Nash Core in Multiwinner Election

Abstract

In the approval-based committee selection problem, a committee is said to be in the core if no subset of voters has an incentive to deviate by selecting a \emph{blocking} committee of proportional size, such that every voter in the deviating group strictly prefers the blocking committee. We consider the setting where candidates can be selected fractionally. Under a mild regularity assumption, we show that there always exists a weighting of candidates such that the fractional committee maximizing the candidate-weighted Nash Social Welfare is in the core. We refer to such a solution as being in the \emph{Nash core}. Additionally, we show that a Nash core solution admits a payment assignment between voters and candidates, where each voter pays a candidate they approve in proportion to the weight. For the discrete setting, where each candidate is either included or excluded from the committee, we prove that every approval-based committee election with at most eight equally weighted voters has a core committee by rounding the fractional Nash core solution. Although the non-emptiness of the core in this setting remains an open question and checking core membership is coNP-hard, we extend the notion of the Nash core to the discrete case, yielding a formulation that is efficiently verifiable and offers a promising path toward establishing core existence in discrete settings. Finally, we test our approach on real voting data using a payment-guided heuristic. We empirically show that the Nash core solution can be efficiently computed through an iterative algorithm in both the fractional and discrete settings.