Abstract
We present here a direct elementary construction of continuous utility functions on perfectly separable totally preordered sets that does not make use of the well-known Debreu’s open gap lemma. This new construction leans on the concept of a separating countable decreasing scale. Starting from a perfectly separable totally ordered structure, we give an explicit construction of a separating countable decreasing scale, from which we show how to get a continuous utility map.
Similar content being viewed by others
References
Beardon A.F. (1992), Debreu’s gap theorem. Economic Theory 2: 150–152
Beardon A.F. (1994), Totally ordered subsets of Euclidean space. Journal of Mathematical Economics 23: 391–393
Beardon A.F., Candeal J.C., Herden G., Induráin E., Mehta G.B. (2002a), The non-existence of a utility function and the structure of non-representable preference relations. Journal of Mathematical Economics 37(1): 17–38
Beardon A.F., Candeal J.C., Herden G., Induráin E., Mehta G.B., (2002b), Lexicographic decomposition of chains and the concept of a planar chain. Journal of Mathematical Economics 37(2): 95–104
Birkhoff G. (1967), Lattice Theory (Third edition). American Mathematical Society, Providence, RI
Bosi G., Mehta G.B. (2002), Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof. Journal of Mathematical Economics 38: 311–328
Bowen R. (1968), A new proof of a theorem in utility theory. International Economic Review 9: 374
Bridges D.S., Mehta G.B. (1995), Representations of Preference Orderings. Springer, Berlin, Heidelberg, New York
Burgess D.C.J., Fitzpatrick M. (1977), On separation axioms for certain types of ordered topological space. Mathematical Proceedings of the Cambridge Philosophical Society 82: 59–65
Candeal J.C., Induráin E. (1993), Utility representations from the concept of measure. Mathematical Social Sciences 26: 51–62
Candeal J.C., Induráin E. (1994), Utility representations from the concept of measure: a corrigendum. Mathematical Social Sciences 28: 67–69
Cantor G. (1895), Beiträge zur Begründung der transfinite Mengenlehre I. Mathematische Annalen 46: 481–512
Cantor G. (1897), Beiträge zur Begründung der transfinite Mengenlehre II. Mathematische Annalen 49: 207–246
Deak, E. (1972), Theory and Application of Directional Structures, Colloquia Mathematica Societatis János Bolyai 8, Topics in Topology, Keszthely, Hungary.
Debreu, G. (1954), Representation of a preference ordering by a numerical function, in Decission processes, Thrall, R., Coombs, C. and Davis, R. (eds.), New York: Wiley, pp. 159–166.
Debreu G. (1964), Continuous properties of Paretian utility. International Economic Review 5: 285–293
Droste M. (1987), Ordinal scales in the theory of measurement. Journal of Mathematical Psychology 31: 60–82
Dubra, J. and Echenique, F. (2001), Monotone preferences over information, Topics in Theoretical Economics 1(1), article 1 (electronic).
Gillman J.L., Jerison M. (1960), Rings of Continuous Functions. D. van Nostrand, New York
Herden G. (1989a), On the existence of utility functions. Mathematical Social Sciences 17: 297–313
Herden G. (1989b), On the existence of utility functions II. Mathematical Social Sciences 18: 107–117
Herden G. (1995), On some equivalent approaches to mathematical utility theory. Mathematical Social Sciences 29: 19–31
Herden G., Mehta G.B. (1996), Open gaps, metrization and utility. Economic Theory 7: 541–546
Jaffray J.Y. (1975), Existence of a continuous utility function: an elementary proof. Econometrica 43: 981–983
Kamke E. (1950). Theory of Sets. Dover, New York
Kelley J.L. (1955), General Topology. D. van Nostrand, New York
Mehta G.B. (1997), A remark on a utility representation theorem of Rader. Economic Theory 9: 367–370
Nachbin L. (1965), Topology and Order. Van Nostrand Reinhold, New York
Rader T. (1963), On the existence of utility functions to represent preferences. Review of Economic Studies 30: 229–232
Uryshon P. (1925), Über die Mächtigkeit der zusammenhängenden Mengen. Mathematische Annalen 94: 262–295
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Alcantud, J.C.R., Bosi, G., Campión, M.J. et al. Continuous Utility Functions Through Scales. Theory Decis 64, 479–494 (2008). https://doi.org/10.1007/s11238-007-9025-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11238-007-9025-7