Network geometry and the scope of the median voter theorem

It is shown that the median voter theorem for committee-decisions holds over a full unimodal preference domain whenever (i) the underlying median interval space satisfi?es interval antiexchange and (ii) unimodality is defi?ned with respect to the incidence-geometry of the relevant outcome space or network. Thus, in particular, the interval spaces canonically induced by trees do support the median voter theorem on their own full unimodal preference domains. Conversely, validity of the median voter theorem on the full unimodal preference domain of a certain median interval space on a discrete outcome space requires that the graph canonically induced by that interval space be precisely a tree.