How To Benefit From Braess’s Paradox?

Braess’s paradox originates from the observation that overall traffic efficiency in a road network decreases under certain conditions after one or more roads are added to the network.

Dietrich Braess was a mathematician at Ruhr University in Germany. When he was doing research on traffic modelling in 1968, he discovered that the flow in a road network could be hindered by adding a new road.

At first thought, this idea sounds counter-intuitive. Why would an extra road aimed at increasing connectivity slow traffic down? In this article, I will be answering this question using a mathematical demonstration of Braess’s paradox. Following that, I will briefly touch upon a few real-world applications of the paradox and show how the applications can be extended beyond road networks.

For the mathematical analysis, let us start by designing a sample road network we can work with.

This essay is supported by Generatebg

Setting Up a Sample Road Network for Braess’s paradox

Let us consider a road network as shown in the below image. A total of 1000 cars need to travel from a town named ‘START’ to a town named ‘END’. There are two paths available to make this trip. One option is to drive through junction ‘A’ (START-A-END). Another option is to drive through junction ‘B’ (START-B-END).

We fix the time taken for any car to travel from ‘START’ to ‘A’ (START-A) at 110 minutes. On the other hand, the time taken for any car to travel from ‘A’ to ‘END’ (A-END) depends upon the traffic. This dependency is expressed by the mathematical relation: (T/10) minutes, where ‘T’ is the total number of cars on the A-END Road.

For the other path, we reverse the situation. The time taken for any car to travel from ‘START’ to ‘B’ (START-B) is given by (T/10) minutes, whereas the time taken for B-END is fixed at 110 minutes. Here, T is the total number of cars on the Start-B Road.

Now that we have set the problem up, let us go ahead and analyze the mathematics that leads to Braess’s paradox.

---

The Math Behind Braess’s paradox

We have four roads in total: START-A, A-END, START-B, and B-END. The time taken for any car that drives through START-A-END is calculated as follows:

Similarly, the time taken for any car that drives through Start-B-END can be calculated as follows:

Assuming that the whole system eventually settles down to an equilibrium state (game theory’s Nash-equilibrium, for example), one-half of the cars drives through START-A-END, and the other half drives through START-B-END. Consequently, the equilibrium travel time taken for each path is calculated as follows:

So, we see that given the current configuration, both paths take the same amount of time (160 minutes) — the Nash equilibrium.

A New Road — More Options

Let us now introduce a new road that connects the junctions: ‘A’ and ‘B’ (A-B). Consequently, any driver has the choice to switch from ‘A’ to ‘B’ and vice-versa at each of the junctions. Let us further suppose that A-B takes approximately 0 minutes to traverse. With this new change introduced, let us see how the road-network behaviour could change.

Let us start from the previous equilibrium state. Suppose that one driver tries out the START-B-A-END route. He discovers that his new driving time is significantly reduced by the following calculation:

He saves 59.9 minutes by taking this new route. Having noticed this, more and more of the 1000 drivers start trying this new route. As a result, the time taken to drive through START-B-A-END starts to rise. When the number of drivers trying the new route reaches 600 (with 400 on the Start-A-END route), the time advantage is neutralised. This is expressed mathematically as follows:

Meanwhile, the drivers on the START-A-END route are being slowed down by the following mathematical effect:

This segment of drivers (START-A-END) is now forced to switch to the new route via B too (START-B-A-END). As a result, the time taken from ‘START’ to ‘END’ settles down to a new (Nash) equilibrium as follows:

At this point, none of the drivers has an incentive to take the START-A route or the B-END route, as trying these would mean taking 210 minutes.

Therefore, we see how opening a new road (A-B) costs everyone 40 minutes more as compared to the original road network configuration (160 minutes versus 200 minutes). If there is one thing that we can take away from this, it is that the (Nash) equilibrium state of road networks need not necessarily lead to the minimum travel time.

This pretty much wraps up the mathematical analysis of Braess’s paradox. Let us look at its practical applications next.

---

Practical Applications of Braess’s paradox

In 2008, Gastner et al. (reference link at the end of the article). demonstrated how closing down specific roads in Boston, New York City, and London could speed up the respective traffic routes. In the following year (2009), New York experimented with shutting down Broadway at Times Square and Herald Square for traffic. This led to such an improvement in traffic flow that New York decided to keep the changes (source: The New York Times). It also resulted in permanent pedestrian plazas.

In 2012, a similar effect was observed in Rouen, France. This time, a bridge had burned down by accident. Contrary to what the locals had expected, the traffic flow significantly improved during the following two years of bridge restoration. It turned out that the other bridges were used more and the total number of cars crossing bridges was reduced (source: Le Monde).

Physicists have extended the concept of Braess’s paradox to power delivery speed in electric grids. It turns out that extra network capacity can negatively affect power delivery speeds just like in road networks (reference link at the end of the article).

Similarly, Motter et al. (reference link at the end of the article) demonstrated how Braess’s paradox can be extended to biological and ecological systems. For instance, selective removal of a doomed species in a food chain could help prevent a series of species extinctions in the food chain.

Final Remarks

Asis often the case with mathematics, Braess’s paradox demonstrates how a mathematically significant observation/discovery could find beneficial applications in different fields.

Having said this, empirical relevance is key in implementing any application of this concept. This is because real-life network systems are seldom deterministic. They are usually highly parameter sensitive. This means that an initially efficient network could significantly change behaviour depending upon a minor dynamic shift in initial or boundary conditions. Game theory has its limitations as well.

One way to be prepared for such shifts could be to model optionality in the network system(s). Then, one could dynamically open or shut down optional pathways based on a combination of mathematical simulations and empirical results.

---

References: Gastner et al. (scientific paper), Max Planck Institute (scientific article), and Motter et al. (scientific paper).

I hope you found this article interesting and useful. If you’d like to get notified when interesting content gets published here, consider subscribing.

Further reading that might interest you: How To Solve The Three 3s Problem? and How To Really Use Mathematical Induction?