  Some approaches to solving challenging dynamic programming problems, such as
Q-learning, begin by transforming the Bellman equation into an alternative
functional equation, in order to open up a new line of attack. Our paper
studies this idea systematically, with a focus on boosting computational
efficiency. We provide a characterization of the set of valid transformations
of the Bellman equation, where validity means that the transformed Bellman
equation maintains the link to optimality held by the original Bellman
equation. We then examine the solutions of the transformed Bellman equations
and analyze correspondingly transformed versions of the algorithms used to
solve for optimal policies. These investigations yield new approaches to a
variety of discrete time dynamic programming problems, including those with
features such as recursive preferences or desire for robustness. Increased
computational efficiency is demonstrated via time complexity arguments and
numerical experiments.
