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:

A Sleeping Beauty paradox

Imagine that one Sunday afternoon, Sleeping Beauty is taking part in a mysterious science experiment. The experimenter tells her:

“I’m going to put you to sleep tonight, and wake you up on Monday. Then, out of your sight, I’m going to flip a fair coin. If it lands Heads, I will send you home. If it lands Tails, I’ll put you back to sleep and wake you up again on Tuesday, and then send you home. But I will also, if the coin lands Tails, administer a drug to you while you’re sleeping that will erase your memory of waking up on Monday.”

So when she wakes up, she doesn’t know what day it is, but she does know that the possibilities are:

  • It’s Monday, and the coin will land either Heads or Tails.
  • It’s Tuesday, and the coin landed Tails.

We can rewrite the possibilities as:

  • Heads, Monday
  • Tails, Monday
  • Tails, Tuesday

I’d argue that since it’s a fair coin, you should place 1/2 probability on the coin being Heads and 1/2 on the coin being  Tails. So the probability on (Heads, Monday) should be 1/2. I’d also argue that since Tails means she wakes up once on Monday and once on Tuesday, and since those two wakings are indistinguishable from each other, you should split the remaining 1/2 probability evenly between (Tails, Monday) and (Tails, Tuesday). So you end up with:

  • Heads, Monday  (P = 1/2)
  • Tails, Monday (P = 1/4)
  • Tails, Tuesday  (P = 1/4)

So, is that the answer? It seems indisputable, right? Not so fast. There’s something troubling about this result. To see what it is, imagine that Beauty is told, upon waking, that it’s Monday. Given that information, what probability should she assign to the coin landing Heads? Well, if you look at the probabilities we’ve assigned to the three scenarios, you’ll see that conditional on it being Monday, Heads is twice as likely as Tails. And why is that so troubling? Because the coin hasn’t been flipped yet. How can Beauty claim that a fair coin is twice as likely to come up Heads as Tails?

Can you figure out what’s wrong with the reasoning in this post?

Easy Math Puzzle – Or is it?

How good are you at basic math? Can you solve this simple logic puzzle? Here, give it a go and let me know how long it took you to answer:

Got it yet?

It looks easier than it is. The options are presented beautifully to cause maximum mental confusion.

As my dad put it, the answer depends on the answer. If the answer is 60%, it’s 25%. If the answer is 25% it’s 50%. If the answer is 50% it’s 25%. There’s an endless loop with no correct answer.

Don’t lose sleep, I “found” an answer, it was hidden: [edited for clarity]

Yes, I photoshopped this. I’m either cheating or engaging in outside-the-box thinking. Sometimes it’s tough to tell the difference.

My preferred set of answers would be:

  • A) 25%
  • B) 50%
  • C) 75%
  • D) 50%

Though I’m tempted to throw a “0%” in for good measure…

(Puzzle via PostSecret by way of Spencer of Ask a Mathematician/Ask a Physicist)

[Edited for clarity]

The mirror paradox

Have you ever wondered why mirrors flip your image horizontally, but not vertically? It’s one of those curious things that doesn’t actually seem curious at first — it seems perfectly natural. But that’s simply because we’re so used to it. Once you reflect on it (sorry!), it becomes much less obvious why that’s the case.

After all, a mirror doesn’t know or care which way is “up” — in fact, there is no such thing as “up” built into the world. It’s just a term we have for a direction pointing away from some plane we’ve designated as the “ground,” and we only bother to define these concepts because we humans happen to be very concerned with gravity and its effects on us. The light bouncing off of a mirror, one assumes, has no such concerns — it could care less about our concepts of verticality and horizontality. But then why is there this asymmetry in the way a mirror reflects our image, turning us around left-to-right but not top-to-bottom?

The first step towards an answer is to realize that the question is flawed. The mirror doesn’t actually reverse your image either left-to-right or top-to-bottom — it reverses your image front-to-back, that is, along the axis perpendicular to the mirror. Imagine you had a hollow Halloween mask, and you turned it inside out. That’s exactly what a mirror does: it “turns you inside out,” so that you’re facing the opposite direction without having been rotated.

But if the mirror is just flipping our image front to back, why does it look like we’re being flipped left to right? It’s because the left and right sides of our bodies are almost identical. So the inside-out person you see in the mirror looks a lot like what you would see if someone created a clone of your body and rotated it 180 degrees so it was facing you. That’s why you have the strong feeling that the mirror is rotating you in the horizontal plane, even though it’s actually just turning you inside out.

The illusion wouldn’t be nearly as strong if our bodies didn’t have left-right symmetry. Let’s say you had a tentacle in place of your right arm. Then your mirror image wouldn’t look like a 180-degree rotation of yourself, because the tentacle would be on the wrong side. The mirror’s ability to make us feel like an image has been turned around only works with a symmetrical axis.

Another experiment you can do to drive this point home is to try lying down on the floor, on your left side, facing a mirror. Now your mirror-image will be lying on his right side with his left side on top, while you’re lying on your left side with your right side on top, creating the illusion that the mirror is flipping your image vertically, as opposed to when you were standing in front of the mirror and it looked like your image was being flipped horizontally. This proves that the nature of the illusion is different than we’d originally thought: it’s not that the mirror seems to rotate you horizontally rather than vertically, but that the mirror seems to rotate you around your symmetrical axis rather than your asymmetrical axis.


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