Dynamic Multi-Agent Contracting with Relative Performance Evaluation and Competition

This paper develops a continuous-time model of contracting between a principal and many agents under moral hazard, linking dynamic contract theory to the mean field game (MFG) framework. Outputs depend on effort, competition, and both idiosyncratic and common shocks. The principal designs contracts using filtered signals that benchmark agents against peers. We show that the first-order approach remains valid: effort is linear in exposure with slope determined by the informativeness of filtered outputs. The Hamilton-Jacobi-Bellman equation implies that optimal incentives depend jointly on marginal revenue, signal variance, and intertemporal insurance. The optimal filter is the generalized least squares residual, which asymptotically eliminates common shocks as group size grows. In a linear-quadratic specialization, equilibrium exists, is unique, and is globally stable, with the uniqueness result coinciding with the Lasry-Lions MFG fixed-point condition. The framework unifies tournaments, RPE, and dynamic contracts, and explains empirical puzzles in executive pay, sales teams, and finance.