Implicit Intransitivity Under Majority Rule with Mixed Motions

When several persons organized as a group are to choose one from among several specified alternatives they may make use of majority voting or some other decision-making process. This paper considers the case in which the decision process includes among the alternatives some that are probability mixtures of others. Examples are presented to illustrate the way in which a situation that is transitive under majority voting rule becomes intransitive when mixtures of the same alternatives are permitted. If utility product maximization is used to determine the group choice, as an agreed upon arbitration process when utilities are assigned to the alternatives by each group member but without interpersonal comparability, some examples are presented to illustrate the results of this method. It is reasoned that some such method is often clearly preferable to majority voting in order both to allot more appropriate influence to minorities and to accommodate for differences in importance of alternatives beyond those represented by individual rank orderings. The utility product maximization section includes several numerical examples to illustrate relationships between the findings of this paper and familiar work in game theory by Nash, and by Luce and Raiffa, where this approach has been suggested as a possible method for "arbitration." Other numerical examples utilize inequalities to illustrate intransitivity effects like those discussed previously by Zeckhauser and Shepsle, and also in the context of Arrow's celebrated Impossibility Theorem.