抄録
It is sometimes argued that although Arrow's proof of the general impossibility theorem remains mathematically valid, its practical application will seldom be of much importance when a large number of people are involved relative to the number of alternative issues to be voted upon. Such an assertion is based, in part, on the calculation of the probabilities of there being no majority winner in a direct voting system.
However, real-world democracy is almost always representative (indirect) democracy, particularly when a large number of people are involved. Therefore, inferring outcomes of real-world democracies from the results of analyses of models based on direct democracies is not warranted.
This paper, first, proves that the proba-bilities of there being no majority winner are always greater under a representative than a direct democracy of the same voters and, second, shows numerically that the probabilities increse dramatically when an indirect instead of direct voting method is used, and suggests that the cyclicity of the majority decisions may well be the rule rath-er than the exception in most real-world democracies. Thirdly, it shows that the problem of cyclical majority arising under an indirect democracy is not related to the fact that the underlying individuals' preference orders of that society would produce a cyclical majority under a direct democracy. This absence of a unique preference order independent of the patterns that partition citizen-voters into electoral districts casts doubt on the rationale of direct as well as representative democracy. However, a direct democracy may have practical merit over an indirect democracy. Namely, the former have less chances of being confronted by the embarrassing problem of a cyclical majority.
