Strategy-proof interval-social choice correspondences over extended single-peaked domains

We consider a social choice model where voters have single-peaked preferences over the alternatives that are aggregated to produce ‘intervals’ of fixed cardinality, 𝐿. This is applicable in situations where the alternatives can be arranged in a line (e.g. plots of land) and a contiguous set of these are required (e.g. a hospital or a school). We define interval-social choice correspondences (I-SCCs) on profiles of single-peaked preferences which select intervals. We extend single-peaked preferences to intervals using responsiveness. We show that generalized median-interval rules are the only strategy-proof, anonymous and interval efficient I-SCCs. An I-SCC is interval efficient if no other interval can make every voter strictly better-off. We show that replacing interval efficiency with a stronger notion, Pareto efficiency, characterizes a sub-class of these rules called the 𝑡-th interval rules.