Procedures & Applications:


There can be as many election procedures as there are elections!  Election procedures view voters’ preferences for candidates as inputs, aggregate the preferences, and return a rank-ordering of the candidates as an output.  If a single winner is desired, then the top-ranked of these candidates is the winner.  If two candidates are to be elected, then the top-two candidates are elected, etc.  Below are short descriptions of some of the more frequently used election procedures.

Plurality Rule and U.S. Presidential Elections

Presidential Seal

Under plurality rule, each voter casts a vote for a single candidate.  The candidate with the most votes wins the election.  For two candidates, plurality rule is the same as majority rule.  For three or more candidates, the plurality rule winner, the candidate with the most first-place votes, may not have a majority of the first-place votes.  As the number n of candidates increase, it is possible for a candidate to win with one vote more than 1/n of all votes cast! 

In U.S. presidential elections, voters cast votes as under plurality rule, but a candidate wins the election if he or she wins a majority of the electoral votes.  In such elections, it is not only possible for a candidate with less than a majority of the popular vote to win the election, but also for the candidate with the most popular votes to lose the election, as has happened in three U.S. presidential elections since 1872.

Click on this link to learn more about plurality rule and how it relates to U.S. presidential elections.

Voting Vectors and Major League Baseball’s Most Valuable Players

Major League BaseballUnder a voting vector or weighted-scoring rule, each voter ranks all possible candidates and gives a score of w1 to his or her top-ranked candidate, w2 to his or her second-ranked candidate, etc., under the condition that w1w2 ≥ … ≥ wn (for n candidates).  The voting vector is denoted by w1, w2, …, wn).  A candidate’s cumulative score is the sum of the candidate’s scores from all voters.  The outcome of the election is determined by the cumulative scores:  the candidate with the highest score comes in first place, the candidate with the second-highest score comes in second place, etc. 

The most popular of voting vectors is the Borda count, named after French mathematician, political scientist, and engineer Jean-Charles de Borda (1733-1799).  Under the Borda count, the voting vector for n candidates is (n-1, n-2, n-3, …, 2, 1, 0). The NCAA Football Division 1A Coaches Poll uses voting vectors to determine the top 25 teams.  Each coach uses a truncated Borda count (25, 24, 23, … 3, 2, 1) in which his top-25 teams receive the corresponding points and all other teams receive zero points.  The Baseball Writers Association of America uses a voting vector to select winners for the most valuable player (MVP) awards for Major League Baseball.  In this case, a writer only gives positive scores to his or her top-ten candidates using the vector (14, 9, 8, 7, 6, 5, 4, 3, 2, 1). 

Click on this link to learn more about voting vectors and how they are applied to determine the Most Valuable Players of the American and National Leagues, as well as how college football coaches rank NCAA football teams.

Single Transferable Vote and the Academy Award


In 1861, Thomas Hare (1806-1891) introduced an election procedure that came to be known as the Hare system or single transferable vote (STV).  This is often used to elect more than one candidate, as in determining nominees for the Academy Awards.  Under STV, when a candidate has in excess of a minimum number or quota of first-place votes, then it is among the winners.  Because the candidate may have received more than the quota, each ballot with this candidate ranked in first place has a portion of its vote transferred to the second-ranked candidate.  As such this procedure encourages voters to rank the candidates sincerely by minimizing the penalty for voting for a heavily favored candidate.

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STV is used in elections in England, Ireland, and Malta.  In England, the Electoral Reform Society promotes the use of STV in elections.  The Academy of Motion Pictures Arts and Sciences uses a variation of single transferable vote to determine nominees for its Academy Awards (including Best Picture, Best Actor, and Best Actress).

Click on this link to learn more about single transferable vote and how it is applied to determine Oscar winners.

Single Transferable Vote to Elect a Single Candidate, The Instant Runoff Procedure

The instant runoff procedure is an implementation of single transferable vote to elect a single candidate.  The winner of an election under the instant runoff procedure is determined through a series of rounds.  In each round, the candidate with the fewest first-place votes is eliminated.  Voters who voted for the eliminated candidate have their votes transferred to their second-place candidates.  This repeats until there is a candidate that receives a majority of the votes.  Recently, Voter Choice Act-House Resolution 2690 (in the U.S. House of Representatives) included a provision to use the instant runoff procedure for all single-winner federal elections.  The instant runoff method has gained popularity and support.  There are websites (e.g., and much of praising instant runoff as the fairest way to run an election.

The instant runoff procedure is used for elections in the following U.S. cities:  San Francisco, CA, Burlington, VT, and Takoma Park, MD.  Click on this link to learn more about the instant runoff procedure. 

Approval Voting/Public Choice Society
Approval voting generalizes plurality rule by allowing voters to vote for, or to approve of, as many of the candidates as possible.  But, unlike plurality rule, in which a voter assigns one point to a most-preferred candidate, a voter may assign one point each to any subset of the candidates.  The winner of an election under approval voting is the candidate that receives the most votes or points. 

Just as instant runoff has garnered popular support, so has approval voting.  There is a political action committee (, as well as a non-profit organization (, that promotes the use of approval voting in single winner elections.  Interestingly, approval voting is used to elect the president of the Public Choice Society, a professional society which consists of economists, political scientists, and sociologists that apply economic and mathematical methods to the study of political science.  Besides the Public Choice Society, approval voting is used to elect the secretary-general of the United Nations and presidents-elect of the Institute of Electrical and Electronics Engineers (IEEE) and the Mathematical Association of America (MAA).  Click on this link to learn more about approval voting and its use in the election of the president of the Public Choice Society.