  We consider the two-sided matching market with bandit learners. In the
standard matching problem, users and providers are matched to ensure incentive
compatibility via the notion of stability. However, contrary to the core
assumption of the matching problem, users and providers do not know their true
preferences a priori and must learn them. To address this assumption, recent
works propose to blend the matching and multi-armed bandit problems. They
establish that it is possible to assign matchings that are stable (i.e.,
incentive-compatible) at every time step while also allowing agents to learn
enough so that the system converges to matchings that are stable under the
agents' true preferences. However, while some agents may incur low regret under
these matchings, others can incur high regret -- specifically, $\Omega(T)$
optimal regret where $T$ is the time horizon. In this work, we incorporate
costs and transfers in the two-sided matching market with bandit learners in
order to faithfully model competition between agents. We prove that, under our
framework, it is possible to simultaneously guarantee four desiderata: (1)
incentive compatibility, i.e., stability, (2) low regret, i.e., $O(\log(T))$
optimal regret, (3) fairness in the distribution of regret among agents, and
(4) high social welfare.
