Do all Tournaments Admit Irrelevant Matches?

We consider tournaments played by a set of agents in order to establish a ranking among them. We introduce the notion of irrelevant match, as a match that does not influence the ultimate ranking of the involved parties. After discussing the basic properties of this notion, we seek out tournaments that have no irrelevant matches, focusing on the class of tournaments where each agent challenges each other exactly once. We prove that tournaments with a static schedule and at least 5 agents always include irrelevant matches. Conversely, dynamic schedules can be devised in ways that avoid irrelevant matches, at least for one of the involved agents.