  We make three contributions to the theory of k-armed adversarial bandits.
First, we prove a first-order bound for a modified variant of the INF strategy
by Audibert and Bubeck [2009], without sacrificing worst case optimality or
modifying the loss estimators. Second, we provide a variance analysis for
algorithms based on follow the regularised leader, showing that without
adaptation the variance of the regret is typically {\Omega}(n^2) where n is the
horizon. Finally, we study bounds that depend on the degree of separation of
the arms, generalising the results by Cowan and Katehakis [2015] from the
stochastic setting to the adversarial and improving the result of Seldin and
Slivkins [2014] by a factor of log(n)/log(log(n)).
