Voting in Combinatorial Domains Jérôme Lang a and Lirong Xia b 10.1 Motivations and classes of problems This chapter addresses preference aggregation and voting on domains which are the Cartesian product (or sometimes, a subset of the Cartesian product) of finite domain values, each corresponding to an issue, a variable, or an attribute. As seen in other chapters of this handbook, voting rules map a profile (usually, a collection of rankings, see Chapter 1 (Zwicker, 2015)) into an alternative or a set of alternatives. A key question has to do with the structure of the set of alternatives. Sometimes, this set has a simple structure and a small cardinality (e.g., in a presidential election). But in many contexts, it has a complex combinatorial structure. We give here three typical examples: