A Note on the Multi-Agent Contracts in Continuous Time

TL;DR

This paper extends continuous-time dynamic incentive contracts to multi-agent settings, deriving conditions for optimal contracts and linking the principal's problem to solving PDEs from BSDEs.

Contribution

It introduces a multi-agent extension of the single-agent model, providing conditions for optimal contracts and a PDE-based framework for solving the principal's problem.

Findings

Derived sufficient conditions for optimal contracts in multi-agent settings.

Established the connection between the principal's problem and HJB equations.

Presented a PDE framework based on BSDEs for solving the contract design problem.

Abstract

Dynamic contracts with multiple agents is a classical decentralized decision-making problem with asymmetric information. In this paper, we extend the single-agent dynamic incentive contract model in continuous-time to a multi-agent scheme in finite horizon and allow the terminal reward to be dependent on the history of actions and incentives. We first derive a set of sufficient conditions for the existence of optimal contracts in the most general setting and conditions under which they form a Nash equilibrium. Then we show that the principal's problem can be converted to solving Hamilton-Jacobi-Bellman (HJB) equation requiring a static Nash equilibrium. Finally, we provide a framework to solve this problem by solving partial differential equations (PDE) derived from backward stochastic differential equations (BSDE).