Preference expression and misrepresentation in points voting schemes

Concluding remarks

This paper focuses on two properties of points voting schemes: their vulnerability to preference misrepresentation and their potential for preference expression. In the spirit of the seminal work of Buchanan and Tullock [2], this dual criterion might be used for the choice of an optimal points voting scheme when the above two aspects only are considered. Unlike recent treatment of the manipulability phenomenon, Gardenfors [3], Gibbard [4], Satterthwaite [8], we do suggest measures for the immunity to mispresentation of a given method. The first measure I[X, f)] is interpreted as a lower bound for δ-manipulation. In other words, I[(X, f)] = c implies that there exists at least one preference profile [x _ij] ∈ X, such that δ-manipulation is possible for δ > c. The second measure K ranges from 0 to 1, and it is interpreted as the relative frequency of preference profiles in X under which the scheme is immune. Another measure D is proposed for the potential of preference expression under each variation. This measure counts the number of possible preference configurations under the scheme (X, f). The use of this measure for the comparison of two voting schemes (T, f), (Z, f) is especially meaningful whenever T ⊂ Z. This approach is illustrated by five variations of the points voting scheme: the well-known points voting method, the Borda scores method, the plurality rule, the federative plurality rule and a dictatorial type voting scheme. It is shown that, as these variations become more restrictive in terms of preference expression (D decreases), they tend to be more immune to individual preference misrepresentation (I increases). We conjecture, first, that a similar relationship will be generally obtained for variations of the points voting method, and, second, that such a relationship also exists between K and D.

Finally, two comments with respect to the commonly used plurality rule are now in order. First, this rule is associated with moderate values of both measures, D(V ₃) = m ⁿ and I(V ₃) = 1/2. Roughly speaking, it can be considered a ‘reasonable’ voting scheme for a society, or committee, which is ‘equally demanding’ in terms of both criteria. Second, the definition of misrepresentation used in this paper, takes into account misrepresentation of preferences with no restriction on the type of motivations for such a behavior. An alternative, and more common, definition deals only with preference misrevelations which are consistent with the sincere preferences of the manipulator, i.e., the winning alternative under the reported preference profile has to be preferred to the outcome of the sincere profile from the non sincere voter's point of view. Obviously, vulnerability to manipulation implies vulnerability to misrepresentation but the converse is not true. Also notice that if the definition of misrepresentation is replaced by that of manipulation, then plurality becomes perfectly immune, since no voter can act contrary to his sincere preferences and alter the social choice in his favor.