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it when a resource is contested is it
better to work together or look out for
yourself we're going to 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 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
treats so they're going to 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 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 3/4 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 blabs reproduce
they'll use the first unit of energy to
create one Offspring for sure then the
remaining 3/4s 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 going to start eating the
mangoes once the team blob realizes is
what's going on it'll start eating two
trying to get what it can before the
solo blob gets everything we're going to
have to make up a number here to 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 of 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 ones 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 3/4 of Omega 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 you 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
[Music]
else
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 going to have a different
starting mixture of Team blobs and solo
blobs this first world is going to start
with 10% team blobs and 90% solo blobs
the next World will start with 20% team
blobs and 80% solo blobs and will
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
[Music]
so this is interesting in some of these
simulations the team blobs take over 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 name 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 going to 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 kind
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
going to be the same whether you also
act like a team blob or like a solo blob
teamwork is still technically a nasty
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 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 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 going to
happen the solo blobs just have an
advantage but we might as well
[Music]
check 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
prisoners dilemma when two solo blobs
meet they each get 3/4 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 going to be
pretty much the same just with the roll
switched or will something else
[Music]
happen 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 prisoners 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 NES equilibrium I know I
keep saying you should try to make a
prediction but here's one I genuinely
think is tough I'm going to make this
one multiple choice and I'm not going to
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 going to 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 and
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 going to happen
[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 an ash 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 rolls 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
[Music]
happen here neither strategy leads to a
nashy 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 a
Nash equilibrium and an evolutionarily
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 going to
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 want to discuss these
questions or share your answers with me
or others or if you're already pretty
familiar with this stuff and want to
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 if
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the video and I'll see you next time