In 1973–75 Gibbard and Satterthwaite published a fundamental impossibility theorem which states that every non-dictatorial social choice function, whose range contains at least three alternatives, at certain profiles can be manipulated by a single individual voter [6, 15]. After that, the natural question arose: if there are no perfect rules, which ones are the best, i.e. least manipulable? To this question there can be no absolute answer – it depends both on the behaviour of the voters, and on the measure used to quantify the term “manipulability”. Among models of voter behaviour, the following two have gained the most attention ([3,14]). The Impartial Culture (IC) model assumes that voters are independent, and that each voter is equally likely to vote for any candidate. The Impartial Anonymous Culture (IAC) model assumes some degree of dependency. This paper concerns itself with the IC model. Among measures of manipulability, the most popular is the probability that the votes fall in such a way as to create the (coalitional or individual) “logical possibility of manipulation” ([4,7,8,9,12,13,14]). This means that some coalition of voters (or individual voter) with incentive to do so can change the election result by voting insincerely. Counterthreats are not considered – we assume that the manipulator(s) are not opposed by the other, naive voters. The probability of manipulability has been especially well-studied for the important class of positional (scoring) voting rules, and significant progress has been made in comparing them. In his seminal paper [14], Saari showed that in his “geometric” model, Borda’s rule is the least manipulable for the three-alternative case in relation to micro manipulation, but that this does not extend to the case of four alternatives. In this paper, we further refine this notion of manipulability by considering the sizes of the coalitions involved. Intuitively, a rule is more resistant to manipulation if many voters must be recruited to assemble the manipulating coalition, and less resistant if only a few voters are required. We may thus consider the probability that a coalition of at most k voters can manipulate (k = 1, 2, . . .). Equivalently, we study the probability distribution of the size of
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