Finding Preference Profiles of Condorcet Dimension kk via SAT

Condorcet winning sets are a set-valued generalization of the well-known concept of a Condorcet winner. As supersets of Condorcet winning sets are always Condorcet winning sets themselves, an interesting property of preference profiles is the size of the smallest Condorcet winning set they admit. This smallest size is called the Condorcet dimension of a preference profile. Since little is known about profiles that have a certain Condorcet dimension, we show in this paper how the problem of finding a preference profile that has a given Condorcet dimension can be encoded as a satisfiability problem and solved by a SAT solver. Initial results include a minimal example of a preference profile of Condorcet dimension 3, improving previously known examples both in terms of the number of agents as well as alternatives. Due to the high complexity of such problems it remains open whether a preference profile of Condorcet dimension 4 exists.Comment: Corrected typos, updated references, and added conclusio

Arguing about Voting Rules

When the members of a group have to make a decision, they can use a voting rule to aggregate their preferences. But which rule to use is a difficult question. Different rules have different properties, and social choice theorists have found arguments for and against most of them. These arguments are aimed at the expert reader, used to mathematical formalism. We propose a logic-based language to instantiate such arguments in concrete terms in order to help people understand the strengths and weaknesses of different voting rules. Our approach allows us to automatically derive a justification for a given election outcome or to support a group in arguing over which voting rule to use. We exemplify our approach with an in-depth study of the Borda rule.nonouirechercheInternationa

Formal Verification of Voting Schemes

Fundamental trust and credibility in democratic systems is commonly established through the existence and execution of democratic elections. The vote-counting of an election, usually formalised by a voting scheme, essentially boils down to a mechanism that aggregates individual preferences of the voters to reach a decision. For this matter, there are various differing voting schemes in use throughout the world, commonly based on high expectations and means to ensure a sensible democratic process. However, incidents such as the ruling by the German federal constitutional court which led to a change of the German legislation in 2013 manifest that it is difficult for a voting scheme to meet these legitimate expectations. In fact, there is no general notion of correctness for a voting scheme and thus no universal mechanism as shown in Kenneth J. Arrow’s Impossibility Theorem in 1951. As a consequence, designing a real-world voting scheme without flaws, which still gives significant democratic guarantees, is a difficult task as a trade-off between desirable properties is non-trivial and error-prone. The approach in this thesis is based on the idea to tackle this issue by proposing an incremental and iterative development process for voting schemes based on automated formal reasoning methods using program verification. We analyse two different forms of verification considering their role in this development process in order to achieve formal correctness of voting schemes. We perform a comprehensive set of case studies by applying ``medium-weight\u27\u27 and ``light-weight\u27\u27 verification techniques. The ``medium- weight\u27\u27 approach uses the annotation-based deductive verification tool VCC based on an auto-active methodology and the ``light-weight\u27\u27 technique is performed with the bounded model checking tool LLBMC. Our analysis covers a set of well-known voting schemes combined with a set of prominent voting scheme criteria. In addition to giving precise formalisations for these criteria adapted to the specific voting schemes and tools used, we advance the efficiency of the ``light-weight\u27\u27 approach by exploiting fundamental symmetric properties. Furthermore, we investigate on encountered challenges posed by the auto-active verification methodology, which lies in-between automatic and interactive verification methodologies, with respect to specific characteristics in voting schemes and also explore the potential of bounded verification techniques to produce precise counterexamples in order to enhance the capability of our envisioned development process to give early feedback. This thesis gives fundamental insights in general challenges and the potential of automated formal reasoning with the goal of correct voting schemes

A Generalized Probabilistic Gibbard-Satterthwaite Theorem

Friedgut, Kalai, and Nisan have proved that social choice functions can be successfully manipulated by random preference reordering with non- negligible probability. However, their results require two restrictions: the social choice function must be neutral, and the election must have at most 3 alternatives. In this thesis we focus on removing the latter restriction and generalizing the results to elections with any number of candidates. We also provide a survey of related work analyzing and comparing results from a number of authors

Computational aspects of voting: a literature survey

Preference aggregation is a topic of study in different fields such as philosophy, mathematics, economics and political science. Recently, computational aspects of preference aggregation have gained especial attention and “computational politics” has emerged as a marked line of research in computer science with a clear concentration on voting protocols. The field of voting systems, rooted in social choice theory, has expanded notably in both depth and breadth in the last few decades. A significant amount of this growth comes from studies concerning the computational aspects of voting systems. This thesis comprehensively reviews the work on voting systems (from a computing perspective) by listing, classifying and comparing the results obtained by different researchers in the field. This survey covers a wide range of new and historical results yet provides a profound commentary on related work as individual studies and in relation to other related work and to the field in general. The deliverables serve as an overview where students and novice researchers in the field can start and also as a depository that can be referred to when searching for specific results. A comprehensive literature survey of the computational aspects of voting is a task that has not been undertaken yet and is initially realized here. Part of this research was dedicated to creating a web-depository that contains material and references related to the topic based on the survey. The purpose was to create a dynamic version of the survey that can be updated with latest findings and as an online practical reference

Voting theory: cui bono?

The theory of voting has largely developed independently of the mechanism design research, but with the introduction of the concept of strategic voting the two traditions found a common ground. This happened some fifty years ago. Yet, despite the voluminous literature that has emerged since then, the impact of voting theory to the design of political institutions remains marginal. Often the assumptions are deemed too simplistic or too abstract or plainly `out of this world'. It looks as if there is a demand for research that aims at building bridges over the wide gap that exists between the abstract social choice results and the behavioral-institutional realities characterizing political systems of today and tomorrow.  We illustrate the applicability problems by discussing a relatively recent proposal for electoral reform of the single-member constituency system in electing the members for the House of Representatives in the United States. The proposed reform would seem to solve a major flaw in the existing system.  As is often the case, this comes with a price, though: the proposal is plagued with problems of its own. However, the voting theory results have a wide area of applicability beyond voting. Yet the applicability of the voting theory results in these areas have remained largely unexplored. This article aims at suggesting some applications.  Most straight-forward ones pertain to multiple-criterion decision making. </p

Rank Aggregation Using Scoring Rules

To aggregate rankings into a social ranking, one can use scoring systems such as Plurality, Veto, and Borda. We distinguish three types of methods: ranking by score, ranking by repeatedly choosing a winner that we delete and rank at the top, and ranking by repeatedly choosing a loser that we delete and rank at the bottom. The latter method captures the frequently studied voting rules Single Transferable Vote (aka Instant Runoff Voting), Coombs, and Baldwin. In an experimental analysis, we show that the three types of methods produce different rankings in practice. We also provide evidence that sequentially selecting winners is most suitable to detect the "true" ranking of candidates. For different rules in our classes, we then study the (parameterized) computational complexity of deciding in which positions a given candidate can appear in the chosen ranking. As part of our analysis, we also consider the Winner Determination problem for STV, Coombs, and Baldwin and determine their complexity when there are few voters or candidates.Comment: 47 pages including appendi

Manipulation of elections by minimal coalitions

Social choice is the study of the issues arising when a population of individuals attempts to combine its views with the objective of determining a collective policy. Recent research in artificial intelligence raises concerns of articial intelligence agents applying computational resources to attack an election. If we think of voting as a way to combine honest preferences, it would be undesirable for some voters cast ballots that differ from their true preferences and achieve a better result for themselves at the expense of the general social welfare. Such an attack is called manipulation. The Gibbard-Satterthwaite theorem holds that all reasonable voting rules will admit a situation in which some voter achieves a better result for itself by misrepresenting its preferences. Bartholdi and Orlin showed that finding a beneficial manipulation under the single transferable vote rule is NP-Complete. Our work explores the practical dificulty of the coalitional manipulation problem. We computed the minimum sizes of successful manipulating coalitions, and compared this to theoretical results

Multi-Winner Voting with Approval Preferences

Approval-based committee (ABC) rules are voting rules that output a fixed-size subset of candidates, a so-called committee. ABC rules select committees based on dichotomous preferences, i.e., a voter either approves or disapproves a candidate. This simple type of preferences makes ABC rules widely suitable for practical use. In this book, we summarize the current understanding of ABC rules from the viewpoint of computational social choice. The main focus is on axiomatic analysis, algorithmic results, and relevant applications.Comment: This is a draft of the upcoming book "Multi-Winner Voting with Approval Preferences