Modularity and monotonicity of games

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 $$v$$v, its Möbius inverse is equal to zero within the framework of the $$k$$k-modularity of $$v$$v for $$k \ge 2$$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 $$v$$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 $$k$$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).

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