Self, social and monopoly optimization in observable queues

Naor?s [8] celebrated paper studies customer decisions in an observable M/M/1 queue where customers utility from joining the system is is a linear decreasing function of the joined position in queue. Naor derives the optimal threshold strategies for the individual, social planner and monopoly. The optimal threshold imposed by a monopoly is not greater than the socially optimal threshold, which is not greater than the individual?s threshold. Studies show that this triangular relation holds in a more general setup where the utility function is not necessarily linear. Many of these extensions share common features. We point out conditions that imply the aforementioned result, and apply them to a new model motivated by order-driven markets. In the new model, customers choose between joining and balking when they might be forced to abandon the system before service completion, and the expected value of joining depends on the service completion probability, which is not linear in the observed queue size.