# Introduction to Formal Preference Spaces

Open access

## Summary

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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

• [1] Kenneth J. Arrow. Social Choice and Individual Values. Yale University Press 1963.

• [2] Robert J. Aumann. Utility theory without the completeness axiom. Econometrica 30(3): 445-462 1962.

• [3] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics 1(2):377-382 1990.

• [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics 1(1):91-96 1990.

• [5] Czesław Bylinski. Partial functions. Formalized Mathematics 1(2):357-367 1990.

• [6] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics 1(1):47-53 1990.

• [7] Agata Darmochwał. Finite sets. Formalized Mathematics 1(1):165-167 1990.

• [8] Klaus E. Grue and Artur Korniłowicz. Basic operations on preordered coherent spaces. Formalized Mathematics 15(4):213-230 2007. doi:10.2478/v10037-007-0025-4.

• [9] Sören Halldén. On the Logic of Better. Lund: Library of Theoria 1957.

• [10] Emil Panek. Podstawy ekonomii matematycznej. Uniwersytet Ekonomiczny w Poznaniu 2005. In Polish.

• [11] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics 1(3):441-444 1990.

• [12] George F. Schumm. Transitivity preference and indifference. Philosophical Studies 52: 435-437 1987.

• [13] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics 1(1): 115-122 1990.

• [14] Andrzej Trybulec. Enumerated sets. Formalized Mathematics 1(1):25-34 1990.

• [15] Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics 1(2):313-319 1990.

• [16] Zinaida Trybulec. Properties of subsets. Formalized Mathematics 1(1):67-71 1990.

• [17] Freek Wiedijk. Arrow’s impossibility theorem. Formalized Mathematics 15(4):171-174 2007. doi:10.2478/v10037-007-0020-9.

• [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] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics 1 (1):73-83 1990.

• [20] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics 1(1):181-186 1990.

• [21] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics 1(1):85-89 1990.

### Formalized Mathematics

Search
###### Journal information
Impact Factor

CiteScore 2018: 0.42

SCImago Journal Rank (SJR) 2018: 0.111
Source Normalized Impact per Paper (SNIP) 2018: 0.169

Target audience:

researchers in the fields of formal methods and computer-checked mathematics

###### Metrics
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 228 91 4