Separable games
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We introduce the notion of separable games, which refines and generalizes that of graphical games. We prove that there exists a minimal splitting with respect to which a game is separable. Moreover we prove that in every strategic equivalence class, there is a game separable with respect to the minimal splitting in the class. This game is also graphical with respect to the smallest graph in the class, which represent a minimal complexity graphical description for the game. We prove a symmetry property of the minimal splitting of potential games and we describe how this property reflects to a decomposition of the potential function. In particular, these last results strengthen the ones recently proved for graphical potential games. Finally, we study the interplay between separability and the classical decomposition of games proposed by [5], characterizing the separability properties of each part of the decomposition.
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