By Clay Shentup & Warren D. Smith 2008
The Boston Tea Party, an offshoot of the USA's Libertarian Party, has adopted Approval Voting as its voting method... except they made some changes to enforce "majority approval" via an elimination process.
Quoting from Article 11 ("conventions") section O of their bylaws:
o) As soon as practicable after the published opening date and time for the convention, member polls for the Party's nominations to the presidency and vice-presidency of the United States shall open and remain open for 24 hours. These polls shall include each of the prospective nominees, as well as "None of the Above," in alphabetical order. The method of voting shall be "approval voting." Each voting member may cast one vote for every candidate which he or she deems an acceptable nominee to the office. At the close of the poll, the candidate receiving the most votes, provided that he or she has also received a vote from a majority of voting members, shall be the nominee of the Party to the office.If no candidate has received a vote from a majority of voting members, then successive 24-hour polls shall be conducted in the same manner, with the candidate receiving the fewest votes in the previous poll eliminated, until such time as a candidate emerges with both the largest number of votes and votes from a majority of voting members. If, at the conclusion of any poll in this process, "None of the Above" receives both the largest number of votes and receives votes from a majority of voting members, the process shall end and the Party shall not nominate a candidate to the office in question for that election.
Here is a copy of the online poll they used to choose their first (2008) US presidential nominee.
This poll will open at 9pm Eastern on June 15th and remain open for 24 hours. All party members may vote. The method is "approval voting" – members may vote for as many of the candidates as they want. For further details, see Article 11, Section O of the bylaws. The winner of this process (or NOTA) will be the Boston Tea Party's 2008 presidential nominee.
Results: The winner was Jay with 28 votes, beating Milnes with 9 and NOTA with 5 (in total 42 approvals cast by 37 voters). In the simultaneous vice-presidential contest the winner was Thomas L. Knapp with 22, beating T.A.Barnett (11), C.Bennett (6), and NOTA (6) – in total 45 approvals cast by 34 voters.