Proportionality in Approval-Based Elections With a Variable Number of Winners

We study proportionality in approval-based multi-winner elections with a variable number of winners, where both the size and identity of the winning committee are informed by voters' opinions. While proportionality has been studied in multiwinner elections with a fixed number of winners, it has not been considered in the variable number of winners setting. The measure of proportionality we consider is average satisfaction (AS), which intuitively measures the number of agreements on average between sufficiently large and cohesive groups of voters and the output of the voting rule. First, we show an upper bound on AS that any deterministic rule can provide, and that straightforward adaptations of deterministic rules from the fixed number of winners setting do not achieve better than a 1/2 approximation to AS even for large numbers of candidates. We then prove that a natural randomized rule achieves a 29/32 approximation to AS.