Breadth of loss due to manipulation

We identify the maximal number of people who are harmed when a social choice rule is manipulated. If there are n individuals, then for any k greater than 0 and less than n, there is a neutral rule for which the maximal number is k, and there is an anonymous rule for which the maximum number is k. If the rule selects the majority winner when there is one and satisfies the Pareto criterion, then the maximal number is (n - 1)/2. Maximal numbers are also calculated for the Borda rule and other scoring rules, and for rules for which the selection depends only on which individuals have which alternatives ranked at the top. We also calculate the maximum number for rules that satisfy the Pareto criterion, but not the Condorcet criterion, and for rules that satisfy the Condorcet criterion, but not the Pareto criterion.