The terminology can be confusing, so this is a brief explanation of the meaning of each of the terms above and the relationships between them.
Preferential Voting (PV), also known as Ranked Choice Voting (RCV), allows the voter to indicate preferences on the ballot by ranking a first choice, second choice, et cetera, for as many choices as the voter has. Some implementations of this may limit the maximum number of choices that may be indicated, usually due to constraints of the particular voting technology in use. Here is an example:
1st 2nd 3rd Candidate A X Candidate B X Candidate C X
Linguistically, "Ranked Choice Voting" better describes the manner in which voters mark their ballots. The traditional mark-one-choice ballot could be thought of as indicating your (one) preference, and thus all voting is preferential, but the term is never used for these mark-one-choice ballots. Indeed, Robert's Rules of Order Newly Revised (RONR), the most widely used parliamentary authority in the USA, uses the term "Preferential Voting" for any voting system where voters can indicate multiple choices in a given race. (Robert's Rules of Order has gone by three names over 10 editions from 1876 to 2000, with PV originally mentioned in 1970 (7th edition) when the "Newly Revised" version of the name was first used.)
Note that some other parliamentary authorities (The Standard Code of Parliamentary Procedure by Alice Sturgis and AIP (2001), and Meeting Procedures by James Lochrie (2003)) use the PV terminology in the IRV sense (see below).
In the RONR and linguistic sense, RCV = PV, and either term only conveys in its meaning the manner in which the votes are cast, not the manner in which they are counted to determine the winner(s).
San Francisco adopted what they called "Ranked Choice Voting" and first voted with this system in November, 2004. The Canadian Wheat Board has used for several years what they call "Preferential Voting." Both SF and the CWB count the votes per the Instant Runoff Voting method described below.
The same manner of marking ballots with ranked choices can be counted in different ways to determine a winner. The most common ways to determine the winner(s) from a given set of ranked-choice ballots are: instant runoff voting, Borda count, and Condorcet.
Instant runoff voting (IRV) is descriptively named, and uses the ranked-choice ballots to simulate a series of runoff elections, each time dropping the lowest vote-getter from the round before, and re-casting each ballot for the highest ranked choice of those still in the running. This is repeated until a candidate receives a majority vote. If in a runoff a ballot has no choice marked for a remaining candidate, the voter in essence is abstaining from voting in that runoff, and is not counted in computing a majority, because a majority is defined as more than half of the votes cast.
RONR illustrates how PV can be done by describing what is called IRV, but never uses the IRV terminology. It also shows how the IRV method can be extended to elect a multi-member body. To elect 3 at-large members to a city council with 7 candidates running, use a series of instant runoffs to eliminate lowest vote-getters until only 3 candidates remain.
The Borda count (BC) assigns more "points" to higher-ranked choices, and the candidate with the most points wins. E.g.,
5 voters rank A, then B, then C;
4 voters rank C, then A, then B;
3 voters rank B, then C, then A.
Each ballot's first choice gets 2 points, second choice gets 1 point, and last choice gets zero. (In general, with N candidates including all write-ins, the first choice gets N-1 points, and each subsequent choice gets one less point.) Thus:
5 voters give A a total of 10 points, and B a total of 5 points;
4 voters give C a total of 8 points, and A a total of 4 points;
3 voters give B a total of 6 points, and C a total of 3 points.
A gets 10+4 = 14 points;
B gets 5+6 = 11 points;
C gets 8+3 = 11 points.
Candidate A got the most points and wins. Borda devised this system in 1770.
A Condorcet Method (CM) winner is a candidate that pairwise beats every other candidate. X pairwise beats Y if more voters rank X higher than Y.
Proportional Representation (PR) also uses PV/RCV ballots, but is specifically for a multi-member body, such as electing 10 at-large members to a city council. If 10 are to be elected and 20 are running, a voter ranks the 20 as 1st to 20th choice. Counting first choices, any candidate getting above a certain threshold (quota) is elected. If the total number of first-choice votes is 1000, then the threshold is set such that if the 10 winners each barely got above the threshold count, all the remaining votes would be less than the minimum winning amount, and even one candidate getting them all would not be elected.
In general, if there are P positions to be elected and F first-choice votes cast, the threshold T that a winner must get more than is:
T = F / (P + 1), and the next whole number more than T is what is required to win.
(If there is just one position, T = F / 2, and a winner must get more than half, the very definition of majority. "Majority" is at times defined erroneously as 50% + 1 or 51%, and due to similar problems, the general threshold formula above should be used instead of any variation, such as needing to get (F/(P+1))+1 to win.)
If winning candidates kept their excess (surplus) votes (those more than necessary to win), then it may be that not all positions are filled. If this is the case, the excess votes are transferred to non-winners as indicated by secondary rankings, until all positions are filled.
One way to transfer excess votes is as follows. Let W = the total number of votes a winner received. Let E = the number of excess votes. Let B = the number of ballots ranking a candidate highest among all non-initial-winners. Transfer (E x (B/W)) votes to that candidate. If done with calculator or computer, fractional vote transfers can be made.
Secondary winners are those who thus obtain more than the threshold, but this may still not fill all the positions. At this point an IRV-style series of instant runoffs are simulated between the still-unelected candidates, dropping the lowest vote-getter in each round until there remain only as many as there are positions still to be filled.
Cumulative Voting also is use for electing to a multi-member body, such as electing 10 at-large members to a city council. It allows voters in such a case to cast 10 votes, and multiple votes can be cast per candidate, such as 10 for one candidate, or 7 for A, 2 for B and 1 for C.
With Approval Voting for an election of a single person from a field of 5, a voter may vote for as many of the 5 as desired (approved), but only one vote per candidate. The one with the most votes wins.
Instant Runoff Voting
Borda Count
Condorcet Method
Proportional Representation
Cumulative Voting
Approval Voting
Miscellaneous
Amy, Douglas J., Behind the Ballot Box: A Citizen's Guide to Voting Systems. (Praeger Publishing, 2000). [Search inside]
Amy, Douglas J., Proportional Representation: The Case for a Better Election System. (Crescent Street Press).
Amy, Douglas J., Real Choices / New Voices: : The Case for Proportional Representation Elections in the United States, 2nd ed. (Columbia University Press, 1993). [Search inside]
Hill, Steven, Fixing Elections: The Failure of America's Winner-Take-All Politics. (Routledge, 2003).
Lochrie, James, Meeting Procedures. (Lanham, MD: Scarecrow Press, 2003).
Robert, Henry M. et al, Robert's Rules of Order Newly Revised, 10th ed. (Cambridge Mass.: Perseus Publishing, 2000). [Sample pages]
Saari, Donald G., Basic Geometry of Voting. (Heidelberg, Ger.: Springer-Verlag, 1995). [Search inside]
Saari, Donald G., Chaotic Elections! A Mathematician Looks at Voting. (American Mathematical Society, 2001). [Search inside]