On Maximum Income Spanning Forest Games

Cooperative games, an important branch of game theory, are mainly to distribute the profits brought by cooperation, which is to find solutions to them. Considering the difficulties in computing the solutions and the concise proof in graph theory, contacting cooperative games with graph theory, we study cooperative games on graph. Since Bird introduced minimum cost spanning tree games in 1976, there have been plenties of literatures to study cooperative games on graph. This paper defines the maximum income spanning forest games which are a new kind of games on graph and have no root. Similarly to the minimum cost spanning tree games, we present the properties of the core, the kernel and the nucleolus in the maximum income spanning forest games. The main content of this paper can be summarized as follows:1. Defining the maximum income spanning forest games on graph firstly, and demonstrating the rationality of the definition, then analyzing the difference between them and the minimum cost spanning tree games.2. Proving the non-emptiness and the representation of the core, and presenting the simple definitions of the core and the nucleolus.3. Describing the relationship among the core, the kernel and the nucleolus, and showing the necessary and sufficient condition of that an imputation in the core belongs to the kernel, then proving that the intersection of the core and the kernel is equal to the nucleolus by iteration independently.4. Studying the maximum income spanning forest games on tree and developing a way to compute the nucleolus (which coincides with the kernel) on tree, then extending the obtained conclusions to the inessential graph.5. Showing the decomposition of the nucleolus on graph containing cut vertexes.