People often express political opinions in starkly dichotomous terms, such as “Trump will either trigger a ruinous trade war or save U.S. factory workers from disaster.” This mode of communication promotes polarization into ideological in-groups and out-groups. We explore the power of an emerging methodology, forecasting tournaments, to encourage clashing factions to do something odd: to translate their beliefs into nuanced probability judgments and track accuracy over time and questions. In theory, tournaments advance the goals of “deliberative democracy” by incentivizing people to be flexible belief updaters whose views converge in response to facts, thus depolarizing unnecessarily polarized debates. We examine the hypothesis that, in the process of thinking critically about their beliefs, tournament participants become more moderate in their own political attitudes and those they attribute to the other side. We view tournaments as belonging to a broader class of psychological inductions that increase epistemic humility and that include asking people to explore alternative perspectives, probing the depth of their cause-effect understanding and holding them accountable to audiences with difficult-to-guess views.