Deducing simple majorities from approval voting ballot data

An approval voting election allows each voter to vote for as many candidates as he wishes. The winner of such an election is the candidate with the most approval votes. Ballot data aggregated for individual voters tell how many voters voted for each subset of candidates. Given reasonable assumptions about what voters' ballots reveal about their preferences, this report shows what must be true of the ballot data in order to conclude that the electorate has a majority preference for one candidate over another. A simple corollary tells what must be true of the ballot data to conclude that the winner is preferred by a majority of voters to every other candidate. Both hypothetical and reconstructed ballot data from a mayoral election in Boston are used to illustrate the results. It is also shown that the likelihood of being able to draw conclusions about majority candidates tends to increase as voters tend to vote for fewer candidates.