Abstract
This paper characterizes continuity and upper and lower semicontinuity of preference relations, which may or may not be representable by utility functions, on arbitrary topological spaces. One characterization is by the existence of an appropriate chain of sets. This approach can be used to generate preference relations that fulfill predetermined conditions, to obtain examples or counterexamples. The second characterization of continuity is closely related to the concept of scale, but, in contrast to previous work, does not rely on the existence of a utility function.
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Notes
Estévez Toranzo and Hervés Beloso (1995) show that for every non-separable metric space there is a continuous preference relation which cannot be represented by a utility function.
(Monteiro (1987), Corollary 2) shows that if such a preference relation is continuous and defined on a path connected space, it has in fact a continuous utility representation.
A chain of sets is also called a nest or a tower (Kelley 1975, p. 32).
Henceforth, \(\subseteq \) will denote (weak) set inclusion and \(\subset \) will denote proper set inclusion.
See Andrikopoulos (2013) for a recent, related result.
Alcantud et al. (2008) deal with representability of preorders, i.e. reflexive and transitive binary relations. This result corresponds to the particularization to preferences of their Theorem 2.5 (see also their Remark 2.6).
The concept of countable scale has been used with slight modifications in the literature. In the original definition, due to Burgess and Fitzpatrick (1977), a countable decreasing (increasing) scale was defined as a family of subsets \(b_q\) indexed in a dense subset \(S\) of \([0,1]\) containing 1 with \(b_1=X\), and fulfilling (CDS.1) and (the analogous property to) (CDS.2). The words “scale” and “pseudoscale” have also received specific meanings for the special cases where order or topological requirements are dropped, respectively.
This particular implication is also an implication of Proposition 2(ii).
References
Alcantud, J.C.R., Bosi, G., Campión, M.J., Candeal, J.C., Induráin, E., Rodríguez-Palmero, C.: Continuous utility functions through scales. Theory Decis. 64, 479–494 (2008)
Andrikopoulos, A.: Compactness in the choice and game theories: a characterization of rationality. Econ. Theory Bull. 1, 105–110 (2013)
Beardon, A.F., Candeal, J.C., Herden, G., Induráin, E., Mehta, G.B.: Lexicographic decomposition of chains and the concept of a planar chain. J. Math. Econ. 37(2), 95–104 (2002a)
Beardon, A.F., Candeal, J.C., Herden, G., Induráin, E., Mehta, G.B.: The non-existence of a utility function and the structure of non-representable preference relations. J. Math. Econ. 37(1), 17–38 (2002b)
Bergstrom, T.C.: Maximal elements of acyclic relations on compact sets. J. Econ. Theory 10, 403–404 (1975)
Bosi, G., Mehta, G.B.: Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof. J. Math. Econ. 38(2), 311–328 (2002)
Burgess, D., Fitzpatrick, M.: On separation axioms for certain types of ordered topological space. Math. Proc. Cambridge Philos. Soc. 82, 59–65 (1977)
Debreu, G.: Representation of a preference ordering by a numerical function. In: Thrall, R., Coombs, C., Davis, R. (eds.) Decis. Process. Wiley, New York (1954)
Eilenberg, S.: Ordered topological spaces. Am. J. Math. 63, 39–45 (1941)
Estévez Toranzo, M., Hervés Beloso, C.: On the existence of continuous preference orderings without utility representations. J. Math. Econ. 24(4), 305–309 (1995)
Gutiérrez, J.M.: A characterization of compactness through preferences. Math. Soc. Sci. 57, 131–133 (2009)
Herden, G.: On the existence of utility functions. Math. Soc. Sci. 17, 297–313 (1989)
Kelley, J.: General Topology, Graduate Texts in Mathematics, vol. 27. Springer, Berlin, Heidelberg, New York (1975)
Monteiro, P.K.: Some results on the existence of utility functions on path connected spaces. J. Math. Econ. 16(2), 147–156 (1987)
Munkres, J.: Topology, 2nd edn. Pearson (Prentice Hall), Upper Saddle River (2000)
Walker, M.: On the existence of maximal elements. J. Econ. Theory 16, 470–474 (1977)
Acknowledgments
We thank José Carlos Rodríguez Alcantud, Johannes Buckenmaier, Johannes Kern, and two anonymous referees for their helpful comments.
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We gratefully acknowledge the financial support from the German Research Foundation (DFG) and the Austrian Science Fund (FWF) under projects Al-1169/1 and I338-G16, respectively.
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Alós-Ferrer, C., Ritzberger, K. On the characterization of preference continuity by chains of sets. Econ Theory Bull 3, 115–128 (2015). https://doi.org/10.1007/s40505-014-0048-2
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DOI: https://doi.org/10.1007/s40505-014-0048-2