Demonstrating Braess’s Paradox with Springs
Many others have already discussed the applications of Braess’s paradox from city traffic and sports to social networks and power grids. However, Braess’s paradox can also be demonstrated through a simple physics experiment using some springs, ropes, and a weight.
The above image shows the set-up of the experiment, where a weight is held up by two springs and three ropes. From the top, the green rope is tied to the lower spring, which supports the weight. Another spring attached to the top is tied to the red rope which is tied to the weight. The two springs are connected by a short blue rope. In this equilibrium, all of the weight is on the two springs and blue rope; the green and red ropes are completely slack and hence share none of the weight.
So what will the new equilibrium look like when the blue rope is cut? Intuitively, one might expect that the weight drops because when the blue rope is cut, all of the tension in the blue rope is released. This may cause the two strings to lengthen and the remaining ropes to tighten, lowering the weight. However, it turns out that cutting the blue string will instead cause the weight to rise (Braess’s paradox).
To find out why this happens, it is important to model this set-up with a graph. The graph from the 9/13/21 Traffic Lecture Slide works perfectly:
Here, the value of the weight is represented by the number of cars on the road and how far the weight moves down is relative to the travel time. The ropes are like roads with a constant travel time per person (it doesn’t matter how many weights are applied to the rope, the length remains constant). Meanwhile, the springs are similar to roads with varying travel times. The more weight added to the springs (or the more people traveling the road), the longer the string extends (the longer the travel time). The graph above matches the initial state of the springs because the constant roads are irrelevant: all traffic travels on the time varying roads/springs.
In the spring example, the two springs are initially in series (the strings take none of the weight and thus have no effect) so the overall spring extends twice as far as one spring. After cutting the blue string, the two springs become parallel (two separate paths) and the weight is equally shared between the two springs. This decreases the force on each spring, which in turn decreases the stretch of each spring and raises the weight. Thus, the experiment matches the traffic example covered in class. When the blue string is cut, the road between the two separate paths is destroyed and counter-intuitively, the travel time per person results in a decrease. This phenomenon demonstrates Braess’s Paradox.
Reference: “The Spring Paradox” by Steve Mould (https://youtu.be/Cg73j3QYRJc)