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# Procedures & Applications :

## Approval Voting and Its Use in Professional Societies

In the early 1970’s, Robert J. Weber (an economist, now of Northwestern University’s Kellogg School of Management) proposed approval voting as a generalization of plurality rule to elections of three or more candidates.  As is often the case in mathematics, other people had the same idea around the same time, making it difficult to attribute the idea to a single person.  However, Weber coined the term “approval voting” and political scientist Steven J. Brams (New York University) and mathematician Peter C. Fishburn (now retired from AT&T Bell Labs) are credited with championing its use in part through the publication of their 1983 book, Approval Voting (re-issued by Springer in 2007).  Brams and Fishburn also contributed extensively to the analysis of approval voting and its properties and effectiveness.

It is not surprising that the Public Choice Society began using approval voting to elect its president-elect in 2006.  The 2004-2006 president was Steven J. Brams.  Before the 2006 election, the president was selected every two years by an executive committee with two-year terms alternating between political scientists and economists.  It was surprising that a society dedicated, in part, to the analysis of politics with the tools of economics and mathematics never held elections!

###### Approval Voting:  Approving candidates by drawing a line in the rank ordering of the candidates.

The six ways to approve of candidates when a voter’s preferences for A, B, C, D, and E are in alphabetical order.

 -- A A A A A A -- B B B B B B -- C C C C C C -- D D D D D D -- E E E E E E --

Under approval voting, a voter divides the candidates into two groups:  those which he or she approves and those which he or she does not.  A voter can rank order the candidates, as in ballots for plurality rule, Borda count, etc., but also can draw a line, to indicate which candidates are approved (above the line) and which are not (below the line). For each ballot, approved candidates each receive one vote or point, while candidates that are not approved receive zero votes or points.

For example, a voter who rank orders candidates A, B, C, D, and E in alphabetical ordering can place the approval line in one of six positions:  above A, between A and B, between B and C, between C and D, between D and E, and below E.  Placing the line above A is equivalent to not voting for any of the candidates, while placing the line below E is equivalent to voting for all of the candidates.  We will disregard a voter that approves of none or all of the candidates as such a vote does not influence the outcome of the election – each candidate would receive the same number of votes or points. Even though two voters may rank order candidates the same way, there are four ways in which to then approve of a subset of the candidates.  Unlike other procedures (plurality rule, Borda count, etc.), knowing all of the voters’ preferences for candidates does not determine the election outcome.

Of course, a voter need only divide the candidates into approve and disapprove sets.  Because the rank orderings are cumbersome, we will represent a preference list of candidates with a line indicating approval, such as A B | C D E (listed horizontally to conserve space) instead by AB, the two candidates which are approved.  One argument for approval voting is that it may be difficult for voters to rank-order a large number of candidates, but easier to approve of, or disapprove of, a candidate.  An argument against approval voting is that approving of multiple candidates does not distinguish between them, even though one may be more preferred than another.

As an example of approval voting, we review the analysis by Brams, Hansen, and Orrison (2006) of 37 votes cast in the 2006 Public Choice Society election for president.  To preserve anonymity, the candidates are listed as A, B, C, D, and E.  The following list indicates the ballots cast and their number.  For example, ABC = 3 means that three ballots approved of candidates A, B, and C.

 none = 1 _ _ _ _ _ A = 2 B = 4 C = 4 D = 3 E = 2 _ AB = 2 AC = 2 AD = 1 BC = 1 BD = 1 DE =3 ABC = 3 ACD = 2 ACE = 1 BDE = 1 _ _ ABCD = 2 ABCE = 1 _ _ _ _ ABCDE = 1 _ _ _ _ _

 Approval Voting and Professional Societies The Public Choice Society began collecting dues for 2006 and the 37 members that paid before the December 2005 deadline were allowed to vote for the 2006-2008 president.  The society views its members as those that attend its annual conference which sees approximate 300 attendees.  However, other larger societies have also used approval voting for its elections, including the American Mathematical Society, AMS (30,000 members); the American Statistical Association, ASA (15,000); the Institute for Electrical and Electronics Engineers, IEEE (377,000); the Institute for Operations Research and Management Sciences, INFORMS (12,000); and the Mathematical Association of America , MAA(32,000).

 Al Gore If voters who voted for Green Party candidate Ralph Nader were able to also approve of Democratic Party candidate Albert A. Gore, then Gore would have defeated George W. Bush to become the 43rd President of the US.

More Election Data:  Institute for Electrical and Electronics Engineers (IEEE)
The IEEE publishes the vote tallies of their elections on their website (see
http://www.ieee.org/web/aboutus/corporate/election/results.html).  Brams and Nagel had access to the 54,204 ballots cast in the 1988 IEEE election for president-elect; the following data appears as Table 3 in a 2005 article by Brams and Fishburn (see " References and Links"). To preserve anonymity of the candidates, we will refer to them as A, B, C, and D.  The following describes how the votes were distributed.

 None = 1,100 _ _ _ _ A = 10,738 B = 6,561 C = 7,626 D = 8,521 _ AB = 3,578 AC = 659 AD = 6,679 BC = 1,425 CD = 608 ABC = 148 ABD = 5,605 ACD = 143 BCD = 89 _ All = 523 _ _ _ _

For example, 3,578 voters cast ballots in which they approved of candidates A and B, but did not approve of C and D.  Notice that A is the sum of the ballots for A, AB, AC, AD, ABC, ABD, and ACD, or 28,073.  Similarly, B, C, and D receive 19,753, 11,221, and 23,992 votes, respectively.  Hence, A wins the election.

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