Axiomatic structure of k-additive capacities

In this paper we deal with the problem of axiomatizing the preference relations modelled through Choquet integral with respect to a $k$-additive capacity, i.e. whose Mobius transform vanishes for subsets of more than $k$ elements. Thus, $k$-additive capacities range from probability measures ($k=1$) to general capacities ($k=n$). The axiomatization is done in several steps, starting from symmetric 2-additive capacities, a case related to the Gini index, and finishing with general $k$-additive capacities. We put an emphasis on 2-additive capacities. Our axiomatization is done in the framework of social welfare, and complete previous results of Weymark, Gilboa and Ben Porath, and Gajdos.

[1]  Michel Grabisch,et al.  Alternative representations of OWA operators , 1997 .

[2]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decision-making , 1988 .

[3]  D. Schmeidler.  Integral representation without additivity , 1986 .

[4]  Bernard De Baets,et al.  Aggregation Operators Defined by k-Order Additive/Maxitive Fuzzy Measures , 1998, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[5]  Peter L. Hammer,et al.  Approximations of pseudo-Boolean functions; applications to game theory , 1992, ZOR Methods Model. Oper. Res..

[6]  B. Baets,et al.  A decomposition of k -additive choquet and k -maxitive sugeno integrals , 2001 .

[7]  Christophe Labreuche,et al.  The Choquet integral for the aggregation of interval scales in multicriteria decision making , 2003, Fuzzy Sets Syst..

[8]  G. Rota.  On the foundations of combinatorial theory I. Theory of Möbius Functions , 1964 .

[9]  Jacques Duchêne,et al.  Subjective Evaluation of Discomfort in Sitting Positions , 2002, Fuzzy Optim. Decis. Mak..

[10]  Alain Chateauneuf,et al.  Modeling attitudes towards uncertainty and risk through the use of choquet integral , 1994, Ann. Oper. Res..

[11]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[12]  M. Sugeno,et al.  Some quantities represented by the Choquet integral , 1993 .

[13]  J. Kacprzyk,et al.  The Ordered Weighted Averaging Operators: Theory and Applications , 1997 .

[14]  Endre Pap,et al.  Non-additive measures and integrals , 2006 .

[15]  Ronald R. Yager,et al.  On ordered weighted averaging aggregation operators in multicriteria decisionmaking , 1988, IEEE Trans. Syst. Man Cybern..

[16]  Bernard De Baets,et al.  A Decomposition of k-Additive Choquet and k-Maxitive Sugeno Integrals , 2001, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[17]  I. Gilboa,et al.  Linear Measures, the Gini Index, and The Income-Equality Trade-off , 1994 .

[18]  Michel Grabisch,et al.  K-order Additive Discrete Fuzzy Measures and Their Representation , 1997, Fuzzy Sets Syst..

[19]  Michel Grabisch,et al.  Characterizing k-additive fuzzy measures , 2002 .

[20]  A. Tversky,et al.  Foundations of Measurement, Vol. I: Additive and Polynomial Representations , 1991 .

[21]  H. Zimmermann,et al.  On the suitability of minimum and product operators for the intersection of fuzzy sets , 1979 .

[22]  G. Choquet.  Theory of capacities , 1954 .

[23]  Peter P. Wakker,et al.  Additive Representations of Preferences , 1989 .

[24]  Christophe Labreuche,et al.  On the extension of pseudo-Boolean functions for the aggregation of interacting criteria , 2003, Eur. J. Oper. Res..

[25]  Michel Grabisch,et al.  Subjective Evaluation of Discomfort in Sitting Positions , 2002, Fuzzy Optim. Decis. Mak..

[26]  Thibault Gajdos,et al.  Measuring Inequalities without Linearity in Envy: Choquet Integrals for Symmetric Capacities , 2002, J. Econ. Theory.

[27]  János C. Fodor,et al.  Characterization of the ordered weighted averaging operators , 1995, IEEE Trans. Fuzzy Syst..

[28]  John A. Weymark,et al.  GENERALIZED GIN 1 INEQUALITY INDICES , 2001 .

[29]  菅野 道夫,et al.  Theory of fuzzy integrals and its applications , 1975 .

[30]  D. Schmeidler.  Subjective Probability and Expected Utility without Additivity , 1989 .

[31]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.