🚦 Congestion Pricing — Wardrop Equilibrium

At the user equilibrium, each driver minimizes only their own travel time, ignoring the extra delay their presence adds to every other driver sharing the congested road. The system optimum instead minimizes total travel time across all drivers, which requires fewer cars on the congestible road than selfish behaviour would produce: here 35 cars/min rather than all 70, cutting total travel time from 1260 to 1015 car-minutes, a saving of about 19 percent. This gap between individually rational and collectively optimal outcomes is a textbook example of a negative externality, similar to overfishing a shared resource.

Braess's paradox, described by Dietrich Braess in 1968, shows that adding a new road (or route capacity) to a network can increase everyone's travel time if it changes the equilibrium routing in an unfavourable way. In this simulation, toggling the shortcut link lets some traffic bypass part of the free route's delay, but because more drivers pile onto that link it becomes congested itself, and the aggregate equilibrium travel time can rise even though a new option was added. Removing certain roads in cities including Seoul and New York has measurably reduced congestion for exactly this reason.

Yes. Singapore's Electronic Road Pricing (1998, evolved from a 1975 paper-license scheme) reduced peak-hour traffic by roughly 10-15%. London's Congestion Charge (2003) cut traffic entering the charging zone by about 15% in its first years. Stockholm ran a seven-month trial congestion tax in 2006 that reduced traffic by 20% and, after a referendum, made the charge permanent; follow-up studies found the reduction persisted for years, evidence that tolling reshapes long-run travel behaviour rather than just short-term route choice.

Because with no toll it is the faster road for any individual driver at every flow up to the full demand: the congestible road takes 4 + 0.2x minutes, which stays below the constant 18 minutes of the free road until it is carrying all 70 cars/min. Each driver therefore has a private incentive to join it, and nobody is left on the free road. That is Pigou's point: the selfish equilibrium wastes time even though no single driver is behaving irrationally. Once a toll is charged the equilibrium is self-correcting - too many drivers on the tolled road makes it slow and expensive, so traffic flows back to the free road until the two generalized costs are equal, which at the Pigouvian toll is exactly the 50/50 optimum.

This is a genuine equity concern raised in the transportation-economics literature: a flat toll represents a larger share of income for lower-income drivers, potentially pricing them off faster routes regardless of how urgently they need to travel. Many real-world schemes address this with revenue recycling — using toll income to fund public transit, provide rebates, or lower other taxes — which can make the overall policy progressive even though the toll itself is regressive in isolation. This simulation does not model income heterogeneity directly but the value-of-time slider approximates how differently drivers might weigh the same toll.

William Vickrey's 1969 bottleneck model is a complementary framework that tolls congestion in the time dimension rather than across parallel routes: drivers choose when to depart, and a time-varying toll spreads departures to eliminate queueing delay while preserving the same equilibrium travel cost for all users. This simulation's route-choice framework and Vickrey's departure-time framework are the two classic building blocks of congestion-pricing theory, and modern dynamic tolling systems (like variably priced express lanes) combine elements of both.

Active areas include distance- and emissions-based dynamic tolling that adjusts in real time using traffic sensor data, equity-aware pricing schemes that pair tolls with targeted rebates for low-income commuters, integration with autonomous-vehicle routing where fleets can be centrally coordinated toward system-optimal rather than user-optimal patterns, and mobility-as-a-service platforms that internalize the congestion externality by pricing trips dynamically across modes rather than per road segment.