Braess's paradox: When More is Less

Dietrich Braess's paradox is a good example of Mathematics which is simple and easy to understand. The result is however, surprising to most people, and counter-intuitive. It shows that adding new roads can make traffic slower! More is less. It is rather similar to the more well known Fred Brooke's law (The Mythical Man-month), "Adding manpower to a late software project makes it later." The following is a simplified example of the paradox.
Assume all cars going from A to D, with the choice of either ABD or ACD. The distances are given below.
A car can travel at maximal speed of 100 km/h, when the number of cars is less than or equal to the road capacity.

path	AB	AC	BD	CD	ABD	ACD
distance(km)	10	50	50	10
capacity	1000	3000	3000	1000
cars	3000	3000	3000	3000
travel time(h)	0.3	0.5	0.5	0.3	0.8	0.8

When the number of cars is greater than the road capacity, the speed decreases by the number of cars divided by capacity. For example, AB has 3000 cars, capacity 1000, distance 10 km. The speed becomes 100/3 km/h and the traveling time is 0.3 h.

Assuming there is no preference for ABD or ACD, the travel time is equal for both paths, namely 0.8 h.
This is an equilibrium, if instead only 2000 cars go the path ABD and 4000 chose ACD, then the travel times are 0.7 and 1.067 h each. Drivers will notice, and more will go ABD until the equilibrium is reached.

Now some road planner has decided to add the stretch BC with the capacity of 2000 cars and distance 20 km.
Let us see what happens:

path	AB	AC	BC	BD	CD	ABD/ ACD	ABCD
distance(km)	10	50	20	50	10
capacity	1000	3000	2000	3000	1000
cars CASE I	4000	2000	2000	2000	4000
travel time(h)	0.4	0.5	0.2	0.5	0.4	0.900	1
cars CASE II	3500	2500	1000	2500	3500
travel time(h)	0.35	0.5	0.2	0.5	0.35	0.850	0.9
cars CASE III	3250	2750	500	2750	3250
travel time(h)	0.325	0.5	0.2	0.5	0.325	0.825	0.85
cars CASE IV	3000	3000	0	3000	3000
travel time(h)	0.3	0.5	0	0.5	0.3	0.800	0

In case I, the distribution of cars is assumed to be ABD (2000), ACD (2000) and ABCD (2000). Path ACBD is omitted, because obviously nobody will select it. The times are worse than before the road BC was built.

In cases II, III we have different car distributions, but again the times are worse than 0.8 h. However, you notice that the times are better, the less cars uses the new road BC.
Case IV is the case when the new road BC is simply ignored, which is really the best.

This result is not related to the systems thinking discussed in Systems Thinking meets Traffic Jams.
There we discussed how new roads can lead to more congestion because the roads will open up new areas of activities, and/or people will buy more cars.
In contrast, in the Braess's paradox, the number of cars is fixed.

Links:

• Braess's Homepage
• Wikipedia on Braess's paradox
• Original paper (German)
• English Translation