Whose opinion counts? Implementation by sortition

Sortition is the process of selecting decision makers or senators by a lottery. We introduce sortition in implementation theory by augmenting a mechanism with a kleroterion or lottery machine p that selects the senators. An outcome is implemented after consulting only the opinions of the senators. We call the corresponding notion of implementation as “p-implementation”, and provide necessary and sufficient conditions for p-implementation. Our main result is that in “economic” environments, every Nash implementable social choice rule (SCR) is also p-implementable if p selects every quartet of players with positive probability and always selects at least three senators. We apply this result to two kleroteria: “oligarchic democracy” and “random sampling”. In economic environments, every Nash implementable SCR can be implemented by oligarchic democracy of three oligarchs. In economic environments, every Nash implementable SCR can be implemented by randomly selecting four senators.