522 research outputs found
Computational Aspects of Nearly Single-Peaked Electorates
Manipulation, bribery, and control are well-studied ways of changing the
outcome of an election. Many voting rules are, in the general case,
computationally resistant to some of these manipulative actions. However when
restricted to single-peaked electorates, these rules suddenly become easy to
manipulate. Recently, Faliszewski, Hemaspaandra, and Hemaspaandra studied the
computational complexity of strategic 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. In case the single-peaked axis is given, we show
that determining the distance is always possible in polynomial time.
Furthermore, we explore the relations between the new notions introduced in
this paper and existing notions from the literature.Comment: Published in the Journal of Artificial Intelligence Research (JAIR).
A short version of this paper appeared in the proceedings of the
Twenty-Seventh AAAI Conference on Artificial Intelligence (AAAI 2013). An
even earlier version appeared in the proceedings of the Fourth International
Workshop on Computational Social Choice 2012 (COMSOC 2012
Reinstating Combinatorial Protections for Manipulation and Bribery in Single-Peaked and Nearly Single-Peaked Electorates
Understanding when and how computational complexity can be used to protect
elections against different manipulative actions has been a highly active
research area over the past two decades. A recent body of work, however, has
shown that many of the NP-hardness shields, previously obtained, vanish when
the electorate has single-peaked or nearly single-peaked preferences. In light
of these results, we investigate whether it is possible to reimpose NP-hardness
shields for such electorates by allowing the voters to specify partial
preferences instead of insisting they cast complete ballots. In particular, we
show that in single-peaked and nearly single-peaked electorates, if voters are
allowed to submit top-truncated ballots, then the complexity of manipulation
and bribery for many voting rules increases from being in P to being
NP-complete.Comment: 28 pages; A shorter version of this paper will appear at the 30th
AAAI Conference on Artificial Intelligence (AAAI-16
Testing Top Monotonicity
Top monotonicity is a relaxation of various well-known domain restrictions
such as single-peaked and single-crossing for which negative impossibility
results are circumvented and for which the median-voter theorem still holds. We
examine the problem of testing top monotonicity and present a characterization
of top monotonicity with respect to non-betweenness constraints. We then extend
the definition of top monotonicity to partial orders and show that testing top
monotonicity of partial orders is NP-complete
Structure in Dichotomous Preferences
Many hard computational social choice problems are known to become tractable
when voters' preferences belong to a restricted domain, such as those of
single-peaked or single-crossing preferences. However, to date, all algorithmic
results of this type have been obtained for the setting where each voter's
preference list is a total order of candidates. The goal of this paper is to
extend this line of research to the setting where voters' preferences are
dichotomous, i.e., each voter approves a subset of candidates and disapproves
the remaining candidates. We propose several analogues of the notions of
single-peaked and single-crossing preferences for dichotomous profiles and
investigate the relationships among them. We then demonstrate that for some of
these notions the respective restricted domains admit efficient algorithms for
computationally hard approval-based multi-winner rules.Comment: A preliminary version appeared in the proceedings of IJCAI 2015, the
International Joint Conference on Artificial Intelligenc
Divergent Platforms
A robust feature of models of electoral competition between two opportunistic, purely office-motivated parties is that both parties become indistinguishable in equilibrium. I this short note, I show that this strong connection between the office motivation of parties and their equilibrium choice of identical platforms depends on the following two - possibly counterfactual - assumptions: 1. Issue spaces are uni-dimensional and 2. Parties are unitary actors whose preferences can be represented by expected utility functions. The main goal here is to provide an example of a two-party model in which parties offer substantially different platforms in equilibrium even though no exogenous asymmetries are assumed. In this example, some voters’ preferences over the 2-dimensional issue space are assumed to exhibit non-convexities and parties evaluate their actions with respect to a set of beliefs on the electorate.Downs model, Games with Incomplete Preferences, Knightian Uncertainty, Uncertainty Aversion, Platform Divergence
Divergent platforms
Models of electoral competition between two opportunistic, office-motivated parties typically predict that both parties become indistinguishable in equilibrium. I show that this strong connection between the office motivation of parties and their equilibrium choice of identical platforms depends on two—possibly false—assumptions: (1) Issue spaces are uni-dimensional and (2) Parties are unitary actors whose preferences can be represented by expected utilities. I provide an example of a two-party model in which parties offer substantially different equilibrium platforms even though no exogenous differences between parties are assumed. In this example, some voters’ preferences over the 2-dimensional issue space exhibit non-convexities and parties evaluate their actions with respect to a set of beliefs on the electorate
The Shield that Never Was: Societies with Single-Peaked Preferences are More Open to Manipulation and Control
Much work has been devoted, during the past twenty years, to using complexity
to protect elections from manipulation and control. Many results have been
obtained showing NP-hardness shields, and 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 societal 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.Comment: 38 pages, 2 figure
Single-Peaked Consistency for Weak Orders Is Easy
In economics and social choice single-peakedness is one of the most important
and commonly studied models for preferences. It is well known that
single-peaked consistency for total orders is in P. However in practice a
preference profile is not always comprised of total orders. Often voters have
indifference between some of the candidates. In a weak preference order
indifference must be transitive. We show that single-peaked consistency for
weak orders is in P for three different variants of single-peakedness for weak
orders. Specifically, we consider Black's original definition of
single-peakedness for weak orders, Black's definition of single-plateaued
preferences, and the existential model recently introduced by Lackner. We
accomplish our results by transforming each of these single-peaked consistency
problems to the problem of determining if a 0-1 matrix has the consecutive ones
property.Comment: In Proceedings TARK 2015, arXiv:1606.0729
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