WEBVTT 00:01.710 --> 00:05.580 Professor Ben Polak: Okay, so last time we looked at 00:05.576 --> 00:07.106 and played this game. 00:07.110 --> 00:10.840 You had to choose grades, so you had to choose Alpha and 00:10.841 --> 00:14.641 Beta, and this table told us what outcome would arise. 00:14.640 --> 00:16.950 In particular, what grade you would get and 00:16.952 --> 00:18.772 what grade your pair would get. 00:18.770 --> 00:21.770 So, for example, if you had chosen Beta and your 00:21.771 --> 00:25.161 pair had chosen Alpha, then you would get a C and your 00:25.157 --> 00:26.687 pair would get an A. 00:26.690 --> 00:29.950 One of the first things we pointed out, is that this is not 00:29.951 --> 00:31.021 quite a game yet. 00:31.020 --> 00:32.410 It's missing something. 00:32.409 --> 00:38.109 This has outcomes in it, it's an outcome matrix, 00:38.108 --> 00:44.048 but it isn't a game, because for a game we need to 00:44.048 --> 00:47.748 know payoffs. Then we looked at some possible 00:47.750 --> 00:49.390 payoffs, and now it is a game. 00:49.390 --> 00:51.890 So this is a game, just to give you some more 00:51.888 --> 00:53.988 jargon, this is a normal-form game. 00:53.990 --> 00:57.440 And here we've assumed the payoffs are those that arise if 00:57.442 --> 01:00.962 players only care about their own grades, which I think was 01:00.955 --> 01:02.525 true for a lot of you. 01:02.530 --> 01:04.970 It wasn't true for the gentleman who's sitting there 01:04.971 --> 01:07.031 now, but it was true for a lot of people. 01:07.030 --> 01:09.240 We pointed out, that in this game, 01:09.238 --> 01:11.378 Alpha strictly dominates Beta. 01:11.380 --> 01:12.470 What do we mean by that? 01:12.469 --> 01:16.769 We mean that if these are your payoffs, no matter what your 01:16.769 --> 01:20.999 pair does, you attain a higher payoff from choosing Alpha, 01:20.995 --> 01:23.585 than you do from choosing Beta. 01:23.590 --> 01:26.030 Let's focus on a couple of lessons of the class before I 01:26.031 --> 01:26.921 come back to this. 01:26.920 --> 01:30.490 One lesson was, do not play a strictly 01:30.486 --> 01:32.506 dominated strategy. 01:32.510 --> 01:34.250 Everybody remember that lesson? 01:34.250 --> 01:36.430 Then much later on, when we looked at some more 01:36.430 --> 01:38.800 complicated payoffs and a more complicated game, 01:38.800 --> 01:42.050 we looked at a different lesson which was this: 01:42.049 --> 01:45.929 put yourself in others' shoes to try and figure out what 01:45.934 --> 01:47.634 they're going to do. 01:47.629 --> 01:49.579 So in fact, what we learned from that is, 01:49.581 --> 01:52.171 it doesn't just matter what your payoffs are -- that's 01:52.166 --> 01:55.136 obviously important -- it's also important what other people's 01:55.141 --> 01:57.541 payoffs are, because you want to try and 01:57.543 --> 02:00.423 figure out what they're going to do and then respond 02:00.420 --> 02:03.410 appropriately. So we're going to return to 02:03.414 --> 02:05.604 both of these lessons today. 02:05.599 --> 02:08.949 Both of these lessons will reoccur today. 02:08.949 --> 02:12.369 Now, a lot of today is going to be fairly abstract, 02:12.371 --> 02:16.341 so I just want to remind you that Game Theory has some real 02:16.341 --> 02:19.421 world relevance. Again, still in the interest of 02:19.416 --> 02:22.436 recapping, this particular game is called the Prisoners' 02:22.444 --> 02:24.044 Dilemma. It's written there, 02:24.039 --> 02:25.289 the Prisoners' Dilemma. 02:25.290 --> 02:27.760 Notice, it's Prisoners, plural. 02:27.759 --> 02:30.439 And we mentioned some examples last time. 02:30.439 --> 02:32.989 Let me just reiterate and mention some more examples which 02:32.987 --> 02:35.487 are actually written here, so they'll find their way into 02:35.489 --> 02:37.049 your notes. So, for example, 02:37.046 --> 02:39.816 if you have a joint project that you're working on, 02:39.815 --> 02:41.915 perhaps it's a homework assignment, 02:41.919 --> 02:44.229 or perhaps it's a video project like these guys, 02:44.233 --> 02:46.303 that can turn into a Prisoners' Dilemma. 02:46.300 --> 02:49.910 Why? Because each individual might 02:49.910 --> 02:52.740 have an incentive to shirk. 02:52.740 --> 02:56.110 Price competition -- two firms competing with one another in 02:56.113 --> 02:59.433 prices -- can have a Prisoners' Dilemma aspect about it. 02:59.430 --> 03:01.940 Why? Because no matter how the other 03:01.935 --> 03:04.565 firm, your competitor, prices you might have an 03:04.571 --> 03:06.291 incentive to undercut them. 03:06.289 --> 03:09.569 If both firms behave that way, prices will get driven down 03:09.569 --> 03:12.849 towards marginal cost and industry profits will suffer. 03:12.849 --> 03:15.459 In the first case, if everyone shirks you end up 03:15.462 --> 03:16.632 with a bad product. 03:16.629 --> 03:19.359 In the second case, if both firms undercut each 03:19.356 --> 03:22.786 other, you end up with low prices, that's actually good for 03:22.793 --> 03:24.693 consumers but bad for firms. 03:24.690 --> 03:26.560 Let me mention a third example. 03:26.560 --> 03:29.880 Suppose there's a common resource out there, 03:29.881 --> 03:34.131 maybe it's a fish stock or maybe it's the atmosphere. 03:34.129 --> 03:36.499 There's a Prisoners' Dilemma aspect to this too. 03:36.500 --> 03:39.210 You might have an incentive to over fish. 03:39.210 --> 03:41.560 Why? Because if the other countries 03:41.558 --> 03:44.188 with this fish stock--let's say the fish stock is the 03:44.190 --> 03:47.430 Atlantic--if the other countries are going to fish as normal, 03:47.430 --> 03:51.120 you may as well fish as normal too. 03:51.120 --> 03:53.460 And if the other countries aren't going to cut down on 03:53.455 --> 03:55.435 their fishing, then you want to catch the fish 03:55.437 --> 03:57.297 now, because there aren't going to 03:57.301 --> 03:58.501 be any there tomorrow. 03:58.500 --> 04:02.410 Another example of this would be global warming and carbon 04:02.407 --> 04:04.517 emissions. Again, leaving aside the 04:04.522 --> 04:07.652 science, about which I'm sure some of you know more than me 04:07.646 --> 04:10.336 here, the issue of carbon emissions is a Prisoners' 04:10.339 --> 04:13.259 Dilemma. Each of us individually has an 04:13.262 --> 04:15.932 incentive to emit carbons as usual. 04:15.930 --> 04:18.130 If everyone else is cutting down I don't have too, 04:18.125 --> 04:20.495 and if everyone else does cut down I don't have to, 04:20.500 --> 04:23.830 I end up using hot water and driving a big car and so on. 04:23.829 --> 04:27.149 In each of these cases we end up with a bad outcome, 04:27.148 --> 04:29.228 so this is socially important. 04:29.230 --> 04:31.980 This is not just some abstract thing going on in a class in 04:31.978 --> 04:33.328 Yale. We need to think about 04:33.327 --> 04:35.407 solutions to this, right from the start of the 04:35.409 --> 04:37.629 class, and we already talked about something. 04:37.629 --> 04:41.989 We pointed out, that this is not just a failure 04:41.989 --> 04:43.789 of communication. 04:43.790 --> 04:46.910 Communication per se will not get you out of a Prisoners' 04:46.912 --> 04:48.852 Dilemma. You can talk about it as much 04:48.846 --> 04:51.406 as you like, but as long as you're going to go home and 04:51.408 --> 04:54.348 still drive your Hummer and have sixteen hot showers a day, 04:54.350 --> 04:56.100 we're still going to have high carbon emissions. 04:56.100 --> 04:59.500 You can talk about working hard on your joint problem sets, 04:59.502 --> 05:02.442 but as long as you go home and you don't work hard, 05:02.436 --> 05:04.756 it doesn't help. In fact, if the other person is 05:04.761 --> 05:06.251 working hard, or is cutting back on their 05:06.247 --> 05:08.267 carbon emissions, you have every bit more 05:08.268 --> 05:11.488 incentive to not work hard or to keep high carbon emissions 05:11.492 --> 05:13.672 yourself. So we need something more and 05:13.674 --> 05:16.444 the kind of things we can see more: we can think about 05:16.441 --> 05:18.521 contracts; we can think about treaties 05:18.522 --> 05:21.782 between countries; we can think about regulation. 05:21.779 --> 05:25.609 All of these things work by changing the payoffs. 05:25.610 --> 05:29.360 Not just talking about it, but actually changing the 05:29.357 --> 05:33.397 outcomes actually and changing the payoffs, changing the 05:33.397 --> 05:35.597 incentives. Another thing we can do, 05:35.598 --> 05:38.018 a very important thing, is we can think about changing 05:38.024 --> 05:40.914 the game into a game of repeated interaction and seeing how much 05:40.907 --> 05:43.627 that helps, and we'll come back and revisit 05:43.627 --> 05:45.227 that later in the class. 05:45.230 --> 05:48.090 One last thing we can think of doing but we have to be a bit 05:48.094 --> 05:50.184 careful here, is we can think about changing 05:50.182 --> 05:51.592 the payoffs by education. 05:51.589 --> 05:53.479 I think of that as the "Maoist" strategy. 05:53.480 --> 05:55.910 Lock people up in classrooms and tell them they should be 05:55.906 --> 05:57.756 better people. That may or may not work -- I'm 05:57.756 --> 05:59.656 not optimistic -- but at least it's the same idea. 05:59.660 --> 06:03.170 We're changing payoffs. 06:03.170 --> 06:08.710 So that's enough for recap and I want to move on now. 06:08.709 --> 06:12.279 And in particular, we left you hanging at the end 06:12.278 --> 06:14.608 last time. We played a game at the very 06:14.613 --> 06:16.623 end last time, where each of you chose a 06:16.624 --> 06:19.724 number -- all of you chose a number -- and we said the winner 06:19.718 --> 06:22.758 was going to be the person who gets closest to two-thirds of 06:22.760 --> 06:24.410 the average in the class. 06:24.410 --> 06:26.690 Now we've figured that out, we figured out who the winner 06:26.690 --> 06:29.130 is, and I know that all of you have been trying to see if you 06:29.133 --> 06:30.073 won, is that right? 06:30.069 --> 06:32.169 I'm going to leave you in suspense. 06:32.170 --> 06:33.480 I am going to tell you today who won. 06:33.480 --> 06:35.960 We did figure it out, and we'll get there, 06:35.963 --> 06:38.753 but I want to do a little bit of work first. 06:38.750 --> 06:40.000 So we're just going to leave it in suspense. 06:40.000 --> 06:46.870 That'll stop you walking out early if you want to win the 06:46.868 --> 06:49.298 prize. So there's going to be lots of 06:49.295 --> 06:51.555 times in this class when we get to play games, 06:51.560 --> 06:54.130 we get to have classroom discussions and so on, 06:54.126 --> 06:57.416 but there's going to be some times when we have to slow down 06:57.418 --> 07:00.138 and do some work, and the next twenty minutes are 07:00.139 --> 07:01.039 going to be that. 07:01.040 --> 07:06.660 So with apologies for being a bit more boring for twenty 07:06.661 --> 07:12.181 minutes, let's do something we'll call formal stuff. 07:12.180 --> 07:18.780 In particular, I want to develop and make sure 07:18.782 --> 07:25.682 we all understand, what are the ingredients of a 07:25.679 --> 07:27.849 game? So in particular, 07:27.845 --> 07:30.675 we need to figure out what formally makes something into a 07:30.681 --> 07:34.201 game. The formal parts of a game are 07:34.196 --> 07:37.556 this. We need players -- and while 07:37.558 --> 07:41.468 we're here let's develop some notation. 07:41.470 --> 07:46.050 So the standard notation for players, I'm going to use things 07:46.054 --> 07:48.274 like little i and little j. 07:48.269 --> 07:51.449 So in that numbers game, the game when all of you wrote 07:51.453 --> 07:54.643 down a number and handed it in at the end of last time, 07:54.636 --> 07:56.106 the players were who? 07:56.110 --> 07:56.960 The players were you. 07:56.960 --> 07:58.490 You'all were the players. 07:58.490 --> 08:02.990 Useful text and expression meaning you plural. 08:02.990 --> 08:16.000 In the numbers game, you'all, were the players. 08:16.000 --> 08:22.280 Second ingredient of the game are strategies. 08:22.280 --> 08:23.170 (There's a good clue here. 08:23.170 --> 08:31.710 If I'm writing you should be writing.) Notation: 08:31.711 --> 08:40.621 so I'm going to use little "s_i" to be a 08:40.617 --> 08:47.157 particular strategy of Player i. 08:47.159 --> 08:53.049 So an example in that game might have been choosing the 08:53.050 --> 08:56.560 number 13. Everyone understand that? 08:56.559 --> 09:00.279 Now I need to distinguish this from the set of possible 09:00.283 --> 09:03.733 strategies of Player I, so I'm going to use capital 09:03.732 --> 09:05.872 "S_i" to be what? 09:05.870 --> 09:13.710 To be the set of alternatives. 09:13.710 --> 09:24.460 The set of possible strategies of Player i. 09:24.460 --> 09:26.980 So in that game we played at the end last time, 09:26.984 --> 09:28.854 what were the set of strategies? 09:28.850 --> 09:38.060 They were the sets 1,2, 3, all the way up to 100. 09:38.059 --> 09:41.209 When distinguishing a particular strategy from the set 09:41.212 --> 09:42.702 of possible strategies. 09:42.700 --> 09:45.420 While we're here, our third notation for 09:45.417 --> 09:49.247 strategy, I'm going to use little "s" without an "i," 09:49.250 --> 09:59.330 (no subscripts): little "s" without an "i," to 09:59.331 --> 10:07.621 mean a particular play of the game. 10:07.620 --> 10:08.700 So what do I mean by that? 10:08.700 --> 10:13.070 All of you, at the end last time, wrote down this number and 10:13.066 --> 10:17.426 handed them in so we had one number, one strategy choice for 10:17.433 --> 10:19.583 each person in the class. 10:19.580 --> 10:23.070 So here they are, here's my collected in, 10:23.066 --> 10:25.416 sort of strategy choices. 10:25.419 --> 10:27.439 Here's the bundle of bits of paper you handed in last time. 10:27.440 --> 10:30.220 This is a particular play of the game. 10:30.220 --> 10:34.150 I've got each person's name and I've got a number from each 10:34.147 --> 10:36.717 person: a strategy from each person. 10:36.720 --> 10:38.870 We actually have it on a spreadsheet as well: 10:38.865 --> 10:41.055 so here it is written out on a spreadsheet. 10:41.059 --> 10:44.859 Each of your names is on this spreadsheet and the number you 10:44.864 --> 10:46.914 chose. So that's a particular play of 10:46.905 --> 10:48.935 the game and that has a different name. 10:48.940 --> 10:56.610 We sometimes call this "a strategy profile." 10:56.610 --> 10:59.180 So in the textbook, you'll sometimes see the term a 10:59.178 --> 11:02.258 strategy profile or a strategy vector, or a strategy list. 11:02.260 --> 11:03.060 It doesn't really matter. 11:03.059 --> 11:07.209 What it's saying is one strategy for each player in the 11:07.211 --> 11:11.941 game. So in the numbers game this is 11:11.943 --> 11:20.473 the spreadsheet -- or an example of this is the spreadsheet. 11:20.470 --> 11:26.130 (I need to make it so you can still see that, 11:26.127 --> 11:31.397 so I'm going to pull down these boards. 11:31.399 --> 11:37.719 And let me clean something.) So you might think we're done 11:37.722 --> 11:40.712 right? We've got players. 11:40.710 --> 11:43.040 We've got the choices they could make: that's their 11:43.035 --> 11:45.315 strategy sets. We've got those individual 11:45.319 --> 11:47.469 strategies. And we've got the choices they 11:47.468 --> 11:49.768 actually did make: that's the strategy profile. 11:49.769 --> 11:51.419 Seems like we've got everything you could possibly want to 11:51.421 --> 11:52.031 describe in a game. 11:52.030 --> 11:54.610 What are we missing here? 11:54.610 --> 11:56.830 Shout it out. "Payoffs." 11:56.830 --> 11:58.130 We're missing payoffs. 11:58.129 --> 12:05.389 So, to complete the game, we need payoffs. 12:05.389 --> 12:07.859 Again, I need notation for payoffs. 12:07.860 --> 12:11.910 So in this course, I'll try and use "U" for utile, 12:11.910 --> 12:14.060 to be Player i's payoff. 12:14.059 --> 12:18.639 So "U_i" will depend on Player 1's choice … 12:18.644 --> 12:22.064 all the way to Player i's own choice … 12:22.063 --> 12:25.253 all the way up to Player N's choices. 12:25.250 --> 12:28.560 So Player i's payoff "U_i," depends on all 12:28.560 --> 12:31.030 the choices in the class, in this case, 12:31.027 --> 12:32.907 including her own choice. 12:32.909 --> 12:37.379 Of course, a shorter way of writing that would be 12:37.375 --> 12:41.835 "U_i(s)," it depends on the profile. 12:41.840 --> 12:44.820 So in the numbers game what is this? 12:44.820 --> 12:49.220 In the numbers game "U_i(s)" can be two 12:49.215 --> 12:54.365 things. It can be 5 dollars minus your 12:54.365 --> 12:58.685 error in pennies, if you won. 12:58.690 --> 13:00.600 I guess it could be something if there was a tie, 13:00.598 --> 13:01.948 I won't bother writing that now. 13:01.950 --> 13:10.320 And it's going to be 0 otherwise. 13:10.320 --> 13:15.490 So we've now got all of the ingredients of the game: 13:15.488 --> 13:18.628 players, strategies, payoffs. 13:18.629 --> 13:22.719 Now we're going to make an assumption today and for the 13:22.718 --> 13:25.968 next ten weeks or so; so for almost all the class. 13:25.970 --> 13:32.090 We're going to assume that these are known. 13:32.090 --> 13:35.990 We're going to assume that everybody knows the possible 13:35.988 --> 13:39.958 strategies everyone else could choose and everyone knows 13:39.958 --> 13:41.978 everyone else's payoffs. 13:41.980 --> 13:44.330 Now that's not a very realistic assumption and we are going to 13:44.328 --> 13:46.328 come back and challenge it at the end of semester, 13:46.330 --> 13:49.730 but this will be complicated enough to give us a lot of 13:49.725 --> 13:51.795 material in the next ten weeks. 13:51.799 --> 13:58.269 I need one more piece of notation and then we can get 13:58.266 --> 14:01.496 back to having some fun. 14:01.500 --> 14:06.020 So one more piece of notation, I'm going to write 14:06.018 --> 14:09.028 "s_-i" to mean what? 14:09.029 --> 14:12.949 It's going to mean a strategy choice for everybody except 14:12.950 --> 14:15.600 person "i." It's going to be useful to have 14:15.597 --> 14:16.827 that notation around. 14:16.830 --> 14:30.050 So this is a choice for all except person "i" or Player i. 14:30.049 --> 14:32.839 So, in particular, if you're person 1 and then 14:32.841 --> 14:35.321 "s_-i" would be "s_2, 14:35.323 --> 14:37.433 s_3, s_4" up to 14:37.433 --> 14:41.283 "s_n" but it wouldn't include "s_1." 14:41.280 --> 14:45.240 It's useful why? Because sometimes it's useful 14:45.235 --> 14:50.445 to think about the payoffs, as coming from "i's" own choice 14:50.445 --> 14:53.225 and everyone else's choices. 14:53.230 --> 14:57.310 It's just a useful way of thinking about things. 14:57.309 --> 15:01.089 Now this is when I want to stop for a second and I know that 15:01.088 --> 15:04.358 some of you, from past experience, are somewhat math 15:04.355 --> 15:06.545 phobic. You do not have to wave your 15:06.552 --> 15:09.752 hands in the air if you're math phobic, but since some of you 15:09.751 --> 15:12.631 are, let me just get you all to take a deep breath. 15:12.629 --> 15:15.609 This goes for people who are math phobic at home too. 15:15.610 --> 15:18.490 So everyone's in a slight panic now. 15:18.490 --> 15:19.420 You came here today. 15:19.419 --> 15:20.729 You thought everything was going to fine. 15:20.730 --> 15:22.650 And now I'm putting math on the board. 15:22.650 --> 15:23.590 Take a deep breath. 15:23.590 --> 15:25.950 It's not that hard, and in particular, 15:25.946 --> 15:29.636 notice that all I'm doing here is writing down notation. 15:29.639 --> 15:31.689 There's actually no math going on here at all. 15:31.690 --> 15:33.610 I'm just developing notation. 15:33.610 --> 15:36.370 I don't want anybody to quit this class because they're 15:36.374 --> 15:38.324 worried about math or math notation. 15:38.320 --> 15:40.590 So if you are in that category of somebody who might quit it 15:40.594 --> 15:41.984 because of that, come and talk to me, 15:41.982 --> 15:43.102 come and talk to the TAs. 15:43.100 --> 15:45.380 We will get you through it. 15:45.380 --> 15:46.460 It's fine to be math phobic. 15:46.460 --> 15:48.490 I'm phobic of all sorts of things. 15:48.490 --> 15:50.970 Not necessarily math, but all sorts of things. 15:50.970 --> 15:53.260 So a serious thing, a lot of people get put off by 15:53.260 --> 15:55.130 notation, it looks scarier than it is, 15:55.129 --> 16:00.879 there's nothing going on here except for notation at this 16:00.878 --> 16:04.588 point. So let's have an example to 16:04.585 --> 16:07.145 help us fix some ideas. 16:07.149 --> 16:14.299 (And again, I'll have to clean the board, so give me a second.) 16:14.299 --> 16:21.099 I think an example might help those people who are disturbed 16:21.103 --> 16:27.383 by the notation. So here's a game which we're 16:27.381 --> 16:31.071 going to discuss briefly. 16:31.070 --> 16:38.040 It involves two players and we'll call the Players I and II 16:38.043 --> 16:43.753 and Player I has two choices, top and bottom, 16:43.745 --> 16:49.535 and Player II has three choices left, center, 16:49.538 --> 16:52.248 and right. It's just a very simple 16:52.254 --> 16:53.394 abstract example for now. 16:53.389 --> 16:55.039 And let's suppose the payoffs are like this. 16:55.039 --> 16:56.619 They're not particularly interesting. 16:56.620 --> 17:01.640 We're just going to do it for the purpose of illustration. 17:01.639 --> 17:08.279 So here are the payoffs: (5, -1), (11,3), 17:08.283 --> 17:13.103 (0,0), (6,4), (0,2), (2,0). 17:13.099 --> 17:17.869 Let's just map the notation we just developed into this game. 17:17.869 --> 17:21.179 So first of all, who are the players here? 17:21.180 --> 17:24.210 Well there's no secret there, the players are -- let's just 17:24.213 --> 17:25.733 write it down why don't we. 17:25.730 --> 17:32.400 The players here in this game are Player I and Player II. 17:32.400 --> 17:40.540 What about the strategy sets or the strategy alternatives? 17:40.539 --> 17:46.789 So here Player I's strategy set, she has two choices top or 17:46.787 --> 17:53.567 bottom, represented by the rows, which are hopefully the top row 17:53.572 --> 17:56.052 and the bottom row. 17:56.049 --> 18:00.119 Player II has three choices, this game is not symmetric, 18:00.122 --> 18:04.272 so they have different number of choices, that's fine. 18:04.269 --> 18:07.739 Player II has three choices left, center, 18:07.736 --> 18:11.376 and right, represented by the left, center, 18:11.376 --> 18:14.406 and right column in the matrix. 18:14.410 --> 18:17.530 Just to point out in passing, up to now, we've been looking 18:17.532 --> 18:19.042 mostly at symmetric games. 18:19.039 --> 18:21.109 Notice this game is not symmetric in the payoffs or in 18:21.114 --> 18:23.394 the strategies. There's no particular reason 18:23.390 --> 18:25.230 why games have to be symmetric. 18:25.230 --> 18:30.060 Payoffs: again, this is not rocket science, 18:30.056 --> 18:32.926 but let's do it anyway. 18:32.930 --> 18:35.070 So just an example of payoffs. 18:35.069 --> 18:39.629 So Player I's payoff, if she chooses top and Player 18:39.632 --> 18:43.922 II chooses center, we read by looking at the top 18:43.922 --> 18:49.442 row and the center column, and Player I's payoff is the 18:49.438 --> 18:53.008 first of these payoffs, so it's 11. 18:53.009 --> 18:56.379 Player II's payoff, from the same choices, 18:56.375 --> 18:59.735 top for Player I, center for Player II, 18:59.740 --> 19:02.660 again we go along the top row and the center column, 19:02.656 --> 19:05.226 but this time we choose Player II's payoff, 19:05.230 --> 19:12.300 which is the second payoff, so it's 3. 19:12.299 --> 19:16.489 So again, I'm hoping this is calming down the math phobics in 19:16.491 --> 19:18.111 the room. Now how do we think this game 19:18.113 --> 19:18.793 is going to be played? 19:18.789 --> 19:20.679 It's not a particularly interesting game, 19:20.680 --> 19:23.000 but while we're here, why don't we just discuss it 19:22.996 --> 19:25.686 for a second. If our mike guys get a little 19:25.694 --> 19:28.884 bit ready here. So how do we think this game 19:28.882 --> 19:30.262 should be played? 19:30.259 --> 19:34.269 Well let's ask somebody at random perhaps. 19:34.269 --> 19:39.249 Ale, do you want to ask this guy in the blue shirt here, 19:39.254 --> 19:43.064 does Player I have a dominated strategy? 19:43.059 --> 19:44.939 Student: No, Player I doesn't have a 19:44.940 --> 19:45.880 dominated strategy. 19:45.880 --> 19:48.480 For instance, if Player II picks left then 19:48.476 --> 19:52.276 Player I wants to pick bottom, but if Player II picks center, 19:52.276 --> 19:54.426 Player I wants to pick center. 19:54.430 --> 19:54.920 Professor Ben Polak: Good. 19:54.920 --> 19:55.760 Excellent. Very good. 19:55.760 --> 19:56.580 I should have had you stand up. 19:56.580 --> 19:57.380 I forgot that. Never mind. 19:57.380 --> 19:59.890 But that was very clear, thank you. 19:59.890 --> 20:01.370 Was that loud enough so people could hear it? 20:01.370 --> 20:03.010 Did people hear that? 20:03.009 --> 20:04.529 People in the back, did you hear it? 20:04.529 --> 20:06.959 So even that wasn't loud enough okay, so we we really need to 20:06.961 --> 20:09.401 get people--That was very clear, very nice, but we need people 20:09.402 --> 20:11.202 to stand up and shout, or these people at the back 20:11.197 --> 20:13.547 can't hear. So your name is? 20:13.550 --> 20:14.170 Student: Patrick. 20:14.170 --> 20:16.780 Professor Ben Polak: What Patrick said was: 20:16.778 --> 20:19.438 no, Player I does not have a dominated strategy. 20:19.440 --> 20:22.300 Top is better than bottom against left -- sorry, 20:22.304 --> 20:25.784 bottom is better than top against left because 6 is bigger 20:25.779 --> 20:28.039 than 5, but top is better than bottom 20:28.039 --> 20:30.449 against center because 11 is bigger than 0. 20:30.450 --> 20:32.310 Everyone see that? 20:32.309 --> 20:36.179 So it's not the case that top always beats--it's not the case 20:36.177 --> 20:40.107 that top always does better than bottom, or that bottom always 20:40.108 --> 20:41.718 does better than top. 20:41.720 --> 20:45.300 What about, raise hands this time, what about Player II? 20:45.299 --> 20:50.409 Does Player II have a dominated strategy? 20:50.410 --> 20:56.330 Everyone's keeping their hands firmly down so as not to get 20:56.329 --> 20:58.989 spotted here. Ale, can we try this guy in 20:58.987 --> 21:01.347 white? Do you want to stand up and 21:01.351 --> 21:05.571 wait until Ale gets there, and really yell it out now. 21:05.569 --> 21:09.159 Student: I believe right is a dominated strategy because 21:09.156 --> 21:12.696 if Player I chooses top, then Player II will choose 21:12.697 --> 21:15.907 center, and if-- I'm getting confused now, 21:15.910 --> 21:18.340 it looks better on my paper. 21:18.339 --> 21:21.499 But yeah, right is never the best choice. 21:21.500 --> 21:22.610 Professor Ben Polak: Okay, good. 21:22.609 --> 21:24.019 Let's be a little bit careful here. 21:24.020 --> 21:26.230 So your name is? Student: Thomas. 21:26.230 --> 21:28.840 Professor Ben Polak: So Thomas said something which was 21:28.844 --> 21:31.244 true, but it doesn't quite match with the definition of a 21:31.244 --> 21:32.234 dominated strategy. 21:32.230 --> 21:36.930 What Thomas said was, right is never a best choice, 21:36.925 --> 21:39.795 that's true. But to be a dominated strategy 21:39.801 --> 21:41.051 we need something else. 21:41.049 --> 21:45.529 We need that there's another strategy of Player II that 21:45.528 --> 21:47.268 always does better. 21:47.269 --> 21:49.219 That turns out also to be true in this case, 21:49.218 --> 21:50.168 but let's just see. 21:50.170 --> 21:56.800 So in this particular game, I claim that center dominates 21:56.802 --> 21:58.832 right. So let's just see that. 21:58.829 --> 22:02.909 If Player I chose top, center yields 3, 22:02.913 --> 22:06.893 right yields 0: 3 is bigger than 0. 22:06.890 --> 22:11.030 And if Player I chooses bottom, then center yields 2, 22:11.028 --> 22:14.448 right yields 0: 2 is bigger than 0 again. 22:14.450 --> 22:18.590 So in this game, center strictly dominates 22:18.591 --> 22:20.891 right. What you said was true, 22:20.889 --> 22:24.189 but I wanted something specifically about domination 22:24.191 --> 22:27.331 here. So what we know here, 22:27.333 --> 22:33.333 we know that Player II should not choose right. 22:33.329 --> 22:35.029 Now, in fact, that's as far as we can get 22:35.027 --> 22:37.057 with dominance arguments in this particular game, 22:37.063 --> 22:38.893 but nevertheless, let's just stick with it a 22:38.888 --> 22:41.548 second. I gave you the definition of 22:41.548 --> 22:45.838 strict dominance last time and it's also in the handout. 22:45.839 --> 22:47.539 By the way, the handout on the web. 22:47.539 --> 22:50.779 But let me write that definition again, 22:50.784 --> 22:55.314 using or making use of the notation from the class. 22:55.309 --> 23:06.619 So definition so: Player i's strategy 23:06.617 --> 23:25.457 "s'_i" is strictly dominated by Player i's strategy 23:25.462 --> 23:33.872 "s_i, " and now we can use our 23:33.873 --> 23:38.933 notation, if "U_I" from choosing "s_i," 23:38.930 --> 23:42.730 when other people choose "s_-i," is strictly 23:42.727 --> 23:45.587 bigger than U_I(s'_i) 23:45.593 --> 23:48.893 when other people choose "s_-i," 23:48.890 --> 23:54.080 and the key part of the definition is, 23:54.081 --> 23:57.871 for all "s_-i." 23:57.869 --> 24:00.529 So to say it in words, Player i's strategy 24:00.531 --> 24:03.971 "s'_i" is strictly dominated by her strategy 24:03.971 --> 24:07.311 "s_i," if "s_i" always does 24:07.312 --> 24:11.602 strictly better -- always yields a higher payoff for Player i -- 24:11.602 --> 24:14.192 no matter what the other people do. 24:14.190 --> 24:17.600 So this is the same definition we saw last time, 24:17.600 --> 24:21.450 just being a little bit more nerdy and putting in some 24:21.446 --> 24:23.826 notation. People panicking about that, 24:23.832 --> 24:26.102 people look like deer in the headlamps yet? 24:26.099 --> 24:31.779 No, you look all right: all rightish. 24:31.779 --> 24:35.979 Let's have a look at another example. 24:35.980 --> 24:40.960 People okay, I can move this? 24:40.960 --> 24:44.810 All right, so it's a slightly more exciting example now. 24:44.809 --> 24:48.729 So imagine the following example, an invader is thinking 24:48.727 --> 24:52.957 about invading a country, and there are two ways -- there 24:52.963 --> 24:57.293 are two passes if you like -- through which he can lead his 24:57.292 --> 24:59.812 army. You are the defender of this 24:59.807 --> 25:03.917 country and you have to decide which of these passes or which 25:03.915 --> 25:06.375 of these routes into the country, 25:06.380 --> 25:08.450 you're going to choose to defend. 25:08.450 --> 25:11.970 And the catch is, you can only defend one of 25:11.966 --> 25:13.516 these two routes. 25:13.519 --> 25:15.709 If you want a real world example of this, 25:15.705 --> 25:18.815 think about the third Century B.C., someone can correct me 25:18.820 --> 25:20.980 afterwards. I think it's the third Century 25:20.977 --> 25:23.137 B.C. when Hannibal is thinking of 25:23.142 --> 25:24.462 crossing the Alps. 25:24.460 --> 25:27.260 Not Hannibal Lecter: Hannibal the general in the 25:27.259 --> 25:28.509 third Century B.C.. 25:28.510 --> 25:29.850 The one with the elephants. 25:29.849 --> 25:34.979 Okay, so the key here is going to be that there are two passes. 25:34.980 --> 25:37.870 One of these passes is a hard pass. 25:37.870 --> 25:39.040 It goes over the Alps. 25:39.039 --> 25:41.129 And the other one is an easy pass. 25:41.130 --> 25:43.870 It goes along the coast. 25:43.869 --> 25:49.269 If the invader chooses the hard pass he will lose one battalion 25:49.268 --> 25:53.708 of his army simply in getting over the mountains, 25:53.710 --> 25:56.690 simply in going through the hard pass. 25:56.690 --> 25:59.970 If he meets your army, whichever pass he chooses, 25:59.968 --> 26:04.198 if he meets your army defending a pass, then he'll lose another 26:04.203 --> 26:06.213 battalion. I haven't given you--I've given 26:06.213 --> 26:08.053 you roughly the choices, the choice they're going to be 26:08.047 --> 26:09.437 for the attacker which pass to choose, 26:09.440 --> 26:11.940 and for the defender which pass to defend. 26:11.940 --> 26:15.580 But let's put down some payoffs so we can start talking about 26:15.575 --> 26:17.465 this. So in this game, 26:17.465 --> 26:22.405 the payoffs for this game are going to be as follows. 26:22.410 --> 26:25.090 It's a simple two by two game. 26:25.089 --> 26:29.219 This is going to be the attacker, this is Hannibal, 26:29.219 --> 26:32.439 and this is going to be the defender, 26:32.440 --> 26:34.550 (and I've forgotten which general was defending and 26:34.545 --> 26:36.015 someone's about to tell me that). 26:36.019 --> 26:38.909 And there are two passes you could defend: 26:38.906 --> 26:41.226 the easy pass or the hard pass. 26:41.230 --> 26:44.190 And there's two you could use to attack through, 26:44.186 --> 26:47.236 easy or hard. (Again, easy pass here just 26:47.241 --> 26:50.531 means no mountains, we're not talking about 26:50.527 --> 26:54.987 something on the New Jersey Turnpike.) So the payoffs here 26:54.985 --> 26:59.975 are as follows, and I'll explain them in a 26:59.983 --> 27:01.893 second. So his payoff, 27:01.891 --> 27:05.051 the attacker's payoff, is how many battalions does he 27:05.052 --> 27:07.122 get to bring into your country? 27:07.119 --> 27:10.799 He only has two to start with and for you, it's how many 27:10.803 --> 27:13.083 battalions of his get destroyed? 27:13.079 --> 27:17.269 So just to give an example, if he goes through the hard 27:17.268 --> 27:20.058 pass and you defend the hard pass, 27:20.059 --> 27:22.229 he loses one of those battalions going over the 27:22.231 --> 27:24.641 mountains and the other one because he meets you. 27:24.640 --> 27:28.440 So he has none left and you've managed to destroy two of them. 27:28.440 --> 27:31.490 Conversely, if he goes on the hard pass and you defend the 27:31.491 --> 27:34.491 easy pass, he's going to lose one of those battalions. 27:34.490 --> 27:36.070 He'll have one left. 27:36.070 --> 27:37.730 He lost it in the mountains. 27:37.730 --> 27:40.940 But that's the only one he's going to lose because you were 27:40.943 --> 27:42.443 defending the wrong pass. 27:42.440 --> 27:46.160 Everyone understand the payoffs of this game? 27:46.160 --> 27:49.340 So now imagine yourself as a Roman general. 27:49.339 --> 27:50.799 This is going to be a little bit of a stretch for 27:50.802 --> 27:52.602 imagination, but imagination yourself as a Roman general, 27:52.599 --> 27:54.889 and let's figure out what you're going to do. 27:54.890 --> 27:56.020 You're the defender. 27:56.020 --> 27:58.710 What are you going to do? 27:58.710 --> 28:00.750 So let's have a show of hands. 28:00.750 --> 28:06.390 How many of you think you should defend the easy pass? 28:06.390 --> 28:08.000 Raise your hands, let's raise your hands so Jude 28:08.004 --> 28:08.954 can see them. Keep them up. 28:08.950 --> 28:10.690 Wave them in the air with a bit of motion. 28:10.690 --> 28:12.310 Wave them in the air. 28:12.309 --> 28:14.799 We should get you flags okay, because these are the Romans 28:14.802 --> 28:15.942 defending the easy pass. 28:15.940 --> 28:20.230 And how many of you think you're going to defend the hard 28:20.233 --> 28:22.423 pass? We have a huge number of people 28:22.421 --> 28:24.661 who don't want to be Roman generals here. 28:24.660 --> 28:26.110 Let's try it again, no abstentions, 28:26.109 --> 28:27.649 right? I'm not going to penalize you 28:27.654 --> 28:28.864 for giving the wrong answer. 28:28.859 --> 28:30.899 So how many of you think you're going to defend the easy pass? 28:30.900 --> 28:33.030 Raise your hands again. 28:33.029 --> 28:36.539 And how many think you're going to defend the hard pass? 28:36.539 --> 28:40.689 So we have a majority choosing easy pass -- had a large 28:40.685 --> 28:42.685 majority. So what's going on here? 28:42.690 --> 28:48.400 Is it the case that defending the easy pass dominates 28:48.403 --> 28:51.263 defending the hard pass? 28:51.260 --> 28:51.820 Is that the case? 28:51.819 --> 28:54.009 Is it the case that defending the easy pass dominates 28:54.006 --> 28:55.096 defending the hard pass? 28:55.100 --> 28:56.590 You can shout out. 28:56.590 --> 28:59.440 No, it's not. In fact, we could check that if 28:59.440 --> 29:03.920 the attacker attacks through the easy pass, not surprisingly, 29:03.920 --> 29:07.500 you do better if you defend the easy pass than the hard pass: 29:07.503 --> 29:09.833 1 versus 0. But if the attacker was to 29:09.825 --> 29:12.915 attack through the hard pass, again not surprisingly, 29:12.916 --> 29:16.716 you do better if you defend the hard pass than the easy pass. 29:16.720 --> 29:19.770 So that's not an unintuitive finding. 29:19.769 --> 29:23.029 It isn't the case that defending easy dominates 29:23.033 --> 29:25.893 defending hard. You just want to match with the 29:25.888 --> 29:27.318 attacker. Nevertheless, 29:27.320 --> 29:29.400 almost all of you chose easy. 29:29.400 --> 29:30.970 What's going on? Can someone tell me what's 29:30.965 --> 29:32.485 going on? Let's get the mikes going a 29:32.494 --> 29:34.384 second. So can we catch the guy with 29:34.382 --> 29:36.722 the--can we catch this guy with the beard? 29:36.720 --> 29:38.830 Just wait for the mike to get there. 29:38.829 --> 29:42.989 If you could stand up: stand up and shout. 29:42.990 --> 29:44.980 There you go. Student: Because you 29:44.982 --> 29:49.192 want to minimize the amount of enemy soldiers that reach Rome 29:49.187 --> 29:51.287 or whatever location it is. 29:51.289 --> 29:53.429 Professor Ben Polak: You want to minimize the number of 29:53.434 --> 29:54.844 soldiers that reach Rome, that's true. 29:54.839 --> 29:56.459 On the other hand, we've just argued that you 29:56.460 --> 29:57.860 don't have a dominant strategy here; 29:57.859 --> 30:00.359 it's not the case that easy dominates hard. 30:00.360 --> 30:01.320 What else could be going on? 30:01.319 --> 30:03.419 While we've got you up, why don't we get the other guy 30:03.418 --> 30:05.158 who's got his hand up there in the middle. 30:05.160 --> 30:06.940 Again, stand up and shout in that mike. 30:06.940 --> 30:08.160 Point your face towards the mike. 30:08.160 --> 30:11.770 Good. Student: It seems as 30:11.772 --> 30:16.312 though while you don't have a dominating strategy, 30:16.309 --> 30:20.209 it seems like Hannibal is better off attacking through--It 30:20.210 --> 30:23.700 seems like he would attack through the easy pass. 30:23.700 --> 30:25.080 Professor Ben Polak: Good, why does it seem like 30:25.080 --> 30:25.660 that? That's right, 30:25.662 --> 30:26.592 we're on the right lines now. 30:26.589 --> 30:29.139 Why does it seem like he's going to attack through the easy 30:29.142 --> 30:31.012 pass? Student: Well if you're 30:31.007 --> 30:33.757 not defending the easy pass, he doesn't lose anyone, 30:33.759 --> 30:38.159 and if he attacks through the hard pass he's going to lose at 30:38.164 --> 30:39.784 least one battalion. 30:39.779 --> 30:42.129 Professor Ben Polak: So let's look at it from--Let's do 30:42.133 --> 30:44.263 the exercise--Let's do the second lesson I emphasized at 30:44.255 --> 30:46.235 the beginning. Let's put ourselves in 30:46.241 --> 30:48.711 Hannibal's shoes, they're probably boots or 30:48.706 --> 30:50.306 something. Whatever you do when you're 30:50.312 --> 30:51.532 riding an elephant, whatever you wear. 30:51.529 --> 30:54.449 Let's put ourselves in Hannibal's shoes and try and 30:54.451 --> 30:57.141 figure out what Hannibal's going to do here. 30:57.140 --> 31:01.480 So it could be--From Hannibal's point of view he doesn't know 31:01.478 --> 31:05.668 which pass you're going to defend, but let's have a look at 31:05.672 --> 31:09.142 his payoffs. If you were to defend the easy 31:09.143 --> 31:13.773 pass and he goes through the easy pass, he will get into your 31:13.770 --> 31:18.550 country with one battalion and that's the same as he would have 31:18.551 --> 31:21.791 got if he went through the hard pass. 31:21.789 --> 31:25.109 So if you defend the easy pass, from his point of view, 31:25.107 --> 31:28.667 it doesn't matter whether he chooses the easy pass and gets 31:28.671 --> 31:32.671 one in there or the hard pass, he gets one in there. 31:32.670 --> 31:36.430 But if you were to defend the hard pass, if you were to defend 31:36.432 --> 31:39.382 the mountains, then if he chooses the easy 31:39.382 --> 31:44.182 pass, he gets both battalions in and if he chooses the hard pass, 31:44.180 --> 31:46.610 he gets no battalions in. 31:46.610 --> 31:49.030 So in this case, easy is better. 31:49.029 --> 31:50.279 We have to be a little bit careful. 31:50.279 --> 31:55.369 It's not the case that for Hannibal, choosing the easy pass 31:55.366 --> 31:59.296 to attack through, strictly dominates choosing the 31:59.301 --> 32:03.381 hard pass, but it is the case that there's a weak notion of 32:03.382 --> 32:06.962 domination here. It is the case -- to introduce 32:06.955 --> 32:11.045 some jargon -- it is the case that the easy pass for the 32:11.054 --> 32:15.604 attacker, weakly dominates the hard pass for the attacker. 32:15.599 --> 32:17.099 What do I mean by weakly dominate? 32:17.099 --> 32:21.879 It means by choosing the easy pass, he does at least as well, 32:21.877 --> 32:25.937 and sometimes better, than he would have done had he 32:25.939 --> 32:27.929 chosen the hard pass. 32:27.930 --> 32:32.150 So here we have a second definition, a new definition for 32:32.150 --> 32:35.240 today, and again we can use our jargon. 32:35.240 --> 32:43.510 Definition- Player i's strategy, "s'_i" is 32:43.509 --> 32:53.399 weakly dominated by her strategy "s_i" if--now we're 32:53.399 --> 33:03.129 going to take advantage of our notation--if Player i's payoff 33:03.127 --> 33:12.207 from choosing "s_i" against "s_-i" is 33:12.207 --> 33:20.157 always as big as or equal, to her payoff from choosing 33:20.160 --> 33:25.220 "s'_i" against "s_-i" and this has to 33:25.220 --> 33:29.920 be true for all things that anyone else could do. 33:29.920 --> 33:34.060 And in addition, Player i's payoff from choosing 33:34.056 --> 33:38.546 "s_i" against "s_-i" is strictly 33:38.545 --> 33:43.115 better than her payoff from choosing "s'_i" 33:43.121 --> 33:48.331 against "s_-i," for at least one thing that 33:48.327 --> 33:50.527 everyone else could do. 33:50.529 --> 33:54.339 Just check, that exactly corresponds to the easy and hard 33:54.335 --> 33:56.165 thing we just had before. 33:56.170 --> 33:58.560 I'll say it again, Player i's strategy 33:58.558 --> 34:01.848 "s'_i" is weakly dominated by her strategy 34:01.851 --> 34:05.271 "s_i" if she always does at least as well by 34:05.273 --> 34:08.763 choosing "s_i" than choosing "s'_i" 34:08.760 --> 34:11.730 regardless of what everyone else does, 34:11.730 --> 34:15.270 and sometimes she does strictly better. 34:15.269 --> 34:17.739 It seems a pretty powerful lesson. 34:17.739 --> 34:20.099 Just as we said you should never choose a strictly 34:20.101 --> 34:22.461 dominated strategy, you're probably never going to 34:22.463 --> 34:24.683 choose a weakly dominated strategy either, 34:24.680 --> 34:28.230 but it's a little more subtle. 34:28.230 --> 34:29.800 Now that definition, if you're worried about what 34:29.797 --> 34:31.657 I've written down here and you want to see it in words, 34:31.659 --> 34:35.169 on the handout I've already put on the web that has the summary 34:35.170 --> 34:38.560 of the first class, I included this definition in 34:38.561 --> 34:41.671 words as well. So compare the definition of 34:41.668 --> 34:45.748 words with what's written here in the nerdy notation on the 34:45.747 --> 34:47.887 board. Now since we think that 34:47.888 --> 34:51.298 Hannibal, the attacker, is not going to play a weakly 34:51.297 --> 34:54.607 dominated strategy, we think Hannibal is not going 34:54.605 --> 34:56.155 to choose the hard pass. 34:56.159 --> 34:59.069 He's going to attack on the easy pass. 34:59.070 --> 35:02.560 And given that, what should we defend? 35:02.559 --> 35:04.879 We should defend easy which is what most of you chose. 35:04.880 --> 35:07.680 So be honest now: was that why most of you chose 35:07.682 --> 35:09.192 easy? Yeah, probably was. 35:09.190 --> 35:10.620 We're able to read this. 35:10.619 --> 35:13.499 So, by putting ourselves in Hannibal's shoes, 35:13.500 --> 35:17.360 we could figure out that his hard attack strategy was weakly 35:17.363 --> 35:20.203 dominated. He's going to choose easy, 35:20.199 --> 35:22.119 so we should defend easy. 35:22.119 --> 35:25.759 Having said that of course, Hannibal went through the 35:25.755 --> 35:28.825 mountains which kind of screws up the lesson, 35:28.831 --> 35:30.301 but too late now. 35:30.300 --> 35:38.310 Now then, I promised you we'd get back to the game from last 35:38.309 --> 35:40.749 time. So where have we got to so far 35:40.746 --> 35:43.686 in this class. We know from last time that you 35:43.692 --> 35:47.812 should not choose a dominated strategy, and we also know we 35:47.813 --> 35:52.293 probably aren't going to choose a weakly dominated strategy, 35:52.289 --> 35:54.609 and we also know that you should put yourself in other 35:54.608 --> 35:57.188 people's shoes and figure out that they're not going to play 35:57.190 --> 35:59.640 strongly or strictly or weakly dominated strategies. 35:59.639 --> 36:02.409 That seems a pretty good way to predict how other people are 36:02.409 --> 36:04.659 going to play. So let's take those ideas and 36:04.661 --> 36:06.941 go back to the numbers game from last time. 36:06.940 --> 36:09.660 Now before I do that, I don't need the people at home 36:09.661 --> 36:12.541 to see this, but how many of you were here last time? 36:12.540 --> 36:13.550 How many of you were not. 36:13.550 --> 36:14.330 I asked the wrong question. 36:14.329 --> 36:16.299 How many of you were not here last time? 36:16.300 --> 36:18.990 So we handed out again that game. 36:18.989 --> 36:21.879 We handed out again the game with the numbers, 36:21.881 --> 36:24.841 but just in case, let me just read out the game 36:24.836 --> 36:28.266 you played. This was the game you played. 36:28.269 --> 36:31.069 "Without showing your neighbor what you are doing, 36:31.067 --> 36:34.547 put it in the box below a whole number between 1 and a 100. 36:34.550 --> 36:37.660 We will (and in fact have) calculated the average number 36:37.661 --> 36:40.661 chosen in the class and the winner of this game is the 36:40.659 --> 36:43.709 person who gets closest to two-thirds times the average 36:43.714 --> 36:46.434 number. They will win five dollars 36:46.425 --> 36:49.005 minus the difference in pennies." 36:49.010 --> 36:53.060 So everybody filled that in last time and I have their 36:53.055 --> 36:55.975 choices here. So before we reveal who won, 36:55.976 --> 36:58.066 let's discuss this a little bit. 36:58.070 --> 37:00.950 Let me come down hazardously off this stage, 37:00.954 --> 37:04.714 and figure out--Let's get the mics up a bit for a second, 37:04.710 --> 37:06.790 we can get some mics ready. 37:06.789 --> 37:10.759 So let me find out from people here and see what people did a 37:10.763 --> 37:13.873 second. You can be honest here since 37:13.871 --> 37:17.101 I've got everything in front of me. 37:17.099 --> 37:22.989 So how many of you chose some number like 32,33, 37:22.993 --> 37:24.993 34? One hand. 37:24.989 --> 37:26.449 Actually I can tell you, nine of you did. 37:26.450 --> 37:27.560 So should I read out the names? 37:27.560 --> 37:30.800 Should I embarrass people? 37:30.800 --> 37:33.940 We've got Lynette, Lukucin, we've Kristin, 37:33.935 --> 37:35.535 Bargeon; there's nine of you here. 37:35.539 --> 37:39.839 Let's try it again.How many of you chose numbers between 32 and 37:39.842 --> 37:41.012 34? Okay, a good number of you. 37:41.010 --> 37:43.010 Now we're seeing some hands up. 37:43.010 --> 37:46.270 So keep your hands up a second, those people. 37:46.270 --> 37:48.060 So let me ask people why? 37:48.059 --> 37:50.809 Can you get your hand into the guy? 37:50.810 --> 37:51.250 What's your name? 37:51.250 --> 37:52.870 If we can get him to stand up. 37:52.869 --> 37:54.699 Stand up a second and shout out to the class. 37:54.700 --> 37:55.780 What's your name? 37:55.780 --> 37:56.800 Student: Chris. 37:56.800 --> 37:58.760 Professor Ben Polak: Chris, you're on this list 37:58.756 --> 37:59.786 somewhere. Maybe you're not on this list 37:59.787 --> 38:01.537 somewhere. Never mind, what did you choose? 38:01.539 --> 38:02.949 Student: I think I chose 30. 38:02.949 --> 38:04.219 Professor Ben Polak: Okay 30, so that's pretty close. 38:04.220 --> 38:06.430 So why did you choose 30? 38:06.429 --> 38:09.039 Student: Because I thought everyone was going to be 38:09.044 --> 38:11.434 around like the 45 range because 66 is two-thirds, 38:11.429 --> 38:14.199 or right around of 100, and they were going to go 38:14.195 --> 38:17.705 two-thirds less than that and I did one less than that one. 38:17.710 --> 38:19.880 Professor Ben Polak: Okay, thank you. 38:19.880 --> 38:21.720 Let's get one of the others. 38:21.720 --> 38:22.630 There was another one in here. 38:22.630 --> 38:24.960 Can you just raise your hands again, the people who were 38:24.963 --> 38:27.443 around 33,34. There's somebody in here. 38:27.440 --> 38:29.910 Can we get you to stand up (and you're between mikes). 38:29.910 --> 38:35.170 So that would be--Yep, go ahead. 38:35.170 --> 38:36.770 Shout it out. What's your name first of all? 38:36.770 --> 38:37.460 Student: Ryan. 38:37.460 --> 38:39.070 Professor Ben Polak: Ryan, I must have you here as 38:39.072 --> 38:39.622 well, never mind. 38:39.620 --> 38:40.740 What did you choose? 38:40.740 --> 38:41.680 Student: 33, I think. 38:41.680 --> 38:42.200 Professor Ben Polak: 33. 38:42.200 --> 38:44.120 Oh you did. You are Ryan Lowe? 38:44.120 --> 38:44.580 Student: Yeah. 38:44.579 --> 38:46.479 Professor Ben Polak: You are Ryan Lowe, 38:46.483 --> 38:48.113 okay. Good, go ahead. 38:48.110 --> 38:51.780 Student: I thought similar to Chris actually and I 38:51.783 --> 38:55.333 also thought that if we got two-thirds and everyone was 38:55.325 --> 38:59.125 choosing numbers in between 1 and 100 ends up with 33, 38:59.130 --> 39:01.790 would be around the number (indiscernible). 39:01.789 --> 39:03.789 Professor Ben Polak: So just to repeat the argument that 39:03.793 --> 39:05.083 we just heard. Again, you have to shout it out 39:05.078 --> 39:06.538 more because I'm guessing people didn't hear that in room. 39:06.539 --> 39:09.339 So I'll just repeat it to make sure everyone hears it. 39:09.340 --> 39:13.140 A reason for choosing a number like 33 might go as follows. 39:13.139 --> 39:18.209 If people in the room choose randomly between 1 and 100, 39:18.206 --> 39:23.916 then the average is going to be around 50 say and two-thirds of 39:23.917 --> 39:28.267 50 is around 33, 33 1/3 actually. 39:28.269 --> 39:31.219 So that's a pretty good piece of reasoning. 39:31.219 --> 39:33.269 What's wrong with that reasoning? 39:33.270 --> 39:34.820 What's wrong with that? 39:34.820 --> 39:38.870 Can we get the guy, the woman in the striped shirt 39:38.872 --> 39:41.092 here, sorry. We haven't had a woman for a 39:41.087 --> 39:42.217 while, so let's have a woman. 39:42.220 --> 39:44.470 Thank you. Student: That even if 39:44.474 --> 39:46.604 everyone else had the same reasoning as you, 39:46.601 --> 39:48.531 it's still going to be way too high. 39:48.530 --> 39:50.030 Professor Ben Polak: So in particular, 39:50.034 --> 39:51.614 if everyone else had the same reasoning as you, 39:51.607 --> 39:52.767 it's going to be way too high. 39:52.769 --> 39:56.429 So if everyone else reasons that way then everyone in the 39:56.425 --> 40:00.205 room would choose a number like 33 or 34, and in that case, 40:00.211 --> 40:02.171 the average would be what? 40:02.170 --> 40:06.890 Sorry, that two-thirds of the average would be what? 40:06.890 --> 40:08.410 Something like 22. 40:08.409 --> 40:12.409 So the flaw in the argument that Chris and Ryan had -- it 40:12.407 --> 40:15.777 isn't a bad argument, it's a good starting point -- 40:15.778 --> 40:19.348 but the flaw in the argument, the mistake in the argument was 40:19.354 --> 40:21.624 the first sentence in the argument. 40:21.619 --> 40:24.339 The first sentence in the argument was, 40:24.343 --> 40:27.143 if the people in the room choose random, 40:27.139 --> 40:29.719 then they will choose around 50. 40:29.720 --> 40:33.010 That's true. The problem is that people in 40:33.008 --> 40:35.778 the room aren't going to choose at random. 40:35.780 --> 40:36.820 Look around the room a second. 40:36.820 --> 40:38.020 Look around yourselves. 40:38.019 --> 40:41.769 Do any of you look like a random number generator? 40:41.769 --> 40:44.729 Actually, from here I can see some of the people, 40:44.726 --> 40:46.386 but I'm not going to put. 40:46.389 --> 40:50.299 Actually looking at some of your answers maybe some of you 40:50.298 --> 40:51.478 are. On the whole, 40:51.479 --> 40:54.319 Yale students are not random number generators. 40:54.320 --> 40:56.830 They're trying to win the game. 40:56.829 --> 41:02.849 So they're unlikely to choose numbers at random. 41:02.849 --> 41:05.519 As a further argument, if in fact everyone thought 41:05.518 --> 41:08.998 that way, and if you figured out everyone was going to think that 41:09.004 --> 41:11.384 way, then you would expect everyone 41:11.381 --> 41:15.391 to choose a number like 33 and in that case you should choose a 41:15.394 --> 41:17.224 number like 22. How many of you, 41:17.222 --> 41:18.342 raise your hands a second. 41:18.340 --> 41:22.880 How many of you chose numbers in the range 21 through 23? 41:22.880 --> 41:23.970 There's way more of you than that. 41:23.969 --> 41:26.499 I'll start reading you out as well. 41:26.500 --> 41:28.810 Actually about twelve of you, raise your hands. 41:28.809 --> 41:30.759 There should be twelve hands going up somewhere. 41:30.760 --> 41:32.220 There's two, three hands going up, 41:32.220 --> 41:33.460 four, five hands going up. 41:33.460 --> 41:37.680 There's actually 12 people who chose exactly 22, 41:37.679 --> 41:41.629 so considerably more if include 23 and 21. 41:41.630 --> 41:43.740 So those people, I'm guessing, 41:43.735 --> 41:46.635 were thinking this way, is that right? 41:46.639 --> 41:50.759 Let me get one of my 22's up again. 41:50.760 --> 41:54.220 Here's a 22. You want to get this guy? 41:54.220 --> 41:57.180 What's your name sir? 41:57.180 --> 41:59.640 Stand up and shout. 41:59.639 --> 42:01.979 Student: Ryan Professor Ben Polak: You 42:01.976 --> 42:04.406 chose 22? Student: I chose 22 42:04.409 --> 42:08.339 because I thought that most people would play the game 42:08.343 --> 42:11.613 dividing by two-thirds a couple of times, 42:11.610 --> 42:16.720 and give numbers averaging around the low 30's. 42:16.719 --> 42:18.619 Professor Ben Polak: So if you think people are going to 42:18.616 --> 42:20.806 play a particular way, in particular if you think 42:20.806 --> 42:24.106 people are going to choose the strategy of Ryan and Chris, 42:24.110 --> 42:28.180 and choose around 33, then 22 seems a great answer. 42:28.179 --> 42:31.659 But you underestimate your Yale colleagues. 42:31.660 --> 42:36.950 In fact, 22 was way too high. 42:36.949 --> 42:39.759 Now, again, let's just iterate the point here. 42:39.760 --> 42:41.710 Let me just repeat the point here. 42:41.710 --> 42:43.820 The point here is when you're playing a game, 42:43.822 --> 42:46.752 you want to think about what other people are trying to do, 42:46.750 --> 42:49.250 to try and predict what they're trying to do, 42:49.249 --> 42:52.429 and it's not necessarily a great starting point to assume 42:52.431 --> 42:55.841 that the people around you are random number generators. 42:55.840 --> 43:00.880 They have aims- trying to win, and they have strategies too. 43:00.880 --> 43:03.590 Let me take this back to the board a second. 43:03.590 --> 43:07.190 So, in particular, are there any strategies here 43:07.193 --> 43:09.113 we can really rule out? 43:09.110 --> 43:11.680 We said already people are not random. 43:11.679 --> 43:14.779 Are there any choices we can just rule out? 43:14.780 --> 43:19.160 We know people are not going to choose those choices. 43:19.160 --> 43:20.090 Let's have someone here. 43:20.090 --> 43:21.400 Can we have the guy in green? 43:21.400 --> 43:23.050 Wait for Ale, there we go. 43:23.050 --> 43:24.710 Good. Stand up. 43:24.710 --> 43:26.090 Give me your name. 43:26.090 --> 43:27.160 Student: My name's Nick. 43:27.159 --> 43:28.549 Professor Ben Polak: Shout it out so people can hear. 43:28.550 --> 43:31.070 Student: No one is going to choose a number over 50. 43:31.070 --> 43:32.130 Professor Ben Polak: No one is going to choose a number 43:32.126 --> 43:34.796 over 50. Okay, I was going--okay that's 43:34.804 --> 43:38.994 fair enough. Some people did. 43:38.990 --> 43:41.620 That's fair enough. 43:41.619 --> 43:42.919 I was thinking of something a little bit less, 43:42.924 --> 43:44.894 that's fine. I was thinking of something a 43:44.888 --> 43:46.338 little bit less ambitious. 43:46.340 --> 43:47.820 Somebody said 66. 43:47.820 --> 43:49.690 So let's start analyzing this. 43:49.690 --> 44:00.440 44:00.440 --> 44:06.600 So, in particular, there's something about these 44:06.600 --> 44:13.940 strategy choices that are greater than 67 at any rate. 44:13.940 --> 44:17.630 Certainly, I mean 66 let's go up a little bit, 44:17.631 --> 44:20.421 so these numbers bigger than 67. 44:20.420 --> 44:24.250 What's wrong with numbers bigger than 67? 44:24.250 --> 44:26.010 What's wrong with--Raise your hands if you have answer. 44:26.010 --> 44:28.390 What's wrong? Can we get the guy in red who's 44:28.393 --> 44:29.633 right close to the mike? 44:29.630 --> 44:30.520 Stand up, give me your name. 44:30.520 --> 44:32.200 Stand up. Shout it out to the crowd. 44:32.200 --> 44:33.000 Student: Peter. 44:33.000 --> 44:33.740 Professor Ben Polak: Yep. 44:33.739 --> 44:37.729 Student: If everyone chooses a 100 it would be 67. 44:37.730 --> 44:40.440 Professor Ben Polak: Good, so even if everyone in the 44:40.436 --> 44:43.136 number--everyone in the room didn't choose randomly but they 44:43.142 --> 44:45.762 all chose a 100, a very unlikely circumstance, 44:45.755 --> 44:48.675 but even if everyone had chosen 100, the highest, 44:48.679 --> 44:51.519 the average, sorry, the highest two-thirds 44:51.516 --> 44:54.626 of the average could possibly be is 66 2/3, 44:54.630 --> 44:57.050 hence 67 would be a pretty good choice in that case. 44:57.050 --> 45:00.290 So numbers bigger than 67 seem pretty crazy choices, 45:00.285 --> 45:03.325 but crazy isn't the word I'm looking for here. 45:03.329 --> 45:07.719 What can we say about those choices, those strategies 67 and 45:07.718 --> 45:10.468 above, bigger than 67,68 and above? 45:10.469 --> 45:12.199 What can we say about those choices? 45:12.199 --> 45:14.729 Somebody right behind you, the woman right behind you, 45:14.733 --> 45:16.193 shout it out. Student: They have no 45:16.192 --> 45:17.412 payoffs for… Professor Ben Polak: 45:17.413 --> 45:18.053 They have no payoffs. 45:18.050 --> 45:19.650 What's the jargon here? 45:19.650 --> 45:21.040 Let's use our jargon. 45:21.039 --> 45:22.929 Somebody shout it out, what's the jargon about that? 45:22.930 --> 45:24.590 They're dominated. 45:24.590 --> 45:27.260 So these strategies are dominated. 45:27.260 --> 45:30.200 Actually, they're only weakly dominated but that's okay, 45:30.197 --> 45:31.797 they're certainly dominated. 45:31.800 --> 45:37.110 In particular, a strategy like 80 is dominated 45:37.112 --> 45:41.222 by choosing 67. You will always get a higher 45:41.223 --> 45:45.283 payoff from choosing 67, at least as high and sometimes 45:45.278 --> 45:47.808 higher, than the payoff you would have 45:47.808 --> 45:51.008 got, had you chosen 80, no matter what else happened in 45:51.008 --> 45:53.338 the room. So these strategies are 45:53.343 --> 45:55.753 dominated. We know, from the very first 45:55.753 --> 45:59.313 lesson of the class last time, that no one should choose these 45:59.313 --> 46:02.863 strategies. They're dominated strategies. 46:02.860 --> 46:07.760 So did anyone choose strategies bigger than 67? 46:07.760 --> 46:10.910 Okay, I'm not going to read out names here, but, 46:10.912 --> 46:12.792 turns out four of you did. 46:12.789 --> 46:15.699 I'm not going to make you wave your--okay. 46:15.699 --> 46:20.679 So okay, for the four of you who did, never mind, 46:20.679 --> 46:23.899 but … well mind actually, 46:23.896 --> 46:27.526 yeah. So once we've eliminated the 46:27.533 --> 46:32.843 possibility that anyone in the room is going to choose a 46:32.843 --> 46:38.223 strategy bigger than 67, it's as if those numbers 68 46:38.218 --> 46:41.088 through 100 are irrelevant. 46:41.090 --> 46:45.160 It's really as if the game is being played where the only 46:45.163 --> 46:48.803 choices available on the table are 1 through 67. 46:48.800 --> 46:51.800 Is that right? We know no one's going to 46:51.803 --> 46:53.813 choose 68 and above, so we can just forget them. 46:53.809 --> 46:58.529 We can delete those strategies and once we delete those 46:58.531 --> 47:03.341 strategies, all that's left are choices 1 through 67. 47:03.340 --> 47:04.610 So can somebody help me out now? 47:04.610 --> 47:07.540 What can I conclude, now I've concluded that the 47:07.543 --> 47:11.043 strategies 68 through 100 essentially don't exist or have 47:11.038 --> 47:14.118 been deleted. What can I conclude? 47:14.119 --> 47:15.589 Let me see if I can get a mike in here. 47:15.590 --> 47:19.200 Stand up and wait for the mike. 47:19.200 --> 47:19.860 And here comes the mike. 47:19.860 --> 47:21.450 Good. Shout out. 47:21.449 --> 47:23.539 Student: That all strategies 45 and above are 47:23.538 --> 47:24.478 hence also ruled out. 47:24.480 --> 47:26.460 Professor Ben Polak: Good, so your name is? 47:26.460 --> 47:29.050 Student: Henry Professor Ben Polak: So 47:29.052 --> 47:31.902 Henry is saying once we've figured out that no one should 47:31.899 --> 47:33.779 choose a strategy bigger than 67, 47:33.780 --> 47:38.650 then we can go another step and say, if those strategies never 47:38.652 --> 47:43.132 existed, then the same argument rules out -- or a similar 47:43.126 --> 47:47.276 argument rules out -- strategies bigger than 45. 47:47.280 --> 47:48.760 Let's be careful here. 47:48.760 --> 47:54.330 The strategies that are less than 67 but bigger than 45, 47:54.332 --> 47:59.802 I think these strategies are not, they're not dominated 47:59.802 --> 48:03.452 strategies in the original game. 48:03.449 --> 48:05.819 In particular, we just argued that if everyone 48:05.820 --> 48:08.460 in the room chose a 100, then 67 would be a winning 48:08.455 --> 48:11.555 strategy. So it's not the case that the 48:11.560 --> 48:16.150 strategies between 45 and 67 are dominated strategies. 48:16.150 --> 48:21.030 But it is the case that they're dominated once we delete the 48:21.034 --> 48:25.344 dominated strategies: once we delete 67 and above. 48:25.340 --> 48:33.370 So these strategies -- let's be careful here with the word 48:33.369 --> 48:42.099 weakly here -- these strategies are not weakly dominated in the 48:42.104 --> 48:50.144 original game. But they are dominated -- 48:50.142 --> 49:03.132 they're weakly dominated -- once we delete 68 through 100. 49:03.130 --> 49:09.190 So all of the strategies 45 through 67, are gone now. 49:09.190 --> 49:11.510 So okay, let's have a look. 49:11.510 --> 49:14.920 Did anyone choose -- raise your hands, Be brave here. 49:14.920 --> 49:22.970 Did anyone choose a strategy between 45 and 67? 49:22.970 --> 49:24.990 Or between 46 and 67? 49:24.989 --> 49:26.619 No one's raising their hand, but I know some of you did 49:26.616 --> 49:29.086 because I got it in front of me, at least four of you did and I 49:29.085 --> 49:32.015 won't read out those names yet, but I might read them out next 49:32.024 --> 49:35.854 time. So four more people chose those 49:35.848 --> 49:39.208 strategies. Now notice, there's a different 49:39.205 --> 49:41.095 part of this, this argument. 49:41.099 --> 49:44.509 The argument that eliminates strategies 67 and above, 49:44.513 --> 49:47.473 or 68 upwards, that strategy just involves the 49:47.467 --> 49:50.817 first lesson of last time: do not choose a dominated 49:50.815 --> 49:54.685 strategy, admittedly weak here, but still. 49:54.690 --> 49:58.830 But the second slice, strategies 45 through 67, 49:58.834 --> 50:04.334 getting rid of those strategies involves a little bit more. 50:04.329 --> 50:08.469 You've got to put yourself in the shoes of your fellow 50:08.473 --> 50:12.303 classmen and figure out, that they're not going to 50:12.303 --> 50:14.183 choose 67 and above. 50:14.179 --> 50:16.199 So the first argument, that's a straight forward 50:16.199 --> 50:17.789 argument, the second argument says, 50:17.789 --> 50:20.859 I put myself in other peoples shoes, I realize they're not 50:20.864 --> 50:22.864 going to play a dominated strategy, 50:22.860 --> 50:24.390 and therefore, having realized they're not 50:24.389 --> 50:25.769 going to play a dominated strategy, 50:25.769 --> 50:29.839 I shouldn't play a strategy between 45 and 67. 50:29.840 --> 50:36.940 So this argument is an 'in shoes' argument. 50:36.940 --> 50:39.930 Now what? Where can we go now? 50:39.929 --> 50:44.449 Yeah, so let's have the guy in the beard, but let the mike get 50:44.452 --> 50:46.522 to him. Yell out your name. 50:46.519 --> 50:49.059 Student: You just repeat the same reasoning again and 50:49.055 --> 50:50.855 again, and you eventually get down to 1. 50:50.860 --> 50:53.220 Professor Ben Polak: We'll do that but let's go one 50:53.224 --> 50:55.054 step at a time. So now we've ruled out the 50:55.052 --> 50:57.562 possibility that anyone's going to choose a strategy 68 and 50:57.556 --> 50:59.366 above because they're weakly dominated, 50:59.369 --> 51:01.979 and we've ruled out the possibility that anyone's going 51:01.984 --> 51:03.974 to choose a strategy between 46 and 67, 51:03.969 --> 51:06.479 because those strategies are dominated, once we've ruled out 51:06.480 --> 51:07.630 the dominated strategies. 51:07.630 --> 51:10.490 So we know no one's choosing any strategies above 45., 51:10.485 --> 51:13.175 It's as if the numbers 46 and above don't exist. 51:13.179 --> 51:17.209 So we know that the highest anyone could ever choose is 45, 51:17.210 --> 51:21.310 and two-thirds of 45 is roughly … someone help me out 51:21.309 --> 51:24.019 here … 30 right: roughly 30. 51:24.019 --> 51:31.389 So we know that all the numbers between 45 and 30, 51:31.392 --> 51:37.112 these strategies were not dominated. 51:37.110 --> 51:40.540 And they weren't dominated even after deleting the dominated 51:40.535 --> 51:42.755 strategies. But they are dominated once we 51:42.761 --> 51:45.411 deleted not just the dominated strategies, but also the 51:45.410 --> 51:48.310 strategies that were dominated once we deleted the dominated 51:48.305 --> 51:50.395 strategies. I'm not going to try and write 51:50.402 --> 51:52.722 that, but you should try and write it in your notes. 51:52.719 --> 51:55.699 So without writing that argument down in detail, 51:55.695 --> 51:59.235 notice that we can rule out the strategies 30 through 45, 51:59.240 --> 52:01.900 not by just examining our own payoffs; 52:01.900 --> 52:04.360 not just by putting ourselves in other people's shoes and 52:04.364 --> 52:07.054 realizing they're not going to choose a dominated strategy; 52:07.050 --> 52:10.160 but by putting our self in other people's shoes while 52:10.159 --> 52:13.329 they're putting themselves in someone else's shoes and 52:13.328 --> 52:15.838 figuring out what they're going to do. 52:15.840 --> 52:21.240 So this is an 'in shoes', be careful where we are here, 52:21.243 --> 52:27.053 this is an 'in shoes in shoes' argument, at which point you 52:27.047 --> 52:30.447 might want to invent the sock. 52:30.450 --> 52:31.480 Now, where's this going? 52:31.480 --> 52:32.850 We were told where it's going. 52:32.849 --> 52:35.189 We're able to rule out 68 and above. 52:35.190 --> 52:37.180 Then we were able to rule out 46 and above. 52:37.179 --> 52:41.429 Now we're able to rule out 31 and above. 52:41.429 --> 52:45.779 By the next slice down we'll be able to eliminate -- what is it 52:45.777 --> 52:51.687 -- about 20 and above, so 30 down to above 20, 52:51.685 --> 53:01.235 and this will be an 'in shoes, in shoes, in shoes'. 53:01.239 --> 53:03.359 These strategies aren't dominated, nor are they 53:03.363 --> 53:05.813 dominated once you delete the dominated strategies, 53:05.809 --> 53:07.639 nor once we dominated the strategies dominated once we've 53:07.637 --> 53:08.777 deleted the dominated strategies, 53:08.780 --> 53:11.660 but they are dominated once we delete the strategies that have 53:11.664 --> 53:14.174 been dominated in the--you get what I'm doing here. 53:14.170 --> 53:16.310 So where is this argument going to go? 53:16.309 --> 53:19.469 Where's this argument going to go? 53:19.469 --> 53:22.729 It's going to go all the way down to 1: all the way down to 53:22.725 --> 53:23.805 1. We could repeat this argument 53:23.808 --> 53:24.488 all the way down to 1. 53:24.489 --> 53:27.769 Notice that once we've deleted the dominated strategies, 53:27.767 --> 53:30.987 you know I had said before about four people chose this 53:30.985 --> 53:32.735 strategy, and in here, 53:32.742 --> 53:37.492 about four people chose this strategy, but in this range 30 53:37.493 --> 53:40.583 through 45, I had lots of people. 53:40.579 --> 53:44.309 How many of you chose a number between 30 and 45? 53:44.310 --> 53:45.230 Well more than that. 53:45.230 --> 53:48.170 I can guarantee you more than that chose a number between 30 53:48.171 --> 53:50.311 and 45. In fact, the people who chose 53:50.309 --> 53:52.929 where we started off 33 chose in that range. 53:52.929 --> 53:55.679 A lot more of you chose numbers between 20 and 30, 53:55.681 --> 53:59.051 so we're really getting into the meat of the distribution. 53:59.050 --> 54:03.090 But we're seeing that these are choices, that perhaps, 54:03.089 --> 54:06.289 are ruled out by this kind of reasoning. 54:06.289 --> 54:09.449 Now, I'm still not going to quite reveal yet who won. 54:09.449 --> 54:12.819 I want to take this just one step more abstract. 54:12.820 --> 54:14.740 So I want to just discuss this a little bit more. 54:14.739 --> 54:18.599 I want to discuss the consequence of rationality in 54:18.601 --> 54:21.841 playing games, slightly philosophical for a 54:21.844 --> 54:24.654 few minutes. So I claim that if you are a 54:24.647 --> 54:27.277 rational player, by which I mean somebody who is 54:27.278 --> 54:30.748 trying to maximize their payoffs by their play of the game, 54:30.750 --> 54:37.890 that simply being rational, just being a rational player, 54:37.894 --> 54:43.894 rules out playing these dominated strategies. 54:43.889 --> 54:47.919 So the four of you who chose numbers bigger than 67, 54:47.916 --> 54:52.646 whose names I'm not going to read out, maybe they were making 54:52.654 --> 54:57.204 a mistake. However, the next slice down 54:57.201 --> 55:01.751 requires more than just rationality. 55:01.750 --> 55:05.840 What else does it require? 55:05.840 --> 55:08.250 Yes, can I get this guy again, sorry? 55:08.250 --> 55:09.850 Shout out your name again, I've forgotten it. 55:09.850 --> 55:10.690 Student: Nick. 55:10.690 --> 55:11.470 Professor Ben Polak: Shout it out. 55:11.470 --> 55:12.130 Student: Nick. 55:12.130 --> 55:12.710 Professor Ben Polak: Yep. 55:12.710 --> 55:14.430 Student: The assumption that your opponents are being 55:14.427 --> 55:14.977 rational as well. 55:14.980 --> 55:16.510 Professor Ben Polak: Good. 55:16.510 --> 55:25.370 To rule out the second slice, I need to be rational myself, 55:25.366 --> 55:32.386 and I need to know that others are rational. 55:32.389 --> 55:35.589 That's illegible, but what it says is rational 55:35.593 --> 55:38.943 and knowledge that other people are rational. 55:38.940 --> 55:41.030 Now how about the next slice after that? 55:41.030 --> 55:45.520 Well now I need to be rational, I need to know that other 55:45.517 --> 55:49.627 people are rational, and I need to know that other 55:49.631 --> 55:53.201 people know that other people are rational. 55:53.199 --> 55:56.259 So to get this slice, this next slice here, 55:56.260 --> 55:58.750 I need rationality; as some of you know that's 55:58.747 --> 56:00.697 widely criticized in the social sciences these days. 56:00.699 --> 56:02.999 Are we right to assume that people are rational? 56:03.000 --> 56:08.530 To get this slice I need rationality, I need knowledge of 56:08.531 --> 56:13.871 rationality, let's call that KR and I need knowledge of 56:13.865 --> 56:16.725 knowledge of rationality. 56:16.730 --> 56:19.240 As I go down further, I'm going to need rationality, 56:19.239 --> 56:21.059 I need to know people are rational; 56:21.059 --> 56:24.469 I need to know that people know that people are rational, 56:24.465 --> 56:27.865 and I need to know that people know that people know that 56:27.870 --> 56:29.330 people are rational. 56:29.329 --> 56:31.529 Now let's just make this more concrete for you. 56:31.530 --> 56:38.080 These people, the four people who chose this, 56:38.076 --> 56:41.346 they made a mistake. 56:41.349 --> 56:45.259 What about the four people who chose numbers between 45 and 67? 56:45.260 --> 56:47.390 What can we conclude about those people? 56:47.389 --> 56:50.489 The people who chose between 45 and 67? 56:50.490 --> 56:52.280 Should I read out their names? 56:52.280 --> 56:55.150 No, I won't, perhaps I better not. 56:55.150 --> 56:56.970 What can we conclude about these people? 56:56.970 --> 56:58.500 Yeah. We're never going to get the 56:58.504 --> 56:59.944 mike to this -- try and get the mike in there. 56:59.940 --> 57:02.580 Come forward as far as you can and then really shout, 57:02.584 --> 57:03.704 yep. Student: They think 57:03.704 --> 57:04.774 their classmates are pretty dumb. 57:04.769 --> 57:05.949 Professor Ben Polak: Right, right. 57:05.949 --> 57:10.919 It's not necessarily that the four people who chose between 46 57:10.919 --> 57:15.969 and 67 are themselves "thick," it's that they think the rest of 57:15.970 --> 57:19.150 you are "thick." Down here, this doesn't require 57:19.153 --> 57:21.663 people to be thick, or to think the rest of you are 57:21.659 --> 57:23.509 thick, they're just people who think 57:23.507 --> 57:25.677 that you think, sorry, they're just people who 57:25.675 --> 57:28.225 think that you think that they're thick and so on. 57:28.230 --> 57:30.920 But again, all the way to 1 we're going to need very, 57:30.921 --> 57:33.151 very many rounds of knowledge, of knowledge, 57:33.147 --> 57:35.267 of knowledge … of rationality. 57:35.269 --> 57:38.409 Does anyone know what we call it if we assume an infinite 57:38.410 --> 57:41.660 sequence of "I know that you know that I know that you know 57:41.662 --> 57:45.422 that I know that you know that I know that you know" something? 57:45.420 --> 57:47.820 What's the expression for that? 57:47.820 --> 57:50.760 Believe it or not, technical expression. 57:50.760 --> 57:55.970 The technical expression of that in philosophy is common 57:55.968 --> 58:01.838 knowledge, which I can never spell, so I'm going to wing it. 58:01.840 --> 58:03.690 Common knowledge is: "I know something, 58:03.686 --> 58:05.436 you know it, you know that I know it, 58:05.436 --> 58:07.906 I know that you know it, I know that you know that I 58:07.914 --> 58:09.134 know it, etc., etc. 58:09.130 --> 58:13.660 etc.: an infinite sequence. 58:13.659 --> 58:19.369 But if we had common knowledge of rationality in this class, 58:19.370 --> 58:23.630 then the optimal choice would have been 1. 58:23.630 --> 58:26.890 How many of you chose 1? 58:26.890 --> 58:27.390 Look around the room. 58:27.390 --> 58:28.180 Let's just pan the room. 58:28.180 --> 58:29.300 Keep your hands up a second. 58:29.300 --> 58:32.500 How many of you chose 1? 58:32.500 --> 58:34.210 So actually a lot of you chose 1. 58:34.210 --> 58:36.150 1 was the modal answer in this class. 58:36.150 --> 58:39.080 A lot of you chose 1. 58:39.080 --> 58:40.260 So those people did pretty well. 58:40.260 --> 58:45.200 They must have done--they must be thinking they're about to 58:45.202 --> 58:48.102 win… but they didn't win. 58:48.099 --> 58:51.949 So it turns out that the average in this class, 58:51.947 --> 58:57.047 the average choice was about 13 1/3, which means two-thirds of 58:57.049 --> 58:58.889 the average was 9. 58:58.889 --> 59:02.609 Two-thirds of the average was 9 and some of you chose 9, 59:02.612 --> 59:04.712 so if you are here, stand up. 59:04.710 --> 59:07.230 The following people chose 9, that's not right, 59:07.225 --> 59:09.135 where are the people who chose 9? 59:09.140 --> 59:10.780 I've got them here somewhere? 59:10.780 --> 59:11.940 I'm sorry there's so many pages of people. 59:11.940 --> 59:13.730 Here we go. The following people chose 9. 59:13.730 --> 59:17.350 So stand up if you're here and if you're that person's roommate 59:17.346 --> 59:18.626 if they're not here. 59:18.630 --> 59:20.940 So Leesing Chang: is Leesing Chang here? 59:20.940 --> 59:22.810 Stand up if you're here. 59:22.809 --> 59:24.179 A G. Christopher Berrera: 59:24.184 --> 59:26.194 you can stand up, if you're here. 59:26.190 --> 59:30.450 And William Fischel: are you here? 59:30.450 --> 59:31.260 I don't know if he is here. 59:31.260 --> 59:32.840 Jed Glickstein: are you here? 59:32.840 --> 59:35.170 Jed Glickstein: stand up if you're here. 59:35.170 --> 59:38.500 And Jeffrey Green: stand up if you're here. 59:38.500 --> 59:42.430 And Allison Hoyt: stand up if you're here. 59:42.430 --> 59:44.410 No Allison Hoyt, okay. 59:44.410 --> 59:48.750 There's John Robinson. 59:48.750 --> 59:52.710 All right so these people, stay up a second so the camera 59:52.710 --> 59:54.660 can see you. There you go, 59:54.661 --> 59:56.141 all the way around. 59:56.140 --> 59:59.420 Wave. Wave to mom at home. 59:59.420 --> 1:00:01.070 Can we get a round of applause for our winners? 1:00:01.070 --> 1:00:09.630 1:00:09.630 --> 1:00:12.880 So Jude has trustworthily brought back the five dollars. 1:00:12.880 --> 1:00:14.640 I've got to focus for a second just to get it. 1:00:14.639 --> 1:00:16.719 Here is the five dollars, we're going to tear this into 1:00:16.723 --> 1:00:18.923 nine pieces, except I'd get arrested and deported if I did 1:00:18.922 --> 1:00:20.732 that, so we're going to find a way to 1:00:20.728 --> 1:00:22.138 break this into change later. 1:00:22.139 --> 1:00:26.839 Come and claim it afterwards, but you're all entitled to 1:00:26.840 --> 1:00:30.860 whatever a ninth, whatever that fraction of five 1:00:30.857 --> 1:00:34.607 dollars is. Okay, so why was it after all 1:00:34.610 --> 1:00:39.550 that work -- why was it that 1 wasn't the winning answer? 1:00:39.550 --> 1:00:41.070 Why wasn't 1 the winning answer? 1:00:41.070 --> 1:00:42.320 Let's have someone we haven't had before. 1:00:42.320 --> 1:00:45.430 Can we get the mike in way in the back there? 1:00:45.429 --> 1:00:48.329 Can we get the mike in there on the row you're on? 1:00:48.330 --> 1:00:48.930 See if you can point. 1:00:48.930 --> 1:00:50.680 Actually good. Stand up. 1:00:50.680 --> 1:00:53.600 Shout. Shout away. 1:00:53.599 --> 1:00:54.679 Student: 1 would have been the winning answer 1:00:54.677 --> 1:00:55.477 [inaudible] Professor Ben Polak: 1:00:55.479 --> 1:00:56.049 Louder, louder, louder. 1:00:56.050 --> 1:00:59.690 Student: 1 would have been the winning answer had 1:00:59.694 --> 1:01:02.874 everyone assumed that the average would have been 1:01:02.874 --> 1:01:07.514 constantly compounded down to 1, but since a couple of people 1:01:07.511 --> 1:01:11.851 chose the, I mean not incorrect answers, but the higher 1:01:11.854 --> 1:01:15.424 averages, then it was pushed up to 13. 1:01:15.420 --> 1:01:16.920 Professor Ben Polak: Right, so to get all the way, 1:01:16.923 --> 1:01:18.453 -- good -- so to get all the way -- thank you -- So to get 1:01:18.453 --> 1:01:19.423 all the way to 1, we need a lot. 1:01:19.420 --> 1:01:21.120 We need not just that you're all rational players, 1:01:21.122 --> 1:01:22.652 not just that you know each other's rational, 1:01:22.650 --> 1:01:24.110 but you know everyone else's rational. 1:01:24.110 --> 1:01:25.910 I mean I know you all know each other because you've met at 1:01:25.912 --> 1:01:27.722 Yale, but you also know each other well enough to know that 1:01:27.715 --> 1:01:28.985 not everyone in the room is rational, 1:01:28.989 --> 1:01:30.979 and you're pretty sure that not everyone knows that you're 1:01:30.981 --> 1:01:32.171 rational and so on and so forth. 1:01:32.170 --> 1:01:34.730 It's asking a lot to get to 1 here, and in fact, 1:01:34.734 --> 1:01:35.884 we didn't get to 1. 1:01:35.880 --> 1:01:39.580 In previous years we were even higher, so this was low this 1:01:39.583 --> 1:01:42.093 year. In 2003, the average was 1:01:42.088 --> 1:01:43.888 eighteen and a half. 1:01:43.889 --> 1:01:45.879 And in 2004, it was twenty-one and a half. 1:01:45.880 --> 1:01:47.820 And in 2005, we had a class that didn't 1:01:47.818 --> 1:01:50.618 trust each other at all I guess, because the average was 1:01:50.623 --> 1:01:52.053 twenty-three. And this year, 1:01:52.053 --> 1:01:53.223 it was thirteen and a third. 1:01:53.219 --> 1:01:56.369 We're getting better there I think. 1:01:56.369 --> 1:01:58.549 One nice thing, by the way -- this is just 1:01:58.545 --> 1:02:01.725 chance I think -- the median answer in the class was nine, 1:02:01.730 --> 1:02:05.270 which is spot on, so the median hit this bang on. 1:02:05.269 --> 1:02:08.879 Now what I wanted you to do, is I want you all to play 1:02:08.875 --> 1:02:10.645 again. We haven't got time to do this 1:02:10.650 --> 1:02:12.660 properly, even though I've given you the sheets. 1:02:12.659 --> 1:02:18.409 So write down -- don't tell this to your neighbors -- write 1:02:18.412 --> 1:02:20.802 down a number. Don't talk among yourselves 1:02:20.797 --> 1:02:25.017 that's cheating. Write down a number. 1:02:25.019 --> 1:02:26.269 If you haven't got a sheet in front of you, 1:02:26.268 --> 1:02:27.218 just write it on your notepad. 1:02:27.220 --> 1:02:28.960 Write down a number. 1:02:28.960 --> 1:02:30.960 Has everyone written down a number? 1:02:30.960 --> 1:02:33.100 I'm going to do a show of hands now. 1:02:33.099 --> 1:02:38.829 How many -- we'll get the camera on you -- how many of you 1:02:38.832 --> 1:02:42.052 chose a number higher than 67? 1:02:42.050 --> 1:02:44.120 Oh there's some spoil makers in the class. 1:02:44.119 --> 1:02:48.959 How many of you chose a number higher than 20? 1:02:48.960 --> 1:02:53.860 How many of you chose a number higher than 10? 1:02:53.860 --> 1:02:58.260 How many chose a number between 5 and 10? 1:02:58.260 --> 1:03:04.380 How many chose a number between 0 -- I'm sorry -- between 1 and 1:03:04.375 --> 1:03:05.685 5? How many of you, 1:03:05.686 --> 1:03:09.006 excluding the people who chose 1 last time, how many of you 1:03:09.008 --> 1:03:12.388 chose a number that was lower than the number you chose last 1:03:12.387 --> 1:03:15.517 time? Now keep your hands up a second. 1:03:15.520 --> 1:03:18.750 So almost all of you came down. 1:03:18.750 --> 1:03:21.530 Why? Why are seeing this massive 1:03:21.527 --> 1:03:23.977 contraction? I'm guessing the average number 1:03:23.979 --> 1:03:25.949 in the class now is probably about 3 or 4, 1:03:25.949 --> 1:03:26.909 maybe even lower. 1:03:26.909 --> 1:03:29.969 Why are we seeing this massive contraction in the numbers being 1:03:29.966 --> 1:03:31.776 chosen? The woman in green, 1:03:31.784 --> 1:03:34.464 I've forgotten your name, I'm sorry? 1:03:34.460 --> 1:03:36.930 Student: Because we've just sat in lecture and you've 1:03:36.934 --> 1:03:39.454 told us we're not being rational if we pick a high number. 1:03:39.449 --> 1:03:41.859 Professor Ben Polak: So part of it is, 1:03:41.863 --> 1:03:44.283 you yourselves have figured out, some of you, 1:03:44.276 --> 1:03:46.686 that you shouldn't choose a high number. 1:03:46.690 --> 1:03:48.530 What else though? 1:03:48.530 --> 1:03:50.120 What else is going on here? 1:03:50.120 --> 1:03:52.480 Let's get somebody. 1:03:52.480 --> 1:03:54.670 There's a guy waving an arm out there. 1:03:54.670 --> 1:03:56.140 Do you want to stand up behind the hat? 1:03:56.140 --> 1:04:00.750 You. Student: Because we've 1:04:00.754 --> 1:04:01.574 repeated the game. 1:04:01.570 --> 1:04:03.110 Professor Ben Polak: It's true we've repeated it. 1:04:03.110 --> 1:04:05.610 It's true we repeated it but what is it about repeating it? 1:04:05.610 --> 1:04:08.140 What is it about talking about this game that makes a 1:04:08.137 --> 1:04:09.967 difference? Let me hazard a guess here. 1:04:09.969 --> 1:04:13.089 I think what makes a difference is not only do you, 1:04:13.092 --> 1:04:16.342 yourselves, know better how to play this game now, 1:04:16.340 --> 1:04:20.110 but you also know that everybody around you knows 1:04:20.112 --> 1:04:22.472 better how to play the game. 1:04:22.469 --> 1:04:26.229 Discussing this game raised not just each person's 1:04:26.232 --> 1:04:29.462 sophistication, but it raised what you know 1:04:29.458 --> 1:04:32.528 about other people's sophistication, 1:04:32.530 --> 1:04:35.960 and you know that other people now know that you understand how 1:04:35.959 --> 1:04:37.009 to play the game. 1:04:37.010 --> 1:04:40.160 So the main lesson I want you to get from this is that not 1:04:40.159 --> 1:04:43.249 only did it matter that you need to put yourself in other 1:04:43.254 --> 1:04:46.464 people's shoes and think about what their payoffs are. 1:04:46.460 --> 1:04:50.780 You also need to put yourself into other people's shoes and 1:04:50.780 --> 1:04:55.100 think about how sophisticated are they at playing games. 1:04:55.099 --> 1:04:58.889 And you need to think about how sophisticated do they think you 1:04:58.892 --> 1:05:00.302 are at playing games. 1:05:00.300 --> 1:05:03.380 And you need to think about how sophisticated do they think that 1:05:03.379 --> 1:05:05.969 you think that they are at playing games and so on. 1:05:05.969 --> 1:05:09.289 This level of knowledge, these layers of knowledge, 1:05:09.290 --> 1:05:12.080 lead to very different play in the game. 1:05:12.079 --> 1:05:15.939 And to make this more concrete, if a firm is competing against 1:05:15.939 --> 1:05:18.279 a competitor it can be pretty sure, 1:05:18.280 --> 1:05:21.440 that competitor is a pretty sophisticated game player and 1:05:21.442 --> 1:05:23.252 knows that the firm is itself. 1:05:23.250 --> 1:05:26.820 If a firm is competing against a customer -- let's say for a 1:05:26.823 --> 1:05:30.763 non-prime loan -- perhaps that assumption is not quite so safe. 1:05:30.760 --> 1:05:33.990 It matters in how we take games through to the real world, 1:05:33.991 --> 1:05:37.451 and we're going to see more of this as the term progresses. 1:05:37.449 --> 1:05:41.549 Now I've got five minutes, do I have five minutes left? 1:05:41.550 --> 1:05:45.130 So I've got five minutes to take a little small aside here. 1:05:45.130 --> 1:05:48.310 We've been talking about knowledge and about common 1:05:48.307 --> 1:05:50.627 knowledge. I just want to do a very quick 1:05:50.630 --> 1:05:52.920 experiment, so everyone stay in their seat. 1:05:52.920 --> 1:05:56.990 I'm going to get two T.A.'s up here, why don't I get Ale and 1:05:56.993 --> 1:05:59.653 Kaj up here. And I wanted to show that 1:05:59.646 --> 1:06:02.946 common knowledge is not such an obvious a concept, 1:06:02.953 --> 1:06:05.453 as I've made it seem on the board. 1:06:05.450 --> 1:06:07.520 Come up on the stage a second. 1:06:07.520 --> 1:06:10.200 You can leave the mike its okay. 1:06:10.199 --> 1:06:13.349 Here we have two of our T.A.'s, actually these are the two head 1:06:13.345 --> 1:06:16.335 T.A.'s, and I want you to face forward so you don't see what 1:06:16.339 --> 1:06:22.219 I'm doing. I'm about to put on their heads 1:06:22.222 --> 1:06:25.692 a hat. Here's a hat on Ale's head, 1:06:25.687 --> 1:06:28.407 and here's a hat on Kaj's head. 1:06:28.409 --> 1:06:31.609 Let's move them this way so they're in focus. 1:06:31.610 --> 1:06:36.070 Now you can all see these hats, and if they turn around to each 1:06:36.072 --> 1:06:38.882 other, they can see each other's hat. 1:06:38.880 --> 1:06:41.830 Now I want to ask you the question here. 1:06:41.829 --> 1:06:47.629 Here is a fact, so is it common knowledge that 1:06:47.627 --> 1:06:55.607 -- is it common knowledge that at least one of these people has 1:06:55.614 --> 1:06:59.354 a pink hat on their head? 1:06:59.350 --> 1:07:01.470 Is it common knowledge? 1:07:01.469 --> 1:07:04.169 So I claim it's not common knowledge. 1:07:04.170 --> 1:07:06.000 What is known here? 1:07:06.000 --> 1:07:11.410 Well I'll reveal the facts now: that in fact Ale knows that Kaj 1:07:11.409 --> 1:07:13.939 has a pink hat on his head. 1:07:13.940 --> 1:07:16.400 So it's true that Ale knows that at least one person in the 1:07:16.401 --> 1:07:17.931 room has a pink hat on their head. 1:07:17.929 --> 1:07:21.329 And it's true that Kaj knows that Ale has a pink hat on his 1:07:21.329 --> 1:07:23.409 head. They both look absurd, 1:07:23.407 --> 1:07:27.397 but never mind. But notice that Ale doesn't 1:07:27.404 --> 1:07:31.844 know the color of the hat on his own head. 1:07:31.840 --> 1:07:35.760 So even though both people know, even though it is mutual 1:07:35.764 --> 1:07:39.904 knowledge that there's at least one pink hat in the room, 1:07:39.900 --> 1:07:43.100 Ale doesn't know what Kaj is seeing. 1:07:43.099 --> 1:07:47.189 So Ale does not know that Kaj knows that there's a pink hat in 1:07:47.192 --> 1:07:49.572 the room. In fact, from Ale's point of 1:07:49.572 --> 1:07:51.452 view, this could be a blue hat. 1:07:51.449 --> 1:07:55.309 So again, they both know that someone in the room has a pink 1:07:55.312 --> 1:07:58.262 hat on their head: it is mutual knowledge that 1:07:58.258 --> 1:08:00.548 there's a pink hat in the room. 1:08:00.550 --> 1:08:05.380 But Ale does not know that Kaj knows that he is wearing a blue, 1:08:05.376 --> 1:08:09.966 a pink hat, and Kaj does not know that Ale knows that Kaj is 1:08:09.969 --> 1:08:11.759 wearing a pink hat. 1:08:11.760 --> 1:08:15.870 Each of their hats -- each of their own hats -- might be blue. 1:08:15.869 --> 1:08:19.059 So notice that common knowledge -- thanks guys -- common 1:08:19.057 --> 1:08:21.837 knowledge is a rather subtle thing, thank you. 1:08:21.840 --> 1:08:23.430 Common knowledge is a subtle thing. 1:08:23.430 --> 1:08:26.660 Mutual knowledge doesn't imply common knowledge. 1:08:26.659 --> 1:08:29.489 Common knowledge is a statement about not just what I know. 1:08:29.489 --> 1:08:32.129 It's about what do I know the other person knows that I know 1:08:32.129 --> 1:08:34.589 that the other person …and so on and so forth. 1:08:34.590 --> 1:08:37.030 Even in this simple example, while you might think it's 1:08:37.029 --> 1:08:39.649 obviously common knowledge, it wasn't common knowledge that 1:08:39.648 --> 1:08:41.318 there was a pink hat in the room. 1:08:41.319 --> 1:08:44.389 Does anybody have smaller siblings or children of their 1:08:44.389 --> 1:08:45.999 own. They can have a pink hat at the 1:08:45.995 --> 1:08:46.725 end of the class? 1:08:46.730 --> 1:08:49.000 We'll see you on Wednesday.