- The paper establishes that as the number of cooperative agents increases, the principal's value function converges to a McKean-Vlasov control problem.
- It introduces methods combining BSDEs and n-player approximations to construct approximately optimal contracts in complex cooperative and MV settings.
- The work connects multi-agent and multitask frameworks, highlighting how task correlation influences the principal's utility and contract performance.
Principal-Agent Problems with Multiple Agents and McKean-Vlasov Dynamics
This paper explores the connections between three distinct but related Principal-Agent (PA) problems: a principal interacting with multiple cooperative agents, a PA problem with McKean-Vlasov (MV) dynamics, and a multitask PA problem. The analysis focuses on characterizing the limiting behavior of the multi-agent problem as the number of agents tends to infinity, and its relationship to the MV dynamics and multitask settings.
Multi-Agent Principal-Agent Problem and its McKean-Vlasov Limit
The paper begins by formulating a PA problem with a principal and n cooperative agents. The agents' cooperation is modeled through a Pareto equilibrium, where the agents collectively optimize their criteria. A key result is that as the number of agents n approaches infinity, the principal's value function converges to that of a McKean-Vlasov optimal control problem. This convergence result provides a constructive method for deriving approximately optimal contracts for the principal in the multi-agent problem when the number of agents is sufficiently large. The problem is reformulated using dynamic programming and Backward Stochastic Differential Equations (BSDEs). Techniques developed in previous work are then used to show the convergence of the principal's value function and to construct approximately optimal contracts. The cooperation setting contrasts with existing literature that focuses on competition among agents via Nash equilibria. In the Nash equilibrium framework, a deviating agent has a negligible effect on the contract, which is not the case in a cooperative setting.
Principal-Agent Problem with McKean-Vlasov Dynamics
The paper then investigates a novel Principal-Agent problem where the agent-controlled production follows MV dynamics, meaning the distribution of the production affects the dynamics themselves. The contract offered by the principal can depend not only on the production (moral hazard) but also on the distribution of the production. This type of problem has not been extensively studied in the literature. It is shown that this MV Principal-Agent problem emerges as the limit of the multi-agent problem as n goes to infinity. Moreover, the principal's value function in this setting is equivalent to that of the McKean-Vlasov control problem from the multi-agent scenario. An optimal contract can be constructed using the solution to the McKean-Vlasov control problem. The distribution dependence creates challenges for the classical BSDE approach, which is addressed using an n-player approximation of the production/value function combined with the weak limit of convergent sequences of BSDEs.
Connection to Multitask Principal-Agent Problem
The paper concludes by exploring a connection to the Multitask Principal-Agent problem, where a principal delegates multiple, potentially correlated tasks to a single agent. In a simplified example with homogeneous tasks, it is shown that the multitask problem is equivalent to the problem of a principal with multiple cooperative agents. As a result, the solution to the multitask problem, when the number of tasks goes to infinity, can be obtained via the McKean-Vlasov control problem. This connection highlights the importance of the correlation/interdependence of tasks, with positive correlation increasing the utility of the principal and negative correlation decreasing it.
Conclusion
This paper establishes a connection between three different Principal-Agent problems. It rigorously demonstrates that the multi-agent problem, in the limit, becomes a Principal-Agent problem with McKean-Vlasov dynamics. This convergence allows for the construction of approximately optimal contracts in the multi-agent setting using the solution of the McKean-Vlasov control problem. Finally, the link to the multitask Principal-Agent problem highlights the role of task correlation and its impact on the principal's utility. The introduction of a Principal-Agent problem with McKean-Vlasov dynamics represents a significant contribution to the contract theory literature. Future work could explore extensions to more general multitask problems with heterogeneous tasks and the design of implementable, model-free RL algorithms for solving these interconnected problems. The analysis also raises questions about rates of convergence and the trade-offs between model complexity and tractability in these interconnected Principal-Agent problems.