Ranked Choice Including Pairwise Elimination

Ranked Choice Including Pairwise Elimination (abbreviated as RCIPE which is pronounced "recipe") is an election vote-counting method that uses ranked ballots and eliminates pairwise losing candidates (elimination-round-specific Condorcet losers) when they occur, and otherwise eliminates the candidate who currently has the smallest top-choice count.

This method modifies instant runoff voting (IRV) by adding the elimination of pairwise losing candidates. This addition reduces the failure rate for the Independence of Irrelevant Alternatives (IIA), which is the kind of failure that occurred in the 2009 mayoral election in Burlington, Vermont. This check for pairwise losing candidates considers all the marks on all the ballots, which contrasts with IRV, which does not consider all the marks on all the ballots.

This method further modifies simple IRV by specifying how to handle ballots on which the voter has marked more than one candidate at the same ranking level.

The RCIPE STV method extends the single-winner RCIPE method to fill multiple equivalent seats.

Description
Voters rank the candidates using as many ranking levels as there are candidates, or at least five ranking levels. Using fewer ranking levels than candidates is useful on paper ballots where space is limited, and in elections where there are numerous candidates who are unlikely to win.

This method eliminates one candidate at a time. The final remaining (not-yet-eliminated) candidate is declared to be the winner.

If an elimination round has a Condorcet loser (a pairwise losing candidate), this candidate is eliminated as the least-popular candidate. The Condorcet loser is the candidate who would lose every one-on-one contest against each and every other candidate.

If an elimination round does not have a Condorcet loser, the candidate who has the smallest top-choice count is eliminated. A candidate's top-choice count is the count of how many ballots rank that candidate highest compared to the other remaining candidates.

Unlike instant-runoff voting, which ends when a candidate reaches majority support, the eliminations continue until only a single candidate remains.

The last candidate to be eliminated is the runner-up candidate. If this counting method is used in the primary election of a major political party, and if the runoff or "general" election is counted in a way that is not vulnerable to vote splitting, then ideally the runner-up candidate would move to the runoff or general election along with the primary-election winner. Very small political parties would not qualify to move their runner-up candidate to the runoff or general election.

Importantly, the runner-up candidate does not deserve to win any kind of elected seat. Instead, the RCIPE STV version should be used for elections that fill multiple seats, such as on a non-partisan city council or a dual-member legislative district.

Ballot Robustness
To avoid spoiled ballots in elections where the voter uses a pen or marker to mark their paper ballot, more than one candidate can be marked at the same ranking level. When an elimination round involves a ballot that has two or more remaining highest-ranked candidates, that ballot's single vote is split equally among these candidates. This splitting of a single vote can be done using fractions or decimal numbers that do not exceed a total of one vote per ballot. If a law does not permit the use of fractions or decimal numbers, the ballots that have the same shared ranking can be distributed uniformly among the same-ranked candidates, such as alternating which candidate gets each successive ballot on which the same two candidates are highest-ranked at the same level. Regardless of which method is used, each elimination round re-calculates which ballots support which candidates.

Also to avoid spoiled ballots, if a voter marks more than one ranking level for the same candidate, only the highest-marked ranking level is used during counting.

The choice of how to handle a ballot on which a voter does not mark any ovals for a candidate depends on how write-in candidates are handled. If write-in candidates are not allowed, an unmarked candidate can be ranked at the ranking level below the lowest ranking level shown on the ballot. If write-in candidates are allowed, an unmarked candidate can be ranked at the lowest ranking level shown on the ballot, and that level also would be used for a write-in candidate whose name does not appear on that ballot.

Tie breaker
If two or more candidates have the same smallest top-choice count, this tie is resolved by eliminating the candidate with the largest pairwise opposition count, which is determined by counting on each ballot the number of not-yet-eliminated tied candidates who are ranked above that candidate, and adding these numbers across all the ballots.

If there is a tie for the largest pairwise opposition count, this tie is resolved by eliminating the candidate with the smallest pairwise support count, which is determined by counting on each ballot the number of not-yet-eliminated tied candidates who are ranked above that candidate, and adding these numbers across all the ballots.

Note that the pairwise opposition count and pairwise support count are calculated using only the candidates who are currently tied. This means that ballot information about eliminated candidates and not-tied candidates is ignored when resolving ties.

If there is also a tie for the smallest pairwise support count, then another tie-breaking method is needed to identify which of the still-tied candidates to eliminate.

Example
These ballot preferences are converted into pairwise counts and displayed in the following tally table.

In the first elimination round, Memphis is eliminated because it is the pairwise loser. This means Memphis loses the pairwise contest against Nashville (42% to 58%), and Memphis loses the pairwise contest against Chattanooga (42% to 58%), and Memphis loses the pairwise contest against Knoxville (42% to 58%).

If there had not been a pairwise loser, Chattanooga would have been eliminated (instead of Memphis) because it has the smallest number of ballots that rank Chattanooga as the first choice.

In the second elimination round, Knoxville is eliminated because it is the pairwise loser. This means Knoxville loses the pairwise contest against Nashville (32% to 68%), and Knoxville loses the pairwise contest against Chattanooga (32% to 68%).

If there had not been a Condorcet loser, Nashville would have been eliminated. The voters who marked Chattanooga as their first choice marked Knoxville as their second choice so in this elimination round Knoxville has 32% support (15% plus 17%). Memphis continues to have 42% support, and Nashville continues to have 26% support, which leaves Nashville with the smallest top-choice support.

In the third and final elimination round, Chattanooga is eliminated because it is the pairwise loser. Specifically Chattanooga loses its pairwise contest against Nashville (17% to 83%).

When there are only two candidates, and they are not tied, the Condorcet loser is always the same as the candidate who has the fewest top-choice votes.

The only remaining candidate is Nashville, so it is declared the winner.

Chattanooga is the runner-up candidate because it was the last to be eliminated.

Mathematical criteria
The frequency with which this method passes or fails each of the following criteria have not been estimated. This method is similar to the Arrow-Raynaud method so their frequency values are likely to be similar.

This method always passes the following criteria.


 * Condorcet loser: pass
 * Majority: pass
 * Majority loser: pass
 * Mutual majority: pass
 * Resolvable: pass
 * Polytime: pass

This method sometimes fails the following criteria.


 * Condorcet: fail
 * Smith/ISDA: fail
 * Cloneproof: fail
 * LIIA: fail
 * IIA: fail
 * Monotone: fail
 * Consistency: fail
 * Reversal symmetry: fail
 * Later no harm: fail
 * Later no help: fail
 * Burying: fail
 * Participation: fail
 * No favorite betrayal: fail
 * Summable: fail

RCIPE STV
RCIPE STV is the multi-winner version of the RCIPE method, which means it functions like the Single Transferable Vote (STV) for electing multiple legislators within the same district, and electing non-partisan members of a city council. RCIPE STV offers these advantages over plain STV:

These advantages occur because:
 * A voter can mark two or more candidates at the same ranking level. This flexibility allows voters to fully rank all the candidates, including the ability to rank the voter's most-disliked candidate lower than all other candidates, even when the number of ranking levels is fewer than the number of candidates.
 * The counting process considers all the marks on all the ballots. This deeper counting is done when identifying pairwise losing candidates.  It prevents a voter's ballot transfer from getting stuck on an unpopular pairwise-losing candidate while other ballots determine which other candidates win seats and which other candidates get eliminated.
 * Changing the ballot-counting sequence does not change who wins. In contrast, plain STV can elect different winners if the ballots are supplied in a different sequence.
 * Vote transfer counts are re-calculated after each candidate is elected.
 * If a counting round does not elect a candidate, the pairwise losing candidate is eliminated.  If there is no pairwise losing candidate, the candidate with the lowest vote transfer count is eliminated.


 * During pairwise counting all the ballots are counted, but the ballots that have zero influence do not contribute any votes to either side of the one-on-one matches.
 * If a full-influence ballot ranks two or more remaining (not-yet-elected and not-yet-eliminated) candidates at the same preference level, and if there are not any remaining candidates ranked higher on this ballot, then this ballot is grouped with other similar (although not necessarily identical) ballots and their influence counts are equally split among the remaining candidates who are ranked at that shared preference level.  For example, if candidates A and B have been elected or eliminated, and a ballot ranks candidate A highest and ranks candidates B, C, and D at the next-highest level, and another ballot ranks candidate B highest and ranks candidates A, C, and D at the next-highest level, then one of these two ballots transfers to candidate C and the other ballot transfers to candidate D.
 * In a counting round that ends with a candidate getting elected, the specific supporting ballots that are changed from full influence to zero influence are chosen to be equally spaced from one another in the supplied ballot sequence, without including the already-zero-influence ballots in the equal-spacing calculations. This rule causes the calculations to yield the same winners if the same ballots were supplied in a different sequence.
 * Ties are resolved using pairwise elimination.

If a jurisdiction has laws that allow a ballot to have decimal influence amounts that range between zero and one, the above rules can be simplified to use decimal influence values.