Distance restricted manipulation in voting

Abstract

We introduce the notion of Distance Restricted Manipulation, where colluding manipulator(s) need to compute if there exist votes which make their preferred alternative win the election when their knowledge about the others' votes is a little inaccurate. We use the Kendall-Tau distance to model the manipulators' confidence in the non-manipulators' votes. To this end, we study this problem in two settings - one where the manipulators need to compute a manipulating vote that succeeds irrespective of perturbations in others' votes (Distance Restricted Pessimistic Manipulation), and the second where the manipulators need to compute a manipulating vote that succeeds for at least one possible vote profile of the others (Distance Restricted Optimistic Manipulation). We show that Distance Restricted Pessimistic Manipulation admits polynomial-time algorithms for every scoring rule, maximin, Bucklin, and simplified Bucklin voting rules for a single manipulator, and for the k-approval rule for any number of manipulators, but becomes intractable for the Copelandα voting rule for every α∈[0,1] even for a single manipulator. In contrast, Distance Restricted Optimistic Manipulation is intractable for almost all the common voting rules, with the exception of the plurality rule. For a constant number of alternatives, we show that both the problems are polynomial-time solvable for every anonymous and efficient voting rule.