IDEAS home Printed from https://ideas.repec.org/
MyIDEAS: Log in (now much improved!) to save this paper

Additive representation of separable preferences over infinite products

  • Pivato, Marcus

Let X be a set of states, and let I be an infinite indexing set. Our first main result states that any separable, permutation-invariant preference order (>) on X^I admits an additive representation. That is: there exists a linearly ordered abelian group A and a `utility function' u:X-->A such that, for any x,y in X^I which differ in only finitely many coordinates, we have x>y if and only if the sum of [u(x_i)-u(y_i)] over all i in I is positive. Our second result states: If (>) also satisfies a weak continuity condition, then, for any x,y in X^I, we have x>y if and only if the `hypersum' of [u(x_i)-u(y_i)] over all i in I is positive. The `hypersum' is an infinite summation operator defined using methods from nonstandard analysis. Like an integration operator or series summation operator, it allows us to define the sum of an infinite set of values. However, unlike these operations, the hypersum does not depend on some form of convergence (recall: A has no topology) ---it is always well-defined. Also, unlike an integral, the hypersum does not depend upon a sigma-algebra or measure on the indexing set I. The hypersum takes values in a linearly ordered abelian group A*, which is an ultrapower extension of A. These results are applicable to infinite-horizon intertemporal choice, choice under uncertainty, and variable-population social choice.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL: https://mpra.ub.uni-muenchen.de/28262/1/MPRA_paper_28262.pdf
File Function: original version
Download Restriction: no

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 28262.

as
in new window

Length:
Date of creation: 19 Jan 2011
Date of revision:
Handle: RePEc:pra:mprapa:28262
Contact details of provider: Postal:
Ludwigstraße 33, D-80539 Munich, Germany

Phone: +49-(0)89-2180-2459
Fax: +49-(0)89-2180-992459
Web page: https://mpra.ub.uni-muenchen.de

More information through EDIRC

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

as in new window
  1. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Choice under Uncertainty," Econometrica, Econometric Society, vol. 59(1), pages 61-79, January.
  2. KIRMAN, Alan P. & SONDERMANN, Dieter, . "Arrow's theorem, many agents, and indivisible dictators," CORE Discussion Papers RP 118, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  3. Lehmann, Daniel, 2001. "Expected Qualitative Utility Maximization," Games and Economic Behavior, Elsevier, vol. 35(1-2), pages 54-79, April.
  4. Lauwers, Luc, 1998. "Intertemporal objective functions: Strong pareto versus anonymity," Mathematical Social Sciences, Elsevier, vol. 35(1), pages 37-55, January.
  5. Herzberg, Frederik, 2009. "Elementary non-Archimedean utility theory," Mathematical Social Sciences, Elsevier, vol. 58(1), pages 8-14, July.
  6. Luc LAUWERS, 2009. "Ordering infinite utility streams comes at the cost of a non-Ramsey set," Working Papers Department of Economics ces09.05, KU Leuven, Faculty of Economics and Business, Department of Economics.
  7. Fishburn, Peter C. & LaValle, Irving H., 1998. "Subjective expected lexicographic utility with infinite state sets," Journal of Mathematical Economics, Elsevier, vol. 30(3), pages 323-346, October.
  8. W. M. Gorman, 1968. "The Structure of Utility Functions," Review of Economic Studies, Oxford University Press, vol. 35(4), pages 367-390.
  9. Joseph Halpern, 2009. "A nonstandard characterization of sequential equilibrium, perfect equilibrium, and proper equilibrium," International Journal of Game Theory, Springer;Game Theory Society, vol. 38(1), pages 37-49, March.
  10. Walter Bossert & David Donaldson & Charles Blackorby, 1998. "Uncertainty and critical-level population principles," Journal of Population Economics, Springer;European Society for Population Economics, vol. 11(1), pages 1-20.
  11. Geir Asheim & Bertil Tungodden, 2004. "Resolving distributional conflicts between generations," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 24(1), pages 221-230, 07.
  12. Zame, William R., 2007. "Can intergenerational equity be operationalized?," Theoretical Economics, Econometric Society, vol. 2(2), June.
  13. Toyotaka Sakai, 2010. "Intergenerational equity and an explicit construction of welfare criteria," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 35(3), pages 393-414, September.
  14. Geir B. Asheim & Claude d’Aspremont & Kuntal Banerjee, 2008. "Generalized time-invariant overtaking," Working Papers 08004, Department of Economics, College of Business, Florida Atlantic University.
  15. Fuhrken, Gebhard & Richter, Marcel K, 1991. "Additive Utility," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 1(1), pages 83-105, January.
  16. FLEURBAEY, Marc & MICHEL, Philippe, 1997. "Intertemporal equity and the extension of the Ramsey criterion," CORE Discussion Papers 1997004, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  17. Blackorby, Charles & Bossert, Walter & Donaldson, David, 2002. "Utilitarianism and the theory of justice," 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 11, pages 543-596 Elsevier.
  18. K. J. Arrow & A. K. Sen & K. Suzumura (ed.), 2002. "Handbook of Social Choice and Welfare," Handbook of Social Choice and Welfare, Elsevier, edition 1, volume 1, number 1.
  19. Lauwers, Luc & Vallentyne, Peter, 2004. "Infinite Utilitarianism: More Is Always Better," Economics and Philosophy, Cambridge University Press, vol. 20(02), pages 307-330, October.
  20. Kaushik Basu & Tapan Mitra, 2003. "Aggregating Infinite Utility Streams with InterGenerational Equity: The Impossibility of Being Paretian," Econometrica, Econometric Society, vol. 71(5), pages 1557-1563, 09.
  21. Gottinger, Hans W., 1982. "Foundations of lexicographic utility," Mathematical Social Sciences, Elsevier, vol. 3(4), pages 363-371, December.
  22. Luc Lauwers, 1997. "Topological aggregation, the case of an infinite population," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 14(2), pages 319-332.
  23. Basu, Kaushik & Mitra, Tapan, 2007. "Utilitarianism for infinite utility streams: A new welfare criterion and its axiomatic characterization," Journal of Economic Theory, Elsevier, vol. 133(1), pages 350-373, March.
  24. Peter C. Fishburn, 1974. "Exceptional Paper--Lexicographic Orders, Utilities and Decision Rules: A Survey," Management Science, INFORMS, vol. 20(11), pages 1442-1471, July.
  25. Halpern, Joseph Y., 2010. "Lexicographic probability, conditional probability, and nonstandard probability," Games and Economic Behavior, Elsevier, vol. 68(1), pages 155-179, January.
  26. Herden, G. & Mehta, G. B., 2004. "The Debreu Gap Lemma and some generalizations," Journal of Mathematical Economics, Elsevier, vol. 40(7), pages 747-769, November.
  27. Kuntal Banerjee, 2006. "On the Extension of the Utilitarian and Suppes–Sen Social Welfare Relations to Infinite Utility Streams," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 27(2), pages 327-339, October.
  28. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, vol. 59(1), pages 81-98, January.
  29. Kannai, Yakar, 1992. "Non-standard concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 21(1), pages 51-58.
  30. John C. Harsanyi, 1955. "Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility," Journal of Political Economy, University of Chicago Press, vol. 63, pages 309.
  31. Seidenfeld, Teddy & Schervish, Mark J. & Kadane, Joseph B., 2009. "Preference for equivalent random variables: A price for unbounded utilities," Journal of Mathematical Economics, Elsevier, vol. 45(5-6), pages 329-340, May.
  32. Streufert, P.A., 1992. "A General Theory of Separability for Preferences Defined on a Countably Infinite Product Space," Working papers 9203, Wisconsin Madison - Social Systems.
  33. Blackorby,Charles & Bossert,Walter & Donaldson,David J., 2005. "Population Issues in Social Choice Theory, Welfare Economics, and Ethics," Cambridge Books, Cambridge University Press, number 9780521532587, November.
  34. Sakai, Toyotaka, 2010. "A characterization and an impossibility of finite length anonymity for infinite generations," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 877-883, September.
  35. 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.
  36. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
Full references (including those not matched with items on IDEAS)

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

When requesting a correction, please mention this item's handle: RePEc:pra:mprapa:28262. 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: (Joachim Winter)

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 references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link 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 profile, as there may be some citations waiting for confirmation.

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

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.