Collective Choice Rules on Convex Restricted Domains

We study sets of preferences that are convex with respect to the betweeness relation induced by the Kemeny distance for preferences. It appears that these sets consist of all preferences containing a certain partial ordering and the other way around all preferences containing a given partial ordering form a convex set. Next we consider restricted domains where each agent has a convex set of preferences. Necessary and sufficient conditions are formulated under which a restricted domain admits unanimous, strategy-proof and non-dictatorial choice rules. Loosely speaking it boils down to admitting monotone and non-image-dictatorial decision rules on two alternatives where the other alternatives are completely disregarded.