A Comparison of the GAI Model and the Choquet Integral w.r.t. a k-ary Capacity

This paper proposes a comparison between a GAI model and the Choquet integral w.r.t. a k-ary capacity. We show that these two models are much closer than one would expect. Based on this comparison, we show a new result on the GAI models: any 2-additive GAI model can be rewritten in such a way that all utility terms in the GAI decomposition are non-negative and monotone. This is very important in practice since it allows reducing the number of monotonicity constraints to be enforced in the elicitation process, from an exponential number (of the number of attributes) to a quadratic number.

[1]  M. Grabisch.  The application of fuzzy integrals in multicriteria decision making , 1996 .

[2]  Eyke Hüllermeier,et al.  Efficient Learning of Classifiers Based on the 2-Additive Choquet Integral , 2013 .

[3]  Elías F. Combarro,et al.  Extreme points of some families of non-additive measures , 2006, Eur. J. Oper. Res..

[4]  Peter C. Fishburn,et al.  INTERDEPENDENCE AND ADDITIVITY IN MULTIVARIATE, UNIDIMENSIONAL EXPECTED UTILITY TIHEORY* , 1967 .

[5]  Patrice Perny,et al.  Decision making with multiple objectives using GAI networks , 2011, Artif. Intell..

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

[7]  Craig Boutilier,et al.  Minimax regret based elicitation of generalized additive utilities , 2007, UAI.

[8]  Jean-Luc Marichal,et al.  Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices , 2006, Games Econ. Behav..

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

[10]  Peter C. Fishburn,et al.  Utility theory for decision making , 1970 .

[11]  Christophe Labreuche,et al.  Capacities on lattices and k-ary capacities , 2003, EUSFLAT Conf..

[12]  Christophe Labreuche,et al.  Bipolarization of posets and natural interpolation , 2008, ArXiv.

[13]  Craig Boutilier,et al.  Local Utility Elicitation in GAI Models , 2005, UAI.

[14]  Michel Grabisch,et al.  A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid , 2010, Ann. Oper. Res..

[15]  Fahiem Bacchus,et al.  Graphical models for preference and utility , 1995, UAI.

[16]  Salvatore Greco,et al.  Robust ordinal regression for value functions handling interacting criteria , 2014, Eur. J. Oper. Res..

[17]  Christophe Labreuche,et al.  Use of the GAI model in multi-criteria decision making: inconsistency handling, interpretation , 2013, EUSFLAT Conf..

[18]  Jun Okamoto,et al.  Inclusion-Exclusion Integral and Its Application to Subjective Video Quality Estimation , 2010, IPMU.

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

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