Chaos in multidimensional issue spaces

(Black 1958) showed that if alternatives can be represented as points along one line and if voters' preferences are single-peaked, indicating resemblance, then a majority-rule equilibrium results. The position of the median voter on the line will beat any other alternative in majority-rule voting. This is normatively attractive because a central alternative prevails. In Figure 8.1 there are five voters with preferences over alternatives A, B, C, D, and E. Each voter's preference curve has only a single peak. Voter 3 has the median preference C, and C will beat by majority vote any alternative that it faces. For example, if D is pitted against C, voters 1 and 2 prefer D > C, voters 3, 4, and 5 prefer C > D, and thus C wins by majority vote. There are no cycles. The preference orders need not be so neat as portrayed in the figure; each only needs to be single-peaked.

Figure 8.2 shows a profile of preferences that is not single-peaked because one of the voter's rankings, in this portrayal #2, has two peaks (single-peakedness and its lack are impervious to rearrangements of the labels). Recall that given three alternatives, there are six possible strong-preference rankings, and that given three voters, one each with cyclical rankings 1 (A > B > C), 3 (C > A > B), and 5 (B > C > A), or with rankings 2, 4, and 6 together, the result of majority voting is inconsistent, that is, A beats B, B beats C, and C beats A.