Star-shapedness of the kernel for homogeneous games

Abstract

Homogeneous games and weighted majority games were introduced by von Neumann-Morgenstern ( Theory of Games and Economic Behavior, 1944) in the constant-sum case. Peleg ( Illinois Journal of Mathematics, 1966, 10, 39–48; SIAM Journal of Applied Mathematics, 1968, 16, 527–532) studied the kernel and nucleolus for these classes of games. The general theory of homogeneous, not necessarily constant-sum, games was developed by Ostmann ( International Journal of Game Theory, 1987a, 16, 69–81), Rosenmüller (On homogeneous weights for simple games, Working Paper 115, 1982; Zeitschrift für Operations Research, 1984, 12, 123–141; Mathematics of Operations Research, 1987a, 12, 309–330), and Sudhölter ( International Journal of Game Theory, 1989, 18, 433–469). Peleg-Rosenmüller ( Games and Economic Behaviour, 1992, 4, 588–605) used it to discuss several solution concepts for homogeneous games without steps. A reduction theorem for the nucleolus and kernel of homogeneous games with steps was proved by Rosenmüller-Sudhölter ( Discrete Applied Mathematics, 1994, 50, 53–76) and Peleg et al. (The kernel of homogeneous games with steps, Essays in Game Theory in Honor of Michael Maschler, 1994), respectively. On the basis of these results, this paper shows that the kernel of each homogeneous game without winning players is star-shaped. Moreover, each of these games possesses a truncated game that is uniquely determined and homogeneous itself. The normalized vector of weights of the minimal representation of the truncated game is a center of both of the kernels of the original game and the truncated game. Moreover, this preimputation is Lorenz maximal within the kernel and can be characterized as the unique minimizer of a social welfare ordering in the kernel. To be more precise, the center solution uniquely minimizes a weighted Gini coefficient. Every weighted majority game occurs as a reduced game of certain homogeneous games called homogeneous extensions. The kernel of a weighted majority game arises from that of each of its homogeneous extensions in a very simple way. Moreover, the kernels of partition games are shown to be singletons.