Distance Restricted Manipulation in Voting
This addresses the problem of ensuring election integrity against strategic manipulation under uncertainty for computational social choice researchers, though it is incremental as it extends existing manipulation models with distance-based constraints.
The paper tackles the problem of election manipulation under uncertainty by introducing Distance Restricted Manipulation, where manipulators must compute votes that ensure their preferred candidate wins despite limited knowledge of others' votes, modeled using Kendall-Tau distance. It shows that strong manipulation is polynomial-time solvable for many voting rules like scoring rules and maximin, but intractable for Copeland, while weak manipulation is generally intractable except for plurality, with both problems becoming tractable for constant alternatives in anonymous and efficient rules.
We introduce the notion of {\em 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 ({\em Distance Restricted Strong 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 ({\em Distance Restricted Weak Manipulation}). We show that {\em Distance Restricted Strong 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 Copeland$^α$ voting rule for every $α\in[0,1]$ even for a single manipulator. In contrast, {\em Distance Restricted Weak 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.