- Look at this arrangement of springs and rope.
These two springs are joined together
by this short bit of blue rope
and there's a weight on the bottom.
There's also a red rope and a green rope.
The red and green ropes are slack.
They're not taking any of the weight.
So the question is what will happen
when I cut the blue rope?
I mean, look, you could think about the forces and energy
and try to work out if the weight will go up or down.
Or you could just consider the fact
that I'm making a video about it.
So it's probably the counter-intuitive answer.
Like, this blue rope is in tension.
So when I cut it, I will release that tension.
So surely the weight will go down.
Let's see what happens.
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(weight thumps)
Isn't that crazy?
The weight goes up.
Why does it do that?
Well, when you add a weight to a spring, the spring extends.
And of course, if the spring were twice as long,
it would extend twice as far like this.
But actually if you consider the initial setup
before the blue rope was cut,
that's exactly what we have.
The blue and green ropes are slack.
They're not doing anything.
They might as well not be there.
And so we have a spring that is twice as long,
plus this little bit.
So these two springs together will extend twice as far
as one spring would on its own.
You could say that these springs are in series,
but what would happen if you put the springs in parallel?
Okay, so start again with a mass on a single spring
and then add a second spring like this.
Now the weight is shared between the two springs.
This spring takes half the weight.
If you half the weight on a spring,
it should extend half as far.
And look, that's what we see.
Actually, it's not exactly what we see
because my springs aren't ideal springs.
They don't follow Hooke's Law.
But anyway what's interesting is springs in parallel
is what we have after the blue rope is cut.
Look, two springs in parallel.
It's just that they've both got this extension rope
on them as well.
So the reason the weight moves up when you cut the blue rope
is because you're switching from two springs in series
to two springs in parallel.
But then I was like, "Hold on, hold on.
If I were to slowly extend the blue rope,
instead of cutting it, surely the weight would go down."
Like my brain wasn't happy, still.
So look, here's the set up here.
I can slowly release the blue rope
and watch what happens as I do that.
Actually the weight does go down.
But it only goes down
until the red and green ropes become taut.
And then as I release more and more of the blue rope,
the weight goes up again.
It's at that point when the red and green ropes become taut,
that you begin to transition from one regime to the other,
from springs in series to springs in parallel.
What's really cool is that this arrangement
of springs and rope is actually an analogy
for a traffic-based paradox called Braess's Paradox.
Imagine we've got two towns, A and B,
with a network of roads between them.
The red and green roads here are really wide and quite long.
The silver roads are really narrow and quite short.
The blue road is really wide and really short.
The red, green and blue roads are so wide,
they never get congested.
It doesn't matter how many cars you put on those roads,
it always takes 11 minutes to travel down the green road
and 11 minutes to travel down the red road
and one minute to travel down the blue road.
The silver roads are so narrow,
the travel time depends on how many cars are on the roads,
because they get congested.
For simplicity, let's say
that the silver roads take one minute to travel down
for every 100 cars that are on those roads.
Let's imagine there are 800 cars trying to get from A to B
and they all take this route.
So it's eight minutes here, one minute here
and eight minutes here.
That's 17 minutes in total.
But is that what drivers would actually do?
Would they all take this same route?
Or would some drivers switch to a different route?
Well, a driver could switch to this route or this route.
But both those routes take 19 minutes,
which is more than 17 minutes.
So no individual driver would take that route.
What we have is an equilibrium.
All drivers taking this same route is an equilibrium
because it doesn't make sense for any individual driver
to change route.
How does this relate to the springs and ropes?
Well, the weight is like the number of cars on the road
and how far down the weight goes is like the travel time.
The green, blue, and red roads
are like the green, blue and red ropes.
It doesn't matter how much weight you put on them.
They don't get any longer.
Just like it doesn't matter how many cars you put
on those roads,
the travel time doesn't go up on those roads.
The silver roads are like the springs.
The more weight you balance to the springs,
the lower the weight goes.
Just like the more cars you add to the silver roads,
the longer the travel time gets on those roads.
In this setup, the green and red ropes are slack.
They're not being used in the same way
that the green and red roads aren't being used.
So what happens when you cut the blue road?
Well, there are now two possible routes between A and B.
And assuming that drivers
can have up-to-date information on congestion,
maybe because they've got Google Maps or whatever,
then a new equilibrium will be reached,
where half the cars take one route
and half the cars take the other route.
That means there are now 400 cars
on each of those silver roads.
So the total travel time for any one car
is now four plus 11 equals 15 minutes.
And that's less than the 17 minutes we had
before the blue road was cut.
And that's Braess's Paradox.
By removing a short fast road from a network,
you might actually decrease travel time.
It's almost like a multi-player version
of the prisoner's dilemma.
And this isn't just an academic curiosity.
It's happened in real life.
We know that in Seoul and Stuttgart it's happened
where the removal of a road has actually caused travel times
to go down without the number of cars
on the road going down.
But surely if you added the blue road back in at this point,
drivers would choose to continue using the better,
faster outer routes.
But with everyone using those outer routes,
look how attractive the blue route is.
Four minutes plus one minute plus four minutes
equals nine minutes,
compared to the 15 minutes of the outer routes.
So it makes sense for any one individual driver
to defect to the blue route.
And unfortunately, every driver that defect
makes the blue route a little bit worse,
but it also makes the outer routes a little bit worse
as well.
In fact, the blue route is always preferable.
So all drivers will switch to the blue route
and it will be worse.
You know, if all the drivers could agree
to not use the blue road as if it wasn't there,
then we could keep it for people who want to go
from here to here.
Wouldn't that be nice?
Unfortunately, humans don't form hive minds,
so it can't happen.
Except maybe one day, we will solve the problem
with a hive mind.
Like if self-driving cars take over the majority of driving
and these self-driving cars can act cooperatively,
then maybe one day we can experience this utopia
of optimization.
It's never going to happen, is it?
Like even if all the car companies could agree
to write software that acts cooperatively with each other,
there's always going to be a human in the car,
ready to override the software and say,
"Do you know what?
Take me the fastest route this time."
Anyway, thanks to George Maveron Nicholas
for the idea for this video.
I appreciate it.
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