no one asked LMEEEEEEEEEEEEEEEEEEEEEEEEEEEEE; We're gonna use some simulations and some ideas from a field of math called "game theory". 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. Alright 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 of food so it'll have a 50% chance of surviving to the next day and the hawk ends its day with 1 and 1/2 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 the use of 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 hot creature to our simulation and see what happens. Now was a good time to pause and predict what you think will happen. Alright 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. 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 had 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 1/2 of food and the hawk gets one and a 1/2 or 3/2 food and if we reverse perspectives, if hawk faces a dove they'll get 3/2 and 1/2. And when a hawk faces another hawk they'll each end up with zero after they waste all 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 Duff 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 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 is 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 in equilibrium If doves and hawks have the same expected average score in a contest. Right? Equilibrium is when, on average, we don't expect to 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 1/2 of food so overall when a dove runs into another creature when the rest of the population is 90 percent 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 tough part again Okay, just like before the rest of the population is 90% doves and against a dove a hawk gets 1 and a 1/2 or 3/2 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 ninety percent 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 guess again 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 1/2 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 energy, not all of it and go home with a score of 1/4 Plugging that in we see the population moved toward 1/3 doves And again, we can see this born 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 payoff gets less and less bad say getting to three-fourths Now the graph lines don't cross at all There's no equilibrium. At this point even if you know you're facing a hawk the 3/4 food you get from fighting is better than the 1/2 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 would 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 and want more like it Then I really think you might like Brilliant's computational biology course. In it you learn things like how to analyze genetic information, map ancestry and predict the structure of proteins Videos are a great way to get excited about a topic but to really learn deeply you have to engage in active problem-solving and that's what's so great about Brilliant Their courses are built around answering questions and some of the exercises even have you run code like this script that analyzes protein folds super cool If you'd like to give Brilliant a try you can go to Brilliant.org/Primer (link in the description) to let them know you came from here, and the first 200 people to use that link get 20% off the annual premium subscription. Check it out :)
Big fan of high-quality explanations to get people into maths, showing how inviting it can be.
This really is a great video on game theory. It gets right what so many get wrong: an equilibrium strategy isn't a winning strategy, it's an equilibrium. It's a fixed point of certain dynamic systems.
Have you seen this game? Pretty nice and accessible way to learn some game theory.
https://ncase.me/trust/
Awesome video — i’ve always been turned off from applied math due to a poor experience with statistics, but can definitely see how game theory leads to cool results like the speaker was discussing in the end
Supposed to be taking game theory next year but up until I had no idea what it was about 😅
I love this video. Only if there were similar videos for GT.
Watched their whole evolution Playlist because of that post. Pretty well made and understandable even for non-natives
he's dryer but I like william spanial better.