Abstract
The present paper gives a topological solution to representability problems related to multi-utility, in the field of Decision Theory. Necessary and sufficient topologies for the existence of a semicontinuous and finite Richter–Peleg multi-utility for a preorder are studied. It is well known that, given a preorder on a topological space, if there is a lower (upper) semicontinuous Richter–Peleg multi-utility, then the topology of the space must be finer than the Upper (resp. Lower) topology. However, this condition fails to be sufficient. Instead of search for properties that must be satisfied by the preorder, we study finer topologies which are necessary or/and sufficient for the existence of semicontinuous representations. We prove that Scott topology must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter–Peleg multi-utility. However, the existence of this representation cannot be guaranteed. A sufficient condition is given by means of Alexandroff’s topology, for that, we prove that more order implies less Alexandroff’s topology, as well as the converse. Finally, the paper is implemented with a topological study of the maximal elements.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Here, it is used that any subnet of a convergent net converges to the same point.
References
Alcantud, J. C. R., Bosi, G., & Zuanon, M. (2016). Richter–Peleg multi-utility representations of preorders. Theory and Decision, 80, 443–450.
Arenas, F. G. (1999). Alexandroff spaces. Bratislava: Acta Math. Univ. Comenianae.
Bosi, G., Estevan, A., Gutiérrez García, J., & Induráin, E. (2015). Continuous representability of interval orders, the topological compatibility setting. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systemss, 23(3), 345–365.
Bosi, G., Estevan, A., & Zuanon, M. (2018). Partial representations of orderings. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 26(03), 453–473.
Bosi, G., & Herden, G. (2016). On continuous multi-utility representations of semi-closed and closed preorders. Mathematical Social Sciences, 79, 20–29.
Bridges, D. S., & Mehta, G. B. (1995). Representations of preference orderings. Berlin: Springer.
Davey, B. A., & Priestley, H. A. (2002). Introduction to lattices and order (2nd ed.). Cambridge: Cambridge University Press. (ISBN 0-521-78451-4).
Debreu, G. (1954). Representation of a preference ordering by a numerical function. In R. Thrall, C. Coombs, & R. Davies (Eds.), Decision processes. New York: Wiley.
Debreu, G. (1964). Continuity properties of paretian utility. International Economic Review, 5, 285–293.
Evren, O., & Ok, E. A. (2011). On the multi-utility representation of preference relations. Journal of Mathematical Economics, 47, 554–563.
Gensemer, S. H. (1987). Continuous semiorder representations. Journal of Mathematical Economics, 16, 275–289.
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M., & Scott, D. S. (2003). Continuous lattices and domains (encyclopedia of mathematics and its applications) (1st ed.). Cambridge: Cambridge University Press.
Kaminski, B. (2007). On quasi-orderings and multi-objective functions. European Journal of Operational Research, 177, 1591–1598.
Mashburn, J. D. (1995). A note on reordering ordered topological spaces and the existence of continuous, strictly increasing functions. Topology Proceedings, 20, 207–250.
Minguzzi, E. (2013). Normally preordered spaces and utilities. Order, 30, 137–150.
Ok, E. A. (2002). Utility representation of an incomplete preference relation. Journal of Economic Theory, 104, 429–449.
Romaguera, S., & Valero, O. (2010). Domain theoretic characterisations of quasi-metric completeness in terms of formal balls. Mathematical Structures in Computer Science, 20, 45–472.
Scott, D., & Suppes, P. (1958). Foundational aspects of theories of measurement. Journal of Symbolic Logic, 23, 113–128.
Speer, T. (2007). A short study of Alexandroff spaces. arXiv:0708.2136v1.
Szpilrajn, E. (1930). Sur l’extension de l’ordre partiel. Fundamenta Mathematicae, 16, 386–389.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Asier Estevan acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under Grants MTM2015-63608-P and ECO2015-65031. Gianni Bosi acknowledges financial support from the Istituto Nazionale di Alta Matematica “F. Severi” (Italy). Armajac Raventos acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under Grant ECO2015-65031.
Rights and permissions
About this article
Cite this article
Bosi, G., Estevan, A. & Raventós-Pujol, A. Topologies for semicontinuous Richter–Peleg multi-utilities. Theory Decis 88, 457–470 (2020). https://doi.org/10.1007/s11238-019-09730-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11238-019-09730-7