New Approximations for Coalitional Manipulation in General Scoring Rules

TL;DR

This paper introduces new approximation algorithms for coalitional manipulation in general scoring rules, especially Borda, providing stronger guarantees and handling weighted and unweighted cases effectively.

Contribution

It develops a unified approximation framework for manipulation under general scoring rules, improving upon previous methods by offering better guarantees and applicability to weighted settings.

Findings

Provides an additive approximation for any scoring rule based on a configuration LP.

Offers a randomized algorithm with an $O(k \sqrt{m \log m})$ guarantee for unweighted cases.

Extends to weighted cases with an $O(W \\sqrt{m \\log m})$ approximation, handling weights effectively.

Abstract

We study the problem of coalitional manipulation---where $k$ manipulators try to manipulate an election on $m$ candidates---under general scoring rules, with a focus on the Borda protocol. We do so both in the weighted and unweighted settings. We focus on minimizing the maximum score obtainable by a non-preferred candidate. In the strongest, most general setting, we provide an algorithm for any scoring rule as described by a vector $\vec{\alpha}=(\alpha_1,\ldots,\alpha_m)$: for some $\beta=O(\sqrt{m\log m})$, it obtains an additive approximation equal to $W\cdot \max_i \lvert \alpha_{i+\beta}-\alpha_i \rvert$, where $W$ is the sum of voter weights. For Borda, both the weighted and unweighted variants are known to be $NP$-hard. For the unweighted case, our simpler algorithm provides a randomized, additive $O(k \sqrt{m \log m} )$ approximation; in other words, if there exists a strategy enabling the preferred candidate to win by an $\Omega(k \sqrt{m \log m} )$ margin, our method, with high probability, will find a strategy enabling her to win (albeit with a possibly smaller margin). It thus provides a somewhat stronger guarantee compared to the previous methods, which implicitly implied a strategy that provides an $\Omega(m)$-additive approximation to the maximum score of a non-preferred candidate. For the weighted case, our generalized algorithm provides an $O(W \sqrt{m \log m} )$-additive approximation, where $W$ is the sum of voter weights. This is a clear advantage over previous methods: some of them do not generalize to the weighted case, while others---which approximate the number of manipulators---pose restrictions on the weights of extra manipulators added. Our methods are based on carefully rounding an exponentially-large configuration linear program that is solved by using the ellipsoid method with an efficient separation oracle.