The Direct Democracy Deficit in Two-tier Voting

A large population of citizens have single-peaked preferences over a one-dimensional convex policy space with independently and identically distributed ideal points. They are partitioned into disjoint constituencies C1, . . . , Cm. Collective decisions are determined either in a directly democratic fashion and the implemented policy XD corresponds to the ideal point of the (issue-specific) population median. Or decisions are taken in a two-tier voting system: one representative of each constituency Ci, with his ideal point matching the constituency median, has a voting weight wi; the collective decision XR equals the ideal point of the assembly pivot defined by weight vector (w1, . . . , wm) and a 50%-quota. Monte-Carlo simulations indicate that proportionality of voting weights and the square root of population sizes minimizes the ‘direct democracy deficit’ of two-tier voting, i.e., the expected value of (XD −XR).