Dynamic signaling with stochastic stakes

We study dynamic signaling in a game of stochastic stakes. Each period, a privately informed agent of binary type chooses whether to continue receiving a return that is an increasing function of both her reputation and an exogenous public stakes variable or to irreversibly exit the game. A strong type has a dominant strategy to continue. In the unique perfect Bayesian equilibrium, the weak type plays a mixed strategy that depends only on current stakes and their historical minimum, and she builds a reputation by continuing when the stakes reach a new minimum. We discuss applications to corporate reputation management, online vendor reputation, and limit pricing with stochastic demand.