In cooperative game theory, associated games allow for providing meaningful characterizations of solution concepts. Moreover, generalized values allow computing an influence or power index of each coalition in a game. In this paper, we view associated games through the lens of game maps and we define the novel Shapley-Hodge Game, briefly ``SHoGa''. We characterize SHoGa via an axiomatic approach as a generalized value, and we thoroughly discuss the consistency properties of its associated game. Furthermore, we describe the Hodge decomposition of an oriented graph representing the transitive closure of the Hasse diagram of coalitions in the game. Finally, we show how SHoGa is linked to the solution of the Poisson equation derived from such a decomposition.