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The Computational Complexity of Dominance and Consistency in CPNets
"... We investigate the computational complexity of testing dominance and consistency in CPnets. Previously, the complexity of dominance has been determined for restricted classes in which the dependency graph of the CPnet is acyclic. However, there are preferences of interest that define cyclic depend ..."
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We investigate the computational complexity of testing dominance and consistency in CPnets. Previously, the complexity of dominance has been determined for restricted classes in which the dependency graph of the CPnet is acyclic. However, there are preferences of interest that define cyclic dependency graphs; these are modeled with general CPnets. In our main results, we show here that both dominance and consistency for general CPnets are PSPACEcomplete. We then consider the concept of strong dominance, dominance equivalence and dominance incomparability, and several notions of optimality, and identify the complexity of the corresponding decision problems. The reductions used in the proofs are from STRIPS planning, and thus reinforce the earlier established connections between both areas.
Voting on Multiattribute Domains with Cyclic Preferential Dependencies
"... In group decision making, often the agents need to decide on multiple attributes at the same time, so that there are exponentially many alternatives. In this case, it is unrealistic to ask agents to communicate a full ranking of all the alternatives. To address this, earlier work has proposed decomp ..."
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Cited by 25 (12 self)
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In group decision making, often the agents need to decide on multiple attributes at the same time, so that there are exponentially many alternatives. In this case, it is unrealistic to ask agents to communicate a full ranking of all the alternatives. To address this, earlier work has proposed decomposing such voting processes by using local voting rules on the individual attributes. Unfortunately, the existing methods work only with rather severe domain restrictions, as they require the voters’ preferences to extend acyclic CPnets compatible with a common order on the attributes. We first show that this requirement is very restrictive, by proving that the number of linear orders extending an acyclic CPnet is exponentially smaller than the number of all linear orders. Then, we introduce a very general methodology that allows us to aggregate preferences when voters express CPnets that can be cyclic. There does not need to be any common structure among the submitted CPnets. Our methodology generalizes the earlier, more restrictive methodology. We study whether properties of the local rules transfer to the global rule, and vice versa. We also address how to compute the winning alternatives.
Anonymityproof voting rules
 In Computational Social Systems and the Internet #07271, Dagstuhl Workshop
, 2007
"... In open, anonymous environments such as the Internet, mechanism design must be extended to take new types of manipulation into account—especially, the possibility that an agent participates in the mechanism multiple times. General social choice or voting settings lie at the heart of mechanism design ..."
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Cited by 24 (13 self)
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In open, anonymous environments such as the Internet, mechanism design must be extended to take new types of manipulation into account—especially, the possibility that an agent participates in the mechanism multiple times. General social choice or voting settings lie at the heart of mechanism design and provide a natural starting point. A (randomized, anonymous) voting rule maps any multiset of total orders of (aka. votes over) a fixed set of alternatives to a probability distribution over these alternatives. A voting rule f is neutral if it treats all alternatives symmetrically. It satisfies participation if no voter ever benefits from not casting her vote. It is falsenameproof if no voter ever benefits from casting additional (potentially different) votes. It is anonymityproof if it satisfies participation and it is falsenameproof. We show that the class of anonymityproof neutral voting rules consists exactly of the rules of the following form.
Socially Desirable Approximations for Dodgson’s Voting Rule ∗ ABSTRACT
"... voting rule that today bears his name. Although Dodgson’s rule is one of the most wellstudied voting rules, it suffers from serious deficiencies, both from the computational point of view—it is N Phard even to approximate the Dodgson score within sublogarithmic factors—and from the social choice p ..."
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Cited by 20 (4 self)
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voting rule that today bears his name. Although Dodgson’s rule is one of the most wellstudied voting rules, it suffers from serious deficiencies, both from the computational point of view—it is N Phard even to approximate the Dodgson score within sublogarithmic factors—and from the social choice point of view—it fails basic social choice desiderata such as monotonicity and homogeneity. In a previous paper [Caragiannis et al., SODA 2009] we have asked whether there are approximation algorithms for Dodgson’s rule that are monotonic or homogeneous. In this paper we give definitive answers to these questions. We design a monotonic exponentialtime algorithm that yields a 2approximation to the Dodgson score, while matching this result with a tight lower bound. We also present a monotonic polynomialtime O(log m)approximation algorithm (where
A Computational Analysis of the Tournament Equilibrium Set
, 2009
"... A recurring theme in the mathematical social sciences is how to select the “most desirable ” elements given a binary dominance relation on a set of alternatives. Schwartz’s tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions prop ..."
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Cited by 20 (12 self)
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A recurring theme in the mathematical social sciences is how to select the “most desirable ” elements given a binary dominance relation on a set of alternatives. Schwartz’s tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions proposed so far. Due to its unwieldy recursive definition, little is known about TEQ. In particular, its monotonicity remains an open problem to date. Yet, if TEQ were to satisfy monotonicity, it would be a very attractive solution concept refining both the Banks set and Dutta’s minimal covering set. We show that the problem of deciding whether a given alternative is contained in TEQ is NPhard, and thus does not admit a polynomialtime algorithm unless P equals NP. Furthermore, we propose a heuristic that significantly outperforms the naive algorithm for computing TEQ.
Computing the minimal covering set
 In Proceedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge
, 2007
"... We present the first polynomialtime algorithm for computing the minimal covering set of a (weak) tournament. The algorithm draws upon a linear programming formulation of a subset of the minimal covering set known as the essential set. On the other hand, we show that no efficient algorithm exists fo ..."
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Cited by 17 (11 self)
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We present the first polynomialtime algorithm for computing the minimal covering set of a (weak) tournament. The algorithm draws upon a linear programming formulation of a subset of the minimal covering set known as the essential set. On the other hand, we show that no efficient algorithm exists for two variants of the minimal covering set, the minimal upward covering set and the minimal downward covering set, unless P equals NP. Finally, we observe a strong relationship between von NeumannMorgenstern stable sets and upward covering on the one hand, and the Banks set and downward covering on the other.
Dealing with Incomplete Agents ’ Preferences and an Uncertain Agenda in Group Decision Making via Sequential Majority Voting
"... We consider multiagent systems where agents ’ preferences are aggregated via sequential majority voting: each decision is taken by performing a sequence of pairwise comparisons where each comparison is a weighted majority vote among the agents. Incompleteness in the agents ’ preferences is common i ..."
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Cited by 13 (7 self)
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We consider multiagent systems where agents ’ preferences are aggregated via sequential majority voting: each decision is taken by performing a sequence of pairwise comparisons where each comparison is a weighted majority vote among the agents. Incompleteness in the agents ’ preferences is common in many reallife settings due to privacy issues or an ongoing elicitation process. In addition, there may be uncertainty about how the preferences are aggregated. For example, the agenda (a tree whose leaves are labelled with the decisions being compared) may not yet be known or fixed. We therefore study how to determine collectively optimal decisions (also called winners) when preferences may be incomplete, and when the agenda may be uncertain. We show that it is computationally easy to determine if a candidate decision always wins, or may win, whatever the agenda. On the other hand, it is computationally hard to know whether a candidate decision wins in at least one agenda for at least one completion of the agents ’ preferences. These results hold even if the agenda must be balanced so that each candidate decision faces the same number of majority votes. Such results are useful for reasoning about preference elicitation. They help understand the complexity of tasks such as determining if a decision can be taken collectively, as well as knowing if the winner can be manipulated by appropriately ordering the agenda.
Winner determination in voting trees with incomplete preferences and weighted votes
, 2012
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Aggregation of Attack Relations: A SocialChoice Theoretical Analysis of Defeasibility Criteria ⋆
"... Abstract. This paper analyzes the aggregation of different abstract attack relations over a common set of arguments. Each of those attack relations can be considered as the representation of a criterion of warrant. It is well known in the field of Social Choice Theory that if some “fairness ” condit ..."
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Abstract. This paper analyzes the aggregation of different abstract attack relations over a common set of arguments. Each of those attack relations can be considered as the representation of a criterion of warrant. It is well known in the field of Social Choice Theory that if some “fairness ” conditions are imposed over an aggregation of preferences, it becomes impossible to yield a result. When the criteria lead to acyclic attack relations, a positive result may ensue under the same conditions, namely that if the class of winning coalitions in an aggregation process by voting is a proper prefilter an outcome will exist. This outcome may preserve some features of the competing attack relations, such as the highly desirable property of acyclicity which can be associated with the existence of a single extension of an argumentation system. The downside of this is that, in fact, the resulting attack relation must be a portion common to the “hidden dictators ” in the system, that is, all the attack relations that belong to all the winning coalitions. 1
A New Perspective on Implementation by Voting Trees
"... Voting trees describe an iterative procedure for selecting a single vertex from a tournament. They provide a very general abstract model of decisionmaking among a group of individuals, and it has therefore been studied which voting rules have a tree that implements them, i.e., chooses according to ..."
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Voting trees describe an iterative procedure for selecting a single vertex from a tournament. They provide a very general abstract model of decisionmaking among a group of individuals, and it has therefore been studied which voting rules have a tree that implements them, i.e., chooses according to the rule for every tournament. While partial results concerning implementable rules and necessary conditions for implementability have been obtained over the past forty years, a complete characterization of voting rules implementable by trees has proven surprisingly hard to find. A prominent rule that cannot be implemented by trees is the Copeland rule, which singles out vertices with maximum degree. In this paper, we suggest a new angle of attack and reexamine the implementability of the Copeland solution using paradigms and techniques that are at the core of theoretical computer science. We study the extent to which voting trees can approximate the maximum degree in a tournament, and give upper and lower bounds on the worstcase ratio between the degree of the vertex chosen by a tree and the maximum degree, both for the deterministic model concerned with a single fixed tree, and for randomizations over arbitrary sets of trees. Our main positive result is a randomization over surjective trees of polynomial size that provides an approximation ratio of at least 1/2. The proof is based on a connection between a randomization over caterpillar trees and a rapidly mixing Markov chain.