On strategy-proofness and single-peakedness: median-voting over intervals

We study solutions that choose an interval of alternatives when agents have single-peaked preferences. Similar to Klaus and Storcken (2002), we ordinally extend these preferences over intervals. Loosely speaking, we extend the results of Moulin (1980) to our setting and show that the results of Ching (1997) cannot always be similarly extended. Our main results are the following. First, strategy-proofness and peaks-onliness characterize the class of generalized median solutions. Second, although peaks-onliness cannot be replaced by the "weaker" property of continuity in our first result -as is the case in Ching (1997)- this equivalence is achieved when voter-sovereignty is also required. Finally, if preferences are symmetric and single-peaked, strategy-proofness and voter-sovereignty characterize the class of efficient generalized median solutions