Fuzzy Average Tree Solution for Graph Games with Fuzzy Coalitions
In this chapter, the model of graph games with fuzzy coalitions is proposed based on graph games and cooperative games with fuzzy coalitions. The fuzzy average tree solution of graph games with fuzzy coalitions is given, which can be regarded as the gener
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Abstract In this chapter, the model of graph games with fuzzy coalitions is proposed based on graph games and cooperative games with fuzzy coalitions. The fuzzy average tree solution of graph games with fuzzy coalitions is given, which can be regarded as the generalization of crisp graph games. It is shown that the fuzzy average tree solution is equal to the fuzzy Shapley value for complete graph games with fuzzy coalitions. We extend the notion of link-convexity, under which the fuzzy core is non-empty and the fuzzy average tree solution lies in this core. Keywords Graph game Link-convexity
· Average tree solution · Imputation · Fuzzy coalition ·
1 Introduction In a cooperative game, cooperation is not always possible for the players. Cooperative games with limited communication structure are called graph games introduced by Myerson [1]. The best-known single-valued solution for graph games is the Myerson value characterized by component efficiency and fairness. In [2, 3] the positional value is proposed. The value for such games is characterized by component efficiency and balanced total threats, see Slikker [4]. Herings et al. [5] defines the average tree solution for cycle-free graph games. Moreover, he generalizes this solution to the class of all graph games in [6]. C. Nie (B) · Q. Zhang School of Management and Economics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China e-mail: [email protected] C. Nie Department of Mathematics and Information Science, Shijiazhuang College, Shijiazhuang 050035, People’s Republic of China
B.-Y. Cao and H. Nasseri (eds.), Fuzzy Information & Engineering and Operations Research & Management, Advances in Intelligent Systems and Computing 211, DOI: 10.1007/978-3-642-38667-1_41, © Springer-Verlag Berlin Heidelberg 2014
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There are some situations where players do not fully participate in a coalition but to a certain extent. A fuzzy coalition is introduced by Aubin [7]. Butnariu [8] defines a Shapley value and shows the explicit form of the Shapley value on a limited class of fuzzy games. Tsurumi et al. [9] gives a new class of fuzzy games with integral form. The fuzzy core for fuzzy cooperative games has been researched in [10, 11]. This chapter is organized as follows. In Sect. 2 we give preliminary notions of graph games. In Sect. 3 we discuss graph games with fuzzy coalitions represented by an undirected graph. The fuzzy average tree solution is introduced. We prove that the fuzzy average tree solution is equal to the fuzzy Shapley value for a complete graph game with fuzzy coalitions. In Sect. 4 we give the relationship between the fuzzy average tree solution and the fuzzy core.
2 Preliminaries We consider a cooperative game with limited communication structure, called a graph game. It is represented by (N , v, L) with N = {1, · · · , n} a node set of players, v : 2 N → R a characteristic function and L ⊆ { {i, j}| i = j, i, j ∈ N } a set of edges. Usually, a cooperative game(N , v) is thought as a complete graph
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