Control in the Presence of Manipulators: Cooperative and Competitive Cases

TL;DR

This paper analyzes the complexity of control attacks in elections with manipulators, considering both cooperative and adversarial scenarios, and provides tight bounds for various voting systems.

Contribution

It offers a comprehensive complexity analysis of election control with manipulators, including tight bounds and polynomial-time results for specific voting systems.

Findings

Complexity bounds vary with election control and manipulation scenarios.

Approval and plurality systems allow polynomial-time control with manipulators.

Borda voting control with manipulators can be computationally hard, unless NP=coNP.

Abstract

Control and manipulation are two of the most studied types of attacks on elections. In this paper, we study the complexity of control attacks on elections in which there are manipulators. We study both the case where the "chair" who is seeking to control the election is allied with the manipulators, and the case where the manipulators seek to thwart the chair. In the latter case, we see that the order of play substantially influences the complexity. We prove upper bounds, holding over every election system with a polynomial-time winner problem, for all standard control cases, and some of these bounds are at the second or third level of the polynomial hierarchy, and we provide matching lower bounds to prove these tight. Nonetheless, for important natural systems the complexity can be much lower. We prove that for approval and plurality elections, the complexity of even competitive clashes between a controller and manipulators falls far below those high bounds, even as low as polynomial time. Yet for a Borda-voting case we show that such clashes raise the complexity unless NP = coNP.