Optimal Bribery in Voting

Studying complexity of various bribery problems has been one of the main research focus in computational social choice. In all the models of bribery studied so far, the briber has to pay every voter some amount of money depending on what the briber wants the voter to report and the briber has some budget at her disposal. Although these models successfully capture many real world applications, in many other scenarios, the voters may be unwilling to deviate too much from their true preferences. In this paper, we study the computational complexity of the problem of finding a preference profile which is as close to the true preference profile as possible and still achieves the briber's goal subject to budget constraints. We call this problem Optimal Bribery. We consider three important measures of distances, namely, swap distance, footrule distance, and maximum displacement distance, and resolve the complexity of the optimal bribery problem for many common voting rules. We show that the problem is polynomial time solvable for the plurality and veto voting rules for all the three measures of distance. On the other hand, we prove that the problem is NP-complete for a class of scoring rules which includes the Borda voting rule, maximin, Copeland$^\alpha$ for any $\alpha\in[0,1]$, and Bucklin voting rules for all the three measures of distance even when the distance allowed per voter is $1$ for the swap and maximum displacement distances and $2$ for the footrule distance even without the budget constraints (which corresponds to having an infinite budget). For the $k$-approval voting rule for any constant $k>1$ and the simplified Bucklin voting rule, we show that the problem is NP-complete for the swap distance even when the distance allowed is $2$ and for the footrule distance even when the distance allowed is $4$ even without the budget constraints.