  We study the problem of incentive-compatible online learning with bandit
feedback. In this class of problems, the experts are self-interested agents who
might misrepresent their preferences with the goal of being selected most
often. The goal is to devise algorithms which are simultaneously
incentive-compatible, that is the experts are incentivised to report their true
preferences, and have no regret with respect to the preferences of the best
fixed expert in hindsight. \citet{freeman2020no} propose an algorithm in the
full information setting with optimal $O(\sqrt{T \log(K)})$ regret and
$O(T^{2/3}(K\log(K))^{1/3})$ regret in the bandit setting.
  In this work we propose the first incentive-compatible algorithms that enjoy
$O(\sqrt{KT})$ regret bounds. We further demonstrate how simple loss-biasing
allows the algorithm proposed in Freeman et al. 2020 to enjoy $\tilde
O(\sqrt{KT})$ regret. As a byproduct of our approach we obtain the first bandit
algorithm with nearly optimal regret bounds in the adversarial setting which
works entirely on the observed loss sequence without the need for
importance-weighted estimators. Finally, we provide an incentive-compatible
algorithm that enjoys asymptotically optimal best-of-both-worlds regret
guarantees, i.e., logarithmic regret in the stochastic regime as well as
worst-case $O(\sqrt{KT})$ regret.
