Existence of continuous utility functions for arbitrary binary relations: some sufficient conditions

Bosi, Gianni and Caterino, Alessandro and Ceppitelli, Rita (2009): Existence of continuous utility functions for arbitrary binary relations: some sufficient conditions.

Abstract

We present new sufficient conditions for the existence of a continuous utility function for an arbitrary binary relation on a topological space. Such conditions are basically obtained by using both the concept of a weakly continuous binary relation on a topological space and the concept of a countable network weight. In particular, we are concerned with suitable topological notions which generalize the concept of compactness and do not imply second countability or local compactness.

• Item Type: MPRA Paper
• Language: English
• Keywords: hereditarily Lindeloef space; weakly continuous binary relation; countable network weight; hemicompactness; submetrizability
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60 - General

References

1. Aumann, R.: Utility theory without the completeness axiom. Econometrica 30, 445-462 (1962).

2. Back, K.: Concepts of similarity for utility functions. Journal of Mathematical Economics 15, 129-142 (1986).

3. Bosi, G., Herden, G.: The structure of completely useful and very useful topologies, preprint (2006).

4. Bosi, G., Herden, G.: Continuous utility representations theorems in arbitrary concrete categories. Applied Categorical Structures, 16(5), 629-651 (2008).

5. Bosi, G. and Mehta, G.B.: Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof, Journal of Mathematical Economics, 38 (2002), 311--328

6. Bridges, D.S., Mehta, G.B.: Representations of Preference Orderings, Springer-Verlag, Berlin, 1995.

7. Caterino, A., Ceppitelli, R., Maccarino, F.: Continuous utility on hemicompact submetrizable k-spaces, submitted (2009).

8. Caterino, A., Ceppitelli, R., Mehta, G.B.: Representations of topological preordered spaces, submitted (2009).

9. Chateauneuf, A.: Continuous representation of a preference relation on a connected topological space, Journal of Mathematical Economics 16, 139--146 (1987).

10. Debreu, G.; Representation of a preference ordering by a numerical function. In: R. Thrall, C. Coombs and R. Davies (Eds.), Decision Processes, pp. 159-166, Wiley, New York, (1954).

11. Debreu, G.: Continuity properties of a Paretian utility. International  Economic Review 5, 285-293 (1964).

12. Eilenberg, S.: Ordered topological spaces. {\em American Journal of  Mathematics 63, 39-45 (1941).

13. Engelking, R.: General Topology, Heldermann Verlag, Berlin, 1989.

14. Estevez, M. and Herves, C.: On the existence of continuous preference orderings without utility representations, Journal of Mathematical Economics 24 (1995), 305-309.

15. Herden, G.: On the Existence of Utility Functions. Mathematical Social Sciences 17, 297-313 (1989).

16. Herden, G.: Topological spaces for which every continuous total preorder can be represented by a continuous utility function. Mathematical Social Sciences 22, 123-136 (1991).

17. Herden, G., Pallack, A.: Useful topologies and separable systems. Applied General Topology 1, 61-82 (2000)

18. Herden, G., Pallack, A.: On the continuous analogue of the Szpilrajn Theorem I, Mathematical Social Sciences 43, 115-134 (2002).

19. Levin, V.L.: A continuous utility theorem for closed preorders on a $\sigma$-compact metrizable space. Soviet Math. Dokl. 28, 715-718 (1983).

20. McCoy, R.A.: Functions spaces which are k-spaces, Topology Proceedings 5, 139-146 (1980).

21. Mehta, G.B.: Some General Theorems on the Existence of Order-Preserving Functions. Mathematical Social Sciences 15, 135-146 (1988).

22. Mehta, G.B.: Preference and Utility. In Handbook of Utility Theory, Volume 1, eds. S. Barberà, P. Hammond and C. Seidl, pp. 1-47, Dordrecht: Kluwer Academic Publishers, 1988.

23. Monteiro, P. K.:. Some results on the existence of utility functions on path connected spaces. Journal of Mathematical Economics 16, 147-156 (1987).

24. Nachbin, L.: Topology and order, Van Nostrand, Princeton, 1965.

25. Ok, E.A.: Utility representation of an incomplete preference relation, Journal of Economic Theory 104, 429-449 (2002).

26. Ok, E.A.: Functional representation of rotund-valued proper multifunctions, preprint, New York University, 2002.

27. Peleg, B.: Utility functions for partially ordered topological spaces. Econometrica 38, 93-96 (1970).

28. Steen, L.A., Seebach, J.A.: Counterexamples in Topology, Dover Publications, New York, 1995.