Tuesday, 8 July 2014

Pricing away the Braess Paradox?

Here is an interesting "commons"-type problem: Imagine that it takes one hour to travel on Route 1 between Point A and Point B, and that the road goes via the fair town of Middlemarch (from George Eliot's eponymous novel), in a way such that the segment A-Middlemarch takes 40 minutes and Middlemarch-B takes 20 minutes if fewer than Y motorists travel on it, and 30 minutes otherwise. There is an alternate road, Route 2, which goes between Points A and B via nearby Coketown (from Charles Dickens' Hard Times), where the segment A-Coketown takes 20 minutes if fewer than Y motorists choose it and 30 minutes otherwise, and segment Coketown-B takes 40 minutes regardless. In total, there are fewer than 2Y potential motorists who could travel between Points A and B.

The individual motorist makes up his mind thusly: "If more motorists are likely to take Route 1 when I want to travel, I choose Route 2 to save on time, otherwise I choose Route 1." Since we are all individuals, the outcome is that travelling from Point A to Point B takes one hour. The situation is illustrated below (click to enlarge).

Now Coketown and Middlemarch are fairly close to one another, and an expressway is constructed between them which takes only five minutes to drive, so that one can combine the segments from the two routes in novel ways. What is the outcome with respect to time it takes to travel between Points A and B? The answer is that the new road, an option which nobody is forced to take, lengthens the average travel time by five minutes, from one hour to one hour and five minutes!
Queer as this sounds, it is perfectly logical. This is because the longest time it takes to drive the 20X-segments is 30 minutes, which is always less than the alternative, 40-minute, segments. Consequently, individuals are always better off driving on the 20X segments. Traffic on the 40-minute segments will die completely.
I wish I could take credit for this fun application of strategic behaviour, but it is actually a fairly well-known phenomenon referred to as the Braess Paradox, whose possibility was first proved by German mathematician Dietrich Braess and I can imagine many people have written about it before. However, since I have not seen the example in a long while, I thought I would share it here.
A question that remains is how common the Braess Paradox is in the real world. Notice that a properly set toll for using the Middlemarch-Coketown expressway will ameliorate the situation; the motorist then chooses, as before, between a 40-minute segment or a 20X-minute one plus the five-minute drive on the expressway, and takes the 40-minute segment if the time saved is valued at less than the toll.
Interestingly, if the motorist's decision happens to push the time it takes to drive the 20X-minute segment down to 20 minutes, other motorists are willing to pay much more for the expressway, since it now saves ten minutes more of their time. A high-enough toll that makes just fewer than Y motorists take the 20X-minute segments therefore seems to be optimal in this case: those motorists drive between Points A and B in 45 minutes and the other motorists in one hour. Of course, if a lower toll turns out to bring in more revenue, this situation will not arise, but it is possible. In conclusion, even more choice (so that the road's owner may charge people for using it) may lead to improved outcomes; choice in moderation (no toll but an expressway) is bad.

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