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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.

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

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Date of creation: 19 Jan 2011
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Handle: RePEc:pra:mprapa:28262
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  1. Kirman, Alan P. & Sondermann, Dieter, 1972. "Arrow's theorem, many agents, and invisible dictators," Journal of Economic Theory, Elsevier, vol. 5(2), pages 267-277, October.
  2. ASHEIM, Geir B. & d’ASPREMONT, Claude & BANERJEE, Kuntal, . "Generalized time-invariant overtaking," CORE Discussion Papers RP -2239, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  3. Lauwers, Luc, 1998. "Intertemporal objective functions: Strong pareto versus anonymity," Mathematical Social Sciences, Elsevier, vol. 35(1), pages 37-55, January.
  4. Zame, William R., 2007. "Can intergenerational equity be operationalized?," Theoretical Economics, Econometric Society, vol. 2(2), June.
  5. 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 327-339, October.
  6. Toyotaka Sakai, 2010. "Intergenerational equity and an explicit construction of welfare criteria," Social Choice and Welfare, Springer, vol. 35(3), pages 393-414, September.
  7. 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.
  8. Walter Bossert & David Donaldson & Charles Blackorby, 1998. "Uncertainty and critical-level population principles," Journal of Population Economics, Springer, vol. 11(1), pages 1-20.
  9. Basu, Kaushik & Mitra, Tapan, 2003. "Utilitarianism for Infinite Utility Streams: A New Welfare Criterion and Its Axiomatic Characterization," Working Papers 03-05, Cornell University, Center for Analytic Economics.
  10. 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.
  11. repec:cup:cbooks:9780521825511 is not listed on IDEAS
  12. Geir Asheim & Bertil Tungodden, 2004. "Resolving distributional conflicts between generations," Economic Theory, Springer, vol. 24(1), pages 221-230, 07.
  13. Halpern, Joseph Y., 2010. "Lexicographic probability, conditional probability, and nonstandard probability," Games and Economic Behavior, Elsevier, vol. 68(1), pages 155-179, January.
  14. 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 137-160 World Scientific Publishing Co. Pte. Ltd..
  15. repec:cup:cbooks:9780521532587 is not listed on IDEAS
  16. 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.
  17. Lehmann, Daniel, 2001. "Expected Qualitative Utility Maximization," Games and Economic Behavior, Elsevier, vol. 35(1-2), pages 54-79, April.
  18. 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.
  19. Fuhrken, Gebhard & Richter, Marcel K, 1991. "Additive Utility," Economic Theory, Springer, vol. 1(1), pages 83-105, January.
  20. 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.
  21. 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.
  22. Kannai, Yakar, 1992. "Non-standard concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 21(1), pages 51-58.
  23. 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.
  24. 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.
  25. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
  26. Joseph Halpern, 2009. "A nonstandard characterization of sequential equilibrium, perfect equilibrium, and proper equilibrium," International Journal of Game Theory, Springer, vol. 38(1), pages 37-49, March.
  27. Peter C. Fishburn, 1974. "Exceptional Paper--Lexicographic Orders, Utilities and Decision Rules: A Survey," Management Science, INFORMS, vol. 20(11), pages 1442-1471, July.
  28. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, vol. 59(1), pages 81-98, January.
  29. Lauwers, Luc & Vallentyne, Peter, 2004. "Infinite Utilitarianism: More Is Always Better," Economics and Philosophy, Cambridge University Press, vol. 20(02), pages 307-330, October.
  30. 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.
  31. Luc Lauwers, 1997. "Topological aggregation, the case of an infinite population," Social Choice and Welfare, Springer, vol. 14(2), pages 319-332.
  32. Gottinger, Hans W., 1982. "Foundations of lexicographic utility," Mathematical Social Sciences, Elsevier, vol. 3(4), pages 363-371, December.
  33. Herzberg, Frederik, 2009. "Elementary non-Archimedean utility theory," Mathematical Social Sciences, Elsevier, vol. 58(1), pages 8-14, July.
  34. 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.
  35. 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.
  36. 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.
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