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# Additive representation of separable preferences over infinite products

## Author

Listed:
• Marcus Pivato

(THEMA - Théorie économique, modélisation et applications - CNRS - Centre National de la Recherche Scientifique - CY - CY Cergy Paris Université)

## 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
(This abstract was borrowed from another version of this item.)

## Suggested Citation

• Marcus Pivato, 2014. "Additive representation of separable preferences over infinite products," Post-Print hal-02979672, HAL.
• Handle: RePEc:hal:journl:hal-02979672
DOI: 10.1007/s11238-013-9391-2
Note: View the original document on HAL open archive server: https://hal-cyu.archives-ouvertes.fr//hal-02979672
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## Citations

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Cited by:

1. McCarthy, David & Mikkola, Kalle & Thomas, Teruji, 2020. "Utilitarianism with and without expected utility," Journal of Mathematical Economics, Elsevier, vol. 87(C), pages 77-113.
2. Nehring, Klaus & Pivato, Marcus, 2019. "Majority rule in the absence of a majority," Journal of Economic Theory, Elsevier, vol. 183(C), pages 213-257.
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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### 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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