Generic uniqueness of ladder equilibria in the LQG benchmark

Prove that, in the linear–quadratic–Gaussian benchmark of the dynamic reputational disclosure game along a fixed expert–decision maker path with observable disclosure clocks, generic parameter configurations yield a unique ladder Markov perfect Bayesian equilibrium. Equivalently, establish that, given other agents’ strategies, each intermediary’s impulse-control problem admits a unique value function solving the associated quasi-variational inequality and that the observable disclosure clock renders the intermediary’s best response single-valued.

Background

In the path analysis, the authors show that Markov perfect Bayesian equilibria admit a ladder (threshold) representation via impulse-control problems and QVIs. While existence and structure are established, the possibility of multiple solutions to the coupled boundary-value system leaves open whether the equilibrium ladder is unique.

They explicitly conjecture that in the linear–quadratic–Gaussian (LQG) benchmark, uniqueness is generic—meaning that, for almost all parameter values, each intermediary’s impulse-control problem has a unique value function and the observable clock makes best responses single-valued—thus pinning down a unique ladder equilibrium. Formalizing and proving this generic uniqueness would strengthen comparative statics and selection results.