journal = {Formalized Mathematics},
keywords = {preferences; preference spaces; social choice},
language = {eng},
number = {3},
pages = {223-233},
title = {Introduction to Formal Preference Spaces},
url = {http://eudml.org/doc/267147},
volume = {21},
year = {2013},
}

TY - JOUR
AU - Eliza Niewiadomska
AU - Adam Grabowski
TI - Introduction to Formal Preference Spaces
JO - Formalized Mathematics
PY - 2013
VL - 21
IS - 3
SP - 223
EP - 233
AB - In the article the formal characterization of preference spaces [1] is given. As the preference relation is one of the very basic notions of mathematical economics [9], it prepares some ground for a more thorough formalization of consumer theory (although some work has already been done - see [17]). There was an attempt to formalize similar results in Mizar, but this work seems still unfinished [18]. There are many approaches to preferences in literature. We modelled them in a rather illustrative way (similar structures were considered in [8]): either the consumer (strictly) prefers an alternative, or they are of equal interest; he/she could also have no opinion of the choice. Then our structures are based on three relations on the (arbitrary, not necessarily finite) set of alternatives. The completeness property can however also be modelled, although we rather follow [2] which is more general [12]. Additionally we assume all three relations are disjoint and their set-theoretic union gives a whole universe of alternatives. We constructed some positive and negative examples of preference structures; the main aim of the article however is to give the characterization of consumer preference structures in terms of a binary relation, called characteristic relation [10], and to show the way the corresponding structure can be obtained only using this relation. Finally, we show the connection between tournament and total spaces and usual properties of the ordering relations.
LA - eng
KW - preferences; preference spaces; social choice
UR - http://eudml.org/doc/267147
ER -

References

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  1. [1] Kenneth J. Arrow. Social Choice and Individual Values. Yale University Press, 1963. Zbl0984.91513
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  17. [17] Freek Wiedijk. Arrow’s impossibility theorem. Formalized Mathematics, 15(4):171-174, 2007. doi:10.2478/v10037-007-0020-9.[Crossref] 
  18. [18] Krzysztof Wojszko and Artur Kuzyka. Formalization of commodity space and preference relation in Mizar. Mechanized Mathematics and Its Applications, 4:67-74, 2005. 
  19. [19] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990. 
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  21. [21] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990. 

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