Direct democracy is studied with the example of the Athenian Assembly, Council of Five Hundred, Committee of Fifty, president, magistrates, and juries. The democratic institutions are evaluated using indices of popularity (average percentage of the population represented on a number of issues), universality (frequency of cases when the majority opinion is represented) and goodness (average of the group-represented-to-majority ratio). The indices are geometrically interpreted as projections of the institution’s characteristic vectors onto the characteristic (mainstream) vector of the society, which resembles the interaction of force vectors in physics with respect to the direction of motion. We offer a proof showing that the representativeness of democratic institutions selected by lot, as was practiced in Athens, is fairly high. It is also explained why the various democratic institutions with different functions should have different sizes: for instance, legislative bodies like the Athenian Assembly with its quorum of 6000 and executive bodies like magistrates with boards of ten, despite their great difference in size, they have the same degree of representativeness. Finally, it is shown that, in an unstable society, personal power is more efficient than democracy.
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