TY - JOUR
T1 - Modularity and monotonicity of games
AU - Asano, Takao
AU - Kojima, Hiroyuki
N1 - Funding Information:
We are grateful to the Associate Editor and two anonymous referees whose detailed comments and suggestions have improved the paper substantially. We are also grateful to Takashi Ui for his comments and advice on this work. This research is financially supported by the JSPS KAKENHI Grant Numbers 25380239, 23730299, 23000001 and 22530186, and the Joint Research Program of KIER. This paper was previously circulated under the title “Some Characterizations of Totally Monotone Games”. Of course, we are responsible for any remaining errors.
PY - 2014/8
Y1 - 2014/8
N2 - The purpose of this paper is twofold. First, we generalize Kajii et al. (J Math Econ 43:218-230, 2007) and provide a condition under which for a game v, its Möbius inverse is equal to zero within the framework of the k -modularity of v for k ≥2. This condition is more general than that in Kajii et al. (J Math Econ 43:218-230, 2007). Second, we provide a condition under which for a game v, its Möbius inverse takes non-negative values, and not just zero. This paper relates the study of totally monotone games to that of $$k$$ k -monotone games. Furthermore, this paper shows that the modularity of a game is related to k -additive capacities proposed by Grabisch (Fuzzy Sets Syst 92:167-189, 1997). To illustrate its application in the field of economics, we use these results to characterize a Gini index representation of Ben-Porath and Gilboa (J Econ Theory 64:443-467, 1994). Our results can also be applied to potential functions proposed by Hart and Mas-Colell (Econometrica 57:589-614, 1989) and further analyzed by Ui et al. (Math Methods Oper Res 74:427-443, 2011).
AB - The purpose of this paper is twofold. First, we generalize Kajii et al. (J Math Econ 43:218-230, 2007) and provide a condition under which for a game v, its Möbius inverse is equal to zero within the framework of the k -modularity of v for k ≥2. This condition is more general than that in Kajii et al. (J Math Econ 43:218-230, 2007). Second, we provide a condition under which for a game v, its Möbius inverse takes non-negative values, and not just zero. This paper relates the study of totally monotone games to that of $$k$$ k -monotone games. Furthermore, this paper shows that the modularity of a game is related to k -additive capacities proposed by Grabisch (Fuzzy Sets Syst 92:167-189, 1997). To illustrate its application in the field of economics, we use these results to characterize a Gini index representation of Ben-Porath and Gilboa (J Econ Theory 64:443-467, 1994). Our results can also be applied to potential functions proposed by Hart and Mas-Colell (Econometrica 57:589-614, 1989) and further analyzed by Ui et al. (Math Methods Oper Res 74:427-443, 2011).
KW - Gini index
KW - Möbius inverse
KW - Potential functions
KW - Totally monotone games
KW - k -Additive capacities
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U2 - 10.1007/s00186-014-0468-7
DO - 10.1007/s00186-014-0468-7
M3 - Article
AN - SCOPUS:84906544900
VL - 80
SP - 29
EP - 46
JO - Mathematical Methods of Operations Research
JF - Mathematical Methods of Operations Research
SN - 1432-2994
IS - 1
ER -