The purpose of this paper is twofold. First, we generalize Kajii et al. (J Math Econ 43:218–230, <CitationRef CitationID="CR16">2007</CitationRef>) and provide a condition under which for a game <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$v$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>v</mi> </math> </EquationSource> </InlineEquation>, its Möbius inverse is equal to zero within the framework of the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$k$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>k</mi> </math> </EquationSource> </InlineEquation>-modularity of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$v$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>v</mi> </math> </EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$k \ge 2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math> </EquationSource> </InlineEquation>. This condition is more general than that in Kajii et al. (J Math Econ 43:218–230, <CitationRef CitationID="CR16">2007</CitationRef>). Second, we provide a condition under which for a game <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$v$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>v</mi> </math> </EquationSource> </InlineEquation>, its Möbius inverse takes non-negative values, and not just zero. This paper relates the study of totally monotone games to that of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$k$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>k</mi> </math> </EquationSource> </InlineEquation>-monotone games. Furthermore, this paper shows that the modularity of a game is related to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$k$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>k</mi> </math> </EquationSource> </InlineEquation>-additive capacities proposed by Grabisch (Fuzzy Sets Syst 92:167–189, <CitationRef CitationID="CR12">1997</CitationRef>). 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, <CitationRef CitationID="CR1">1994</CitationRef>). Our results can also be applied to potential functions proposed by Hart and Mas-Colell (Econometrica 57:589–614, <CitationRef CitationID="CR15">1989</CitationRef>) and further analyzed by Ui et al. (Math Methods Oper Res 74:427–443, <CitationRef CitationID="CR26">2011</CitationRef>). Copyright Springer-Verlag Berlin Heidelberg 2014
