Strategic Exit with Random Observations

In the standard optimal stopping problems, actions are artificially restricted to the moments of observations of costs or benefits. In the standard experimentation and learning models based on two-armed Poisson bandits, it is possible to take an action between two sequential observations. The latter models do not recognize the fact that timing of decisions depends not only on the rate of arrival of observations, but also on the dynamics of costs or benefits. We combine together these two strands of literature and consider bandits of “evolving shade of grey” instead of two-armed bandits who are either “white knights” or “black villains.” Stopping decisions in a model with Poisson bandits of “evolving shade of gray” are qualitatively different from those in optimal stopping or Poisson bandit models. We consider a case of two firms operating a technology which may experience costly breakdowns. The cost of breakdowns follows a jump-diffusion process. Breakdowns occur at random times, which follow a Poisson process independent of the cost process. The arrival rate of breakdowns may be high or low, but it is initially unknown. The firms differ by the rates of arrival, recovery rates and costs of breakdowns. We solve for the optimal exit strategy of the players.