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, 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).