1 00:00:00,070 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue to offer high-quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:20,152 --> 00:00:22,360 PROFESSOR: All right, so today what we're going to do 9 00:00:22,360 --> 00:00:25,170 is we're just start with a short review problem 10 00:00:25,170 --> 00:00:27,070 on rugged landscapes, just so that you 11 00:00:27,070 --> 00:00:28,770 get some sense of the kind of thing 12 00:00:28,770 --> 00:00:32,330 that I would expect you to be able to do a week from today. 13 00:00:32,330 --> 00:00:36,680 And then we'll get into the core topic of the class, which 14 00:00:36,680 --> 00:00:39,150 is evolutionary game theory. 15 00:00:39,150 --> 00:00:41,180 And we'll discuss why it is that you 16 00:00:41,180 --> 00:00:43,519 don't need to invoke any notion of rationality, which 17 00:00:43,519 --> 00:00:45,560 is kind of the traditional thing we do when we're 18 00:00:45,560 --> 00:00:47,685 talking about game theory applied to human decision 19 00:00:47,685 --> 00:00:48,820 making. 20 00:00:48,820 --> 00:00:50,806 Then we'll try to understand this difference 21 00:00:50,806 --> 00:00:53,180 to know what a Nash equilibrium is in the context of game 22 00:00:53,180 --> 00:00:56,120 theory versus an evolutionary stable strategy 23 00:00:56,120 --> 00:00:57,924 in this context. 24 00:00:57,924 --> 00:01:00,340 And we'll say something about the evolution of cooperation 25 00:01:00,340 --> 00:01:04,260 and experiments that one can do with microbial populations 26 00:01:04,260 --> 00:01:05,010 in the laboratory. 27 00:01:07,600 --> 00:01:11,045 Are there any questions before I get started? 28 00:01:19,312 --> 00:01:25,330 All right, so just on this question of evolutionary paths, 29 00:01:25,330 --> 00:01:27,780 on Tuesday we discussed the Weinreich paper 30 00:01:27,780 --> 00:01:31,510 where he talked about sort of different models 31 00:01:31,510 --> 00:01:34,350 that you might use to try to make estimates of the path 32 00:01:34,350 --> 00:01:36,480 that evolution might take on that fitness landscape 33 00:01:36,480 --> 00:01:37,930 that he measured. 34 00:01:37,930 --> 00:01:41,560 So he measured this MIC, the minimum inhibitory 35 00:01:41,560 --> 00:01:45,097 concentration, on all 2 to the 5 or 32 different states, 36 00:01:45,097 --> 00:01:47,305 and then tried to say something about the probability 37 00:01:47,305 --> 00:01:49,370 that different paths will be taken. 38 00:01:49,370 --> 00:01:51,870 So I just want to explore this question 39 00:01:51,870 --> 00:01:55,162 about paths in a simpler landscape, where 40 00:01:55,162 --> 00:01:57,620 by construction here, I'm to going to give you some fitness 41 00:01:57,620 --> 00:02:00,130 values just so that we can be clear about why 42 00:02:00,130 --> 00:02:02,194 it is that there might be different paths, 43 00:02:02,194 --> 00:02:03,610 or what determines the probability 44 00:02:03,610 --> 00:02:05,146 that different paths are taken. 45 00:02:05,146 --> 00:02:06,770 So what we want to do is assume that we 46 00:02:06,770 --> 00:02:12,445 are in a population that is experiencing this Moran process 47 00:02:12,445 --> 00:02:18,660 or Moran model, constant population size N equal to, 48 00:02:18,660 --> 00:02:21,780 in this case, we'll say 1,000. 49 00:02:21,780 --> 00:02:27,740 And let's say that the mutation rate is 10 to the minus 6. 50 00:02:27,740 --> 00:02:30,204 So each time that an individual divides, it has a 1 51 00:02:30,204 --> 00:02:32,610 in a million probability of mutating. 52 00:02:32,610 --> 00:02:36,785 And that's a per base pair mutation rate. 53 00:02:36,785 --> 00:02:39,494 And I'll show you what I mean by that. 54 00:02:39,494 --> 00:02:41,535 And in particular, we're going to have genotypes. 55 00:02:44,890 --> 00:02:46,360 Originally when we discussed this, 56 00:02:46,360 --> 00:02:48,845 we were talking about just mutations, maybe A's and B's. 57 00:02:48,845 --> 00:02:50,220 But now, what we're going to have 58 00:02:50,220 --> 00:02:54,920 is just a short genome that's string length 2. 59 00:02:54,920 --> 00:03:02,890 So we might have 0, 0, which has relative fitness 1, 0, 1. 60 00:03:14,880 --> 00:03:17,810 So we're assuming that this is relative fitness 61 00:03:17,810 --> 00:03:19,380 as compared to the 0, 0 state. 62 00:03:19,380 --> 00:03:25,350 We're going to start in the 0, 0 state with 1,000 63 00:03:25,350 --> 00:03:28,390 isogenic individuals, all 0, 0. 64 00:03:28,390 --> 00:03:32,020 And the question is, what's going to happen eventually? 65 00:03:34,660 --> 00:03:55,980 And in particular, what path will be taken on this landscape 66 00:03:55,980 --> 00:03:56,480 here? 67 00:04:03,372 --> 00:04:04,830 In particular, what we want to know 68 00:04:04,830 --> 00:04:07,000 is the probability of taking this path. 69 00:04:33,989 --> 00:04:36,030 You can start thinking about it while I write out 70 00:04:36,030 --> 00:04:38,630 some possibilities that we can vote for, 71 00:04:38,630 --> 00:04:40,626 and I'll give you a minute to think about it. 72 00:04:40,626 --> 00:04:45,090 So don't-- 73 00:05:19,350 --> 00:05:22,644 Are there any questions about what I'm trying to ask here? 74 00:05:22,644 --> 00:05:28,299 AUDIENCE: So this is the long time? 75 00:05:28,299 --> 00:05:30,480 So we assume that in the long time, 76 00:05:30,480 --> 00:05:32,880 it will go from 0, 0 to 1, 1? 77 00:05:32,880 --> 00:05:36,400 PROFESSOR: That's right, yes, so if we wait long enough, 78 00:05:36,400 --> 00:05:39,900 the population will get there, and the 1, 1 genotype 79 00:05:39,900 --> 00:05:42,210 will fix in the population. 80 00:05:42,210 --> 00:05:44,490 We can talk a bit later about how long it's 81 00:05:44,490 --> 00:05:46,432 going to take to get there, and so forth. 82 00:05:46,432 --> 00:05:48,892 AUDIENCE: And we're assuming that from 0, 1, 83 00:05:48,892 --> 00:05:52,342 it can't go back to 0, 0? 84 00:05:52,342 --> 00:05:53,050 PROFESSOR: Right. 85 00:05:53,050 --> 00:05:57,080 Yeah, so we'll discuss the situations 86 00:05:57,080 --> 00:05:58,480 when we have to worry about that, 87 00:05:58,480 --> 00:06:00,010 and when we don't, and so forth. 88 00:06:00,010 --> 00:06:02,320 But for now, if you'd like, we can 89 00:06:02,320 --> 00:06:05,400 say that this is even just mu sub b, the beneficial mutation 90 00:06:05,400 --> 00:06:09,200 rate per base pair, assuming that the 0's can only 91 00:06:09,200 --> 00:06:10,770 turn into 1s. 92 00:06:10,770 --> 00:06:13,130 Then after we think about this, we 93 00:06:13,130 --> 00:06:17,029 could figure out if that's important, 94 00:06:17,029 --> 00:06:18,570 or when it's important, and so forth. 95 00:06:23,130 --> 00:06:23,880 Yes. 96 00:06:23,880 --> 00:06:24,796 AUDIENCE: [INAUDIBLE]? 97 00:06:30,880 --> 00:06:31,540 PROFESSOR: No. 98 00:06:31,540 --> 00:06:34,720 All right, so we're starting with all 1,000 individuals 99 00:06:34,720 --> 00:06:36,770 being in the 0, 0 state, because now we're 100 00:06:36,770 --> 00:06:38,187 allowing some mutation rate. 101 00:06:38,187 --> 00:06:39,141 AUDIENCE: [INAUDIBLE]. 102 00:06:45,579 --> 00:06:47,120 PROFESSOR: And you also have to think 103 00:06:47,120 --> 00:06:51,370 about this first mutation-- will it fix or not? 104 00:06:55,128 --> 00:06:59,000 AUDIENCE: But it's not [INAUDIBLE] first mutation, 105 00:06:59,000 --> 00:07:01,904 the mutation of one element of that population 106 00:07:01,904 --> 00:07:06,141 will go to 0, 1 [INAUDIBLE]? 107 00:07:06,141 --> 00:07:08,390 PROFESSOR: I'm not sure if I understand your question. 108 00:07:08,390 --> 00:07:11,785 AUDIENCE: So if we start out with all 0, 0, and then one 109 00:07:11,785 --> 00:07:12,755 mutation [INAUDIBLE]? 110 00:07:16,150 --> 00:07:19,545 And if that mutation-- are we assuming 111 00:07:19,545 --> 00:07:22,470 that that mutation is 0, 1 and then figuring out [INAUDIBLE]? 112 00:07:22,470 --> 00:07:24,740 PROFESSOR: Well, OK, you're asking kind 113 00:07:24,740 --> 00:07:26,902 of what I mean by path here. 114 00:07:26,902 --> 00:07:28,230 AUDIENCE: Yeah, I guess. 115 00:07:28,230 --> 00:07:31,180 PROFESSOR: Yeah, all right, so I'll say path 116 00:07:31,180 --> 00:07:37,130 means that that this was the dominant probability 117 00:07:37,130 --> 00:07:40,180 trajectory of the population through there. 118 00:07:40,180 --> 00:07:43,140 We'll also discuss whether it somehow 119 00:07:43,140 --> 00:07:46,690 is very likely is going to kind of have to go through one 120 00:07:46,690 --> 00:07:48,930 or the other of them. 121 00:07:48,930 --> 00:07:52,450 The probability of getting both mutations in one generation 122 00:07:52,450 --> 00:07:55,230 is going to be 10 to the minus 12. 123 00:07:55,230 --> 00:08:00,040 So that's going to be a very rare thing, at least given 124 00:08:00,040 --> 00:08:02,157 these parameters, and so forth. 125 00:08:02,157 --> 00:08:03,990 And then there's another question, which is, 126 00:08:03,990 --> 00:08:07,780 will 0, 1 actually fix in the population 127 00:08:07,780 --> 00:08:12,247 before you later fix this population? 128 00:08:12,247 --> 00:08:14,580 And actually, I think the answers to all these questions 129 00:08:14,580 --> 00:08:16,163 are in principal already on the board. 130 00:08:20,330 --> 00:08:22,404 Because there's a question of do we 131 00:08:22,404 --> 00:08:24,070 have to worry about clonal interference? 132 00:08:24,070 --> 00:08:27,060 Are these things neutral or not? 133 00:08:27,060 --> 00:08:30,500 And really, this is in some ways a very simple problem. 134 00:08:30,500 --> 00:08:32,010 But in another way, you have to keep 135 00:08:32,010 --> 00:08:34,110 track of lots of different things, 136 00:08:34,110 --> 00:08:37,080 and which regime we're in and so forth. 137 00:08:37,080 --> 00:08:40,820 So that's what makes it such a wonderful exam problem. 138 00:08:40,820 --> 00:08:44,780 If you understand what's going on, 139 00:08:44,780 --> 00:08:46,477 you can answer it in a minute. 140 00:08:46,477 --> 00:08:48,310 But if you don't understand what's going on, 141 00:08:48,310 --> 00:08:49,590 it'll take you an hour. 142 00:08:52,010 --> 00:08:52,510 Yes? 143 00:08:52,510 --> 00:08:53,010 No? 144 00:08:53,010 --> 00:08:53,610 Maybe? 145 00:08:53,610 --> 00:08:57,269 Well, I'll give you another 20 seconds. 146 00:08:57,269 --> 00:08:58,935 Hopefully, you've been thinking about it 147 00:08:58,935 --> 00:08:59,976 while we've been talking. 148 00:09:19,305 --> 00:09:20,680 All right, do you need more time? 149 00:10:10,000 --> 00:10:11,470 Why don't we go ahead and vote? 150 00:10:11,470 --> 00:10:13,820 I think it's very likely that we will not 151 00:10:13,820 --> 00:10:15,889 be at the kind of 100% mark, in which case 152 00:10:15,889 --> 00:10:18,222 you'll have a chance to talk about it and think about it 153 00:10:18,222 --> 00:10:18,722 some more. 154 00:10:18,722 --> 00:10:19,370 Ready? 155 00:10:19,370 --> 00:10:21,460 Three, two, one. 156 00:10:25,476 --> 00:10:29,710 OK, all right, so we do have a fair range of answers. 157 00:10:29,710 --> 00:10:32,050 I'd say it might be kind of something like 50-50. 158 00:10:32,050 --> 00:10:33,090 And that's great. 159 00:10:33,090 --> 00:10:35,880 It means that there should be something to talk about. 160 00:10:35,880 --> 00:10:37,072 So turn to a neighbor. 161 00:10:37,072 --> 00:10:38,530 You should be able to find somebody 162 00:10:38,530 --> 00:10:40,180 that disagrees with you. 163 00:10:40,180 --> 00:10:44,080 And if everyone around you agrees, you can maybe-- 164 00:10:44,080 --> 00:10:46,690 all right, so there's a group of D's and a group of B's here, 165 00:10:46,690 --> 00:10:48,064 which means that everybody-- 166 00:10:48,064 --> 00:10:48,980 AUDIENCE: Let's fight. 167 00:10:48,980 --> 00:10:49,680 PROFESSOR: All right, so everybody 168 00:10:49,680 --> 00:10:50,950 thinks that everybody agrees with them, 169 00:10:50,950 --> 00:10:53,366 but you just need to look a little bit more long distance. 170 00:10:53,366 --> 00:10:55,270 So turn to a pseudo-neighbor. 171 00:10:55,270 --> 00:10:58,910 You should be able to find somebody there. 172 00:10:58,910 --> 00:11:02,695 It's roughly even here, so you should be able to find someone. 173 00:11:02,695 --> 00:11:06,118 [INTERPOSING VOICES] 174 00:11:46,329 --> 00:11:48,495 So I don't see much in the way of vibrant discussion 175 00:11:48,495 --> 00:11:49,280 and argument. 176 00:11:49,280 --> 00:11:52,882 You guys should be passionately defending your choice here. 177 00:11:52,882 --> 00:11:56,354 [INTERPOSING VOICES] 178 00:12:41,264 --> 00:12:42,680 Yeah, that's a higher order point. 179 00:12:42,680 --> 00:12:43,846 I wouldn't worry about that. 180 00:12:46,089 --> 00:12:46,922 [INTERPOSING VOICES] 181 00:12:59,350 --> 00:13:02,050 All right, it looks like people are having a nice discussion. 182 00:13:02,050 --> 00:13:04,350 But I might still go ahead and cut it short, 183 00:13:04,350 --> 00:13:07,842 just so that we can get on to evolutionary game theory. 184 00:13:07,842 --> 00:13:09,550 But I would like to see where people are. 185 00:13:09,550 --> 00:13:11,730 And we'll discuss it as a group, so don't 186 00:13:11,730 --> 00:13:14,960 be too disappointed if you don't get finished there. 187 00:13:14,960 --> 00:13:17,140 But I do want to see kind of where we are. 188 00:13:17,140 --> 00:13:18,150 Ready? 189 00:13:18,150 --> 00:13:19,970 Three, two, one. 190 00:13:23,748 --> 00:13:30,090 OK, so it still is, maybe, split roughly equally between D 191 00:13:30,090 --> 00:13:35,200 and maybe a B-ish and some C's. 192 00:13:35,200 --> 00:13:38,840 All right, does somebody want to volunteer their explanation? 193 00:13:44,250 --> 00:13:45,468 Yes. 194 00:13:45,468 --> 00:13:47,963 AUDIENCE: I'm not sure how good it is, 195 00:13:47,963 --> 00:13:51,955 but I was thinking about what's the probability of going 196 00:13:51,955 --> 00:13:55,448 to 0, 1 instead of 1, 0. 197 00:13:55,448 --> 00:14:04,929 And I just took it as the ratio of the extra benefit of 0, 1 198 00:14:04,929 --> 00:14:07,460 over the benefit of 1, 0. 199 00:14:07,460 --> 00:14:09,910 PROFESSOR: Sure, OK, and just to start out, 200 00:14:09,910 --> 00:14:11,510 which answer are you arguing for? 201 00:14:11,510 --> 00:14:12,100 AUDIENCE: D. 202 00:14:12,100 --> 00:14:13,130 PROFESSOR: D, OK, all right. 203 00:14:13,130 --> 00:14:15,180 So you're saying D. And you're saying, all right, 204 00:14:15,180 --> 00:14:18,500 maybe because of the extra, that 1, 0 is somehow 205 00:14:18,500 --> 00:14:20,520 more fit than 0, 1. 206 00:14:20,520 --> 00:14:24,290 And you've taken some relative rates or ratios 207 00:14:24,290 --> 00:14:27,218 for which reason? 208 00:14:27,218 --> 00:14:31,130 AUDIENCE: Well, I took 0.02 and then 0.1, which is 1/5. 209 00:14:31,130 --> 00:14:34,176 And then I decided that that should 210 00:14:34,176 --> 00:14:36,509 be around what it is, but slightly less, because there's 211 00:14:36,509 --> 00:14:40,010 also a chance that [INAUDIBLE]. 212 00:14:40,010 --> 00:14:40,810 PROFESSOR: OK, yes. 213 00:14:40,810 --> 00:14:43,320 I think the arguments there-- there's 214 00:14:43,320 --> 00:14:46,730 a lot of truth to the arguments that you're saying. 215 00:14:46,730 --> 00:14:50,580 Yeah, it's a little-- right and another question 216 00:14:50,580 --> 00:14:55,080 is exactly why might it be 1/6 instead of 1/5 is, 217 00:14:55,080 --> 00:14:58,630 I think, a little bit hazy in this here. 218 00:14:58,630 --> 00:15:00,309 It's OK, but it's close. 219 00:15:00,309 --> 00:15:02,100 Does somebody want to offer an explanation? 220 00:15:02,100 --> 00:15:05,580 So here, that was an argument of roughly maybe why it's D-ish. 221 00:15:05,580 --> 00:15:07,720 Because D is very different from B-- 222 00:15:07,720 --> 00:15:09,930 order of magnitude different. 223 00:15:09,930 --> 00:15:13,460 So can somebody offer why their neighbor thought it was B? 224 00:15:13,460 --> 00:15:14,460 Yeah. 225 00:15:14,460 --> 00:15:18,039 AUDIENCE: So I knew that it was B, because I considered the two 226 00:15:18,039 --> 00:15:20,631 paths, both from 0, 0 to 0, 1. 227 00:15:20,631 --> 00:15:23,310 I first checked S, N and it's non-neutral. 228 00:15:23,310 --> 00:15:27,766 So probably [INAUDIBLE] S. So the probability 229 00:15:27,766 --> 00:15:31,378 for that first path would be the S for 0, 1, 230 00:15:31,378 --> 00:15:34,820 so it's 0.02, which is 1/50, multiplied by the probability 231 00:15:34,820 --> 00:15:38,800 that the other [INAUDIBLE] 1, 0 would die out [INAUDIBLE]. 232 00:15:38,800 --> 00:15:41,360 PROFESSOR: Right, so there are two related questions. 233 00:15:41,360 --> 00:15:43,150 And I think that this explanation here 234 00:15:43,150 --> 00:15:45,780 is answering a slightly different question. 235 00:15:45,780 --> 00:15:47,350 OK, so let me try to explain what 236 00:15:47,350 --> 00:15:50,050 the two questions are here. 237 00:15:50,050 --> 00:15:51,830 So the question that you're answering 238 00:15:51,830 --> 00:15:58,410 is, if you have kind of 998 individuals that 239 00:15:58,410 --> 00:16:01,860 are 0, 0 individuals, and you have one that's 0, 1, 240 00:16:01,860 --> 00:16:05,200 and you have one individual that is 1, 0. 241 00:16:05,200 --> 00:16:08,660 So this is like these problems that we did a couple weeks ago, 242 00:16:08,660 --> 00:16:11,324 where we said, you imagine in the population you have 243 00:16:11,324 --> 00:16:13,740 a couple different kinds of mutants that are present maybe 244 00:16:13,740 --> 00:16:15,599 in one copy. 245 00:16:15,599 --> 00:16:17,140 And then we were asking, well, what's 246 00:16:17,140 --> 00:16:19,930 the probability that this individual is going to fix? 247 00:16:19,930 --> 00:16:21,440 And what's the probability this individual is going to fix? 248 00:16:21,440 --> 00:16:23,380 And what's the probability that these guys are 249 00:16:23,380 --> 00:16:26,580 going to go extinct, and this one will therefore fix? 250 00:16:26,580 --> 00:16:30,270 And I think that's the calculation that you're 251 00:16:30,270 --> 00:16:32,440 describing, where you say, OK, well, 252 00:16:32,440 --> 00:16:35,710 in order for this individual to fix, 253 00:16:35,710 --> 00:16:40,840 he has to survive stochastic extinction, which happens 254 00:16:40,840 --> 00:16:43,820 with the probability of 2%. 255 00:16:43,820 --> 00:16:49,190 And the 1, 0 individual has to go extinct, 256 00:16:49,190 --> 00:16:53,220 which happens 90% of the time. 257 00:16:53,220 --> 00:16:57,340 And so this is, indeed, answering the question 258 00:16:57,340 --> 00:17:02,300 that if you had one copy of each of these two mutant individuals 259 00:17:02,300 --> 00:17:06,910 in the population, that's the answer to what 260 00:17:06,910 --> 00:17:10,460 is the probability that this 0, 1 mutant 261 00:17:10,460 --> 00:17:11,880 would fix in the population. 262 00:17:11,880 --> 00:17:13,300 Right? 263 00:17:13,300 --> 00:17:16,290 But that's a slightly different question than if we ask, 264 00:17:16,290 --> 00:17:19,236 we're going to start with an entire population at 0, 0, 265 00:17:19,236 --> 00:17:21,319 and now these mutations will be occurring randomly 266 00:17:21,319 --> 00:17:22,200 at some rate. 267 00:17:22,200 --> 00:17:23,970 And then something's going to happen. 268 00:17:23,970 --> 00:17:25,250 Somehow the population is going to climb up 269 00:17:25,250 --> 00:17:26,349 this fitness landscape. 270 00:17:26,349 --> 00:17:29,230 And we're trying to figure out the relative probability 271 00:17:29,230 --> 00:17:32,010 that it's going to take kind of one path or another. 272 00:17:32,010 --> 00:17:37,090 Do you see the difference between these two questions? 273 00:17:37,090 --> 00:17:44,550 So indeed, this is the correct answer to a different question. 274 00:17:44,550 --> 00:17:47,260 And so it's going to end up being D. 275 00:17:47,260 --> 00:17:50,990 And now we want to try to figure out how to get there. 276 00:17:50,990 --> 00:17:54,310 Because I think it is a bit tricky. 277 00:17:54,310 --> 00:17:56,580 And in order in order to figure out to get there, 278 00:17:56,580 --> 00:17:59,860 we have to make sure we keep track of which 279 00:17:59,860 --> 00:18:01,210 parameter regime we're in. 280 00:18:01,210 --> 00:18:03,390 So there are a couple of questions we have to ask. 281 00:18:03,390 --> 00:18:05,590 First of all, we have to remember 282 00:18:05,590 --> 00:18:07,940 that we start out with everybody, all 1,000 283 00:18:07,940 --> 00:18:09,190 individuals in the 0, 0 state. 284 00:18:09,190 --> 00:18:12,210 So there are initially no mutants in the population. 285 00:18:12,210 --> 00:18:14,240 But they're just replicating at some rate. 286 00:18:14,240 --> 00:18:16,910 And every now and then, mutation's going to occur. 287 00:18:16,910 --> 00:18:20,480 Now one thing we have to answer, we have to think about, 288 00:18:20,480 --> 00:18:22,550 is whether these are nearly neutral mutations. 289 00:18:28,440 --> 00:18:29,350 Verbally yes or no? 290 00:18:29,350 --> 00:18:29,850 Ready? 291 00:18:29,850 --> 00:18:31,610 Three, two, one. 292 00:18:31,610 --> 00:18:32,180 AUDIENCE: No. 293 00:18:32,180 --> 00:18:33,570 PROFESSOR: No, right. 294 00:18:33,570 --> 00:18:35,017 And that's because we want to ask 295 00:18:35,017 --> 00:18:36,850 for, if it's nearly neutral, we want to ask, 296 00:18:36,850 --> 00:18:40,510 is the magnitude of S times N much greater or much less 297 00:18:40,510 --> 00:18:42,040 than 1? 298 00:18:42,040 --> 00:18:44,450 In particular, if they're much greater than 1, 299 00:18:44,450 --> 00:18:49,559 as is the case here, then we're in a nice, simple regime. 300 00:18:49,559 --> 00:18:51,600 And it's easy to get paralyzed in this situation, 301 00:18:51,600 --> 00:18:56,920 because there's more than one S. But in both cases, 302 00:18:56,920 --> 00:18:58,260 S times N is much larger than 1. 303 00:18:58,260 --> 00:19:01,580 We can take the smaller S, which is 2 over 100, 304 00:19:01,580 --> 00:19:03,540 and S times N is 20, right? 305 00:19:03,540 --> 00:19:10,680 So S for the 0, 1 state times N is 20, 306 00:19:10,680 --> 00:19:14,080 which is much greater than 1. 307 00:19:14,080 --> 00:19:15,710 What this tells us is that if we do 308 00:19:15,710 --> 00:19:19,340 get this mutant appearing in the population, 309 00:19:19,340 --> 00:19:23,810 then he or she will have a probability S of surviving 310 00:19:23,810 --> 00:19:24,810 stochastic extinction. 311 00:19:28,050 --> 00:19:34,470 So probability of surviving stochastic extinction 312 00:19:34,470 --> 00:19:38,220 if the individual appears is equal to S 0, 313 00:19:38,220 --> 00:19:42,450 1, which is equal to 2%. 314 00:19:42,450 --> 00:19:57,500 Whereas for the 1, 0 state, that's going to be 0.1. 315 00:19:57,500 --> 00:20:00,660 Now, this is assuming that the mutation appears 316 00:20:00,660 --> 00:20:03,350 in the population, that's the probability it will survive 317 00:20:03,350 --> 00:20:05,300 stochastic extinction. 318 00:20:05,300 --> 00:20:08,730 Now, just as a reminder, surviving stochastic extinction 319 00:20:08,730 --> 00:20:12,540 roughly corresponds to this becoming an established idea. 320 00:20:12,540 --> 00:20:14,600 And becoming established was what again? 321 00:20:21,980 --> 00:20:22,952 AUDIENCE: [INAUDIBLE]. 322 00:20:22,952 --> 00:20:23,910 PROFESSOR: What's that? 323 00:20:23,910 --> 00:20:26,090 AUDIENCE: S1 is [INAUDIBLE]. 324 00:20:26,090 --> 00:20:27,440 PROFESSOR: It's when S1-- 325 00:20:27,440 --> 00:20:28,380 AUDIENCE: [INAUDIBLE]. 326 00:20:28,380 --> 00:20:29,630 PROFESSOR: Yeah, that's right. 327 00:20:29,630 --> 00:20:32,860 All right, so established-- when we say established, 328 00:20:32,860 --> 00:20:37,960 what we mean is that this corresponds 329 00:20:37,960 --> 00:20:42,280 to saying that this probability that we talked about 330 00:20:42,280 --> 00:20:47,050 before this X sub i is approximately equal to 1. 331 00:20:47,050 --> 00:20:49,050 So this question is, how many individuals do you 332 00:20:49,050 --> 00:20:51,910 have to get to in the population before you're 333 00:20:51,910 --> 00:20:54,030 very likely to fix? 334 00:20:54,030 --> 00:21:00,130 And what we found is that that number established went as 1 335 00:21:00,130 --> 00:21:02,570 over the selection coefficient. 336 00:21:02,570 --> 00:21:06,264 So in this case, you would need to have 50 individuals 337 00:21:06,264 --> 00:21:08,430 before you were kind of more likely to fix than not. 338 00:21:08,430 --> 00:21:10,013 So if you want to be much more likely, 339 00:21:10,013 --> 00:21:13,420 you might need twice that or so. 340 00:21:13,420 --> 00:21:16,242 Do you guys remember that? 341 00:21:16,242 --> 00:21:18,450 This is not important for this question, necessarily, 342 00:21:18,450 --> 00:21:22,284 but it might be important at a later date. 343 00:21:22,284 --> 00:21:24,260 AUDIENCE: And so for nearly neutral mutations, 344 00:21:24,260 --> 00:21:25,989 the whole point is that the number needed 345 00:21:25,989 --> 00:21:29,200 to become established is equal to the population. 346 00:21:29,200 --> 00:21:32,430 PROFESSOR: Yeah, so everything kind of works, right? 347 00:21:35,910 --> 00:21:37,710 OK, so the way that we can think about this 348 00:21:37,710 --> 00:21:41,020 is, now we have this population, 1,000 individuals. 349 00:21:41,020 --> 00:21:43,590 They're dividing at some rate. 350 00:21:43,590 --> 00:21:45,060 Mutations are going to appear. 351 00:21:47,640 --> 00:21:51,270 Now we know if they did appear, the probability they would fix. 352 00:21:51,270 --> 00:21:53,785 This is assuming there's no clonal interference, right? 353 00:22:05,375 --> 00:22:07,000 Because if there's clonal interference, 354 00:22:07,000 --> 00:22:09,450 then surviving stochastic extinction 355 00:22:09,450 --> 00:22:11,572 is not the same thing as fixing. 356 00:22:11,572 --> 00:22:13,155 If they both appear in the population, 357 00:22:13,155 --> 00:22:15,780 and they both survive stochastic extinction, 358 00:22:15,780 --> 00:22:19,440 then this mutant loses to this mutant. 359 00:22:19,440 --> 00:22:21,570 That's the clonal interference. 360 00:22:21,570 --> 00:22:24,236 Do we have to worry about clonal interference in this situation? 361 00:22:50,330 --> 00:22:54,340 So remember, this was comparing the two time scales. 362 00:22:54,340 --> 00:22:57,400 This was comparing the time between 363 00:22:57,400 --> 00:23:06,998 successive establishment events, which went as 1 over mu N S. 364 00:23:06,998 --> 00:23:11,200 And the other one is the time between the time to fix, 365 00:23:11,200 --> 00:23:16,130 which went as 1 over S log of NS, right? 366 00:23:16,130 --> 00:23:17,710 So we can ignore clonal interference 367 00:23:17,710 --> 00:23:19,180 if this is much larger than that. 368 00:23:22,730 --> 00:23:24,820 So no clonal interference corresponds 369 00:23:24,820 --> 00:23:32,960 to mu N log NS much less than 1. 370 00:23:32,960 --> 00:23:35,860 No clonal interference, same as this statement. 371 00:23:35,860 --> 00:23:36,480 Is that right? 372 00:23:40,630 --> 00:23:41,386 Did I do it right? 373 00:23:41,386 --> 00:23:43,920 OK. 374 00:23:43,920 --> 00:23:47,650 So and once again, there are multiple S's, and it's 375 00:23:47,650 --> 00:23:50,220 easy to get kind of upset about this. 376 00:23:50,220 --> 00:23:53,096 But you can just use whichever S would be-- 377 00:23:53,096 --> 00:23:54,970 which S would you want to use to be kind of-- 378 00:23:57,221 --> 00:23:58,762 AUDIENCE: Small or large [INAUDIBLE]? 379 00:24:01,400 --> 00:24:04,550 PROFESSOR: To be on the safe or conservative side, 380 00:24:04,550 --> 00:24:06,760 we want to take this to be as big as possible. 381 00:24:06,760 --> 00:24:11,120 So we take S actually as big as we can, right? 382 00:24:11,120 --> 00:24:12,250 It's in the log. 383 00:24:12,250 --> 00:24:13,720 So details, right? 384 00:24:13,720 --> 00:24:20,900 But we can see we have 10 to the minus 6, 10 to the 3, 385 00:24:20,900 --> 00:24:28,430 and then this is the log of maybe 100, which 386 00:24:28,430 --> 00:24:32,012 is, like, 4 or 5. 387 00:24:32,012 --> 00:24:32,970 Is it closer to 4 or 5? 388 00:24:32,970 --> 00:24:35,480 I don't know, but it doesn't matter. 389 00:24:35,480 --> 00:24:36,680 We'll say 5. 390 00:24:36,680 --> 00:24:38,070 This is indeed much less than 1. 391 00:24:43,420 --> 00:24:45,010 So indeed, we don't have to worry 392 00:24:45,010 --> 00:24:46,680 about clonal interference. 393 00:24:46,680 --> 00:24:49,240 This is a wonderful simplification. 394 00:24:49,240 --> 00:24:52,440 What it's saying is that the population is dividing. 395 00:24:52,440 --> 00:24:55,760 Every now and then, a mutation occurs in the population. 396 00:24:55,760 --> 00:24:58,290 It could be either the 0, 1 or the 1, 0. 397 00:24:58,290 --> 00:25:01,000 But in either case, the fate of that mutation 398 00:25:01,000 --> 00:25:04,039 is resolved before the next mutation occurs. 399 00:25:04,039 --> 00:25:05,580 So you don't need to worry about them 400 00:25:05,580 --> 00:25:06,900 competing in the population. 401 00:25:06,900 --> 00:25:10,440 Instead, just at some constant rate they're appearing. 402 00:25:10,440 --> 00:25:13,290 And given that they appear, there's some probability 403 00:25:13,290 --> 00:25:14,610 that they're going to fix. 404 00:25:14,610 --> 00:25:17,420 So that leads to effective rates going to each of those two 405 00:25:17,420 --> 00:25:20,690 steps-- going to 0, 1 or 1, 0. 406 00:25:20,690 --> 00:25:24,570 And in particular, this is like a chemical reaction, where we 407 00:25:24,570 --> 00:25:29,110 have some chemical state here. 408 00:25:29,110 --> 00:25:30,540 We have two rates. 409 00:25:30,540 --> 00:25:35,760 There's the k going to 0, 1, the k going to 1, 0. 410 00:25:39,410 --> 00:25:41,899 And what we know is we know the ratio of those rates. 411 00:25:41,899 --> 00:25:43,440 And that's everything we need to know 412 00:25:43,440 --> 00:25:45,481 to calculate the relative probabilities of taking 413 00:25:45,481 --> 00:25:50,810 those states, because the probability of going through 414 00:25:50,810 --> 00:25:53,960 to 0, 1-- we want to go that direction-- 0, 1, this 415 00:25:53,960 --> 00:26:02,330 is going to be given by k 0, 1 divided by k 0, 1 plus k 1, 0. 416 00:26:02,330 --> 00:26:05,810 So this is how we get 1/6 instead of 1/5. 417 00:26:05,810 --> 00:26:10,950 Because this thing is 1/5 of that. 418 00:26:10,950 --> 00:26:12,870 So it's like 1, and then 1, 5. 419 00:26:21,140 --> 00:26:24,270 So this is actually, in principle, not quite answering 420 00:26:24,270 --> 00:26:26,320 the question that I asked, because this 421 00:26:26,320 --> 00:26:30,250 is talking about the relative probability of the first state, 422 00:26:30,250 --> 00:26:32,120 the first mutant to fix. 423 00:26:32,120 --> 00:26:35,780 In principle, it is possible that from there, there's 424 00:26:35,780 --> 00:26:37,040 some rate of coming back. 425 00:26:37,040 --> 00:26:40,214 Or they might not necessarily move forward on up that hill. 426 00:26:40,214 --> 00:26:42,130 Do you guys understand what I'm talking about? 427 00:26:42,130 --> 00:26:43,588 Because it goes from here to there. 428 00:26:43,588 --> 00:26:47,520 Because we really want to know about this next step, 429 00:26:47,520 --> 00:26:49,290 going to the 1, 1 state. 430 00:26:52,980 --> 00:26:57,760 But in this case do we have to worry about going backwards? 431 00:26:57,760 --> 00:26:58,320 No. 432 00:26:58,320 --> 00:26:58,990 And why not? 433 00:27:02,910 --> 00:27:03,720 It's very unlikely. 434 00:27:03,720 --> 00:27:05,502 And in particular, you could think now 435 00:27:05,502 --> 00:27:07,710 that you're here you can talk about the rate of going 436 00:27:07,710 --> 00:27:11,310 to the 1, 1 state as compared to the rate of going to 0, 1. 437 00:27:11,310 --> 00:27:17,080 And those are going to be exponentially different. 438 00:27:17,080 --> 00:27:19,950 Because just as this was a non-neutral beneficial 439 00:27:19,950 --> 00:27:21,820 mutation, that means that going from 0, 440 00:27:21,820 --> 00:27:24,260 1 back is going to be a non-neutral deleterious 441 00:27:24,260 --> 00:27:25,685 mutation. 442 00:27:25,685 --> 00:27:28,310 So the probability of fixing it in the back direction is not 0, 443 00:27:28,310 --> 00:27:29,726 but it's exponentially suppressed. 444 00:27:36,060 --> 00:27:38,890 I think it's very important to understand 445 00:27:38,890 --> 00:27:41,290 all the different pieces of this kind of puzzle, 446 00:27:41,290 --> 00:27:44,070 because it incorporates many different ideas that we've 447 00:27:44,070 --> 00:27:46,665 talked about over the last few weeks. 448 00:27:46,665 --> 00:27:50,480 If there are questions, please ask now. 449 00:27:50,480 --> 00:27:51,277 Yes. 450 00:27:51,277 --> 00:27:52,818 AUDIENCE: What about the [INAUDIBLE]? 451 00:27:58,285 --> 00:28:09,219 [INAUDIBLE] 0, 0 to 1, 0 to 1, 1? 452 00:28:09,219 --> 00:28:13,195 Then it seems like the benefit of 1, 0 versus [INAUDIBLE]. 453 00:28:15,700 --> 00:28:19,300 PROFESSOR: All right, so you're wondering about-- 454 00:28:19,300 --> 00:28:22,810 so the fitness of the 1, 1 state was 1.2. 455 00:28:22,810 --> 00:28:26,860 So you're pointing out that it's actually easier 456 00:28:26,860 --> 00:28:29,700 to go from the 0, 1 state to the 1, 1 as 457 00:28:29,700 --> 00:28:31,645 compared to the 1, 0 to the 1, 1. 458 00:28:31,645 --> 00:28:34,615 AUDIENCE: Right, which seems like a reason for why 459 00:28:34,615 --> 00:28:38,486 we wouldn't care about [INAUDIBLE]. 460 00:28:38,486 --> 00:28:39,610 PROFESSOR: Yeah, OK, right. 461 00:28:39,610 --> 00:28:43,810 So if anything, in some ways, this 462 00:28:43,810 --> 00:28:46,590 actually provides a bias going towards the 0, 1 state, 463 00:28:46,590 --> 00:28:49,456 because it's saying that if we do get to 0, 1, 464 00:28:49,456 --> 00:28:51,080 it's actually easier to move forward as 465 00:28:51,080 --> 00:28:52,324 compared to this other path. 466 00:28:52,324 --> 00:28:53,990 In practice, it doesn't actually matter, 467 00:28:53,990 --> 00:28:56,890 because this acts as a ratchet. 468 00:28:56,890 --> 00:28:59,220 Because all these mutations are non-neutral, 469 00:28:59,220 --> 00:29:03,660 once you fix this state or this one, you can't go back. 470 00:29:03,660 --> 00:29:05,980 So the population will move forward 471 00:29:05,980 --> 00:29:07,835 once it gets to one of those two states. 472 00:29:07,835 --> 00:29:09,960 Now I mean, it would be a very interesting question 473 00:29:09,960 --> 00:29:14,920 to ask if we instead did a different arrangement. 474 00:29:14,920 --> 00:29:18,060 What would the rate of evolution be, and so forth? 475 00:29:18,060 --> 00:29:21,690 Yeah, but what you're saying is certainly true, 476 00:29:21,690 --> 00:29:24,360 that if this took up all of the benefit going here, 477 00:29:24,360 --> 00:29:25,957 then it may not actually be somehow 478 00:29:25,957 --> 00:29:28,540 an optimal path in terms of the rate of evolution or something 479 00:29:28,540 --> 00:29:29,039 like that. 480 00:29:32,420 --> 00:29:34,510 I'll think about that when designing. 481 00:29:34,510 --> 00:29:36,848 Problems. 482 00:29:36,848 --> 00:29:42,560 AUDIENCE: In this system, 0, 0 eventually becomes 1, 1. 483 00:29:42,560 --> 00:29:43,560 PROFESSOR: That's right. 484 00:29:43,560 --> 00:29:46,590 AUDIENCE: So the probability is 1. 485 00:29:46,590 --> 00:29:50,310 PROFESSOR: That's right, so we are guaranteed 486 00:29:50,310 --> 00:29:53,470 that we will eventually evolve to this peak in the fitness 487 00:29:53,470 --> 00:29:53,970 landscape. 488 00:29:53,970 --> 00:29:57,430 And so what we're asking here is which of these two paths 489 00:29:57,430 --> 00:29:58,621 is going to be taken. 490 00:29:58,621 --> 00:30:02,058 AUDIENCE: Yeah, so how to mathematically prove 491 00:30:02,058 --> 00:30:06,995 that the system will go from 0, 0 to 1, 1? 492 00:30:06,995 --> 00:30:09,120 PROFESSOR: I mean, I feel like I kind of proved it, 493 00:30:09,120 --> 00:30:11,411 although I understand that nothing I said was rigorous. 494 00:30:15,580 --> 00:30:18,670 And of course, there are non-zero probabilities 495 00:30:18,670 --> 00:30:19,620 of going backwards. 496 00:30:19,620 --> 00:30:22,390 It's just that they are reduced. 497 00:30:22,390 --> 00:30:24,510 And actually, you can prove, for those of you 498 00:30:24,510 --> 00:30:26,580 who are interested in such things, 499 00:30:26,580 --> 00:30:31,510 that over long time scales, there's 500 00:30:31,510 --> 00:30:34,260 going to be an equilibrium that distribution 501 00:30:34,260 --> 00:30:40,610 over all these states, where the probability of being 502 00:30:40,610 --> 00:30:47,490 in a particular state will-- it goes as the fitness. 503 00:30:47,490 --> 00:30:52,100 It scales as the relative fitness to the Nth power. 504 00:30:54,620 --> 00:30:57,940 So we talk about these fitness landscapes 505 00:30:57,940 --> 00:30:59,670 as energy landscapes. 506 00:30:59,670 --> 00:31:04,760 And indeed, in this regime where you have small mutation rates, 507 00:31:04,760 --> 00:31:06,550 then it's going to be a detailed balance. 508 00:31:06,550 --> 00:31:08,570 And it's actually a thermodynamic system. 509 00:31:08,570 --> 00:31:11,930 So then in that case, you can make a correspondence 510 00:31:11,930 --> 00:31:14,210 between everything that we normally talk about, 511 00:31:14,210 --> 00:31:18,020 where fitness is like energy and population 512 00:31:18,020 --> 00:31:20,720 size is like temperature. 513 00:31:20,720 --> 00:31:24,610 So the relative amplitude of being in this peak as compared 514 00:31:24,610 --> 00:31:27,670 to the other states is going to be, in this case, 515 00:31:27,670 --> 00:31:30,150 the ratios of those things is, indeed, described 516 00:31:30,150 --> 00:31:33,160 by the ratios of the fitnesses. 517 00:31:33,160 --> 00:31:38,550 And it's going to go as kind of like 1.1 to the 1,000th power, 518 00:31:38,550 --> 00:31:40,690 which is big. 519 00:31:40,690 --> 00:31:42,760 Which means that the population has really 520 00:31:42,760 --> 00:31:47,500 cohered at this peak in the finished landscape. 521 00:31:47,500 --> 00:31:48,726 Yeah. 522 00:31:48,726 --> 00:31:52,386 AUDIENCE: So if you want to calculate a problem going 523 00:31:52,386 --> 00:31:57,022 from 0, 1 to 0, 0, then [INAUDIBLE] 524 00:31:57,022 --> 00:32:01,440 that would just be-- I guess I'm not sure. 525 00:32:01,440 --> 00:32:03,225 PROFESSOR: OK, you want to know the rate 526 00:32:03,225 --> 00:32:04,080 that that's going to happen. 527 00:32:04,080 --> 00:32:04,500 AUDIENCE: Yeah. 528 00:32:04,500 --> 00:32:04,990 PROFESSOR: No, that's fine. 529 00:32:04,990 --> 00:32:05,690 Let's do that. 530 00:32:10,170 --> 00:32:13,110 So for example, let's imagine that we 531 00:32:13,110 --> 00:32:23,650 don't have-- so let's imagine that we just have the 0, 0 532 00:32:23,650 --> 00:32:28,190 and the 0, 1 states, just so we don't have to worry 533 00:32:28,190 --> 00:32:29,980 about going up the landscape. 534 00:32:29,980 --> 00:32:38,090 And so what we have is we have r is relative fitness 1 and 1.02. 535 00:32:38,090 --> 00:32:41,200 Now what we want to do is we want to ask, well, 536 00:32:41,200 --> 00:32:44,390 what is the rate of going back and forth? 537 00:32:44,390 --> 00:32:51,620 Well, so the rate of going forward, 538 00:32:51,620 --> 00:32:55,760 well, we sample mutations at a rate mu. 539 00:32:55,760 --> 00:32:57,980 And this is mu only for this one state, 540 00:32:57,980 --> 00:33:02,270 because pretend that we're not going to mutate this other one. 541 00:33:02,270 --> 00:33:07,580 So rate mu N, you have mutations appearing. 542 00:33:07,580 --> 00:33:10,789 And times this s, 0.02, is the probability 543 00:33:10,789 --> 00:33:12,830 that it'll actually fix in the forward direction. 544 00:33:15,580 --> 00:33:19,700 And now what we want to know is the rate of coming back. 545 00:33:19,700 --> 00:33:21,720 Well, the beginning part's the same, 546 00:33:21,720 --> 00:33:25,640 because we have mu N is the rate that you 547 00:33:25,640 --> 00:33:27,970 get this deleterious mutant in the population. 548 00:33:27,970 --> 00:33:30,595 But then we need to multiply it by the probability of fixation. 549 00:33:33,720 --> 00:33:36,630 And the probability of fixation is-- 550 00:33:36,630 --> 00:33:43,290 there was this thing X1, which was 1 minus-- now this is r, 551 00:33:43,290 --> 00:33:45,670 but it's r in the other direction, so be careful. 552 00:33:45,670 --> 00:33:47,550 Because the general equation was 1 over. 553 00:33:52,620 --> 00:33:55,750 But now r, instead of being 1.02, is 1/1.02. 554 00:34:02,360 --> 00:34:05,000 So which of these terms is going to be dominant? 555 00:34:05,000 --> 00:34:07,540 This thing gets up to be some really big number 556 00:34:07,540 --> 00:34:09,090 is our problem. 557 00:34:09,090 --> 00:34:11,780 So we should be able to figure this out, though. 558 00:34:11,780 --> 00:34:14,389 Because this new r is 1/1.02. 559 00:34:14,389 --> 00:34:21,300 So we for example, have 1 minus 1.02, 1 minus 1.02 560 00:34:21,300 --> 00:34:24,754 to the 1,000. 561 00:34:24,754 --> 00:34:26,420 All right, so this is a negative number, 562 00:34:26,420 --> 00:34:27,878 but this is a negative number, too. 563 00:34:27,878 --> 00:34:37,925 So we end up with 0.02-- 564 00:34:37,925 --> 00:34:39,280 AUDIENCE: 200. 565 00:34:39,280 --> 00:34:41,550 PROFESSOR: Is it 200? 566 00:34:41,550 --> 00:34:44,139 Yeah, you're keeping only the first, 567 00:34:44,139 --> 00:34:47,000 which, since it's much larger than 1, it's bigger than 200. 568 00:34:47,000 --> 00:34:47,500 Right? 569 00:34:47,500 --> 00:34:50,480 I mean do you guys understand what I'm saying? 570 00:34:50,480 --> 00:34:52,550 You can't keep just the first term in a series. 571 00:34:52,550 --> 00:34:57,070 If the terms grow with number. 572 00:34:57,070 --> 00:35:02,340 AUDIENCE: [INAUDIBLE] squared, 3.98 or something. 573 00:35:02,340 --> 00:35:03,948 PROFESSOR: Wait, which one? 574 00:35:03,948 --> 00:35:07,760 AUDIENCE: 1.02 to the 1,000. 575 00:35:07,760 --> 00:35:08,800 PROFESSOR: It's 4? 576 00:35:08,800 --> 00:35:11,440 OK, all right. 577 00:35:11,440 --> 00:35:15,040 OK, so it's 1 minus 4. 578 00:35:15,040 --> 00:35:21,200 So it's 2/300. 579 00:35:21,200 --> 00:35:26,430 OK, so this is teamwork, right? 580 00:35:26,430 --> 00:35:30,859 OK, so there's less than a 1% probability of it fixing. 581 00:35:30,859 --> 00:35:31,650 Is this believable? 582 00:35:37,592 --> 00:35:39,972 AUDIENCE: It's about right. 583 00:35:39,972 --> 00:35:47,055 PROFESSOR: 2, 1,000, 50-- I think 584 00:35:47,055 --> 00:35:52,465 that you did it 1.02 to the 100 rather than 1.02 to the 1,000. 585 00:35:52,465 --> 00:35:54,900 AUDIENCE: OK. 586 00:35:54,900 --> 00:35:56,330 PROFESSOR: No? 587 00:35:56,330 --> 00:35:57,987 Do you not have it in front of you? 588 00:35:57,987 --> 00:35:59,448 AUDIENCE: No, it's 3-- [INAUDIBLE]. 589 00:36:02,857 --> 00:36:05,407 AUDIENCE: 4 times 10 to the 8. 590 00:36:05,407 --> 00:36:07,240 AUDIENCE: I never thought that my calculator 591 00:36:07,240 --> 00:36:08,710 would become so controversial. 592 00:36:08,710 --> 00:36:10,126 AUDIENCE: Oh, 4 times 10 to the 8. 593 00:36:18,440 --> 00:36:21,080 PROFESSOR: Yes, sorry, I was just saying this 594 00:36:21,080 --> 00:36:23,580 doesn't-- so this is why I'm saying you always check to make 595 00:36:23,580 --> 00:36:26,900 sure that your calculation makes any sense at all. 596 00:36:26,900 --> 00:36:29,780 So it's not this. 597 00:36:29,780 --> 00:36:31,817 But it's tiny, right? 598 00:36:31,817 --> 00:36:34,620 AUDIENCE: Yes, [INAUDIBLE]. 599 00:36:34,620 --> 00:36:36,620 PROFESSOR: Yeah, because this didn't make sense, 600 00:36:36,620 --> 00:36:39,520 because this was of the same order as-- well, 601 00:36:39,520 --> 00:36:43,772 this would be larger than 1 over N, so it's totally nonsensical. 602 00:36:43,772 --> 00:36:45,980 Because 1 over N would be the probability of fixation 603 00:36:45,980 --> 00:36:47,050 of a neutral mutation. 604 00:36:47,050 --> 00:36:49,182 This is a deleterious mutation. 605 00:36:49,182 --> 00:36:50,390 It's not even nearly neutral. 606 00:36:50,390 --> 00:36:54,481 So it has to be much less than 1 over N, right? 607 00:36:54,481 --> 00:36:58,317 So this whole thing is 10 to the minus 10, 608 00:36:58,317 --> 00:36:59,275 or something like that? 609 00:37:03,300 --> 00:37:08,309 OK, 4 times 10 to the minus 8. 610 00:37:08,309 --> 00:37:09,100 Well, OK, whatever. 611 00:37:11,620 --> 00:37:12,630 It's 10 to the minus 9. 612 00:37:12,630 --> 00:37:14,440 It's something small. 613 00:37:14,440 --> 00:37:18,080 So this times the probability of fixation, which is 10 minus 9-- 614 00:37:18,080 --> 00:37:20,607 this is how you would calculate the rate of going backwards. 615 00:37:20,607 --> 00:37:22,440 There's some rate that the mutation appears, 616 00:37:22,440 --> 00:37:24,450 and you multiply by the probability that it would fix. 617 00:37:24,450 --> 00:37:25,120 And it's tiny. 618 00:37:29,520 --> 00:37:30,020 OK? 619 00:37:32,960 --> 00:37:34,950 All right, any other questions about how 620 00:37:34,950 --> 00:37:37,120 to think about these sorts of evolutionary dynamics 621 00:37:37,120 --> 00:37:40,594 with presence of mutation, fixation, everything? 622 00:37:40,594 --> 00:37:41,814 Yeah. 623 00:37:41,814 --> 00:37:43,272 AUDIENCE: Can we handle a situation 624 00:37:43,272 --> 00:37:48,350 where [INAUDIBLE] interference is important at this point? 625 00:37:48,350 --> 00:37:51,150 PROFESSOR: Yeah, so this is what you do in your problem set 626 00:37:51,150 --> 00:37:54,090 with simulations. 627 00:37:54,090 --> 00:37:54,590 Yeah. 628 00:37:54,590 --> 00:37:55,923 AUDIENCE: [INAUDIBLE] numerical. 629 00:37:55,923 --> 00:37:59,780 PROFESSOR: You know, I think that it 630 00:37:59,780 --> 00:38:01,988 gets really messy with clonal interference, I'll say. 631 00:38:01,988 --> 00:38:05,059 AUDIENCE: But, like, with basic-- 632 00:38:05,059 --> 00:38:07,308 I guess I was thinking about it and you could probably 633 00:38:07,308 --> 00:38:10,284 imagine that [INAUDIBLE] calculate the probability 634 00:38:10,284 --> 00:38:14,760 that 1, 0 doesn't arise first. 635 00:38:14,760 --> 00:38:17,500 PROFESSOR: Right, yeah, OK, this is an important statement. 636 00:38:17,500 --> 00:38:20,850 In the limit, as you get more and more mutations, when 637 00:38:20,850 --> 00:38:22,600 clonal interference is really significant, 638 00:38:22,600 --> 00:38:24,224 then you're pretty much just guaranteed 639 00:38:24,224 --> 00:38:25,600 to take the 1, 0 path. 640 00:38:25,600 --> 00:38:29,037 Because if you have many mutants, 641 00:38:29,037 --> 00:38:30,870 the definition of clonal interference is you 642 00:38:30,870 --> 00:38:33,332 have multiple mutations that have established. 643 00:38:33,332 --> 00:38:35,040 And once you have multiple mutations that 644 00:38:35,040 --> 00:38:37,350 have established, then it's likely that one of them 645 00:38:37,350 --> 00:38:38,520 is going to be this. 646 00:38:38,520 --> 00:38:41,957 And if it's established, it's going to win. 647 00:38:41,957 --> 00:38:44,540 But the other thing is that as you go up in the mutation rate, 648 00:38:44,540 --> 00:38:47,260 you don't even do successive fixations. 649 00:38:47,260 --> 00:38:50,785 So it may be that neither state ever actually fixes, 650 00:38:50,785 --> 00:38:52,410 because it could be that the 1, 0 state 651 00:38:52,410 --> 00:38:55,100 is growing exponentially, but is a minority of the population. 652 00:38:55,100 --> 00:38:57,740 And it gets another mutation that allows it to go to 1, 1. 653 00:38:57,740 --> 00:39:00,000 So as you increase the mutation rate, 654 00:39:00,000 --> 00:39:02,450 you don't have to actually take single steps. 655 00:39:02,450 --> 00:39:04,730 You can kind of move through states. 656 00:39:04,730 --> 00:39:08,760 And there's a whole literature of the rate at which you 657 00:39:08,760 --> 00:39:10,410 cross fitness valleys. 658 00:39:10,410 --> 00:39:12,650 So this is like tunneling in quantum mechanics or so. 659 00:39:12,650 --> 00:39:14,275 And it has a lot of the same behaviors, 660 00:39:14,275 --> 00:39:17,029 in the sense of exponential suppression of probabilities 661 00:39:17,029 --> 00:39:19,570 as a function of the depth and the width of the valley you're 662 00:39:19,570 --> 00:39:20,390 trying to traverse. 663 00:39:20,390 --> 00:39:22,870 And there's some very nice papers, 664 00:39:22,870 --> 00:39:25,420 if you're interested in looking at this stuff. 665 00:39:25,420 --> 00:39:28,775 And one of them is actually in the syllabus that I mentioned. 666 00:39:28,775 --> 00:39:29,900 I'm trying to remember who. 667 00:39:29,900 --> 00:39:32,020 It was Journal of Theoretical Biology, 668 00:39:32,020 --> 00:39:34,629 but I put it on as optional reading for those of you 669 00:39:34,629 --> 00:39:35,420 who are interested. 670 00:39:40,470 --> 00:39:41,505 All right, OK. 671 00:39:41,505 --> 00:39:43,630 So what I want to do now is I want to switch gears, 672 00:39:43,630 --> 00:39:45,330 so we can think about this evolutionary game theory 673 00:39:45,330 --> 00:39:45,829 business. 674 00:39:49,950 --> 00:39:51,830 And I think the most important thing 675 00:39:51,830 --> 00:39:54,200 to stress when thinking about evolutionary game theory 676 00:39:54,200 --> 00:39:57,721 is just that this point that we don't need to assume anything 677 00:39:57,721 --> 00:39:58,470 about rationality. 678 00:40:02,750 --> 00:40:10,270 Because the puzzles that we like to give each other in your dorm 679 00:40:10,270 --> 00:40:13,600 rooms Friday night, you give these logic 680 00:40:13,600 --> 00:40:15,850 puzzles to each other. 681 00:40:15,850 --> 00:40:17,140 Is that-- I don't know. 682 00:40:17,140 --> 00:40:18,330 [LAUGHTER] 683 00:40:18,330 --> 00:40:21,330 OK, let me just say, back when I was in college, that was, like, 684 00:40:21,330 --> 00:40:23,930 all the cool kids were doing it. 685 00:40:23,930 --> 00:40:27,600 But in these puzzles, you assume that there's hyper-rationality. 686 00:40:27,600 --> 00:40:32,130 You assume that if this guy knows 687 00:40:32,130 --> 00:40:33,896 that I did this, and that, and if I 688 00:40:33,896 --> 00:40:35,020 did that, he would do that. 689 00:40:35,020 --> 00:40:36,645 And then you end up, and then you 690 00:40:36,645 --> 00:40:38,270 have the villagers that are jumping off 691 00:40:38,270 --> 00:40:40,625 cliffs on the seventh day. 692 00:40:40,625 --> 00:40:42,000 Have you guys done these puzzles? 693 00:40:42,000 --> 00:40:42,360 No? 694 00:40:42,360 --> 00:40:42,720 OK. 695 00:40:42,720 --> 00:40:42,980 All right. 696 00:40:42,980 --> 00:40:43,480 Well. 697 00:40:43,480 --> 00:40:45,960 [LAUGHTER] 698 00:40:45,960 --> 00:40:49,620 So the point was that people assume 699 00:40:49,620 --> 00:40:51,740 that when we're talking about game theory, 700 00:40:51,740 --> 00:40:55,320 you have to invoke this hyper-rationality even 701 00:40:55,320 --> 00:40:57,060 humans don't engage in. 702 00:40:57,060 --> 00:40:59,130 And I think that it's just very important 703 00:40:59,130 --> 00:41:01,379 to remember that we're talking about evolutionary game 704 00:41:01,379 --> 00:41:04,990 theory in the case of, well, biological evolution. 705 00:41:04,990 --> 00:41:06,840 You don't assume anything about rationality. 706 00:41:06,840 --> 00:41:11,656 Instead, you simply have mutations that sample 707 00:41:11,656 --> 00:41:12,530 different strategies. 708 00:41:17,550 --> 00:41:26,050 And then you have differences in fitness 709 00:41:26,050 --> 00:41:29,470 that just lead to evolution towards the same solutions 710 00:41:29,470 --> 00:41:31,080 of the game. 711 00:41:31,080 --> 00:41:36,710 So it's evolution to the game solutions, 712 00:41:36,710 --> 00:41:38,710 so the Nash equilibrium, for example. 713 00:41:43,020 --> 00:41:45,840 So it's not that we think that the cells are 714 00:41:45,840 --> 00:41:50,210 engaging in any sort of weird puzzle solving. 715 00:41:50,210 --> 00:41:52,290 Instead, they're just mutations. 716 00:41:52,290 --> 00:41:56,190 And the more fit individuals spread in the population. 717 00:41:56,190 --> 00:41:59,400 And somehow, you evolve to the same or similar solutions, 718 00:41:59,400 --> 00:42:02,360 to these Nash equilibria in the context of game theory. 719 00:42:06,190 --> 00:42:10,040 And we'll see how this plays out in a few concrete examples. 720 00:42:15,130 --> 00:42:20,640 Now, there are always different ways of looking at these games. 721 00:42:20,640 --> 00:42:23,579 One thing I want to stress, though, is that all 722 00:42:23,579 --> 00:42:25,120 the selection that we've been talking 723 00:42:25,120 --> 00:42:29,210 about in the last few weeks, that all 724 00:42:29,210 --> 00:42:31,250 is consistent with game theory in the sense 725 00:42:31,250 --> 00:42:33,500 that the idea of the game theory is 726 00:42:33,500 --> 00:42:36,200 that we allow for the possibility 727 00:42:36,200 --> 00:42:38,240 that the fitness of individuals depends 728 00:42:38,240 --> 00:42:41,060 upon the rest of the population. 729 00:42:41,060 --> 00:42:43,200 Whereas in all these calculations we've been doing, 730 00:42:43,200 --> 00:42:47,210 I told you, all right, I just gave you some fitnesses. 731 00:42:47,210 --> 00:42:50,550 So I said, here we have a 0, 0 state that has some fitness. 732 00:42:50,550 --> 00:42:53,090 0, 1 has a higher fitness, and so forth. 733 00:42:53,090 --> 00:42:56,050 But in general, these fitness values 734 00:42:56,050 --> 00:43:00,990 may depend upon what the population composition is. 735 00:43:00,990 --> 00:43:03,590 And in that situation, then you want to use evolutionary game 736 00:43:03,590 --> 00:43:05,910 theory. 737 00:43:05,910 --> 00:43:08,870 In many cases, people just assume 738 00:43:08,870 --> 00:43:12,950 that you can do something like this-- that you can describe it 739 00:43:12,950 --> 00:43:14,290 as some fitness landscape. 740 00:43:14,290 --> 00:43:17,950 But you can't do that if there's this frequency-dependent 741 00:43:17,950 --> 00:43:20,260 selection-- if there's any sort of evolutionary game 742 00:43:20,260 --> 00:43:22,340 interactions going on. 743 00:43:22,340 --> 00:43:27,410 So it's just important. 744 00:43:27,410 --> 00:43:32,280 If the fitnesses depend on composition-- 745 00:43:32,280 --> 00:43:34,990 this is the population composition-- 746 00:43:34,990 --> 00:43:39,610 then you cannot even define a fitness landscape. 747 00:43:43,250 --> 00:43:44,430 Then no fitness landscape. 748 00:43:47,800 --> 00:43:51,120 For example, you can have situations where the population 749 00:43:51,120 --> 00:43:54,700 evolves to lower fitness. 750 00:43:54,700 --> 00:43:56,510 So you can have a situation where, 751 00:43:56,510 --> 00:44:00,040 if I tell you individual 0, you measure its growth 752 00:44:00,040 --> 00:44:05,760 rate, whatnot, its fitness might be 1. 753 00:44:05,760 --> 00:44:10,680 So this is genome, and this is fitness. 754 00:44:10,680 --> 00:44:13,090 Now, if I go and I measure the fitness 755 00:44:13,090 --> 00:44:17,380 of some other individual, different genome-- so 756 00:44:17,380 --> 00:44:20,480 another strain of bacteria or yeast or whatever-- 757 00:44:20,480 --> 00:44:23,650 and you say, oh, well, its fitness is 1.2. 758 00:44:23,650 --> 00:44:26,730 So this strain has higher fitness than this strain. 759 00:44:26,730 --> 00:44:32,160 Now, it would be very natural to assume that this strain will 760 00:44:32,160 --> 00:44:34,380 out-compete this strain. 761 00:44:34,380 --> 00:44:37,080 And indeed, that's been the assumption in everything 762 00:44:37,080 --> 00:44:38,690 we've been talking about. 763 00:44:38,690 --> 00:44:41,580 But it's not necessarily true. 764 00:44:41,580 --> 00:44:45,840 And that's the basic insight of evolutionary game theory, 765 00:44:45,840 --> 00:44:50,130 is that just knowing the fitness of a pure population 766 00:44:50,130 --> 00:44:53,110 is not actually enough information to know that it's 767 00:44:53,110 --> 00:44:54,600 going to be selected for. 768 00:44:54,600 --> 00:44:57,800 Because it's still possible that in a mixed population, 769 00:44:57,800 --> 00:45:00,880 the genome 0 may actually have higher fitness 770 00:45:00,880 --> 00:45:01,892 than the genome 1. 771 00:45:06,050 --> 00:45:08,760 And once you kind of study these things, 772 00:45:08,760 --> 00:45:10,430 it's kind of clear that it can happen. 773 00:45:10,430 --> 00:45:12,800 But then it's easy to then go back to the lab 774 00:45:12,800 --> 00:45:13,900 and forget that it's true. 775 00:45:17,930 --> 00:45:21,933 And so we'll see how this plays out. 776 00:45:21,933 --> 00:45:25,790 AUDIENCE: On this game theory, [INAUDIBLE]? 777 00:45:25,790 --> 00:45:27,530 PROFESSOR: No. 778 00:45:27,530 --> 00:45:29,710 That's the other thing, is that I 779 00:45:29,710 --> 00:45:31,842 like to just draw these things as graphs, because I 780 00:45:31,842 --> 00:45:33,800 think it's much easier to see what's happening. 781 00:45:33,800 --> 00:45:35,800 And it's clear that things can be non-linear. 782 00:45:35,800 --> 00:45:39,940 But the basic insights are all intact. 783 00:45:39,940 --> 00:45:42,800 From my standpoint as kind of an experimentalist-- 784 00:45:42,800 --> 00:45:47,090 don't forget about the exam-- I think 785 00:45:47,090 --> 00:45:50,689 that the more formal evolutionary game theory 786 00:45:50,689 --> 00:45:52,730 thing-- these two-player games that you guys just 787 00:45:52,730 --> 00:45:56,200 read about-- I think they're important because they tell you 788 00:45:56,200 --> 00:46:00,054 what are the possible outcomes of measurements or of systems, 789 00:46:00,054 --> 00:46:02,220 even in the most simple situation where everything's 790 00:46:02,220 --> 00:46:02,950 linear. 791 00:46:02,950 --> 00:46:04,700 Now, when things are not linear, of course 792 00:46:04,700 --> 00:46:06,530 you can get even richer dynamics. 793 00:46:06,530 --> 00:46:10,391 But in practice, you basically get the categories of outcomes 794 00:46:10,391 --> 00:46:11,140 that we saw there. 795 00:46:22,230 --> 00:46:27,350 So maybe what I'll do is-- so what we're going to do 796 00:46:27,350 --> 00:46:30,380 is think about competition between two individuals 797 00:46:30,380 --> 00:46:38,260 A and B. And often, we talk about these things 798 00:46:38,260 --> 00:46:41,866 in the context of the two-player games, where we have A, B, C, 799 00:46:41,866 --> 00:46:47,390 D. And because this is really importing the kind of approach, 800 00:46:47,390 --> 00:46:51,230 or the nomenclature, from conventional game theory, 801 00:46:51,230 --> 00:46:54,020 and then immediately applying it to populations where you just 802 00:46:54,020 --> 00:46:55,990 assume that all the individuals have 803 00:46:55,990 --> 00:46:58,280 equal probability of interacting with everybody 804 00:46:58,280 --> 00:46:59,317 in the population. 805 00:46:59,317 --> 00:47:00,900 So it's what you would get if you just 806 00:47:00,900 --> 00:47:04,850 had some two-player game like they study in game theory, 807 00:47:04,850 --> 00:47:07,290 but in a population of 1,000 or whatnot. 808 00:47:07,290 --> 00:47:09,790 You just made a bunch of random pairwise interactions. 809 00:47:09,790 --> 00:47:10,930 You had them play the game. 810 00:47:10,930 --> 00:47:12,846 And then you had them do that again over time. 811 00:47:12,846 --> 00:47:18,550 And then this is the payouts that you read about 812 00:47:18,550 --> 00:47:20,577 in Chapter Four are kind of what would happen 813 00:47:20,577 --> 00:47:22,410 in that sort of situation, where everybody's 814 00:47:22,410 --> 00:47:26,020 interacting with everybody else with equal probability. 815 00:47:26,020 --> 00:47:28,450 Now, remember the way that you read 816 00:47:28,450 --> 00:47:33,340 this is that, depending upon the strategy that these guys are 817 00:47:33,340 --> 00:47:37,590 following, you get different payouts. 818 00:47:37,590 --> 00:47:41,990 And normally what we say is that if this could be, 819 00:47:41,990 --> 00:47:45,350 for example, strategy one and two, strategy one and two. 820 00:47:45,350 --> 00:47:47,880 And this is telling us about the payout 821 00:47:47,880 --> 00:47:50,670 that the A individual gets depending on what he does, 822 00:47:50,670 --> 00:47:52,986 and depending upon what his opponent does. 823 00:47:52,986 --> 00:47:55,360 Now, we're not explicitly saying what the payout to the B 824 00:47:55,360 --> 00:47:56,785 individual is, but we're assuming 825 00:47:56,785 --> 00:47:59,160 that this is a symmetric game, so you could figure it out 826 00:47:59,160 --> 00:48:02,290 by looking at the opposite entry. 827 00:48:02,290 --> 00:48:05,710 So if A follows strategy one, B follows strategy one, 828 00:48:05,710 --> 00:48:10,420 then individual A gets little a fitness, 829 00:48:10,420 --> 00:48:13,030 whereas B also gets little a fitness, 830 00:48:13,030 --> 00:48:15,840 because it's a symmetric game. 831 00:48:15,840 --> 00:48:18,930 So the case it's different is when we're in the diagonals. 832 00:48:22,410 --> 00:48:26,240 And from this framework, you can see 833 00:48:26,240 --> 00:48:28,470 that there are going to be already 834 00:48:28,470 --> 00:48:30,550 a bunch of kind of non-trivial things 835 00:48:30,550 --> 00:48:33,470 that can happen, even in this regime where everything's 836 00:48:33,470 --> 00:48:35,750 linear. 837 00:48:35,750 --> 00:48:42,070 And the probably best well-known of these 838 00:48:42,070 --> 00:48:50,500 is this Prisoner's Dilemma, which 839 00:48:50,500 --> 00:48:53,770 is the standard model of cooperation 840 00:48:53,770 --> 00:48:54,949 in the field of game theory. 841 00:48:54,949 --> 00:48:56,740 So there's a story that goes along with it. 842 00:48:56,740 --> 00:49:02,120 It's this idea that-- I'm sure you guys watch these cop shows, 843 00:49:02,120 --> 00:49:05,110 where you have the cops bring in the two accomplices. 844 00:49:05,110 --> 00:49:07,300 And then they put them in separate rooms. 845 00:49:07,300 --> 00:49:09,220 And they tell them that they have 846 00:49:09,220 --> 00:49:11,520 to confess to committing the crime, 847 00:49:11,520 --> 00:49:13,800 because the guy in the other room is confessing, 848 00:49:13,800 --> 00:49:15,300 and if he doesn't confess, then he's 849 00:49:15,300 --> 00:49:16,880 going to be in trouble, et cetera. 850 00:49:16,880 --> 00:49:20,470 You've seen these cop shows? 851 00:49:20,470 --> 00:49:23,900 And incidentally, in these questions, 852 00:49:23,900 --> 00:49:25,510 when cops are questioning witnesses, 853 00:49:25,510 --> 00:49:31,730 they're actually allowed to lie to the person being questioned, 854 00:49:31,730 --> 00:49:34,250 which feels a little bit weird, actually. 855 00:49:34,250 --> 00:49:35,670 Doesn't it? 856 00:49:35,670 --> 00:49:37,630 I know, I know, this is not relevant. 857 00:49:40,459 --> 00:49:42,000 So the idea of the prisoner's dilemma 858 00:49:42,000 --> 00:49:46,787 is that if you set up these jail sentences in the right way, 859 00:49:46,787 --> 00:49:48,870 then it could be the case that each individual has 860 00:49:48,870 --> 00:49:52,260 the incentive to confess, even though both individuals would 861 00:49:52,260 --> 00:49:57,200 be better off if they cooperated. 862 00:49:57,200 --> 00:50:01,132 And you can come up with some reasonable payout structure 863 00:50:01,132 --> 00:50:02,090 that has that property. 864 00:50:18,000 --> 00:50:24,515 And we'll call this-- so this is for individual one, 865 00:50:24,515 --> 00:50:26,010 say and individual two. 866 00:50:26,010 --> 00:50:29,550 So there are different strategies you can follow. 867 00:50:29,550 --> 00:50:32,920 And do you guys remember from the reading 868 00:50:32,920 --> 00:50:35,800 slash my explanation how to read these charts? 869 00:50:39,460 --> 00:50:44,510 All right, now the question is, just to remind ourselves, 870 00:50:44,510 --> 00:50:47,370 what is the Nash equilibrium of this game? 871 00:50:51,290 --> 00:50:54,300 And I know that you read about it last night. 872 00:50:54,300 --> 00:50:55,640 Well, use your cards. 873 00:50:55,640 --> 00:50:59,190 Is it C or is it D? 874 00:50:59,190 --> 00:51:02,910 Or is there no Nash equilibrium, you can flash something else. 875 00:51:10,590 --> 00:51:12,520 AUDIENCE: Are those negative or positive? 876 00:51:12,520 --> 00:51:14,103 PROFESSOR: These are positive fitness. 877 00:51:16,670 --> 00:51:19,970 I kind of don't like the Prisoner's Dilemma as a story, 878 00:51:19,970 --> 00:51:22,164 because it's not very intuitive, because you 879 00:51:22,164 --> 00:51:23,830 have to actually specify the jail terms, 880 00:51:23,830 --> 00:51:25,990 and you have to remember that jail terms are bad, not good. 881 00:51:25,990 --> 00:51:27,370 So these are good things, OK? 882 00:51:27,370 --> 00:51:30,960 These are years off that you get as a result 883 00:51:30,960 --> 00:51:32,210 of doing one thing or another. 884 00:51:35,930 --> 00:51:37,560 You want to get big numbers. 885 00:51:37,560 --> 00:51:38,360 Ready? 886 00:51:38,360 --> 00:51:40,220 Three, two, one. 887 00:51:44,640 --> 00:51:46,980 So at least we have a majority that are D, 888 00:51:46,980 --> 00:51:49,270 but it's not all of them. 889 00:51:49,270 --> 00:51:51,300 And I think this is basically a reflection-- 890 00:51:51,300 --> 00:51:53,300 and D is indeed the Nash equilibrium. 891 00:51:56,550 --> 00:51:58,690 It's to do this strategy D that we're saying here. 892 00:51:58,690 --> 00:52:01,060 All right, now the question is why? 893 00:52:01,060 --> 00:52:03,320 And part of the challenge here is just understanding 894 00:52:03,320 --> 00:52:05,540 how to read these charts. 895 00:52:05,540 --> 00:52:09,670 Now, first of all, the payout that everybody gets if everyone 896 00:52:09,670 --> 00:52:12,130 follows strategy D is what? 897 00:52:12,130 --> 00:52:14,130 Verbally, three, two, one. 898 00:52:14,130 --> 00:52:15,170 AUDIENCE: One. 899 00:52:15,170 --> 00:52:17,440 PROFESSOR: So everybody gets payout one. 900 00:52:17,440 --> 00:52:20,100 Now, if you look at this chart, you 901 00:52:20,100 --> 00:52:23,410 say, well, gee, that is a shame. 902 00:52:23,410 --> 00:52:28,900 Because 1 is just not the biggest number you see here. 903 00:52:28,900 --> 00:52:30,820 And indeed, the important point to note 904 00:52:30,820 --> 00:52:34,670 here is that if both players had followed this strategy 905 00:52:34,670 --> 00:52:39,930 C for cooperate, D for defect, then both individuals would be 906 00:52:39,930 --> 00:52:43,840 getting fitness 3, or payout 3. 907 00:52:43,840 --> 00:52:46,550 So the idea here is that both individuals would 908 00:52:46,550 --> 00:52:49,140 do better if they both played strategy C. 909 00:52:49,140 --> 00:52:51,660 But the problem is that that's not evolutionarily stable. 910 00:52:51,660 --> 00:52:53,590 Or in the context of game theory, 911 00:52:53,590 --> 00:52:56,770 that is cheatable in some ways. 912 00:52:56,770 --> 00:52:59,020 And so the reason that this is a Nash equilibrium 913 00:52:59,020 --> 00:53:01,890 is that you ask-- so a Nash equilibrium, what it means, 914 00:53:01,890 --> 00:53:05,070 if you recall, is that if everyone's 915 00:53:05,070 --> 00:53:07,640 playing that strategy, then nobody has the incentive 916 00:53:07,640 --> 00:53:09,770 to change strategy. 917 00:53:09,770 --> 00:53:11,825 So no incentive to change strategy. 918 00:53:17,902 --> 00:53:19,360 So now you just imagine, let's say, 919 00:53:19,360 --> 00:53:20,390 that you're playing against somebody else, 920 00:53:20,390 --> 00:53:22,100 or in the context of biology, it's 921 00:53:22,100 --> 00:53:24,746 a population of individuals following the D strategy. 922 00:53:24,746 --> 00:53:27,740 The question is whether you as an individual 923 00:53:27,740 --> 00:53:32,040 would have the incentive to switch to the other strategy? 924 00:53:32,040 --> 00:53:34,830 And the answer is no, because what you have control over 925 00:53:34,830 --> 00:53:38,510 is this rows. 926 00:53:38,510 --> 00:53:41,450 The column is specified by the rest of the population. 927 00:53:41,450 --> 00:53:45,095 So if you're in this state, what you have a choice of 928 00:53:45,095 --> 00:53:47,612 is to switch to the cooperate strategy, which 929 00:53:47,612 --> 00:53:48,570 would be to go up here. 930 00:53:48,570 --> 00:53:53,890 So you have a choice to move up to this 0 payout, 931 00:53:53,890 --> 00:53:55,732 but that's not to your advantage. 932 00:53:55,732 --> 00:53:57,940 Now, it's true that your opponent would get payout 5. 933 00:53:57,940 --> 00:54:01,170 So you opponent would actually do wonderfully. 934 00:54:01,170 --> 00:54:02,370 But you would do poorly. 935 00:54:02,370 --> 00:54:05,890 So you'd be selected against, if you imagine 936 00:54:05,890 --> 00:54:07,980 this being in the context of biology-- 937 00:54:07,980 --> 00:54:09,990 that you have a genotype that are playing 938 00:54:09,990 --> 00:54:13,120 D. If you're a mutant that starts following this strategy 939 00:54:13,120 --> 00:54:17,627 C, you have lower fitness, so you're selected against. 940 00:54:17,627 --> 00:54:19,835 So that's saying that the strategy D is noninvadable. 941 00:54:23,050 --> 00:54:25,040 We can also think about what happens if we're 942 00:54:25,040 --> 00:54:28,710 a population of cooperators. 943 00:54:28,710 --> 00:54:31,251 Now everybody has high fitness-- fitness 3. 944 00:54:31,251 --> 00:54:32,750 Question is, what happens if there's 945 00:54:32,750 --> 00:54:35,070 a mutation that leads to one individual following the D 946 00:54:35,070 --> 00:54:35,570 strategy? 947 00:54:41,560 --> 00:54:45,426 Is he selected four or not? 948 00:54:45,426 --> 00:54:46,320 AUDIENCE: Yes. 949 00:54:46,320 --> 00:54:48,570 PROFESSOR: Yes, so the point here 950 00:54:48,570 --> 00:54:53,410 is that you always will have higher fitness, 951 00:54:53,410 --> 00:54:57,010 regardless of what your opponent does in the context of a game 952 00:54:57,010 --> 00:54:59,260 theory situation, or regardless of the distribution 953 00:54:59,260 --> 00:55:00,910 of cooperation and defection in the population. 954 00:55:00,910 --> 00:55:02,410 It's always better to be a defector. 955 00:55:05,280 --> 00:55:11,308 So the problem here is, it's always better to play D. 956 00:55:21,280 --> 00:55:28,911 Now, I really like drawing the graphs of these things, 957 00:55:28,911 --> 00:55:30,660 because I think it's just much more clear. 958 00:55:34,770 --> 00:55:37,940 And you can either draw the fitness 959 00:55:37,940 --> 00:55:41,860 of the two types minus each other, or just the raw fitness. 960 00:55:41,860 --> 00:55:43,086 Yes. 961 00:55:43,086 --> 00:55:45,537 AUDIENCE: So what if instead of 5, you have 7? 962 00:55:45,537 --> 00:55:49,830 Because then the population as a whole's fitness 963 00:55:49,830 --> 00:55:54,123 decreases when you [INAUDIBLE]. 964 00:55:54,123 --> 00:55:56,680 So how does that-- 965 00:55:56,680 --> 00:55:59,236 PROFESSOR: So you're saying if this 5 were a 7 instead? 966 00:55:59,236 --> 00:55:59,930 AUDIENCE: Yeah. 967 00:55:59,930 --> 00:56:01,950 PROFESSOR: Right, then what you're saying 968 00:56:01,950 --> 00:56:04,020 is that-- so it doesn't change. 969 00:56:04,020 --> 00:56:07,940 The Nash equilibrium is still defect. 970 00:56:07,940 --> 00:56:13,600 The subtle thing here is that, in general, 971 00:56:13,600 --> 00:56:17,180 in terms of game theory, we like it when the mean of these two 972 00:56:17,180 --> 00:56:19,437 is smaller than this one. 973 00:56:19,437 --> 00:56:20,770 That's why you're asking, right? 974 00:56:20,770 --> 00:56:23,120 Exactly, because that's right. 975 00:56:23,120 --> 00:56:24,130 Exactly, right. 976 00:56:24,130 --> 00:56:27,630 So yeah, so that's a slightly more complicated situation, 977 00:56:27,630 --> 00:56:31,490 because in that situation, then, if you had two rational agents, 978 00:56:31,490 --> 00:56:34,820 say, playing this game, then you could alternate cooperation 979 00:56:34,820 --> 00:56:35,940 and defection. 980 00:56:35,940 --> 00:56:39,060 And that would actually be the ultimate form of cooperation 981 00:56:39,060 --> 00:56:42,030 in such a game, because you could actually get a higher 982 00:56:42,030 --> 00:56:45,180 payout by alternating. 983 00:56:45,180 --> 00:56:47,390 Right, so we've chosen the numbers 984 00:56:47,390 --> 00:56:54,480 as they are so that this is subtlety is not an issue. 985 00:56:54,480 --> 00:56:56,280 Does everybody understand the issue there? 986 00:57:00,030 --> 00:57:02,850 So in the context of evolutionary game theory, what 987 00:57:02,850 --> 00:57:07,030 we can do is we can plot as a function of the fraction 988 00:57:07,030 --> 00:57:10,610 of the population that's cooperator between 0 and 1, 989 00:57:10,610 --> 00:57:12,770 say. 990 00:57:12,770 --> 00:57:16,420 And we can plot the payout for the cooperator 991 00:57:16,420 --> 00:57:17,350 and for the defector. 992 00:57:23,020 --> 00:57:29,510 For example, I'm going to draw a solid line for the cooperator, 993 00:57:29,510 --> 00:57:32,450 dashed line for the defector. 994 00:57:32,450 --> 00:57:39,150 Now the question is, what should be the y-axes on either ends 995 00:57:39,150 --> 00:57:39,730 and so forth? 996 00:57:39,730 --> 00:57:40,479 Do you understand? 997 00:57:43,030 --> 00:57:45,850 So what should these things look like? 998 00:57:45,850 --> 00:57:47,780 I'd like to encourage you to-- I'll 999 00:57:47,780 --> 00:57:51,055 give you 30 seconds to try to draw 1000 00:57:51,055 --> 00:57:52,180 what this should look like. 1001 00:57:52,180 --> 00:57:59,610 So this is the payout or the expected payout. 1002 00:57:59,610 --> 00:58:03,610 So we're assuming that you're going to interact randomly 1003 00:58:03,610 --> 00:58:05,860 with the other members of the population as a function 1004 00:58:05,860 --> 00:58:07,290 of the fraction cooperator. 1005 00:58:07,290 --> 00:58:09,570 So then 1 minus that will be the fraction defector. 1006 00:58:13,170 --> 00:58:15,321 Do you understand what I'm trying to ask you to do? 1007 00:58:15,321 --> 00:58:17,966 AUDIENCE: So the scale on the right-hand side supposed 1008 00:58:17,966 --> 00:58:20,140 to be for the defectors? 1009 00:58:20,140 --> 00:58:21,759 [INAUDIBLE]? 1010 00:58:21,759 --> 00:58:24,050 PROFESSOR: This is just a legend, or key, or something. 1011 00:58:24,050 --> 00:58:26,630 So I want you to draw something over here that's 1012 00:58:26,630 --> 00:58:28,030 a solid line and a dashed line. 1013 00:58:28,030 --> 00:58:29,380 AUDIENCE: All right, so it's just one scale. 1014 00:58:29,380 --> 00:58:30,005 And you don't-- 1015 00:58:30,005 --> 00:58:31,088 PROFESSOR: It's one scale. 1016 00:58:31,088 --> 00:58:32,230 AUDIENCE: [INAUDIBLE]. 1017 00:58:32,230 --> 00:58:34,614 PROFESSOR: Oh, yeah, sorry. 1018 00:58:34,614 --> 00:58:36,780 I'm just telling you what's going to be a solid line 1019 00:58:36,780 --> 00:58:39,540 and what's going to be a dashed line. 1020 00:58:39,540 --> 00:58:42,290 And I'll give you a hint, that up here is number 5. 1021 00:59:23,450 --> 00:59:27,775 This is going to be the expected payout for a lone individual 1022 00:59:27,775 --> 00:59:29,650 given the rest of the population is following 1023 00:59:29,650 --> 00:59:31,070 some fraction of cooperator. 1024 01:00:05,630 --> 01:00:08,136 Do you guys understand what I'm asking you to do? 1025 01:00:08,136 --> 01:00:10,010 Because I'm a little bit concerned that there 1026 01:00:10,010 --> 01:00:14,706 are very few plots in front. 1027 01:00:14,706 --> 01:00:16,020 AUDIENCE: What is fc? 1028 01:00:16,020 --> 01:00:17,660 PROFESSOR: So this is the fraction of the population 1029 01:00:17,660 --> 01:00:18,409 that's cooperator. 1030 01:00:40,700 --> 01:00:44,530 Well, I was giving you a chance to think about it. 1031 01:00:44,530 --> 01:00:46,930 But from looking around, I think that maybe you're 1032 01:00:46,930 --> 01:00:48,960 not quite sure what I'm trying to ask you to do. 1033 01:00:48,960 --> 01:00:50,964 So I'm trying to plot the expected payout 1034 01:00:50,964 --> 01:00:53,380 for an individual that is either cooperating or defecting, 1035 01:00:53,380 --> 01:00:55,421 based on the fact that the rest of the population 1036 01:00:55,421 --> 01:01:00,590 has some composition between all cooperate or all defect. 1037 01:01:00,590 --> 01:01:03,170 So it's the evolution game theory extension 1038 01:01:03,170 --> 01:01:06,460 of this simple model. 1039 01:01:06,460 --> 01:01:11,360 So first, we can ask, well, if the entire population is 1040 01:01:11,360 --> 01:01:15,460 cooperating, we want to know the fitness for a cooperator 1041 01:01:15,460 --> 01:01:16,000 or defector. 1042 01:01:16,000 --> 01:01:20,710 Well, this is really just saying that we're 1043 01:01:20,710 --> 01:01:25,190 all the way over on here, and we just choose between the two. 1044 01:01:25,190 --> 01:01:27,250 And the defector is the 5 one. 1045 01:01:27,250 --> 01:01:30,390 So this is going to be a dashed line that's 1046 01:01:30,390 --> 01:01:31,880 going to start from here. 1047 01:01:31,880 --> 01:01:33,530 And then this is 2 and 1/2, 3. 1048 01:01:37,080 --> 01:01:40,435 So this is cooperator starts here, and defector starts here. 1049 01:01:40,435 --> 01:01:43,390 Do you understand what I'm? 1050 01:01:43,390 --> 01:01:45,290 Now, OK, let's see. 1051 01:01:45,290 --> 01:01:48,740 So now this is the one where if everybody else is defecting, 1052 01:01:48,740 --> 01:01:51,250 well, now, the cooperator line goes to what? 1053 01:01:51,250 --> 01:01:54,160 Verbally, three, two, one. 1054 01:01:54,160 --> 01:01:54,740 AUDIENCE: 0. 1055 01:01:54,740 --> 01:01:56,900 PROFESSOR: 0. 1056 01:01:56,900 --> 01:01:58,410 The defector line goes to 1. 1057 01:02:09,310 --> 01:02:11,640 OK, that line, I started going the wrong direction, 1058 01:02:11,640 --> 01:02:13,370 but that's supposed to be a line. 1059 01:02:15,940 --> 01:02:18,020 So this is an example of what this 1060 01:02:18,020 --> 01:02:20,914 looks like for the Prisoner's Dilemma. 1061 01:02:20,914 --> 01:02:22,580 And what you see is the defector fitness 1062 01:02:22,580 --> 01:02:25,180 is always above the cooperator fitness. 1063 01:02:25,180 --> 01:02:27,502 So for any population composition, 1064 01:02:27,502 --> 01:02:29,460 defectors have higher fitness than cooperators. 1065 01:02:29,460 --> 01:02:34,630 So evolution brings you to the pure defecting state, 1066 01:02:34,630 --> 01:02:36,170 where you have fitness 1. 1067 01:02:40,001 --> 01:02:41,500 And if you want, you could calculate 1068 01:02:41,500 --> 01:02:45,230 what the mean fitness of the population is, for example. 1069 01:02:45,230 --> 01:02:48,400 And the mean fitness starts out over here, 1070 01:02:48,400 --> 01:02:49,830 and ends up over here. 1071 01:02:49,830 --> 01:02:51,930 So the mean fitness decreases over time. 1072 01:02:54,610 --> 01:03:01,240 Now, you can imagine that in the simple, two-player models, 1073 01:03:01,240 --> 01:03:02,590 all these are lines. 1074 01:03:02,590 --> 01:03:04,970 But you can imagine that the only thing that's important 1075 01:03:04,970 --> 01:03:09,340 are how these lines cross each other. 1076 01:03:09,340 --> 01:03:12,090 So for example, there are only a few different things 1077 01:03:12,090 --> 01:03:14,160 that can happen. 1078 01:03:14,160 --> 01:03:24,260 You can have one strategy that dominates, 1079 01:03:24,260 --> 01:03:26,660 which is what occurred here. 1080 01:03:26,660 --> 01:03:29,240 And surprisingly, that does not mean 1081 01:03:29,240 --> 01:03:32,290 that that strategy is higher fitness, 1082 01:03:32,290 --> 01:03:36,070 in the sense that you may evolve to a state of low fitness. 1083 01:03:36,070 --> 01:03:37,166 That's what's weird. 1084 01:03:37,166 --> 01:03:43,930 You can have coexistence, or you can have bi-stability. 1085 01:03:48,880 --> 01:03:53,175 So I'll give you another example of this. 1086 01:03:59,420 --> 01:04:01,990 So now we're just going to have two strategies. 1087 01:04:01,990 --> 01:04:07,516 The strategies-- we'll just call them A and B. 1088 01:04:11,460 --> 01:04:16,045 And the question is, what is the Nash equilibrium? 1089 01:04:22,620 --> 01:04:34,500 Is it A, B, or C should be neither, D is both. 1090 01:04:34,500 --> 01:04:35,950 Do you understand? 1091 01:04:35,950 --> 01:04:38,950 I'm going to ask, because if this is the game 1092 01:04:38,950 --> 01:04:41,670 and this is the interaction, is the Nash equilibrium A, 1093 01:04:41,670 --> 01:04:43,400 Nash equilibrium B? 1094 01:04:43,400 --> 01:04:47,240 If you vote C, it means neither, D means both. 1095 01:04:47,240 --> 01:04:50,590 Do you understand the question? 1096 01:04:50,590 --> 01:04:53,474 I'll give you 30 seconds to think about it. 1097 01:05:38,660 --> 01:05:44,670 All right, are we ready to vote? 1098 01:05:44,670 --> 01:05:47,935 Ready, three, two, one. 1099 01:05:53,621 --> 01:05:55,370 All right, so we have a fair distribution. 1100 01:05:55,370 --> 01:06:01,250 I may not have us vote, but yeah, in this case, 1101 01:06:01,250 --> 01:06:04,970 they're actually both Nash equilibria. 1102 01:06:04,970 --> 01:06:06,500 So let's see this. 1103 01:06:06,500 --> 01:06:09,290 If both individuals, or an entire population, 1104 01:06:09,290 --> 01:06:13,000 say, is playing A, they're getting fitness 5. 1105 01:06:13,000 --> 01:06:14,790 Question is, as a lone individual, 1106 01:06:14,790 --> 01:06:17,720 you can choose to switch over and get fitness 3. 1107 01:06:17,720 --> 01:06:19,610 Do you want to do that? 1108 01:06:19,610 --> 01:06:21,150 No. 1109 01:06:21,150 --> 01:06:23,510 So that means that A is going to be a Nash equilibrium. 1110 01:06:23,510 --> 01:06:25,010 Incidentally, the difference between 1111 01:06:25,010 --> 01:06:27,780 the so-called regular Nash equilibrium and the strict Nash 1112 01:06:27,780 --> 01:06:32,050 equilibrium is that Nash equilibrium 1113 01:06:32,050 --> 01:06:35,630 means that no individual has the incentive to change strategy. 1114 01:06:35,630 --> 01:06:40,860 A strict Nash equilibrium means that any change in strategy 1115 01:06:40,860 --> 01:06:42,800 leads to an actual decrease in fitness. 1116 01:06:42,800 --> 01:06:45,920 So it's a question of whether you can make neutral changes 1117 01:06:45,920 --> 01:06:47,690 in strategy or not. 1118 01:06:47,690 --> 01:06:48,440 Do you understand? 1119 01:06:51,800 --> 01:06:54,540 So A is a Nash equilibrium. 1120 01:06:54,540 --> 01:06:55,570 What about B? 1121 01:06:55,570 --> 01:06:59,530 Well, in that case, everybody's getting to fitness 1. 1122 01:06:59,530 --> 01:07:02,980 Now, as a lone individual, what can you do? 1123 01:07:02,980 --> 01:07:04,850 All you can do is switch. 1124 01:07:04,850 --> 01:07:07,050 As an individual, you can only choose rows. 1125 01:07:07,050 --> 01:07:08,260 So you go up to 0. 1126 01:07:08,260 --> 01:07:09,950 That's a decrease of fitness. 1127 01:07:09,950 --> 01:07:14,530 So that means that strategy B is also a Nash equilibrium. 1128 01:07:17,410 --> 01:07:19,450 So there are two Nash equilibria in this game. 1129 01:07:19,450 --> 01:07:21,850 And what does that mean about which regime 1130 01:07:21,850 --> 01:07:25,830 you're in here, if you convert this into an evolutionary game 1131 01:07:25,830 --> 01:07:27,075 theory scenario? 1132 01:07:37,460 --> 01:07:41,700 Ready, three, two, one. 1133 01:07:41,700 --> 01:07:43,550 OK, so a majority is saying yes. 1134 01:07:43,550 --> 01:07:47,130 This is indeed a situation in which you have bi-stability. 1135 01:07:50,030 --> 01:07:53,417 So what does that mean in terms of these lines if we draw them? 1136 01:07:57,730 --> 01:08:07,130 So this is payout as a function of the fraction that 1137 01:08:07,130 --> 01:08:08,555 is playing the A strategy. 1138 01:08:12,000 --> 01:08:13,100 Should the lines cross? 1139 01:08:13,100 --> 01:08:13,620 Yes or no? 1140 01:08:13,620 --> 01:08:15,997 Ready, three, two, one. 1141 01:08:15,997 --> 01:08:16,580 AUDIENCE: Yes. 1142 01:08:16,580 --> 01:08:20,898 PROFESSOR: Yes, and indeed, in principle, 1143 01:08:20,898 --> 01:08:22,689 the math that we do in all these situations 1144 01:08:22,689 --> 01:08:25,319 is kind of super simple. 1145 01:08:25,319 --> 01:08:28,540 Yet it's easy to get confused about what's going on 1146 01:08:28,540 --> 01:08:29,770 in all these situations. 1147 01:08:29,770 --> 01:08:33,930 So the idea here is that if the population is A, that 1148 01:08:33,930 --> 01:08:37,044 means that the A here is at 5. 1149 01:08:41,390 --> 01:08:42,694 But then it goes down to 0. 1150 01:08:47,140 --> 01:08:50,448 Whereas over here, B here is 3. 1151 01:08:50,448 --> 01:08:51,364 And then it goes to 1. 1152 01:08:59,200 --> 01:09:01,960 Because these two lines cross, does that 1153 01:09:01,960 --> 01:09:06,970 mean that you have bi-stability? 1154 01:09:06,970 --> 01:09:10,278 Ready, yes or no, three, two, one, 1155 01:09:10,278 --> 01:09:10,819 AUDIENCE: No. 1156 01:09:10,819 --> 01:09:13,780 PROFESSOR: No, and why not? 1157 01:09:13,780 --> 01:09:15,210 AUDIENCE: [INAUDIBLE]. 1158 01:09:15,210 --> 01:09:16,120 PROFESSOR: That's right, because you can also 1159 01:09:16,120 --> 01:09:18,411 do the other thing, and then that leads to coexistence. 1160 01:09:22,310 --> 01:09:26,270 Now, in some ways coexistence is the most subtle 1161 01:09:26,270 --> 01:09:27,580 of the situations. 1162 01:09:27,580 --> 01:09:30,955 And that's for an interesting reason. 1163 01:09:30,955 --> 01:09:32,746 AUDIENCE: Sorry, sir, you said you can also 1164 01:09:32,746 --> 01:09:33,537 do the other thing. 1165 01:09:33,537 --> 01:09:34,830 What is the other thing here? 1166 01:09:37,398 --> 01:09:39,189 PROFESSOR: I'm saying that these things can 1167 01:09:39,189 --> 01:09:42,250 cross in the other orientation. 1168 01:09:42,250 --> 01:09:52,734 Let me put a matrix out there, and then-- 1169 01:09:52,734 --> 01:09:56,130 so this is something that, for example, is what's 1170 01:09:56,130 --> 01:09:57,480 known as a Hawk/Dove game. 1171 01:09:57,480 --> 01:09:58,755 Or it has many other names. 1172 01:10:12,341 --> 01:10:14,840 And we can maybe figure out what would be the Hawk strategy, 1173 01:10:14,840 --> 01:10:16,048 and what's the Dove strategy. 1174 01:10:19,710 --> 01:10:23,080 Now, we want to ask the same question-- is A a Nash 1175 01:10:23,080 --> 01:10:23,630 equilibrium? 1176 01:10:23,630 --> 01:10:25,480 Is B a Nash equilibrium? 1177 01:10:25,480 --> 01:10:26,440 Is it neither? 1178 01:10:26,440 --> 01:10:27,300 Or is it both? 1179 01:10:27,300 --> 01:10:30,104 And maybe I shouldn't have covered this up, 1180 01:10:30,104 --> 01:10:32,020 so you're not influenced, in case you actually 1181 01:10:32,020 --> 01:10:34,290 did do the reading. 1182 01:10:34,290 --> 01:10:37,090 Then I don't want to you to be influenced by this. 1183 01:10:40,960 --> 01:10:42,590 So think about it for 30 seconds. 1184 01:11:09,430 --> 01:11:11,047 Do you need more time? 1185 01:11:11,047 --> 01:11:12,130 Let's go see where we are. 1186 01:11:12,130 --> 01:11:16,650 Ready, three, two, one. 1187 01:11:16,650 --> 01:11:19,200 All right, so most of the group is agreeing in this case, 1188 01:11:19,200 --> 01:11:21,320 neither are the Nash equilibrium. 1189 01:11:24,590 --> 01:11:29,320 So neither are a Nash equilibrium. 1190 01:11:29,320 --> 01:11:32,810 Does that mean that this game has no Nash equilibrium? 1191 01:11:32,810 --> 01:11:36,556 Yes or no, verbally-- ready, three, two, one. 1192 01:11:36,556 --> 01:11:37,270 AUDIENCE: No. 1193 01:11:37,270 --> 01:11:39,140 PROFESSOR: No, it does not mean that. 1194 01:11:39,140 --> 01:11:40,920 This game has a Nash equilibrium. 1195 01:11:40,920 --> 01:11:43,505 And indeed, all games like this have Nash equilibria. 1196 01:11:47,200 --> 01:11:49,270 And this is what Nash won the Nobel Prize for, 1197 01:11:49,270 --> 01:11:53,660 so this is the famous one-page paper published in PNAS. 1198 01:11:53,660 --> 01:11:56,500 If you look at it, I have no idea what it says. 1199 01:12:00,510 --> 01:12:03,930 I mean, he basically just pointed out 1200 01:12:03,930 --> 01:12:07,950 that this theorem implies this, implies that-- done. 1201 01:12:07,950 --> 01:12:14,390 And so it's good that somebody knew what he was saying, 1202 01:12:14,390 --> 01:12:17,050 otherwise we'd be in trouble, all of us. 1203 01:12:17,050 --> 01:12:20,760 So what he proved is that such games, 1204 01:12:20,760 --> 01:12:26,050 even with more players, more options, and so forth, 1205 01:12:26,050 --> 01:12:30,860 they always have such a solution in this sense. 1206 01:12:30,860 --> 01:12:32,870 There exists some strategy such as if everybody 1207 01:12:32,870 --> 01:12:34,870 were playing in, nobody would have the incentive 1208 01:12:34,870 --> 01:12:36,290 to change strategy. 1209 01:12:36,290 --> 01:12:39,330 But you have to include so-called probabilistic or 1210 01:12:39,330 --> 01:12:40,180 mixed strategies. 1211 01:12:46,400 --> 01:12:49,350 And we can draw what this thing is. 1212 01:12:49,350 --> 01:12:53,590 So just like always, so everyone else is following A, 1213 01:12:53,590 --> 01:12:55,045 then A starts here at 3. 1214 01:12:58,170 --> 01:12:59,660 And then it goes to 1. 1215 01:13:04,010 --> 01:13:07,830 Whereas the B individuals start at 5, and they go to 0. 1216 01:13:14,220 --> 01:13:17,530 So this looks very similar to that, 1217 01:13:17,530 --> 01:13:19,720 but they're rather different, in the sense 1218 01:13:19,720 --> 01:13:26,120 that in this situation, we had bi-stability. 1219 01:13:26,120 --> 01:13:31,410 So if you look at the direction of evolution, 1220 01:13:31,410 --> 01:13:32,970 depending upon where you start, you 1221 01:13:32,970 --> 01:13:36,330 go to either all B or all A. Whereas 1222 01:13:36,330 --> 01:13:41,210 in this situation over here, we have coexistence. 1223 01:13:41,210 --> 01:13:42,840 Does not matter where you start. 1224 01:13:42,840 --> 01:13:44,940 So long as you have some members of both A and B 1225 01:13:44,940 --> 01:13:47,810 in the population, you'll always evolve to the same equilibrium. 1226 01:13:47,810 --> 01:13:50,860 Now, the important thing here that's, I think, 1227 01:13:50,860 --> 01:13:54,220 interesting is that in a population, 1228 01:13:54,220 --> 01:13:57,430 if you have genetic A's and genetic B's that are each 1229 01:13:57,430 --> 01:14:00,840 giving birth to their own type, then 1230 01:14:00,840 --> 01:14:05,020 you evolve to some coexistence of genotypes. 1231 01:14:05,020 --> 01:14:08,020 So here, this is some fraction. 1232 01:14:08,020 --> 01:14:17,677 f a star is the equilibrium fraction, fraction 1233 01:14:17,677 --> 01:14:18,635 of A in the population. 1234 01:14:21,240 --> 01:14:25,320 So this is a case where you have genetic diversity that 1235 01:14:25,320 --> 01:14:28,590 leads to phenotypic diversity in the population. 1236 01:14:28,590 --> 01:14:34,430 Whereas the mixed Nash equilibrium-- this 1237 01:14:34,430 --> 01:14:37,560 is a situation where you have, in principle, 1238 01:14:37,560 --> 01:14:41,020 genetic homogeneity. 1239 01:14:41,020 --> 01:14:48,849 So this is a single genotype that is implementing 1240 01:14:48,849 --> 01:14:49,890 phenotypic heterogeneity. 1241 01:14:58,174 --> 01:14:59,840 And indeed, one of the things that we've 1242 01:14:59,840 --> 01:15:01,860 been excited about exploring in my group 1243 01:15:01,860 --> 01:15:09,160 is this distinction here, where it's known that in many cases, 1244 01:15:09,160 --> 01:15:12,360 isogenic populations of microbes can exhibit 1245 01:15:12,360 --> 01:15:15,100 a diversity of phenotypes as a result of, for example, 1246 01:15:15,100 --> 01:15:17,555 stochastic gene expression and bi-stability. 1247 01:15:21,150 --> 01:15:23,230 So that's a molecular mechanism for how 1248 01:15:23,230 --> 01:15:25,400 you might get heterogeneity. 1249 01:15:25,400 --> 01:15:28,280 Another question is, what is the evolution explanation 1250 01:15:28,280 --> 01:15:31,140 for why that behavior might have evolved? 1251 01:15:31,140 --> 01:15:33,820 Now in general, we cannot prove why something evolved, 1252 01:15:33,820 --> 01:15:37,390 but we can make educated guesses that make experimentally 1253 01:15:37,390 --> 01:15:38,684 testable hypotheses. 1254 01:15:38,684 --> 01:15:40,100 And for example, in the experiment 1255 01:15:40,100 --> 01:15:44,360 that we've been doing, we've been looking at bi-modality 1256 01:15:44,360 --> 01:15:48,230 in expression of the galactose genes in yeast. 1257 01:15:48,230 --> 01:15:53,815 And that was still a problem set in early-- oh, 1258 01:15:53,815 --> 01:15:55,700 no, we removed that one this year. 1259 01:15:55,700 --> 01:16:02,880 Well, so experimentally yeast, in some environments, 1260 01:16:02,880 --> 01:16:04,840 bimodally or stochastically activate 1261 01:16:04,840 --> 01:16:07,470 the genes required to break down the sugar galactose. 1262 01:16:07,470 --> 01:16:09,600 And what we've demonstrated is that if you 1263 01:16:09,600 --> 01:16:13,590 make the mutants that always turn on or always don't 1264 01:16:13,590 --> 01:16:16,160 turn on these genes, then they're actually playing game 1265 01:16:16,160 --> 01:16:20,050 where you actually get this exact thing-- where 1266 01:16:20,050 --> 01:16:22,640 you get evolution towards coexistence of those two 1267 01:16:22,640 --> 01:16:23,740 strategies. 1268 01:16:23,740 --> 01:16:26,070 So that's saying that maybe the wild type that 1269 01:16:26,070 --> 01:16:29,270 follows this stochastic, mixed strategy-- it 1270 01:16:29,270 --> 01:16:32,210 may be implementing the solution of some game that 1271 01:16:32,210 --> 01:16:34,640 is a result of such frequency dependence. 1272 01:16:34,640 --> 01:16:36,720 There are other possible explanations to this. 1273 01:16:36,720 --> 01:16:38,760 In the coming weeks, we'll talk about this idea 1274 01:16:38,760 --> 01:16:41,801 of bet hedging-- that given uncertain or fluctuating 1275 01:16:41,801 --> 01:16:43,300 environments, it may be advantageous 1276 01:16:43,300 --> 01:16:46,520 for clonal populations to have a variety of different strategies 1277 01:16:46,520 --> 01:16:47,890 to cope with that uncertainty. 1278 01:16:47,890 --> 01:16:50,225 So we'll talk about those models later. 1279 01:16:50,225 --> 01:16:52,350 But since we're talking about mixed strategies now, 1280 01:16:52,350 --> 01:16:53,647 I wanted to mention that. 1281 01:16:53,647 --> 01:16:54,810 Yeah. 1282 01:16:54,810 --> 01:16:56,658 AUDIENCE: So f a star, just to be sure, 1283 01:16:56,658 --> 01:16:59,440 is going to converge to this probability [INAUDIBLE]? 1284 01:16:59,440 --> 01:17:04,380 PROFESSOR: Exactly, so the Nash equilibrium mixed strategy 1285 01:17:04,380 --> 01:17:08,680 plays A with probability-- so it's p should 1286 01:17:08,680 --> 01:17:10,020 be equal to f a star. 1287 01:17:13,260 --> 01:17:16,710 Exactly, so indeed, the heterogeneity there 1288 01:17:16,710 --> 01:17:19,980 can be implemented either way. 1289 01:17:19,980 --> 01:17:22,516 It's either coexistence of genotypes following 1290 01:17:22,516 --> 01:17:23,890 different strategies, or it could 1291 01:17:23,890 --> 01:17:25,587 be one genotype implementing both, 1292 01:17:25,587 --> 01:17:27,420 or it could be a mixture of those, actually. 1293 01:17:27,420 --> 01:17:30,670 Indeed, a characteristic of these situations 1294 01:17:30,670 --> 01:17:38,070 is that, let's say that you have a genotype-- a population has 1295 01:17:38,070 --> 01:17:40,810 this genotype that is implementing the mixed Nash 1296 01:17:40,810 --> 01:17:44,680 equilibrium choosing strategy A with probability p. 1297 01:17:44,680 --> 01:17:46,400 That's that equilibrium fraction. 1298 01:17:46,400 --> 01:17:49,530 What's interesting is that any individual 1299 01:17:49,530 --> 01:17:54,850 in the population following any strategy has the same fitness. 1300 01:17:54,850 --> 01:17:58,280 And of course, that's kind of why this was in equilibrium. 1301 01:17:58,280 --> 01:18:01,295 This equilibrium is when the two strategies have equal fitness. 1302 01:18:01,295 --> 01:18:02,920 But the funny thing is, what that means 1303 01:18:02,920 --> 01:18:05,137 is, it doesn't matter what you do at the equilibrium. 1304 01:18:05,137 --> 01:18:06,970 Depending on how you look at it, it's either 1305 01:18:06,970 --> 01:18:09,030 super deep or super trivial. 1306 01:18:09,030 --> 01:18:11,630 But it's a weird thing that if your at the equilibrium, 1307 01:18:11,630 --> 01:18:13,520 or if the population or the opponent 1308 01:18:13,520 --> 01:18:15,942 is playing this Nash equilibrium in these games, 1309 01:18:15,942 --> 01:18:17,650 then it just does not matter what you do. 1310 01:18:17,650 --> 01:18:21,920 You can do A. You can do B, actually, in any fraction. 1311 01:18:21,920 --> 01:18:23,810 So since A and B have the same fitness, 1312 01:18:23,810 --> 01:18:26,480 you can choose between them at any frequency you want, 1313 01:18:26,480 --> 01:18:29,270 and you have the same fitness, if the rest of the population 1314 01:18:29,270 --> 01:18:33,330 is playing this mixed Nash equilibrium. 1315 01:18:33,330 --> 01:18:37,490 And indeed, in this there are nice conditions 1316 01:18:37,490 --> 01:18:42,070 for what makes it this Nash equilibrium. 1317 01:18:42,070 --> 01:18:46,770 And I'm going to just highlight that you should make sense 1318 01:18:46,770 --> 01:18:49,910 of why it means what it is. 1319 01:18:49,910 --> 01:18:54,690 So if the payout, the expected fitness or payout-- 1320 01:18:54,690 --> 01:18:56,820 if you're following the Nash equilibrium 1321 01:18:56,820 --> 01:19:03,890 against Nash equilibrium-- is equal to this guy. 1322 01:19:03,890 --> 01:19:07,230 So that's what I just said-- that it 1323 01:19:07,230 --> 01:19:09,700 doesn't matter what you do. 1324 01:19:09,700 --> 01:19:11,490 If everyone else is doing p star, 1325 01:19:11,490 --> 01:19:14,370 you have the same fitness. 1326 01:19:14,370 --> 01:19:16,360 So that's saying it's a Nash equilibrium. 1327 01:19:19,430 --> 01:19:22,980 Whereas there's another interesting kind of statement 1328 01:19:22,980 --> 01:19:25,010 here, that-- 1329 01:19:25,010 --> 01:19:27,435 AUDIENCE: [INAUDIBLE]? 1330 01:19:27,435 --> 01:19:30,587 That you can't unilaterally increase your fitness 1331 01:19:30,587 --> 01:19:32,770 by switching. 1332 01:19:32,770 --> 01:19:34,900 PROFESSOR: Right, it's an equality, which 1333 01:19:34,900 --> 01:19:36,250 means it is a Nash equilibrium. 1334 01:19:36,250 --> 01:19:38,599 Because it's saying that you don't have the incentive 1335 01:19:38,599 --> 01:19:39,390 to change strategy. 1336 01:19:39,390 --> 01:19:42,080 It's true that you're not dis-incentivized. 1337 01:19:42,080 --> 01:19:43,810 So it's a Nash equilibrium. 1338 01:19:43,810 --> 01:19:45,914 It's not a strict Nash equilibrium. 1339 01:19:45,914 --> 01:19:46,830 AUDIENCE: [INAUDIBLE]. 1340 01:19:50,754 --> 01:19:52,920 PROFESSOR: Well, it has to be greater than/equal to, 1341 01:19:52,920 --> 01:19:55,084 and it's actually equal to. 1342 01:19:55,084 --> 01:19:57,250 The condition it has to be greater than or equal to, 1343 01:19:57,250 --> 01:20:00,239 but it's equal to, which means it is a Nash equilibrium. 1344 01:20:00,239 --> 01:20:03,017 AUDIENCE: [INAUDIBLE] Nash equilibrium for that situation, 1345 01:20:03,017 --> 01:20:04,410 [INAUDIBLE] strategy. 1346 01:20:04,410 --> 01:20:04,710 PROFESSOR: That's right. 1347 01:20:04,710 --> 01:20:06,293 So this is not the definition of that. 1348 01:20:06,293 --> 01:20:08,070 But this thing is true, which means 1349 01:20:08,070 --> 01:20:09,278 that it's a Nash equilibrium. 1350 01:20:12,500 --> 01:20:18,690 And this other thing that's interesting is that-- 1351 01:20:18,690 --> 01:20:21,690 so this tells us that it's actually one of these ESS's. 1352 01:20:21,690 --> 01:20:28,340 And if you have questions about this, I'm happy to answer it. 1353 01:20:28,340 --> 01:20:30,640 It's explained in the book as well. 1354 01:20:30,640 --> 01:20:32,700 We are out of time, so I should let you go. 1355 01:20:32,700 --> 01:20:35,645 But good luck on the exam next Thursday. 1356 01:20:35,645 --> 01:20:37,770 If you have questions and you want to meet with me, 1357 01:20:37,770 --> 01:20:38,811 I'm available on Tuesday. 1358 01:20:38,811 --> 01:20:40,940 So please let me know.