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. See one candidate. Reilly (2001, 33–34). 12 This is so named because of Smith (1973). 10 See Zavist and Tideman (1989). Schwartz (1986) refers to the Smith set as the Voting matters, Issue 29 2 James Green-Armytage: Condorcet-Hare Hybrid Methods candidate inside the set is preferred by some process repeats until there is such a candidate, majority of the voters to every candidate who is then declared the winner. outside the set. When there is a Condorcet Formally, in each round we determine winner, it is the only member of the Smith set. whether Formally, the Smith set is the set of candidates such that these conditions hold: : ,: 0 0. If so, then , and the process stops. Other- , , . wise, we perform these calculations: :, , . 1 0 , : 0 , , . 3 Method Definitions ∑ ,. Woodall method:13 Score candidates according argmin. ∞. to their elimination scores, and choose the Then, we proceed to the next round. Smith set candidate with best score. 15 That is, define each candidate’s elimination Smith-AV method: Eliminate candidates not in the Smith set, and then conduct an AV tally score as the round in which he is eliminated by among remaining candidates. AV. (The AV winner is not eliminated, so we set his score to .) If the Smith set has only one Tideman method:16 Alternate between elimin- member, then this is the Woodall winner; ating all candidates outside the Smith set, and otherwise, the winner is the candidate from eliminating the plurality loser, until one inside the Smith set who has the best candidate remains. elimination score. That is, as in Smith-AV, we begin by Formally, we begin with the definitions of eliminating all candidates outside the Smith set. the AV method and Smith set as given above. If this leaves only one candidate (a Condorcet Then, Υ 1 Ω ,, and winner), then he is elected. Otherwise, we eliminate the candidate with the fewest first argmaxΥ. choice votes. Then, we recalculate the Smith Benham method:14 Eliminate the plurality set, and eliminate any candidates who were in it loser until there is a Condorcet winner. before but are no longer in it as a result of the That is, if there is a Condorcet winner, he is plurality loser elimination. These two steps also the Woodall winner. Otherwise, the repeat until only one candidate (the winner) method eliminates the candidate with the remains. fewest first-choice votes, and checks to see Formally, in stage 1, we define or re-define whether there is a candidate who beats all other according to the following conditions: non-eliminated candidates pairwise. This , : 0, . , 0. : GETCHA set, and also defines another set called the GOCHA set, which is now also known as the ,: 0, . Schwartz set. The Schwartz set is the union of Then, we make the following adjustment to the minimal undominated sets, where an undominated vector: ∞. set is a set such that no member of the set is In stage 2, we perform the following pairwise-defeated by a non-member.