# Would you open the shortcut?

4,000 drivers. Two routes. 65 minutes to arrive. A very fast new road becomes available. What do you think happens?

At 4,000 drivers, opening the shortcut raises the journey from 65 to 80 minutes. Below 3,000 drivers it helps. At 3,000, both options take 60 minutes.

## Everyone chooses well for themselves. Why do they lose together?

On the upper route, one road gets slower with traffic and the other always takes 45 minutes. On the lower route, that order is reversed. The shortcut connects the two roads that get slower.

The 4,000 drivers divide equally. The variable road takes 2,000 ÷ 100 = 20 minutes. Every journey takes 20 + 45 = 65 minutes.

With the new road open, everyone ends up using both variable roads. Each takes 4,000 ÷ 100 = 40 minutes. The total is 40 + 0 + 40 = 80 minutes. A driver avoiding the shortcut alone would take about 85 minutes.

At equilibrium, nobody gains by changing route alone. That does not guarantee the best journey time for everyone. This is Braess’s paradox.

## How the calculation works

On each variable road, the time in minutes is the number of drivers using it divided by 100. The other two roads each take 45 minutes.

Without the shortcut, time is 45 + N ÷ 200. With it, time is N ÷ 50 up to 4,500 drivers. Between 4,500 and 9,000 drivers, time is 90 minutes.

N is total demand. We use a continuous-flow equilibrium: every used route has the same minimum time. Fractional values are rounded to one decimal place. The moving dots do not simulate real minutes.

## A model with clear limits

This is a calculation for an imaginary network, not a forecast for a city. Demand stays fixed, people seek the fastest route, and the shortcut takes zero minutes. We do not model queues, traffic lights, or people starting to drive because a new road opens.

[Easley and Kleinberg, Networks, Crowds, and Markets, chapter 8, figures 8.1 and 8.2. Variable-demand scenarios are calculated here using the same rules.](https://www.cs.cornell.edu/home/kleinber/networks-book/networks-book-ch08.pdf)

[A project by Marius Comper](https://mariuscomper.uk/en/)