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.
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.
@cite_19 @cite_15
2065358633 2572410607 2952897834 1644339822
The margin of victory of an election, defined as the smallest number k such that k voters can change the winner by voting differently, is an important measurement for robustness of the election outcome. It also plays an important role in implementing efficient post-election audits, which has been widely used in the United States to detect errors or fraud caused by malfunctions of electronic voting machines. In this paper, we investigate the computational complexity and (in)approximability of computing the margin of victory for various voting rules, including approval voting, all positional scoring rules (which include Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs. We also prove a dichotomy theorem, which states that for all continuous generalized scoring rules, including all voting rules studied in this paper, either with high probability the margin of victory is Θ(√n), or with high probability the margin of victory is Θ(n), where n is the number of voters. Most of our results are quite positive, suggesting that the margin of victory can be efficiently computed. This sheds some light on designing efficient post-election audits for voting rules beyond the plurality rule. We study the complexity of the bribery problem with distance restrictions. In particular, in the bribery problem, we are given an election and a distinguished candidate p, and are asked whether we can make p win not win the election by bribing at most k voters to recast their votes. In the bribery problem with distance restrictions, we require that the votes recast by the bribed voters are close to their original votes. To measure the closeness between two votes, we adopt the prevalent Kendall-Tau distance and the Hamming distance. We achieve a wide range of complexity results for this problem under a variety of voting correspondences, including the Borda, Condorcet, Copelandα for every 0 ≤ e ≤ 1 and Maximin. 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. In this paper, we propose a framework to study a general class of strategic behavior in voting, which we call vote operations. We prove the following theorem: if we fix the number of alternatives, generate @math votes i.i.d. according to a distribution @math , and let @math go to infinity, then for any @math , with probability at least @math , the minimum number of operations that are needed for the strategic individual to achieve her goal falls into one of the following four categories: (1) 0, (2) @math , (3) @math , and (4) @math . This theorem holds for any set of vote operations, any individual vote distribution @math , and any integer generalized scoring rule, which includes (but is not limited to) almost all commonly studied voting rules, e.g., approval voting, all positional scoring rules (including Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs. We also show that many well-studied types of strategic behavior fall under our framework, including (but not limited to) constructive destructive manipulation, bribery, and control by adding deleting votes, margin of victory, and minimum manipulation coalition size. Therefore, our main theorem naturally applies to these problems.