17,296 research outputs found
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
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
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
Acyclic domains of linear orders: a survey
Among the many significant contributions that Fishburn made to social choice theory some have focused on what he has called "acyclic sets", i.e. the sets of linear orders where majority rule applies without the "Condorcet effect" (majority relation never has cycles). The search for large domains of this type is a fascinating topic. I review the works in this field and in particular consider a recent one that allows to show the connections between some of them that have been unrelated up to now.acyclic set;alternating scheme;distributive lattice;effet Condorcet;linear order,maximal chain,permutoèdre lattice, single-peaked domain,weak Bruhat order,value restriction.
Possible Winners in Noisy Elections
We consider the problem of predicting winners in elections, for the case
where we are given complete knowledge about all possible candidates, all
possible voters (together with their preferences), but where it is uncertain
either which candidates exactly register for the election or which voters cast
their votes. Under reasonable assumptions, our problems reduce to counting
variants of election control problems. We either give polynomial-time
algorithms or prove #P-completeness results for counting variants of control by
adding/deleting candidates/voters for Plurality, k-Approval, Approval,
Condorcet, and Maximin voting rules. We consider both the general case, where
voters' preferences are unrestricted, and the case where voters' preferences
are single-peaked.Comment: 34 page
Bisymmetric and quasitrivial operations: characterizations and enumerations
We investigate the class of bisymmetric and quasitrivial binary operations on
a given set and provide various characterizations of this class as well as
the subclass of bisymmetric, quasitrivial, and order-preserving binary
operations. We also determine explicitly the sizes of these classes when the
set is finite.Comment: arXiv admin note: text overlap with arXiv:1709.0916
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