Wardrop’s first and second principles, named after English mathematician John Glen Wardrop 1, are pivotal to the understanding of user behavior in transportation systems. Wardrop’s principles enable the systematic analysis of transportation networks. It does this by elegantly linking road performance to system objectives.

In the middle of the 20th century, transportation engineers and planners knew that an increased number of cars result in increased congestion–reduced speed and capacity. They were able to use accurate empirical relationships to model congestion thanks to BD Greenshield 2. They knew that given different route options, people would choose the one that took less time. But there was still an important piece of the puzzle missing. Wardrop formulated the math to quantitatively model route choice behavior.

Under Wardrop’s first principle, users make choices to minimize travel times. This is also referred to as a “user equilibrium” condition. In the example network below, people living in the red and blue neighborhoods, at home location ‘o’, are travelling to work location ‘d’. Under user equilibrium, every user would choose the road with the lowest travel time, even if it resulted in congestion and increased travel time for everyone else.

user equilibrium condition
Figure 1 User equilibrium–everyone uses the wide road, causing congestion.

In contrast, under Wardrop’s second principle, all users make decision to minimize the total network travel time, a “system optimal” condition. In this case, users would avoid congestion even if it results in higher travel time for themselves.

system optimal condition
Figure 2 System optimal–most people use the wide road, but a small number of people travel longer on small streets to avoid causing congestion on the wide road.

We will skip the calculations in this blog post, but the concept of Wardrop’s principles is very intuitive and has a wide range of application beyond transportation. In fact, the mathematical framework behind user equilibrium, i.e. Nash Equilibrium (or in other contexts, the tragedy of the commons), is deeply rooted in game theory.

While in transportation planning we generally assume a user equilibrium network condition, we often experiment with different measures (such as road advisories, congestion charges, ramp metering, tolls) to encourage users to act in a manner that reduces congestion. Sometimes, these measures can be very political and wildly unpopular.

  1. Research by JG Wardrop on Google Scholar, and more on JG Wardrop on Wiki. ↩

  2. Early mathematical models of road performance by BD Greenshield were fundamental to Wardrop’s work. ↩