The majority runoff system is widely used around the world. Yet, our understanding of its properties and of voters’ behavior is limited. In this paper, we fully characterize the set of strictly perfect voting equilibria in large three-candidate majority runoff elections. Considering all possible distributions of preference orderings and intensities, we prove that only two types of equilibria can exist. First, there are always equilibria in which only two candidates receive votes. Second, there may exist an equilibrium in which three candidates receive votes. Its characteristics challenge common beliefs: (i) neither sincere voting by all voters, nor push over tactics (i.e. supporters of the front-runner voting for a less-preferred candidate in order to influence who will face the front-runner in the second round) are supported in equilibrium, and (ii) the winner does not necessarily have democratic legitimacy since the Condorcet winner may not even participate in the second round.