A Rank-Based Reward between a Principal and a Field of Agents: Application to Energy Savings

TL;DR

This paper designs a ranking-based reward system for a large population of agents, analyzing its theoretical properties and applying it to energy savings regulation with real data.

Contribution

It introduces a mean-field framework for ranking incentives, characterizes the optimal reward, and demonstrates its effectiveness in energy savings applications.

Findings

Existence and uniqueness of mean-field equilibrium with ranking rewards

Explicit representation of the equilibrium distribution

Ranking system achieves energy savings targets in a real case study

Abstract

In this paper, we consider the problem of a Principal aiming at designing a reward function for a population of heterogeneous agents. We construct an incentive based on the ranking of the agents, so that a competition among the latter is initiated. We place ourselves in the limit setting of mean-field type interactions and prove the existence and uniqueness of the equilibrium distribution for a given reward, for which we can find an explicit representation. Focusing first on the homogeneous setting, we characterize the optimal reward function using a convex reformulation of the problem and provide an interpretation of its behaviour. We then show that this characterization still holds for a sub-class of heterogeneous populations. For the general case, we propose a convergent numerical method which fully exploits the characterization of the mean-field equilibrium. We develop a case study related to the French market of Energy Saving Certificates based on the use of realistic data, which shows that the ranking system allows to achieve the sobriety target imposed by the regulation.