[Paper Review] Optimal Dynamic Contracts for a Large-Scale Principal-Agent Hierarchy: A Concavity-Preserving Approach

Christopher W. Miller, Insoon Yang|arXiv (Cornell University)|Jun 17, 2015

Auction Theory and Applications4 citations

TL;DR

This paper develops an optimal dynamic contract for a large-scale principal-agent hierarchy using a concavity-preserving dynamic programming approach. It enables a top-level principal to incentivize all agents in the chain through indirect monitoring, reducing the problem to a one-dimensional control and state space regardless of hierarchy size, with explicit construction via iterative linear algebra and homeomorphisms preserving utility concavity.

ABSTRACT

We present a continuous-time contract whereby a top-level player can incentivize a hierarchy of players below him to act in his best interest despite only observing the output of his direct subordinate. This paper extends Sannikov's approach from a situation of asymmetric information between a principal and an agent to one of hierarchical information between several players. We develop an iterative algorithm for constructing an incentive compatible contract and define the correct notion of concavity which must be preserved during iteration. We identify conditions under which a dynamic programming construction of an optimal dynamic contract can be reduced to only a one-dimensional state space and one-dimensional control set, independent of the size of the hierarchy. In this sense, our results contribute to the applicability of dynamic programming on dynamic contracts for a large-scale principal-agent hierarchy.

Motivation & Objective

• To design an optimal dynamic contract in continuous-time for a hierarchical principal-agent system with moral hazard.
• To enable a top-level principal to incentivize all downstream agents despite only observing the output of their immediate subordinate.
• To reduce the dimensionality of the dynamic programming problem to one state variable and one control variable, independent of hierarchy size.
• To preserve utility concavity through iterative construction, ensuring incentive compatibility across the hierarchy.
• To generalize Sannikov’s two-player dynamic programming approach to multi-level hierarchical contracts.

Proposed method

• Uses a recursive, iterative algorithm to construct the agent’s continuation value and optimal control strategy at each level of the hierarchy.
• Applies a generalized two-player principal-agent framework by treating the entire lower chain as a single aggregate agent.
• Employs homeomorphisms ψₖ: ℝ → Γₖ to map scalar values to multi-dimensional control spaces, preserving concavity.
• Constructs the effective utility function μₖ(x) = -½xᵀΛₖx using strictly positive-definite diagonal matrices Λₖ and weight vectors gₖ with 1ᵀgₖ = 1.
• Reduces the Hamilton-Jacobi-Bellman (HJB) equation to a one-dimensional form by leveraging the one-dimensional structure of the control space.
• Uses the stochastic maximum principle and forward-backward SDEs implicitly through the dynamic programming framework, avoiding direct solution of complex equations.

Experimental results

Research questions

• RQ1 Can a top-level principal design an optimal dynamic contract that incentivizes all agents in a large-scale hierarchy, even when only observing the output of the first subordinate?
• RQ2 Under what conditions can the dynamic programming solution for a hierarchical principal-agent problem be reduced to a one-dimensional state and control space?
• RQ3 How can concavity of the agent’s utility be preserved through recursive construction across multiple levels of the hierarchy?
• RQ4 What mathematical structure enables the reduction of high-dimensional control problems to one-dimensional optimization in hierarchical contracting?
• RQ5 How can the agent’s continuation value and incentive compatibility be maintained across multiple layers of moral hazard?

Key findings

• The optimal contract for the top-level principal can be constructed via an iterative algorithm that preserves concavity of the agent’s utility at each hierarchical level.
• The dynamic programming problem reduces to a one-dimensional HJB equation with a single control variable, regardless of the number of agents in the hierarchy.
• The control space Γ₁ for the top principal is a one-dimensional linear subspace, parametrized by a vector g₁ with 1ᵀg₁ = 1, enabling tractable optimization.
• The effective utility function for the aggregate agent is quadratic and strictly concave, with μ₁(x) = -½xᵀΛ₁x for a strictly positive-definite diagonal matrix Λ₁.
• The HJB equation for the principal simplifies to vₜ + sup_y [½γ(γσ²v_ww + v_w)y² + y] = 0, where γ = g₁ᵀΛ₁g₁, enabling efficient solution.
• The iterative construction relies only on linear algebraic operations, such as matrix inversion and vector scaling, at each step, ensuring computational feasibility.