A comparison of control strategies applied to a pricing problem in
retail

When sales of a product are affected by randomness in demand, retailers can
use dynamic pricing strategies to maximise their profits. In this article the
pricing problem is formulated as a stochastic optimal control problem, where
the optimal policy can be found by solving the associated Bellman equation. The
aim is to investigate Approximate Dynamic Programming algorithms for this
problem. For realistic retail applications, modelling the problem and solving
it to optimality is intractable. Thus practitioners make simplifying
assumptions and design suboptimal policies, but a thorough investigation of the
relative performance of these policies is lacking.
To better understand such assumptions, we simulate the performance of two
algorithms on a one-product system. It is found that for more than half of the
realisations of the random disturbance, the often-used, but approximate,
Certainty Equivalent Control policy yields larger profits than an optimal,
maximum expected-value policy. This approximate algorithm, however, performs
significantly worse in the remaining realisations, which colloquially can be
interpreted as a more risk-seeking attitude by the retailer. Another policy,
Open-Loop Feedback Control, is shown to work well as a compromise between the
Certainty Equivalent Control and the optimal policy.