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On the extension of binary relations in economic and game theories

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  • Athanasios Andrikopoulos

    (University of Patras)

Abstract

Szpilrajn’s extension theorem on binary relations and its strengthening by Dushnik and Miller are fundamental in economic and game theories. Szpilrajn’s result entails that each partial order extends to a linear order. Dushnik and Miller use Szpilrajn’s theorem to show that each partial order has a realizer. Since then, many authors utilize Szpilrajn’s theorem and the well-ordering principle to prove more general theorems on extending binary relations. The original extension theorems of Szpilrajn, Dushnik-Miller and Moulin-Weymark are called: Szpilrajn extension theorem, Dushnik-Miller extension theorem and Moulin-Weymark’s Pareto extension theorem respectively. The generalizations of these theorems are called: Szpilrajn-type extension theorem, Dushnik-Miller-type extension theorem and Moulin-Weymark’s Pareto-type extension theorem respectively. The presented results generalize well-known extension theorems in the literature.

Suggested Citation

  • Athanasios Andrikopoulos, 2019. "On the extension of binary relations in economic and game theories," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 277-285, June.
  • Handle: RePEc:spr:decfin:v:42:y:2019:i:1:d:10.1007_s10203-018-0213-4
    DOI: 10.1007/s10203-018-0213-4
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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. Bossert, Walter & Sprumont, Yves & Suzumura, Kotaro, 2002. "Upper semicontinuous extensions of binary relations," Journal of Mathematical Economics, Elsevier, vol. 37(3), pages 231-246, May.
    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. Paolo Scapparone, 1999. "Existence of a convex extension of a preference relation," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 22(1), pages 5-11, March.
    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. Demuynck, Thomas, 2009. "A general extension result with applications to convexity, homotheticity and monotonicity," Mathematical Social Sciences, Elsevier, vol. 57(1), pages 96-109, January.
    9. Jaffray, Jean-Yves, 1975. "Semicontinuous extension of a partial order," Journal of Mathematical Economics, Elsevier, vol. 2(3), pages 395-406, December.
    10. Podinovski, Vladislav V., 2013. "Non-dominance and potential optimality for partial preference relations," European Journal of Operational Research, Elsevier, vol. 229(2), pages 482-486.
    11. Demuynck, Thomas & Lauwers, Luc, 2009. "Nash rationalization of collective choice over lotteries," Mathematical Social Sciences, Elsevier, vol. 57(1), pages 1-15, January.
    12. Athanasios Andrikopoulos, 2012. "On the construction of non-empty choice sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(2), pages 305-323, February.
    13. 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.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Extension theorems; Consistent binary relations; Intersection of binary relations; Realizer;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • D00 - Microeconomics - - General - - - General
    • D60 - Microeconomics - - Welfare Economics - - - General
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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