\section{Conclusion and open problems}   \label{open_problem}
%It is currently open whether the rate region can be extended beyond Eq.\ \eqref{nec-suff} in the adversarial setting. 
In this paper, we proposed a black-box reduction from the fair bandits problem to the unconstrained bandits problem and bounded the regret and cumulative target rate violations. 
%As a consequence, any improvement in the regret bound for MAB 
Since we use adversarial MAB policies as subroutines, it is reasonable to conjecture that the proposed \textsc{BanditQ} policy would work in the adversarial setting as well. Substantiating this statement would be an interesting research direction \citep{sinha2023playing}.  
Improving the regret and rate violation bounds by, \emph{e.g.,} working with a different Lyapunov function would be practically useful \citep{sinha2024tight}.
Finally, coming up with sharper instance-dependent regret bounds would be interesting as well.
%Extending the \textsc{BanditQ} policy to the bandit information setup would also be of substantial interest. 
%Finally, designing an anytime version of the policy that does not need to know the  horizon length $T$ in advance would be practically useful.