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Additive representation of separable preferences over infinite products

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  • Marcus Pivato

Abstract

Let $$\mathcal{X }$$ X be a set of outcomes, and let $$\mathcal{I }$$ I be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order $$(\succcurlyeq )$$ ( ≽ ) on $$\mathcal{X }^\mathcal{I }$$ X I admits an additive representation. That is: there exists a linearly ordered abelian group $$\mathcal{R }$$ R and a ‘utility function’ $$u:\mathcal{X }{{\longrightarrow }}\mathcal{R }$$ u : X ⟶ R such that, for any $$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$ x , y ∈ X I which differ in only finitely many coordinates, we have $$\mathbf{x}\succcurlyeq \mathbf{y}$$ x ≽ y if and only if $$\sum _{i\in \mathcal{I }} \left[u(x_i)-u(y_i)\right]\ge 0$$ ∑ i ∈ I u ( x i ) - u ( y i ) ≥ 0 . Importantly, and unlike almost all previous work on additive representations, this result does not require any Archimedean or continuity condition. If $$(\succcurlyeq )$$ ( ≽ ) also satisfies a weak continuity condition, then the paper shows that, for any $$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$ x , y ∈ X I , we have $$\mathbf{x}\succcurlyeq \mathbf{y}$$ x ≽ y if and only if $${}^*\!\sum _{i\in \mathcal{I }} u(x_i)\ge {}^*\!\sum _{i\in \mathcal{I }}u(y_i)$$ ∗ ∑ i ∈ I u ( x i ) ≥ ∗ ∑ i ∈ I u ( y i ) . Here, $${}^*\!\sum _{i\in \mathcal{I }} u(x_i)$$ ∗ ∑ i ∈ I u ( x i ) represents a nonstandard sum, taking values in a linearly ordered abelian group $${}^*\!\mathcal{R }$$ ∗ R , which is an ultrapower extension of $$\mathcal{R }$$ R . The paper also discusses several applications of these results, including infinite-horizon intertemporal choice, choice under uncertainty, variable-population social choice and games with infinite strategy spaces. Copyright Springer Science+Business Media New York 2014

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  • Marcus Pivato, 2014. "Additive representation of separable preferences over infinite products," Theory and Decision, Springer, vol. 77(1), pages 31-83, June.
  • Handle: RePEc:kap:theord:v:77:y:2014:i:1:p:31-83
    DOI: 10.1007/s11238-013-9391-2
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    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, February.
    2. 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.
    3. 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.
    4. 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 407-434.
    5. 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, September.
    6. 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.
    7. Kannai, Yakar, 1992. "Non-standard concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 21(1), pages 51-58.
    8. 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.
    9. W. M. Gorman, 1968. "The Structure of Utility Functions," Review of Economic Studies, Oxford University Press, vol. 35(4), pages 367-390.
    10. 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.
    11. 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.
    12. 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..
    13. 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, July.
    14. Halpern, Joseph Y., 2010. "Lexicographic probability, conditional probability, and nonstandard probability," Games and Economic Behavior, Elsevier, vol. 68(1), pages 155-179, January.
    15. 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.
    16. Lehmann, Daniel, 2001. "Expected Qualitative Utility Maximization," Games and Economic Behavior, Elsevier, vol. 35(1-2), pages 54-79, April.
    17. 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.
    18. Lauwers, Luc, 1998. "Intertemporal objective functions: Strong pareto versus anonymity," Mathematical Social Sciences, Elsevier, vol. 35(1), pages 37-55, January.
    19. John C. Harsanyi, 1955. "Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility," Journal of Political Economy, University of Chicago Press, vol. 63, pages 309-309.
    20. Zame, William R., 2007. "Can intergenerational equity be operationalized?," Theoretical Economics, Econometric Society, vol. 2(2), June.
    21. Peter C. Fishburn, 1974. "Exceptional Paper--Lexicographic Orders, Utilities and Decision Rules: A Survey," Management Science, INFORMS, vol. 20(11), pages 1442-1471, July.
    22. 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.
    23. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, vol. 59(1), pages 81-98, January.
    24. 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.
    25. 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.
    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. 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.
    28. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
    29. 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.
    30. Gottinger, Hans W., 1982. "Foundations of lexicographic utility," Mathematical Social Sciences, Elsevier, vol. 3(4), pages 363-371, December.
    31. 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.
    32. 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.
    33. 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, March.
    34. Herzberg, Frederik, 2009. "Elementary non-Archimedean utility theory," Mathematical Social Sciences, Elsevier, vol. 58(1), pages 8-14, July.
    35. Lauwers, Luc & Vallentyne, Peter, 2004. "Infinite Utilitarianism: More Is Always Better," Economics and Philosophy, Cambridge University Press, vol. 20(2), pages 307-330, October.
    36. 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.
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    Cited by:

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    3. Marcus Pivato, 2015. "Social choice with approximate interpersonal comparison of welfare gains," Theory and Decision, Springer, vol. 79(2), pages 181-216, September.
    4. János Flesch & Dries Vermeulen & Anna Zseleva, 2019. "Catch games: the impact of modeling decisions," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(2), pages 513-541, June.
    5. Pivato, Marcus, 2013. "Multiutility representations for incomplete difference preorders," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 196-220.
    6. David McCarthy & Kalle Mikkola & Teruji Thomas, 2019. "Aggregation for potentially infinite populations without continuity or completeness," Papers 1911.00872, arXiv.org.
    7. Han Bleichrodt & Umut Keskin & Kirsten I. M. Rohde & Vitalie Spinu & Peter Wakker, 2015. "Discounted Utility and Present Value—A Close Relation," Operations Research, INFORMS, vol. 63(6), pages 1420-1430, December.
    8. Pivato, Marcus, 2013. "Variable-population voting rules," Journal of Mathematical Economics, Elsevier, vol. 49(3), pages 210-221.
    9. Flesch, János & Vermeulen, Dries & Zseleva, Anna, 2017. "Zero-sum games with charges," Games and Economic Behavior, Elsevier, vol. 102(C), pages 666-686.

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    More about this item

    Keywords

    Additively separable; Intertemporal; Uncertainty; Utilitarian; Nonstandard analysis; Non-Archimedean utility; D81; D90; D61;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General
    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis

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