Diffusion mechanism design, which investigates how to incentivise agents to invite as many colleagues to a multi-agent decision making process as possible, is a new research paradigm at the intersection between microeconomics and computer science. In this paper, we extend traditional facility location games into the model of diffusion mechanism design. Our objective is to completely understand to what extent of anonymity/voter-relevance we can achieve, along with strategy-proofness and Pareto efficiency when voters strategically invite colleagues. We define a series of anonymity properties applicable to the diffusion mechanism design model, as well as parameterized voter-relevance properties for guaranteeing reasonably-fair decision making. We obtained two impossibility theorems and two existence theorems, which partially answer the question we have raised at the beginning of the paper.