On the Separability of Functions and Games

We study the notion of (additive) separability of a function of several variables with respect to a hypergraph (H-graph). We prove the existence of a unique minimal H-graph with respect to which a function is separable and show that the corresponding minimal decomposition of the function can be obtained through a recursive algorithm. We then focus on (strategic form) games and propose a concept of separability for a game with respect to a forward directed hypergraph (FDH-graph). This notion refines and generalizes that of the graphical game and is invariant with respect to strategic equivalence. We show that every game is separable with respect to a minimal FDH-graph. Moreover, for exact potential games, such minimal FDH-graph reduces to the minimal H-graph of the potential function. Our results imply and refine known results on graphical potential games and yield a new proof of the celebrated Hammersely-Clifford theorem on Markov random fields.