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Approval Voting Under Dichotomous Preferences: a Catalogue of Characterizations
Draft – June 25, 2021 Approval Voting under Dichotomous Preferences: A Catalogue of Characterizations Florian Brandl Dominik Peters University of Bonn Harvard University [email protected] [email protected] Approval voting allows every voter to cast a ballot of approved alternatives and chooses the alternatives with the largest number of approvals. Due to its simplicity and superior theoretical properties it is a serious contender for use in real-world elections. We support this claim by giving eight characterizations of approval voting. All our results involve the reinforcement axiom, which requires choices to be consistent across different electorates. In addition, we consider strategyproofness, consistency with majority opinions, consistency under cloning alternatives, and invariance under removing inferior alternatives. We prove our results by reducing them to a single base theorem, for which we give a simple and intuitive proof. 1 Introduction Around the world, when electing a leader or a representative, plurality is by far the most common voting system: each voter casts a vote for a single candidate, and the candidate with the most votes is elected. In pioneering work, Brams and Fishburn (1983) proposed an alternative system: approval voting. Here, each voter may cast votes for an arbitrary number of candidates, and can thus choose whether to approve or disapprove of each candidate. The election is won by the candidate who is approved by the highest number of voters. Approval voting allows voters to be more expressive of their preferences, and it can avoid problems such as vote splitting, which are endemic to plurality voting. Together with its elegance and simplicity, this has made approval voting a favorite among voting theorists (Laslier, 2011), and has led to extensive research literature (Laslier and Sanver, 2010). -
Instructor's Manual
The Mathematics of Voting and Elections: A Hands-On Approach Instructor’s Manual Jonathan K. Hodge Grand Valley State University January 6, 2011 Contents Preface ix 1 What’s So Good about Majority Rule? 1 Chapter Summary . 1 Learning Objectives . 2 Teaching Notes . 2 Reading Quiz Questions . 3 Questions for Class Discussion . 6 Discussion of Selected Questions . 7 Supplementary Questions . 10 2 Perot, Nader, and Other Inconveniences 13 Chapter Summary . 13 Learning Objectives . 14 Teaching Notes . 14 Reading Quiz Questions . 15 Questions for Class Discussion . 17 Discussion of Selected Questions . 18 Supplementary Questions . 21 3 Back into the Ring 23 Chapter Summary . 23 Learning Objectives . 24 Teaching Notes . 24 v vi CONTENTS Reading Quiz Questions . 25 Questions for Class Discussion . 27 Discussion of Selected Questions . 29 Supplementary Questions . 36 Appendix A: Why Sequential Pairwise Voting Is Monotone, and Instant Runoff Is Not . 37 4 Trouble in Democracy 39 Chapter Summary . 39 Typographical Error . 40 Learning Objectives . 40 Teaching Notes . 40 Reading Quiz Questions . 41 Questions for Class Discussion . 42 Discussion of Selected Questions . 43 Supplementary Questions . 49 5 Explaining the Impossible 51 Chapter Summary . 51 Error in Question 5.26 . 52 Learning Objectives . 52 Teaching Notes . 53 Reading Quiz Questions . 54 Questions for Class Discussion . 54 Discussion of Selected Questions . 55 Supplementary Questions . 59 6 One Person, One Vote? 61 Chapter Summary . 61 Learning Objectives . 62 Teaching Notes . 62 Reading Quiz Questions . 63 Questions for Class Discussion . 65 Discussion of Selected Questions . 65 CONTENTS vii Supplementary Questions . 71 7 Calculating Corruption 73 Chapter Summary . 73 Learning Objectives . 73 Teaching Notes . -
Explaining All Three-Alternative Voting Outcomes*
Journal of Economic Theory 87, 313355 (1999) Article ID jeth.1999.2541, available online at http:ÂÂwww.idealibrary.com on Explaining All Three-Alternative Voting Outcomes* Donald G. Saari Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2730 dsaariÄnwu.edu Received January 12, 1998; revised April 13, 1999 A theory is developed to explain all possible three-alternative (single-profile) pairwise and positional voting outcomes. This includes all preference aggregation paradoxes, cycles, conflict between the Borda and Condorcet winners, differences among positional outcomes (e.g., the plurality and antiplurality methods), and dif- ferences among procedures using these outcomes (e.g., runoffs, Kemeny's rule, and Copeland's method). It is shown how to identify, interpret, and construct all profiles supporting each paradox. Among new conclusions, it is shown why a standard for the field, the Condorcet winner, is seriously flawed. Journal of Economic Literature Classification Numbers: D72, D71. 1999 Academic Press 1. INTRODUCTION Over the last two centuries considerable attention has focussed on the properties of positional voting procedures. These commonly used approaches assign points to alternatives according to how each voter posi- tions them. The standard plurality method, for instance, assigns one point to a voter's top-ranked candidate and zero to all others, while the Borda count (BC) assigns n&1, n&2, ..., n&n=0 points, respectively, to a voter's first, second, ..., nth ranked candidate. The significance of these pro- cedures derives from their wide usage, but their appeal comes from their mysterious paradoxes (i.e., counterintuitive conclusions) demonstrating highly complex outcomes. By introducing doubt about election outcomes, these paradoxes raise the legitimate concern that, inadvertently, we can choose badly even in sincere elections. -
Arxiv:1811.06739V3 [Cs.GT] 3 Jun 2019 St
MEASURING MAJORITY POWER AND VETO POWER OF VOTING RULES ALEKSEI Y. KONDRATEV AND ALEXANDER S. NESTEROV Abstract. We study voting rules with respect to how they allow or limit a majority from dominating minorities: whether a voting rule makes a majority powerful, and whether minorities can veto the candidates they do not prefer. For a given voting rule, the minimal share of voters that guarantees a victory to one of their most preferred candidates is the measure of majority power, and the minimal share of voters that allows them to veto each of their least preferred candidates is the measure of veto power. We find tight bounds on these minimal shares for voting rules that are popular in the literature and in real elections. We order these rules according to majority power and veto power. The instant-runoff voting has both the highest majority power and the highest veto power and the plurality rule has the lowest. In general, the higher the majority power of a voting rule is, the higher its veto power. The three exceptions are: voting with proportional veto power, Black’s rule, and Borda rule, which have a relatively low level of majority power and a high level of veto power and thus provide minority protection. Our results can shed light on how voting rules provide different incentives for voter participation and candidate nomination. Keywords: majority tyranny, voting system, plurality voting, two-round system, Borda count, voting paradox, minority protection, electoral participation, voting procedure, voting method, voting rule, preferential voting, electoral system JEL Classification D71, D72 Date: 03 June 2019. -
This Is Chapter 25 of a Forthcoming Book Edited by K. Arrow, A. Sen, and K
This is Chapter 25 of a forthcoming book edited by K. Arrow, A. Sen, and K. Suzumura. It is intended only for participants of my 2007 MAA short course on the “mathematics of voting” in New Orleans. Chapter 25 GEOMETRY OF VOTING1 Donald G. Saari Director, Institute for Mathematical Behavioral Sciences Departments of Mathematics and Economics University of California, Irvine Irvine, California 92698 1. Introduction 2. Simple Geometric Representations 2.1 A geometric profile representation 2.2 Elementary geometry; surprising results 2.3 The source of pairwise voting problems 2.4 Finding other pairwise results 3. Geometry of Axioms 3.1 Arrow’s Theorem 3.2 Cyclic voters in Arrow’s framework? 3.3 Monotonicity, strategic behavior, etc. 4. Plotting all election outcomes 4.1 Finding all positional and AV outcomes 4.2 The converse; finding election relationships 5. Finding symmetries – and profile decompositions 6. Summary Abstract. I show how to use simple geometry to analyze pairwise and posi- tional voting rules as well as those many other decision procedures, such as runoffs and Approval Voting, that rely on these methods. The value of us- ing geometry is introduced with three approaches, which depict the profiles along with the election outcomes, that help us find new voting paradoxes, compute the likelihood of disagreement among various election outcomes, and explain problems such as the “paradox of voting.” This geometry even extends McGarvey’s theorem about possible pairwise election rankings to 1This research was supported by NSF grant DMI-0233798. My thanks to K. Arrow, N. Baigent, H. Nurmi, T. Ratliff, M. -
Comparative Survey of Multiwinner Election Methods
Smith typeset 18:38 18 Jun 2006 multiwinner survey Comparative survey of multiwinner election methods Warren D. Smith∗ [email protected] June 18, 2006 Abstract — After ≈ 100 years of relative sterility, recent to explain that. progress has both invented several new multiwinner vot- ing systems, and allowed us to begin to judge which of them is the best. We define and compare them here and 2 Statement of the problem come to the tentative conclusion that W.D.Smith’s uncon- ventional “asset voting” system is the best multiwinner There are V > 1 voters and C > 1 candidates. The voters election system known, and can be argued, for fundamen- cast some sort of “votes” that somehow provide information tal reasons, to be unmatchable by any conventional voting about their preferences among the candidates. Those votes system. are somehow processed by the election system, with the result being the selection of W “winning candidates,” 1 W < C. ≤ The question is what the votes and election system ought to 1 Multiwinner versus single-winner be. voting systems In my opinion, the situation with single-winner elections is 3 Properties for multiwinner elec- comparatively simple because there is a clear yardstick, called “Bayesian regret,” (which is a rigorously mathematically and tion systems statistically defined quantity) for deciding whether election As either an election system design tool, or as a method of system A is “better” than election system B. My computer discriminating better from worse election systems, one can try measured Bayesian regrets in a vast number of simulated elec- demanding the election system have certain properties. -
Descriptions of Single-Winner Voting Systems
Smith typeset 12:24 12 Jul 2006 voting systems Descriptions of single-winner voting systems Warren D. Smith∗ [email protected] July 12, 2006 Abstract — We describe over 40 single-winner voting sys- 6.18 Loring’s and Cretney’s Condorcet-IRV systems 17 tems, and elucidate their most important properties. 6.19 BTR-IRV: Bottom Two Runoff IRV . 17 6.20 CoombsSTVsystem(1954)[14] . 18 Contents 6.21 Nanson-Baldwin elimination (1882) [55] . 18 6.22 Rouse’s elimination method . 19 1 Notation 2 6.23 H.Raynaud’s elimination method . 19 2 Which vote-count and margins matrices are 6.24 Arrow-Raynaud pairwise elimination . 19 possible? 2 6.25Bucklin ...................... 20 3 Some combinatorics: the number of kinds of 6.26 Woodall’sDAC[83] . 21 votes 4 6.27 SPl: Sarvo-plurality(2004)[71] . 21 4 Systems that ignore the votes 4 6.28Smithset ..................... 21 4.1 Randomwinner.................. 4 6.29 Fishburnset[26] . 22 4.2 Optimumwinner ................. 4 4.3 Worstwinner ................... 5 6.30 Banksset[3][52] . 22 5 Systems in which each vote is the name of a 7 Systems in which each vote is a real N-vector 22 single candidate 5 7.1 Dabagh“voteand a half”(1934)[19] . 22 5.1 Plurality...................... 5 7.2 “Vote-for-and-against”(2004) . 22 5.2 Anti-plurality (sometimes called “Bullet vot- ing”; perhaps better would be “veto-voting”) . 5 7.3 Signed: G.A.W.Boehm’s“negative voting”(1976) 22 5.3 GR1: Gibbard’s Random Dictator (1977) [33] . 5 7.4 Cumulative voting (continuum version) . -
Table of Contents
TABLE OF CONTENTS Chapter 1. Introduction ......................................................................................................................3 1.1 This Report ...............................................................................................................................3 1.2 Our Goal ..................................................................................................................................4 Chapter 2. Single-Winner Systems ......................................................................................................5 2.1 Plurality – The American Status Quo .........................................................................................6 2.2 Delayed Runoff ........................................................................................................................8 2.3 Rating Election Methods ......................................................................................................... 10 2.3.1 Range Voting ............................................................................................................................. 10 2.3.2 Majority Judgment Voting ......................................................................................................... 11 2.3.3 Approval Voting ......................................................................................................................... 12 2.3.4 Benefits and Criticisms of Rating Systems ............................................................................... -
GROUP DECISION MAKING UNDER MULTIPLE CRITERIA SOCIAL CHOICE THEORY Content
GROUP DECISION MAKING UNDER MULTIPLE CRITERIA SOCIAL CHOICE THEORY Content Arrow's conditions for social walfare functions Criteria to evaluate the election methods Single transferable vote The Coombs method Arrow's conditions for social walfare functions Axiom 1: For all x and y, either x R y or y R x. [connected ] Axiom 2: For all x, y, and z, x R y and y R z imply x R z. [transitive] x R y means «x is preferred or indifferent to y» Conditions: 1. domain properties: The number of alternatives in A is greater than or equal to three. The social welfare function f is defined for all possible profiles of individual orderings. There are at least two individuals. 2. Positive Association of Social and Individual Values 3. Independence of Irrelevant Alternatives 4. Citizen's Sovereignty 5. Non-dictatorship Arrow's conditions POSITIVE ASSOCIATION OF SOCIAL AND INDIVIDUAL VALUES If the welfare function asserts that x is preferred to y for a given profile of individual preferences, it shall assert the same when the profile is modified as follows: (a) The individual paired comparisons between alternatives other than x are not changed, (b) Each individual paired comparison between x and any other alternative either remains unchanged or it is modified in x's favor. Monotonocity failure!* Upward monotonicity failure: Given the use of voting method and a ballot profile in which candidate X is the winner, X may nevertheless lose if some voters change their preference in the favor of X. Downward monotonicity failure: Given the use of voting method and a ballot profile B in which candidate X is a loser, X may nevertheless win if some voters change their preference against the favor of X. -
Alternative Systems Presented During the Public Consultation Phase
February 13, 2007 Dear Assembly Member, During the learning phase, members were introduced to the major families of electoral systems. These families which we categorized as Plurality (First past the post, block vote and single non-transferable vote); Majority Family (Two-Round System and Alternative Vote); Proportional Family (List Proportional Representation and Single Transferable Vote); Mixed Family (Mixed Member Proportional and Parallel) make up all the electoral systems used in democratic countries around the world today. In our consultation phase, we heard from many who recommended variations of these families. We also heard several unique systems not presently used in any nation to elect politicians. Since these were not discussed in the learning phase and are not included in the texts we used, George and I thought that it would be useful to create a document that summarizes each of them in the authors’ own words. It is attached to this letter. Should a group of you want to consider any of these models, as one of the two alternative models we design, we will have an opportunity to consider that after we have designed a first alternative in weekend two. If you have any questions about these systems, please do not hesitate to get in touch with me or any member of the learning team. Sincerely, Jonathan Rose Academic Director www.citizensassembly.gov.on.ca Alternative Systems presented during the Public Consultation Phase www.citizensassembly.gov.on.ca Table of Contents Section A: Approval Voting (Page 4-7) James Ronback, "One Candidate -
An Optimal Single-Winner Preferential Voting System Based on Game Theory
An Optimal Single-Winner Preferential Voting System Based on Game Theory Ronald L. Rivest and Emily Shen Abstract We describe an optimal single-winner preferential voting system, called the \GT method" because of its use of symmetric two-person zero-sum game theory to deter- mine the winner. Game theory is used not to describe voting as a multi-player game between voters, but rather to define when one voting system is better than another one. The cast ballots determine the payoff matrix, and optimal play corresponds to picking winners optimally. The GT method is a special case of the \maximal lottery methods" proposed by Fishburn [14], when the preference strength between two candidates is measured by just the margin between them. We suggest that such methods have been somewhat underappreciated and deserve further study. The GT system, essentially by definition, is optimal: no other preferential voting system can produce election outcomes that are preferred more by the voters, on the average, to those of the GT system. We also look at whether the GT system has several standard properties, such as monotonicity, Condorcet consistency, etc. We also briefly discuss a deterministic variant of GT, which we call GTD. We present empirical data comparing GT and GTD against other voting systems on simulated data. The GT system is not only theoretically interesting and optimal, but simple to use in practice. We feel that it can be recommended for practical use. 1 Introduction Voting systems have a rich history and are still being vigorously researched. We refer the reader to surveys and texts, such as B¨orgers[1], Brams [2], Brams and Fishburn [3], Fishburn [13], Kelly [18], and Tideman [34], for overviews. -
Approval Voting Under Dichotomous Preferences: a Catalogue of Characterizations
Draft – July 21, 2021 Approval Voting under Dichotomous Preferences: A Catalogue of Characterizations Florian Brandl Dominik Peters University of Bonn Harvard University [email protected] [email protected] Approval voting allows every voter to cast a ballot of approved alterna- tives and chooses the alternatives with the largest number of approvals. Due to its simplicity and superior theoretical properties it is a serious contender for use in real-world elections. We support this claim by giving eight char- acterizations of approval voting. All our results involve the reinforcement axiom, which requires choices to be consistent across different electorates. In addition, we consider strategyproofness, consistency with majority opin- ions, consistency under cloning alternatives, and invariance under removing inferior alternatives. We prove our results by reducing them to a single base theorem, for which we give a simple and intuitive proof. 1 Introduction Around the world, when electing a leader or a representative, plurality is by far the most common voting system: each voter casts a vote for a single candidate, and the candidate with the most votes is elected. In pioneering work, Brams and Fishburn (1983) proposed an alternative system: approval voting. Here, each voter may cast votes for an arbitrary number of candidates, and can thus choose whether to approve or disapprove of each candidate. The election is won by the candidate who is approved by the highest number of voters. Approval voting allows voters to be more expressive of their preferences, and it can avoid problems such as vote splitting, which are endemic to plurality voting.