Counting Methods Help

OpaVote implements a wide variety of methods for counting votes. Available methods include traditional methods where a voter simply checks a box to select a candidate and more sophisticated methods where a voter can rank the candidates in order of preference.

For ease of navigation, the available methods are grouped into categories:

  • Traditional Methods — Methods where the voter simply checks a box to select a candidate.
  • Ranked-Choice Voting — Methods where voters rank candidates instead of selecting them.
  • Instant Runoff Voting — Ranked-choice voting for electing a single candidate.
  • Single Transferable Vote — Ranked-choice voting for electing multiple candidates with proportional representation.
  • Other Methods — Other methods where voters rank candidates that do not fit into the above categories and also approval voting.

For people who are unsure of which counting method to use, OpaVote provides the following recommendations:

  • For electing one candidate, we recommend instant runoff voting for most organizations. If the members of your organization are comfortable with a more complicated counting system, then we recommend Condorcet voting.
  • For electing more than one candidate, we recommend Scottish STV for most organizations. If the members of your organization are comfortable with a more complicated counting system, then we recommend Meek STV.

An exception to the above, if there are only two candidates in the election, then use plurality voting because there is no benefit to using a ranked method.

Traditional Methods

There are two types of traditional methods. The first is for electing a single candidate, and the second is for electing a group of candidates (such as a committee or council).

Plurality/FPTP/SNTV

This is the most common type of voting. Each voter selects one candidate, and the candidate with the largest number of votes is the winner. This method is known as plurality voting, first past the post, and the single non-transferable vote (in contrast to transferable vote methods below).

Although commonly used, this is not a good method for counting votes. When there are more than two candidates, the winner of the election may have much less than a majority of votes, and other counting methods below will generally provide a better result. For example, FairVote has a comparison of plurality voting and instant runoff voting.

Plurality at Large/Block Voting/MNTV

This method can be used to elect a council or committee. Suppose a committee of four is to be elected. Each voter selects up to four candidates, and the four candidates receiving the largest number of votes are the winners. This method is known as plurality at large voting, block (or bloc) voting, and the multiple non-transferable vote.

This is also not a good method for counting votes because it allows a majority of voters to control the entire committee. Suppose that a city is electing a council of 10 and that 51% of the city is party A and 49% is party B. Under this method, party A can elect its candidates to all 10 seats of the council and party B will be unrepresented even though it is supported by 49% of the voters.

Ranked-Choice Voting (RCV)

Ranked-choice voting (RCV) is a style of voting where voters rank candidates instead of selecting a candidate (e.g., checking a box). RCV includes both instant runoff voting (IRV) and the single transferable vote (STV), which are discussed in more detail below. Generally, IRV is for electing a single candidate and STV is used for electing multiple candidates with proportional representation.

Some of the RCV methods described below have options, and these options are described at the bottom of this page. Note that the counting options are only available when doing an OpaVote count. If you are doing an OpaVote election or poll and you need to specify counting options, then you need to download the ballots and create an OpaVote count to access the counting options.

Instant Runoff Voting (IRV)

A runoff election is where a second election is held at a later time when none of the candidates in a first election receive a majority of the votes. Instant runoff voting (IRV) provides the same benefit of a runoff election (a majority winner) but does in a single election through the use of a ranked ballot.

With IRV, a voter ranks the candidates in order of preference (e.g., 1, 2, 3, etc.). The first step in the counting is to count the first place votes. If a candidate receives a majority of first place votes, then he or she is the winner. If not, the candidate with the fewest first place votes is eliminated and those votes are transferred to their second choices. This elimination process is repeated until a majority winner is obtained.

OpaVote provides a general version of IRV with several counting options, and specific versions of IRV as used in San Francisco and Oakland, California.

Instant Runoff Voting

The general version of IRV has several options (but note that these options are available only for counts and not for elections and polls). By default, IRV uses the following options:

  • Zero for Candidate Elimination
  • Backward for Tie Breaking
  • Skip for Remove Overvotes
  • Skip for Remove Undervotes

San Francisco RCV

This method implements IRV as used by the city of San Francisco (which calls it RCV). San Francisco enacted IRV in 2002, its first election with IRV was in 2004, and it has been used annually since then.

The differences with the general method are the following options:

  • Losers for Candidate Elimination
  • Random for Tie Breaking
  • Stop for Remove Overvotes
  • Skip for Remove Undervotes

Oakland RCV

This method implements IRV as used by the cities of Oakland, San Leandro, and Berkeley. Oakland approved use of IRV in 2006, and San Leandro and Berkeley approved use of IRV in 2010. All three cities had their first IRV elections in 2010.

The differences with the general method are the following options:

  • Zero for Candidate Elimination
  • Random for Tie Breaking
  • Stop for Remove Overvotes
  • Skip for Remove Undervotes

Single Transferable Vote (STV)

The single transferable vote (STV) is used to elect a group of candidates, such as a council or committee, and provides proportional representation of the voters.

STV also uses ranked ballots and also eliminates last place candidates similar to IRV. In addition, STV transfers surplus votes from candidates who have too many votes. How a candidate can have too many votes is best exaplined with an example. Suppose that three candidates are competing for two seats, and the votes have the following distribution:

  • 95% of the voters rank A first and B second
  • 2% of the voters rank B first
  • 3% of the voters rank C first

If IRV were used to select the two winners, then A and C would be the winners since B has the fewest first place votes. Looking at the votes, however, 97% of the voters prefer B over C. To get a better outcome, one could say that A has too many (or surplus) votes, and some of those surplus votes can be transferred to their second choices. If this happens, then B will have more votes than C, and B will be the second winner instead of C.

To determine whether a candidate has surplus votes, a winning threshold is used. The winning threshold (called a Droop threshold) is computed as

threshold = number of votes number of seats + 1 + 1

and then dropping and fraction. When the number of seats is 1, this is the same as requiring a majority to elect a single winner.

Counting votes with STV generally proceeds as follows:

  1. Count the first place votes.
  2. Transfer votes from one candidate to other candidates:
    • If one or more candidates have surplus votes (votes in excess of the winning threshold), then transfer surplus votes to their next choices.
    • Otherwise, eliminate the last place candidate and transfer those votes.
  3. If not all seats have been filled, then go to step 2.

Each of the STV methods below specify additional details (or modify) these three steps.

Scottish STV

The Scottish STV rules are recommended for most organizations because the rules are well defined (by Scottish law) and provide a straightforward implementation of STV that is easier to understand. Scotland enacted STV in 2007 and had its first election that year.

Scottish STV has the following features:

  • Single for Candidate Elimination
  • Static and Whole Threshold
  • No delayed transfer of surplus votes
  • Precision of 5
  • Backward for Tie Breaking

Meek STV

Meek STV is recommended for organizations whose members are comfortable with a more complicated counting method. Meek STV provides more accurate proportional representation than other STV methods, but takes more effort to understand.

One advantage of Meek STV is that when a candidate is eliminated from the election, the votes are counted as if the candidate was never in the election at all and the order of elimination cannot effect the outcome. With other STV methods, the order of elimination can effect the outcome.

Another advantage of Meek STV is that surplus votes are transferred in a better way. With other STV methods, surplus votes are not ever transferred to a candidate who has already won. With Meek STV, surplus votes are always transferred to the next candidate on the ballot.

For additional details about Meek STV, see the first issue of Voting Matters.

ERS97 STV

The Electoral Reform Society of the United Kingdom has been providing STV rules since at least as early as 1955 and its latest rules from 1997 are commonly referred to as the ERS97 rules. These rules are widely used in the UK. The ERS97 rules are the most complicated rules of all the STV rules provided by OpaVote. For this reason, we do not recommend their use.

Minneapolis STV

Minneapolis enacted STV in 2006 and had its first STV election in 2009. The Minneapolis rules are very similar to the Scottish rules.

Minneapolis STV has the following features:

  • Losers for Candidate Elimination
  • Static and Whole Threshold
  • Delayed transfer of surplus votes
  • Precision of 4
  • Random Tie Breaking

N. Ireland STV

The N. Ireland STV rules (statute) are similar to the ERS97 rules, but significantly simpler.

Warren STV

Warren STV is very similar to Meek STV. To learn more about the differences, see the first issue of Voting Matters.

Cambridge STV

The City of Cambridge, Massachusetts has used the single transferable vote to elect its city council and school committee since 1941 (statute). Note that the statute allows Cambridge to use any method for transfering surplus votes that was in use in 1938, and Cambridge has chosen to use the Cincinnati method.

Since candidates with fewer than 50 votes are eliminated, this method should not be used with a small number of ballots.

The City of Cambridge describes the Cincinnati method as follows:

The ballots of the candidate who has a surplus are numbered sequentially in the order in which they have been counted (that is, in the sequence dictated by the random draw of precincts) and then every nth ballot is drawn and transferred to a continuing candidate until the original candidate is credited with ballots equaling no more than quota. n is nearest whole number computed by the formula

n = Candidate's Total Ballots
Surplus Ballots.

A ballot selected by this method that does not show a preference for a continuing candidate is skipped and remains with the original candidate. If not enough ballots are removed when ballots n, 2n, 3n, .... have been transferred, the sequence starts again with n+1, 2n+1, 3n+1, ....

Fractional Transfer STV

The fractional transfer STV method is a generalization of the Scottish STV rules, and you can use the options to customize your counting rules (but note that these options are available only for counts and not for elections and polls).

Random Transfer STV

Except for Cambridge STV, all of the STV counting rules above use fractional votes for transferring surplus votes. This is a general method that allows you transfer surplus votes as whole votes instead of fractional votes. This method is not recommended for actual elections, but you may find it interesting to compare with other methods.

Note that the transfers of votes are not actually random, but the outcome of an election can depend on the order of the ballots. If you shuffle the ballots and recount, you could obtain a different winner. All of the methods above (except Cambridge STV) will always produce the same winners after shuffling ballots.

Other Methods

OpaVote also provides other counting methods that are not typically classified as IRV or STV methods, though all of them, except for approval voting, use ranked ballots.

Approval Voting

Approval voting is a very simple method, both in terms of the voter casting a vote and counting all of the votes. Each voter can approve (by checking a box) any number of candidates. The candidate who receives the largest number of approvals is the winner.

Condorcet Voting

Condorcet voting is for electing a single candidate using a ranked ballot. If the members of your organization are comfortable with a more complicated counting system, then we recommend Condorcet voting. The winner in Condorcet voting is the candidate who beats all other candidate in pairwise elections. Since we have ranked ballots, it is straightforward to determine for each pair of candidates which one would win an election between the two.

With Condorcet voting, it can happen that there is no outright pairwise winner. It is possible that candidate A beats B, B beats C, and C beats A. In this situation, a Condorcet completion method (similar to a tie breaker) can be used to select the winner. The set of candidates in the cycle is commonly referred to as the Smith set, and OpaVote provides several Condorcet variants for choosing a winner from the Smith set:

  • Condorcet Beathpath — See here.
  • Condorcet IRV — The winner is chosen from the Smith set using IRV.
  • Condorcet Borda — The winner is chosen from the Smith set using the Borda Count.
  • Condorcet Copeland — See here.

Coombs Method

The Coombs method is just like IRV, except that a different technique is used to select a candidate to eliminate. With IRV, the candidate with the fewest votes is eliminated at a round of the count. With Coombs, the candidate with the most last place votes is eliminated.

Borda Count

The Borda count uses ranked ballots, but votes are not transferable. Instead, a score is generated for each candidate from the ranked ballots, and the candidate with the highest score is the winner. If there are N candidates in the election, then each candidate gets N-1 points for each first place vote, N-2 points for each second place vote, and so forth.

The Borda Count has an option called Ballot Completion. Because of the nature of the Borda Count, voters have incentive to rank only their first choice candidate so the second choice cannot cause the first choice to lose. Turning ballot completion on reduces this incentive. With ballot completion, a ballot that ranks fewer than all the candidates (an imcomplete ballot) is completed by adding the remaining candidates. For example, if there are 5 candidates running and a voter specifies only a first choice, the other four candidates will be added to the ballot as all tied for second.

Bucklin System

The Bucklin system is also known as the Grand Junction system (where it was once used) and American preferential voting. The voters rank the candidates and a candidate receiving a majority of first choices is declared the winner. If no candidate has a majority of first choices, then a candidate receiving a majority of first and second choices is the winner. If more than one candidate has a majority of first and second choices, then the candidate having the most first choices is the winner. This process is repeated for further choices as necessary.

Counting Options

  • Remove Overvotes — If a voter is casting a vote on paper, it is possible that the voter might accidentally give two or more candidates the same ranking. E.g., the voter might rank Alice first, both Bob and Charlie second, and Denise third. This is called an overvote. When running an election or poll with OpaVote, voters can only cast valid votes and there will not be any overvotes. If you collect ballots on your own, however, then you may have overvotes. OpaVote provides three options for handling overvotes.
    • Skip — The overvote is skipped over. E.g., for the ballot above, it would be treated as Alice was ranked first and Denise was ranked second.
    • Stop — Counting is stopped at the overvote. E.g., for the ballot above, it would be treated as Alice was ranked first and with no other rankings.
    • Void — The entire ballot is treated as void and is not used.
  • Remove Undervotes — If a voter is casting a vote on paper, it is also possible that the voter might accidentally skip a ranking. E.g., the voter might rank Alice first, rank Bob third, and not rank anyone second. This is called an undervote. When running an election or poll with OpaVote, voters can only cast valid votes and there will not be any undervotes. If you collect ballots on your own, however, then you may have undervotes. OpaVote provides three options for handling undervotes
    • Skip — The undervote is skipped over. E.g., for the ballot above, it would be treated as Alice was ranked first and Bob was ranked second.
    • Stop — Counting is stopped at the undervote. E.g., for the ballot above, it would be treated as Alice was ranked first and with no other rankings.
    • Void — The entire ballot is treated as void and is not used.
  • Candidate Elimination — With RCV counting methods, candidates with the smallest number of votes are eliminated. This can be done in several ways.
    • Single — Candidates are always eliminated one at a time.
    • Zero — The first time that candidates are eliminated, all candidates with zero votes are eliminated simultaneously. Afterwards, candidates are eliminated one by one.
    • Losers — All candidates who cannot win the election are eliminated simultaneously. At a given round of counting, one can look at the distribution of votes, and determine mathematically which candidates still have a chance of winning and which candidates cannot possibly win.
  • Tie Breaking — As with any election, ties may occur in RCV elections. Ties may occur when selecting candidates to eliminate, selecting surplus votes for transfer, or selecting winners. Several options for breaking ties are available.
    • Random — The tie is broken randomly (like flipping a coin).
    • Backward — The vote count at the previous round is used to break the tie. Suppose we are selecting a candidate to eliminate at round 4 and Alice and Bob are tied for last place. The tie would be broken by comparing Alice and Bob's vote counts at round 3. If they are still tied at round 3, then round 2 is used and so forth. If the candidates are tied at all rounds, then the tie is broken randomly.
    • Forward — The vote count at round 1 is used to break the tie. Suppose we are selecting a candidate to eliminate at round 4 and Alice and Bob are tied for last place. The tie would be broken by comparing Alice and Bob's vote counts at round 1. If they are tied at round 1, then round 2 is used and so forth. If they are still tied at round 3, then round 2 is used and so forth. If the candidates are tied at all rounds, then the tie is broken randomly.
  • Precision — Some STV methods use fractional votes. To use fractional votes, a precision (the number of digits after the decimal point) must be specified.
  • Fractional Threshold — STV methods have a winning threshold (the Droop quota). A candidate who reaches this threshold is declared elected. The winning threshold may be a whole number (to simplify the results) or a fraction (to obtain more precision, which may be useful with a small number of voters).
  • Dynamic Threshold — The winning threshold in an STV election may be static or dynamic. A static threshold is determined at the beginning of the election and does not change. A dynamic threshold is computed at each round and may decrease as votes become exhausted (when votes do not count towards any candidate because the voter did not rank all of the candidates).
  • Delay Surplus Transfer — At each round of an STV count, it is decided whether to eliminate a candidate or to transfer suplus votes.
    • Off — Transferring surplus votes is not delayed. At each round, if any candidate has a surplus, then surplus votes are transferred in the round. Candidates are only eliminated if no candidates have surplus votes.
    • On — Transferring surplus votes is delayed if there is a canddiate in the election who cannot win (see Candidate Elimination above). At each round, the decision proceeds in three steps: (i) if there are candidates who cannot win, then those candidates are eliminated, (ii) if all candidates could still win, then surplus votes are transferred, and (iii) if there are no surplus votes, then the last place candidate is eliminated. Delaying transfers of surplus votes simplifies the counting process because transferring surplus votes requires splitting votes into fractions and eliminating candidates does not.