Abstract
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).
| Original language | English |
|---|---|
| Pages (from-to) | 29-46 |
| Number of pages | 18 |
| Journal | Mathematical Methods of Operations Research |
| Volume | 80 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2014 |
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Keywords
- Gini index
- k -Additive capacities
- Möbius inverse
- Potential functions
- Totally monotone games
ASJC Scopus subject areas
- Mathematics(all)
- Software
- Management Science and Operations Research
Cite this
Modularity and monotonicity of games. / Asano, Takao; Kojima, Hiroyuki.
In: Mathematical Methods of Operations Research, Vol. 80, No. 1, 2014, p. 29-46.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Modularity and monotonicity of games
AU - Asano, Takao
AU - Kojima, Hiroyuki
PY - 2014
Y1 - 2014
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 - k -Additive capacities
KW - Möbius inverse
KW - Potential functions
KW - Totally monotone games
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U2 - 10.1007/s00186-014-0468-7
DO - 10.1007/s00186-014-0468-7
M3 - Article
VL - 80
SP - 29
EP - 46
JO - Mathematical Methods of Operations Research
JF - Mathematical Methods of Operations Research
SN - 1432-2994
IS - 1
ER -