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 ). Copyright Springer-Verlag Berlin Heidelberg 2014