@MISC{PREFER_1.MIZ,
  AUTHOR = {Niewiadomska, Eliza and Grabowski, Adam},
  TITLE = {Introduction to Formal Preference Spaces},
  SECTION1 = {Preliminaries},
  SECTION2 = {Properties of Binary Relations},
  SECTION3 = {Preference Structures},
  SECTION4 = {Constructing Examples},
  SECTION5 = {Characteristic Relation of a Preference Space},
  SECTION6 = {Generating Preference Space from Arbitrary (Characteristic) Relation},
  SECTION7 = {Flat Preference Spaces},
  SECTION8 = {Tournament Preference Spaces},
  SECTION9 = {Total Preference Spaces},
  DAY = {7},
  MONTH = {October},
  YEAR = {2013},
  ADDRESS1 = {Institute of Mathematics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok\\Poland},
  ADDRESS1 = {Institute of Informatics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok\\Poland},
  SUMMARY = {In the article the formal characterization of
  preference spaces \cite{Arrow}
  is given. As the preference relation is one
  of the very basic notions of mathematical economics \cite{Hallden},
  it prepares
  some ground for the more thorough formalization of consumer theory
  (although some work was already done -- see \cite{ARROW.ABS}.
  There was an attempt to formalize similar results in Mizar, but
  this work seems still unfinished \cite{Kuzyka}). \par
  There are many approaches to preferences in the literature.
  We modelled them in a rather illustrative way (similar structures
  were considered in \cite{PCS_0.ABS}):
  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 \cite{Aumann} which is more
  general \cite{Schumm}.
  Additionally we assume all three relations are
  disjoint and their set-theoretic union gives a whole universe of
  alternatives. \par
  We constructed some positive and negative examples of preference
  structures; main aim of the article however is to give the characterization
  of consumer preference structures in terms of a binary relation, called
  characteristic relation \cite{Panek}, 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.}}

@ARTICLE{Kuzyka,
 AUTHOR={Wojszko, Krzysztof and Kuzyka, Artur},
 TITLE={Formalization of Commodity Space and Preference Relation
   in {M}izar},
 JOURNAL={Mechanized Mathematics and Its Applications},
 YEAR=2005,
 VOLUME=4,
 PAGES={67--74}}

@ARTICLE{Aumann,
 AUTHOR={Aumann, Robert J.},
 TITLE={Utility Theory Without the Completeness Axiom},
 JOURNAL={Econometrica},
 YEAR=1962,
 VOLUME=30,
 NUMBER=3,
 PAGES={445--462}}

@ARTICLE{Schumm,
 AUTHOR={Schumm, George F.},
 TITLE={Transitivity, Preference, and Indifference},
 JOURNAL={Philosophical Studies},
 YEAR=1987,
 VOLUME=52,
 PAGES={435--437}}

@BOOK{Arrow,
 AUTHOR={Arrow, Kenneth J.},
 TITLE={Social Choice and Individual Values},
 YEAR=1963,
 PUBLISHER = {Yale University Press}}

@BOOK{Hallden,
 AUTHOR={Halld\'en, S\"oren},
 TITLE={On the Logic of Better},
 YEAR=1957,
 PUBLISHER = {Lund: Library of Theoria}}

@BOOK{Panek,
 AUTHOR={Panek, Emil},
 TITLE={Podstawy ekonomii matematycznej},
 PUBLISHER = {Uniwersytet Ekonomiczny w Poznaniu},
 YEAR=2005,
 NOTE={In Polish}}

