  The model of a non-Bayesian agent who faces a repeated game with incomplete
information against Nature is an appropriate tool for modeling general
agent-environment interactions. In such a model the environment state
(controlled by Nature) may change arbitrarily, and the feedback/reward function
is initially unknown. The agent is not Bayesian, that is he does not form a
prior probability neither on the state selection strategy of Nature, nor on his
reward function. A policy for the agent is a function which assigns an action
to every history of observations and actions. Two basic feedback structures are
considered. In one of them -- the perfect monitoring case -- the agent is able
to observe the previous environment state as part of his feedback, while in the
other -- the imperfect monitoring case -- all that is available to the agent is
the reward obtained. Both of these settings refer to partially observable
processes, where the current environment state is unknown. Our main result
refers to the competitive ratio criterion in the perfect monitoring case. We
prove the existence of an efficient stochastic policy that ensures that the
competitive ratio is obtained at almost all stages with an arbitrarily high
probability, where efficiency is measured in terms of rate of convergence. It
is further shown that such an optimal policy does not exist in the imperfect
monitoring case. Moreover, it is proved that in the perfect monitoring case
there does not exist a deterministic policy that satisfies our long run
optimality criterion. In addition, we discuss the maxmin criterion and prove
that a deterministic efficient optimal strategy does exist in the imperfect
monitoring case under this criterion. Finally we show that our approach to
long-run optimality can be viewed as qualitative, which distinguishes it from
previous work in this area.
