Majority Decision-Making with Partial Unidimensionality

A major dilemma for majority decision-making occurs when the summation of transitive individual preference orderings results in an intransitive social ordering. The problems posed by this phenomenon, which is known as the paradox of voting, can be seen in the following standard example. Suppose there are three individuals, one with each of the following preference orders of three alternatives: ABC, BCA, CAB. Under majority rule, A would defeat B, B would defeat C, and C would defeat A, so there is no majority winner. Most voting procedures, of course, yield a unique result whether or not the paradox occurs. But from this example it is apparent that when the paradox does occur, a majority of the voters prefer an alternative other than the one which is selected. Moreover, if a typical voting procedure is used, which of the alternatives is selected depends on the order in which the alternatives are voted on. Clearly these results have important implications, whether one is concerned with normative questions about majority rule or with the practical politics of legislative decision-making. In the burgeoning literature on the voting paradox, surely one of the most impressive and well-known findings is Black's and Arrow's demonstration that the paradox cannot occur if the set of individual preference orderings is single-peaked. Since single-peakedness implies that the individuals and alternatives can be arrayed on a single dimension, their finding has a meaningful substantive interpretation. Namely, complete agreement on a dimension for judging the alternatives ensures that majority voting will yield a transitive social ordering of the alternatives.