1982 年 9 巻 12 号 p. 97-109
“Social preference orders” obtained by voting rules may be evaluated in terms of how individual preferences expressed by voters have been properly reflected in the obtained orders. To represent the degree of reflection of the preferences by voter i, Pi, in a given social order, P, we defined the matching score, tip by the number of those pairs of alternatives the orders of which match for Pi and P. Then we proposed using the mean and variance of the tip to evaluate outcomes of voting, i.e., social orders, given a profile of individual preferences expressed by voters. A new rule, Varmin rule, is proposed which yields the order that minimizes the variance of these matching scores. Various social orders obtained by a number of voting rules, such as, Majority, Borda, Double Plurality, and Varmin, were evaluated in terms of means and variances of matching scores through a computer simulation of 15, 000 profiles. Results generally indicated as follows: The Majority rule tends to yield rather large variances, although it theoretically yields in largest mean. The Borda rule yielded rather small variances, keeping the means at rather high levels. The Double Plurality rule yielded large variances with unpredictable fluctuations of the means. The Varmin rule yielded too small means, although it theoretically yields the least possible variance.
A method to modify social orders to reduce the variances most effectively was proposed (called, “Maximum Covariance Method”), showing that only a small modification of the Majority orders would be sufficient to yield very small variances with only a minor decrease of means.
