This paper is concerned with the existence of solutions to a dynamic network equilibrium problem modeled as an infinite dimensional variational inequality. Our results are based on properties of operators that map path flow departure rates to consistent time-dependent path flows and other link performance functions. The existence result requires the introduction of a novel concept that strengthens the familiar concept of First-In-First-Out (FIFO).

The continuous dynamic network loading problem (CDNLP) consists in determining, on a congested network, time-dependent arc volumes, together with arc and path travel times, given the time-varying path flow departue rates over a finite time horizon. This problem constitutes an intrinsic part of the dynamic traffic assignment problem. In this paper, we present a formulation of the CDNLP where travel delays may be nonlinear functions of arc traffic volumes. We prove, under a boundedness condition, that there exists a unique solution to the problem and propose for its solution a finite-step algorithm. Some computational results are reported for a discretized version of the algorithm.

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The rapid development of smartphone technology has led to the increased popularity of dynamic ridesharing apps used to organize ad hoc ridesharing trips between strangers on short notice. To support such real-time on-demand services, cost-sharing between drivers and riders is commonly centrally determined by ridesharing apps according to prescribed rules. To highlight the impacts of appropriate cost-sharing strategies on the success of ridesharing programs, this paper models the mode choices of a group of heterogeneous travelers with continuously distributed values of time in a single-corridor network, considering the complex interactions between travelers’ mode choices and the attractiveness of ridesharing in terms of rider/driver waiting/detouring times and matching probabilities. The equilibrium state under any given cost-sharing strategy is described by a system of variational inequalities based on which the existence of equilibria is established. With the proposed modeling framework, various cost-sharing strategies are examined to avoid mode shifts among transit users to autos and/or reduce vehicular traffic in the short run; the necessary conditions for cost-sharing strategies to sustain participation and/or reduce vehicle usage are explicitly provided. It is shown that when driving alone is faster but more expensive than public transit, no cost-sharing strategy exists to sustain an active ridesharing platform without inducing transit users to join the ridesharing program. Moreover, the existence of cost-sharing strategies capable of reducing vehicular traffic on the road is not always guaranteed, depending on the costs of driving alone and taking public transit in the considered corridor, fuel prices, and travelers’ prioritization of safety and privacy. Furthermore, it is found that the initial state with no ridesharing participants is an equilibrium under any cost-sharing strategy if the additional cost incurred by a traveler through participating in a ridesharing program is nonnegative. This explains the difficulty of initiating a ridesharing program and implies the initial necessity of subsidizing all intended riders and/or drivers to encourage participation. The online appendix is available at https://doi.org/10.1287/trsc.2017.0801 .

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