Many ranking-and-selection (R&S) problems arise in settings where information is noisy, structured, and expensive. In multi-fidelity experimentation, one can query cheap but biased proxies or expensive high-fidelity measurements; in dueling bandits, feedback arrives only through pairwise comparisons rather than direct rewards. These models are increasingly natural in engineering design, simulation optimization, preference learning for LLMs, and human-in-the-loop evaluation, where absolute scores are often unavailable or prohibitively costly and decisions must be made with a prescribed level of confidence. What makes these settings especially challenging is that the usual single-winner template is no longer sufficient. First, the answer map may be set-valued: in good-alternative or subset-selection problems, several answers can be simultaneously correct. Second, even when the true instance is answerable, a noisy estimate may temporarily fall outside the answerable set.

Recent work on deriving O(log T) anytime regret bounds for stochastic dueling bandit problems has considered mostly Condorcet winners, which do not always exist, and more recently, winners defined by the Copeland set, which do always exist. In this work, we consider a broad notion of winners defined by tournament solutions in social choice theory, which include the Copeland set as a special case but also include several other notions of winners such as the top cycle, uncovered set, and Banks set, and which, like the Copeland set, always exist. We develop a family of UCB-style dueling bandit algorithms for such general tournament solutions, and show O(log T) anytime regret bounds for them. Experiments confirm the ability of our algorithms to achieve low regret relative to the target winning set of interest.

We study best-arm identification in stochastic dueling bandits under the sole assumption that a Condorcet winner exists, i.e., an arm that wins each noisy pairwise comparison with probability at least $1/2$. We introduce a new identification procedure that exploits the full gap matrix $Δ_{i,j}=q_{i,j}-\tfrac12$ (where $q_{i,j}$ is the probability that arm $i$ beats arm $j$), rather than only the gaps between the Condorcet winner and the other arms. We derive high-probability, instance-dependent sample-complexity guarantees that (up to logarithmic factors) improve the best known ones by leveraging informative comparisons beyond those involving the winner. We complement these results with new lower bounds which, to our knowledge, are the first for Condorcet-winner identification in stochastic dueling bandits. Our lower-bound analysis isolates the intrinsic cost of locating informative entries in the gap matrix and estimating them to the required confidence, establishing the optimality of our non-asymptotic bounds. Overall, our results reveal new regimes and trade-offs in the sample complexity that are not captured by asymptotic analyses based only on the expected budget.

When the optimality gap is negligible, we propose another algorithm that outperforms our first algorithm, highlighting the subtlety of this dueling bandit problem.

The action is to choose a set of arms, whereupon feedback for each arm in the chosen set is received.