The vulnerability of point-voting schemes to preference variation and strategic manipulation

This essay measures and analyzes for a special class of point-voting schemes (the Borda method, plurality rule and the unrestricted point-voting scheme) sensitivity to preference variation (a simple change in the socially winning alternative resulting from alteration of a single voter's preferences) and vulnerability to individual strategic manipulation (a change in the winning alternative that benefits the voter whose preferences are altered). Assuming that society (n voters with linear preference orders on a finite set of m alternatives) satisfies the impartial-culture assumption, that is, each randomly selected voter is equally likely to hold any one of the randomly picked possible preference orders on the alternatives, we demonstrate: (i) for a given rule and a fixed number of voters, the sensitivity to individual preference variation and the vulnerability to individual strategic manipulation are greater, the larger the total number of alternatives. (ii) For a given rule and a fixed number of alternatives, the vulnerability to individual strategic manipulation, in general, is not greater the smaller the total number of voters. Such a relationship does hold, however, if n is sufficiently large. (iii) For any given combination of number of voters and number of alternatives, the unrestricted point-voting scheme is more sensitive to preference variation than the Borda method, which, in turn, is more exposed to such variation relative to the plurality rule. A similar conclusion does not hold with respect to vulnerability to individual strategic manipulation, unless the number of voters is sufficiently small.