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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.

Suggested Citation

  • Andrikopoulos, Athanasios, 2009. "Szpilrajn-type theorems in economics," MPRA Paper 14345, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:14345
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    File URL: https://mpra.ub.uni-muenchen.de/14345/1/MPRA_paper_14345.pdf
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    References listed on IDEAS

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    1. Stephen A. Clark, 1988. "An extension theorem for rational choice functions," Review of Economic Studies, Oxford University Press, vol. 55(3), pages 485-492.
    2. Klaus Nehring & Clemens Puppe, 1998. "Extended partial orders:A unifying structure for abstract choice theory," Annals of Operations Research, Springer, vol. 80(0), pages 27-48, January.
    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. 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.
    5. Sophie Bade, 2005. "Nash equilibrium in games with incomplete preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(2), pages 309-332, August.
    6. Duggan, John, 1999. "A General Extension Theorem for Binary Relations," Journal of Economic Theory, Elsevier, vol. 86(1), pages 1-16, May.
    7. 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.
    8. Walter Bossert & David Donaldson & Charles Blackorby, 1999. "Rationalizable solutions to pure population problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 16(3), pages 395-407.
    9. Suzumura, Kataro, 1976. "Remarks on the Theory of Collective Choice," Economica, London School of Economics and Political Science, vol. 43(172), pages 381-390, November.
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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.

    More about this item

    Keywords

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

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • D60 - Microeconomics - - Welfare Economics - - - General
    • D00 - Microeconomics - - General - - - General
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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