×

Generalized Debreu’s open gap lemma and continuous representability of biorders. (English) Zbl 1355.06002

In this paper, the author proves a generalization of the key open gap lemma by G. Debreu, issued in [Int. Econ. Rev. 5, 285–293 (1964; Zbl 0138.16301)]. In the original result, Debreu proved that given a subset \(S\) of the real line we can find a strictly increasing real-valued function \(f\) such that all the gaps of \(f(S)\) are open. This is a crucial fact in utility theory because, using it, it is possible to prove that given a total ordered set \(X\), if there exists a real-valued utility function defined on it, it is also possible to find another utility function with the additional property of being continuous as regards the order topology on \(X\) and the usual Euclidean topology on the real line.
In the present manuscript, the author introduces a generalization of Debreu’s open gap lemma to the case of \(n\) subsets of the real line, proving in some special suitable situations the existence of a strictly increasing real function such that the gaps on the image of all of each of those original sets is open. It is shown that this generalization is not equivalent to Debreu’s lemma working on the union of the given sets, neither on its intersection.
The new generalization is then used to find continuous representations of biorders, in utility theory and conjoint measurement.
Finally, some cases of order structures where Debreu’s open gap lemma does not look helpful are also considered and analyzed, mentioning that a good combination of the (generalized) open gap lemma and Abel’s functional equation could be successful at this stage.

MSC:

06A06 Partial orders, general
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
91B16 Utility theory
06A75 Generalizations of ordered sets

Citations:

Zbl 0138.16301
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abrísqueta, F.J., Candeal, J.C., Catalán, R.G., De Miguel, J.R., Induráin, E.: Generalized Abel functional equations and numerical representability of semiorders. Publ. Math. Debrecen 78(3-4), 557-568 (2011) · Zbl 1240.39047 · doi:10.5486/PMD.2011.4784
[2] Agaev, R., Aleskerov, F.: Interval choice: Classic and general cases. Math. Social Sci. 26, 249-272 (1993) · Zbl 0802.90002 · doi:10.1016/0165-4896(93)90022-B
[3] Beardon, A.F.: Debreu’s gap theorem. Economic Theory 2, 150-152 (1992) · Zbl 0808.90010 · doi:10.1007/BF01213257
[4] Bosi, G.: Continuous representations of interval orders based on induced preorders. Rivista di Matematica per le Scienze Economiche e Soziali 18(1), 75-82 (1995) · Zbl 0870.90020
[5] Bosi, G., Campión, M.J., Candeal, J.C., Induráin, E.: Interval-valued representability of qualitative data: The continuous case. Int. J. Uncertainty Fuzziness Knowledge Based Syst. 15(3), 299-319 (2007) · Zbl 1298.68264 · doi:10.1142/S0218488507004698
[6] Bosi, G., Candeal, J.C., Induráin, E.: Continuous representability of interval orders and biorders. J. Math. Psych. 51, 122-125 (2007) · Zbl 1114.06001 · doi:10.1016/j.jmp.2006.10.005
[7] Bosi, G., Estevan, A., Gutiérrez García, J., Induráin, E.: Continuous representability of interval orders, the topological compatibility setting. Int. J. Uncertainty Fuzziness Knowledge Based Syst. 23(3), 345-365 (2015) · Zbl 1328.06001
[8] Bosi, G., Herden, G.: Continuous multi-utility representations of preorders. J. Math. Econ. 48, 212-218 (2012) · Zbl 1250.91045 · doi:10.1016/j.jmateco.2012.05.001
[9] Bosi, G., Mehta, G.B.: Existence of a semicontinuous or continuous utility function: A unified approach and an elementary proof. J. Math. Econ. 38, 311-328 (2002) · Zbl 1033.91005 · doi:10.1016/S0304-4068(02)00058-7
[10] Bosi, G., Zuanon, M.: Upper semicontinuous representations of interval orders. Math. Social Sci. 60, 60-63 (2014) · Zbl 1315.91020 · doi:10.1016/j.mathsocsci.2013.12.005
[11] Bowen, R.: A new proof of a theorem in utility theory. Int. Econ. Rev. 9(3), 374 (1968) · Zbl 0164.50307 · doi:10.2307/2556234
[12] Bridges, D.S., Mehta, G.B.: Representations of Preference Orderings. Springer-Verlag, Berlin-Heidelberg-New York (1995) · Zbl 0836.90017 · doi:10.1007/978-3-642-51495-1
[13] Candeal, J.C., Estevan, A., Gutiérrez-García, J., Induráin, E.: Semiorders with separability properties. J. Math. Psych. 56, 444-451 (2012) · doi:10.1016/j.jmp.2013.01.003
[14] Candeal, J.C., Induráin, E., Zudaire, M.: Numerical representability of semiorders. Math. Social Sci. 43(1), 61-77 (2002) · Zbl 1013.06004 · doi:10.1016/S0165-4896(01)00082-8
[15] Candeal, J.C., Induráin, E., Zudaire, M.: Continuous representability of interval orders. Appl. Gen. Topol. 5(2), 213-230 (2004) · Zbl 1069.54022 · doi:10.4995/agt.2004.1971
[16] Debreu, G.; Thrall, R. (ed.); Coombs, C. (ed.); Davies, R. (ed.), Representation of a preference ordering by a numerical function (1954), New York · Zbl 0058.13803
[17] Debreu, G.: Continuity properties of Paretian utility. Int. Econ. Rev. 5, 285-293 (1964) · Zbl 0138.16301 · doi:10.2307/2525513
[18] Doignon, J.P., Ducamp, A., Falmagne, J.C.: On realizable biorders and the biorder dimension of a relation. J. Math. Psych. 28, 73-109 (1984) · Zbl 0562.92018 · doi:10.1016/0022-2496(84)90020-8
[19] Ducamp, A., Falmagne, J.C.: Composite measurement. J. Math. Psych. 6, 359-390 (1969) · Zbl 0184.45501 · doi:10.1016/0022-2496(69)90012-1
[20] Estevan, A., Gutiérrez García, J., Induráin, E.: Numerical representation of semiorders. Order 30, 455-462 (2013) · Zbl 1273.54029 · doi:10.1007/s11083-012-9255-3
[21] Estevan, A., Gutiérrez García, J., Induráin, E.: Further results on the continuous representability of semiorders. Int. J. Uncertainty Fuzziness Knowledge Based Syst. 21(5), 675-694 (2013) · Zbl 1326.06002 · doi:10.1142/S0218488513500323
[22] Fishburn, P.C.: Semiorders and risky choices. J. Math. Psych. 5, 358-361 (1968) · Zbl 0157.26603 · doi:10.1016/0022-2496(68)90080-1
[23] Gensemer, S.H.: Continuous semiorder representations. J. Math. Econom. 16, 275-289 (1987) · Zbl 0637.90010 · doi:10.1016/0304-4068(87)90013-9
[24] Jaffray, J.: Existence of a continuous utility function: An elementary proof. Econometrica 43, 981-983 (1975) · Zbl 0321.90006 · doi:10.2307/1911340
[25] Krantz, D.H.: Extensive measurement in semiorders. Philos. Sci. 34, 348-362 (1967) · doi:10.1086/288173
[26] Luce, R.D.: Semiorders and a theory of utility discrimination. Econometrica 24, 178-191 (1956) · Zbl 0071.14006 · doi:10.2307/1905751
[27] Olóriz, E., Candeal, J.C., Induráin, E.: Representability of interval orders. J. Econ. Theory 78(1), 219-227 (1998) · Zbl 0895.90023 · doi:10.1006/jeth.1997.2346
[28] Ouwehand, P.: A simple proof of Debreu’s gap lemma. ORiOn 26(1), 17-20 (2010) · doi:10.5784/26-1-83
[29] Scott, D., Suppes, P.: Foundational aspects of theories of measurement. J. Symb. Log. 23, 113-128 (1958) · Zbl 0084.24603 · doi:10.2307/2964389
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.