Some Characterizations of k-Monotonicity Through the Bipolar Möbius Transform in Bi-Capacities
Katsushige Fujimoto*, and Toshiaki Murofushi**
*College of Symbiotic Systems Science, Fukushima University, 1 Kanayagawa, Fukushima 960-1296, Japan
**Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, 4259-G3-47 Nagatsuta, Midori-ku, Yokohama 226-8502, Japan
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 3N, 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.