Margin of victory for tournament solutions

Brill, Markus; Schmidt-Kraepelin, Ulrike; Suksompong, Warut

doi:10.1016/j.artint.2021.103600

Cited by 6 publications

(3 citation statements)

References 43 publications

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“…The phase transition behavior for CM also happens for MoV [Mossel et al, 2013, Xia, 2015. Previous work has also investigated expected MoV [Pritchard and Slinko, 2006] and MoV for tournament rules [Brill et al, 2022]. The likelihood of MoV = 1 has been used to measure the decisiveness of voting, sometimes called voting power, which plays an important role in the paradox of voting [Downs, 1957] and in power indices in cooperative game theory.…”

Section: Our Contributionsmentioning

confidence: 99%

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How Likely A Coalition of Voters Can Influence A Large Election?

Xia¹

2022

Preprint

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For centuries, it has been widely believed that the influence of a small coalition of voters is negligible in a large election. Consequently, there is a large body of literature on characterizing the asymptotic likelihood for an election to be influence, especially by the manipulation of a single voter, establishing an 𝑂 ( 1 √ 𝑛) upper bound and an Ω( 1 𝑛 67 ) lower bound for many commonly studied voting rules under the i.i.d. uniform distribution, known as Impartial Culture (IC) in social choice, where 𝑛 is the number is voters.In this paper, we extend previous studies in three aspects: (1) we consider a more general and realistic semi-random model that resembles the model in smoothed analysis, (2) we consider many coalitional influence problems, including coalitional manipulation, margin of victory, and various vote controls and bribery, and(3) we consider arbitrary and variable coalition size 𝐵. Our main theorem provides asymptotically tight bounds on the semi-random likelihood of the existence of a size-𝐵 coalition that can successfully influence the election under a wide range of voting rules. Applications of the main theorem and its proof techniques resolve long-standing open questions about the likelihood of coalitional manipulability under IC, by showing that the likelihood is Θ min 𝐵 √ 𝑛, 1 for many commonly studied voting rules. The main technical contribution is a characterization of the semi-random likelihood for a Poisson multinomial variable (PMV) to be unstable, which we believe to be a general and useful technique with independent interest.

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Section: Our Contributionsmentioning

confidence: 99%

“…Results in [Mossel et al, 2013, Xia, 2015 discussed above also apply to MoV. Brill et al [2022] studies the distribution of MoV for some tournament rules, under a probability distribution on tournament graphs, where the direction of each edge is drawn independently and uniformly.…”

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confidence: 99%

How Likely A Coalition of Voters Can Influence A Large Election?

Xia¹

2022

Preprint

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“…While these are natural as well, our definition is particularly appealing if each district may be only associated with its most frequent color if its margin of victory is too high. Notably, margin of victory is also a popular concept in other domains such as group identification [9], tournament solutions [12], and voting theory [17,49].…”

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confidence: 99%

A refined complexity analysis of fair districting over graphs

Boehmer

Koana

Niedermeier

2022

Auton Agent Multi-Agent Syst

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We study the NP-hard Fair Connected Districting problem recently proposed by Stoica et al. [AAMAS 2020]: Partition a vertex-colored graph into k connected components (subsequently referred to as districts) so that in every district the most frequent color occurs at most a given number of times more often than the second most frequent color. Fair Connected Districting is motivated by various real-world scenarios where agents of different types, which are one-to-one represented by nodes in a network, have to be partitioned into disjoint districts. Herein, one strives for “fair districts” without any type being in a dominating majority in any of the districts. This is to e.g. prevent segregation or political domination of some political party. We conduct a fine-grained analysis of the (parameterized) computational complexity of Fair Connected Districting. In particular, we prove that it is polynomial-time solvable on paths, cycles, stars, and caterpillars, but already becomes NP-hard on trees. Motivated by the latter negative result, we perform a parameterized complexity analysis with respect to various graph parameters including treewidth, and problem-specific parameters, including, the numbers of colors and districts. We obtain a rich and diverse, close to complete picture of the corresponding parameterized complexity landscape (that is, a classification along the complexity classes FPT, XP, W[1]-hard, and para-NP-hard).

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Generalized kings and single-elimination winners in random tournaments

Manurangsi

Suksompong

2022

Auton Agent Multi-Agent Syst

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Tournaments can be used to model a variety of practical scenarios including sports competitions and elections. A natural notion of strength of alternatives in a tournament is a generalized king: an alternative is said to be a k-king if it can reach every other alternative in the tournament via a directed path of length at most k. In this paper, we provide an almost complete characterization of the probability threshold such that all, a large number, or a small number of alternatives are k-kings with high probability in two random models. We show that, perhaps surprisingly, all changes in the threshold occur in the range of constant k, with the biggest change being between $$k=2$$ k = 2 and $$k=3$$ k = 3 . In addition, we establish an asymptotically tight bound on the probability threshold for which all alternatives are likely able to win a single-elimination tournament under some bracket.

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Margin of victory for tournament solutions

Cited by 6 publications

References 43 publications

How Likely A Coalition of Voters Can Influence A Large Election?

How Likely A Coalition of Voters Can Influence A Large Election?

A refined complexity analysis of fair districting over graphs

Generalized kings and single-elimination winners in random tournaments

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