  Two-sided matching markets have been widely studied in the literature due to
their rich applications. Since participants are usually uncertain about their
preferences, online algorithms have recently been adopted to learn them through
iterative interactions. An existing work initiates the study of this problem in
a many-to-one setting with responsiveness. However, their results are far from
optimal and lack guarantees of incentive compatibility. We first extend an
existing algorithm for the one-to-one setting to this more general setting and
show it achieves a near-optimal bound for player-optimal regret. Nevertheless,
due to the substantial requirement for collaboration, a single player's
deviation could lead to a huge increase in its own cumulative rewards and a
linear regret for others. In this paper, we aim to enhance the regret bound in
many-to-one markets while ensuring incentive compatibility. We first propose
the adaptively explore-then-deferred-acceptance (AETDA) algorithm for
responsiveness setting and derive an upper bound for player-optimal stable
regret while demonstrating its guarantee of incentive compatibility. To the
best of our knowledge, it constitutes the first polynomial player-optimal
guarantee in matching markets that offers such robust assurances without known
$\Delta$, where $\Delta$ is some preference gap among players and arms. We also
consider broader substitutable preferences, one of the most general conditions
to ensure the existence of a stable matching and cover responsiveness. We
devise an online DA (ODA) algorithm and establish an upper bound for the
player-pessimal stable regret for this setting.
