Shortcut
A traffic thought experiment
A free shortcut can add 15 minutes to everyone's trip.
Same 4,000 drivers. Same destination. One new road that takes zero minutes. Open it and watch the commute get worse.
The shortcut is closed. Drivers split evenly between the two routes.
4,000 drivers / shortcut closed
A hypothetical network. All roads run toward End; the central link runs A to B.
Why doesn't anyone take the empty road?
The road can be empty while the whole route is slower. Inspect a route to highlight the roads it shares with everyone else.
With no shortcut, each route carries 2,000 drivers. The busy road takes 20 minutes, then the fixed road takes 45. No driver can improve their time by changing routes.
But a shortcut is not always a mistake.
Change the number of drivers. Compare the two networks at the same demand, after everyone has chosen their fastest available route.
The shortcut hurts between 3,000 and 9,000 drivers. At very low demand it helps; above 9,000 drivers nobody uses it.
This control also updates the road diagram above. Opening and closing the link keeps this driver count fixed.
Inspect the 19 calculated scenarios and download them
These are calculated model outputs, not observed travel data. The selected slider value can fall between table rows. Times shown to two decimal places.
| Drivers | Closed min | Open min | Change min | Shortcut users | Best average min |
|---|
Can you beat everyone's best choice?
Back to the 4,000-driver case. You are the dispatcher: decide how many use the shortcut. Split everyone else evenly between the two outside routes. Try to minimize the average trip.
Average time across all 4,000 drivers
A low average can be unstable.
Moving a driver changes the delays other drivers face. An individual sees their own travel time. A dispatcher counts everyone's time.
With everyone on the shortcut, no one wants to switch. Both alternatives are slower.
The target is 64.7 minutes.
The best dispatch sends only 500 drivers through the shortcut and 1,750 along each outside route. Their times are unequal: 45 minutes on the shortcut, 67.5 outside. The weighted average is 64.6875.
That arrangement needs coordination. If outside drivers switch to the faster route, they overload both shared roads, pulling the system back toward 80 minutes.
The dispatch exercise has its own fixed demand of 4,000. It does not change the demand slider or network above.
Take the idea to another junction.
Three quick checks. You can revise your answer after reading the explanation.
0 of 3 understood
The model, made explicit
Adapted from the textbook network by David Easley and Jon Kleinberg. All figures describe a hypothetical traffic model. The headline compares its 4,000-driver equilibria.
We use continuous flow: a single driver's effect is treated as negligible. Congestible roads take x/100 minutes for x users; the other two take 45 minutes each. The one-way shortcut takes zero. There are no queues evolving over time and no new trips generated by opening it.
Reproduce the calculations
Let N be total flow, z the shortcut flow, and split N-z equally between the outside routes.
Shared road flow = (N + z) / 2
Outside route time = 45 + (N + z) / 200
Shortcut route time = (N + z) / 100
Average = [(N + z)² / 200 + 45(N - z)] / N
With the link closed, z = 0. With it open, equilibrium z = min(N, max(0, 9000 - N)). The minimum-average dispatch uses z = min(N, max(0, 4500 - N)). The slider, curve, table and export use these equations.
The best dispatch follows by differentiating total travel time with respect to z. Symmetry minimizes congestion for a given z, so an equal split of the outside routes loses no better solution.
Sources & context
Easley & Kleinberg, Chapter 8 (2010)
University-hosted source for the network and 65-to-80-minute example, pages 230-232. The demand sweep and dispatch calculations here extend those assumptions.
Braess's original paper and English translation
Coauthor Anna Nagurney's university page. Braess identified the paradox in 1968; the translation appeared in 2005.
UK Department for Transport research (28 May 2026)
Why this question still matters: capacity changes can alter travel demand and estimated benefits. That is a separate mechanism from the fixed-demand paradox explored here.
This model shows why a shortcut can hurt. It does not say that a specific real road should be closed.