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! 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! 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... 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. CuriosityStream gives you unlimited access
to over 2,400 brain-feeding documentaries starting at just $2.99 per month. 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.
