A controller-stopper-game with hidden controller type

TL;DR

This paper models a stochastic dynamic game involving an owner and a manager with hidden effectiveness, analyzing equilibrium strategies using filtering methods and exploring threshold-based solutions.

Contribution

It introduces a novel game framework with hidden manager types and derives equilibrium strategies under both strong and weak control formulations.

Findings

Threshold equilibrium strategies are identified in the strong formulation.

The threshold equilibrium remains valid in the more general weak formulation.

Filtering methods are used to analyze the hidden type and control dynamics.

Abstract

We consider a continuous time stochastic dynamic game between a stopper (Player $1$, the \textit{owner} of an asset yielding an income) and a controller (Player $2$, the \textit{manager} of the asset), where the manager is either effective or non-effective. An effective manager can choose to exert low or high effort which corresponds to a high or a low positive drift for the accumulated income of the owner with random noise in terms of Brownian motion; where high effort comes at a cost for the manager. The manager earns a salary until the game is stopped by the owner, after which also no income is earned. A non-effective manager cannot act but still receives a salary. For this game we study (Nash) equilibria using stochastic filtering methods; in particular, in equilibrium the manager controls the learning rate (regarding the manager type) of the owner. First, we consider a strong formulation of the game which requires restrictive assumptions for the admissible controls, and find an equilibrium of (double) threshold type. Second, we consider a weak formulation, where a general set of admissible controls is considered. We show that the threshold equilibrium of the strong formulation is also an equilibrium in the weak formulation.