Gradual adjustment and equilibrium uniqueness under noisy monitoring

We study the implications of flexible adjustment in strategic interactions using a class of finite-horizon models in continuous time. Players take costly actions to affect the evolution of state variables that are commonly observable and perturbed by Brownian noise. The values of these state variables influence players' terminal payoffs at the deadline, as well as their flow payoffs. In contrast to the static case, the equilibrium is unique under a general class of terminal payoff functions. Our characterization of the equilibrium builds on recent developments in the theory of backward stochastic differential equations (BSDEs). We use this tool to analyze applications, including team production, hold-up problems, and dynamic contests. In a team production model, the unique equilibrium selects an effcient outcome when frictions vanish.