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Computational Aspects of Voting: a Literature Survey Fatima Talib

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Computational Aspects of Voting: a Literature Survey Fatima Talib View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RIT Scholar Works Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 2007 Computational aspects of voting: a literature survey Fatima Talib Follow this and additional works at: http://scholarworks.rit.edu/theses Recommended Citation Talib, Fatima, "Computational aspects of voting: a literature survey" (2007). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Fatima Al-Raisi May 16th, 2007 Abstract 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 the- ory, has expanded notably in both depth and breadth in the last few decades. A significant amount of this growth comes from studies con- cerning 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 re- sults 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 computa- tional aspects of voting is a task that has not been undertaken yet and is initially realized here. Part of this research was dedicated to creat- ing 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. Preface The notions of preference and preference-aggregation are fundamental notions in many dis- ciplines such as philosophy, social choice theory, mathematical decision theory, economics, applied mathematics, and computer science. In computer science, the topics of \prefer- ence" and \preference aggregation" are studied in Artificial Intelligence for coordinating a society of software agents, in Databases for filtering the information retrieved in response to queries, in Natural Language Processing for classifying different tokens in a text, and in Complexity Theory for analyzing the computational complexity of various preference aggregation methods. Among Preference aggregation methods is the naturally appealing method of voting. Recently, computational aspects of preference aggregation through voting have gained es- pecial 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 sys- tems, 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 extensively reviews the work on voting systems (from a computing perspec- tive) by listing, classifying and comparing the results obtained by the researchers in the field. This survey covers a wide range of new and historical results yet provides a com- mentary on related work as individual studies and in relation to other related work and to the field in general. This work serves 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. Each chapter is concluded with comments and bibliographic notes and is appended with separate bibliography. The entire thesis is appended with a comprehensive index of subjects/terms and an index of authors that contains the names of over two hun- dred authors of works in computational aspects of voting cited here1. At the end is a list of all references sorted in alphabetical order by author (last) name. The list of references con- tains more than two hundred items including books, book chapters, conference proceedings, journal articles, technical reports, and other forms of published and unpublished work. A comprehensive literature survey of the computational aspects of voting is a task that has not been undertaken yet and is initially realized here. 1Names of authors of cited works appear in the authors index whether or not the names are explicitly mentioned in the text. 1 2 Part of this research was dedicated to creating a web-depository that contains material and references related to the topic of the survey. The web-depository is a dynamic version of the survey that can be updated with latest findings and used as an online practical reference. Both the URL of the dynamic survey and the electronic copy of this thesis can be accessed at: http://www.cs.rit.edu:8080/ms/static/index.html. Acknowledgements He who does not thank people is not grateful to God. Prophet Muhammad (pbuh) Most of all, I would like to thank my advisor, and teacher, Edith Hemaspaandra. I thank Edith for triggering my initial interest in Computational Politics and in Complexity Theory in general. Throughout my thesis development, Edith has been a great advisor and teacher. She encouraged me to work in this area, provided advice and guidance, promptly responded to my queries, and taught me countless things from the basic concepts to the details of structuring and presenting the work. I am grateful for the opportunity of being Edith's advisee and student. I also thank the other members of my committee, Chris Homan and Piotr Faliszewski. I thank Chris for agreeing to serve on the committee on a short notice, and then for his valuable comments on the text. His feedback has contributed to improved quality of this work. Special thanks go to Piotr Faliszewski for pointing out some related works that I was not aware of, and for clarifying some points. I have been very fortunate to have such a great committee. I would also like to extend my thanks to Stanislaw Radziszowski for insightful discus- sions on the recommended depth of the chapter on Electronic Voting. I must acknowledge as well my colleagues at RIT, Nathan Russell and Eric Brelsford for providing information and answering questions regarding their theses, so that I can include their work in the survey, and my colleagues and friends at both RIT and SQU (Sultan Qaboos University, Oman) for reading various parts of this relatively long Master's thesis: Jeremy Paskali, Fatou Bagayogo, Youcef Baghdadi, and especially Anisa Al-Hafeedh. I am always indebted to the people whom without I wouldn't be where I am and who I am today, my family. I thank my family, and especially my parents, for their support, love, and patience while I was thousands of miles away pursuing my Masters in Computer Science. Finally, to God is all praise and ultimate thanks. 3 4 Contents Preface 3 1 Preliminaries 15 1.0 Prologue . 15 1.1 Mathematical Preliminaries . 15 1.1.1 Elementary Concepts of Set Theory . 15 1.1.2 Some Data Structures . 16 Graphs . 16 Binary Trees . 17 1.1.3 Binary Relations and Rankings . 17 1.2 Social Choice Theory - an Introduction . 18 1.3 Elections and Voting Systems . 21 1.3.1 Definitions and Terminology . 21 1.3.2 Some Voting Rules . 22 1.3.3 Properties of Voting Rules: Criteria for Evaluating Voting Rules . 27 1.4 Comments and Bibliographic Notes . 29 Chapter 1 Bibliography . 29 2 Classification of Related Work 35 2.0 Prologue . 35 2.1 Mathematics-based Classification . 35 2.2 Direction . 36 2.3 Analysis versus Design . 36 2.4 MCDM vs. DMU vs. Social Choice . 36 2.5 This Classification . 37 Chapter 2 Bibliography . 37 3 Preferences: Representation and Elicitation 41 3.0 Prologue . 41 3.1 Three Pictures Depicting Preferences . 43 3.1.1 Preference Graphs . 43 3.1.2 Preference Curves . 44 3.1.3 Preference Matrices . 45 5 6 CONTENTS 3.2 Preference Representation . 47 3.2.1 Motivation . 47 3.2.2 The Domain . 48 3.2.3 Types of Preferences . 48 3.2.4 Preference Representation Languages . 49 Examples of Preference Representation Languages . 50 3.2.5 Single-peaked Preferences . 52 Recognizing Single-peaked Preferences . 55 Implications and Results . 56 3.3 Preference Elicitation . 58 3.3.1 Incomparability and Incompleteness of Preferences . 58 3.3.2 What is Preference Elicitation? . 59 3.3.3 Vote Elicitation . 60 3.3.4 Extending Voting Rules to Handle Incomplete Preferences . 61 3.3.5 Effective Elicitation . 62 3.3.6 Possible and Necessary Winners . 63 3.3.7 Vote Elicitation and Manipulation . 64 3.3.8 Some Results . 65 3.4 Comments and Bibliographic Notes . 65 Chapter 3 Bibliography . 66 4 Complexity-Theoretic Aspects of Voting Systems 71 4.0 Prologue . 71 4.1 Complexity Theory: an Introduction . 71 4.1.1 Problems as Languages . 72 4.1.2 The Computational Complexity of a Problem . 73 4.1.3 Some Important Complexity Classes . 74 4.1.4 Operations on Problems and Complexity Classes . 76 4.1.5 The Notion of Completeness . 77 Some NP-complete Problems . 77 4.1.6 The Polynomial Hierarchy . 78 4.1.7 Complexity Theory and the Study of Voting .
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