A Few Good Choices

Condorcet winning sets address the Condorcet paradox by selecting a small set of candidates—rather than a single winner—such that a majority prefers no unselected alternative over all members of the set. This notion extends to \(\alpha\)-undominated sets, which require the same property to hold for any \(\alpha\)-fraction of voters. Such sets are guaranteed to exist with constant size for any fixed \(\alpha\). However, the requirement that an outsider be preferred to every member of the set can be overly restrictive and difficult to justify in many applications. Motivated by this, we introduce a more flexible notion: \((t, \alpha)\)-undominated sets. Here, each voter compares an outsider to their \(t\)-th most preferred member of the set, and the set is undominated if no outsider is preferred by more than an \(\alpha\)-fraction of voters. This framework subsumes prior definitions, recovering Condorcet winning sets when \((t = 1, \alpha = 1/2)\) and \(\alpha\)-undominated sets when \(t = 1\), and introduces a new, tunable notion of collective acceptability for \(t \gt 1\).