A city opens a shortcut. Nobody is required to use it. Everyone still has the old roads available. Yet the new pattern of individually sensible choices can leave every traveler taking longer than before.
This possibility is called Braess’s paradox. It is a statement about how choices interact in a congested network. It does not say that all new roads make traffic worse. Its value is that a small, inspectable example can overturn an apparently safe assumption: adding an option cannot hurt if people are free to ignore it.
Build a city small enough to calculate
Our fictional network has a start S, destination T, and two junctions, A and B. There are initially two routes: S–A–T and S–B–T. One segment on each route has a constant modeled time of forty-five minutes. The other takes x/100 minutes when its flow is x travelers per chosen interval.
The variable segments are S–A and B–T. The fixed segments are A–T and S–B. Think of the numbers as a deliberately simplified cost rule. A constant forty-five-minute segment is an assumption for this mathematical example, not a realistic promise that a particular road ignores congestion.
Send 4,000 travelers through. With 2,000 on each route, each variable segment costs twenty minutes. Each full route therefore costs sixty-five minutes. If a traveler moves from one otherwise identical route to the other, the destination route becomes slightly more congested. The balanced split is the familiar equilibrium.
| Route | Flow | Time |
|---|---|---|
| S–A–T | 2,000 | 20 + 45 = 65 minutes |
| S–B–T | 2,000 | 45 + 20 = 65 minutes |
Add one almost irresistible link
Now open a one-way link from A to B with zero modeled time. The third route is S–A–B–T. It connects the two variable segments and avoids both forty-five-minute segments. At the old flow levels, that looks like a forty-minute route.
The trouble is that those old flow levels do not stay fixed. As travelers move onto the new route, more use both variable segments. When all 4,000 use it, each variable segment takes forty minutes. The shortcut route now costs eighty minutes.
Can one traveler improve by returning to an old route? S–A–T costs about forty plus forty-five, or eighty-five minutes at the prevailing flows. S–B–T likewise costs about eighty-five. A unilateral deviation does not help. Everyone has arrived at an eighty-minute journey even though the old sixty-five-minute arrangement remains physically possible.
Open the shortcut
Change modeled demand and compare no-link and all-shortcut equilibrium travel times within the stated range.
Try a lighter rush hour
The calculator changes total demand while keeping this network’s rules. At 2,000 travelers, the original equal split costs fifty-five minutes. If everyone takes the new route, its two variable segments total forty minutes. At that demand, the link helps.
At 3,000 travelers, both arrangements cost sixty minutes. At 4,000, the added link leads to eighty instead of sixty-five. Demand matters. A diagram of the roads alone cannot tell you whether this particular paradox occurs; you also need the flow and cost relationships.
The control deliberately stops at 4,400 travelers. Beyond the range where all-shortcut flow remains an equilibrium, another calculation is required. A model should not continue displaying a convenient formula after its assumptions stop supporting that answer.
The missing cost is imposed on other people
Each traveler considers the journey available to that traveler. Entering a congestible segment also increases the cost of that segment for others. In the fictional network, switching to the shortcut is attractive under the initial conditions, but a collective migration changes the conditions everyone faces.
The English translation of Braess’s 1968 paper distinguishes an individually optimized flow from a flow that minimizes the system’s cost. The two objectives need not select the same arrangement. The paradox is therefore not a failure of arithmetic or proof that the travelers are individually foolish.
An organizer could preserve the original split by coordinating choices. In a real setting, pricing, access rules, information, or different network design might change the incentives. Those possibilities require their own analysis. The classroom network cannot tell a particular city which intervention it should adopt.
What would count as evidence in an actual city?
A before-and-after travel-time change by itself is not enough to identify this mechanism. Traffic demand may change, construction may end, transit service may improve, or a measurement method may differ. A serious study needs the relevant network, route choices, changing flows, and appropriate comparison conditions.
Anna and Ladimer Nagurney’s research chapter places the paradox among congested networks with decentralized decisions. The University of Massachusetts research collection links the original work and later studies. Those are routes into the evidence, while our forty-five-minute example is explicitly fictional.
There is another practical qualification. The word “selfish” often appears in the mathematics of routing. It describes an optimization assumption: each participant selects a personally favorable route. It is not a diagnosis of a driver’s character or an accusation about anyone who follows a navigation app.
The useful question after the surprise
Return to the opening claim that people can simply ignore the new option. They can, but they must make decisions in a network reshaped by everyone else’s decisions. The old road still exists; the old distribution of traffic does not automatically persist.
That is the idea worth carrying away. When an added choice changes how people share a limited resource, compare the resulting equilibrium with the previous one. Do not evaluate the new option in isolation while quietly freezing the rest of the world.
Follow the evidence: Braess, Nagurney and Wakolbinger: original paper in translation; Anna and Ladimer Nagurney: The Braess Paradox; University of Massachusetts: Braess research collection.
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