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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$k$$</EquationSource> </InlineEquation> greater than 0 and less than n, there is a neutral rule for which the maximal number is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$k$$</EquationSource> </InlineEquation>, and there is an anonymous rule for which the maximum...</equationsource></inlineequation></equationsource></inlineequation>

There exist social choice rules for which every manipulation benefits everyone. This paper constructs a large variety of rules with this property and provides two characterizations of such rules. Copyright Springer-Verlag Berlin Heidelberg 2014