Ergodic control of a heterogeneous population and application to electricity pricing

We consider a control problem for a heterogeneous population composed of customers able to switch at any time between different contracts, depending not only on the tariff conditions but also on the characteristics of each individual. A provider aims to maximize an average gain per time unit, supposing that the population is of infinite size. This leads to an ergodic control problem for a "mean-field"MDP in which the state space is a product of simplices, and the population evolves according to a controlled linear dynamics. By exploiting contraction properties of the dynamics in Hilbert's projective metric, we show that the ergodic eigenproblem admits a solution. This allows us to obtain optimal strategies, and to quantify the gap between steady-state strategies and optimal ones. We illustrate this approach on examples from electricity pricing, and show in particular that the optimal policies may be cyclic -alternating between discount and profit taking stages.