Markov Decision Problems Where Means Bound Variances

We identify a rich class of finite-horizon Markov decision problems (MDPs) for which the variance of the optimal total reward can be bounded by a simple affine function of its expected value. The class is characterized by three natural properties: reward boundedness, existence of a do-nothing action, and optimal action monotonicity. These properties are commonly present and typically easy to check. Implications of the class properties and of the variance bound are illustrated by examples of MDPs from operations research, operations management, financial engineering, and combinatorial optimization.