Whose Opinion Counts? Political Processes and the Implementation Problem

The mechanism used in Nash implementation is a form of direct democracy, taking everyone's opinion into account. We augment this mechanism with a political process that selects the opinions of a subset of the individuals. We study three such processes - oligarchy, oligarchic democracy and random sampling - and compare the social choice rules (SCRs) that can be implemented using each of these processes with those that can be Nash implemented. In oligarchy, only the opinions of a fixed subset of the individuals - the oligarchs - determine the implemented alternative. We obtain a negative result for oligarchies: there exist Nash implementable SCRs that cannot be implemented by any oligarchy. Oligarchic democracy is a perturbation of oligarchy, in which the opinions of the oligarchs “almost always" determine the implemented alternative but sometimes, everyone's opinions are considered. In a sharp contrast to the negative result for oligarchies, we show that in economic environments, every Nash implementable SCR can be implemented by an oligarchic democracy in which any three individuals act as oligarchs. In random sampling, opinions of a fixed number of individuals are selected randomly, which then determine the implemented alternative. We show that in economic environments, every Nash implementable SCR can be implemented by randomly sampling opinions of four individuals.