Why Blocking Roads Can Speed Up Traffic

It’s so counter-intuitive that it’s called Braess’ Paradox: How can closing a road actually make everyone’s commute shorter? You would think that blocking a route would be an inconvenience, but under some circumstances it’s actually for the best.

Doesn’t sound right, does it?  Here’s the situation: Assume drivers are rational and intelligent.  I know, that’s a stretch – I grew up around DC.  But bear with me.  If there are multiple paths that people can take, they should in theory find an equilibrium between them.  If one path has less traffic and takes less time, more people will switch to it until it loses its advantage.  If one path starts longer than the others, nobody will use it until the other paths get congested enough to make it worth it.

So how can an extra path actually make the average commute time longer?  Shouldn’t an extra path just give people more options to choose from, and ultimately find the best equilibrium?

The Situation:

It turns out that when some roads are more prone to traffic than others, it can create Braess’ Paradox.  Imagine that some roads aren’t as affected by traffic – I picture these as the local roads with traffic lights. They add a fixed amount of time to your commute, say 45 minutes. The other roads are heavily dependent on traffic – these highways can either be wonderfully fast or a mess of stop-and-go congestion, depending on how many other people are on them. The average time it takes to drive on them is the number of cars over 100.

(Image modified from Wikipedia)

Let’s say there are 4000 cars driving from the start to finish. Without the connector (dotted in the diagram), an equilibrium forms where half the drivers (2000 cars) take the top route through A, and half take the bottom route through B.  The highway takes 2000/100 = 20 minutes, and the local road takes 45 minutes. So half the population spends 45 minutes on a local street, followed by 20 minutes on a highway, and the other half of the drivers spend 20 minutes on a highway, followed by 45 minutes on a local street. Everyone gets to their destination in 65 minutes. Nobody has any incentive to switch.

But what if a new connector is opened between A and B, allowing people to go straight from one highway to the other? Now everyone thinks to themselves, “Hey, why spend 45 minutes on a local street when I could spend 20 minutes on the highway? I’m going to take the route Start –> A –> B –> Finish, and shave 25 minutes off of my commute time!”

Of course, if everyone thinks that way, there are now double the cars on each highway than there were before, and it’s half as fast: now each highway takes 40 minutes, not 20 minutes. That’s still 5 minutes less than the 45 minutes it takes to drive on the local street, though, so everyone still has an incentive to take the highway.

So in the end, how has the connector affected people’s commutes? Everyone’s commute used to be 65 minutes; now, everyone’s commute is 80 minutes. And to make it stranger, there’s no better path to take – anyone considering switching to their original route would be looking at an 85 minute drive.

How does this happen?

How can opening a new, super-fast connector make commutes worse? It comes down to the price of anarchy and people’s selfish motivations.  With the connector open, each set of cars has the option to clog up the other half’s highways – saving themselves 5 minutes but adding 20 minutes to the other guys’ commute.

It’s like the prisoner’s dilemma: Each driver has the motivation to take the highways, even though it damages the overall system. Without the connector, nobody is allowed to “defect” for personal gain. In the traditional prisoner’s dilemma, it would be like a mafia boss keeping all his criminals anonymous. Without the option to rat each other out, criminals would avoid the selfish temptation and the entire system is better off.

Braess’ Paradox isn’t purely hypothetical – it has real-world implications in city planning. According to this New York Times article titled What if They Closed 42d Street and Nobody Noticed?, “When a network is not congested, adding a new street will indeed make things better. But in the case of congested networks, adding a new street probably makes things worse at least half the time, mathematicians say.”  That’s shocking. My intuitions about how traffic works were way off.

Lastly, via Presh Talkwalkar’s fantastic game theory blog, Mind Your Decisions, (which brought Braess’ paradox to my attention) there’s a great video of the paradox physically in action with springs. Check it out:

10 Responses to Why Blocking Roads Can Speed Up Traffic

  1. Andrew T says:

    That spring demo is super nifty.

    This kind of thing is why central planning and government coordination isn’t necessarily Evil Communism taking our god given rights to drive as selfishly as possible. But then, so is a lot of rational thinking :/

    • J says:

      Ah, but now selfishly-driven technology has given us adaptive navigation on our smartphones that lets us load-balance across all available paths. So once the shortest route starts to get slower than the others, drivers automatically get routed to the current best route. It’s a great counterpoint to your comment: people have just-so stories about why central planning is best, and defend it even when technology can make things better for everyone.

      • Tyrrell McAllister says:

        On the contrary. Adaptive navigation, or something equivalent, is precisely how each driver knows that he or she can save 5 minutes (individually) by “defecting”, so everyone will defect, so everyone will be worse off.

        Indeed, the scenario in the OP works *only if* everyone has adaptive navigation, or an equivalent source of information. The OP *assumes* that all drivers have this information. That’s how the OP infers how the traffic will distribute itself.

        More information does not help these selfish agents. It’s just like the prisoner’s dilemma. The only way for the agents to help themselves is to commit to cooperating, even though it will be in each individual agent’s self-interest to defect (by using the new road). They will have to coordinate to remove the option of defecting. In that sense, the collective will have to act to constrain the actions of the individual.

        You have to grant certain premises to the scenario, such as that each agent always chooses the immediately fastest route available, and that there are no “outside the box” solutions like flying cars. But, granting these premises, central planning of some kind really is the only option.

        (Or you can just bite the bullet and say that the longer commute time is worth it, if it means avoiding this kind of collectivism.)

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  5. That is interesting, but rather than being a strict paradox, it suggests a low limit to efficacy in complex relations in this particular case. More becomes more hassle as a certain point, and reduced efficiency. One need to pick where the point may be in different cases, if this is a general rule.

  6. Been scratching my head why I like this and I now know. It may relate to pillars of knowledge, like main roads, being connected by supposed reconciliations with jargon to further explain knowledge, when in fact they are blocking up the works, adding hassle.

  7. Dennis Gentry says:

    This isn’t *exactly* the same effect, but it brought to mind traffic on the San Francisco Peninsula during the month the Bay Bridge was closed after the Loma Prieta earthquake. As soon as they re-opened the Bay Bridge, it took noticeably longer to get anywhere on the Peninsula. It’s as if the road network had a certain local capacity, then they overlaid an additional bunch of non-local traffic on top of the already fullish network.

  8. Alrenous says:

    The capacity of the linked network is higher. Confirmed by the spring model: in serial, each is extended less and thus that system can carry more weight overall.

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