This chapter discusses different types of domain restrictions. We begin by analyzing various qualitative conditions on preference profiles. Value-restricted preferences (with single-peaked preferences as one of its subcases), limited agreement as well as antagonistic and dichotomous preferences are relatively easy to interpret. In our view, the property of single-peakedness stands out in particular. It proves to be a central restriction under majority voting. However, it also plays an essential role in the context of strategy-proof voting rules (which is the topic of another chapter in this Handbook). Furthermore, we consider quantitative or number-specific requirements on the distribution of voters over different preference orderings, and we shall see that some of those requirements are logically related to the qualitative conditions such as extremal restriction and value-restricted preferences. While the latter restrictions are requirements on combinations of individual orderings, the domains of individual orderings that admit n-person nondictatorial social welfare functions a la Arrow result from restrictions on permissible preferences for individuals. While the first five sections study the aggregation problem within the framework of arbitrary finite sets of discrete alternatives, the final section discusses continuous choice rules; the alternatives are assumed to be n-dimensional vectors in Euclidean space. Contractibility as a condition on the topological space of preferences proves to be necessary and sufficient for the existence of continuous aggregation rules.
