Preservation of Proximity and Merging Functions

In this paper we are interested in a property due to Baigent (1987) called proximity preserving. In the conventional model of voting theory it was proposed by Baigent that aggregation procedures should be proximity preserving in the sense that given three preference profiles if the second is closer to the first than the third according to an additively separable metric then the second social ranking should also be closer to the first social ranking compared to the third social ranking. In this framework distance between profiles is measured as the sum of distances between the preferences of each agent. In this paper we assume a metric on the space of alternatives and thereby generate an entire family of metrics on ballot profiles. An interesting special case of our family of metrics is the distance between two profiles measured as the sum of the distances between the candidates of each agent on the two ballot profiles. In this framework we obtain the result that there is no merging function which satisfies anonymity and the proximity preservation property. A particular case of a merging function is when the universal set of alternatives is finite and the elected outcome is required to be a candidate who must have received at least one vote. We call such merging functions vote aggregators. It therefore follows as a consequence of the above result that there is no vote aggregator which satisfies anonymity and the proximity preservation property. Two similar results, one about social welfare functions and the other about social decision functions can be found in Baigent (1987). However, not only is the context of our analysis different, but the method of proof bears little resemblance to the ones available in the work just cited.