Vsauce, Kevin here, with a slot car race track to visualize how fast I can safely drive without
getting pulled over.
What’s my maximum speed without wreaking havoc on the justice system?
And how can we use game theory to find the answer?
Here we go.
Oops!
Alright that didn’t really work out very well.
Honestly, I don’t even have enough room to properly setup this track on this table
so...
Um, I'm gonna need some help.
Luckily, I partnered with CuriosityStream, and I know just the guy...
Hey, Sean!
As host of the awesome new CuriosityStream show, Speed, I was hoping that you could help
me visualize speeding but.. it looks like... you already have the track here.
Yeah, well.
Kevin, to do just that, I have magically manifested exactly the same track that you have, right
here on my table.
That's super convenient.
Yeah, boom!
Okay so.
Let's race.
I will be the cop car and you will be the civilian driver.
I wouldn't have it any other way.
Okay, do I get a head start?
Yes.
I'm gonna pull you over.
Let's go!
Let's go.
Okay, so I'm racing ahead.
There's a turbo button, which you can use kinda in the straightaways, that's helping
me gain on you, actually.
Oh...
But...
Ahhh....
If you turbo in the corner it's too much speed and you fly right off.
Why did that happen, though?
Okay, that happens because, in this case, you exceeded the coefficient of friction between
the tires, the slot, and the top of the track.
Now, these cars, they can go around the corner a lot faster than a normal car can.
Because they have this little peg that's riding in that slot.
Yeah.
So when they go around the corner, they're resisting centripetal force which wants to
sling it off... the car wants to keep going in a straight line.
Right.
But this slot helps drag it around the corner.
Now in our cars, we don't have a slot.
So if we go around a corner too fast we slide off when we exceed the coefficient of friction
of our tires.
And our tires, although they look kinda big, they really don't have that much space on
the road.
And I'm gonna show ya that.
I brought one of the tires off one of my motorbikes here.
I'm just gonna sharpie the top here and this looks like a pretty big, round tire.
Right?
Rides down the road...
Only when it's riding down the road, on the street, it's really only got about 1 inch.
Wow!
Right?
If we take a look at that, there.
There's really only about 1 inch wide and maybe 3 inches long.
That's the contact between me, and the motorcycle, and the ground.
The same is true on the tires on your car, right?
The tire's all the way around but the only part that's actually touching the ground is
that little bit on the bottom.
And if you have your tire's overinflated, you're getting a much smaller contact patch.
That makes sense.
Because your tire's not deforming.
Yeah, yeah!
So that little contact patch is important to understand.
You got four of these little contact patches that are maybe 4 inches by 5 inches wide on
your four tires and the weight of that car pushing down on that contact patch is the
only thing that keeps you from sliding off the road when you go around a corner.
The weight of the car, the inertia of the car, wants it to keep going straight, when
you turn the car, the front wheels move off in one direction and the friction makes those
tires track around the corner.
The reason that's important, is because you're also using that same friction, not only to
go around a corner but you're using it to accelerate and to brake.
If you accelerate too fast, you overcome the coefficient of friction between your tire
and the ground with that little contact patch and you do a burnout.
Mmm hmm.
Which can actually be a lot of fun, in a controlled environment.
But!
The more important thing for all of us driving around, that's really, really pertinent to
our safety is that the faster you go, you're asking more and more of that little tiny contact
patch to help slow you down when you hit the brakes.
So if you're going way faster, you have to be easier on the brakes.
If you lock up the brakes, you'll slide the whole car forward and that's what's happening
here.
Is you're just having it carrying too much speed into the turn, tires and the slot can't
take it, and it's sliding right off the track.
Okay.
Wanna try it again?
Yeah, yeah!
Umm.. where's the cop car?
Let's restore the...
Go ahead!
Okay!
Here we go.
So that's a technical reason why you don't wanna speed but there's lots of other reasons
why you don't wanna speed too, right?
Yeah, I mean, that's why it's dangerous to speed but also, in society, y'know, we have
to have..
Oops.
There we go.
In society, we can't have that happening.
No, can't have that happening.
Y'know, we want everyone to be safe and we have rules, regulations, and law enforcement
in place to make sure that everyone is acting safely.
The thing we don't really think about is that driving in a system governed by law enforcement
is actually a two player game.
And both the driver and the police officer are playing this game.
Let's say that the speed limit is 65 miles per hour.
What's the fastest you've ever driven because I have a feeling it's a lot more than that.
I've drive about 130 miles per hour.
Okay.
Yeah.
But my question is, what is a sensible speed to drive on the highway?
The answer to that question is somewhat complicated because it really depends on the highway,
and the conditions, and if it's raining, and the state of your car.
Right?
But, in general, for roads that are, we collectively deem we need to go slower in because either
it's near a neighborhood, or because the road itself is twistier, we have lower speed limits.
Right?
Yeah.
And we have big, straight superhighways where we can all do 75 or whatever the limit is
in your area.
But, the point is that, cops can't pull over single driver just for going one mile per
hour over whatever that limit is because, not only is that logistically impossible,
but that level of law enforcement would probably make everyone pretty angry.
So, when we drive and the police monitor us, we're playing a game of recursion, to get
the best result.
Both of the police and the drivers are sort of playing this game from different sides.
Exactly, exactly.
We've got a couple of basic rules in this game.
We have a legal speed limit of, let's say 65 miles per hour, and we have common sense
rules that tell us that we can only go so fast before it gets dangerous and we're flying
off the edge of the road and the police officer has a threshold at which he or she will ticket
the driver.
So each player is deciding what to do based on what they think the other player will do.
Yeah, there's a gray area, isn't there?
Yeah, it basically works like this.
As the cop, I'm thinking how fast does Sean think he can go without getting pulled over.
Right, and as the driver, I'm thinking, how fast can I get away with, without getting
pulled over.
Exactly!
And as the cop, I'm also thinking how fast does Sean think I think he thinks he can go?
And he's thinking, how fast do I think he thinks he can go?
And that's recursion.
At the end of this process, which can be infinite, both the driver and the cop decide their best
course of action.
For the driver, that's, like you said, how fast he can go safely without getting a ticket,
and for the cop, that's how fast he'll let the driver go before pulling him over.
Right, and this whole game takes place in that gray area.
I mean I know if I stay under 65, I'll never get pulled over.
That's the easy way to do it.
Right?
And I know if I go 130, I'm gonna get pulled over.
So there's this strip there, and that's the gray area where that game you're talking about
is getting played.
Ultimately, so, what do you think would be an exact value if the limit was 65 miles per
hour that we could get.
That you could drive, without getting pulled over, and so that you're happy with the speed
that you're going.
I think it's general wisdom that you can get away with about 5 miles over the speed limit,
especially on the highway when there's no chance of coming across a stop sign or someone
on the road, right?
So when there's... in a safe environment like that I think everybody pretty much feel like
they can go 5 miles over the speed limit and not get pulled over or somewhere in that gray
area.
Both the police and the speeder, are playing this game from their own sides, right?
The speeder trying to figure out how fast they can get away with, the police trying
to figure out how much they'll allow, but they both have an incentive to make the whole
system work.
So when they both play the game and they find this place where they get to speed a little
and don't get pulled over they're in that gray area and they find a balance.
And that balance is called a Nash Equilibrium.
What Sean described is the best reply to another player’s best action, and when neither player
has an incentive to deviate from their decision, they’ve reached a Nash Equilibrium.
Okay, I need to talk more about Nash Equilibrium but I need my whiteboard table, so, thank
you so much, Sean for all of your help.
Make sure you watch his show, Speed on CuriosityStream, I know that you guys will love it.
But for now...
I have to...
I have to go.
Okay, now one of the most famous Nash Equilibrium scenarios comes in the form of the well-known
Prisoner’s Dilemma.
The scene is simple: two people get arrested for a crime.
They’re separated and can’t communicate with each other.
The police can convict both on a lesser charge, but they need more proof to convict them on
the big one.
That means one has to testify that the other did the crime.
So, is it in each prisoner’s best interest to rat on the other one or to stay quiet?
Here’s how it plays out:
If both Prisoners A and B rat on each other, they’ll each get 2 years in prison.
If A blames B, but B stays quiet, A goes free and B gets 3 years in prison -- and vice versa.
If both A and B stay quiet, they’ll each get 1 year in prison.
The rational prisoner rats, because that’s the only way he’s going to go free… and
when he rats, it always gives him a better result than if he doesn’t rat regardless
of what the other prisoner does.
That’s called a dominant strategy, and the Nash Equilibrium in the Prisoner’s Dilemma
is achieved by… betrayal.
But if the other prisoner also behaves rationally, he would rat as well, and they’d both get
2 years.
The best course of action here is cooperation where both stay quiet, serve a lesser sentence,
and have the least bad outcome.
In the Prisoner’s Dilemma, not betraying the other guy goes against the Nash Equilibrium
-- but it gives the best possible result for both players combined.
Because if one rats and the other doesn’t, 3 years are served, if both rat 4 years are
served, and through cooperation only 2 years are served.
Political scientist Robert Axelrod asked 14 game theory experts to submit computer simulations
of the Prisoner’s Dilemma.
The ones that advocated more self-interest by betrayal were noted as “not nice” players,
while the ones that focused on cooperation, they were “nice.”
After running them all, the top half of the performers were all “nice” models, while
the bottom half were all “not nice.”
So, it pays to be nice… as long as the other player thinks it pays to be nice, too.
We play these 2-player games all day long, not just with the police.
When we give a gift, we’re playing a game.
For example, say you’ve just met someone, and you kinda like them.
Giving them a spatula is probably not a great gift.
Giving them a diamond ring on the second date is probably a bit much.
Pump the brakes, there...
Billy.
In the gift-giving game, you want to send the right message based on what you feel and
the recursive process that happens with what you think they’ll feel.
You want to feel good about the gift you gave and they want to feel good about the gift
they’ve received.
And whether you’re driving down the highway or trying to impress your crush, you’re
almost always in some kind of two-player game, and sometimes you’re in several at once.
The best way to play is to be smart.
To pay attention.
And keep your best interest in mind.
But you’ll usually win the most when you make sure to be nice.
I guess grandma was right.
And as always -- thanks for watching!
Hey gigantic thanks to Sean Riley for opening up his shop to me.
Go check out his brand new show Speed.
It’s a four-part docuseries where Sean blasts through humanity’s need for speed.
On land, wow!
On sea, through the skies and into space.
CuriosityStream and I worked out a deal for you so that can sign up and get 30 days completely
free.
Just click the link below and go to CuriosityStream.com/Vsauce2 and enter the promo code vsauce2, all lowercase.
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But for you, you can watch Sean’s show free by using the promo code vsauce2.
So you should do that.
The show is awesome.
Honestly.
You’ll like it.
Go watch it.
Go watch it now.
Okay. now.