Majority rule: Second thoughts and refutations

Abstract

We have observed that some recent results cast serious doubts uponthe applicability of majority rule as a device for forming a group-willout of individual wills. On the other hand, majority rule has manyproperties to recommend it. Especially, the decisiveness propertyshould please anyone interested in efficiency considerations. How-ever, when it comes to the second major criterion, consistency, major-ity rule with pairwise comparison of alternatives often leads to diffi-culties. Indeed, we have seen that there is typically no core or Con-dorcet winner when the policy space is multidimensional real space.And when there is no core, the situation looks grim indeed for major-ity rule: literally any outcome may become the majority winner. Somerelief of this very strong result is provided by the fact that no majoritycycles can go through the outcomes that belong to the M2 bargainingset. This finding, of course, solves the problem facing majority rule onlyinsofar as a method can be devised by which the M2 bargaining set out-comes can be determined in a reliable way. Here, thus, we are con-fronted with the sincerity problem. The absence of the core can lead to chaotic trajectories of votingoutcomes in multidimensional space. However, there are ways of over-coming the difficulties if a sufficient amount of flexibility is allowedin institutional design. If we consider the various nonequilibrium solu-tion concepts, and the minimax set in particular, we can always designa decision rule which (under mild restrictions) makes the vote-maximiz-ing outcome trajectory enter the solution set. The flexibility needed isgreat, indeed: the decision rule, λ-majority, must be alterable for eachvoting situation. The determination of the winner in a pair-wise comparison of alter-natives is not the main source of the difficulties surrounding majorityrule. As a matter of fact, the method of building the agenda is the mostcrucial part of the problem of cyclical majorities. In multicandidatecontests the occurrence of cyclical majorities has motivated the studyof other.decision procedures in addition to the majority comparisonof alternatives. The most recent proposal, approval voting, is morestrategyproof than its competitors in the class of single-ballot, non-ranked voting systems. At the moment the relationships betweenapproval voting and majority rule are not clear. It seems that furtherstudy of the interrelationships of these procedures is called for. Thefeatures in which approval voting is superior to majority comparisonof pairs of alternatives, however, are not, strictly speaking, procedural,but are external to the method of making collective decisions.