My bibliography  Save this paper

# On the extension of a preorder under translation invariance

## Author

Listed:
• 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 or under a Pareto axiom.

## Suggested Citation

• Mabrouk, Mohamed, 2018. "On the extension of a preorder under translation invariance," MPRA Paper 86313, University Library of Munich, Germany.
• Handle: RePEc:pra:mprapa:86313
as

File URL: https://mpra.ub.uni-muenchen.de/86313/1/MPRA_paper_86313.pdf
File Function: original version
---><---

## References listed on IDEAS

as
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.
Full references (including those not matched with items on IDEAS)

## Most related items

These are the items that most often cite the same works as this one and are cited by the same works as this one.
1. Mabrouk, Mohamed, 2018. "On the Extension and Decomposition of a Preorder under Translation Invariance," MPRA Paper 90537, University Library of Munich, Germany.
2. Asheim, Geir B. & d'Aspremont, Claude & Banerjee, Kuntal, 2010. "Generalized time-invariant overtaking," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 519-533, July.
3. Adachi, Tsuyoshi & Cato, Susumu & Kamaga, Kohei, 2014. "Extended anonymity and Paretian relations on infinite utility streams," Mathematical Social Sciences, Elsevier, vol. 72(C), pages 24-32.
4. Kohei Kamaga & Takashi Kojima, 2010. "On the leximin and utilitarian overtaking criteria with extended anonymity," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 35(3), pages 377-392, September.
5. Geir B. Asheim & Kuntal Banerjee, 2010. "Fixed‐step anonymous overtaking and catching‐up," International Journal of Economic Theory, The International Society for Economic Theory, vol. 6(1), pages 149-165, March.
6. José Carlos R. Alcantud & María D. García-Sanz, 2013. "Evaluations of Infinite Utility Streams: Pareto Efficient and Egalitarian Axiomatics," Metroeconomica, Wiley Blackwell, vol. 64(3), pages 432-447, July.
7. Ram Sewak Dubey & Tapan Mitra, 2011. "On equitable social welfare functions satisfying the Weak Pareto Axiom: A complete characterization," International Journal of Economic Theory, The International Society for Economic Theory, vol. 7(3), pages 231-250, September.
8. Zuber, Stéphane & Asheim, Geir B., 2012. "Justifying social discounting: The rank-discounted utilitarian approach," Journal of Economic Theory, Elsevier, vol. 147(4), pages 1572-1601.
9. Chiaki Hara & Tomoichi Shinotsuka & Kotaro Suzumura & Yongsheng Xu, 2008. "Continuity and egalitarianism in the evaluation of infinite utility streams," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 31(2), pages 179-191, August.
10. Marcus Pivato, 2014. "Additive representation of separable preferences over infinite products," Theory and Decision, Springer, vol. 77(1), pages 31-83, June.
11. Kohei Kamaga & Takashi Kojima, 2009. "$${\mathcal{Q}}$$ -anonymous social welfare relations on infinite utility streams," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 33(3), pages 405-413, September.
12. 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.
13. M. Ali Khan & Metin Uyanik, 2020. "Binary Relations in Mathematical Economics: On the Continuity, Additivity and Monotonicity Postulates in Eilenberg, Villegas and DeGroot," Papers 2007.01952, arXiv.org.
14. Geir Asheim & Stéphane Zuber, 2013. "A complete and strongly anonymous leximin relation on infinite streams," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(4), pages 819-834, October.
15. 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.
16. Geir Asheim & Stéphane Zuber, 2013. "A complete and strongly anonymous leximin relation on infinite streams," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(4), pages 819-834, October.
17. Jonsson, Adam & Voorneveld, Mark, 2018. "The limit of discounted utilitarianism," Theoretical Economics, Econometric Society, vol. 13(1), January.
18. Henrik Petri, 2019. "Asymptotic properties of welfare relations," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(4), pages 853-874, June.
19. Charles Figuières & Mabel Tidball, 2016. "Sustainable Exploitation of a Natural Resource: A Satisfying Use of Chichilnisky’s Criterion," Studies in Economic Theory, in: Graciela Chichilnisky & Armon Rezai (ed.), The Economics of the Global Environment, pages 207-229, Springer.
20. Luc Lauwers, 2016. "Intergenerational Equity, Efficiency, and Constructibility," Studies in Economic Theory, in: Graciela Chichilnisky & Armon Rezai (ed.), The Economics of the Global Environment, pages 191-206, Springer.

### 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

## Corrections

All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:pra:mprapa:86313. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: https://edirc.repec.org/data/vfmunde.html .

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Joachim Winter (email available below). General contact details of provider: https://edirc.repec.org/data/vfmunde.html .

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.