ISSN 1745-6231 Voting matters To advance the understanding of preferential voting systems published by The McDougall Trust Issue 29 October 2011 Contents Contents Editorial . i 1 James Green-Armytage: Four Condorcet-Hare Hybrid Methods for Single-Winner Elections 1 2 I.D. Hill: Party Lists and Preference Voting 15 3 Peter Emerson: The Matrix Vote: Electing an All-Party Coalition Cabinet 20 4 Svante Janson: Another Note on the Droop Quota and Rounding 32 5 Markus Schulze: Review — Voting Theory for Democracy 35 About the McDougall Trust (reg. charity no. 212151) The McDougall Trust is a charitable trust formed in 1948. The charity’s purposes as stated in its governing scheme of 1959 are to advance knowledge of and encourage the study of and research into: • political or economic science and functions of government and the services pro- vided to the community by public and voluntary organisations; and • methods of election of and the selection and government of representative organisations whether national, civic, commercial, industrial or social. The Trust’s work includes the maintenance and development of the Lakeman Library for Electoral Studies, a unique research resource, the production and publication of Representation: The Journal of Representative Democracy, and, of course, this publica- tion Voting matters, that examines the technical issues of the single transferable vote and related electoral systems. For further information on the Trust, please contact: The Secretary, McDougall Trust, 6 Chancel Street, London SE1 0UX, UK. Telephone: +44 (0)20 7620 1080 Facsimile: +44 (0)20 7928 1528 Email: [email protected] Web: www.mcdougall.org.uk For further information on this publication, please contact Nicolaus Tideman, the Editor, at Economics Department, Virginia Tech, Blacksburg VA 24061, USA, or by email at [email protected]. CONTENTS Editorial There are five items in this issue: • The first paper, by James Green-Armytage, describes four voting procedures for electing a single candidate from ranked preferences of voters. The four procedures differ very slightly from one another, and all are notable for electing the Condorcet winner when there is one and for strongly limiting the opportun- ity to benefit from strategic voting. • In the second paper, David Hill explains the virtues, in terms of representativeness and the minimization of wasted votes, of having voters rank parties and using transfers in the vote counting, if party lists are to be used to elect a set of representatives. • In the third paper, Peter Emerson makes a case for using the matrix vote to elect a collection of leaders and, with the same ballot, to name a person to each leadership position. • The fourth paper is a discussion by Svante Janson of the virtues of using an exact Droop quota rather than a rounded Droop quota. • The fifth and final item is Markus Schulze’s review of Voting Theory for Democracy, by Thomas Colignatus. Readers are reminded that views expressed in Voting matters by contributors do not neces- sarily reflect those of the McDougall Trust or its trustees. Voting matters, Issue 29 i Four Condorcet-Hare Hybrid Methods for Single-Winner Elections James Green-Armytage The weakness of this guideline is that it does [email protected] not specify what majority rule requires when there is no Condorcet winner. For these Abstract situations, the Smith set provides a useful This paper examines four single-winner generalization of the Condorcet winner election methods, denoted here as concept. The Smith set is the smallest set Woodall, Benham, Smith-AV, and such that any candidate in would win a one- Tideman, that all make use of both on-one race against any candidate not in . Condorcet’s pairwise comparison Thus the Smith principle, which requires voting principle and Hare’s elimination and rules to select winning candidates from the reallocation principle used in the Smith set, is an extension of the Condorcet alternative vote. These methods have principle that is applicable to all election many significant properties in common, 2 including Smith efficiency and relatively outcomes. For example, suppose that A is strong resistance to strategic manipu- preferred by a majority to B, B is preferred by a lation, though they differ with regard to majority to C, C is preferred by a majority to A, the minor properties of ‘Smith-IIA’ and and all three of these candidates are preferred ‘mono-add-plump’. by majorities to D. In this case, electing A, B, or C is consistent with the majority rule 1 Introduction guideline provided by the Smith principle, but electing D is not. Several election methods have been The concept of majority rule is trickier than proposed that satisfy the Smith principle. most people realize. When there are only two Among them are ranked pairs,3 beatpath,4 candidates in an election, then its meaning is river,5 Kemeny,6 Nanson,7 and Copeland.8 quite clear: it tells us that the candidate with the However, the four methods on which this paper most votes is elected. However, when there are focuses possess another property, in addition to more than two candidates, and no single Smith efficiency, that makes them particularly candidate is the first choice of a majority, the interesting: they appear to be unusually meaning is no longer obvious. resistant to strategic manipulation. Therefore, The Condorcet principle1 offers a plausible if a society wishes to choose among multiple guideline for the meaning of majority rule in options by majority rule given one balloting, multi-candidate elections: if voters rank and if it wishes to minimize the probability that candidates in order of preference, and these rankings indicate that there is a candidate who ––––––––––––––––– would win a majority of votes in a one-on-one 2 Smith (1973) refers to his idea as a generalization race against any other candidate on the ballot (a of Condorcet consistency. 3 Defined in Tideman (1987). Condorcet winner), then we may interpret 4 ‘majority rule’ as requiring his election. Defined in Schulze (2003). 5 Defined in Heitzig (2004). 6 Defined in Kemeny (1959). ––––––––––––––––– 7 See Tideman (2006), page 201–203. 1 Condorcet (1785) defines this principle. 8 Defined in Copeland (1951). 1 James Green-Armytage: Condorcet-Hare Hybrid Methods voters will have an incentive to behave voters who rank candidate ahead of candidate strategically, these methods are worthy of . If , then pairwise-beats . strong consideration. These four methods also share the character- Condorcet winner: A candidate who wins all istic of employing the ‘Hare principle’, that is, of his pairwise comparisons. Formally, is a the principle of eliminating the candidate with Condorcet winner if and only if the fewest first-choice votes and reallocating ,. 9 those votes to other candidates. Condorcet method: Any single-winner voting I will use the names Woodall, Benham, rule that always elects the Condorcet winner Smith-AV, and Tideman to refer to these rules, when one exists. as they do not have standard names. They are deeply similar to one another and will choose Majority rule cycle: A situation in which each the same winner in the vast majority of cases, of the candidates suffers at least one pairwise but they are not identical. The purpose of this defeat, so that there is no Condorcet winner. paper is to provide a solid understanding of Formally, , : . how these methods work, how they differ from The Alternative Vote (AV):11 The candidate one another, and how they compare to other with the fewest first choice votes (ballots single-winner methods. ranking the candidate above all others in the race) is eliminated. The process is repeated 2 Preliminary Definitions until only one candidate remains. Formally, in each round 1,…,1, we Assume that there are candidates and perform the following operations: voters. Let be a tiebreaking vector that gives 1 0 a unique score 0,1 to each candidate , : 0 ,,. 1,…,; can be random, predetermined, or determined by a tie-breaking ranking of ∑ ,. candidates.10 Let be a vector of candidate argmin. ∞. Ω . eliminations, such that is initially set to zero for each candidate 1,…,. Let denote After round 1, argmin, and Ω . the winning candidate. Let be the utility of Here, is a by matrix indicating voter for candidate . Let be the ranking that voter gives to candidate (such that individual voters’ top choices. is a length- lower-numbered rankings are better). All vector of the candidates’ first choice vote totals, voting methods described in this paper, with the which incorporates the unique fractional values exception of approval voting and range voting, in the tiebreaking vector in order to ensure begin with the voters ranking the candidates in that there will not be a tie for plurality loser. order of preference. Infinity can be added to the values of eliminated candidates to prevent them from Pairwise comparison: An imaginary head-to- being identified as the plurality loser in head contest between two candidates, in which subsequent rounds. The vector Ω gives an each voter is assumed to vote for the candidate ‘elimination score’ for each candidate, which whom he gives a better ranking to. Formally, will be used by the Woodall method. let ∑ 1 be the number of Smith set:12 Or, the ‘minimal dominant set’. The smallest set of candidates such that every ––––––––––––––––– 9 Thomas Hare offered the first voting procedure that ––––––––––––––––– included the iterative transfer of votes from plurality 11 Also known as instant runoff voting (IRV) and as losers to candidates ranked next on ballots. See the Hare method, the alternative vote (AV) is the Hoag and Hallett (1926, 162–95). The first person application of proportional representation by the to apply the ‘Hare principle’ to the election of a single transferable vote (STV) to the case of electing single candidate was Robert Ware, in 1871.