Stochastics and StatisticsGames on fuzzy communication structures with Choquet players
Section snippets
Preliminary
A transferable utility cooperative game is a pair (N, v) where N is a finite set and is a function with v(∅) = 0. Cooperative games was introduced by von Neumann and Morgensten (1944). The elements of N = {1, 2, … , n} are called players, the subsets S ⊆ N coalitions and v(S) is the worth (benefits) of S. A game (N, v) is superadditive iff for all S, T ⊂ N with S ∩ T = ∅ happen v(S ∪ T) ⩾ v(S) + v(T). In a superadditive game disjoint groups of players always prefer to join in a bigger coalition. We are only
Fuzzy graphs
We introduce now some concepts about fuzzy graphs for which the reader can see Mordeson and Nair (2000). In this paper we use the operators ∧, ∨ as the minimum and the maximum respectively.
Let N be a finite set. A fuzzy set in N is an application τ: N → [0, 1]. We denote by [0, 1]N the family of fuzzy sets in N. If τ ∈ [0, 1]N then a fuzzy relation over τ is ρ ∈ [0, 1]N×N such that ρ(i, j) ⩽ τ(i) ∧ τ(j) for all i, j ∈ N. The fuzzy relation ρ over τ is reflexive if ρ(i, i) = τ(i) for each i ∈ N, and it is symmetric
Fuzzy communication structures
Let (N, v) be a superadditive cooperative game. We consider a fuzzy graph which changes the initial situation. The number ρ(i, i) is interpreted as the real level of involvement of player i ∈ N in the game and ρ(i, j) represents the maximum level which the link (i, j) can be used. Therefore, we understand a fuzzy communication structure over N as a fuzzy graph without loops over N. For each fuzzy communication structure ρ over N we would take the different possibilities of communication
Particular cases
Returning to the Choquet extension (4) introduced by Tsurumi et al. (2001), we are going to establish the relationship between this extension and the fuzzy Myerson value. Definition 7 A fuzzy graph is complete by links iff it satisfies that ρ(i, j) = ρ(i, i) ∧ ρ(j, j) for all i, j ∈ N.
Each link in ρ complete by links can be used at maximum permitted from its vertices. Hence, we can identify the complete by links fuzzy communication structures with the fuzzy coalitions in N by the relation ρ(i, i) = τ(i) for
Acknowledgments
This research has been partially supported by the Spanish Ministry of Education an Science and the European Regional Development Funf, under Grant SEJ2006-00706, and by the FQM237 Grant of the Andalusian Government.
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