Ergodic control of a heterogeneous population and application to electricity pricing

TL;DR

This paper develops an ergodic control framework for a large heterogeneous population, applying it to electricity pricing, and demonstrates how cyclic policies can outperform constant strategies through theoretical analysis and numerical algorithms.

Contribution

It introduces a novel ergodic control approach for mean-field Markov decision processes with applications to electricity pricing, including existence of solutions and cyclic policy strategies.

Findings

Cyclic policies can outperform constant-price strategies in certain cases.

The proposed policy iteration algorithm reduces memory usage for decomposable models.

Numerical results show cyclic promotions emerge in electricity pricing scenarios.

Abstract

We consider a control problem for a heterogeneous population composed of agents able to switch at any time between different options. The controller 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" Markov Decision Process in which the state space is a product of simplices, and the population evolves according to controlled linear dynamics. By exploiting contraction properties of the dynamics in Hilbert's projective metric, we prove that the infinite-dimensional ergodic eigenproblem admits a solution and show that the latter is in general non unique. This allows us to obtain optimal strategies, and to quantify the gap between steady-state strategies and optimal ones. In particular, we prove in the one-dimensional case that there exist cyclic policies -- alternating between discount and profit taking stages -- which secure a greater gain than constant-price policies. On numerical aspects, we develop a policy iteration algorithm with "on-the-fly" generated transitions, specifically adapted to decomposable models, leading to substantial memory savings. We finally apply our results on realistic instances coming from an electricity pricing problem encountered in the retail markets, and numerically observe the emergence of cyclic promotions for sufficient inertia in the customer behavior.