Is Approval Voting Optimal Given Approval Votes?

Some crowdsourcing platforms ask workers to express their opinions by approving a set of k good alternatives. It seems that the only reasonable way to aggregate these k-approval votes is the approval voting rule, which simply counts the number of times each alternative was approved. We challenge this assertion by proposing a probabilistic framework of noisy voting, and asking whether approval voting yields an alternative that is most likely to be the best alternative, given k-approval votes. While the answer is generally positive, our theoretical and empirical results call attention to situations where approval voting is suboptimal.