Approval elections with a variable number of winners

Multi‐winner elections, for example, the election of members to a committee, are now quite common, and include the interesting subclass of elections with a variable number of winners, or VNW elections. In VNW elections, voters determine how many winners there are, as well as which candidates win. Common VNW elections include elections to bestow honorary status, such as enshrinement in a hall of fame, and elections to determine a shortlist of, say, job candidates for interviews. Such elections are VNW elections whenever the number of winners is not specified in advance. Multi‐winner elections are often conducted by adapting standard procedures for single‐winner elections. Approval balloting is particularly appropriate for multi‐winner elections, as every ballot, and the set of winners, are subsets of the set of all candidates. Many methods of counting approval ballots appropriate to VNW elections are reviewed and illustrated, and their properties–desirable and undesirable–discussed.