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at the water's edge
Demarchy is the rule by the randomly selected. Obviously, no modern country would ever adopt this form of government, but it should be noted that randomness has been used as part of selecting leaders in the past. A similar system is often used to form juries as well. In particular, the electors for the Doge of Venice were selected in the following way:
“”Thirty members of the Great Council, chosen by lot, were reduced by lot to nine; the nine chose forty and the forty were reduced by lot to twelve, who chose twenty-five. The twenty-five were reduced by lot to nine and the nine elected forty-five. Then the forty-five were once more reduced by lot to eleven, and the eleven finally chose the forty-one who actually elected the doge."
Randomness, in this case, was to eliminate the possibilities of great families or politicking polluting the selection, thus ensuring the greatest possible Doge for the city.