  Consider a multi-phase project management problem where the decision maker
needs to deal with two issues: (a) how to allocate resources to projects within
each phase, and (b) when to enter the next phase, so that the total expected
reward is as large as possible. We formulate the problem as a multi-armed
bandit problem with precedence relations. In Chan, Fuh and Hu (2005), a class
of asymptotically optimal arm-pulling strategies is constructed to minimize the
shortfall from perfect information payoff. Here we further explore optimality
properties of the proposed strategies. First, we show that the efficiency
benchmark, which is given by the regret lower bound, reduces to those in Lai
and Robbins (1985), Hu and Wei (1989), and Fuh and Hu (2000). This implies that
the proposed strategy is also optimal under the settings of aforementioned
papers. Secondly, we establish the super-efficiency of proposed strategies when
the bad set is empty. Thirdly, we show that they are still optimal with
constant switching cost between arms. In addition, we prove that the Wald's
equation holds for Markov chains under Harris recurrent condition, which is an
important tool in studying the efficiency of the proposed strategies.
