Principal agent mean field games in REC markets

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.

• The paper introduces a novel model integrating principal-agent and mean field game theories to optimize REC market design.
• It derives agents’ optimal REC generation and trading strategies using a forward-backward stochastic differential equation framework.
• The regulator’s optimal strategy simplifies complex market designs to a linear penalty system resembling a tax-and-rebate approach.

Principal-Agent Mean Field Games in REC Markets

The paper "Principal agent mean field games in Renewable Energy Certificate (REC) markets" (2112.11963) examines the application of principal-agent mean field games to the market design and regulation of Renewable Energy Certificates (RECs). This novel approach is used to analyze how the interactions between regulators (principals) and energy firms (agents) can be optimized, particularly in the context of promoting renewable energy generation through market-based incentives.

Introduction

The paper addresses the complex issue of climate change and the need for effective emissions regulation policies. It emphasizes the need for countries and regions to implement market-based solutions like carbon taxes, cap-and-trade markets, and REC markets to reduce emissions. These markets are designed to incentivize firms to generate electricity from renewable sources by providing RECs - tradable assets that represent proof of electricity generated via clean energy sources. In particular, the paper focuses on the design of optimal REC markets, analyzing the behavior of regulated firms and the incentives provided by the regulators.

Model Formulation

The model adopted in the paper integrates principal-agent games with mean field games (MFGs) to address the market dynamics between numerous energy firms and a regulatory principal. The principal-agent setup allows the regulator to design contracts that aim to maximize environmental benefits while optimizing revenue through penalties for non-compliance. The agents (firms) operate under MFG settings to find Nash equilibria among themselves and interact with the principal in a Stackelberg equilibrium.

Agent's Problem

Each firm aims to optimize its REC generation and trading behavior, considering short-term generation plans, trading rates, and long-term capacity expansion. The objective is to minimize their operational costs while complying with the REC requirements set by the principal. The cost functional includes terms for generation costs, trading frictions, market trading prices, and penalties for non-compliance.

Principal's Problem

The regulator seeks to choose penalty functions that influence the firms’ behavior optimally. The principal identifies penalty functions such that the firms’ mean field equilibrium is attainable and their costs do not exceed a predefined reservation utility. The regulator’s utility function balances environmental impact and generated revenue from penalties.

Solution Approach

The paper applies advanced mathematical techniques involving mean field games, stochastic control, and principal-agent theories to determine optimal strategies for both agents and the principal. By solving the agents’ and principal's optimization problems in the infinite population limit, the authors derive the equilibrium strategy and market conditions.

Agent's Strategy

The agents’ optimal generation, trading, and capacity decisions are derived using a forward-backward stochastic differential equation (FBSDE), reflecting the interaction between the firms’ state, market prices, and penalty structures.

Principal's Strategy

Using a reduction of the problem to a standard stochastic control with extended McKean-Vlasov dynamics, the principal’s optimization problem is solved by leveraging sufficient conditions for optimality, revealing the optimal linear penalty functions that emulate tax-like regulations instead of complex market structures.

Implications

The study concludes that, under the given assumptions, the most efficient market design resembles a tax-and-rebate system, questioning the efficacy of more complex REC market designs. The principal optimally imposes a linear penalty, which suggests a fixed marginal cost for REC acquisition, resulting in constant market trading rates and deterministic market prices. This insight could profoundly impact regulatory policy for emissions markets, although further exploration of real-world considerations is necessary.

Conclusion

The paper provides a mathematical framework for analyzing REC market designs using principal-agent MFGs. It posits that optimal emissions regulation can be akin to a tax system rather than a traditional market structure, with implications for future regulatory strategies. The findings underscore the need for further studies incorporating market realities and extending the theoretical model to multi-period settings and other complexities inherent in real-world applications.