This article uncovers dynamic properties of the von Neumann–Morgenstern solution in weak tournaments and majoritarian games. We propose a new procedure for the construction of choice sets from weak tournaments, based on dynamic stability criteria. The idea is to analyze dynamic versions of tournament games. The exploration of a specific class of Markov perfect equilibria in these “dynamic tournament games” yields a new solution concept for weak tournaments—the A-stable set. The alternatives in an A-stable set constitute persistent, long-run policy outcomes in the corresponding dynamic tournament games. We find that, in any weak tournament, the class of A-stable sets coincides with that of von Neumann–Morgenstern stable sets.