The margin of victory of an election is a useful measure to capture the robustness of an election outcome. It also plays a crucial role in determining the sample size of various algorithms in post election audit, polling etc. In this work, we present efficient sampling based algorithms for estimating the margin of victory of elections. More formally, we introduce the problem, where given an election @math on @math voters, the goal is to estimate the margin of victory @math of @math within an additive factor of @math . We study the problem for many commonly used voting rules including scoring rules, approval, Bucklin, maximin, and Copeland @math We observe that even for the voting rules for which computing the margin of victory is NP-Hard, there may exist efficient sampling based algorithms, as observed in the cases of maximin and Copeland @math voting rules.
@cite_19: We study the complexity of influencing elections through bribery: How computationally complex is it for an external actor to determine whether by a certain amount of bribing voters a specified candidate can be made the election's winner? We study this problem for election systems as varied as scoring protocols and Dodgson voting, and in a variety of settings regarding homogeneous-vs.-nonhomogeneous electorate bribability, bounded-size-vs.-arbitrary-sized candidate sets, weighted-vs.-unweighted voters, and succinct-vs.-nonsuccinct input specification. We obtain both polynomial-time bribery algorithms and proofs of the intractability of bribery, and indeed our results show that the complexity of bribery is extremely sensitive to the setting. For example, we find settings in which bribery is NP-complete but manipulation (by voters) is in P, and we find settings in which bribing weighted voters is NP-complete but bribing voters with individual bribe thresholds is in P. For the broad class of elections (including plurality, Borda, k-approval, and veto) known as scoring protocols, we prove a dichotomy result for bribery of weighted voters: We find a simple-to-evaluate condition that classifies every case as either NP-complete or in P.
@cite_15: We study the complexity of influencing elections through bribery: How computationally complex is it for an external actor to determine whether by paying certain voters to change their preferences a specified candidate can be made the election's winner? We study this problem for election systems as varied as scoring protocols and Dodgson voting, and in a variety of settings regarding homogeneous-vs.-nonhomogeneous electorate bribability, bounded-size-vs.-arbitrary-sized candidate sets, weighted-vs.-unweighted voters, and succinct-vs.-nonsuccinct input specification. We obtain both polynomial-time bribery algorithms and proofs of the intractability of bribery, and indeed our results show that the complexity of bribery is extremely sensitive to the setting. For example, we find settings in which bribery is NP-complete but manipulation (by voters) is in P, and we find settings in which bribing weighted voters is NP-complete but bribing voters with individual bribe thresholds is in P. For the broad class of elections (including plurality, Borda, k- approval,and veto) known as scoring protocols, we prove a dichotomy result for bribery of weighted voters: We find a simple-to-evaluate condition that classifies every case as either NP-complete or in P.
The margin of victory problem is the same as the optimization version of the destructive bribery problem introduced by @cite_19 @cite_15 . However, to the best of our knowledge, there is no prior work on estimating the cost of bribery by sampling votes.