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Braess’ Paradox: Close Down Roads, Reduce Congestion

Fun & everyday life

In 2010, when the mayor of New York City at the time, Michael Bloomberg, closed Times Square and Herald Square to car traffic, citizens were anxious that the city’s traffic situation would become even worse than it already was. However, NYC’s streets were generally less clogged as a result, with northbound cab rides now taking 17 percent less time.1

According to Youn et al. (2008), this was by no means coincidental.2 Principal roads in NYC, Boston and London have been identified which, if closed down, improve congestion and reduce drivers’ travel time.

Why is that? To understand this counter-intuitive phenomenon (called Braess’ Paradox after the German mathematician who discovered it) basic game theory lends a hand.

Imagine the following simple setup: 100 car drivers want to go from city A to city B in the least amount of time possible. To reach city B, each driver selects either itinerary 1 (‘city A – north junction – city B’) or 2 (‘city A – south junction – city B’). If a given driver chooses itinerary 1, they drive on a highway for 20 minutes until they reach the north junction. From the north junction onwards, the driver then takes a single-lane road until they reach city B. Note that, while the multilane highway is never clogged, the single-lane road is in principle shorter (it takes the driver about 5 minutes to reach city B, if they are the only driver), but easily congested (each driver on the road increases the driver’s travel time by 6 seconds). If the driver chooses itinerary 2, they first drive on a single-lane road and then on a highway (both road and highway in the south are equivalent to their northern counterparts).

How long does it take any single driver to go from city A to city B? Well, this depends on how many cars cS drive on the single-lane road in the south and how many cars cN take the northern, single-lane road. With drivers always choosing the route where they expect less traffic, on average 50 cars follow itinerary 1 and itinerary 2 respectively (cS=cN= 50). Thus, it takes the average driver 20 minutes on the highway and 10 minutes on the single-lane road (5 min + (0.1 × 50) min), totaling 30 minutes.

Road planners now map out a connecting road enabling drivers to get from the north to the south junction and vice versa in 2 minutes flat. The connecting road introduces a third itinerary (itinerary 3: ‘city A – south junction – north junction – city B’) which, leaving its effect on congestion aside, decreases average travel time.

This does not, however, allow for congestion. Let us consider the travel plans of a driver who previously chose itinerary 1. Obviously, the driver should take advantage of itinerary 3, if no other driver does (cN= 49, cS= 50). In this case, the travel time decreases from 30 minutes to 22 minutes and 6 seconds (5 min + (0.1 × (50+1)) min + 2 min + 5 min + (0.1 × (49+1)) min).

Crucially, even if all other drivers decide to follow the third itinerary via the connecting road (cN= 99, cS= 99), it is still time-efficient for the respective driver to do so too. How so? Using the connecting road as well, the driver spends a lot of time stuck in traffic and reaches city B after 32 min (5 min + (0.1 × (99+1)) min + 2 min + 5 min + (0.1 × (99+1)) min). Alternatively, they may choose either itinerary 1 or 2. Inevitably, they are still stuck in traffic on one of the single-lane roads caused by drivers having selected itinerary 3. Hence, the driver reaches city B only after 35 min (5 min + (0.1 × (99+1)) min + 20 min).

It follows that choosing itinerary 3 is a strictly dominant strategy for all drivers. Since all drivers take the connecting road, they are relatively worse off, spending 32 minutes to arrive in city B instead of 30 minutes without the connecting road being available.

As you might have already figured out at this point, this illustration of Braess’ Paradox reflects the Prisoner’s Dilemma: while it would be beneficial to all drivers, if everyone circumvented the connecting road and minimized congestion, each driver individually has an incentive to use the connecting road anyway and leave other drivers stuck in traffic longer. Closing the connecting road (or not building it in the first place) is an effective tool to break up the prisoner’s dilemma. Or to put it simply, less roads sometimes mean less congestion.

In practice, it is not easily determined which roads are detrimental and which ones are beneficial to traffic. Mistakes will inescapably be made: in 1969 the city administration of Stuttgart built an additional street to ease downtown traffic. Shortly afterwards, it was demolished because congestion only got worse.3

 

SOURCES:

1 Gelinas, N. (2012), Ungridlocked, retrieved 18 September 2018, from https://www.city-journal.org/html/ungridlocked-13472.html

2 Youn, H., Gastner, M. T., & Jeong, H. (2008), Price of anarchy in transportation networks: efficiency and optimality control, Physical review letters, 101(12): 128701

3 Kolata, G. (1990), What if They Closed 42d Street and Nobody Noticed? retrieved 18 September  2018, from https://www.nytimes.com/1990/12/25/health/what-if-they-closed-42d-street-and-nobody-noticed.html