We present an axiomatic characterization of entropies with properties of branching, continuity, and weighted additivity. We deliberately do not assume that the entropies are symmetric. The resulting entropies are generalizations of the entropies of degree α, including the Shannon entropy as the case α = 1. Such “weighted” entropies have potential applications to the “utility of gambling” problem.
@article{bwmeta1.element.doi-10_2478_s11533-010-0026-3,
author = {Bruce Ebanks},
title = {Weighted entropies},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {602-615},
zbl = {1204.39025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0026-3}
}
Bruce Ebanks. Weighted entropies. Open Mathematics, Tome 8 (2010) pp. 602-615. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0026-3/
[1] Aczél J., personal communication, March 13, 2007
[2] de Bruijn N.G., Functions whose differences belong to a given class, Nieuw Arch. Wiskd. (2), 1951, 23, 194–218 | Zbl 0042.28805
[3] Ebanks B.R., The branching property in generalized information theory, Adv. in Appl. Probab., 1978, 10, 788–802 http://dx.doi.org/10.2307/1426659 | Zbl 0401.94013
[4] Ebanks B.R., Branching and generalized-recursive inset entropies, Proc. Amer. Math. Soc., 1980, 79, 260–267 http://dx.doi.org/10.2307/2043247 | Zbl 0401.94014
[5] Ebanks B.R., Functional equations for weighted entropies, Results Math., 2009, 55, 329–357 http://dx.doi.org/10.1007/s00025-009-0435-4 | Zbl 1184.39013
[6] Ebanks B.R., Sahoo P.K., Sander W., Characterizations of Information Measures, World Scientific, Singapore/New Jersey/London/Hong Kong, 1998
[7] Havrda J., Charvát F., Quantification method of classification processes. Concept of structural α-entropy, Kybernetika (Prague), 1967, 3, 30–35 | Zbl 0178.22401
[8] Kemperman J.H.B., A general functional equation, Trans. Amer. Math. Soc., 1957, 86, 28–56 http://dx.doi.org/10.2307/1992864 | Zbl 0079.33402
[9] Luce R.D., Marley A.A.J., Ng C.T., Entropy-related measures of the utility of gambling, In: Brams S., Gehrlein W., Roberts F. (Eds.), The Mathematics of Preference, Choice, and Order: Essays in Honor of Peter C. Fishburn, Springer, New York, 2009, 5–25 http://dx.doi.org/10.1007/978-3-540-79128-7_1
[10] Luce R.D., Ng C.T., Marley A.A.J., Aczél J., Utility of gambling I: Entropy-modified linear weighted utility, Economic Theory, 2008, 36, 1–33 http://dx.doi.org/10.1007/s00199-007-0260-5 | Zbl 1136.91008
[11] Luce R.D., Ng C.T., Marley A.A.J., Aczél J., Utility of gambling II: Risk, paradoxes, and data, Economic Theory, 2008, 36, 165–187 http://dx.doi.org/10.1007/s00199-007-0259-y | Zbl 1137.91010
[12] Marley A.A.J., Luce R.D., Independence properties vis-à-vis several utility representations, Theory and Decision, 2005, 58, 77–143 http://dx.doi.org/10.1007/s11238-005-2460-4 | Zbl 1137.91382
[13] Ng C.T., Measures of information with the branching property over a graph and their representations, Inform. and Control, 1979, 41, 214–231 http://dx.doi.org/10.1016/S0019-9958(79)90571-0 | Zbl 0423.94006
[14] Ng C.T., Luce R.D., Marley A.A.J., Utility of gambling when events are valued: An application of inset entropy, Theory and Decision, 2009, 67, 23–63 http://dx.doi.org/10.1007/s11238-007-9065-z | Zbl 1168.91336
[15] Ng C.T., Luce R.D., Marley A.A.J., Utility of gambling under p-additive joint receipt and segregation or duplex decomposition, J. Math. Psych., 2009, 53, 273–286 http://dx.doi.org/10.1016/j.jmp.2009.04.001 | Zbl 1187.91067