VoteFair representation ranking

VoteFair representation ranking is a Proportional-representation (PR) vote-counting method that uses ranked ballots and selects a candidate to win the second seat in a two-seat legislative district. The second-seat winner represents the voters who are not well-represented by the first-seat winner. Any single-winner election method that uses ranked ballots and pairwise counting can be used for the popularity calculations.

This method can be repeated to fill additional seats. For this purpose the STV version would be used for non-partisan elections, such as electing members of a city council. The default (non-STV) version assumes that other elections in the same region still use plurality voting and therefore causes politics to be dominated by two large political parties.

Description
This method first identifies which voters are well-represented by the first-seat winner. Then a reduced influence is calculated for these ballots. Their influence is determined by the extent to which they exceed the 50% majority minimum that is needed to elect the first-seat winner. The remaining ballots have full influence. Using these adjusted influence levels, the most popular of the remaining candidates becomes the second-seat winner.

This method ignores which political party each candidate is in, yet the second-seat winner is typically from a political party that is different from the first-seat winner.

In the default (non-STV) version, if a district has 5 seats, the third-seat winner and the fourth-seat winner are identified using the same steps that were used to fill the first two seats. In this case the fifth-seat winner would be determined by asking voters to indicate their favorite political party, calculating which party is most under-represented, looking at just the ballots that indicate that party as their favorite, and identifying the most popular candidate from that party.

Calculation steps
After the winner of the district's first seat is identified, the following steps calculate which candidate wins the second seat.


 * 1) Identify the ballots that rank the first-seat winner as their first — highest-ranked — choice.  (If there are no such ballots, no ballots will be ignored in the next step.)
 * 2) Completely ignore the ballots identified in step 1, and use the remaining ballots to identify the most popular candidate from among the remaining candidates. (If no ballots were identified in step 1, then use all the ballots.) This candidate will not necessarily be the second-seat winner.  Instead, this candidate is used in step 4 to identify which ballots are from voters who are well-represented by the first-seat winner.
 * 3) Again consider all the ballots.
 * 4) Identify the ballots in which the first-seat winner is preferred over the candidate identified in step 2.  This step identifies the ballots from voters who are well-represented by the first-seat winner.  Note that the only way for a voter to avoid having his or her ballot identified in this step is to express a preference that significantly reduces the chances that the preferred candidate will be ranked as most popular.
 * 5) Proportionally reduce the influence of the ballots identified in step 4. (This step reduces the influence of the voters who are well-represented by the first-most representative choice.) This calculation uses the following sub-steps:
 * 6) Count the number of ballots that were identified in step 4.
 * 7) Subtract half the number of total ballots.
 * 8) The result represents the ballot-number-based influence deserved for the ballots identified in step 4.
 * 9) Divide the ballot-number-based influence number by the number of ballots identified in step 4.
 * 10) The result is the fraction of a vote that is allowed for each ballot identified in step 4.
 * 11) Based on all the ballots, but with reduced influence for the ballots identified in step 4, identify the most popular candidate from among the remaining candidates. This candidate becomes the second-seat winner.

STV version
This calculation method can be extended to fill multiple seats. This usage produces proportional results such as achieved when using the Single Transferable Vote (STV). Unlike STV this method provides better protection against tactical voting.

For these purposes:


 * Use the method described above to fill the first two seats.
 * In step 1, the ballots to be ignored include any ballot on which any of the already-elected candidates appear at the top preference level.
 * In step 5-2 the word half is changed to two-thirds when filling the third seat, three-fourths when filling the fourth seat, four-fifths when filling the fifth seat, and so on.
 * When filling the third seat, a ballot is given zero influence if that ballot ranks both of the already-elected candidates higher than the candidate identified in step 1.  Similarly, when filling the fourth and fifth seats, a ballot is given zero influence if that ballot ranks all three or four (respectively) of the already-elected candidates higher than the candidate identified in step 1.  For this purpose, if a ballot ranks a not-yet-elected candidate at the same preference level as an already-elected candidate then that ballot is given zero influence during steps 4 and 5.

Example
The ballots below are interpreted as if the four cities were competing for two seats in a legislature.The Condorcet-Kemeny method identifies Nashville as the most popular candidate, meaning it wins the first seat.

VoteFair representation ranking identifies Memphis as the winner of the second seat.

The following details show how the second-seat winner is identified.


 * 26% of the ballots rank the most popular candidate (Nashville) as their first choice.
 * Looking at only the remaining 74% of the ballots, the most popular candidate (according to the Condorcet-Kemeny method) is Memphis.
 * 58% of the ballots rank Nashville higher than Memphis.
 * 58% exceeds 50% (the minimum majority) by 8% (the excess beyond majority).
 * 8% divided by 58% equals 0.1379 which is used as the weight for each of the 58% of the ballots that rank Nashville higher than Memphis.
 * Full weight for the ballots that do not rank Nashville higher than Memphis, combined with a weight of 0.1379 (about 14%) for the remaining ballots (that do rank Nashville higher than Memphis), identifies (according to the Condorcet-Kemeny method) the most popular candidate to be Memphis.

Memphis is declared the winner of the second seat. This candidate represents the voters who are not well-represented by the first-seat winner (Nashville).

History
VoteFair representation ranking was created by Richard Fobes while writing the book titled Ending The Hidden Unfairness In U.S. Elections, and is described in that book as part of the full VoteFair Ranking system.

This method has been used anonymously by non-governmental organizations that conduct their elections using the VoteFair.org website.