On the Egalitarian Weights of Nations

Voters from m disjoint constituencies (regions, federal states, etc.) are indirectly represented in an assembly which applies a weighted voting rule. All agents have single-peaked preferences over an interval and each delegate’s preferences match the constituency’s median voter. The collective decision corresponds to the assembly’s Condorcet winner. Which voting weights w1, . . . ,wm ought to be selected if constituency sizes differ and all voters are to have a priori equal influence on collective decisions? It is shown that representation is approximately egalitarian for weights proportional to the square root of constituency sizes if all ideal points are i.i.d. If, however, preferences are polarized along constituency lines then weights should induce a Shapley value linear in size.