Myerson values for games with fuzzy communication structure

In 1977, Myerson considered cooperative games with communication structure. A communication structure is an undirected graph describing the bilateral relationships among the players. He introduced the concept of allocation rule for a game as a function obtaining an outcome for each communication structure among the players of the game. The Myerson value is a specific allocation rule extending the Shapley value of the game. More recently, the authors studied games with fuzzy communication structures using fuzzy graph-theoretic ideas. Now we propose a general framework in order to define fuzzy Myerson values. Players in a coalition need to measure their profit using their real individual and communication capacities at every moment because these attributes are fuzzy when the game is proposed. So, they look for forming connected coalitions working at the same level. The different ways to obtain these partitions by levels determine different Myerson values for the game. Several interesting examples of these ways are studied in the paper, following known models in games with fuzzy coalitions: the proportional model and the Choquet model.