Many electoral bribery, control, and manipulation problems (which we will refer to in general as "manipulative actions" problems) are NP-hard in the general case. It has recently been noted that many of these problems fall into polynomial time if the electorate is single-peaked (i.e., is polarized along some axis issue). However, real-world electorates are not truly single-peaked. There are usually some mavericks, and so real-world electorates tend to merely be nearly single-peaked. This paper studies the complexity of manipulative-action algorithms for elections over nearly single-peaked electorates, for various notions of nearness and various election systems. We provide instances where even one maverick jumps the manipulative-action complexity up to @math -hardness, but we also provide many instances where a reasonable number of mavericks can be tolerated without increasing the manipulative-action complexity.
The four papers most related to the present one are the following. insightfully raised the idea that general complexity results may change in single-peaked societies. His manipulative-action example (STV) actually provides a case where single-peakedness fails to lower manipulation complexity, but in a different context he did find a lowering of complexity for single-peakedness. The papers , the first of which was in the TARK conference, then broadly explored the effect of single-peakedness on manipulative actions. These three papers are all in the model of (perfect) single-peakedness. , in the context of preference elicitation, raised and experimentally studied the issue of single-peaked societies. also discussed nearness to single-peakedness, and the papers @cite_3 @cite_2 both raise as open issues whether shield-evaporation (complexity) results for single-peakedness will withstand near-single-peakedness. The present paper seeks to bring the nearly single-peaked'' lens to the study of manipulative actions.
@cite_3 @cite_2
1983312130 2233988666
Manipulation, bribery, and control are well-studied ways of changing the outcome of an election. Many voting systems are, in the general case, computationally resistant to some of these manipulative actions. However when restricted to single-peaked electorates, these systems suddenly become easy to manipulate. Recently, Faliszewski, Hemaspaandra, and Hemaspaandra (2011b) studied the complexity of dishonest behavior in nearly single-peaked electorates. These are electorates that are not single-peaked but close to it according to some distance measure. In this paper we introduce several new distance measures regarding single-peakedness. We prove that determining whether a given profile is nearly single-peaked is NP-complete in many cases. For one case we present a polynomial-time algorithm. Furthermore, we explore the relations between several notions of nearly single-peakedness. This paper is devoted to the proportional representation (PR) problem when the preferences are clustered single-peaked. PR is a "multi-winner" election problem, that we study in Chamberlin and Courant's scheme [6]. We define clustered single-peakedness as a form of single-peakedness with respect to clusters of candidates, i.e. subsets of candidates that are consecutive (in arbitrary order) in the preferences of all voters. We show that the PR problem becomes polynomial when the size of the largest cluster of candidates (width) is bounded. Furthermore, we establish the polynomiality of determining the single-peaked width of a preference profile (minimum width for a partition of candidates into clusters compatible with clustered single-peakedness) when the preferences are narcissistic (i.e., every candidate is the most preferred one for some voter). Many electoral control and manipulation problems-which we will refer to in general as ''manipulative actions'' problems-are NP-hard in the general case. It has recently been noted that many of these problems fall into polynomial time if the electorate is single-peaked, i.e., is polarized along some axis issue. However, real-world electorates are not truly single-peaked. There are usually some mavericks, and so real-world electorates tend merely to be nearly single-peaked. This paper studies the complexity of manipulative-action algorithms for elections over nearly single-peaked electorates. We do this for many notions of nearness and for a broad range of election systems. We provide instances where even one maverick jumps the manipulative-action complexity up to NP-hardness, but we also provide many instances where some number of mavericks can be tolerated without increasing the manipulative-action complexity. Much work has been devoted, during the past 20years, to using complexity to protect elections from manipulation and control. Many ''complexity shield'' results have been obtained-results showing that the attacker's task can be made NP-hard. Recently there has been much focus on whether such worst-case hardness protections can be bypassed by frequently correct heuristics or by approximations. This paper takes a very different approach: We argue that when electorates follow the canonical political science model of societ al preferences the complexity shield never existed in the first place. In particular, we show that for electorates having single-peaked preferences, many existing NP-hardness results on manipulation and control evaporate.