The Surname Wardrop
The surname Wardrop occurs more frequently in the United States than in any other country or territory. It is also a common last name in England.
Wardrop began his career at CSIR in 1945 and moved to La Trobe University in 1964, where he was Foundation Professor of Botany. He was a distinguished student of plant cell walls, especially those of fibers and tracheids. Visit https://moneywars.com/wardrop/ to learn more.
John Glen Wardrop, an English mathematician, developed two important principles that allow us to quantitatively model route choice behavior in traffic networks. These two principles, known as the first and second principles of traffic assignment, enable us to link road performance to system objectives, such as minimizing total network travel time.
The first of Wardrop’s principles, also referred to as the user equilibrium principle (UE) or dynamic route choice principle (DUE), states that for each origin-destination pair (OD), the routes used by travelers must have equal and minimal travel times. This assumes that the demand at each OD is perfectly known and that each traveler is independent, non-cooperative, and rational.
While this equilibrium is not optimal in the sense that it does not maximize network utility (travel time), it is generally considered to be a fair trade-off between travel times and congestion and therefore is often used as a reference point for model calibration. In the presence of informational attacks, however, the UE can be shifted by poisoning the traffic data, misguiding users, and manipulating traffic patterns.
The second of Wardrop’s principles, also referred to as the “social Wardrop equilibrium” or “generalized route choice principle,” requires that all users choose routes that minimize total network travel time. This is a more ambitious and desirable condition, but it may not be achievable in practice since it implies that all travelers cooperate to avoid congestion, even if it results in a higher travel time for themselves.
Wardrop’s first principle
Named after English mathematician John Glen Wardrop, these principles allow the systematic analysis of route choice behavior in transportation networks. They enable engineers and planners to link network performance with system objectives and identify the optimum way to allocate traffic across a network. They have also helped to understand how traffic flow changes when a new route opens or a road is improved, and they can be used to analyze the effects of congestion management strategies on network performance.
The first principle, known as user equilibrium (UE), states that the travel times on routes actually used are equal and less than those of any unused path between an origin-destination pair. This assumption is based on the assumption that each driver is identical, non-cooperative, and always seeks to minimize his or her travel costs. Traffic flows satisfying this principle are often referred to as “user-optimized” (UE) or “stochastic user equilibrium” (SUE).
Traffic assignment models based on this principle are commonly used in modeling travel demand and traffic assignments. They provide an alternative to a more traditional model that assigns traffic based on link capacities and considers travel time as the only input variable. The UE model is more realistic in that it takes into account travel costs and delays due to congestion, which are often ignored by traditional models.
However, it is important to note that UE cannot be applied in all scenarios. For example, if a route has a high induced cost and thus attracts more travelers than the shortest route, the overall system travel time will increase, and equilibrium may no longer be reached. This is why UE is generally only considered in a limited number of cases where the benefits of increased efficiency outweigh the cost of increasing congestion.
While Wardrop did not provide algorithms for finding UE, many studies have found that the existence of UE implies the existence of Nash equilibria in the same setting. The solution to a Nash equilibrium can be found through iterative simulation, with each agent choosing their route given the choices of others. This approach is very computationally intensive, however. The more efficient Frank-Wolfe algorithm combines the iterative approach with dynamic programming to solve problems more quickly.
Wardrop’s Second Principle
In transport planning, Wardrop’s second principle is a mathematical model that links transportation system objectives and road performance. It explains route choice and allocation processes by linking them to network efficiency goals. The concept is similar to shortest path analysis in graph theory. The goal is to find a route between two vertices (or nodes) in such a way that the total weight of the paths is minimized.
The model is used to determine the minimum travel time between two points in a network. It also identifies the optimum allocation of traffic on different routes. The shortest route is determined by finding the path with the least overall weight or load, which is the sum of the weights of all edges in the network. It is also important to note that the shortest path does not necessarily guarantee the lowest travel times. This is because the quality of the road surface, speed limits, and congestion may influence the actual travel time.
While the second principle is an excellent tool for analyzing transportation systems, it does not explain why travelers choose specific routes. To understand the underlying reasons for route choice, it is necessary to consider the behavioral assumptions outlined in Wardrop’s first principle.
Travelers in a deterministic user equilibrium select routes that maximize their personal utility, which is defined as the budget or income minus the travel time. However, there is a limit to this approach because it assumes that all drivers perceive costs in an identical manner. In real-world networks, this is unrealistic and results in a socially optimal solution that is far from reality.
Stochastic user equilibrium is a more realistic and useful model of route choice that allows for uncertainty in travel costs. It is based on the assumption that travelers are stochastically motivated to minimize their travel costs. The travel cost functions used in stochastic user equilibrium models include a deterministic measured travel cost and a random term that represents schedule delay or perceived disutility.
As a result, incorporating the concept of wardrop into transportation assignment models enables engineers to solve a wide range of problems, including optimizing departure time for a single commute route or minimizing total travel from multiple origins to one destination. The model also helps with modal split analyses with small sample sizes.
Wardrop’s generalized principle
During the past several decades, researchers have studied the issue of traffic network equilibrium. Pigou considered a two-node, two-link transportation network in 1920, but it was not until 1952 that John Glen Wardrop laid the foundation for this field of study by proposing the first and second principles that are named after him.
The first principle, referred to as the “user equilibrium” (UE) route choice principle, states that for every origin-destination pair, all routes actually used must have equal and minimal travel times. It assumes that each individual traveler chooses his or her own route based on an evaluation of utility, with the goal being to maximize this utility. In the simplest case, a traveler’s total utility is the sum of his or her budget or income minus the travel time.
A second principle, referred to as the “social Wardrop equilibrium” or “system optimal” (SO) route choice condition, requires that all individuals make their decisions in such a way as to minimize overall network travel time. This would result in a situation where travelers are cooperative and able to avoid congestion, even if this results in a higher travel time for some of them. SO conditions are generally viewed by economists and traffic modelers as desirable, though it is not clear whether this can be achieved through market-based pricing or a central routing authority dictating route choices for all users.
Given the continuing improvements in real-time information for drivers regarding alternative routes, it is important to investigate how these new capabilities can be best exploited. Specifically, can a Wardrop equilibrium be attained in a realistic traffic simulation with complete and up-to-date travel-time information for each individual?
This paper considers a simple, two-route scenario to answer this question. It is assumed that each driver is fully informed about the travel time of both the main line and a low-speed bypass. The analysis utilizes a Nagel-Schreckenberg cellular automaton model to analyze the behavior of drivers in this environment. It is found that an ideal dynamic user optimum is not attained, but the steady state is close to both Wardrop and SO.