Additive representation of separable preferences over infinite products
Let X be a set of states, and let I be an infinite indexing set. Our first main result states that any separable, permutationinvariant 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 welldefined. Also, unlike an integral, the hypersum does not depend upon a sigmaalgebra 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 infinitehorizon intertemporal choice, choice under uncertainty, and variablepopulation 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.
Length:  
Date of creation:  19 Jan 2011 
Date of revision:  
Handle:  RePEc:pra:mprapa:28262 
Contact details of provider:  Postal: Ludwigstraße 33, D80539 Munich, Germany Phone: +49(0)8921802459 Fax: +49(0)892180992459 Web page: https://mpra.ub.unimuenchen.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.:
 Lauwers, Luc, 2010.
"Ordering infinite utility streams comes at the cost of a nonRamsey set,"
Journal of Mathematical Economics,
Elsevier, vol. 46(1), pages 3237, January.
 Luc LAUWERS, 2009. "Ordering infinite utility streams comes at the cost of a nonRamsey set," Center for Economic Studies  Discussion papers ces09.05, Katholieke Universiteit Leuven, Centrum voor Economische Studiën.
 Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217237.
 ASHEIM, Geir B. & dâ€™ASPREMONT, Claude & BANERJEE, Kuntal, 2008.
"Generalized timeinvariant overtaking,"
CORE Discussion Papers
2008065, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
 Asheim, Geir B. & d'Aspremont, Claude & Banerjee, Kuntal, 2010. "Generalized timeinvariant overtaking," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 519533, July.
 Geir B. Asheim & Claude d’Aspremont & Kuntal Banerjee, 2008. "Generalized timeinvariant overtaking," Working Papers 08004, Department of Economics, College of Business, Florida Atlantic University.
 ASHEIM, Geir B. & dâ€™ASPREMONT, Claude & BANERJEE, Kuntal, . "Generalized timeinvariant overtaking," CORE Discussion Papers RP 2239, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
 Geir B, ASHEIM & Claude, Dâ€™ASPREMONT & Kuntal, BANERJEE, 2008. "Generalized timeinvariant overtaking," Discussion Papers (ECON  Département des Sciences Economiques) 2008044, Université catholique de Louvain, Département des Sciences Economiques.
 Asheim, Geir B. & d'Aspremont, Claude & Banerjee, Kuntal, 2008. "Generalized timeinvariant overtaking," PIE/CIS Discussion Paper 394, Center for Intergenerational Studies, Institute of Economic Research, Hitotsubashi University.
 Geir B. , ASHEIM & Claude, DASPREMONT & Kuntal, BANERJEE, 2008. "Generalized timeinvariant overtaking," Discussion Papers (ECON  Département des Sciences Economiques) 2008039, Université catholique de Louvain, Département des Sciences Economiques.
 Sakai, Toyotaka, 2010. "A characterization and an impossibility of finite length anonymity for infinite generations," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 877883, September.
 Basu, Kaushik & Mitra, Tapan, 2003.
"Aggregating Infinite Utility Streams with Intergenerational Equity: The Impossibility of Being Paretian,"
Working Papers
0303, Cornell University, Center for Analytic Economics.
 Kaushik Basu & Tapan Mitra, 2003. "Aggregating Infinite Utility Streams with InterGenerational Equity: The Impossibility of Being Paretian," Econometrica, Econometric Society, vol. 71(5), pages 15571563, 09.
 Fuhrken, Gebhard & Richter, Marcel K, 1991. "Additive Utility," Economic Theory, Springer, vol. 1(1), pages 83105, January.
 Blackorby,Charles & Bossert,Walter & Donaldson,David J., 2005.
"Population Issues in Social Choice Theory, Welfare Economics, and Ethics,"
Cambridge Books,
Cambridge University Press, number 9780521825511, 1.
 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, 1.
 Fishburn, Peter C. & LaValle, Irving H., 1998. "Subjective expected lexicographic utility with infinite state sets," Journal of Mathematical Economics, Elsevier, vol. 30(3), pages 323346, October.
 M. Fleurbaey & P. Michel, 1997.
"Intertemporal equity and the extension of the Ramsey criterion,"
THEMA Working Papers
9711, THEMA (THéorie Economique, Modélisation et Applications), Université de CergyPontoise.
 Fleurbaey, Marc & Michel, Philippe, 2003. "Intertemporal equity and the extension of the Ramsey criterion," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 777802, September.
 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).
 Halpern, Joseph Y., 2010. "Lexicographic probability, conditional probability, and nonstandard probability," Games and Economic Behavior, Elsevier, vol. 68(1), pages 155179, January.
 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.
 Peter C. Fishburn, 1974. "Exceptional PaperLexicographic Orders, Utilities and Decision Rules: A Survey," Management Science, INFORMS, vol. 20(11), pages 14421471, July.
 Walter Bossert & David Donaldson & Charles Blackorby, 1998. "Uncertainty and criticallevel population principles," Journal of Population Economics, Springer, vol. 11(1), pages 120.
 Kannai, Yakar, 1992. "Nonstandard concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 21(1), pages 5158.
 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 543596 Elsevier.
 Kirman, Alan P. & Sondermann, Dieter, 1972.
"Arrow's theorem, many agents, and invisible dictators,"
Journal of Economic Theory,
Elsevier, vol. 5(2), pages 267277, October.
 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).
 Luc Lauwers, 1997. "Topological aggregation, the case of an infinite population," Social Choice and Welfare, Springer, vol. 14(2), pages 319332.
 Zame, William R., 2007. "Can intergenerational equity be operationalized?," Theoretical Economics, Econometric Society, vol. 2(2), June.
 Streufert, P. A., 1995.
"A general theory of separability for preferences defined on a countably infinite product space,"
Journal of Mathematical Economics,
Elsevier, vol. 24(5), pages 407434.
 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.
 Kuntal Banerjee, 2006. "On the Extension of the Utilitarian and Suppes–Sen Social Welfare Relations to Infinite Utility Streams," Social Choice and Welfare, Springer, vol. 27(2), pages 327339, October.
 Lehmann, Daniel, 2001. "Expected Qualitative Utility Maximization," Games and Economic Behavior, Elsevier, vol. 35(12), pages 5479, April.
 Lawrence Blume & Adam Brandenburger & Eddie Dekel, 2014.
"Lexicographic Probabilities and Choice Under Uncertainty,"
World Scientific Book Chapters,
in: The Language of Game Theory Putting Epistemics into the Mathematics of Games, chapter 6, pages 137160
World Scientific Publishing Co. Pte. Ltd..
 Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Choice under Uncertainty," Econometrica, Econometric Society, vol. 59(1), pages 6179, January.
 Toyotaka Sakai, 2010.
"Intergenerational equity and an explicit construction of welfare criteria,"
Social Choice and Welfare,
Springer, vol. 35(3), pages 393414, September.
 Sakai, Toyotaka, 2008. "Intergenerational equity and an explicit construction of welfare criteria," PIE/CIS Discussion Paper 395, Center for Intergenerational Studies, Institute of Economic Research, Hitotsubashi University.
 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(56), pages 329340, May.
 Lauwers, Luc & Vallentyne, Peter, 2004. "Infinite Utilitarianism: More Is Always Better," Economics and Philosophy, Cambridge University Press, vol. 20(02), pages 307330, October.
 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 350373, March.
 Basu, Kaushik & Mitra, Tapan, 2003. "Utilitarianism for Infinite Utility Streams: A New Welfare Criterion and Its Axiomatic Characterization," Working Papers 0305, Cornell University, Center for Analytic Economics.
 Gottinger, Hans W., 1982. "Foundations of lexicographic utility," Mathematical Social Sciences, Elsevier, vol. 3(4), pages 363371, December.
 W. M. Gorman, 1968. "The Structure of Utility Functions," Review of Economic Studies, Oxford University Press, vol. 35(4), pages 367390.
 Lauwers, Luc, 1998. "Intertemporal objective functions: Strong pareto versus anonymity," Mathematical Social Sciences, Elsevier, vol. 35(1), pages 3755, January.
 Herzberg, Frederik, 2009. "Elementary nonArchimedean utility theory," Mathematical Social Sciences, Elsevier, vol. 58(1), pages 814, July.
 Basu, Kaushik & Mitra, Tapan, 2005. "On the Existence of Paretian Social Welfare Relations for Infinite Utility Streams with Extended Anonymity," Working Papers 0506, Cornell University, Center for Analytic Economics.
 Geir Asheim & Bertil Tungodden, 2004. "Resolving distributional conflicts between generations," Economic Theory, Springer, vol. 24(1), pages 221230, 07.
 Herden, G. & Mehta, G. B., 2004. "The Debreu Gap Lemma and some generalizations," Journal of Mathematical Economics, Elsevier, vol. 40(7), pages 747769, November.
 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.
 Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, vol. 59(1), pages 8198, January.
 Joseph Halpern, 2009. "A nonstandard characterization of sequential equilibrium, perfect equilibrium, and proper equilibrium," International Journal of Game Theory, Springer, vol. 38(1), pages 3749, March.
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.