The ostrogorski paradox and its relation to nontransitive choice

Let Q,I,J be sets of social goods, aspects and individuals, respectively, and consider the functions v: Q×I×J ? {‐1,1} and w: Q×J ? {‐1,1}. We study two 2‐stage methods for ordering Q characterized by majority vote and, respectively, amalgamating v first over I and then over J (method IP) or vice versa (method PI). An Ostrogorski paradox occurs when IP and PI give different outcomes. Conditions for this paradox and its relations to Condorcet's paradox are investigated, particularly via a scalogram structure of w. While PI seems more like direct democracy than IP, for decisions on k ≥ 3 social goods PI is transitive whereas IP may give a cyclic outcome. Extensions to richer than binary data and to decision systems different from majority vote are explored.