The main real world issue with tolls and congestion charges as opposed to actual road closure is one of ethics. The mathematical model presented here clearly demonstrates that reducing traffic flow on the faster route decreases the average time for a journey, the problem with using a pricing system to reduce traffic is that it creates a situation where the poorer sections of society are forced into longer commutes then the more affluent. It is because of this that closing specific junctions seems, to me, the better response

julian havil (2008) also devotes a chapter to Braess rather vividly in his book: “impossible”

link:

http://books.google.com/books?id=yOQuyUthek0C&printsec=frontcover&dq=havil+impossible&ei=WFJ3SpPVForWyATcqsSDAw&hl=nl#v=onepage&q=&f=false

The history goes like this: in 1920, in the first edition of his important and influential _Economics of Welfare_ (I can’t find this edition online), Pigou had an example like yours where too many cars traveled on the narrow, good road and too few on the broad, but poor road. The problem is that each driver ignores the costs imposed on others, called an externality, and this leads to a suboptimal allocation of trucks. He christened this a “market failure” and said government regulation was needed. By applying the optimal tax (or toll), the government could fix the inefficiency of the market because the drivers would be forced to take the costs of their additional congestion into account. Pigou was British and trusted government and experts to get it right.

Knight was an American who distrusted authority and government. In 1926, in the Quarterly Journal of Economics (http://www.jstor.org/stable/1884592), he said this was hardly a market failure because the problem was that no one owned the road! Knight argued that private ownership would fix the problem — the profit maximizing price equaled the optimal toll — and, best of all, without government intervention.

You ended your nice post with “Lest you think this is an academic exercise,” so I will too. Many people think private ownership of roads is silly, but Knight would have smiled when Illinois sold the Chicago Skyway (Knight became famous at U of Chicago) and I live in Indiana which sold its toll road for $4 billion a few years ago. Private roads are no mere theoretical abstraction. There’s lots more to talk about, but I gotta read about the basketball example that originally brought me here!

P.S. Knight’s idea lay dormant and Pigou’s optimal tax/subsidy (some externalities are positive — they grant benefits to others, like painting your house or education) approach was all there was for decades until Coase in 1960. His work led eventually to marketable pollution permits and cap and trade . . .

While it’s certainly true that Braess’s paradox is important, much of this analysis assumes that roads are static. It ignores the problems of accidents, construction, and road maintenance, when road capacity is different from normal. Removing an exit or road tends to make thing more brittle. Tolling is a much more robust solution. (And optimal tolling, while difficult, is superior.)

Let’s take your simplified example. Suppose we have removed the exit at C. Now suppose that there’s an accident or road construction that closes the alley from C to B. Now, if everyone knows about this, then everyone will take the bottom roads, since the top roads are impassible. Total travel time for everyone: 20 minutes. If the exit at C still exists, but there is no tolls or other pricing, everyone will still take the bottom roads, total travel time, 20 minutes. However, if there’s tolls placed on the alleys, then you can not only achieve the optimal result with normal flow, but with the alley from C to B closed you can also have half the drivers take the highway from A to C, then switch at the exit and do highway from C to B, and the other half take the bottom roads the whole way. Tolls provide a better result.

The effect is more dramatic if you assume that the total traffic flow in your example is not 10 but 16. Without exit C closed, the naive equilibrium is 6 people on highway, 10 people on alley each time (with 4 switching at exit C), for everyone traveling in 20 minutes. (People will choose the alley until it fills to 10 cars.) The optimal solution is 5 people on alley, 11 people on each highway (with 6 switching at C), for everyone traveling in 15 minutes. The Cost of Anarchy is 5 minutes per person. With exit C closed, there is some improvement, but not optimally. The result would be that 8 people choose either pair of routes, and everyone travels in 18 minutes. Still a savings, but not as good as tolling.

But now let’s imagine what happens in the 16 traffic flow example if the alley from C to B is closed due to accident. Without exit C closed, 10 people travel on the alley from A to C, 6 on the highway, then all 16 go on the highway. Total travel time for all: 20 minutes. The optimal solution is 5 on the alley from A to C, 11 on the highway, and then all 16 on the highway from C to B. Total travel time for all: 15 minutes. Now let’s see what closing exit C does. All 16 people must take alley from A to C, then highway from C to B. Total travel time for everyone: 26 minutes. Wow, closing the exit really made things worse. With tolls to discourage more than 5 people from taking the alleys, the system remains robust and people still take the optimal route.

Conclusion: Removing exits and roads may improve things for given weights and road capacities, but have a higher probability of making things worse if road capacities change due to accident or construction, and a new optimal path presents itself. It removes robustness and ability for the system to correct, making things more brittle.