Optimal queue length information disclosure when service quality is uncertain

We investigate a server's best queue disclosure strategy in a single-server service system with an uncertain quality level (which is assumed to be binary). We consider this problem from the perspective of a Bayesian persuasion game. The server first commits to a possibly mixed strategy stating the probability that the queue length will be revealed to customers on their arrival given a realized quality level. The service quality level is then realized, and the server's corresponding queue-disclosure action is observed by customers, who then update their beliefs regarding service quality and decide whether to join the service system. We reformulate the server's decision problem as looking for the best Bayes-plausible distribution of posterior beliefs regarding service quality. We demonstrate that the maximal expected effective arrival rate, as a function of the prior belief, can be graphed as the upper envelope of all convex combinations of any two arbitrary points on the two effective arrival rate functions of the revealed and concealed queues. We show that when the market size is sufficiently small (large), the server always conceals (reveals) the queue, regardless of the realized service quality. Numerically, we find that in a medium-sized market, the server's optimal commitment strategy is often hybrid or mixed, that is, randomizing queue concealment and revelation. We also extend our analysis to a situation in which the server aims to maximize social welfare. We show that under certain conditions, it is always beneficial for the welfare-maximizing social planner to randomize queue concealment and revelation, regardless of the market size.