  Under the expected total reward criterion, the optimal value of a
finite-horizon Markov decision process can be determined by solving the Bellman
equations. The equations were extended by D. J. White to processes with vector
rewards in 1982. Using a counterexample, we show that the assumptions
underlying this extension fail to guarantee its validity. Analysis of the
counterexample leads us to articulate a sufficient condition for White's
functional equations to be valid. The condition is shown to be true when the
policy space has been refined to include a special class of non-Markovian
policies, or when the dynamics of the model are deterministic, or when the
decision making horizon does not exceed three time steps. The paper
demonstrates that, in general, the solutions to White's equations are sets of
Pareto efficient policy returns over the refined policy space. Our results are
illustrated with an example.
