Fairness and Public Goods: the Case of Multiwinner Rules

An instance of a multiwinner election consists of a set of alternatives, a population of voters---each voter approves a subset of alternatives, and the desired committee size k; the goal is to select a committee (a subset) of k alternatives according to the preferences (approvals) of the voters. What does it mean that the committee is fair to the voters? We will review a few most prominent concepts of fairness. The talk will be almost entirely concentrated around the open questions that boil down to pure combinatorics. Specifically, I will explain the notion of the core. Intuitively, a committee S is core-stable if no subgroup of voters can deviate by selecting a proportionally smaller committee S' such that each member of the deviating group would prefer S' and S. The following question: “does there always (for each preference profile) exist a core-stable committee” is considered important and challenging. I will also cover the most recent advancement in the subject. My goal will be to attract the audience with a few interesting, important and challenging open questions.