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On the extension of a preorder under translation invariance

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  • Mabrouk, Mohamed

Abstract

This paper proves the existence, for a translation-invariant preorder on a divisible commutative group, of a complete preorder extending the preorder in question and satisfying translation invariance. We also prove that the extension may inherit a property of continuity. As an application, we prove the existence of a complete translation-invariant strict preorder on ℝ which transgresses scalar invariance and also the existence of a complete translation-invariant preorder satisfying the social choice axioms strong Pareto and fixed--step-anonymity on a set X^{ℕ₀}, where X is a divisible commutative group. Moreover, the two extension results are used to make scalar invariance appear as a consequence of translation invariance under a continuity requirement.

Suggested Citation

  • Mabrouk, Mohamed, 2018. "On the extension of a preorder under translation invariance," MPRA Paper 86564, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:86564
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    References listed on IDEAS

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    1. Mohamed Ben Ridha Mabrouk, 2011. "Translation invariance when utility streams are infinite and unbounded," International Journal of Economic Theory, The International Society for Economic Theory, vol. 7(4), pages 317-329, December.
    2. Fleurbaey, Marc & Michel, Philippe, 2003. "Intertemporal equity and the extension of the Ramsey criterion," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 777-802, September.
    3. d'Aspremont, Claude & Gevers, Louis, 2002. "Social welfare functionals and interpersonal comparability," Handbook of Social Choice and Welfare,in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 1, chapter 10, pages 459-541 Elsevier.
    4. Zame, William R., 2007. "Can intergenerational equity be operationalized?," Theoretical Economics, Econometric Society, vol. 2(2), June.
    5. Lauwers, Luc, 2010. "Ordering infinite utility streams comes at the cost of a non-Ramsey set," Journal of Mathematical Economics, Elsevier, vol. 46(1), pages 32-37, January.
    6. Monteiro, Paulo Klinger, 1987. "Some results on the existence of utility functions on path connected spaces," Journal of Mathematical Economics, Elsevier, vol. 16(2), pages 147-156, April.
    7. Jaffray, Jean-Yves, 1975. "Semicontinuous extension of a partial order," Journal of Mathematical Economics, Elsevier, vol. 2(3), pages 395-406, December.
    8. Demuynck, Thomas & Lauwers, Luc, 2009. "Nash rationalization of collective choice over lotteries," Mathematical Social Sciences, Elsevier, vol. 57(1), pages 1-15, January.
    9. Kenneth J. Arrow, 1950. "A Difficulty in the Concept of Social Welfare," Journal of Political Economy, University of Chicago Press, vol. 58, pages 328-328.
    10. Basu, Kaushik & Mitra, Tapan, 2005. "On the Existence of Paretian Social Welfare Relations for Infinite Utility Streams with Extended Anonymity," Working Papers 05-06, Cornell University, Center for Analytic Economics.
    11. Jörgen W. Weibull, 1985. "Discounted-Value Representations of Temporal Preferences," Mathematics of Operations Research, INFORMS, vol. 10(2), pages 244-250, May.
    12. Svensson, Lars-Gunnar, 1980. "Equity among Generations," Econometrica, Econometric Society, vol. 48(5), pages 1251-1256, July.
    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.
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    More about this item

    Keywords

    Order extension; Translation invariance;

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • D7 - Microeconomics - - Analysis of Collective Decision-Making
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • D9 - Microeconomics - - Micro-Based Behavioral Economics

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