Clone Structures in Voters' Preferences

In elections, a set of candidates ranked consecutively (though possibly in different order) by all voters is called a clone set, and its members are called clones. A clone structure is a family of all clone sets of a given election. In this paper we study properties of clone structures. In particular, we give an axiomatic characterization of clone structures, show their hierarchical structure, and analyze clone structures in single-peaked and single-crossing elections. We give a polynomial-time algorithm that finds a minimal collection of clones that need to be collapsed for an election to become single-peaked, and we show that this problem is NP-hard for single-crossing elections.Comment: 35 pages, 3 figure

The Complexity of Fully Proportional Representation for Single-Crossing Electorates

We study the complexity of winner determination in single-crossing elections under two classic fully proportional representation rules---Chamberlin--Courant's rule and Monroe's rule. Winner determination for these rules is known to be NP-hard for unrestricted preferences. We show that for single-crossing preferences this problem admits a polynomial-time algorithm for Chamberlin--Courant's rule, but remains NP-hard for Monroe's rule. Our algorithm for Chamberlin--Courant's rule can be modified to work for elections with bounded single-crossing width. To circumvent the hardness result for Monroe's rule, we consider single-crossing elections that satisfy an additional constraint, namely, ones where each candidate is ranked first by at least one voter (such elections are called narcissistic). For single-crossing narcissistic elections, we provide an efficient algorithm for the egalitarian version of Monroe's rule.Comment: 23 page

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

Condorcet Domains, Median Graphs and the Single Crossing Property

Condorcet domains are sets of linear orders with the property that, whenever the preferences of all voters belong to this set, the majority relation has no cycles. We observe that, without loss of generality, such domain can be assumed to be closed in the sense that it contains the majority relation of every profile with an odd number of individuals whose preferences belong to this domain. We show that every closed Condorcet domain is naturally endowed with the structure of a median graph and that, conversely, every median graph is associated with a closed Condorcet domain (which may not be a unique one). The subclass of those Condorcet domains that correspond to linear graphs (chains) are exactly the preference domains with the classical single crossing property. As a corollary, we obtain that the domains with the so-called `representative voter property' (with the exception of a 4-cycle) are the single crossing domains. Maximality of a Condorcet domain imposes additional restrictions on the underlying median graph. We prove that among all trees only the chains can induce maximal Condorcet domains, and we characterize the single crossing domains that in fact do correspond to maximal Condorcet domains. Finally, using Nehring's and Puppe's (2007) characterization of monotone Arrowian aggregation, our analysis yields a rich class of strategy-proof social choice functions on any closed Condorcet domain

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

Are there any nicely structured preference~profiles~nearby?

We investigate the problem of deciding whether a given preference profile is close to having a certain nice structure, as for instance single-peaked, single-caved, single-crossing, value-restricted, best-restricted, worst-restricted, medium-restricted, or group-separable profiles. We measure this distance by the number of voters or alternatives that have to be deleted to make the profile a nicely structured one. Our results classify the problem variants with respect to their computational complexity, and draw a clear line between computationally tractable (polynomial-time solvable) and computationally intractable (NP-hard) questions

Combinatorial Voter Control in Elections

Voter control problems model situations such as an external agent trying to affect the result of an election by adding voters, for example by convincing some voters to vote who would otherwise not attend the election. Traditionally, voters are added one at a time, with the goal of making a distinguished alternative win by adding a minimum number of voters. In this paper, we initiate the study of combinatorial variants of control by adding voters: In our setting, when we choose to add a voter~$v$, we also have to add a whole bundle $\kappa(v)$ of voters associated with $v$. We study the computational complexity of this problem for two of the most basic voting rules, namely the Plurality rule and the Condorcet rule.Comment: An extended abstract appears in MFCS 201

Recognising Multidimensional Euclidean Preferences

Euclidean preferences are a widely studied preference model, in which decision makers and alternatives are embedded in d-dimensional Euclidean space. Decision makers prefer those alternatives closer to them. This model, also known as multidimensional unfolding, has applications in economics, psychometrics, marketing, and many other fields. We study the problem of deciding whether a given preference profile is d-Euclidean. For the one-dimensional case, polynomial-time algorithms are known. We show that, in contrast, for every other fixed dimension d > 1, the recognition problem is equivalent to the existential theory of the reals (ETR), and so in particular NP-hard. We further show that some Euclidean preference profiles require exponentially many bits in order to specify any Euclidean embedding, and prove that the domain of d-Euclidean preferences does not admit a finite forbidden minor characterisation for any d > 1. We also study dichotomous preferencesand the behaviour of other metrics, and survey a variety of related work.Comment: 17 page

Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges

Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multi-agent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problem-specific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context

Spoiler effects in proportional representation systems: evidence from eight Polish parliamentary elections, 1991–2015

I consider a model of multiple winner elections with several types of spoilers. In single-office elections, a “classic” spoiler turns a winner into a non-winner and a non-winner into a winner. Such spoilers rarely appear in multi-office elections. In such elections, spoilers include a “Kingmaker”, who turns a non-winner into a winner; a “Kingslayer”, who turns a winner into a non-winner; a “Valuegobbler”, who subtracts from some competitor more seats than it receives; and “Selfspoilers”, who may be hurt by competing separately rather than creating an electoral coalition. Various strategic spoilers, such as fake parties, are possible as well. I look for spoilers in eight Polish parliamentary elections that have taken place since the fall of communism in 1989. In two elections, the consequences of spoilers were massive. In 1993, multiple spoilers on the right helped the two post-communist parties return to power, slow down decommunization and create strong institutional obstacles to further democratization. In 2015, a spoiler manufactured a majority for the largest party (PiS) and, as a consequence, enabled PiS quickly to implement radical reforms. In other elections, spoilers had smaller, but noticeable consequences. The results suggest that parliamentary elections using PR party-list systems are vulnerable to spoiler problems that may cause significant political effects