Single-peaked preferences and probabilities of cyclical majorities

Most investigations of multidimensional single-peaked preferences presume that the set of decision alternatives is a product set. In the present paper, it is assumed that alternatives exhibit two or more salient attributes, but no product structure is presumed. It is assumed that each voter can conditionally order the alternatives on the basis of each salient attribute, that the sets of voters' preference orders conditioned on each attribute are singlepeaked in the traditional sense, and that a voter unconditionally prefers alternative x to alternative y when he prefers x to y conditionally for every attribute. The analysis strongly supports the assertion that, with three alternatives and two salient attributes, the probability of getting a simple majority cycle becomes vanishingly small as the number of voters becomes large. The cases of more than three alternatives and/or more than two attributes are also discussed.