We propose Compatibility Weighted Voting Games, a variant of Weighted Voting Games in which some pairs of agents are compatible and some are not. In a Weighted Voting Game each agent has a weight, and a set of agents can form a winning coalition if the sum of their weights is at least a given quota. Whereas the original Weighted Voting Game model assumes that all agents are compatible, we consider a model in which the agents’ compatibility is described by a compatibility graph. We consider power indices, which measure the power of each agent to affect the outcome of the game, and show that their computation is tractable under certain restrictions (chiefly that the agents’ compatibilities have spectral structure). Through simulations we investigate the effect an agent’s compatibility restrictions has on its power.