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

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 behavior. We then show that this characterization still holds for a specific type of heterogeneous population. For the general case, we propose a 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 one to achieve the sobriety target imposed by the regulation.