This paper explores implications for one-stage and two-stage decision processes of a theory of choice tha t accommodates nontransitive preferences. It focuses on probabilistic convexification of finite base sets and on choice from convex sets. The one-stage formulation always has a maximally-preferred element in the convex set. Two-stage processes allow not only a holistic procedure for the entire problem, but also give rise to naive and sophisticated sequential procedures. All three have unambiguous solutions, but they can be radically different under intransitivities. The thre e two-stage solutions coincide when preferences are transitive. Copyright 1988 by The Econometric Society.