Iterative Deliberation via Metric Aggregation

We investigate an iterative deliberation process for an agent community wishing to make a joint decision. We develop a general model consisting of a community of n agents, each with their initial ideal point in some metric space (X, d), such that in each iteration of the iterative deliberation process, all agents move slightly closer to the current winner, according to some voting rule R. For several natural metric spaces and suitable voting rules for them, we identify conditions under which such an iterative deliberation process is guaranteed to converge.