  We study dynamic allocation problems for discrete time multi-armed bandits
under uncertainty, based on the the theory of nonlinear expectations. We show
that, under strong independence of the bandits and with some relaxation in the
definition of optimality, a Gittins allocation index gives optimal choices.
This involves studying the interaction of our uncertainty with controls which
determine the filtration. We also run a simple numerical example which
illustrates the interaction between the willingness to explore and uncertainty
aversion of the agent when making decisions.
