- [Justin] This video is sponsored by Blob Shirt.
It's a soft, comfy shirt with blobs on it.
When a resource is contested,
is it better to work together, or look out for yourself?
We're gonna analyze this question
from an evolutionary perspective using simulations
and game theory.
Each of these mango trees has two mangoes
on the bottom layer.
So if a blob finds a tree on its own,
it gets to eat both mangoes.
Then when it goes home,
it uses the energy from each mango to make one offspring.
So two offspring in this case. And also, it dies.
The blobs only live for one day.
On the next day,
the newly born blobs will spread out
to each get their own tree, and then again,
they'll reproduce twice before dying.
So now there are four blobs, but still only two trees.
So they're gonna have to deal with that somehow.
We'll start out with blobs that work together as a team.
They'll notice that there are two more mangoes
higher up in the tree.
And so they'll work together to shake the tree,
causing those mangoes to drop,
and then they'll share the total.
So that comes out to two mangoes each when they cooperate.
It does cost them a little bit of energy
to shake the tree though.
So let's subtract a quarter mango worth of energy
from each blob,
leaving them with one and three quarters when they go home.
You might notice that this number is completely made up.
Later on, we'll look at what happens
when we change some of these numbers.
But for now, we're just picking something specific
to get a simulation running.
When these blobs reproduce,
they'll use the first unit of energy
to create one offspring for sure,
then the remaining three quarters
will convert into a probability for one more offspring.
Okay, two of them reproduced once,
and two of them reproduced twice.
So now we have six blobs total.
But we still only have two trees.
The blobs avoid going to a tree that already has two blobs.
So if the world has more than twice as many blobs
as it does trees,
then the last blobs to go out
will end up wandering around until they die.
So two times the number of trees
will be the carrying capacity for any of these worlds.
All right, at this stage, let's introduce a mutation.
Instead of a team blob, this red blob is a solo blob.
Instead of cooperating,
it'll fight trying to get as many mangoes for itself
as it can.
Just like team blobs,
they can also get a tree to themselves.
And when a solo blob and a team blob meet,
the team blob will still offer to cooperate
and start shaking the tree.
But the solo blob is just gonna start eating the mangoes.
Once the team blob realizes what's going on,
it'll start eating too, trying to get what it can
before the solo blob gets everything.
We're gonna have to make up a number here too,
which again, we can change later.
But let's start by saying the solo blob
gets one and a half mangoes
and the team blob gets the remaining half a mango.
Then when they go home to reproduce,
the solo blob will produce one offspring for sure,
and then have a 50% chance to produce a second.
And the team blob will just have a 50% chance
to produce one offspring.
Okay, now that we have two solo blobs,
we can see what happens when they meet.
They each go for the other one's mango,
but then they fight a little bit.
They end up stopping and eating their one mango,
but not before wasting some energy.
Let's say the energy penalty is pretty minor for now.
So each solo blob goes home with a net reward
of three quarters of a mango worth of energy,
which then leads to a 75% chance to reproduce.
Okay, now that we have all the rewards laid out,
let's make the world a bit bigger
and set up a graph to track the relative abundance
of each type.
Before we start this simulation,
what do you think will happen?
Will one strategy do a better job of reproducing
and then be more common in the population?
And if so, will it be so extreme
that the other one will go extinct?
Something else?
(bright music)
(bright music continues)
This blue area represents the fraction
of team blobs over time.
And the red area represents the fraction of solo blobs.
And since those are the only two kinds,
the two fractions have to add up to one.
So it looks like teamwork went extinct.
That could have been luck though.
So let's double check by adding some more worlds.
And to mix things up,
each world is gonna have a different starting mixture
of team blobs and solo blobs.
This first world is gonna start with 10% team blobs
and 90% solo blobs.
The next world will start with 20% team blobs
and 80% solo blobs,
and we'll increase the fraction of team blobs
by 10% all the way up to the final one.
That starts with 90% team blobs and 10% solo blobs.
And here's another opportunity to make a prediction. Got it?
All right, let's see how it goes.
(bright music)
(bright music continues)
So this is interesting.
In some of these simulations, the team blobs takeover,
and in others, the solo blobs take over.
But it doesn't look random.
It looks like the solo blobs do better
when there are many other solo blobs,
and the team blobs do better when
there are many other team blobs.
So why might this be?
Now it's time for some game theory, the math.
And also some evolutionary game theory.
As you might expect from the names, they're related,
but not quite the same.
In normal game theory,
we imagine the game is between two players
and each player can choose whichever strategy they want,
trying to get the best reward they can.
In evolutionary game theory though,
the players don't really have a choice.
They just do what their genes tell them to do
and their reproduction is determined by the reward.
Before we really get into analyzing these rewards,
there's a couple things we can do to make it easier.
First, we can ignore this case
where the blobs find their own tree.
That case is important for helping the population
of blobs fill up the carrying capacity of the world,
but it doesn't have anything to do with the competition,
which is what we're trying to focus on.
And this thing we're left with is called a reward matrix.
Second, it'll be a little bit easier
to compare the rewards if we write them as simple fractions
with the same denominator.
Okay, so these rewards are for the player on the left,
so let's think of ourselves in that position
with the top player as our opponent.
If your opponent is going to cooperate,
then you're better off cooperating too.
Seven is bigger than six.
And we can record this decision with an arrow.
And since this game is symmetrical,
the same is true from the opponent's perspective.
This can be a little easier to see
if we extend each entry in the table
to also show the rewards for the opponent.
There's no new information here,
but it makes it a little bit easier to see
that the opponent should also cooperate
if we're going to cooperate.
Again, because seven is bigger than six.
This kind of situation,
where neither player benefits from switching,
is called a Nash equilibrium.
In my humble opinion,
the reward matrix looks a bit cluttered this way.
And these two arrows are really the same arrow,
so we only need one of them to see that this situation
with both blobs working together as a team
is a Nash equilibrium.
So let's go back to just showing each reward one time.
So what if your opponent is gonna go solo?
Well, you're better off also going solo,
even though you'll get in a fight.
And again, the same is true from the opponent's perspective.
So the case where both players fight
is also a Nash equilibrium.
Usually the best plan is to play toward the Nash equilibrium
if there is one.
But in this case, there are two of them.
So which one should you choose?
You're best off doing the same thing your opponent
is going to do, but it's not clear which one they'll choose,
so it's not clear which one you should choose.
To decide, you might have to resort to psychology,
or some other aspect of the situation.
Now for evolutionary game theory,
we saw in the simulation that when the population
is full of one kind of creature, that kinda does well.
The population essentially ends up at a Nash equilibrium
by accident, rather than by analysis.
If a mutant shows up
and starts playing a different strategy,
it doesn't gain an advantage.
A strategy with this property
is called an evolutionarily stable strategy.
This is a pretty significant concept.
For a genetic behavior to take over a population
and stay dominant,
it needs to be part of an evolutionarily stable strategy.
In this case, both strategies are evolutionarily stable,
so whichever one of them gets a lead
ends up dominating the population.
There's some crossover point
where the situation goes from favoring one type
to favoring the other.
And we can actually calculate that crossover point
based on the reward matrix.
But before we do that more general calculation,
it'll be useful to try a few more specific variations
and see what other kinds of possibilities there are.
So let's copy this matrix and make a variation.
One thing we could do is to make the mangoes harder
to shake out of the trees,
reducing the payoff of working as a team.
Now if you're matched up with the team blob,
your reward is gonna be the same,
whether you also act like a team blob or like a solo blob.
Teamwork is still technically a Nash equilibrium here
because if both are cooperating,
neither benefits from switching.
But even though you don't do better by switching,
you don't do worse either.
This is called a weak Nash equilibrium.
And if you do do worse by switching,
it's called a strong Nash equilibrium.
Frankly, I think these names are a little bit confusing,
but you do get used to them over time.
All right, time for a couple more simulations.
But first,
try pausing to guess what the results will look like.
(gentle music)
Okay, it looks like if the population is full of solo blobs,
it stays that way.
So in this case,
being a solo blob is an evolutionarily stable strategy.
And for teamwork, it mostly looks like
it's not an evolutionarily stable strategy,
though there is this one world
that started with 90% team blobs,
and it pretty much stayed that way for the 50 days.
Since teamwork is still a Nash equilibrium,
if a weak one,
solo blobs in a sea of team blobs
don't really have much of an advantage.
But the more solo blobs there are,
the more of an advantage they have.
So as long as it's possible for a team blob
to produce an offspring that's a solo blob
through a mutation,
then it's just a matter of time
until the solo blobs take over.
So teamwork isn't an evolutionarily stable strategy here,
even though it's kind of close.
So that's one variation.
And we could make another variation,
making cooperation even less beneficial.
So for this matrix, you're better off going solo,
regardless of what your opponent is doing.
Both blobs fighting is still a strong Nash equilibrium
and teamwork isn't even a weak Nash equilibrium.
For this one, it seems kind of obvious what's gonna happen.
The solo blobs just have an advantage,
but we might as well check.
(gentle music)
Okay, yep.
Going solo is still an evolutionarily stable strategy
and the team blobs get their butts kicked.
This kind of situation has a special name.
The prisoner's dilemma.
When two solo blobs meet,
they each get three quarters of a mango as a net reward.
But if they would just work as a team,
they could each get five quarters of a mango.
But in any given case, you're better off going solo,
so the solo blobs take over.
So now we have three matrices
that progressively make cooperation
less and less beneficial.
And these encompass the three possibilities
for what gets you the best reward when you meet a team blob.
Either you should team up with them, you should go solo,
or it doesn't matter.
The other thing we could change is to also increase the cost
of getting in a fight.
First, let's reduce it until the team blobs
and the solo blobs get the same reward.
This is a sort of mirror image of this other one,
but this time teamwork is a strong Nash,
and fighting is a weak Nash.
So what would you expect here?
Is it gonna be pretty much the same,
just with the role switched?
Or will something else happen?
(gentle music)
Okay, it turns out it is pretty much the same as before,
but flipped.
Cooperation is an evolutionarily stable strategy,
and going solo isn't.
Even though the world that started out
with mostly solo blobs still seems to be hanging on.
Next, let's make fighting even more costly.
And this one is a mirror image
of the prisoner's dilemma situation.
So as you might expect, the results are opposite.
Teamwork is an evolutionarily stable strategy,
and solo blobs get their butts kicked.
We've only changed one of the payoffs at a time,
but we could combine these changes to get four more cases
and these are worth looking at as well.
In this middle case here,
both strategies lead to a weak Nash equilibrium.
I know I keep saying you should try to make a prediction,
but here's one I genuinely think is tough.
I'm gonna make this one multiple choice.
And I'm not gonna grade it, but if you pause and pick,
I think you'll learn more.
So the first possibility is it's just total chaos.
The lines are gonna be squiggling everywhere.
Another possibility is that the starting populations
will tend to stay about where they are.
So we'll see roughly straight lines all across the graph.
Or maybe the team blobs will do better in more of the worlds
because there's just better rewards
when everyone's cooperating.
Or, it could be that I'm baiting you with all of these
and it's none of them, in which case,
say what you think is gonna happen.
(gentle music)
Okay, looking at the overall average,
it does seem to favor solo blobs slightly,
but looking at individual cases,
it does seem pretty clear it's just chaos.
With this matrix,
your reward is entirely dependent on your opponent
and it has nothing to do with what you decide to do.
It's pure luck.
So here, neither strategy is evolutionarily stable.
The population also doesn't stay at a consistent mixture
because there's nothing to correct
lucky bounces one way or the other.
And even though the overall rewards are higher
when the population is full of team blobs,
the solo blobs get those rewards too.
So there's no advantage for the team blobs.
All right, moving on.
From here, let's make fighting more costly.
This one's not a mirror image of anything we've done so far.
So try determining whether either strategy
leads to a Nash equilibrium,
and whether it's evolutionarily stable.
Okay, here's another one
where teamwork is an evolutionarily stable strategy.
The interesting thing here
is that this is the first time we've seen
an evolutionarily stable strategy
that just corresponds to a weak Nash equilibrium.
When the population is mostly team blobs,
they don't have much of an advantage.
So in most of the sims,
the solo blobs didn't actually go extinct.
But if there are a bunch of solo blobs,
they hurt each other.
So even though they can linger around,
they can never actually take over the population.
And when we look at this mirror image case,
we again see something similar, but with the roles reversed.
All right, one more case,
where both cooperation and fighting have a high cost.
This situation also has a special name.
It's called a Hawk-Dove game.
The solo blobs are called hawks
and the teamwork blobs are called doves.
Anyway, what do you expect to happen?
(gentle music)
Here, neither strategy leads to a Nash equilibrium,
not even a weak one, and neither strategy is an ESS either.
It turns out there's a fraction of hawks versus doves
that the population keeps trying to return to.
There's still a little bit of chaos,
but if one strategy starts to take over,
suddenly it's advantageous to be the other strategy
and the population moves back
toward that fraction of cooperators.
And just like that crossover point
for the very first situation we looked at,
we can calculate what that number will be
for any reward matrix.
But before we do that, let's see if we can look back,
and summarize the relationship between the Nash equilibrium
and an evolutionary stable strategy.
It looks like a strategy
is always an evolutionarily stable strategy
if it leads to a strong Nash equilibrium.
And a strategy can also be an ESS
if it leads to a weak Nash equilibrium,
as long as the other strategy doesn't lead
to a Nash equilibrium, strong or weak.
This is the definition of an evolutionarily stable strategy
that you'd find in a textbook.
But for me at least,
it makes a lot more sense
after going through all these examples.
And these really are all of the examples,
at least for a symmetrical game
where the players can't communicate.
Of course, there could be all kinds of different numbers
and there could be more than two strategies.
But any strategy with any rewards
will always lead to either a strong Nash equilibrium,
a weak Nash equilibrium, or no Nash equilibrium.
And evolutionary stability
will still always follow this rule.
There may be an upcoming video
with more than two strategies,
but you're just gonna have to wait.
At this point, there is a little bit more we could do.
We could derive a formula that'll give us
the stable equilibrium mixture
in a Hawk-Dove game for any values in the reward matrix.
We could do the same thing,
but for the crossover point in that first situation
where both strategies were evolutionarily stable.
Another question which turns out
to have an interesting answer is,
what if we allow the blobs to flip a coin
to decide which strategy to play?
Consider those questions homework.
I know, I said we would do those calculations,
but I didn't say in this video.
Videos are great and all,
but you learn the most when you do things.
And when you connect with other people
who are also thinking about the same stuff.
So if you wanna discuss these questions,
or share your answers with me or others,
or if you're already pretty familiar with this stuff
and wanna help out,
there's a Discord link in the video description.
I also stream myself working on Twitch a few times per week
and I'm happy to talk about it there.
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I appreciate very much that you watch all the way
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And I'll see you next time!