[The results of A. K. Sen and P. K. Pattanaik on sufficient conditions for the transitivity of simple majorities are extended to the case where it is not assumed that the individual's indifference relations are transitive.]

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.

[Social choice lottery rules are analyzed for two-candidate elections with voters who may be uncertain about whom they prefer. A voter's uncertainty is reflected by a nonobservable choice probability of voting for candidate A rather than candidate B, given that he votes. Lottery rules are based on the votes for A and B; they are to be monotonic and symmetric in voters and in candidates. Given n voters, all lottery rules are convex combinations of about n/2 basic rules ranging from the coin-flip rule to simple majority. Candidate A's win probability and two measures of expected voter satisfaction are examined as functions of the individuals' choice probabilities and the lottery rules. Comparisons are made between simple majority and the proportional lottery rule which assigns social choice probability of j/n to A when A gets j of n votes. Each of simple majority and the proportional lottery rule satisfies attractive properties that are not satisfied by the other rule.]