Weighted Quasi-Arithmetic Means and the Domain Translations

Yuji Yoshida

doi:10.20965/jaciii.2012.p0148

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JACIII Vol.16 No.1 pp. 148-153

doi: 10.20965/jaciii.2012.p0148

(2012)

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Weighted Quasi-Arithmetic Means and the Domain Translations

Yuji Yoshida

Faculty of Economics and Business Administration, University of Kitakyushu, 4-2-1 Kitagata, Kokuraminami, Kitakyushu 802-8577, Japan

Received:

January 15, 2011

Accepted:

October 3, 2011

Published:

January 20, 2012

Keywords:

weighted quasi-arithmetic mean, utility functions, weighting functions, domain translations, indices translation

Abstract

The monotonicity of the weighted quasi-arithmetic means is discussed from the viewpoint of utility functions and weighting functions. Observing the indices given by utility functions and weighting functions, we find that these indices give similarmonotonicity for the weighted quasi-arithmetic means, and this paper investigates the reason using the direct domain translation adapted to the weighted quasi-arithmetic means. Further, general domain translations are introduced and they are discussed in relation to these indices. Some examples are given for the domain translations and as a special case the translation invariance is presented, and a lot of examples with various utility functions and weighting functions are shown, and their indices and the translated forms are derived.

Cite this article as:

Y. Yoshida, “Weighted Quasi-Arithmetic Means and the Domain Translations,” J. Adv. Comput. Intell. Intell. Inform., Vol.16 No.1, pp. 148-153, 2012.

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References

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[2] P. C. Fishburn, “Utility Theory for Decision Making,” John Wiley and Sons, New York, 1970.
[3] Y. Yoshida, “Aggregated mean ratios of an interval induced from aggregation operations,” V. Torra and Y. Narukawa (Eds.), Modeling Decisions for Artificial Intelligence – MDAI 2008, Lecture Notes in Artificial Intelligence, Vol.5285, pp. 26-37, Springer, Oct. 2008.
[4] Y. Yoshida, “Quasi-arithmetic means and ratios of an interval induced from weighted aggregation operations,” Soft Computing, Vol.14, pp. 473-485, 2010.
[5] A. N. Kolmogoroff, “Sur la notion de la moyenne,” Acad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez., Vol.12, pp. 388-391, 1930.
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[8] Y. Yoshida, “Weighted quasi-arithmetic means and conditional expectations,” V. Torra, Y. Narukawa, and M. Daumas (Eds.), Modeling Decisions for Artificial Intelligence – MDAI 2010, Lecture Notes in Artificial Intelligence, Vol.6408, Springer, pp. 31-42, Oct. 2010.
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[12] J. W. Pratt, “Risk Aversion in the Small and the Large,” Econometrica, Vol.32, pp. 122-136, 1964.
[13] Y. Yoshida, “Weighted quasi-arithmetic means and a risk index for stochastic environments,” Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.19, Issue1, pp. 1-16, 2011.
[14] V. Torra and Y. Narukawa, “Modeling Decisions – Information Fusion and Aggregation Operators,” Springer, 2002.
[15] J. Lázaro, T. Rückschlossová, and T. Calvo, “Shift invariant binary weighted aggregation operators,” Fuzzy Sets and Systems, Vol.142, pp. 51-62, 2004.
[16] R.Mesiar and T. Rückschlossová, “Characterization of invariant aggregation operators,” Fuzzy Sets and Systems, Vol.142, pp. 63-73, 2004.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] J. J. Dujmović, “Weighted Conjunctive and disjunctive means and their application in system evaluation,” Univ. Beograd. Publ. Elektotech. Fak. Ser. Mat. Fiz., Vol.483, pp. 147-158, 1974.

[2] [2] P. C. Fishburn, “Utility Theory for Decision Making,” John Wiley and Sons, New York, 1970.

[3] [3] Y. Yoshida, “Aggregated mean ratios of an interval induced from aggregation operations,” V. Torra and Y. Narukawa (Eds.), Modeling Decisions for Artificial Intelligence – MDAI 2008, Lecture Notes in Artificial Intelligence, Vol.5285, pp. 26-37, Springer, Oct. 2008.

[4] [4] Y. Yoshida, “Quasi-arithmetic means and ratios of an interval induced from weighted aggregation operations,” Soft Computing, Vol.14, pp. 473-485, 2010.

[5] [5] A. N. Kolmogoroff, “Sur la notion de la moyenne,” Acad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez., Vol.12, pp. 388-391, 1930.

[6] [6] K. Nagumo, “Über eine Klasse der Mittelwerte,” Japanese J. of Mathematics, Vol.6, pp. 71-79, 1930.

[7] [7] J. Aczél, “On weighted mean values,” Bulletin of the American Math. Society, Vol.54, pp. 392-400, 1948.

[8] [8] Y. Yoshida, “Weighted quasi-arithmetic means and conditional expectations,” V. Torra, Y. Narukawa, and M. Daumas (Eds.), Modeling Decisions for Artificial Intelligence – MDAI 2010, Lecture Notes in Artificial Intelligence, Vol.6408, Springer, pp. 31-42, Oct. 2010.

[9] [9] G. Gollier, “The Economics of Risk and Time,” MIT Publishers, 2001.

[10] [10] L. Eeckhoudt, G. Gollier, and H. Schkesinger, “Economic and Financial Decisions under Risk,” Princeton University Press, 2005.

[11] [11] K. J. Arrow, “Essays in the Theory of Risk-Bearing,” Markham, Chicago, 1971.

[12] [12] J. W. Pratt, “Risk Aversion in the Small and the Large,” Econometrica, Vol.32, pp. 122-136, 1964.

[13] [13] Y. Yoshida, “Weighted quasi-arithmetic means and a risk index for stochastic environments,” Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.19, Issue1, pp. 1-16, 2011.

[14] [14] V. Torra and Y. Narukawa, “Modeling Decisions – Information Fusion and Aggregation Operators,” Springer, 2002.

[15] [15] J. Lázaro, T. Rückschlossová, and T. Calvo, “Shift invariant binary weighted aggregation operators,” Fuzzy Sets and Systems, Vol.142, pp. 51-62, 2004.

[16] [16] R.Mesiar and T. Rückschlossová, “Characterization of invariant aggregation operators,” Fuzzy Sets and Systems, Vol.142, pp. 63-73, 2004.