[Paper Review] Principal agent mean field games in REC markets

Dena Firoozi, Arvind Shrivats+1 more|arXiv (Cornell University)|Dec 22, 2021

Climate Change Policy and Economics4 citations

TL;DR

This paper formulates a principal-agent mean field game (PA-MFG) model for Renewable Energy Certificate (REC) markets, where a regulator (principal) sets a non-compliance penalty function and multiple heterogeneous firms (agents) optimize their REC generation and trading strategies in response. Using mean field game theory and extended McKean-Vlasov control, it shows that the optimal penalty function is linear in agents’ state variables, implying that optimal emissions regulation resembles a tax or rebate system regardless of the principal’s utility function.

ABSTRACT

Principal agent games are a growing area of research which focuses on the optimal behaviour of a principal and an agent, with the former contracting work from the latter, in return for providing a monetary award. While this field canonically considers a single agent, the situation where multiple agents, or even an infinite amount of agents are contracted by a principal are growing in prominence and pose interesting and realistic problems. Here, agents form a Nash equilibrium among themselves, and a Stackelberg equilibrium between themselves as a collective and the principal. We apply this framework to the problem of implementing Renewable Energy Certificate (REC) markets, where the principal requires regulated firms (power generators) to pay a non-compliance penalty which is inversely proportional to the amount of RECs they have. RECs can be obtained by generating electricity from clean sources or purchasing on the market. The agents react to this penalty and optimize their behaviours to navigate the system at minimum cost. In the agents' model we incorporate market clearing as well as agent heterogeneity. For a given market design, we find the Nash equilibrium among agents using techniques from mean field games. We then use techniques from extended McKean-Vlasov control problems to solve the principal (regulators) problem, who aim to choose the penalty function in such a way that balances environmental and revenue impacts optimally. We find through these techniques that the optimal penalty function is linear in the agents' state, suggesting the optimal emissions regulation market is more akin to a tax or rebate, regardless of the principal's utility function.

Motivation & Objective

• To model the strategic interaction between a regulator (principal) and a large number of heterogeneous power generators (agents) in Renewable Energy Certificate (REC) markets.
• To analyze how agents collectively form a Nash equilibrium under a given market design, while reacting to a penalty function set by the principal.
• To derive the optimal penalty function from the principal’s perspective, balancing environmental and revenue objectives using stochastic control techniques.
• To extend mean field game theory to principal-agent frameworks in environmental markets, particularly for REC and emissions regulation.
• To demonstrate that the optimal penalty structure is linear in agents’ state variables, suggesting a tax-like mechanism is optimal.

Proposed method

• Models the agents’ behavior using a mean field game (MFG) framework, where each agent optimizes under the influence of the aggregate distribution of other agents’ states and controls.
• Incorporates market clearing conditions endogenously to determine the REC price as a function of the mean field distribution of agent inventories and generation capacities.
• Uses a stochastic control formulation with forward-backward stochastic differential equations (FBSDEs) to characterize the Nash equilibrium among agents under a given penalty structure.
• Applies extended McKean-Vlasov control techniques to solve the principal’s optimization problem, where the penalty function is chosen to balance environmental and financial objectives.
• Imposes regularity conditions on the penalty function (non-increasing, convex, C¹ with bounded Lipschitz-continuous derivative) to ensure well-posedness of the agents’ problem.
• Defines the mean field distribution of states as the product of the distribution of REC inventories (μ) and generation capacities (ν), forming the state flow θ = μ × ν.

Experimental results

Research questions

• RQ1 How do heterogeneous agents in an REC market optimally respond to a non-compliance penalty set by a central regulator?
• RQ2 What is the structure of the Nash equilibrium among a large number of agents when they face a common penalty function and interact through a clearing price mechanism?
• RQ3 What penalty function should the principal (regulator) choose to optimally balance environmental impact and revenue generation?
• RQ4 How does the optimal penalty function depend on the agents’ state variables, and does it vary with the principal’s utility function?
• RQ5 Can the principal’s optimal policy be derived using extended McKean-Vlasov control in a mean field game setting?

Key findings

• The optimal penalty function for the principal is linear in the agents’ state variables, regardless of the principal’s specific utility function.
• This linearity implies that optimal emissions regulation in REC markets is structurally equivalent to a tax or rebate system, rather than a complex, nonlinear penalty.
• The agents’ Nash equilibrium is characterized by a system of forward-backward stochastic differential equations (FBSDEs) driven by the mean field distribution of states.
• Market clearing is enforced in the infinite-population limit by requiring the average trading rate across agents to vanish almost surely.
• The solution to the principal’s problem is well-posed under standard regularity conditions: the penalty function must be convex, non-increasing, continuously differentiable, with bounded and Lipschitz-continuous first derivative.
• The mean field distribution of states is defined as the product of the distribution of REC inventories and generation capacities, capturing agent heterogeneity and market dynamics.