WEBVTT Kind: captions Language: en 00:00:00.180 --> 00:00:02.550 - [Justin] This video is sponsored by Blob Shirt. 00:00:02.550 --> 00:00:05.073 It's a soft, comfy shirt with blobs on it. 00:00:06.390 --> 00:00:08.250 When a resource is contested, 00:00:08.250 --> 00:00:11.910 is it better to work together, or look out for yourself? 00:00:11.910 --> 00:00:13.530 We're gonna analyze this question 00:00:13.530 --> 00:00:16.440 from an evolutionary perspective using simulations 00:00:16.440 --> 00:00:17.373 and game theory. 00:00:20.040 --> 00:00:22.230 Each of these mango trees has two mangoes 00:00:22.230 --> 00:00:23.550 on the bottom layer. 00:00:23.550 --> 00:00:25.560 So if a blob finds a tree on its own, 00:00:25.560 --> 00:00:27.510 it gets to eat both mangoes. 00:00:27.510 --> 00:00:28.740 Then when it goes home, 00:00:28.740 --> 00:00:31.860 it uses the energy from each mango to make one offspring. 00:00:31.860 --> 00:00:36.210 So two offspring in this case. And also, it dies. 00:00:36.210 --> 00:00:38.400 The blobs only live for one day. 00:00:38.400 --> 00:00:39.270 On the next day, 00:00:39.270 --> 00:00:41.370 the newly born blobs will spread out 00:00:41.370 --> 00:00:43.710 to each get their own tree, and then again, 00:00:43.710 --> 00:00:46.110 they'll reproduce twice before dying. 00:00:46.110 --> 00:00:49.950 So now there are four blobs, but still only two trees. 00:00:49.950 --> 00:00:52.500 So they're gonna have to deal with that somehow. 00:00:52.500 --> 00:00:55.620 We'll start out with blobs that work together as a team. 00:00:55.620 --> 00:00:57.630 They'll notice that there are two more mangoes 00:00:57.630 --> 00:00:58.830 higher up in the tree. 00:00:58.830 --> 00:01:01.290 And so they'll work together to shake the tree, 00:01:01.290 --> 00:01:02.670 causing those mangoes to drop, 00:01:02.670 --> 00:01:04.860 and then they'll share the total. 00:01:04.860 --> 00:01:08.130 So that comes out to two mangoes each when they cooperate. 00:01:08.130 --> 00:01:10.080 It does cost them a little bit of energy 00:01:10.080 --> 00:01:11.520 to shake the tree though. 00:01:11.520 --> 00:01:13.980 So let's subtract a quarter mango worth of energy 00:01:13.980 --> 00:01:14.880 from each blob, 00:01:14.880 --> 00:01:18.000 leaving them with one and three quarters when they go home. 00:01:18.000 --> 00:01:21.030 You might notice that this number is completely made up. 00:01:21.030 --> 00:01:22.500 Later on, we'll look at what happens 00:01:22.500 --> 00:01:24.000 when we change some of these numbers. 00:01:24.000 --> 00:01:26.040 But for now, we're just picking something specific 00:01:26.040 --> 00:01:27.900 to get a simulation running. 00:01:27.900 --> 00:01:29.520 When these blobs reproduce, 00:01:29.520 --> 00:01:31.170 they'll use the first unit of energy 00:01:31.170 --> 00:01:33.360 to create one offspring for sure, 00:01:33.360 --> 00:01:34.890 then the remaining three quarters 00:01:34.890 --> 00:01:38.200 will convert into a probability for one more offspring. 00:01:38.200 --> 00:01:40.440 Okay, two of them reproduced once, 00:01:40.440 --> 00:01:42.240 and two of them reproduced twice. 00:01:42.240 --> 00:01:44.520 So now we have six blobs total. 00:01:44.520 --> 00:01:46.890 But we still only have two trees. 00:01:46.890 --> 00:01:49.890 The blobs avoid going to a tree that already has two blobs. 00:01:49.890 --> 00:01:52.380 So if the world has more than twice as many blobs 00:01:52.380 --> 00:01:53.430 as it does trees, 00:01:53.430 --> 00:01:54.990 then the last blobs to go out 00:01:54.990 --> 00:01:57.870 will end up wandering around until they die. 00:01:57.870 --> 00:02:00.000 So two times the number of trees 00:02:00.000 --> 00:02:03.180 will be the carrying capacity for any of these worlds. 00:02:03.180 --> 00:02:06.570 All right, at this stage, let's introduce a mutation. 00:02:06.570 --> 00:02:10.710 Instead of a team blob, this red blob is a solo blob. 00:02:10.710 --> 00:02:11.850 Instead of cooperating, 00:02:11.850 --> 00:02:14.280 it'll fight trying to get as many mangoes for itself 00:02:14.280 --> 00:02:15.450 as it can. 00:02:15.450 --> 00:02:16.830 Just like team blobs, 00:02:16.830 --> 00:02:19.170 they can also get a tree to themselves. 00:02:19.170 --> 00:02:21.720 And when a solo blob and a team blob meet, 00:02:21.720 --> 00:02:23.880 the team blob will still offer to cooperate 00:02:23.880 --> 00:02:25.230 and start shaking the tree. 00:02:25.230 --> 00:02:28.530 But the solo blob is just gonna start eating the mangoes. 00:02:28.530 --> 00:02:30.870 Once the team blob realizes what's going on, 00:02:30.870 --> 00:02:33.090 it'll start eating too, trying to get what it can 00:02:33.090 --> 00:02:35.670 before the solo blob gets everything. 00:02:35.670 --> 00:02:37.680 We're gonna have to make up a number here too, 00:02:37.680 --> 00:02:39.450 which again, we can change later. 00:02:39.450 --> 00:02:41.670 But let's start by saying the solo blob 00:02:41.670 --> 00:02:43.140 gets one and a half mangoes 00:02:43.140 --> 00:02:46.050 and the team blob gets the remaining half a mango. 00:02:46.050 --> 00:02:48.090 Then when they go home to reproduce, 00:02:48.090 --> 00:02:50.790 the solo blob will produce one offspring for sure, 00:02:50.790 --> 00:02:54.090 and then have a 50% chance to produce a second. 00:02:54.090 --> 00:02:56.670 And the team blob will just have a 50% chance 00:02:56.670 --> 00:02:58.500 to produce one offspring. 00:02:58.500 --> 00:03:00.690 Okay, now that we have two solo blobs, 00:03:00.690 --> 00:03:03.390 we can see what happens when they meet. 00:03:03.390 --> 00:03:05.250 They each go for the other one's mango, 00:03:05.250 --> 00:03:06.960 but then they fight a little bit. 00:03:06.960 --> 00:03:09.330 They end up stopping and eating their one mango, 00:03:09.330 --> 00:03:11.550 but not before wasting some energy. 00:03:11.550 --> 00:03:14.010 Let's say the energy penalty is pretty minor for now. 00:03:14.010 --> 00:03:16.470 So each solo blob goes home with a net reward 00:03:16.470 --> 00:03:19.200 of three quarters of a mango worth of energy, 00:03:19.200 --> 00:03:23.100 which then leads to a 75% chance to reproduce. 00:03:23.100 --> 00:03:25.380 Okay, now that we have all the rewards laid out, 00:03:25.380 --> 00:03:26.932 let's make the world a bit bigger 00:03:26.932 --> 00:03:29.610 and set up a graph to track the relative abundance 00:03:29.610 --> 00:03:30.930 of each type. 00:03:30.930 --> 00:03:32.520 Before we start this simulation, 00:03:32.520 --> 00:03:34.080 what do you think will happen? 00:03:34.080 --> 00:03:36.660 Will one strategy do a better job of reproducing 00:03:36.660 --> 00:03:38.760 and then be more common in the population? 00:03:38.760 --> 00:03:41.040 And if so, will it be so extreme 00:03:41.040 --> 00:03:42.990 that the other one will go extinct? 00:03:42.990 --> 00:03:44.574 Something else? 00:03:44.574 --> 00:03:47.157 (bright music) 00:03:54.565 --> 00:03:57.982 (bright music continues) 00:04:00.660 --> 00:04:03.390 This blue area represents the fraction 00:04:03.390 --> 00:04:05.460 of team blobs over time. 00:04:05.460 --> 00:04:09.090 And the red area represents the fraction of solo blobs. 00:04:09.090 --> 00:04:11.400 And since those are the only two kinds, 00:04:11.400 --> 00:04:13.920 the two fractions have to add up to one. 00:04:13.920 --> 00:04:17.190 So it looks like teamwork went extinct. 00:04:17.190 --> 00:04:18.660 That could have been luck though. 00:04:18.660 --> 00:04:21.480 So let's double check by adding some more worlds. 00:04:21.480 --> 00:04:22.680 And to mix things up, 00:04:22.680 --> 00:04:24.840 each world is gonna have a different starting mixture 00:04:24.840 --> 00:04:26.760 of team blobs and solo blobs. 00:04:26.760 --> 00:04:29.700 This first world is gonna start with 10% team blobs 00:04:29.700 --> 00:04:31.590 and 90% solo blobs. 00:04:31.590 --> 00:04:34.200 The next world will start with 20% team blobs 00:04:34.200 --> 00:04:36.030 and 80% solo blobs, 00:04:36.030 --> 00:04:38.070 and we'll increase the fraction of team blobs 00:04:38.070 --> 00:04:40.320 by 10% all the way up to the final one. 00:04:40.320 --> 00:04:44.550 That starts with 90% team blobs and 10% solo blobs. 00:04:44.550 --> 00:04:48.870 And here's another opportunity to make a prediction. Got it? 00:04:48.870 --> 00:04:50.673 All right, let's see how it goes. 00:04:50.673 --> 00:04:53.256 (bright music) 00:05:00.449 --> 00:05:04.680 (bright music continues) 00:05:04.680 --> 00:05:06.360 So this is interesting. 00:05:06.360 --> 00:05:09.240 In some of these simulations, the team blobs takeover, 00:05:09.240 --> 00:05:11.550 and in others, the solo blobs take over. 00:05:11.550 --> 00:05:12.780 But it doesn't look random. 00:05:12.780 --> 00:05:14.580 It looks like the solo blobs do better 00:05:14.580 --> 00:05:16.530 when there are many other solo blobs, 00:05:16.530 --> 00:05:18.060 and the team blobs do better when 00:05:18.060 --> 00:05:19.980 there are many other team blobs. 00:05:19.980 --> 00:05:22.020 So why might this be? 00:05:22.020 --> 00:05:25.380 Now it's time for some game theory, the math. 00:05:25.380 --> 00:05:28.110 And also some evolutionary game theory. 00:05:28.110 --> 00:05:30.660 As you might expect from the names, they're related, 00:05:30.660 --> 00:05:32.250 but not quite the same. 00:05:32.250 --> 00:05:33.750 In normal game theory, 00:05:33.750 --> 00:05:35.820 we imagine the game is between two players 00:05:35.820 --> 00:05:38.610 and each player can choose whichever strategy they want, 00:05:38.610 --> 00:05:40.620 trying to get the best reward they can. 00:05:40.620 --> 00:05:42.540 In evolutionary game theory though, 00:05:42.540 --> 00:05:44.670 the players don't really have a choice. 00:05:44.670 --> 00:05:47.250 They just do what their genes tell them to do 00:05:47.250 --> 00:05:50.820 and their reproduction is determined by the reward. 00:05:50.820 --> 00:05:53.400 Before we really get into analyzing these rewards, 00:05:53.400 --> 00:05:55.920 there's a couple things we can do to make it easier. 00:05:55.920 --> 00:05:57.450 First, we can ignore this case 00:05:57.450 --> 00:05:59.576 where the blobs find their own tree. 00:05:59.576 --> 00:06:02.250 That case is important for helping the population 00:06:02.250 --> 00:06:04.740 of blobs fill up the carrying capacity of the world, 00:06:04.740 --> 00:06:07.260 but it doesn't have anything to do with the competition, 00:06:07.260 --> 00:06:09.450 which is what we're trying to focus on. 00:06:09.450 --> 00:06:13.140 And this thing we're left with is called a reward matrix. 00:06:13.140 --> 00:06:14.850 Second, it'll be a little bit easier 00:06:14.850 --> 00:06:17.700 to compare the rewards if we write them as simple fractions 00:06:17.700 --> 00:06:19.083 with the same denominator. 00:06:19.950 --> 00:06:23.880 Okay, so these rewards are for the player on the left, 00:06:23.880 --> 00:06:26.010 so let's think of ourselves in that position 00:06:26.010 --> 00:06:28.110 with the top player as our opponent. 00:06:28.110 --> 00:06:30.300 If your opponent is going to cooperate, 00:06:30.300 --> 00:06:32.430 then you're better off cooperating too. 00:06:32.430 --> 00:06:34.410 Seven is bigger than six. 00:06:34.410 --> 00:06:37.320 And we can record this decision with an arrow. 00:06:37.320 --> 00:06:39.270 And since this game is symmetrical, 00:06:39.270 --> 00:06:41.910 the same is true from the opponent's perspective. 00:06:41.910 --> 00:06:43.710 This can be a little easier to see 00:06:43.710 --> 00:06:45.660 if we extend each entry in the table 00:06:45.660 --> 00:06:48.090 to also show the rewards for the opponent. 00:06:48.090 --> 00:06:49.800 There's no new information here, 00:06:49.800 --> 00:06:51.600 but it makes it a little bit easier to see 00:06:51.600 --> 00:06:53.520 that the opponent should also cooperate 00:06:53.520 --> 00:06:55.200 if we're going to cooperate. 00:06:55.200 --> 00:06:58.170 Again, because seven is bigger than six. 00:06:58.170 --> 00:06:59.310 This kind of situation, 00:06:59.310 --> 00:07:01.290 where neither player benefits from switching, 00:07:01.290 --> 00:07:03.630 is called a Nash equilibrium. 00:07:03.630 --> 00:07:05.130 In my humble opinion, 00:07:05.130 --> 00:07:07.740 the reward matrix looks a bit cluttered this way. 00:07:07.740 --> 00:07:10.140 And these two arrows are really the same arrow, 00:07:10.140 --> 00:07:12.780 so we only need one of them to see that this situation 00:07:12.780 --> 00:07:15.120 with both blobs working together as a team 00:07:15.120 --> 00:07:16.680 is a Nash equilibrium. 00:07:16.680 --> 00:07:19.473 So let's go back to just showing each reward one time. 00:07:20.430 --> 00:07:22.980 So what if your opponent is gonna go solo? 00:07:22.980 --> 00:07:25.230 Well, you're better off also going solo, 00:07:25.230 --> 00:07:26.820 even though you'll get in a fight. 00:07:26.820 --> 00:07:29.850 And again, the same is true from the opponent's perspective. 00:07:29.850 --> 00:07:31.650 So the case where both players fight 00:07:31.650 --> 00:07:33.870 is also a Nash equilibrium. 00:07:33.870 --> 00:07:37.080 Usually the best plan is to play toward the Nash equilibrium 00:07:37.080 --> 00:07:38.130 if there is one. 00:07:38.130 --> 00:07:39.930 But in this case, there are two of them. 00:07:39.930 --> 00:07:42.240 So which one should you choose? 00:07:42.240 --> 00:07:44.460 You're best off doing the same thing your opponent 00:07:44.460 --> 00:07:47.490 is going to do, but it's not clear which one they'll choose, 00:07:47.490 --> 00:07:50.490 so it's not clear which one you should choose. 00:07:50.490 --> 00:07:53.160 To decide, you might have to resort to psychology, 00:07:53.160 --> 00:07:56.040 or some other aspect of the situation. 00:07:56.040 --> 00:07:58.050 Now for evolutionary game theory, 00:07:58.050 --> 00:08:00.480 we saw in the simulation that when the population 00:08:00.480 --> 00:08:03.810 is full of one kind of creature, that kinda does well. 00:08:03.810 --> 00:08:06.720 The population essentially ends up at a Nash equilibrium 00:08:06.720 --> 00:08:09.360 by accident, rather than by analysis. 00:08:09.360 --> 00:08:10.350 If a mutant shows up 00:08:10.350 --> 00:08:12.030 and starts playing a different strategy, 00:08:12.030 --> 00:08:14.040 it doesn't gain an advantage. 00:08:14.040 --> 00:08:15.480 A strategy with this property 00:08:15.480 --> 00:08:18.750 is called an evolutionarily stable strategy. 00:08:18.750 --> 00:08:21.120 This is a pretty significant concept. 00:08:21.120 --> 00:08:24.030 For a genetic behavior to take over a population 00:08:24.030 --> 00:08:25.380 and stay dominant, 00:08:25.380 --> 00:08:28.710 it needs to be part of an evolutionarily stable strategy. 00:08:28.710 --> 00:08:31.860 In this case, both strategies are evolutionarily stable, 00:08:31.860 --> 00:08:33.810 so whichever one of them gets a lead 00:08:33.810 --> 00:08:35.970 ends up dominating the population. 00:08:35.970 --> 00:08:37.500 There's some crossover point 00:08:37.500 --> 00:08:39.840 where the situation goes from favoring one type 00:08:39.840 --> 00:08:41.190 to favoring the other. 00:08:41.190 --> 00:08:43.470 And we can actually calculate that crossover point 00:08:43.470 --> 00:08:45.570 based on the reward matrix. 00:08:45.570 --> 00:08:47.850 But before we do that more general calculation, 00:08:47.850 --> 00:08:50.760 it'll be useful to try a few more specific variations 00:08:50.760 --> 00:08:53.460 and see what other kinds of possibilities there are. 00:08:53.460 --> 00:08:56.190 So let's copy this matrix and make a variation. 00:08:56.190 --> 00:08:58.500 One thing we could do is to make the mangoes harder 00:08:58.500 --> 00:08:59.820 to shake out of the trees, 00:08:59.820 --> 00:09:02.580 reducing the payoff of working as a team. 00:09:02.580 --> 00:09:04.380 Now if you're matched up with the team blob, 00:09:04.380 --> 00:09:05.940 your reward is gonna be the same, 00:09:05.940 --> 00:09:09.750 whether you also act like a team blob or like a solo blob. 00:09:09.750 --> 00:09:12.090 Teamwork is still technically a Nash equilibrium here 00:09:12.090 --> 00:09:13.920 because if both are cooperating, 00:09:13.920 --> 00:09:15.720 neither benefits from switching. 00:09:15.720 --> 00:09:18.030 But even though you don't do better by switching, 00:09:18.030 --> 00:09:19.890 you don't do worse either. 00:09:19.890 --> 00:09:22.380 This is called a weak Nash equilibrium. 00:09:22.380 --> 00:09:24.180 And if you do do worse by switching, 00:09:24.180 --> 00:09:26.190 it's called a strong Nash equilibrium. 00:09:26.190 --> 00:09:28.980 Frankly, I think these names are a little bit confusing, 00:09:28.980 --> 00:09:31.800 but you do get used to them over time. 00:09:31.800 --> 00:09:34.290 All right, time for a couple more simulations. 00:09:34.290 --> 00:09:35.123 But first, 00:09:35.123 --> 00:09:38.289 try pausing to guess what the results will look like. 00:09:38.289 --> 00:09:40.872 (gentle music) 00:09:42.930 --> 00:09:46.170 Okay, it looks like if the population is full of solo blobs, 00:09:46.170 --> 00:09:47.490 it stays that way. 00:09:47.490 --> 00:09:48.323 So in this case, 00:09:48.323 --> 00:09:51.960 being a solo blob is an evolutionarily stable strategy. 00:09:51.960 --> 00:09:54.270 And for teamwork, it mostly looks like 00:09:54.270 --> 00:09:56.910 it's not an evolutionarily stable strategy, 00:09:56.910 --> 00:09:58.470 though there is this one world 00:09:58.470 --> 00:10:00.780 that started with 90% team blobs, 00:10:00.780 --> 00:10:03.573 and it pretty much stayed that way for the 50 days. 00:10:04.710 --> 00:10:06.930 Since teamwork is still a Nash equilibrium, 00:10:06.930 --> 00:10:07.950 if a weak one, 00:10:07.950 --> 00:10:10.290 solo blobs in a sea of team blobs 00:10:10.290 --> 00:10:12.420 don't really have much of an advantage. 00:10:12.420 --> 00:10:14.160 But the more solo blobs there are, 00:10:14.160 --> 00:10:16.230 the more of an advantage they have. 00:10:16.230 --> 00:10:18.540 So as long as it's possible for a team blob 00:10:18.540 --> 00:10:20.760 to produce an offspring that's a solo blob 00:10:20.760 --> 00:10:21.810 through a mutation, 00:10:21.810 --> 00:10:23.250 then it's just a matter of time 00:10:23.250 --> 00:10:25.320 until the solo blobs take over. 00:10:25.320 --> 00:10:28.710 So teamwork isn't an evolutionarily stable strategy here, 00:10:28.710 --> 00:10:31.320 even though it's kind of close. 00:10:31.320 --> 00:10:33.000 So that's one variation. 00:10:33.000 --> 00:10:34.680 And we could make another variation, 00:10:34.680 --> 00:10:37.380 making cooperation even less beneficial. 00:10:37.380 --> 00:10:39.780 So for this matrix, you're better off going solo, 00:10:39.780 --> 00:10:42.060 regardless of what your opponent is doing. 00:10:42.060 --> 00:10:45.000 Both blobs fighting is still a strong Nash equilibrium 00:10:45.000 --> 00:10:48.210 and teamwork isn't even a weak Nash equilibrium. 00:10:48.210 --> 00:10:51.210 For this one, it seems kind of obvious what's gonna happen. 00:10:51.210 --> 00:10:53.730 The solo blobs just have an advantage, 00:10:53.730 --> 00:10:55.544 but we might as well check. 00:10:55.544 --> 00:10:58.650 (gentle music) 00:10:58.650 --> 00:10:59.910 Okay, yep. 00:10:59.910 --> 00:11:03.210 Going solo is still an evolutionarily stable strategy 00:11:03.210 --> 00:11:05.724 and the team blobs get their butts kicked. 00:11:05.724 --> 00:11:08.580 This kind of situation has a special name. 00:11:08.580 --> 00:11:10.200 The prisoner's dilemma. 00:11:10.200 --> 00:11:11.640 When two solo blobs meet, 00:11:11.640 --> 00:11:14.550 they each get three quarters of a mango as a net reward. 00:11:14.550 --> 00:11:16.320 But if they would just work as a team, 00:11:16.320 --> 00:11:18.540 they could each get five quarters of a mango. 00:11:18.540 --> 00:11:21.150 But in any given case, you're better off going solo, 00:11:21.150 --> 00:11:23.460 so the solo blobs take over. 00:11:23.460 --> 00:11:25.620 So now we have three matrices 00:11:25.620 --> 00:11:27.510 that progressively make cooperation 00:11:27.510 --> 00:11:29.100 less and less beneficial. 00:11:29.100 --> 00:11:31.110 And these encompass the three possibilities 00:11:31.110 --> 00:11:34.260 for what gets you the best reward when you meet a team blob. 00:11:34.260 --> 00:11:36.840 Either you should team up with them, you should go solo, 00:11:36.840 --> 00:11:38.340 or it doesn't matter. 00:11:38.340 --> 00:11:41.250 The other thing we could change is to also increase the cost 00:11:41.250 --> 00:11:42.780 of getting in a fight. 00:11:42.780 --> 00:11:44.820 First, let's reduce it until the team blobs 00:11:44.820 --> 00:11:47.190 and the solo blobs get the same reward. 00:11:47.190 --> 00:11:49.710 This is a sort of mirror image of this other one, 00:11:49.710 --> 00:11:51.750 but this time teamwork is a strong Nash, 00:11:51.750 --> 00:11:54.000 and fighting is a weak Nash. 00:11:54.000 --> 00:11:55.530 So what would you expect here? 00:11:55.530 --> 00:11:57.030 Is it gonna be pretty much the same, 00:11:57.030 --> 00:11:58.740 just with the role switched? 00:11:58.740 --> 00:12:00.821 Or will something else happen? 00:12:00.821 --> 00:12:03.404 (gentle music) 00:12:04.560 --> 00:12:07.530 Okay, it turns out it is pretty much the same as before, 00:12:07.530 --> 00:12:09.000 but flipped. 00:12:09.000 --> 00:12:11.700 Cooperation is an evolutionarily stable strategy, 00:12:11.700 --> 00:12:13.680 and going solo isn't. 00:12:13.680 --> 00:12:15.270 Even though the world that started out 00:12:15.270 --> 00:12:19.080 with mostly solo blobs still seems to be hanging on. 00:12:19.080 --> 00:12:21.690 Next, let's make fighting even more costly. 00:12:21.690 --> 00:12:23.070 And this one is a mirror image 00:12:23.070 --> 00:12:25.301 of the prisoner's dilemma situation. 00:12:25.301 --> 00:12:28.470 So as you might expect, the results are opposite. 00:12:28.470 --> 00:12:31.320 Teamwork is an evolutionarily stable strategy, 00:12:31.320 --> 00:12:33.483 and solo blobs get their butts kicked. 00:12:34.590 --> 00:12:37.470 We've only changed one of the payoffs at a time, 00:12:37.470 --> 00:12:40.500 but we could combine these changes to get four more cases 00:12:40.500 --> 00:12:42.750 and these are worth looking at as well. 00:12:42.750 --> 00:12:44.130 In this middle case here, 00:12:44.130 --> 00:12:47.700 both strategies lead to a weak Nash equilibrium. 00:12:47.700 --> 00:12:50.100 I know I keep saying you should try to make a prediction, 00:12:50.100 --> 00:12:52.830 but here's one I genuinely think is tough. 00:12:52.830 --> 00:12:54.660 I'm gonna make this one multiple choice. 00:12:54.660 --> 00:12:57.660 And I'm not gonna grade it, but if you pause and pick, 00:12:57.660 --> 00:12:59.103 I think you'll learn more. 00:12:59.940 --> 00:13:03.210 So the first possibility is it's just total chaos. 00:13:03.210 --> 00:13:05.730 The lines are gonna be squiggling everywhere. 00:13:05.730 --> 00:13:08.760 Another possibility is that the starting populations 00:13:08.760 --> 00:13:11.070 will tend to stay about where they are. 00:13:11.070 --> 00:13:14.580 So we'll see roughly straight lines all across the graph. 00:13:14.580 --> 00:13:17.520 Or maybe the team blobs will do better in more of the worlds 00:13:17.520 --> 00:13:19.410 because there's just better rewards 00:13:19.410 --> 00:13:21.393 when everyone's cooperating. 00:13:21.393 --> 00:13:25.230 Or, it could be that I'm baiting you with all of these 00:13:25.230 --> 00:13:27.008 and it's none of them, in which case, 00:13:27.008 --> 00:13:29.230 say what you think is gonna happen. 00:13:29.230 --> 00:13:31.813 (gentle music) 00:13:34.410 --> 00:13:36.450 Okay, looking at the overall average, 00:13:36.450 --> 00:13:39.060 it does seem to favor solo blobs slightly, 00:13:39.060 --> 00:13:40.920 but looking at individual cases, 00:13:40.920 --> 00:13:43.470 it does seem pretty clear it's just chaos. 00:13:43.470 --> 00:13:44.340 With this matrix, 00:13:44.340 --> 00:13:47.190 your reward is entirely dependent on your opponent 00:13:47.190 --> 00:13:50.010 and it has nothing to do with what you decide to do. 00:13:50.010 --> 00:13:51.420 It's pure luck. 00:13:51.420 --> 00:13:55.080 So here, neither strategy is evolutionarily stable. 00:13:55.080 --> 00:13:58.080 The population also doesn't stay at a consistent mixture 00:13:58.080 --> 00:13:59.490 because there's nothing to correct 00:13:59.490 --> 00:14:01.440 lucky bounces one way or the other. 00:14:01.440 --> 00:14:03.990 And even though the overall rewards are higher 00:14:03.990 --> 00:14:06.060 when the population is full of team blobs, 00:14:06.060 --> 00:14:08.400 the solo blobs get those rewards too. 00:14:08.400 --> 00:14:11.370 So there's no advantage for the team blobs. 00:14:11.370 --> 00:14:12.870 All right, moving on. 00:14:12.870 --> 00:14:15.330 From here, let's make fighting more costly. 00:14:15.330 --> 00:14:18.030 This one's not a mirror image of anything we've done so far. 00:14:18.030 --> 00:14:20.280 So try determining whether either strategy 00:14:20.280 --> 00:14:21.720 leads to a Nash equilibrium, 00:14:21.720 --> 00:14:23.733 and whether it's evolutionarily stable. 00:14:29.520 --> 00:14:30.690 Okay, here's another one 00:14:30.690 --> 00:14:33.510 where teamwork is an evolutionarily stable strategy. 00:14:33.510 --> 00:14:34.710 The interesting thing here 00:14:34.710 --> 00:14:36.420 is that this is the first time we've seen 00:14:36.420 --> 00:14:38.310 an evolutionarily stable strategy 00:14:38.310 --> 00:14:41.328 that just corresponds to a weak Nash equilibrium. 00:14:41.328 --> 00:14:44.220 When the population is mostly team blobs, 00:14:44.220 --> 00:14:46.170 they don't have much of an advantage. 00:14:46.170 --> 00:14:47.490 So in most of the sims, 00:14:47.490 --> 00:14:50.790 the solo blobs didn't actually go extinct. 00:14:50.790 --> 00:14:52.830 But if there are a bunch of solo blobs, 00:14:52.830 --> 00:14:53.760 they hurt each other. 00:14:53.760 --> 00:14:55.890 So even though they can linger around, 00:14:55.890 --> 00:14:58.770 they can never actually take over the population. 00:14:58.770 --> 00:15:00.870 And when we look at this mirror image case, 00:15:00.870 --> 00:15:04.710 we again see something similar, but with the roles reversed. 00:15:04.710 --> 00:15:06.420 All right, one more case, 00:15:06.420 --> 00:15:09.780 where both cooperation and fighting have a high cost. 00:15:09.780 --> 00:15:11.850 This situation also has a special name. 00:15:11.850 --> 00:15:14.040 It's called a Hawk-Dove game. 00:15:14.040 --> 00:15:15.750 The solo blobs are called hawks 00:15:15.750 --> 00:15:19.410 and the teamwork blobs are called doves. 00:15:19.410 --> 00:15:21.695 Anyway, what do you expect to happen? 00:15:21.695 --> 00:15:24.278 (gentle music) 00:15:26.250 --> 00:15:29.280 Here, neither strategy leads to a Nash equilibrium, 00:15:29.280 --> 00:15:33.630 not even a weak one, and neither strategy is an ESS either. 00:15:33.630 --> 00:15:36.150 It turns out there's a fraction of hawks versus doves 00:15:36.150 --> 00:15:38.880 that the population keeps trying to return to. 00:15:38.880 --> 00:15:40.410 There's still a little bit of chaos, 00:15:40.410 --> 00:15:42.630 but if one strategy starts to take over, 00:15:42.630 --> 00:15:45.450 suddenly it's advantageous to be the other strategy 00:15:45.450 --> 00:15:47.070 and the population moves back 00:15:47.070 --> 00:15:49.290 toward that fraction of cooperators. 00:15:49.290 --> 00:15:51.030 And just like that crossover point 00:15:51.030 --> 00:15:53.280 for the very first situation we looked at, 00:15:53.280 --> 00:15:55.260 we can calculate what that number will be 00:15:55.260 --> 00:15:57.120 for any reward matrix. 00:15:57.120 --> 00:15:59.610 But before we do that, let's see if we can look back, 00:15:59.610 --> 00:16:02.340 and summarize the relationship between the Nash equilibrium 00:16:02.340 --> 00:16:05.100 and an evolutionary stable strategy. 00:16:05.100 --> 00:16:06.180 It looks like a strategy 00:16:06.180 --> 00:16:08.850 is always an evolutionarily stable strategy 00:16:08.850 --> 00:16:11.820 if it leads to a strong Nash equilibrium. 00:16:11.820 --> 00:16:14.370 And a strategy can also be an ESS 00:16:14.370 --> 00:16:16.710 if it leads to a weak Nash equilibrium, 00:16:16.710 --> 00:16:18.900 as long as the other strategy doesn't lead 00:16:18.900 --> 00:16:21.570 to a Nash equilibrium, strong or weak. 00:16:21.570 --> 00:16:24.780 This is the definition of an evolutionarily stable strategy 00:16:24.780 --> 00:16:26.700 that you'd find in a textbook. 00:16:26.700 --> 00:16:27.720 But for me at least, 00:16:27.720 --> 00:16:29.010 it makes a lot more sense 00:16:29.010 --> 00:16:31.530 after going through all these examples. 00:16:31.530 --> 00:16:34.140 And these really are all of the examples, 00:16:34.140 --> 00:16:35.790 at least for a symmetrical game 00:16:35.790 --> 00:16:37.950 where the players can't communicate. 00:16:37.950 --> 00:16:40.050 Of course, there could be all kinds of different numbers 00:16:40.050 --> 00:16:42.120 and there could be more than two strategies. 00:16:42.120 --> 00:16:44.130 But any strategy with any rewards 00:16:44.130 --> 00:16:47.160 will always lead to either a strong Nash equilibrium, 00:16:47.160 --> 00:16:50.880 a weak Nash equilibrium, or no Nash equilibrium. 00:16:50.880 --> 00:16:52.440 And evolutionary stability 00:16:52.440 --> 00:16:54.600 will still always follow this rule. 00:16:54.600 --> 00:16:56.190 There may be an upcoming video 00:16:56.190 --> 00:16:57.630 with more than two strategies, 00:16:57.630 --> 00:16:59.547 but you're just gonna have to wait. 00:16:59.547 --> 00:17:02.550 At this point, there is a little bit more we could do. 00:17:02.550 --> 00:17:04.740 We could derive a formula that'll give us 00:17:04.740 --> 00:17:06.690 the stable equilibrium mixture 00:17:06.690 --> 00:17:10.500 in a Hawk-Dove game for any values in the reward matrix. 00:17:10.500 --> 00:17:11.700 We could do the same thing, 00:17:11.700 --> 00:17:14.550 but for the crossover point in that first situation 00:17:14.550 --> 00:17:17.670 where both strategies were evolutionarily stable. 00:17:17.670 --> 00:17:19.260 Another question which turns out 00:17:19.260 --> 00:17:21.030 to have an interesting answer is, 00:17:21.030 --> 00:17:23.070 what if we allow the blobs to flip a coin 00:17:23.070 --> 00:17:25.800 to decide which strategy to play? 00:17:25.800 --> 00:17:28.140 Consider those questions homework. 00:17:28.140 --> 00:17:30.570 I know, I said we would do those calculations, 00:17:30.570 --> 00:17:33.000 but I didn't say in this video. 00:17:33.000 --> 00:17:34.440 Videos are great and all, 00:17:34.440 --> 00:17:37.080 but you learn the most when you do things. 00:17:37.080 --> 00:17:38.760 And when you connect with other people 00:17:38.760 --> 00:17:41.070 who are also thinking about the same stuff. 00:17:41.070 --> 00:17:43.050 So if you wanna discuss these questions, 00:17:43.050 --> 00:17:45.360 or share your answers with me or others, 00:17:45.360 --> 00:17:47.490 or if you're already pretty familiar with this stuff 00:17:47.490 --> 00:17:48.720 and wanna help out, 00:17:48.720 --> 00:17:51.229 there's a Discord link in the video description. 00:17:51.229 --> 00:17:54.540 I also stream myself working on Twitch a few times per week 00:17:54.540 --> 00:17:56.610 and I'm happy to talk about it there. 00:17:56.610 --> 00:17:58.290 If you get value from these videos, 00:17:58.290 --> 00:18:00.090 and you'd like to support monetarily, 00:18:00.090 --> 00:18:01.770 there's a few ways you can do that. 00:18:01.770 --> 00:18:04.410 The newest way is to pick up one of these shirts. 00:18:04.410 --> 00:18:07.140 Another great way is to support monthly on Patreon. 00:18:07.140 --> 00:18:09.420 Patreons get early access to videos, 00:18:09.420 --> 00:18:10.920 some behind-the-scenes updates, 00:18:10.920 --> 00:18:13.170 and discounts on shirts and blob plushies 00:18:13.170 --> 00:18:14.640 and other things in the store. 00:18:14.640 --> 00:18:16.860 And even if you're not able to support monetarily, 00:18:16.860 --> 00:18:19.050 I appreciate very much that you watch all the way 00:18:19.050 --> 00:18:20.280 to the end of the video. 00:18:20.280 --> 00:18:21.630 And I'll see you next time!