The design of deterministic and fair mechanisms for selection among a set of self-motivated agents based solely on these agents' input is a major challenge in multiagent systems. This challenge is especially difficult when the agents can only communicate via a broadcast channel. We propose the notion of selection games: a special case of zero-sum games where the only possible outcomes are selections of a single agent among the set of agents. We assume the lack of an external coordinator, and therefore we focus on mechanisms which have a solution where the agents play weakly dominant strategies. Our first major result shows that dominated strategies could be added to any selection mechanism, so that the resulting mechanism becomes quasi-symmetric. For fairness, we require the mechanism to be non-imposing; that is, the mechanism should allow any agent to be selected in such a solution. We first show that such mechanisms do not exist when there are two or three agents in the system....