To measure whether a plan is proportionately fair -- giving both urban and rural dwellers representation roughly proportional to their population -- this attached article now introduces an objective, nonpartisan population density fairness (PDF) measure for evaluating when a plan produces legislative representation approximately proportional to its relative share of regions having diverse population densities.
I worked really hard to simplify the measure after deriving it in a much more complicated fashion. I have posted the revised paper, with its fairly simple new population density fairness measure described on pp. 20-24, here: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1945879 ---- Election-Methods mailing list - see http://electorama.com/em for list info
