A great challenge for democracy is the resolution of conflict between divergent individual preferences, views or interests. 1 Sometimes it is held that democracy is about finding “the will of the people,” but if the “wills” of different individuals bear little resemblance to each other, it may be hard to extract anything from the set of individual “wills” that can plausibly be considered “the will of the people.” Condorcet’s famous paradox captures some of these problems. If there are three individuals, where one prefers option x to option y to option z, the second prefers option y to option z to option x, and the third prefers option z to option x to option y, then there exist a majority for x against y, a majority for y against z, and a majority for z against x. This means that, if pairwise majority voting is the method for aggregating individual preferences into collective ones, or more grandly, for extracting “the will of the people,” then the resulting collective preferences can be cyclical and thus useless for reaching consistent collective outcomes. More generally, Arrow’s celebrated impossibility theorem (1951/1963) shows that, if the domain of admissible individual preference input is unrestricted (the universal domain condition), there exists no procedure for aggregating individual preferences in this domain into collective ones in accordance with a set of arguably undemanding minimal conditions—conditions capturing the requirement, in a nutshell, that collective preferences be both minimally responsive to individual preferences and consistent.

The difficulties posed by aggregation depend crucially on how divergent the preferences, views or interests of individuals are. In the (rare) limiting case of unanimity the difficulties obviously disappear. If everybody had exactly the same preferences, views or interests, there would be no conflict to resolve. But while unanimity is sufficient for the disappearance of the famous Condorcet and Arrow aggregation problems, it is not necessary. Since Duncan Black’s seminal work (1948), it is well known that Condorcet’s paradox can be traced back to a “lack of structure” in the relevant set, also called profile, of individual preferences across individuals. Black himself proved that single-peakedness, a structure condition to be discussed more formally below,(jointly with the (harmless) technical condition that the number