Multiple Referenda and Multiwinner Elections Using Hamming Distances: Complexity and Manipulability

We study multiple referenda and committee elections, when the ballot of each voter is simply a set of approved binary issues (or candidates). Two well-known rules under this model are the commonly used candidate-wise majority, also called the minisum rule, as well as the minimax rule. In the former, the elected committee consists of the candidates approved by a majority of voters, whereas the latter picks a committee minimizing the maximum Hamming distance to all ballots.

As these rules are in some ways extreme points in the whole spectrum of solutions, we consider a general family of rules, using the Ordered Weighted Averaging (OWA) operators. Each rule is parameterized by a weight vector, showing the importance of the i-th highest Hamming distance of the outcome to the voters. The objective then is to minimize the weighted sum of the (ordered) distances. We study mostly computational, but also manipulability properties for this family. We first exhibit that for many rules, it is NP-hard to find a winning committee. We then proceed to identify cases where the problem is either efficiently solvable, or approximable with a small approximation factor. Finally, we investigate the issue of manipulating such rules and provide conditions that make this possible.