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Continuous multi-utility representations of preorders

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  • Bosi, Gianni
  • Herden, Gerhard

Abstract

Let (X,t) be a topological space. Then a preorder ≾ on (X,t) has a continuous multi-utility representation if there exists a family F of continuous and isotonic real-valued functions f on (X,≾,t) such that for all x∈X and all y∈X the inequalities x≾y mean that for all f∈F the inequalities f(x)≤f(y) hold. We discuss the existence of a continuous multi-utility representation by using suitable concepts of continuity of a preorder. In addition, we clarify in detail the relation between the concept of a continuous multi-utility representation and Nachbin’s concept of a normally preordered space.

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  • Bosi, Gianni & Herden, Gerhard, 2012. "Continuous multi-utility representations of preorders," Journal of Mathematical Economics, Elsevier, vol. 48(4), pages 212-218.
    • Handle: RePEc:eee:mateco:v:48:y:2012:i:4:p:212-218
    DOI: 10.1016/j.jmateco.2012.05.001
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    Cited by:

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    2. Candeal, Juan C., 2022. "Bi-utility representable orderings on a countable set," Economics Letters, Elsevier, vol. 217(C).
    3. Cosimo Munari, 2020. "Multi-utility representations of incomplete preferences induced by set-valued risk measures," Papers 2009.04151, arXiv.org.
    4. A. Zapata & A. M. Mármol & L. Monroy & M. A. Caraballo, 2019. "A Maxmin Approach for the Equilibria of Vector-Valued Games," Group Decision and Negotiation, Springer, vol. 28(2), pages 415-432, April.
    5. Cosimo Munari, 2021. "Multi-utility representations of incomplete preferences induced by set-valued risk measures," Finance and Stochastics, Springer, vol. 25(1), pages 77-99, January.
    6. Pedro Hack & Daniel A. Braun & Sebastian Gottwald, 2023. "The classification of preordered spaces in terms of monotones: complexity and optimization," Theory and Decision, Springer, vol. 94(4), pages 693-720, May.
    7. Alcantud, José Carlos R. & Bosi, Gianni & Zuanon, Magalì, 2013. "Representations of preorders by strong multi-objective functions," MPRA Paper 52329, University Library of Munich, Germany.
    8. José Carlos R. Alcantud & Gianni Bosi & Magalì Zuanon, 2016. "Richter–Peleg multi-utility representations of preorders," Theory and Decision, Springer, vol. 80(3), pages 443-450, March.
    9. Uyanik, Metin & Khan, M. Ali, 2022. "The continuity postulate in economic theory: A deconstruction and an integration," Journal of Mathematical Economics, Elsevier, vol. 101(C).
    10. Andreas H. Hamel & Sophie Qingzhen Wang, 2017. "A set optimization approach to utility maximization under transaction costs," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 40(1), pages 257-275, November.
    11. Paolo Leonetti, 2022. "Expected multi-utility representations of preferences over lotteries," Papers 2210.04739, arXiv.org, revised Jan 2024.
    12. M. Ali Khan & Metin Uyanik, 2020. "Binary Relations in Mathematical Economics: On the Continuity, Additivity and Monotonicity Postulates in Eilenberg, Villegas and DeGroot," Papers 2007.01952, arXiv.org.
    13. Shaofang Qi, 2016. "A characterization of the n-agent Pareto dominance relation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 46(3), pages 695-706, March.
    14. Gianni Bosi & Magalì Zuanon, 2021. "Topologies for the Continuous Representability of All Continuous Total Preorders," Journal of Optimization Theory and Applications, Springer, vol. 188(2), pages 420-431, February.
    15. Dan Qin, 2021. "A Note on Numerical Representations of Nested System of Strict Partial Orders," Games, MDPI, vol. 12(3), pages 1-9, July.
    16. Bosi, Gianni & Herden, Gerhard, 2014. "Topological spaces for which every closed and semi-closed preorder respectively admits a continuous multi-utility representation," MPRA Paper 53404, University Library of Munich, Germany.
    17. Nishimura, Hiroki & Ok, Efe A., 2016. "Utility representation of an incomplete and nontransitive preference relation," Journal of Economic Theory, Elsevier, vol. 166(C), pages 164-185.

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