The Complexity of Online Manipulation of Sequential Elections

TL;DR

This paper explores the computational complexity of online manipulation in sequential elections, revealing that complexity varies greatly depending on election rules, coalition size, and whether the model assumes unique or non-unique winners.

Contribution

It introduces a new framework for online coalitional manipulation in sequential elections and provides a detailed complexity classification across various election systems and models.

Findings

Manipulation can be PSPACE-complete depending on coalition size.

Online weighted manipulation for plurality is easy in non-unique-winner models but hard in unique-winner models.

First P^NP[1]- and P^NP-completeness results in computational social choice.

Abstract

Most work on manipulation assumes that all preferences are known to the manipulators. However, in many settings elections are open and sequential, and manipulators may know the already cast votes but may not know the future votes. We introduce a framework, in which manipulators can see the past votes but not the future ones, to model online coalitional manipulation of sequential elections, and we show that in this setting manipulation can be extremely complex even for election systems with simple winner problems. Yet we also show that for some of the most important election systems such manipulation is simple in certain settings. This suggests that when using sequential voting, one should pay great attention to the details of the setting in choosing one's voting rule. Among the highlights of our classifications are: We show that, depending on the size of the manipulative coalition, the online manipulation problem can be complete for each level of the polynomial hierarchy or even for PSPACE. We obtain the most dramatic contrast to date between the nonunique-winner and unique-winner models: Online weighted manipulation for plurality is in P in the nonunique-winner model, yet is coNP-hard (constructive case) and NP-hard (destructive case) in the unique-winner model. And we obtain what to the best of our knowledge are the first P^NP[1]-completeness and P^NP-completeness results in the field of computational social choice, in particular proving such completeness for, respectively, the complexity of 3-candidate and 4-candidate (and unlimited-candidate) online weighted coalition manipulation of veto elections.