A few good choices

A Condorcet winning set addresses the Condorcet paradox by selecting a few candidates--rather than a single winner--such that no unselected alternative is preferred to all of them by a majority of voters. This idea extends to $\alpha$-undominated sets, which ensure the same property for any $\alpha$-fraction of voters and are guaranteed to exist in constant size for any $\alpha$. 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 $t > 1$. We establish three main results: 1. We prove that a $(t, \alpha)$-undominated set of size $O(t/\alpha)$ exists for all values of $t$ and $\alpha$. 2. We show that as $t$ becomes large, the minimum size of such a set approaches $t/\alpha$, which is asymptotically optimal. 3. In the special case $t = 1$, we improve the bound on the size of an $\alpha$-undominated set given by Charikar, Lassota, Ramakrishnan, Vetta, and Wang (STOC 2025). As a consequence, we show that a Condorcet winning set of five candidates exists, improving their bound of six.