We introduce a class of voting rules called generalized scoring rules. Under such a rule, each vote generates a vector of k scores, and the outcome of the voting rule is based only on the sum of these vectors—more specifically, only on the order (in terms of score) of the sum’s components. This class is extremely general: we do not know of any commonly studied rule that is not a generalized scoring rule. We then study the coalitional manipulation problem for gener-alized scoring rules. We prove that under certain natural assump-), then tions, if the number of manipulators is O(n p) (for any p < 1 2 the probability that a random profile is manipulable is O(n p − 1 2), where n is the number of voters. We also prove that under another set of natural assumptions, if the number of manipulators is Ω(n p) (for any p> 1) and o(n), then the probability that a random pro-2 file is manipulable (to any possible winner under the voting rule) is 1 − O(e −Ω(n2p−1)). We also show that common voting rules satisfy these conditions (for the uniform distribution). These results generalize earlier results by Procaccia and Rosenschein as well as even earlier results on the probability of an election being tied.

We investigate the problem of coalitional manipulation in elections, which is known to be hard in a variety of voting rules. We put forward efficient algorithms for the problem in Scoring rules, Maximin and Plurality with Runoff, and analyze their windows of error. Specifically, given an instance on which an algorithm fails, we bound the additional power the manipulators need in order to succeed. We finally discuss the implications of our results with respect to the popular approach of employing computational hardness to preclude manipulation. 1

Recent results have established that a variety of voting rules are computationally hard to manipulate in the worst-case; this arguably provides some guarantee of resistance to manipulation when the voters have bounded computational power. Nevertheless, it has become apparent that a truly dependable obstacle to manipulation can only be provided by voting rules that are average-case hard to manipulate. In this paper, we analytically demonstrate that, with respect to a wide range of distributions over votes, the coalitional manipulation problem can be decided with overwhelming probability of success by simply considering the ratio between the number of truthful and untruthful voters. Our results can be employed to significantly focus the search for that elusive average-case-hard-to-manipulate voting rule, but at the same time these results also strengthen the case against the existence of such a rule. 1

We provide an overview of more than two decades of work, mostly in AI, that studies computational complexity as a barrier against manipulation in elections.

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New measures of the difficulty of manipulation of voting rules

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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), wherenis 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.

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 n votes i.i.d. according to a distribution π, and let n go to infinity, then for any ɛ> 0, with probability at least 1 − ɛ, 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) Θ ( √ n), (3) Θ(n), and (4) ∞. This theorem holds for any set of vote operations, any individual vote distribution π, 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. Keywords: operations Computational social choice; generalized scoring rules; vote