# Rating RCV as Voting Method

Scholars rate Voting Methods using mathematically-derived Voting Method criteria, which describe desirable features of a method. No Ranked-Choice Voting (RCV) preference method can meet all of the criteria.

Many of the mathematical criteria by which Voting methods are compared were formulated for Voters with Ordinal preferences. If Voters vote according to the same Ordinal preferences in both rounds, criteria can be applied to Two-Round systems of Runoffs, and in that case, each of the criteria failed by RCV is also failed by the two-round system as they relate to Automatic Elimination of Trailing Candidates. Partial results exist for other models of Voter behavior in the Two-Round method.

The criteria that RCV meets, and those that it does not, are listed below.

Majority Criterion - This states that "if one candidate is preferred by an absolute majority of voters, then that candidate must win". RCV meets this Criterion.

Mutual Majority Criterion - This states that "if an absolute majority of voters prefer every member of a group of candidates to every candidate not in that group, then one of the preferred group must win". RCV meets this Criterion.

Later-No-Harm Criterion - This states that "if a voter alters the order of candidates lower in his/her preference (e.g. swapping the second and third preferences), then that does not affect the chances of the most preferred candidate being elected". IRV meets this Criterion.

Resolvability Criterion - This states that "the probability of an exact tie must diminish as more votes are cast". RCV meets this Criterion.

Condorcet Winner Criterion - This states that "if a candidate would win a head-to-head competition against every other candidate, then that candidate must win the overall election". It is incompatible with the Later-No-Harm Criterion, so RCV does not meet this Criterion.

RCV is more likely to Elect the Condorcet winner than Plurality Voting and Traditional Runoff Elections. The California Cities of Oakland, San Francisco, and San Leandro in 2010 provide an example; there were a total of four Elections in which the Plurality Voting Leader in first Choice Rankings was Defeated, and in each case the RCV Winner was the Condorcet Winner, including a San Francisco Election in which the RCV Winner was in Third Place in First choice rankings.

Condorcet Loser Criterion - This states that "if a candidate would lose a head-to-head competition against every other candidate, then that candidate must not win the overall election". RCV meets this Criterion.

Consistency Criterion - This states that "if dividing the electorate into two groups and running the same election separately with each group returns the same result for both groups, then the election over the whole electorate should return this result". IRV, like all Preferential Voting Methods which are not Positional, does not meet this Criterion.

Monotonicity Criterion - This states that "a voter can't harm a candidate's chances of winning by voting that candidate higher, or help a candidate by voting that candidate lower, while keeping the relative order of all the other candidates equal." RCV does not meet this criterion. Allard (Crispin Allard (January 1996). "Estimating the Probability of Monotonicity Failure in a UK General Election".) claims failure is unlikely, at a less than 0.03% chance per election. Some critics argue in turn that Allard's calculations are wrong and the probability of Monotonicity failure is much greater, at 14.5% under the Impartial Culture Election model in the Three-Candidate case, or 7-10% in the case of a Left-Right Spectrum. Lepelley et al. find a 2%-5% probability of Monotonicity failure under the same Election model as Allard.

Participation Criterion - This states that "the best way to help a candidate win must not be to abstain". RCV does not meet this Criterion: in some cases, the Voter's preferred Candidate can be best helped if the Voter does not vote at all. Depankar Ray finds a 50% probability that, when RCV Elects a different Candidate than Plurality, some Voters would have been better off not showing up.

Reversal Symmetry Criterion - This states that "if candidate A is the unique winner, and each Voter's Individual preferences are inverted, then A must not be Elected". RCV does not meet this Criterion: it is possible to construct an Election where Reversing the Order of every Ballot paper does not alter the Final Winner.

Independence of Irrelevant Alternatives Criterion - This states that "the election outcome remains the same even if a candidate who cannot win decides to run." RCV does not meet this Criterion In the general case, RCV can be susceptible to Strategic Nomination, whether or not a Candidate decides to Run at all can affect the Result even if the new Candidate cannot themselves win. This is much less likely to happen than under Plurality.

Independence of Clones Criterion - This states that "the election outcome remains the same even if an identical candidate who is equally preferred decides to run." RCv meets this Criterion.

NYC Wins When Everyone Can Vote! Michael H. Drucker

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Rating RCV as Voting Method

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