  This paper investigates the regret associated with the Distributionally
Robust Control (DRC) strategies used to address multistage optimization
problems where the involved probability distributions are not known exactly,
but rather are assumed to belong to specified ambiguity families. We quantify
the price (distributional regret) that one ends up paying for not knowing the
exact probability distribution of the stochastic system uncertainty while
aiming to control it using the DRC strategies. The conservatism of the DRC
strategies for being robust to worst-case uncertainty distribution in the
considered ambiguity set comes at the price of lack of knowledge about the true
distribution in the set. We use the worst case Conditional Value-at-Risk to
define the distributional regret and the regret bound was found to be
increasing with tighter risk level. The motive of this paper is to promote the
design of new control algorithms aiming to minimize the distributional regret.
