Some Characterizations of <i>k</i>-Monotonicity Through the Bipolar Möbius Transform in Bi-Capacities

Abstract

This paper first proposes the bipolar Möbius transform as an extension of dividends of cooperative games to that of bi-cooperative games (bi-capacities) defined on 3 N , which is different from the Möbius transform defined by Grabisch and Labreuche. The k -monotonicity of bi-capacities is characterized through each of the following notions: the bipolar and ordinary Möbius transforms, discrete derivatives, and partial derivatives of the piecewise multilinear extension of the ternary pseudo-Boolean function corresponding to the bi-capacities.