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Simulating the Evolution of Aggression

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Jul 27, 2019

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Simulating the Evolution of Aggression
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  • - [Justin] In this video, we're gonna start
  • exploring conflict between creatures.
  • To try to build some understanding here,
  • we're gonna use some simulations and some ideas
  • from a field of math called Game Theory.
  • (light music)
  • Okay, so in our simulation, food will appear each day,
  • and then blobs will appear and go out to eat the food.
  • We'll use the same survival and reproduction rules
  • as in previous videos.
  • Eating one piece of food lets a creature
  • survive to the next day,
  • and eating two pieces of food allows a creature
  • to both survive and reproduce.
  • What's different in this simulation though
  • is that food will come in pairs.
  • Each creature randomly picks a pair of food to walk to,
  • so it might get the pair all to itself
  • and get to go home with two food and then reproduce,
  • or another creature might find the pair at the same time.
  • And when this happens, they have to somehow figure out
  • how to split things up.
  • We'll start out by having only one possible strategy
  • for creatures who run into each other.
  • They'll just share, each taking a piece of food
  • and going home to survive to the next day.
  • And because this strategy is so nice,
  • we'll give it the name "dove."
  • All right, let's let things run for a bit.
  • (light music)
  • All right, now let's add a new strategy
  • called the hawk strategy.
  • Hawks are more aggressive.
  • If a hawk meets a dove,
  • the hawk will go for the same piece of food as the dove,
  • eat half of it, and then quickly eat
  • the other piece of food, taking it for itself.
  • This half food does complicate our survival
  • and reproduction rules a little bit.
  • So in this situation, a dove ends the day with half a food,
  • so it'll have a 50% chance of surviving to the next day,
  • and the hawk ends its day with one-and-a-half food,
  • so it'll survive for sure,
  • and also have a 50% chance of reproducing.
  • So it looks good to be a hawk, but it's also risky.
  • If two hawks meet, they'll fight,
  • and fighting is taxing.
  • At the very least, they use a lot of energy,
  • and they might also get injured.
  • So, when hawks fight, each one gets a piece of food,
  • but they spend so much energy fighting
  • that they use up all the benefit of the food right away
  • and effectively go home with zero food,
  • meaning they won't survive.
  • So, now let's try adding a hawk creature to our simulation,
  • and see what happens.
  • Now is a good time to pause and predict
  • what you think will happen.
  • (light music)
  • All right, it looks like we have a mixture
  • that fluctuates roughly around half and half.
  • And, there are also fewer creatures overall,
  • even with the same amount of food.
  • Here's an example of how natural selection
  • doesn't necessarily act for the good of the species.
  • And, to cover our bases, let's try starting with all hawks.
  • (light music)
  • Okay, not too surprisingly,
  • they're tearing each other apart,
  • and their max population size didn't even reach
  • half of the population size of the doves.
  • Now, if we add a dove to the mix in the next day,
  • what do you think will happen?
  • Okay, so it took the doves a little while
  • to gain a foothold here,
  • but eventually we end up in a similar situation,
  • with a fluctuating mixture of hawks and doves.
  • So why do we care?
  • Well, this is a situation where survival of the fittest
  • doesn't help us understand what's going on.
  • There isn't one fittest strategy.
  • We can get a better sense for why this is
  • by translating our conflict rules from before into a table.
  • If two doves face each other, they'll each get one food.
  • If a dove faces a hawk, the dove gets half a food,
  • and the hawk gets one and a half or three-halves food.
  • And if we reverse perspectives, if a hawk faces a dove,
  • they'll get three halves and one half.
  • And when a hawk faces another hawk,
  • they'll each end up with zero
  • after they waste a lot of that energy fighting each other.
  • Now that we have this table, let's imagine blobs
  • that can choose which strategy they want to play.
  • Say I control the blob on top,
  • and you control the blob on the left.
  • Say you know that I'm going to play a hawk strategy,
  • which of course I am, what should you do?
  • Well, you're better off just backing down
  • and taking your half food.
  • That might be annoying since it feels
  • like I'm winning somehow,
  • and you might be tempted to challenge me and also play hawk
  • to teach me that I can't just push you around.
  • This could make sense if we were gonna play this game
  • against each other over and over again,
  • as two humans might do,
  • and that is something we'll talk about in future videos.
  • But, in this situation,
  • we're just these simple blobs with no social structure,
  • interacting once, and even if we do see each other again,
  • we won't remember it.
  • So, all that matters is how much food
  • we take home right now.
  • And if you want to maximize your chances of surviving
  • and reproducing, you'll play dove.
  • Discretion is the better part of valor here.
  • Let's record this by drawing an arrow.
  • If we're in the right-hand column because I'm playing hawk,
  • the situation in the upper-right square
  • is the best you can do.
  • Okay, in the other case where I'm not so mean,
  • you know that I'm going to play the dove strategy.
  • In this case, you'll do better playing hawk.
  • And here again, because you're a very smart human,
  • you might be tempted to think about the future,
  • and want to reward me for playing nice
  • and play dove yourself,
  • but we're just these really simple blob creatures
  • who might never see each other again.
  • So, if you want to maximize your chance of reproducing,
  • you'll play hawk.
  • And, we can record this with another arrow.
  • So now, to complete this table,
  • we can reverse perspectives and think about
  • what I should do in response to you,
  • which I won't go through in detail, it's the same reasoning.
  • But, we'd get similar arrows in the rows here.
  • These arrows all point to more advantageous strategies,
  • and the interesting thing to notice
  • is that there are two stable situations:
  • either you play hawk and I play dove,
  • or you play dove and I play hawk.
  • If we're in one of those two situations,
  • either one of us would be worse off
  • if we pick a different strategy.
  • And by the way, this way of analyzing choices
  • is called Game Theory,
  • which is a whole field of math.
  • In a situation where nobody benefits
  • from changing their strategy,
  • it's called a Nash Equilibrium, named after John Nash,
  • who some would say had a beautiful mind.
  • So, the best strategy isn't hawk or dove.
  • It's to do the opposite of what your opponent is doing.
  • When there are a lot of doves, it's better to be a hawk,
  • and when there are a lot of hawks, it's better to be a dove.
  • There's some equilibrium fraction of doves
  • that the population is always pulled toward.
  • Great, so we have the main conceptual point down,
  • but we can deepen our understanding
  • by calculating what that equilibrium fraction should be.
  • The population will be an equilibrium
  • if doves and hawks have the same
  • expected average score in a contest.
  • Right?
  • Equilibrium is when, on average,
  • we don't expect a change one way or the other,
  • so we can't have one strategy doing better.
  • They're equal.
  • Our goal is to find the fraction of doves
  • that makes this condition true.
  • On our way there, let's first calculate
  • the expected average score for a dove
  • in a hypothetical example.
  • Say, where the rest of the population is 90% doves.
  • So let's see, a dove will have a 90% chance
  • of facing another dove,
  • in which case it gets the dove
  • versus dove payoff of one food.
  • And a dove also has a 10% chance of facing a hawk, right?
  • That's just the rest of the creatures.
  • In which case it only gets a half a food.
  • So overall, when a dove runs into another creature,
  • when the rest of the population is 90% doves,
  • it'll come away with 0.95 food on average.
  • This number is pretty meaningless on its own.
  • But, once we calculate the expected hawk score,
  • we can compare the two to see
  • whether the equilibrium condition is met.
  • So let's do that; let's find the expected hawk score.
  • It could be good to pause and try to do this yourself
  • to make sure it all makes sense.
  • Maybe even rewinding to watch the dove part again.
  • Okay, just like before,
  • the rest of the population is 90% doves,
  • and against a dove, a hawk gets one-and-a-half
  • or three-halves pieces of food.
  • And again, there's a 10% chance
  • of running into another hawk,
  • in which case our hawk goes home with zero food.
  • And this comes out to 1.35 food on average.
  • Now, notice that 1.35 is more than 0.95.
  • So at 90% doves, hawks will do better,
  • and we'd expect the fraction of hawks to increase
  • in the next generation.
  • So, it's not equilibrium.
  • Not 90%.
  • Now to find out what fraction of doves
  • does meet the equilibrium condition,
  • we can write the fractions of doves and hawks
  • as variables instead of just guessing at specific numbers.
  • And you might be saying right now,
  • "Wow, that's a lot of letters,"
  • which is a fair point, but we're almost there,
  • and our next step is actually to get rid
  • of one of those letters.
  • So, there's a nice treat already.
  • Doves and hawks make up all the creatures,
  • so their fractions have to add up to one.
  • And, this means we can replace the small h
  • with one minus small d.
  • And now, the expected dove and hawk scores
  • are both written as functions of one variable.
  • And the same variable.
  • So, we can graph them on top of each other.
  • The expected scores are equal when the graphed lines cross.
  • And, indeed, the equilibrium condition is met at 50% doves.
  • And, if we run a simulation
  • with way more creatures than before,
  • unfortunately too many to animate,
  • the randomness smooths out a bit,
  • and we can see that the prediction is true.
  • Okay, so, it might feel like that was kind of a lot of work
  • just to verify what we already thought.
  • But, the fraction of doves
  • isn't always going to be one half.
  • It depends on the numbers in our payoff grid.
  • The most interesting number to play with here
  • is the hawk-versus-hawk payoff.
  • So far, we've been saying that the hawks
  • each get one piece of food,
  • but waste all the energy of the food on fighting.
  • But, what if instead, they only waste
  • most of the energy, not all of it,
  • and go home with a score of 1/4th?
  • Plugging that in, we see the population
  • move toward 1/3rd doves.
  • And again, we can see this borne out in the simulation.
  • At this point, congratulations,
  • we have a pretty detailed understanding
  • of how populations of hawks and doves work.
  • And as basic as this model is,
  • with only two simple strategies,
  • it's a powerful starting point
  • for analyzing behavior in the real world.
  • And, before we go, I want to give you some teasers
  • for how we'll build on this
  • to get closer to reality in future videos.
  • First, creatures in the real world
  • can play more than one strategy.
  • So instead of having their behavior
  • completely determined by a single gene,
  • our creatures could have several genes
  • affecting their behavior,
  • causing them to have different chances
  • of playing hawk or dove.
  • And the Game Theory term for this is mixed strategies.
  • There can also be more complex, conditional strategies
  • that act differently depending who they're facing.
  • For example, there could be a strategy
  • that fights with hawks, but is nice to doves.
  • And, there could also be a strategy
  • that tries to threaten a fight,
  • but runs away if things get serious.
  • And, seeing what happens with these kinds of strategies
  • can help us understand why some animals
  • put on threatening displays while rarely actually fighting,
  • or have somewhat ritualistic fights
  • that usually don't harm anyone.
  • Next, most conflicts are actually asymmetric.
  • So far, we've been assuming that everyone has
  • the same amount to gain and lose,
  • and that all the creatures are on equal footing.
  • But when this changes, we can start to understand things
  • like territorial behavior and dominance hierarchies.
  • And last, let's go back to our equations
  • and see what happens as the hawk payout
  • gets less and less bad.
  • Say, getting to three fourths.
  • Now the graphed lines don't cross at all.
  • There's no equilibrium.
  • At this point, even if you know you're facing a hawk,
  • the three-fourths food you get from fighting
  • is better than the one-half you get from being nice.
  • So these arrows should actually flip,
  • and, it only ever makes sense to play hawk.
  • We end up in this tragic situation
  • where everyone's fighting all the time,
  • even though they would do better
  • if they could just cooperate.
  • This kind of situation has a special name.
  • It's called the prisoner's dilemma.
  • It can feel kind of grim, but there are ways out of it,
  • which we'll talk about in future videos.
  • And, I'll see you then.
  • Okay, so now I have some people to thank.
  • First, thanks to you for watching to the end.
  • Second, thanks to everyone who's become a patron on Patreon.
  • Your support is what makes me feel
  • like people actually get value from these videos,
  • and gives me the confidence that they'll be funded
  • into the future.
  • Third, I want to thank the channel 3Blue1Brown,
  • who shared the last video
  • and really gave this channel a kick.
  • If you like this channel,
  • you really should go check out 3Blue1Brown.
  • And finally, this video was supported in part by Brilliant.
  • If you like how I treat biology as a quantitative subject
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