Efficiency of correlation in a bottleneck game

We consider a model of bottleneck congestion in discrete time with a penalty cost for being late. This model can be applied to several situations where agents need to use a capacitated facility in order to complete a task before a hard deadline. A possible example is a situation where commuters use a train service to go from home to office in the early morning. Trains run at regular intervals, take always the same time to cover their itinerary, and have a fixed capacity. Commuters must reach their office in time. This is a hard constraint whose violation involves a heavy penalty. Conditionally on meeting the deadline, commuters want to take the train as late as possible. With the intent of considering strategic choices of departure, we model this situation as a game and we show that it does not have pure Nash equilibria. Then we characterize the best and worst mixed Nash equilibria, and show that they are both inefficient with respect to the social optimum. We then show that there exists a correlated equilibrium that approximates the social optimum when the penalty for missing the deadline is sufficiently large