Multi-Commodity Nash Flows Over Time

TL;DR

This paper extends the theory of Nash flows over time to multiple commodities with different origins and destinations, proving their existence and analyzing their structural properties in dynamic traffic models.

Contribution

It introduces a multi-commodity model for Nash flows over time and proves their existence using multi-commodity thin flows with resetting, providing new structural insights.

Findings

Existence of multi-commodity Nash flows over time established.

Structural properties of equilibrium strategies analyzed.

Special cases reduce to single-commodity scenarios.

Abstract

Motivated by the dynamic traffic assignment problem, we consider flows over time model with deterministic queuing. Dynamic equilibria, called Nash flows over time, have been studied intensively since their introduction by Koch and Skutella in 2009. Unfortunately, the original model can only handle a single commodity, i.e., all flow goes from one origin to one destination, therefore, this theory is not applicable to every-day traffic scenarios. For this reason we consider Nash flows over time in a setting with multiple commodities, each with a individual origin and destination. We prove their existence with the help of multi-commodity thin flows with resetting. This proof gives some interesting structural insights into the strategy profiles underlying the equilibria, as they emphasis the inter-commodity dependencies. Finally, we show that the two special cases of a common origin or a common destination can be reduced to the single-commodity case.