Transcription
[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.
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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.
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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, 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.