Equilibrium behavioural strategies in an M/M/1 queue

For an M/M/1 system, we analyse the strategic interactions of the social optimiser, the service provider and customers and their consequences on the system. The social optimiser chooses the type of information to make available to customers (make the system observable or unobservable), the service provider chooses the service rate with which he performs the service, and customers decide, according to the strategic choices of the first two agents, to use or not the system. As these agents are interacting in a common environment with respect to their objectives, we model the problem as a three-stage game between them. A resolution of the different stages will be made, which will give the overall solution to the considered problem, corresponding to the subgame perfect Nash equilibrium in behavioural strategies. A numerical analysis will be made where one can see the graphical solution of the game, comparisons and interpretations will be well established.