Szpilrajn-type theorems in economics
The Szpilrajn "constructive type" theorem on extending binary relations, or its generalizations by Dushnik and Miller , is one of the best known theorems in social sciences and mathematical economics. Arrow , Fishburn , Suzumura , Donaldson and Weymark  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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- Weymark, John A., 2000. "A generalization of Moulin's Pareto extension theorem," Mathematical Social Sciences, Elsevier, vol. 39(2), pages 235-240, March.
- 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.
- Klaus Nehring & Clemens Puppe & Selva Demiralp, 2003. "Extended Partial Orders: A Unifying Structure For Abstract Choice Theory," Working Papers 976, University of California, Davis, Department of Economics.
- Klaus Nehring & Clemens Puppe, . "Extended Partial Orders: A Unifying Structure For Abstract Choice Theory," Department of Economics 97-06, California Davis - Department of Economics.
- 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.
- Duggan, John, 1999. "A General Extension Theorem for Binary Relations," Journal of Economic Theory, Elsevier, vol. 86(1), pages 1-16, May.
- 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.
- Charles Blackorby, & Walter Bossert & David Donaldson,, . "Rationalizable Solutions to Pure Population Problems," Discussion Papers 97/12, University of Nottingham, School of Economics.
- 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.
- Stephen A. Clark, 1988. "An extension theorem for rational choice functions," Review of Economic Studies, Oxford University Press, vol. 55(3), pages 485-492.
- 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, 08.
- 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.
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