  We adress the problem of dueling bandits defined on partially ordered sets,
or posets. In this setting, arms may not be comparable, and there may be
several (incomparable) optimal arms. We propose an algorithm, UnchainedBandits,
that efficiently finds the set of optimal arms of any poset even when pairs of
comparable arms cannot be distinguished from pairs of incomparable arms, with a
set of minimal assumptions. This algorithm relies on the concept of decoys,
which stems from social psychology. For the easier case where the
incomparability information may be accessible, we propose a second algorithm,
SlicingBandits, which takes advantage of this information and achieves a very
significant gain of performance compared to UnchainedBandits. We provide
theoretical guarantees and experimental evaluation for both algorithms.
