Analytical equilibrium of bicriterion choices with heterogeneous user preferences: application to the morning commute problem

The morning commute problem, introduced in Vickrey [1969. "Congestion Theory and Transport Investment." American Economic Review 56: 251-260], addresses the equilibrium trip schedule of the users for commuting through a single bottleneck with fixed capacity over the morning peak. In this paper, we adapt the concept of the efficient frontier from Portfolio Theory [Markowitz, H. 1952. "Portfolio Selection." The Journal of Finance 7 (1): 77-91] to propose an analytical solution to the morning commute problem with a general distribution of the schedule preferences over time and a continuous joint distribution of schedule penalty preferences over the population of the commuters. On this basis, we analytically derive the equilibrium arrivals of the heterogeneous commuters to the bottleneck given the independent probability distributions of earliness and lateness penalty factors. We also propose a method to retrieve independent probability distributions of the schedule penalty factors in the equilibrium condition from a joint distribution. The proposed model can be inversely used to approximate the independent distribution of schedule penalty preferences of the user for the early and late commuters using empirical arrival time data from the network. The result is also used to propose a dynamic pricing strategy to optimize the system by avoiding the formation of a queue, which can also be extended to design a dynamic pricing strategy on the network level using the macroscopic fundamental diagram (MFD). To provide an example, the approach is employed to derive a closed-form solution when the probability distribution of the preferences is uniform. A numerical example is also presented using the proposed model to compare the solutions for different distributions of the schedule deviation penalty preferences.