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Szpilrajn-type theorems in economics

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  • Andrikopoulos, Athanasios

Abstract

The Szpilrajn "constructive type" theorem on extending binary relations, or its generalizations by Dushnik and Miller [10], is one of the best known theorems in social sciences and mathematical economics. Arrow [1], Fishburn [11], Suzumura [22], Donaldson and Weymark [8] and others utilize Szpilrajn's Theorem and the Well-ordering principle to obtain more general "existence type" theorems on extending binary relations. Nevertheless, we are generally interested not only in the existence of linear extensions of a binary relation R, but in something more: the conditions of the preference sets and the properties which $R$ satisfies to be "inherited" when one passes to any member of some \textquotedblleft interesting\textquotedblright family of linear extensions of R. Moreover, in extending a preference relation $R$, the problem will often be how to incorporate some additional preference data with a minimum of disruption of the existing structure or how to extend the relation so that some desirable new condition is fulfilled. The key to addressing these kinds of problems is the szpilrajn constructive method. In this paper, we give two general "constructive type" theorems on extending binary relations, a Szpilrajn type and a Dushnik-Miller type theorem, which generalize and give a "constructive type" version of all the well known extension theorems in the literature.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 14345.

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Date of creation: 26 Feb 2009
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Handle: RePEc:pra:mprapa:14345

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Keywords: Consistent binary consistent binary relations; extension theorems; intersection of binary relations;

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  1. Charles Blackorby, & Walter Bossert & David Donaldson,, . "Rationalizable Solutions to Pure Population Problems," Discussion Papers 97/12, University of Nottingham, School of Economics.
  2. Klaus Nehring & Clemens Puppe, . "Extended Partial Orders: A Unifying Structure For Abstract Choice Theory," Department of Economics 97-06, California Davis - Department of Economics.
  3. Weymark, John A., 2000. "A generalization of Moulin's Pareto extension theorem," Mathematical Social Sciences, Elsevier, vol. 39(2), pages 235-240, March.
  4. Clark, Stephen A, 1988. "An Extension Theorem for Rational Choice Functions," Review of Economic Studies, Wiley Blackwell, vol. 55(3), pages 485-92, July.
  5. Sholomov, Lev A., 2000. "Explicit form of neutral social decision rules for basic rationality conditions," Mathematical Social Sciences, Elsevier, vol. 39(1), pages 81-107, January.
  6. Sophie Bade, 2005. "Nash equilibrium in games with incomplete preferences," Economic Theory, Springer, vol. 26(2), pages 309-332, 08.
  7. Duggan, John, 1999. "A General Extension Theorem for Binary Relations," Journal of Economic Theory, Elsevier, vol. 86(1), pages 1-16, May.
  8. Suzumura, Kataro, 1976. "Remarks on the Theory of Collective Choice," Economica, London School of Economics and Political Science, vol. 43(172), pages 381-90, November.
  9. Herden, Gerhard & Pallack, Andreas, 2002. "On the continuous analogue of the Szpilrajn Theorem I," Mathematical Social Sciences, Elsevier, vol. 43(2), pages 115-134, March.
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Cited by:
  1. T. Demuynck, 2009. "Common ordering extensions," Working Papers of Faculty of Economics and Business Administration, Ghent University, Belgium 09/593, Ghent University, Faculty of Economics and Business Administration.

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