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On the Egalitarian Weights of Nations

Voters from m disjoint constituencies (regions, federal states, etc.) are represented in an assembly which contains one delegate from each constituency and applies a weighted voting rule. All agents are assumed to have single-peaked preferences over an interval; each delegate's preferences match his constituency's median voter; and the collective decision equals the assembly's Condorcet winner. We characterize the asymptotic behavior of the probability of a given delegate determining the outcome (i.e., being the weighted median of medians) in order to address a contentious practical question: which voting weights w_1, ..., w_m 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 if ideal point distributions have identical median M and are suitably continuous, the probability for a given delegate i's ideal point λ_i being the Condorcet winner becomes asymptotically proportional to i's voting weight w_i times λ_i's density at M as $m\to \infty$. Indirect representation of citizens is approximately egalitarian for weights proportional to the square root of constituency sizes if all individual ideal points are i.i.d. In contrast, weights that are linear in-- or, better, induce a Shapley value linear in-- size are egalitarian when preferences are sufficiently strongly affiliated within constituencies.