A note on the direct democracy deficit in two-tier voting

A large population of voters with single-peaked preferences are partitioned into disjoint constituencies. Collective decisions are taken by their representatives, one from each constituency, according to a weighted voting rule. It is assumed that each representative’s ideal point perfectly matches that of the respective constituency median and that top-tier decisions are in the voting game’s core. The resulting representative-democratic voting outcomes generally differ from those of a direct-democratic, single-tier system. The expected discrepancy varies with the voting weight allocation. It is minimized by weights proportional to constituency population sizes only if citizens differ sufficiently more between than within constituencies. Weights proportional to the square root of population sizes perform better if all citizens have independent and identically distributed ideal points.