Nonemptiness of the Hare core for arbitrary approval elections

Establish that every approval election has a nonempty Hare core, without restricting the number of candidates, seats, voter types, or the sizes of voters’ disapproval sets.

Background

The paper studies Hare-core stability in approval-based committee elections, where a coalition meeting the Hare quota blocks a committee if all coalition members strictly improve by moving to another candidate set. The main theorem proves nonemptiness when every voter disapproves at most two candidates, allowing arbitrary positive rational voter weights and unbounded numbers of candidates, seats, and voter types.

The unrestricted deterministic nonemptiness problem remains unresolved. Existing results cited in the paper cover only bounded numbers of seats or candidates, bounded numbers of voter types, or special preference structures. Thus, the open problem is to remove the paper’s co-rank-two restriction and prove nonemptiness for general approval elections.