WEBVTT 00:00:00.000 --> 00:00:04.348 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:04.348 --> 00:00:10.420 align:middle line:90% 00:00:10.420 --> 00:00:12.922 align:middle line:84% IAN BALL: So I know last class was quite theoretical. 00:00:12.922 --> 00:00:14.380 align:middle line:84% It was kind of on the mathier side. 00:00:14.380 --> 00:00:16.225 align:middle line:84% So I hope people who like math enjoyed that. 00:00:16.225 --> 00:00:17.600 align:middle line:84% Today, the lecture is going to be 00:00:17.600 --> 00:00:20.005 align:middle line:84% a bit more on the economic side, so a little less math. 00:00:20.005 --> 00:00:21.380 align:middle line:84% I mean, there'll be some algebra, 00:00:21.380 --> 00:00:23.780 align:middle line:84% but it's a little more applied, a little more practical. 00:00:23.780 --> 00:00:26.140 align:middle line:84% So today we're going to talk about cartels. 00:00:26.140 --> 00:00:29.360 align:middle line:90% 00:00:29.360 --> 00:00:33.580 align:middle line:84% And I think it's helpful to start with some examples. 00:00:33.580 --> 00:00:37.480 align:middle line:84% So the classic example of a cartel would be OPEC. 00:00:37.480 --> 00:00:40.800 align:middle line:84% OPEC is the Organization of Petroleum Exporting Countries. 00:00:40.800 --> 00:00:45.560 align:middle line:84% And what OPEC does is they get together, 00:00:45.560 --> 00:00:49.200 align:middle line:84% and they try to coordinate the production of oil 00:00:49.200 --> 00:00:51.440 align:middle line:84% to ensure that oil prices are high, 00:00:51.440 --> 00:00:54.663 align:middle line:84% and therefore, that all these countries can make more money. 00:00:54.663 --> 00:00:56.080 align:middle line:84% So the general idea of the cartels 00:00:56.080 --> 00:01:00.280 align:middle line:84% we're going to look at today, maybe I'll say the goal-- 00:01:00.280 --> 00:01:02.020 align:middle line:84% there's many different kinds of cartels. 00:01:02.020 --> 00:01:04.478 align:middle line:84% So I'm going to talk about the goal of these specific kinds 00:01:04.478 --> 00:01:05.519 align:middle line:90% of cartel-- 00:01:05.519 --> 00:01:16.800 align:middle line:84% is generally to coordinate on lower production 00:01:16.800 --> 00:01:18.700 align:middle line:90% in order to raise market prices. 00:01:18.700 --> 00:01:29.480 align:middle line:90% 00:01:29.480 --> 00:01:32.760 align:middle line:84% Do people know any other examples of cartels 00:01:32.760 --> 00:01:38.840 align:middle line:84% that maybe you've come across in your experience 00:01:38.840 --> 00:01:42.020 align:middle line:84% or in other econ classes, or just things you've heard of? 00:01:42.020 --> 00:01:50.920 align:middle line:90% 00:01:50.920 --> 00:01:52.220 align:middle line:90% Any ideas? 00:01:52.220 --> 00:01:52.720 align:middle line:90% Yeah? 00:01:52.720 --> 00:01:53.700 align:middle line:90% STUDENT: Are drug cartels-- 00:01:53.700 --> 00:01:54.658 align:middle line:90% IAN BALL: Drug cartels. 00:01:54.658 --> 00:01:56.400 align:middle line:90% Great example. 00:01:56.400 --> 00:02:05.040 align:middle line:84% So drug cartels often maybe interact a little bit 00:02:05.040 --> 00:02:05.540 align:middle line:90% differently. 00:02:05.540 --> 00:02:07.040 align:middle line:84% But they're certainly-- drug cartels 00:02:07.040 --> 00:02:10.680 align:middle line:84% are interested in controlling what's on the market 00:02:10.680 --> 00:02:13.060 align:middle line:84% and controlling how much revenue they get. 00:02:13.060 --> 00:02:15.320 align:middle line:84% Often, drug cartels maybe operate-- 00:02:15.320 --> 00:02:18.040 align:middle line:84% maybe the way that they limit production or limit competition 00:02:18.040 --> 00:02:20.060 align:middle line:84% is often they segment the market. 00:02:20.060 --> 00:02:21.700 align:middle line:84% So it's common for one drug cartel 00:02:21.700 --> 00:02:24.200 align:middle line:84% to, say, operate in the north of the country, another cartel 00:02:24.200 --> 00:02:25.825 align:middle line:84% to operate in the south of the country. 00:02:25.825 --> 00:02:28.280 align:middle line:84% And they have this agreement that only one cartel is going 00:02:28.280 --> 00:02:29.740 align:middle line:84% to produce in one side of the country, 00:02:29.740 --> 00:02:31.900 align:middle line:84% and only the other cartel will produce in the other side. 00:02:31.900 --> 00:02:33.710 align:middle line:84% And that's one way of reducing production 00:02:33.710 --> 00:02:35.960 align:middle line:84% because you only have one side selling in each market. 00:02:35.960 --> 00:02:36.460 align:middle line:90% Yes? 00:02:36.460 --> 00:02:38.053 align:middle line:84% STUDENT: What if a new group comes in? 00:02:38.053 --> 00:02:38.720 align:middle line:90% IAN BALL: Right. 00:02:38.720 --> 00:02:41.480 align:middle line:90% That can often lead to violence. 00:02:41.480 --> 00:02:43.660 align:middle line:84% And even without new groups coming in, 00:02:43.660 --> 00:02:45.100 align:middle line:90% there's often a fight over turf. 00:02:45.100 --> 00:02:46.580 align:middle line:90% So I like to use this example. 00:02:46.580 --> 00:02:48.140 align:middle line:84% This is a good example of a cartel. 00:02:48.140 --> 00:02:49.900 align:middle line:84% It's not really going to fit in with our models 00:02:49.900 --> 00:02:52.525 align:middle line:84% because our models aren't going to have the threat of violence. 00:02:52.525 --> 00:02:55.000 align:middle line:84% So I think this is an important economic question, how 00:02:55.000 --> 00:02:58.142 align:middle line:84% criminal organizations use violence to sustain cooperation. 00:02:58.142 --> 00:02:59.600 align:middle line:84% But we're going to think of cartels 00:02:59.600 --> 00:03:02.380 align:middle line:84% that threaten to produce more of the good, 00:03:02.380 --> 00:03:04.340 align:middle line:90% not to kill the other cartels. 00:03:04.340 --> 00:03:07.120 align:middle line:84% So this has violence, which will not really 00:03:07.120 --> 00:03:09.440 align:middle line:84% be covered in our models, but it's 00:03:09.440 --> 00:03:10.880 align:middle line:90% a great example of a cartel. 00:03:10.880 --> 00:03:15.220 align:middle line:84% Other examples of cartels that people have come across? 00:03:15.220 --> 00:03:17.920 align:middle line:90% 00:03:17.920 --> 00:03:22.040 align:middle line:84% So there's a famous one, the famous vitamin cartel. 00:03:22.040 --> 00:03:24.280 align:middle line:84% In the US, these producers of-- it's 00:03:24.280 --> 00:03:28.560 align:middle line:84% maybe not the sexiest industry, but these firms that 00:03:28.560 --> 00:03:31.120 align:middle line:84% produced various vitamins, they colluded 00:03:31.120 --> 00:03:32.863 align:middle line:90% to try to keep prices low. 00:03:32.863 --> 00:03:34.280 align:middle line:84% I think they also split up exactly 00:03:34.280 --> 00:03:37.120 align:middle line:84% which vitamins they each produced. 00:03:37.120 --> 00:03:40.240 align:middle line:84% One that I think maybe people don't think of as much 00:03:40.240 --> 00:03:45.440 align:middle line:84% that I think is a nice example is actually 00:03:45.440 --> 00:03:46.860 align:middle line:90% the production of doctors. 00:03:46.860 --> 00:03:50.105 align:middle line:90% 00:03:50.105 --> 00:03:52.480 align:middle line:84% So I don't know if any of you are applying to med school. 00:03:52.480 --> 00:03:56.760 align:middle line:84% The number of slots in the entering classes at med schools 00:03:56.760 --> 00:03:58.760 align:middle line:90% is very restricted. 00:03:58.760 --> 00:04:02.760 align:middle line:84% Medical schools cannot enroll more students than 00:04:02.760 --> 00:04:06.280 align:middle line:84% they've agreed to with various organizations, like the AMA. 00:04:06.280 --> 00:04:10.960 align:middle line:84% Why do you think doctors have reached this agreement that med 00:04:10.960 --> 00:04:13.728 align:middle line:84% schools cannot enroll too many students in the United States? 00:04:13.728 --> 00:04:15.020 align:middle line:90% Why might they want to do that? 00:04:15.020 --> 00:04:15.500 align:middle line:90% Yeah? 00:04:15.500 --> 00:04:16.800 align:middle line:84% STUDENT: To be able to [INAUDIBLE] salaries. 00:04:16.800 --> 00:04:17.550 align:middle line:90% IAN BALL: Exactly. 00:04:17.550 --> 00:04:19.560 align:middle line:90% So it's exactly the same idea. 00:04:19.560 --> 00:04:22.880 align:middle line:84% OPEC wants to underproduce oil so that the price of oil 00:04:22.880 --> 00:04:23.960 align:middle line:90% is high. 00:04:23.960 --> 00:04:25.733 align:middle line:90% How do we produce doctors? 00:04:25.733 --> 00:04:27.900 align:middle line:84% Doctors are also something that we produce and sell. 00:04:27.900 --> 00:04:30.220 align:middle line:84% I mean, it's a bit weird to think about selling your labor, 00:04:30.220 --> 00:04:31.220 align:middle line:90% but that's what's happening. 00:04:31.220 --> 00:04:33.240 align:middle line:84% Well, the production of doctors is med school. 00:04:33.240 --> 00:04:35.365 align:middle line:84% And if medical schools reduce the number of doctors 00:04:35.365 --> 00:04:37.350 align:middle line:84% they produce, then the price you can 00:04:37.350 --> 00:04:39.600 align:middle line:84% sell a doctor for, that is, the wage that doctors get, 00:04:39.600 --> 00:04:40.500 align:middle line:90% will be higher. 00:04:40.500 --> 00:04:42.780 align:middle line:84% And that's exactly what we see here. 00:04:42.780 --> 00:04:45.520 align:middle line:84% And classical guilds also fall into this. 00:04:45.520 --> 00:04:48.960 align:middle line:84% Maybe some people would say that the structure of PhD programs 00:04:48.960 --> 00:04:51.680 align:middle line:84% also fits into an academic cartel. 00:04:51.680 --> 00:04:54.840 align:middle line:84% And then maybe one last example I'll 00:04:54.840 --> 00:04:58.040 align:middle line:84% give that also doesn't quite fit with our model today 00:04:58.040 --> 00:05:02.360 align:middle line:84% is actually the Quebec maple syrup cartel. 00:05:02.360 --> 00:05:04.098 align:middle line:84% Probably not well known to people. 00:05:04.098 --> 00:05:05.640 align:middle line:84% This one is different because this is 00:05:05.640 --> 00:05:07.640 align:middle line:90% a government-sponsored cartel. 00:05:07.640 --> 00:05:10.160 align:middle line:84% So the government of Quebec and the government of Canada 00:05:10.160 --> 00:05:14.680 align:middle line:84% explicitly allow this cartel because they 00:05:14.680 --> 00:05:21.280 align:middle line:84% think that producing maple syrup is a valuable cultural practice 00:05:21.280 --> 00:05:22.920 align:middle line:84% in Canada, and they want to sustain 00:05:22.920 --> 00:05:25.940 align:middle line:84% the profits and the revenues of producers of maple syrup. 00:05:25.940 --> 00:05:31.620 align:middle line:84% But because this is explicit, it's contractual, 00:05:31.620 --> 00:05:33.460 align:middle line:84% it has explicit government support, 00:05:33.460 --> 00:05:36.960 align:middle line:84% this also won't really fit within our models today. 00:05:36.960 --> 00:05:38.880 align:middle line:84% So we're going to think about models 00:05:38.880 --> 00:05:42.440 align:middle line:84% that have no violence, and also no explicit government 00:05:42.440 --> 00:05:43.060 align:middle line:90% endorsement. 00:05:43.060 --> 00:05:44.980 align:middle line:90% So these are implicit cartels. 00:05:44.980 --> 00:05:47.320 align:middle line:84% And I'll say more about what that means. 00:05:47.320 --> 00:05:49.045 align:middle line:84% So the goal of these cartels tells 00:05:49.045 --> 00:05:50.920 align:middle line:84% is to coordinate on lower production in order 00:05:50.920 --> 00:05:52.180 align:middle line:90% to raise the market prices. 00:05:52.180 --> 00:05:55.240 align:middle line:90% What is the challenge? 00:05:55.240 --> 00:05:57.240 align:middle line:84% Suppose you're an oil-producing nation. 00:05:57.240 --> 00:05:59.400 align:middle line:84% You get together with the other oil producers, 00:05:59.400 --> 00:06:00.852 align:middle line:90% and you propose a plan. 00:06:00.852 --> 00:06:02.560 align:middle line:84% We're each going to produce less in order 00:06:02.560 --> 00:06:03.960 align:middle line:90% to keep the market price high. 00:06:03.960 --> 00:06:04.940 align:middle line:90% That sounds great. 00:06:04.940 --> 00:06:06.060 align:middle line:90% Everyone does better. 00:06:06.060 --> 00:06:06.938 align:middle line:90% The prices are high. 00:06:06.938 --> 00:06:07.980 align:middle line:90% That's good for everyone. 00:06:07.980 --> 00:06:09.063 align:middle line:90% What's the challenge here? 00:06:09.063 --> 00:06:10.500 align:middle line:90% Why might this not be so easy? 00:06:10.500 --> 00:06:11.000 align:middle line:90% Yeah? 00:06:11.000 --> 00:06:11.960 align:middle line:90% STUDENT: Defectors. 00:06:11.960 --> 00:06:12.793 align:middle line:90% IAN BALL: Defectors. 00:06:12.793 --> 00:06:14.560 align:middle line:90% And how might they defect? 00:06:14.560 --> 00:06:16.920 align:middle line:84% STUDENT: They start selling more than they agreed to 00:06:16.920 --> 00:06:19.660 align:middle line:84% because the price is high, so they make more profit. 00:06:19.660 --> 00:06:20.680 align:middle line:90% IAN BALL: Exactly right. 00:06:20.680 --> 00:06:22.860 align:middle line:84% So collectively, this is good for us. 00:06:22.860 --> 00:06:25.160 align:middle line:84% But the problem is that each firm or each country 00:06:25.160 --> 00:06:26.680 align:middle line:84% will have an individual incentive 00:06:26.680 --> 00:06:28.720 align:middle line:90% to deviate by overproducing. 00:06:28.720 --> 00:06:39.360 align:middle line:84% So the challenge is that there's an individual incentive 00:06:39.360 --> 00:06:41.987 align:middle line:90% to overproduce. 00:06:41.987 --> 00:06:44.320 align:middle line:84% And we can frame this actually in the classical economic 00:06:44.320 --> 00:06:45.880 align:middle line:90% language of externalities. 00:06:45.880 --> 00:06:49.360 align:middle line:84% If a single firm unilaterally deviates and overproduces, 00:06:49.360 --> 00:06:51.680 align:middle line:84% they impose an externality on the other firms. 00:06:51.680 --> 00:06:54.200 align:middle line:84% What is the externality that I impose on the other firms 00:06:54.200 --> 00:06:56.640 align:middle line:90% if I produce more? 00:06:56.640 --> 00:06:59.680 align:middle line:84% How do I affect the other firms when I produce more? 00:06:59.680 --> 00:07:00.200 align:middle line:90% Yeah? 00:07:00.200 --> 00:07:01.320 align:middle line:84% STUDENT: Your increased production 00:07:01.320 --> 00:07:03.330 align:middle line:84% leads to a lower price than what they can sell. 00:07:03.330 --> 00:07:04.080 align:middle line:90% IAN BALL: Exactly. 00:07:04.080 --> 00:07:06.760 align:middle line:84% So when I raise my production, I only 00:07:06.760 --> 00:07:09.247 align:middle line:84% internalize the effect it has on my profits. 00:07:09.247 --> 00:07:11.580 align:middle line:84% But I'm actually hurting the other members of the cartel 00:07:11.580 --> 00:07:14.560 align:middle line:84% because by overproducing I'm lowering the market price, 00:07:14.560 --> 00:07:16.160 align:middle line:90% and other people feel that. 00:07:16.160 --> 00:07:19.377 align:middle line:84% So this is a classic issue of people 00:07:19.377 --> 00:07:20.960 align:middle line:84% don't internalize their externalities, 00:07:20.960 --> 00:07:22.100 align:middle line:90% and they overproduce. 00:07:22.100 --> 00:07:24.333 align:middle line:84% You can go back to pollution or congestion 00:07:24.333 --> 00:07:26.500 align:middle line:84% on roads, all these classical externality questions, 00:07:26.500 --> 00:07:27.880 align:middle line:90% this is what's going on. 00:07:27.880 --> 00:07:29.800 align:middle line:84% So what's the solution from the-- 00:07:29.800 --> 00:07:32.300 align:middle line:84% I want to say solution from a socially efficient standpoint, 00:07:32.300 --> 00:07:36.200 align:middle line:84% but a solution from the cartel standpoint. 00:07:36.200 --> 00:07:38.920 align:middle line:84% So I don't use solution with necessarily positive 00:07:38.920 --> 00:07:40.400 align:middle line:90% connotation. 00:07:40.400 --> 00:07:41.820 align:middle line:90% Well, there's a few solutions. 00:07:41.820 --> 00:07:43.840 align:middle line:84% One is to threaten violence, but we're not 00:07:43.840 --> 00:07:45.520 align:middle line:90% going to talk about that today. 00:07:45.520 --> 00:07:47.300 align:middle line:90% One is to write contracts. 00:07:47.300 --> 00:07:50.880 align:middle line:90% 00:07:50.880 --> 00:07:54.240 align:middle line:84% What we can say is, we're going to write a contract that this 00:07:54.240 --> 00:07:57.140 align:middle line:84% is exactly how much oil you have to produce, 00:07:57.140 --> 00:07:59.320 align:middle line:84% or this is exactly how many units of vitamins 00:07:59.320 --> 00:08:00.880 align:middle line:90% you're allowed to produce. 00:08:00.880 --> 00:08:03.185 align:middle line:84% But often, we don't see contracts written like this. 00:08:03.185 --> 00:08:04.060 align:middle line:90% Does anyone know why? 00:08:04.060 --> 00:08:06.560 align:middle line:84% Why do we not see contracts as the solution 00:08:06.560 --> 00:08:07.860 align:middle line:90% to this kind of problem? 00:08:07.860 --> 00:08:09.960 align:middle line:90% I think there's a few reasons. 00:08:09.960 --> 00:08:10.467 align:middle line:90% Yeah? 00:08:10.467 --> 00:08:12.300 align:middle line:84% STUDENT: I think it's probably frowned upon, 00:08:12.300 --> 00:08:14.760 align:middle line:90% as [? making ?] [? a ?] cartel. 00:08:14.760 --> 00:08:16.560 align:middle line:84% Usually, the government would want 00:08:16.560 --> 00:08:19.580 align:middle line:84% to force some kind of fair competition. 00:08:19.580 --> 00:08:20.280 align:middle line:90% IAN BALL: Right. 00:08:20.280 --> 00:08:22.113 align:middle line:84% So it's not only frowned upon, it's illegal. 00:08:22.113 --> 00:08:23.360 align:middle line:90% It's very frowned upon. 00:08:23.360 --> 00:08:25.280 align:middle line:84% So in a domestic context, I think there's 00:08:25.280 --> 00:08:27.880 align:middle line:90% two reasons we don't see this. 00:08:27.880 --> 00:08:29.500 align:middle line:90% So maybe I'll say but-- 00:08:29.500 --> 00:08:32.559 align:middle line:90% 00:08:32.559 --> 00:08:35.200 align:middle line:84% maybe I'll start a new board here. 00:08:35.200 --> 00:08:38.080 align:middle line:90% But what's the issue? 00:08:38.080 --> 00:08:45.043 align:middle line:84% Well, within a country, this is often illegal. 00:08:45.043 --> 00:08:46.460 align:middle line:84% I mean, it depends on the country. 00:08:46.460 --> 00:08:47.918 align:middle line:84% But certainly in the United States, 00:08:47.918 --> 00:08:49.440 align:middle line:84% if two firms get together, and they 00:08:49.440 --> 00:08:51.280 align:middle line:84% were to write a contract that says, 00:08:51.280 --> 00:08:53.440 align:middle line:84% we're only going to produce this much, that 00:08:53.440 --> 00:08:55.160 align:middle line:84% would go against antitrust regulations, 00:08:55.160 --> 00:08:58.518 align:middle line:90% and they'd get sued. 00:08:58.518 --> 00:09:01.060 align:middle line:84% What about-- but I think there's slightly a different reason. 00:09:01.060 --> 00:09:02.920 align:middle line:90% What about across countries? 00:09:02.920 --> 00:09:05.120 align:middle line:90% So what about OPEC? 00:09:05.120 --> 00:09:07.320 align:middle line:84% It's not necessarily legal if Iran and Saudi 00:09:07.320 --> 00:09:10.200 align:middle line:90% Arabia sign a contract. 00:09:10.200 --> 00:09:14.960 align:middle line:84% It's not really clear what court would say that they 00:09:14.960 --> 00:09:16.720 align:middle line:90% were violating antitrust law. 00:09:16.720 --> 00:09:18.400 align:middle line:84% But I think there's another problem. 00:09:18.400 --> 00:09:20.960 align:middle line:84% So why is it hard for countries to write contracts 00:09:20.960 --> 00:09:21.700 align:middle line:90% with each other? 00:09:21.700 --> 00:09:24.520 align:middle line:90% 00:09:24.520 --> 00:09:25.020 align:middle line:90% Let's see. 00:09:25.020 --> 00:09:25.160 align:middle line:90% Yeah? 00:09:25.160 --> 00:09:25.320 align:middle line:90% [INAUDIBLE] 00:09:25.320 --> 00:09:27.100 align:middle line:84% STUDENT: Is it like an enforcement issue? 00:09:27.100 --> 00:09:27.725 align:middle line:90% IAN BALL: Yeah. 00:09:27.725 --> 00:09:28.762 align:middle line:90% So say more about that. 00:09:28.762 --> 00:09:30.220 align:middle line:84% STUDENT: Like, you have a contract. 00:09:30.220 --> 00:09:32.160 align:middle line:84% Like, the only way that you can enforce it 00:09:32.160 --> 00:09:34.140 align:middle line:84% is through your own oil production. 00:09:34.140 --> 00:09:36.880 align:middle line:84% But like, there's knowing you can snitch on and say 00:09:36.880 --> 00:09:38.177 align:middle line:90% that they overproduced. 00:09:38.177 --> 00:09:40.260 align:middle line:84% IAN BALL: So I think you're getting at two things. 00:09:40.260 --> 00:09:41.802 align:middle line:84% One is I think what you're getting at 00:09:41.802 --> 00:09:43.600 align:middle line:84% is kind of observability, that maybe it's 00:09:43.600 --> 00:09:44.927 align:middle line:90% hard to see who's overproduced. 00:09:44.927 --> 00:09:47.260 align:middle line:84% I guess I was putting at-- but your word "enforcement"-- 00:09:47.260 --> 00:09:49.360 align:middle line:84% I think one basic question is, who enforces 00:09:49.360 --> 00:09:51.093 align:middle line:90% contracts between countries? 00:09:51.093 --> 00:09:52.760 align:middle line:84% If there's two firms in the US, and they 00:09:52.760 --> 00:09:54.860 align:middle line:84% write a contract, and someone breaches the contract, 00:09:54.860 --> 00:09:56.200 align:middle line:90% they can go to court in the US. 00:09:56.200 --> 00:09:58.620 align:middle line:84% If Iran and Saudi Arabia sign a contract, well, 00:09:58.620 --> 00:09:59.540 align:middle line:90% who do they go to? 00:09:59.540 --> 00:10:01.620 align:middle line:84% It's not clear who can enforce that contract. 00:10:01.620 --> 00:10:04.680 align:middle line:84% So across countries, it's not clear 00:10:04.680 --> 00:10:11.880 align:middle line:84% how we enforce contracts, kind of no enforcement mechanism-- 00:10:11.880 --> 00:10:13.860 align:middle line:84% no enforcement, maybe I'll say, through courts. 00:10:13.860 --> 00:10:16.800 align:middle line:90% 00:10:16.800 --> 00:10:21.040 align:middle line:84% So in both the domestic context, where collusion as a cartel 00:10:21.040 --> 00:10:23.440 align:middle line:84% is illegal, and the international context, 00:10:23.440 --> 00:10:26.017 align:middle line:84% where enforcing contracts between countries 00:10:26.017 --> 00:10:27.600 align:middle line:84% is challenging because we don't really 00:10:27.600 --> 00:10:30.400 align:middle line:84% have a court to enforce them, our initial solution 00:10:30.400 --> 00:10:32.560 align:middle line:84% of writing contracts becomes a challenge. 00:10:32.560 --> 00:10:34.480 align:middle line:84% And because of that, what we often turn 00:10:34.480 --> 00:10:38.300 align:middle line:90% to are implicit cartels. 00:10:38.300 --> 00:10:44.800 align:middle line:90% 00:10:44.800 --> 00:10:47.400 align:middle line:84% So by an implicit cartel, I mean we're not writing down 00:10:47.400 --> 00:10:50.620 align:middle line:84% a contract that says, if you produce this much, 00:10:50.620 --> 00:10:51.920 align:middle line:90% this is what happens. 00:10:51.920 --> 00:10:56.600 align:middle line:84% But we're going to try to arrange our behavior in such 00:10:56.600 --> 00:10:58.980 align:middle line:84% a way that if someone were to overproduce, 00:10:58.980 --> 00:11:02.000 align:middle line:84% they would be punished, maybe by the other firms producing 00:11:02.000 --> 00:11:03.760 align:middle line:90% more as well to punish them. 00:11:03.760 --> 00:11:06.840 align:middle line:84% The problem is we have to make sure 00:11:06.840 --> 00:11:08.760 align:middle line:90% that this is self-enforcing. 00:11:08.760 --> 00:11:10.900 align:middle line:90% But it must be-- 00:11:10.900 --> 00:11:13.840 align:middle line:90% 00:11:13.840 --> 00:11:15.620 align:middle line:90% maybe I'll say self-enforcing. 00:11:15.620 --> 00:11:19.280 align:middle line:90% 00:11:19.280 --> 00:11:21.760 align:middle line:84% So the idea is we don't have an external court 00:11:21.760 --> 00:11:24.700 align:middle line:84% to take you to jail if you violate it. 00:11:24.700 --> 00:11:28.017 align:middle line:84% But now, well, if someone overproduces, 00:11:28.017 --> 00:11:30.100 align:middle line:84% then they're going to be punished by another firm. 00:11:30.100 --> 00:11:31.560 align:middle line:84% But then we have to make sure, is that other firm 00:11:31.560 --> 00:11:33.143 align:middle line:84% going to want to actually punish them? 00:11:33.143 --> 00:11:34.980 align:middle line:84% Are they going to want to overproduce? 00:11:34.980 --> 00:11:37.313 align:middle line:84% And then we might need someone to punish them and so on. 00:11:37.313 --> 00:11:40.480 align:middle line:84% So what we want is exactly a Nash equilibrium. 00:11:40.480 --> 00:11:42.987 align:middle line:84% In the absence of courts and contracts, 00:11:42.987 --> 00:11:45.320 align:middle line:84% we want this agreement to be self-enforcing in the sense 00:11:45.320 --> 00:11:47.900 align:middle line:84% that no one can profitably deviate from this. 00:11:47.900 --> 00:11:49.580 align:middle line:84% So what have we reduced the problem to? 00:11:49.580 --> 00:11:52.200 align:middle line:84% Well, this is precisely a repeated game 00:11:52.200 --> 00:11:53.660 align:middle line:90% between strategic players. 00:11:53.660 --> 00:11:55.560 align:middle line:84% And we want to try to understand the Nash 00:11:55.560 --> 00:11:57.153 align:middle line:84% equilibria or the subgame perfect Nash 00:11:57.153 --> 00:11:59.320 align:middle line:84% equilibria of this game, because that will precisely 00:11:59.320 --> 00:12:02.920 align:middle line:84% describe what can be achieved through a self-enforcing 00:12:02.920 --> 00:12:05.500 align:middle line:90% interaction. 00:12:05.500 --> 00:12:09.820 align:middle line:84% So today we're going to look at a model of this, 00:12:09.820 --> 00:12:12.880 align:middle line:84% which is going to be repeated Cournot competition. 00:12:12.880 --> 00:12:19.240 align:middle line:84% So formally, the model today is going to be infinitely 00:12:19.240 --> 00:12:23.080 align:middle line:84% repeated Cournot-- remember, Cournot 00:12:23.080 --> 00:12:24.940 align:middle line:84% refers to quantity-- competition. 00:12:24.940 --> 00:12:32.760 align:middle line:90% 00:12:32.760 --> 00:12:34.580 align:middle line:84% You can think about a very abstract cartel, 00:12:34.580 --> 00:12:37.140 align:middle line:84% but I think this is a pretty good model for OPEC. 00:12:37.140 --> 00:12:40.000 align:middle line:84% So keep that in mind as you're running example 00:12:40.000 --> 00:12:42.040 align:middle line:90% as we go over this. 00:12:42.040 --> 00:12:45.220 align:middle line:84% And we want to formally examine the issues we've brought up. 00:12:45.220 --> 00:12:46.280 align:middle line:90% We want to see-- 00:12:46.280 --> 00:12:48.600 align:middle line:84% we want to formalize this intuitive idea 00:12:48.600 --> 00:12:50.280 align:middle line:84% that firms will have an incentive 00:12:50.280 --> 00:12:52.063 align:middle line:90% to deviate and overproduce. 00:12:52.063 --> 00:12:53.480 align:middle line:84% And then we're going to see how we 00:12:53.480 --> 00:12:58.480 align:middle line:84% can try to dampen that incentive through future threats 00:12:58.480 --> 00:13:00.660 align:middle line:90% to punish by overproduction. 00:13:00.660 --> 00:13:03.800 align:middle line:90% So let's get into that. 00:13:03.800 --> 00:13:05.440 align:middle line:90% So what's the model? 00:13:05.440 --> 00:13:07.280 align:middle line:84% The model in each period is just going 00:13:07.280 --> 00:13:10.060 align:middle line:84% to be standard Cournot quantity competition. 00:13:10.060 --> 00:13:17.040 align:middle line:84% So let's say we have firms, i equals 1 through n. 00:13:17.040 --> 00:13:19.600 align:middle line:90% And we have periods. 00:13:19.600 --> 00:13:22.640 align:middle line:90% Well, 0, 1, 2, 3, and so on. 00:13:22.640 --> 00:13:26.602 align:middle line:84% Again, there's no natural point in time when OPEC breaks down. 00:13:26.602 --> 00:13:28.060 align:middle line:84% Maybe eventually we run out of oil, 00:13:28.060 --> 00:13:29.500 align:middle line:84% but we don't know exactly when that is. 00:13:29.500 --> 00:13:31.167 align:middle line:84% So I think the infinite horizon model is 00:13:31.167 --> 00:13:33.000 align:middle line:90% a pretty good model for that. 00:13:33.000 --> 00:13:35.560 align:middle line:84% And to simplify things, we're going 00:13:35.560 --> 00:13:38.400 align:middle line:84% to assume that each firm has constant marginal cost 00:13:38.400 --> 00:13:39.480 align:middle line:90% production. 00:13:39.480 --> 00:13:56.040 align:middle line:84% So C, where we have 0 C less than 1. 00:13:56.040 --> 00:13:58.680 align:middle line:84% This is 1.1 stylization of the model. 00:13:58.680 --> 00:14:01.420 align:middle line:84% In OPEC, not all firms are probably symmetric. 00:14:01.420 --> 00:14:04.420 align:middle line:84% Some countries have better resources to produce oil. 00:14:04.420 --> 00:14:05.897 align:middle line:84% They can produce oil more cheaply. 00:14:05.897 --> 00:14:07.980 align:middle line:84% You could extend the model to an asymmetric model. 00:14:07.980 --> 00:14:09.120 align:middle line:84% But to keep things simple, we're going 00:14:09.120 --> 00:14:11.000 align:middle line:84% to imagine that each of the firms involved 00:14:11.000 --> 00:14:13.080 align:middle line:90% has the same marginal cost. 00:14:13.080 --> 00:14:16.080 align:middle line:84% And then demand is going to be, as we always say, 00:14:16.080 --> 00:14:23.800 align:middle line:84% the standard demand model, we'll write P of Q equals the max of 1 00:14:23.800 --> 00:14:25.800 align:middle line:90% minus Q and 0. 00:14:25.800 --> 00:14:28.880 align:middle line:84% Remember here, P of Q is the market price 00:14:28.880 --> 00:14:30.300 align:middle line:90% that's faced by all the firms. 00:14:30.300 --> 00:14:32.840 align:middle line:84% So this would be the market price for oil. 00:14:32.840 --> 00:14:36.080 align:middle line:84% The market price for oil depends on the quantity of oil that's 00:14:36.080 --> 00:14:39.160 align:middle line:84% on the market, and it's decreasing in the quantity 00:14:39.160 --> 00:14:40.340 align:middle line:90% of oil on the market. 00:14:40.340 --> 00:14:42.740 align:middle line:84% The more oil is on the market and is produced, 00:14:42.740 --> 00:14:44.200 align:middle line:90% the lower the price is. 00:14:44.200 --> 00:14:45.860 align:middle line:90% The price can never go negative. 00:14:45.860 --> 00:14:47.900 align:middle line:84% So that's why we have the max with zero. 00:14:47.900 --> 00:14:49.575 align:middle line:90% So if tons of oil is produced-- 00:14:49.575 --> 00:14:52.200 align:middle line:84% well, there was a time recently actually where the price of oil 00:14:52.200 --> 00:14:54.040 align:middle line:84% got negative, but it was only a little bit 00:14:54.040 --> 00:14:56.420 align:middle line:84% negative because no one wanted to store it during COVID. 00:14:56.420 --> 00:14:58.760 align:middle line:84% But in general, the price of oil is generally not 00:14:58.760 --> 00:15:00.080 align:middle line:90% going to be negative. 00:15:00.080 --> 00:15:02.142 align:middle line:84% And how do we interpret this one-- 00:15:02.142 --> 00:15:03.600 align:middle line:84% I mean, one should be like whatever 00:15:03.600 --> 00:15:05.520 align:middle line:84% the total amount of oil that could possibly 00:15:05.520 --> 00:15:07.060 align:middle line:90% be produced in the world is. 00:15:07.060 --> 00:15:09.460 align:middle line:84% So one is a very big amount of oil. 00:15:09.460 --> 00:15:12.560 align:middle line:84% We're just changing our units to make that one unit of oil. 00:15:12.560 --> 00:15:18.522 align:middle line:84% And Q, remember, is going to be q1 plus qn. 00:15:18.522 --> 00:15:20.480 align:middle line:84% So we're going to use the notation from before, 00:15:20.480 --> 00:15:23.000 align:middle line:84% that little q's refer to how much is 00:15:23.000 --> 00:15:25.040 align:middle line:90% produced by a single firm. 00:15:25.040 --> 00:15:28.000 align:middle line:84% The big Q is the total amount produced by all the firms 00:15:28.000 --> 00:15:28.695 align:middle line:90% together. 00:15:28.695 --> 00:15:30.320 align:middle line:84% So the total amount produced is the sum 00:15:30.320 --> 00:15:34.520 align:middle line:84% of the amounts produced by each of the n firms. 00:15:34.520 --> 00:15:37.980 align:middle line:84% And then we're going to use standard discounting. 00:15:37.980 --> 00:15:43.760 align:middle line:84% So everyone's going to have the same discount factor delta. 00:15:43.760 --> 00:15:46.720 align:middle line:90% 00:15:46.720 --> 00:15:49.360 align:middle line:90% So what that means-- 00:15:49.360 --> 00:15:52.460 align:middle line:84% maybe let's try to write this out formally. 00:15:52.460 --> 00:16:00.480 align:middle line:90% 00:16:00.480 --> 00:16:03.260 align:middle line:84% So what I want to write down now using this discount factor is, 00:16:03.260 --> 00:16:07.200 align:middle line:84% let's suppose that this is the sequence of production levels. 00:16:07.200 --> 00:16:08.960 align:middle line:90% Now, there's no subscript here. 00:16:08.960 --> 00:16:12.880 align:middle line:84% So what Q superscript 0 means is the vector of quantities 00:16:12.880 --> 00:16:14.400 align:middle line:90% produced in period 0. 00:16:14.400 --> 00:16:17.280 align:middle line:84% So remember, superscripts are always about time, 00:16:17.280 --> 00:16:19.360 align:middle line:84% and subscripts are always about the firm. 00:16:19.360 --> 00:16:20.960 align:middle line:84% If there's no subscript, this should 00:16:20.960 --> 00:16:22.920 align:middle line:84% be interpreted as a vector that tells me 00:16:22.920 --> 00:16:26.080 align:middle line:84% how much every firm produced in period 0. 00:16:26.080 --> 00:16:28.440 align:middle line:84% Then q1 is another vector that tells me 00:16:28.440 --> 00:16:30.600 align:middle line:84% how much every firm produced in period one. 00:16:30.600 --> 00:16:33.520 align:middle line:84% And I want to compute here what is firm i's 00:16:33.520 --> 00:16:35.720 align:middle line:84% average discounted profit if this 00:16:35.720 --> 00:16:37.960 align:middle line:84% is the sequence of production levels? 00:16:37.960 --> 00:16:39.640 align:middle line:84% Well, it's going to be a summation. 00:16:39.640 --> 00:16:42.380 align:middle line:84% We're going to use 1 minus delta up front, like we always do, 00:16:42.380 --> 00:16:44.640 align:middle line:84% so that we look at average discounted profits instead 00:16:44.640 --> 00:16:46.800 align:middle line:84% of just the sum of discounted profits. 00:16:46.800 --> 00:16:50.480 align:middle line:84% And then we're going to take a sum over all the periods, so 00:16:50.480 --> 00:16:53.000 align:middle line:90% from t equals 0 to infinity. 00:16:53.000 --> 00:16:58.320 align:middle line:84% And then we're going to have delta to the t times the profit 00:16:58.320 --> 00:17:04.160 align:middle line:84% that firm i makes in period t, so times pi i. 00:17:04.160 --> 00:17:05.720 align:middle line:84% Maybe this is little pi i, and this 00:17:05.720 --> 00:17:09.160 align:middle line:84% is big pi I. I don't know if you can really tell the difference. 00:17:09.160 --> 00:17:13.280 align:middle line:84% And here, we're going to have qt. 00:17:13.280 --> 00:17:15.240 align:middle line:90% Remember what is qt? 00:17:15.240 --> 00:17:20.000 align:middle line:90% qt is q1t up to qnt. 00:17:20.000 --> 00:17:23.500 align:middle line:84% It specifies how much each firm produces in period t. 00:17:23.500 --> 00:17:27.520 align:middle line:90% 00:17:27.520 --> 00:17:31.640 align:middle line:84% So we can now say, what is a firm's average discounted profit 00:17:31.640 --> 00:17:33.740 align:middle line:84% from a sequence of production decisions? 00:17:33.740 --> 00:17:35.480 align:middle line:84% We can express it like this, but we still 00:17:35.480 --> 00:17:37.720 align:middle line:84% have to fill in what the per period profit is. 00:17:37.720 --> 00:17:48.440 align:middle line:84% So let's recall where, in general, pi i of q1t up to qnt 00:17:48.440 --> 00:17:50.240 align:middle line:90% is what? 00:17:50.240 --> 00:17:53.500 align:middle line:84% Well, this says in period t, what does firm i get? 00:17:53.500 --> 00:17:55.560 align:middle line:84% Well, we need to, as usual, keep track 00:17:55.560 --> 00:17:57.780 align:middle line:84% of what is their profit per unit, 00:17:57.780 --> 00:17:59.880 align:middle line:84% and then how many units do they produce? 00:17:59.880 --> 00:18:03.680 align:middle line:84% So the number of units they produce is qit. 00:18:03.680 --> 00:18:07.640 align:middle line:84% In the t-th period, firm i is producing qit units. 00:18:07.640 --> 00:18:10.280 align:middle line:84% And their profit is going to be that number of units 00:18:10.280 --> 00:18:14.240 align:middle line:84% times the profit they're getting per unit. 00:18:14.240 --> 00:18:19.504 align:middle line:84% So what is the profit per unit here? 00:18:19.504 --> 00:18:20.004 align:middle line:90% Yeah? 00:18:20.004 --> 00:18:21.879 align:middle line:84% STUDENT: [? Will ?] [INAUDIBLE] get the price 00:18:21.879 --> 00:18:23.810 align:middle line:84% by summing all the quantities at this time. 00:18:23.810 --> 00:18:24.560 align:middle line:90% IAN BALL: Exactly. 00:18:24.560 --> 00:18:26.500 align:middle line:90% STUDENT: And then use pi C. 00:18:26.500 --> 00:18:27.500 align:middle line:90% IAN BALL: Exactly right. 00:18:27.500 --> 00:18:32.920 align:middle line:90% So that's going to be P of q1t-- 00:18:32.920 --> 00:18:34.400 align:middle line:90% try to write small here-- 00:18:34.400 --> 00:18:35.680 align:middle line:90% up to qnt. 00:18:35.680 --> 00:18:37.080 align:middle line:84% This is the sum of the quantities 00:18:37.080 --> 00:18:38.460 align:middle line:90% that are produced at time t. 00:18:38.460 --> 00:18:40.027 align:middle line:90% So that's the market quantity. 00:18:40.027 --> 00:18:41.360 align:middle line:90% That's the amount in the market. 00:18:41.360 --> 00:18:43.088 align:middle line:84% That determines the market price, P. 00:18:43.088 --> 00:18:44.880 align:middle line:84% But of course, that's not all profit for me 00:18:44.880 --> 00:18:47.460 align:middle line:84% because I also have to pay a cost of production. 00:18:47.460 --> 00:18:48.620 align:middle line:90% So we subtract C here. 00:18:48.620 --> 00:18:57.280 align:middle line:90% 00:18:57.280 --> 00:19:02.760 align:middle line:84% So now we've formalized the game. 00:19:02.760 --> 00:19:04.280 align:middle line:84% I guess one final assumption we have 00:19:04.280 --> 00:19:06.447 align:middle line:84% to keep track of is remember, whenever we've studied 00:19:06.447 --> 00:19:08.600 align:middle line:84% infinitely repeated games, we've assumed 00:19:08.600 --> 00:19:10.332 align:middle line:90% that past actions are observed. 00:19:10.332 --> 00:19:12.540 align:middle line:84% And in this context, that may be a strong assumption, 00:19:12.540 --> 00:19:13.940 align:middle line:84% and this goes to a question we had. 00:19:13.940 --> 00:19:15.982 align:middle line:84% Remember, we're going to maintain the assumption. 00:19:15.982 --> 00:19:30.760 align:middle line:84% So I'll say maintained assumption 00:19:30.760 --> 00:19:32.880 align:middle line:84% past actions are observed, which in this case, 00:19:32.880 --> 00:19:36.880 align:middle line:84% past actions are just quantities produced. 00:19:36.880 --> 00:19:42.880 align:middle line:84% Do you think this is a good assumption in this model or not? 00:19:42.880 --> 00:19:43.480 align:middle line:90% Yeah. 00:19:43.480 --> 00:19:43.980 align:middle line:90% Yeah? 00:19:43.980 --> 00:19:45.518 align:middle line:90% I think it's not great. 00:19:45.518 --> 00:19:46.060 align:middle line:90% Not terrible. 00:19:46.060 --> 00:19:47.680 align:middle line:90% It depends on the context. 00:19:47.680 --> 00:19:49.530 align:middle line:84% Sometimes we might be able to see 00:19:49.530 --> 00:19:50.780 align:middle line:90% how much people are producing. 00:19:50.780 --> 00:19:52.360 align:middle line:90% Sometimes it may be a bit hard. 00:19:52.360 --> 00:19:54.280 align:middle line:84% And if you read when OPEC has disagreements, 00:19:54.280 --> 00:19:55.580 align:middle line:84% this is always what happens right. 00:19:55.580 --> 00:19:57.080 align:middle line:84% One country says, oh, you definitely overproduced. 00:19:57.080 --> 00:19:57.830 align:middle line:90% They said, oh, no. 00:19:57.830 --> 00:19:58.600 align:middle line:90% It wasn't me. 00:19:58.600 --> 00:19:59.600 align:middle line:90% And you'll see cartels-- 00:19:59.600 --> 00:20:02.058 align:middle line:84% I mean, if you watched Breaking Bad, this is often a story. 00:20:02.058 --> 00:20:04.080 align:middle line:84% People are trying to secretly sell their product 00:20:04.080 --> 00:20:06.160 align:middle line:84% without other members of the cartel observing 00:20:06.160 --> 00:20:09.000 align:middle line:90% that they're selling it. 00:20:09.000 --> 00:20:11.560 align:middle line:84% And I think maybe a better model of this 00:20:11.560 --> 00:20:14.400 align:middle line:84% would be that we observe noisy signals of how much people 00:20:14.400 --> 00:20:15.060 align:middle line:90% have produced. 00:20:15.060 --> 00:20:17.200 align:middle line:84% Because really, what we observe is the price. 00:20:17.200 --> 00:20:18.880 align:middle line:84% And what happens with OPEC is they see 00:20:18.880 --> 00:20:20.100 align:middle line:90% the market price has dropped. 00:20:20.100 --> 00:20:21.960 align:middle line:84% Everyone sees that, and that suggests 00:20:21.960 --> 00:20:23.520 align:middle line:90% that someone has overproduced. 00:20:23.520 --> 00:20:26.160 align:middle line:84% The hard thing is to figure out who it was who overproduced. 00:20:26.160 --> 00:20:28.520 align:middle line:84% And that's an exciting topic, kind 00:20:28.520 --> 00:20:30.540 align:middle line:84% of getting more modern research ideas, 00:20:30.540 --> 00:20:32.680 align:middle line:84% what we do when we only have imperfect monitoring 00:20:32.680 --> 00:20:33.700 align:middle line:90% of past quantities. 00:20:33.700 --> 00:20:35.860 align:middle line:84% Unfortunately, it's mathematically a lot harder, 00:20:35.860 --> 00:20:36.940 align:middle line:84% so we won't cover it in this class. 00:20:36.940 --> 00:20:38.460 align:middle line:84% But if you continue studying game theory, 00:20:38.460 --> 00:20:39.660 align:middle line:84% I think that's a really interesting topic. 00:20:39.660 --> 00:20:42.143 align:middle line:84% What happens if we see price as gone-- is really low, 00:20:42.143 --> 00:20:43.560 align:middle line:84% and we're trying to figure out who 00:20:43.560 --> 00:20:45.180 align:middle line:84% it was who did the overproduction, 00:20:45.180 --> 00:20:46.920 align:middle line:84% and we don't really know who it is. 00:20:46.920 --> 00:20:50.280 align:middle line:84% But we'll maintain that assumption today. 00:20:50.280 --> 00:20:52.840 align:middle line:84% So before we get into actually solving 00:20:52.840 --> 00:20:55.860 align:middle line:84% for a subgame perfect Nash equilibrium of this game, 00:20:55.860 --> 00:20:59.880 align:middle line:84% let's try to just think about this basic challenge 00:20:59.880 --> 00:21:02.180 align:middle line:84% of each firm having an incentive to overproduce. 00:21:02.180 --> 00:21:04.400 align:middle line:84% So let's just first think of the static-- 00:21:04.400 --> 00:21:06.140 align:middle line:84% let's just first look at the stage game. 00:21:06.140 --> 00:21:09.120 align:middle line:90% 00:21:09.120 --> 00:21:11.600 align:middle line:84% And let's see if we can remember-- 00:21:11.600 --> 00:21:14.540 align:middle line:84% the stage game has a unique Nash equilibrium that we solve for. 00:21:14.540 --> 00:21:18.240 align:middle line:84% I think this is a good thing to keep in mind maybe for exams. 00:21:18.240 --> 00:21:24.412 align:middle line:84% The stage game Nash equilibrium maybe I'll say was q1 equals qn. 00:21:24.412 --> 00:21:26.620 align:middle line:84% Remember, now, we're just thinking of the stage game. 00:21:26.620 --> 00:21:28.495 align:middle line:84% So let's just think about everyone's choosing 00:21:28.495 --> 00:21:31.240 align:middle line:84% how much to produce, and everyone produced maybe 00:21:31.240 --> 00:21:34.040 align:middle line:90% what I'll call qNE. 00:21:34.040 --> 00:21:36.220 align:middle line:84% This is the Nash equilibrium quantity. 00:21:36.220 --> 00:21:38.280 align:middle line:90% So this is a number. 00:21:38.280 --> 00:21:43.880 align:middle line:84% And that number turned out to be 1 minus c over n plus 1. 00:21:43.880 --> 00:21:46.772 align:middle line:84% So it turned out that in equilibrium, 00:21:46.772 --> 00:21:48.480 align:middle line:84% the amount that everyone produced-- well, 00:21:48.480 --> 00:21:50.280 align:middle line:84% it depends on how many firms there are. 00:21:50.280 --> 00:21:52.300 align:middle line:84% The more firms there are in the market, 00:21:52.300 --> 00:21:54.920 align:middle line:84% the less each firm produces, and the higher 00:21:54.920 --> 00:21:59.400 align:middle line:84% the cost c of production, the less each firm produces. 00:21:59.400 --> 00:22:06.840 align:middle line:84% And what that meant was the market price, pNE, was what? 00:22:06.840 --> 00:22:10.340 align:middle line:84% Well, it's 1 minus the total quantity. 00:22:10.340 --> 00:22:12.920 align:middle line:84% So maybe let's keep track of this. 00:22:12.920 --> 00:22:15.040 align:middle line:90% What is qNE? 00:22:15.040 --> 00:22:17.880 align:middle line:84% So I've told you what each individual firm produces. 00:22:17.880 --> 00:22:20.293 align:middle line:84% qNE is the aggregate quantity produced. 00:22:20.293 --> 00:22:21.960 align:middle line:84% Well, that's just the sum of the amounts 00:22:21.960 --> 00:22:24.140 align:middle line:84% that the individual firms produce. 00:22:24.140 --> 00:22:28.180 align:middle line:84% So I have 1 minus c over n plus 1 summed over the n firms. 00:22:28.180 --> 00:22:28.900 align:middle line:90% What is that? 00:22:28.900 --> 00:22:33.400 align:middle line:84% That's just n over n plus 1 times 1 minus c. 00:22:33.400 --> 00:22:36.080 align:middle line:84% I've just multiplied this individual production level 00:22:36.080 --> 00:22:38.800 align:middle line:90% by the total number of firms, n. 00:22:38.800 --> 00:22:41.820 align:middle line:84% And once we see that, we can see the market price, 00:22:41.820 --> 00:22:58.240 align:middle line:84% pNE, is going to be 1 minus qNE, which is 1 minus n 00:22:58.240 --> 00:23:02.520 align:middle line:84% over plus n plus 1 times 1 minus c. 00:23:02.520 --> 00:23:03.818 align:middle line:90% And there's a nice-- 00:23:03.818 --> 00:23:05.360 align:middle line:84% just to help us understand this, this 00:23:05.360 --> 00:23:09.280 align:middle line:84% is kind of a convenient algebra trick. 00:23:09.280 --> 00:23:12.440 align:middle line:90% Let's add c and subtract c. 00:23:12.440 --> 00:23:15.680 align:middle line:84% So if we do that, I'll add c to the front. 00:23:15.680 --> 00:23:17.460 align:middle line:84% And now I'm going to subtract c later on. 00:23:17.460 --> 00:23:24.600 align:middle line:84% So we get 1 minus c minus n over n plus 1, 1 minus c. 00:23:24.600 --> 00:23:27.640 align:middle line:84% So all I've done is I've put in a c here, 00:23:27.640 --> 00:23:29.840 align:middle line:90% and I've subtracted a c here. 00:23:29.840 --> 00:23:32.080 align:middle line:90% So I haven't done anything. 00:23:32.080 --> 00:23:34.240 align:middle line:84% But now we can simplify things a lot 00:23:34.240 --> 00:23:38.720 align:middle line:84% because I have 1 minus c minus n over n plus 1 times 1 minus c. 00:23:38.720 --> 00:23:44.360 align:middle line:84% So if I factor out the 1 minus c, I get c plus 1 00:23:44.360 --> 00:23:46.340 align:middle line:90% minus c over n plus 1. 00:23:46.340 --> 00:23:55.000 align:middle line:90% 00:23:55.000 --> 00:23:57.820 align:middle line:84% And I think this is a bit more easily interpretable. 00:23:57.820 --> 00:24:00.920 align:middle line:84% Can people-- does anyone want to give an interpretation 00:24:00.920 --> 00:24:03.360 align:middle line:90% for this market price here? 00:24:03.360 --> 00:24:07.400 align:middle line:84% So how much is each firm charging in this equilibrium? 00:24:07.400 --> 00:24:09.680 align:middle line:84% Maybe I'll just go-- they're charging-- well, 00:24:09.680 --> 00:24:11.740 align:middle line:90% their cost plus a markup. 00:24:11.740 --> 00:24:15.760 align:middle line:84% So this is the classic markup term. 00:24:15.760 --> 00:24:18.980 align:middle line:84% This tells me that each firm is making positive profits 00:24:18.980 --> 00:24:21.640 align:middle line:84% because each firm is producing a positive amount, 00:24:21.640 --> 00:24:23.720 align:middle line:84% and the price they're getting is strictly higher 00:24:23.720 --> 00:24:25.140 align:middle line:90% than the cost of production. 00:24:25.140 --> 00:24:28.120 align:middle line:84% So that means their profit per unit is strictly positive. 00:24:28.120 --> 00:24:30.480 align:middle line:84% And the scale of the markup is decreasing 00:24:30.480 --> 00:24:31.580 align:middle line:90% in the number of firms. 00:24:31.580 --> 00:24:34.020 align:middle line:84% The more firms there are, the smaller this markup is. 00:24:34.020 --> 00:24:37.930 align:middle line:90% 00:24:37.930 --> 00:24:41.080 align:middle line:84% And now, let's just compute one last thing, 00:24:41.080 --> 00:24:44.920 align:middle line:90% which is what is pi NE? 00:24:44.920 --> 00:24:51.600 align:middle line:84% So what is the profit that each firm makes in this equilibrium? 00:24:51.600 --> 00:24:55.720 align:middle line:84% Well, each firm is producing 1 minus c over n plus 1. 00:24:55.720 --> 00:24:57.640 align:middle line:84% And their profit per unit is exactly 00:24:57.640 --> 00:25:00.600 align:middle line:84% the markup, which is 1 minus c over n plus 1. 00:25:00.600 --> 00:25:05.900 align:middle line:84% So it turns out this is 1 minus c over n plus 1 squared. 00:25:05.900 --> 00:25:09.840 align:middle line:90% 00:25:09.840 --> 00:25:13.600 align:middle line:90% So that's the Nash equilibrium. 00:25:13.600 --> 00:25:15.200 align:middle line:84% Now, the question is, how could they 00:25:15.200 --> 00:25:18.960 align:middle line:90% do better in the stage game? 00:25:18.960 --> 00:25:28.480 align:middle line:84% So I'll ask can firms do better than the Nash equilibrium? 00:25:28.480 --> 00:25:30.100 align:middle line:84% This may not be a Nash equilibrium, 00:25:30.100 --> 00:25:31.808 align:middle line:84% but I'm just saying, is there another way 00:25:31.808 --> 00:25:35.200 align:middle line:84% they could produce to do better than the Nash equilibrium? 00:25:35.200 --> 00:25:37.040 align:middle line:84% And the idea is, so let's say suppose 00:25:37.040 --> 00:25:39.960 align:middle line:84% the firms wanted to make the sum of all their profits 00:25:39.960 --> 00:25:41.028 align:middle line:90% as high as possible. 00:25:41.028 --> 00:25:41.820 align:middle line:90% What would they do? 00:25:41.820 --> 00:25:43.153 align:middle line:90% So let's be a bit more specific. 00:25:43.153 --> 00:25:44.120 align:middle line:90% How they do better? 00:25:44.120 --> 00:25:48.400 align:middle line:84% Well, what they could do is maximize 00:25:48.400 --> 00:25:49.860 align:middle line:90% the sum of their profits. 00:25:49.860 --> 00:25:55.983 align:middle line:90% 00:25:55.983 --> 00:25:56.900 align:middle line:90% How would you do this? 00:25:56.900 --> 00:25:58.740 align:middle line:84% How would the firms maximize the sum of their profits? 00:25:58.740 --> 00:25:59.080 align:middle line:90% Yes? 00:25:59.080 --> 00:26:00.760 align:middle line:84% STUDENT: You could choose exactly one firm 00:26:00.760 --> 00:26:01.740 align:middle line:90% to produce everything. 00:26:01.740 --> 00:26:04.220 align:middle line:84% And the reason is, in that case, n is 1, 00:26:04.220 --> 00:26:06.720 align:middle line:84% so then the market can be higher. 00:26:06.720 --> 00:26:07.800 align:middle line:90% IAN BALL: That's right. 00:26:07.800 --> 00:26:09.300 align:middle line:84% So I think we're getting very close. 00:26:09.300 --> 00:26:10.160 align:middle line:90% Yeah. 00:26:10.160 --> 00:26:12.540 align:middle line:90% They want to think about-- 00:26:12.540 --> 00:26:14.640 align:middle line:84% so if only one firm produced, how much-- 00:26:14.640 --> 00:26:17.680 align:middle line:84% what would be the right amount for that firm to produce? 00:26:17.680 --> 00:26:21.360 align:middle line:84% STUDENT: Then that firm would produce 1 minus c over 2. 00:26:21.360 --> 00:26:22.066 align:middle line:90% IAN BALL: Right. 00:26:22.066 --> 00:26:22.420 align:middle line:90% Exactly. 00:26:22.420 --> 00:26:24.120 align:middle line:84% And we can actually-- there's a few different ways 00:26:24.120 --> 00:26:24.953 align:middle line:90% we can achieve that. 00:26:24.953 --> 00:26:29.023 align:middle line:84% So the key is, if we want to maximize the sum of our profits, 00:26:29.023 --> 00:26:30.440 align:middle line:84% what we want to do-- and this is-- 00:26:30.440 --> 00:26:32.357 align:middle line:84% what you described as a special case of this-- 00:26:32.357 --> 00:26:34.720 align:middle line:84% we want to collectively act as a monopolist. 00:26:34.720 --> 00:26:37.285 align:middle line:84% We want to make sure that the sum of the amounts we produce 00:26:37.285 --> 00:26:39.160 align:middle line:84% is exactly the amount that a monopolist would 00:26:39.160 --> 00:26:43.200 align:middle line:84% produce because we know that a monopolist maximizes profits. 00:26:43.200 --> 00:26:46.280 align:middle line:84% So to maximize the sum of the profits, what we need 00:26:46.280 --> 00:26:52.960 align:middle line:84% is we need q1 plus qn to equal the quantity that maximizes 00:26:52.960 --> 00:26:53.940 align:middle line:90% the sum of the profits. 00:26:53.940 --> 00:26:56.480 align:middle line:84% It turns out that's exactly the monopoly quantity. 00:26:56.480 --> 00:26:59.180 align:middle line:84% So we want this to be the monopoly quantity. 00:26:59.180 --> 00:27:03.840 align:middle line:90% 00:27:03.840 --> 00:27:04.380 align:middle line:90% Why is this? 00:27:04.380 --> 00:27:07.200 align:middle line:84% Well, the trick is if I produce q1 00:27:07.200 --> 00:27:11.040 align:middle line:84% and firm 2 produces q2 all the way up to firm n, producing qn, 00:27:11.040 --> 00:27:14.320 align:middle line:84% the sum of the firm's profits is exactly the same as the profit 00:27:14.320 --> 00:27:17.920 align:middle line:84% that a single firm would make if they produced q1 all the way up 00:27:17.920 --> 00:27:20.080 align:middle line:90% to qn by themselves. 00:27:20.080 --> 00:27:21.895 align:middle line:84% And the way that we maximize that amount 00:27:21.895 --> 00:27:24.520 align:middle line:84% is exactly by setting it equal to the monopoly quantity, which, 00:27:24.520 --> 00:27:28.000 align:middle line:84% in this example, is 1 minus c over 2. 00:27:28.000 --> 00:27:30.000 align:middle line:84% So what you described is one special case 00:27:30.000 --> 00:27:32.618 align:middle line:84% of this, where one firm could produce the monopoly quantity 00:27:32.618 --> 00:27:34.160 align:middle line:84% and everyone else could produce zero. 00:27:34.160 --> 00:27:36.040 align:middle line:84% But I don't think that would be a very effective cartel. 00:27:36.040 --> 00:27:38.160 align:middle line:84% If I said, hey, other countries, I want you to all 00:27:38.160 --> 00:27:40.407 align:middle line:84% to produce zero, and I'm going to produce everything, 00:27:40.407 --> 00:27:42.740 align:middle line:84% I don't think those other countries would be very happy. 00:27:42.740 --> 00:27:45.320 align:middle line:84% So probably we would split it a bit differently. 00:27:45.320 --> 00:27:47.040 align:middle line:84% A natural split would be an even split, 00:27:47.040 --> 00:27:50.440 align:middle line:84% where each of us produces 1 minus c over 2n. 00:27:50.440 --> 00:27:56.440 align:middle line:84% So one thing we could do-- so, e,g., we could do q1-- 00:27:56.440 --> 00:27:58.080 align:middle line:90% I'll go down here. 00:27:58.080 --> 00:28:09.440 align:middle line:84% The even split would be we each produce maybe what I'll call qM. 00:28:09.440 --> 00:28:13.420 align:middle line:84% So before, I described the Nash equilibrium quantity. 00:28:13.420 --> 00:28:14.972 align:middle line:84% Now I'm going to describe-- when I 00:28:14.972 --> 00:28:16.680 align:middle line:84% say monopoly quantity, what I really mean 00:28:16.680 --> 00:28:19.520 align:middle line:84% is each firm's share of the monopoly quantity. 00:28:19.520 --> 00:28:25.600 align:middle line:90% So this is 1 minus c over 2n. 00:28:25.600 --> 00:28:29.403 align:middle line:84% So qM is technically 1 n-th of the monopoly quantity. 00:28:29.403 --> 00:28:31.320 align:middle line:84% Because it's the quantity that each firm needs 00:28:31.320 --> 00:28:33.680 align:middle line:84% to produce to ensure that the total production 00:28:33.680 --> 00:28:36.840 align:middle line:84% is equal to the monopoly quantity. 00:28:36.840 --> 00:28:51.893 align:middle line:84% Now, if we compare these two numbers, what do you see? 00:28:51.893 --> 00:28:54.060 align:middle line:84% Which is bigger, and what's your intuition for that? 00:28:54.060 --> 00:28:56.600 align:middle line:90% 00:28:56.600 --> 00:29:01.560 align:middle line:84% Let's assume n is at least 1-- at least 2. 00:29:01.560 --> 00:29:04.480 align:middle line:90% So which one is bigger and why? 00:29:04.480 --> 00:29:05.580 align:middle line:90% Yeah? 00:29:05.580 --> 00:29:07.560 align:middle line:84% STUDENT: The Nash equilibrium is larger. 00:29:07.560 --> 00:29:09.040 align:middle line:84% And the intuition for that could be 00:29:09.040 --> 00:29:11.960 align:middle line:84% that in order to do better than Nash equilibrium, 00:29:11.960 --> 00:29:16.800 align:middle line:84% the firms kind of need to have a closer collaboration to maximize 00:29:16.800 --> 00:29:19.160 align:middle line:90% as if they were a single firm. 00:29:19.160 --> 00:29:22.320 align:middle line:84% Whereas, each of them in that non-Nash equilibrium 00:29:22.320 --> 00:29:23.880 align:middle line:84% state, a monopolist state, would be 00:29:23.880 --> 00:29:26.940 align:middle line:84% kind of incentivized to produce more to capture profit. 00:29:26.940 --> 00:29:27.940 align:middle line:90% IAN BALL: Exactly right. 00:29:27.940 --> 00:29:31.560 align:middle line:84% So this is exactly what we talked about at the beginning. 00:29:31.560 --> 00:29:33.440 align:middle line:84% In the Nash equilibrium, the firms 00:29:33.440 --> 00:29:36.140 align:middle line:84% are behaving optimally given the way everyone else is behaving. 00:29:36.140 --> 00:29:38.240 align:middle line:84% But that actually turns out that collectively it 00:29:38.240 --> 00:29:40.320 align:middle line:84% hurts them because they're producing too much. 00:29:40.320 --> 00:29:43.600 align:middle line:84% They would actually do better if they each produced less. 00:29:43.600 --> 00:29:46.480 align:middle line:84% If they each produce less, then the market price 00:29:46.480 --> 00:29:51.873 align:middle line:84% is going to be higher, and the payoff of each firm 00:29:51.873 --> 00:29:53.540 align:middle line:84% is actually going to be strictly higher. 00:29:53.540 --> 00:29:55.800 align:middle line:84% The problem is this will not be in equilibrium. 00:29:55.800 --> 00:29:58.380 align:middle line:84% And what will happen is suppose we did this. 00:29:58.380 --> 00:29:59.360 align:middle line:90% Let's just check. 00:29:59.360 --> 00:30:01.720 align:middle line:84% This is not going to be a Nash equilibrium. 00:30:01.720 --> 00:30:02.360 align:middle line:90% And why? 00:30:02.360 --> 00:30:05.420 align:middle line:84% What is the deviation for, say, firm i? 00:30:05.420 --> 00:30:07.745 align:middle line:84% What would they-- how could they do better? 00:30:07.745 --> 00:30:10.120 align:middle line:84% Would they want to increase or decrease their production? 00:30:10.120 --> 00:30:11.495 align:middle line:84% Increase their production, right? 00:30:11.495 --> 00:30:16.120 align:middle line:84% So it's not a Nash equilibrium because firm i 00:30:16.120 --> 00:30:27.980 align:middle line:84% prefers to strictly increase production. 00:30:27.980 --> 00:30:31.158 align:middle line:90% 00:30:31.158 --> 00:30:32.700 align:middle line:84% And we can check that mathematically. 00:30:32.700 --> 00:30:35.320 align:middle line:84% But let's just-- this seems a little puzzling now, 00:30:35.320 --> 00:30:37.680 align:middle line:84% and this is getting at a core game theory idea. 00:30:37.680 --> 00:30:39.760 align:middle line:84% We said that these production levels maximize 00:30:39.760 --> 00:30:41.680 align:middle line:90% the sum of everyone's profits. 00:30:41.680 --> 00:30:44.920 align:middle line:84% But now I'm saying one firm can do better by strictly increasing 00:30:44.920 --> 00:30:46.200 align:middle line:90% how much they produce. 00:30:46.200 --> 00:30:47.307 align:middle line:90% So how is that consistent? 00:30:47.307 --> 00:30:49.140 align:middle line:84% I thought we were already maximizing profit. 00:30:49.140 --> 00:30:52.240 align:middle line:84% So how can one firm do better by deviating? 00:30:52.240 --> 00:30:53.280 align:middle line:90% Yeah, on the front? 00:30:53.280 --> 00:30:55.590 align:middle line:84% STUDENT: You're maximizing the [? sum ?] [INAUDIBLE]. 00:30:55.590 --> 00:30:56.340 align:middle line:90% IAN BALL: Exactly. 00:30:56.340 --> 00:30:57.173 align:middle line:90% So what must happen? 00:30:57.173 --> 00:31:00.900 align:middle line:84% If this firm produces strictly more, this firm is doing better. 00:31:00.900 --> 00:31:04.208 align:middle line:84% But what do we know must also happen? 00:31:04.208 --> 00:31:05.500 align:middle line:90% The other ones are doing worse. 00:31:05.500 --> 00:31:07.000 align:middle line:84% In fact, the amount by which they're doing worse 00:31:07.000 --> 00:31:09.500 align:middle line:84% more than offsets the amount that this firm is doing better. 00:31:09.500 --> 00:31:10.940 align:middle line:90% So I benefit myself. 00:31:10.940 --> 00:31:12.240 align:middle line:90% I hurt everyone else. 00:31:12.240 --> 00:31:14.520 align:middle line:84% On average, the sum of payoffs goes down 00:31:14.520 --> 00:31:16.980 align:middle line:84% because I hurt everyone else more than I help myself. 00:31:16.980 --> 00:31:19.080 align:middle line:84% But I don't care because I'm helping myself. 00:31:19.080 --> 00:31:21.180 align:middle line:84% That's the basic intuition of this. 00:31:21.180 --> 00:31:23.820 align:middle line:90% 00:31:23.820 --> 00:31:29.200 align:middle line:84% So now the question is, can we sustain the monopoly levels 00:31:29.200 --> 00:31:30.420 align:middle line:90% in equilibrium? 00:31:30.420 --> 00:31:33.460 align:middle line:90% So the question is-- 00:31:33.460 --> 00:31:36.880 align:middle line:90% 00:31:36.880 --> 00:31:37.940 align:middle line:90% here's a question. 00:31:37.940 --> 00:31:50.397 align:middle line:90% 00:31:50.397 --> 00:31:52.480 align:middle line:84% So we're all better off if every period we produce 00:31:52.480 --> 00:31:54.480 align:middle line:90% qM rather than qNE. 00:31:54.480 --> 00:31:56.400 align:middle line:90% That would be great for us. 00:31:56.400 --> 00:31:58.200 align:middle line:84% It's not clear, though, how we can sustain 00:31:58.200 --> 00:32:00.360 align:middle line:84% this in a self-enforcing way because we already 00:32:00.360 --> 00:32:02.280 align:middle line:84% said if we try to do this, one firm 00:32:02.280 --> 00:32:04.640 align:middle line:84% can strictly benefit by deviating and producing 00:32:04.640 --> 00:32:06.920 align:middle line:90% slightly more. 00:32:06.920 --> 00:32:07.900 align:middle line:90% So how can we do this? 00:32:07.900 --> 00:32:10.240 align:middle line:90% Well, what's the idea? 00:32:10.240 --> 00:32:13.140 align:middle line:84% We need some way to punish people for overproduction. 00:32:13.140 --> 00:32:14.640 align:middle line:84% And, in fact, we'll just punish them 00:32:14.640 --> 00:32:17.182 align:middle line:84% if they produce anything other than what they're supposed to. 00:32:17.182 --> 00:32:19.620 align:middle line:84% Really, overproduction is the tempting thing to do. 00:32:19.620 --> 00:32:20.740 align:middle line:90% But here's the idea. 00:32:20.740 --> 00:32:28.020 align:middle line:84% We want to punish firms for overproduction. 00:32:28.020 --> 00:32:39.080 align:middle line:90% 00:32:39.080 --> 00:32:43.840 align:middle line:84% So any ideas about how we might be able to do this? 00:32:43.840 --> 00:32:45.340 align:middle line:90% So we're getting together. 00:32:45.340 --> 00:32:46.820 align:middle line:84% We see that one firm has deviated. 00:32:46.820 --> 00:32:48.280 align:middle line:90% Say, they've produced too much. 00:32:48.280 --> 00:32:50.740 align:middle line:84% Now, what might we want to do to try to punish them? 00:32:50.740 --> 00:32:54.280 align:middle line:90% 00:32:54.280 --> 00:32:54.920 align:middle line:90% Yeah? 00:32:54.920 --> 00:32:57.730 align:middle line:84% STUDENT: We could all move to the Nash equilibrium quantities. 00:32:57.730 --> 00:32:58.480 align:middle line:90% IAN BALL: Exactly. 00:32:58.480 --> 00:33:00.147 align:middle line:90% So I think the first-- 00:33:00.147 --> 00:33:01.480 align:middle line:90% that's exactly where were going. 00:33:01.480 --> 00:33:02.660 align:middle line:84% I was expecting a different answer. 00:33:02.660 --> 00:33:03.100 align:middle line:90% So well done. 00:33:03.100 --> 00:33:03.860 align:middle line:90% You jumped ahead. 00:33:03.860 --> 00:33:04.800 align:middle line:84% What I would expect people to say 00:33:04.800 --> 00:33:06.327 align:middle line:84% is we would just flood the market. 00:33:06.327 --> 00:33:08.660 align:middle line:84% The problem with that is that may not be self-enforcing, 00:33:08.660 --> 00:33:09.800 align:middle line:84% because why does the firm actually 00:33:09.800 --> 00:33:10.842 align:middle line:90% want to flood the market? 00:33:10.842 --> 00:33:12.500 align:middle line:84% So indeed, what you're suggesting-- 00:33:12.500 --> 00:33:13.900 align:middle line:84% I mean, we could think about doing that later. 00:33:13.900 --> 00:33:16.525 align:middle line:84% But indeed, the natural thing to do is, the way we could punish 00:33:16.525 --> 00:33:18.460 align:middle line:84% is we could revert to the Nash equilibrium. 00:33:18.460 --> 00:33:21.300 align:middle line:84% Remember, last class when we talked about the folk theorem, 00:33:21.300 --> 00:33:24.380 align:middle line:84% we described these strategies where if anyone deviated, 00:33:24.380 --> 00:33:25.880 align:middle line:84% we reverted to the Nash equilibrium. 00:33:25.880 --> 00:33:34.920 align:middle line:84% In this case-- so by reverting to qNE. 00:33:34.920 --> 00:33:36.800 align:middle line:84% So what this says is we're going to try 00:33:36.800 --> 00:33:38.140 align:middle line:90% to produce the monopoly level. 00:33:38.140 --> 00:33:39.540 align:middle line:84% We're each going to produce a small amount. 00:33:39.540 --> 00:33:42.165 align:middle line:84% We're going to get high profits because the price is very high. 00:33:42.165 --> 00:33:44.560 align:middle line:84% If we observe anyone deviating, well, now we're 00:33:44.560 --> 00:33:46.740 align:middle line:84% all going to produce the Nash equilibrium level. 00:33:46.740 --> 00:33:48.500 align:middle line:84% We're all worse off, but we're actually 00:33:48.500 --> 00:33:50.000 align:middle line:84% willing to carry out this punishment 00:33:50.000 --> 00:33:51.740 align:middle line:84% because this is itself a Nash equilibrium. 00:33:51.740 --> 00:33:52.407 align:middle line:90% That's the idea. 00:33:52.407 --> 00:33:55.480 align:middle line:84% And now we want to formally check this. 00:33:55.480 --> 00:33:57.720 align:middle line:84% I guess first question, will this always work? 00:33:57.720 --> 00:33:59.137 align:middle line:84% What are we going to need for this 00:33:59.137 --> 00:34:01.120 align:middle line:84% to work, for this to be a subgame perfect Nash 00:34:01.120 --> 00:34:01.860 align:middle line:90% equilibrium? 00:34:01.860 --> 00:34:06.027 align:middle line:90% 00:34:06.027 --> 00:34:07.860 align:middle line:84% What's going to need to be true about delta? 00:34:07.860 --> 00:34:12.760 align:middle line:90% 00:34:12.760 --> 00:34:13.340 align:middle line:90% Yeah? 00:34:13.340 --> 00:34:15.382 align:middle line:84% STUDENT: It needs to be sufficiently large enough 00:34:15.382 --> 00:34:17.739 align:middle line:84% that we kind of value the profits in future periods. 00:34:17.739 --> 00:34:18.739 align:middle line:90% IAN BALL: Yeah, exactly. 00:34:18.739 --> 00:34:20.300 align:middle line:84% So our guess-- I mean, we'll formally check this, 00:34:20.300 --> 00:34:22.300 align:middle line:84% but it's always good to make a guess in advance. 00:34:22.300 --> 00:34:29.000 align:middle line:84% Our guess is that this will work if delta is high enough. 00:34:29.000 --> 00:34:32.180 align:middle line:90% 00:34:32.180 --> 00:34:35.840 align:middle line:84% Because the idea is I can benefit today by overproducing. 00:34:35.840 --> 00:34:37.960 align:middle line:84% The only reason I'm not going to do that is I'm 00:34:37.960 --> 00:34:39.780 align:middle line:84% afraid of this punishment in the future. 00:34:39.780 --> 00:34:42.460 align:middle line:84% But I have to be patient enough that I care about that future. 00:34:42.460 --> 00:34:44.675 align:middle line:84% If I only care about profits today, then, of course, 00:34:44.675 --> 00:34:46.300 align:middle line:84% it's not going to be a Nash equilibrium 00:34:46.300 --> 00:34:49.620 align:middle line:84% because all I care about is the benefit that I get today. 00:34:49.620 --> 00:34:52.500 align:middle line:84% So now we're going to actually construct this equilibrium. 00:34:52.500 --> 00:34:59.160 align:middle line:90% So let's start here. 00:34:59.160 --> 00:34:59.660 align:middle line:90% All right. 00:34:59.660 --> 00:35:19.080 align:middle line:90% 00:35:19.080 --> 00:35:28.240 align:middle line:84% So now let's construct the SPNE that has this form. 00:35:28.240 --> 00:35:33.217 align:middle line:84% It's going to be helpful to define two functions, 00:35:33.217 --> 00:35:34.800 align:middle line:84% just because if we don't define these, 00:35:34.800 --> 00:35:36.300 align:middle line:84% the algebra gets really complicated. 00:35:36.300 --> 00:35:42.960 align:middle line:84% So let's just make two definitions-- f of q and g of q. 00:35:42.960 --> 00:35:45.400 align:middle line:84% So for any quantity level q, f of q 00:35:45.400 --> 00:35:49.880 align:middle line:84% is going to say if we all produce q, what is our path? 00:35:49.880 --> 00:35:57.400 align:middle line:84% So mathematically this is q times 1 minus nq minus c. 00:35:57.400 --> 00:36:00.872 align:middle line:84% And technically, this only makes sense if q is less than 1 over 00:36:00.872 --> 00:36:03.580 align:middle line:84% n, so that this number doesn't-- so prices don't become negative. 00:36:03.580 --> 00:36:05.480 align:middle line:84% So this says if we all produce q, 00:36:05.480 --> 00:36:07.720 align:middle line:84% the market price is going to be 1 minus nq 00:36:07.720 --> 00:36:10.120 align:middle line:84% because nq is the market quantity. 00:36:10.120 --> 00:36:12.480 align:middle line:84% So 1 minus nq is the market price. 00:36:12.480 --> 00:36:15.460 align:middle line:84% And if we subtract c from that, we get the profit per unit, 00:36:15.460 --> 00:36:18.520 align:middle line:90% and then we multiply it by q. 00:36:18.520 --> 00:36:20.840 align:middle line:84% And then the other thing we want to understand 00:36:20.840 --> 00:36:26.600 align:middle line:84% is how well can a firm do if everyone else is playing q, 00:36:26.600 --> 00:36:30.338 align:middle line:84% and I choose my best deviation in the stage game? 00:36:30.338 --> 00:36:31.880 align:middle line:84% This is going to be the key deviation 00:36:31.880 --> 00:36:32.620 align:middle line:90% we have to worry about. 00:36:32.620 --> 00:36:33.980 align:middle line:84% So we want to have notation for this. 00:36:33.980 --> 00:36:35.522 align:middle line:84% So this is going to say, what happens 00:36:35.522 --> 00:36:39.880 align:middle line:84% if I choose my quantity level q prime to maximize 00:36:39.880 --> 00:36:44.160 align:middle line:84% my profit given that all the other firms are choosing q? 00:36:44.160 --> 00:36:47.560 align:middle line:84% Well, if I choose q prime, this is how much I produce. 00:36:47.560 --> 00:36:49.800 align:middle line:90% What is the price going to be? 00:36:49.800 --> 00:36:54.940 align:middle line:84% Well, it's going to be 1 minus n minus 1q minus q prime, 00:36:54.940 --> 00:36:57.160 align:middle line:90% and then I subtract c. 00:36:57.160 --> 00:37:00.960 align:middle line:84% So here, the price is 1 minus the total quantity 00:37:00.960 --> 00:37:02.080 align:middle line:90% on the market. 00:37:02.080 --> 00:37:03.735 align:middle line:84% The total quantity of the market is 00:37:03.735 --> 00:37:05.360 align:middle line:84% what's produced by everyone else, which 00:37:05.360 --> 00:37:07.160 align:middle line:90% is q times n minus 1. 00:37:07.160 --> 00:37:09.160 align:middle line:84% Because there's n minus 1 other firms. 00:37:09.160 --> 00:37:11.160 align:middle line:90% They're each producing q. 00:37:11.160 --> 00:37:15.640 align:middle line:84% If I choose q prime, then this gives me 00:37:15.640 --> 00:37:17.100 align:middle line:90% the total amount produced. 00:37:17.100 --> 00:37:18.580 align:middle line:90% This gives me the price. 00:37:18.580 --> 00:37:22.040 align:middle line:90% And then I subtract my cost. 00:37:22.040 --> 00:37:28.743 align:middle line:84% And if we could actually do a little algebra. 00:37:28.743 --> 00:37:30.160 align:middle line:84% I don't expect people to see this, 00:37:30.160 --> 00:37:36.483 align:middle line:84% but I think it's actually going to be this. 00:37:36.483 --> 00:37:37.900 align:middle line:84% The formula doesn't really matter, 00:37:37.900 --> 00:37:39.067 align:middle line:90% but this is what it will be. 00:37:39.067 --> 00:37:43.200 align:middle line:90% 00:37:43.200 --> 00:37:46.380 align:middle line:84% But we'll just stick with these two formulas here. 00:37:46.380 --> 00:37:50.420 align:middle line:90% 00:37:50.420 --> 00:37:54.640 align:middle line:84% And then what is our strategy profile? 00:37:54.640 --> 00:37:56.100 align:middle line:90% Well, it's exactly what we said. 00:37:56.100 --> 00:38:00.360 align:middle line:90% 00:38:00.360 --> 00:38:12.880 align:middle line:90% So one, each firm produces qM. 00:38:12.880 --> 00:38:14.060 align:middle line:90% Until a deviation. 00:38:14.060 --> 00:38:19.440 align:middle line:90% 00:38:19.440 --> 00:38:21.980 align:middle line:90% But two, if anyone has deviated. 00:38:21.980 --> 00:38:31.920 align:middle line:90% 00:38:31.920 --> 00:38:39.058 align:middle line:84% every firm produces qNE So this is just formally writing down 00:38:39.058 --> 00:38:40.600 align:middle line:84% the Nash reversion idea that we said. 00:38:40.600 --> 00:38:43.480 align:middle line:84% We start by producing qM every period. 00:38:43.480 --> 00:38:45.660 align:middle line:84% As long as no one's deviated, that's what we do. 00:38:45.660 --> 00:38:48.000 align:middle line:84% But if any firm deviates, then we all 00:38:48.000 --> 00:38:50.720 align:middle line:84% revert to the Nash equilibrium quantities, qNE. 00:38:50.720 --> 00:38:54.520 align:middle line:90% 00:38:54.520 --> 00:38:58.480 align:middle line:84% So now we want to check that this is a subgame perfect Nash 00:38:58.480 --> 00:39:00.680 align:middle line:90% equilibrium. 00:39:00.680 --> 00:39:03.320 align:middle line:84% As usual, we're going to use the one-shot deviation principle. 00:39:03.320 --> 00:39:04.903 align:middle line:84% And we're going to check every history 00:39:04.903 --> 00:39:08.278 align:middle line:84% and check that no firm has a profitable one-shot deviation. 00:39:08.278 --> 00:39:10.320 align:middle line:84% But we're going to use a different representation 00:39:10.320 --> 00:39:12.720 align:middle line:84% of this strategy called an automaton, which 00:39:12.720 --> 00:39:15.353 align:middle line:84% is going to turn out to be very useful for structuring 00:39:15.353 --> 00:39:17.020 align:middle line:84% the way we think about these equilibria. 00:39:17.020 --> 00:39:22.080 align:middle line:84% So let's look at what's called the automaton representation-- 00:39:22.080 --> 00:39:27.360 align:middle line:90% automaton representation. 00:39:27.360 --> 00:39:29.700 align:middle line:84% So you recall that when we describe these strategies, 00:39:29.700 --> 00:39:32.640 align:middle line:84% the way we analyze them is we often grouped all the histories 00:39:32.640 --> 00:39:33.468 align:middle line:90% into two buckets. 00:39:33.468 --> 00:39:35.760 align:middle line:84% We said they're the histories where someone's deviated, 00:39:35.760 --> 00:39:38.120 align:middle line:84% and the histories where no one's deviated. 00:39:38.120 --> 00:39:40.760 align:middle line:84% And the automaton representation formalizes 00:39:40.760 --> 00:39:43.080 align:middle line:90% that in terms of states. 00:39:43.080 --> 00:39:46.082 align:middle line:84% We think of, what's the current state that we're in, 00:39:46.082 --> 00:39:47.540 align:middle line:84% and how do we behave in that state? 00:39:47.540 --> 00:39:48.748 align:middle line:90% So let me just write it down. 00:39:48.748 --> 00:39:51.960 align:middle line:84% To represent this equilibrium, we're going to need two states. 00:39:51.960 --> 00:39:56.880 align:middle line:84% We'll call it the monopoly state and the stage game Nash 00:39:56.880 --> 00:39:59.480 align:middle line:90% equilibrium state. 00:39:59.480 --> 00:40:02.980 align:middle line:84% And now we need to specify how we behave in each state. 00:40:02.980 --> 00:40:04.480 align:middle line:84% Well, in the monopoly state, we each 00:40:04.480 --> 00:40:05.840 align:middle line:90% produce the monopoly price-- 00:40:05.840 --> 00:40:06.700 align:middle line:90% monopoly quantity. 00:40:06.700 --> 00:40:07.200 align:middle line:90% Sorry. 00:40:07.200 --> 00:40:07.940 align:middle line:90% That's easy. 00:40:07.940 --> 00:40:12.280 align:middle line:90% So qi equals qM for all i. 00:40:12.280 --> 00:40:14.400 align:middle line:84% And in the Nash equilibrium state, 00:40:14.400 --> 00:40:19.020 align:middle line:84% we each produce the Nash equilibrium quantity. 00:40:19.020 --> 00:40:21.080 align:middle line:90% That's pretty easy. 00:40:21.080 --> 00:40:23.040 align:middle line:84% So what do I really mean by these states? 00:40:23.040 --> 00:40:27.120 align:middle line:84% Well, they're this mental construct. 00:40:27.120 --> 00:40:28.940 align:middle line:84% But let's just describe how things work, 00:40:28.940 --> 00:40:30.500 align:middle line:84% and then we can maybe interpret it. 00:40:30.500 --> 00:40:32.680 align:middle line:84% So the idea is we start in this state. 00:40:32.680 --> 00:40:34.500 align:middle line:84% Initially, we're each going to produce qM. 00:40:34.500 --> 00:40:35.875 align:middle line:84% So I'm going to put an arrow here 00:40:35.875 --> 00:40:38.600 align:middle line:84% to say this is the state that we start in. 00:40:38.600 --> 00:40:40.640 align:middle line:84% And in this state, what we're supposed to do 00:40:40.640 --> 00:40:43.400 align:middle line:84% is we're each supposed to produce the monopoly quantity. 00:40:43.400 --> 00:40:49.120 align:middle line:84% Now what we have to describe is how 00:40:49.120 --> 00:40:51.520 align:middle line:90% we transition between states. 00:40:51.520 --> 00:40:54.065 align:middle line:84% So the idea of this equilibrium is 00:40:54.065 --> 00:40:55.440 align:middle line:84% we're going to stay in this state 00:40:55.440 --> 00:40:57.680 align:middle line:90% as long as no one has deviated. 00:40:57.680 --> 00:41:00.720 align:middle line:84% But if someone deviates, then we move to a different state. 00:41:00.720 --> 00:41:05.600 align:middle line:84% So what I'll write is I'll write an arrow here to say, 00:41:05.600 --> 00:41:06.840 align:middle line:90% we're in state m. 00:41:06.840 --> 00:41:09.200 align:middle line:90% And we stay in state m if what? 00:41:09.200 --> 00:41:15.620 align:middle line:90% If qi equals qM for all i. 00:41:15.620 --> 00:41:18.920 align:middle line:90% 00:41:18.920 --> 00:41:22.360 align:middle line:84% That's the only way we're going to stay in this state. 00:41:22.360 --> 00:41:25.360 align:middle line:84% Otherwise, someone has deviated, and then we 00:41:25.360 --> 00:41:28.480 align:middle line:90% move to the other state. 00:41:28.480 --> 00:41:29.900 align:middle line:90% So I'll write otherwise here. 00:41:29.900 --> 00:41:37.310 align:middle line:90% 00:41:37.310 --> 00:41:41.138 align:middle line:84% If we're in this state, well, once someone's deviated 00:41:41.138 --> 00:41:43.430 align:middle line:84% and we're in this state, we actually stay in this state 00:41:43.430 --> 00:41:43.970 align:middle line:90% forever. 00:41:43.970 --> 00:41:45.470 align:middle line:84% There's no way to leave this state. 00:41:45.470 --> 00:41:49.810 align:middle line:84% So we're just going to put an arrow here that says, anything. 00:41:49.810 --> 00:41:53.150 align:middle line:90% 00:41:53.150 --> 00:41:57.750 align:middle line:84% So this simple graph actually fully describes a strategy. 00:41:57.750 --> 00:42:00.030 align:middle line:84% And, in fact, fully describes the strategy profile 00:42:00.030 --> 00:42:01.170 align:middle line:90% in this equilibrium. 00:42:01.170 --> 00:42:03.030 align:middle line:84% And I think the way I like to visualize it-- 00:42:03.030 --> 00:42:05.450 align:middle line:84% I mean, of course, if firms are actually colluding today, 00:42:05.450 --> 00:42:06.170 align:middle line:90% they'd use computers. 00:42:06.170 --> 00:42:07.337 align:middle line:90% They'd keep track of things. 00:42:07.337 --> 00:42:10.230 align:middle line:84% But a very simple way to think about this is all I need to do 00:42:10.230 --> 00:42:12.270 align:middle line:84% is have two positions on my desk-- 00:42:12.270 --> 00:42:14.620 align:middle line:84% the monopoly position and the Nash equilibrium position. 00:42:14.620 --> 00:42:16.370 align:middle line:84% And I like to imagine I just have a stone. 00:42:16.370 --> 00:42:19.070 align:middle line:84% And I start by putting the stone in the monopoly position. 00:42:19.070 --> 00:42:20.050 align:middle line:90% I look at my desk. 00:42:20.050 --> 00:42:22.330 align:middle line:84% I see-- and every firm is doing the same thing. 00:42:22.330 --> 00:42:24.190 align:middle line:84% We start with the monopoly position. 00:42:24.190 --> 00:42:25.810 align:middle line:90% We all say, what do we do here? 00:42:25.810 --> 00:42:27.510 align:middle line:84% We all produce the monopoly level. 00:42:27.510 --> 00:42:30.590 align:middle line:84% Then tomorrow we observe what everyone produced yesterday. 00:42:30.590 --> 00:42:32.550 align:middle line:84% And then we potentially move the stone 00:42:32.550 --> 00:42:35.770 align:middle line:84% from the monopoly position to the Nash equilibrium position 00:42:35.770 --> 00:42:37.670 align:middle line:84% if we observe that someone has deviated. 00:42:37.670 --> 00:42:39.877 align:middle line:84% If no one deviates, we keep the stone there. 00:42:39.877 --> 00:42:41.710 align:middle line:84% And then tomorrow we see where the stone is, 00:42:41.710 --> 00:42:43.030 align:middle line:90% and we decide how to play. 00:42:43.030 --> 00:42:45.510 align:middle line:84% So this is a very simple representation 00:42:45.510 --> 00:42:48.190 align:middle line:90% of the strategy profile. 00:42:48.190 --> 00:42:53.310 align:middle line:84% So now we come over here, and let's 00:42:53.310 --> 00:42:56.490 align:middle line:84% write down the conditions for equilibrium in this game. 00:42:56.490 --> 00:43:07.710 align:middle line:90% 00:43:07.710 --> 00:43:11.350 align:middle line:84% So it turns out the key to analyzing equilibrium 00:43:11.350 --> 00:43:14.630 align:middle line:84% using this automaton representation is 00:43:14.630 --> 00:43:18.890 align:middle line:84% to write down what we'll call the value function. 00:43:18.890 --> 00:43:23.430 align:middle line:90% 00:43:23.430 --> 00:43:25.990 align:middle line:84% And in this case, we need to specify the value function each 00:43:25.990 --> 00:43:26.850 align:middle line:90% of the two states. 00:43:26.850 --> 00:43:32.470 align:middle line:84% So we're going to have vM and vNE. 00:43:32.470 --> 00:43:36.130 align:middle line:84% And what the value function says is, in the first case, 00:43:36.130 --> 00:43:39.430 align:middle line:84% suppose we're in state M today, and everyone 00:43:39.430 --> 00:43:41.390 align:middle line:90% follows this strategy. 00:43:41.390 --> 00:43:44.910 align:middle line:84% What is the average discounted payoff of the firms going to be? 00:43:44.910 --> 00:43:47.870 align:middle line:84% And then vNE says, well, if we're in state NE, 00:43:47.870 --> 00:43:49.630 align:middle line:84% and we all follow that strategy, what 00:43:49.630 --> 00:43:52.950 align:middle line:84% is the average discounted payoff of each firm going to be? 00:43:52.950 --> 00:43:57.205 align:middle line:84% So let's start with M. If we're in state M, 00:43:57.205 --> 00:43:59.830 align:middle line:84% It's a symmetric equilibrium, so we're just going to write down 00:43:59.830 --> 00:44:00.810 align:middle line:90% one firm's payoffs. 00:44:00.810 --> 00:44:02.685 align:middle line:84% All the other firms are going to be the same. 00:44:02.685 --> 00:44:04.910 align:middle line:84% If we're in state M, and we all follow the strategy, 00:44:04.910 --> 00:44:09.930 align:middle line:84% what is the expected discounted average payoff of each firm? 00:44:09.930 --> 00:44:12.412 align:middle line:90% 00:44:12.412 --> 00:44:13.870 align:middle line:84% Well, first let's see what happens. 00:44:13.870 --> 00:44:19.592 align:middle line:84% If we're in M, well today we all produced a monopoly quantity. 00:44:19.592 --> 00:44:22.050 align:middle line:84% We say in M. Tomorrow we all produce the monopoly quantity. 00:44:22.050 --> 00:44:23.090 align:middle line:84% We stay an M. The next day, we all 00:44:23.090 --> 00:44:24.290 align:middle line:90% produce the monopoly quantity. 00:44:24.290 --> 00:44:25.707 align:middle line:84% So we're going to get the monopoly 00:44:25.707 --> 00:44:27.150 align:middle line:90% quantity every single period. 00:44:27.150 --> 00:44:29.010 align:middle line:84% And what is the payoff from that-- 00:44:29.010 --> 00:44:31.590 align:middle line:90% 00:44:31.590 --> 00:44:35.150 align:middle line:84% the average discounted payoff from that? 00:44:35.150 --> 00:44:40.830 align:middle line:90% Well, it's just f of qM. 00:44:40.830 --> 00:44:44.350 align:middle line:84% f Of q is a function that tells me what is my payoff if we all 00:44:44.350 --> 00:44:46.510 align:middle line:90% produce quantity q? 00:44:46.510 --> 00:44:49.610 align:middle line:84% So if we all produce quantity qM, this is exactly my payoff. 00:44:49.610 --> 00:44:55.350 align:middle line:90% 00:44:55.350 --> 00:44:58.790 align:middle line:90% What about vNE? 00:44:58.790 --> 00:45:02.390 align:middle line:90% 00:45:02.390 --> 00:45:06.450 align:middle line:84% Well, here it's going to be f qNE. 00:45:06.450 --> 00:45:09.050 align:middle line:90% 00:45:09.050 --> 00:45:11.630 align:middle line:84% This is what are my average discounted payoff from here 00:45:11.630 --> 00:45:14.950 align:middle line:84% on if we're in state in this state. 00:45:14.950 --> 00:45:17.670 align:middle line:84% So now we're going to use the value functions to write down 00:45:17.670 --> 00:45:19.090 align:middle line:90% the equilibrium conditions. 00:45:19.090 --> 00:45:28.790 align:middle line:90% 00:45:28.790 --> 00:45:31.730 align:middle line:84% And when we use an automaton representation, 00:45:31.730 --> 00:45:34.750 align:middle line:84% we have one equilibrium condition for every state. 00:45:34.750 --> 00:45:40.430 align:middle line:84% So we're going to have state M, and we have state NE. 00:45:40.430 --> 00:45:42.150 align:middle line:84% And what our equilibrium condition says 00:45:42.150 --> 00:45:45.770 align:middle line:84% is if we're in state in this-- let's take this condition. 00:45:45.770 --> 00:45:47.830 align:middle line:84% If we're in state M, then each firm 00:45:47.830 --> 00:45:51.210 align:middle line:84% is better off continuing with their equilibrium strategy 00:45:51.210 --> 00:45:54.150 align:middle line:84% than choosing the best possible one-shot deviation. 00:45:54.150 --> 00:45:58.290 align:middle line:84% So each equilibrium is going to have the following form. 00:45:58.290 --> 00:46:01.590 align:middle line:90% 00:46:01.590 --> 00:46:03.510 align:middle line:84% In state M, we need to check that if I 00:46:03.510 --> 00:46:06.430 align:middle line:84% do what I'm supposed to, well, by definition 00:46:06.430 --> 00:46:07.297 align:middle line:90% I get my payoff vM. 00:46:07.297 --> 00:46:09.630 align:middle line:84% That's what it means if we continue with the equilibrium 00:46:09.630 --> 00:46:12.230 align:middle line:84% strategy in state M. And we have to compare 00:46:12.230 --> 00:46:14.190 align:middle line:84% that to the best possible payoff I could 00:46:14.190 --> 00:46:16.670 align:middle line:90% get from a one-shot deviation. 00:46:16.670 --> 00:46:18.230 align:middle line:84% And then we'll do the same in NE. 00:46:18.230 --> 00:46:19.450 align:middle line:90% Here we are today. 00:46:19.450 --> 00:46:22.170 align:middle line:84% This is what I'll get if I follow the equilibrium strategy. 00:46:22.170 --> 00:46:23.190 align:middle line:84% And on the right-hand side we want 00:46:23.190 --> 00:46:25.310 align:middle line:84% to say, what is the best possible payoff I could 00:46:25.310 --> 00:46:28.750 align:middle line:90% get from a one-shot deviation? 00:46:28.750 --> 00:46:30.950 align:middle line:90% So let's start-- 00:46:30.950 --> 00:46:33.770 align:middle line:84% I think maybe this one is easier. 00:46:33.770 --> 00:46:36.270 align:middle line:90% 00:46:36.270 --> 00:46:36.770 align:middle line:90% Oh, no. 00:46:36.770 --> 00:46:38.062 align:middle line:90% We can start with M. It's fine. 00:46:38.062 --> 00:46:41.030 align:middle line:90% So what is the best possible-- 00:46:41.030 --> 00:46:42.670 align:middle line:84% if I'm going to deviate, how should I 00:46:42.670 --> 00:46:45.310 align:middle line:84% deviate when we're in state M. Yeah? 00:46:45.310 --> 00:46:47.170 align:middle line:90% STUDENT: [INAUDIBLE]. 00:46:47.170 --> 00:46:48.170 align:middle line:90% IAN BALL: Exactly right. 00:46:48.170 --> 00:46:49.290 align:middle line:84% So if I'm going to deviate at all, 00:46:49.290 --> 00:46:50.490 align:middle line:90% I know I'm going to be punished. 00:46:50.490 --> 00:46:52.490 align:middle line:84% I'm going to be punished in exactly the same way 00:46:52.490 --> 00:46:53.702 align:middle line:90% no matter how I deviate. 00:46:53.702 --> 00:46:55.910 align:middle line:84% So I might as well choose the deviation that gives me 00:46:55.910 --> 00:46:57.090 align:middle line:90% the highest payoff today. 00:46:57.090 --> 00:46:58.910 align:middle line:84% And that's exactly-- and my payoff 00:46:58.910 --> 00:47:02.270 align:middle line:84% from that is exactly given by this function g. 00:47:02.270 --> 00:47:04.830 align:middle line:84% So the best possible deviation-- well, 00:47:04.830 --> 00:47:09.710 align:middle line:90% today I'm going to get g of qM. 00:47:09.710 --> 00:47:11.670 align:middle line:84% Remember, this doesn't mean I'm playing qM. 00:47:11.670 --> 00:47:13.910 align:middle line:84% This means if everyone else is playing qM, 00:47:13.910 --> 00:47:16.750 align:middle line:84% and I choose the quantity q prime 00:47:16.750 --> 00:47:19.350 align:middle line:84% that maximizes my flow payoff, this 00:47:19.350 --> 00:47:21.723 align:middle line:90% is the payoff I'm going to get. 00:47:21.723 --> 00:47:23.390 align:middle line:84% And then if I do that, what state are we 00:47:23.390 --> 00:47:24.410 align:middle line:90% going to be in tomorrow? 00:47:24.410 --> 00:47:27.870 align:middle line:90% 00:47:27.870 --> 00:47:29.250 align:middle line:90% Well, we're in this state. 00:47:29.250 --> 00:47:33.910 align:middle line:84% I deviate, which means we move to the Nash equilibrium state. 00:47:33.910 --> 00:47:36.230 align:middle line:84% So from tomorrow onward, we're going 00:47:36.230 --> 00:47:37.850 align:middle line:84% to be in the Nash equilibrium state. 00:47:37.850 --> 00:47:39.670 align:middle line:90% So my payoff is going to be vNE. 00:47:39.670 --> 00:47:42.110 align:middle line:84% But the only thing we're missing is discounting. 00:47:42.110 --> 00:47:44.350 align:middle line:84% Remember, with average discounted payoffs, 00:47:44.350 --> 00:47:46.550 align:middle line:84% how much weight do I put on what I get today, 00:47:46.550 --> 00:47:49.150 align:middle line:84% and how much weight do I put on the infinite future? 00:47:49.150 --> 00:47:52.630 align:middle line:84% Well, we know we always put 1 minus delta today and delta 00:47:52.630 --> 00:47:53.550 align:middle line:90% in the future. 00:47:53.550 --> 00:47:59.030 align:middle line:84% So we're going to have 1 minus delta here plus delta vNE. 00:47:59.030 --> 00:48:01.910 align:middle line:84% And it's really important that we interpret this correctly. 00:48:01.910 --> 00:48:05.270 align:middle line:84% The first term is what's the payoff I get today 00:48:05.270 --> 00:48:08.310 align:middle line:84% if I choose this optimal deviation? 00:48:08.310 --> 00:48:11.070 align:middle line:84% This term is not saying, what's my payoff tomorrow? 00:48:11.070 --> 00:48:14.130 align:middle line:84% This is incorporating my payoff tomorrow, the next day, 00:48:14.130 --> 00:48:15.790 align:middle line:90% the next day, and the next day. 00:48:15.790 --> 00:48:18.750 align:middle line:84% But by our definition of vNE, we've 00:48:18.750 --> 00:48:21.510 align:middle line:84% defined vNE to say, what is my payoff from today 00:48:21.510 --> 00:48:23.870 align:middle line:90% onwards once I'm in state NE? 00:48:23.870 --> 00:48:26.950 align:middle line:84% And that's exactly what we're going to measure on this side. 00:48:26.950 --> 00:48:28.830 align:middle line:84% So this is reflect-- it's a little tricky 00:48:28.830 --> 00:48:31.530 align:middle line:84% because in this case, it happens to agree with my flow payoff. 00:48:31.530 --> 00:48:33.072 align:middle line:84% But it's just important to understand 00:48:33.072 --> 00:48:35.810 align:middle line:84% that this is representing everything from tomorrow onward. 00:48:35.810 --> 00:48:38.430 align:middle line:84% It's not just representing what happens tomorrow. 00:48:38.430 --> 00:48:40.670 align:middle line:84% Because what I do today affects everything 00:48:40.670 --> 00:48:41.910 align:middle line:90% from tomorrow onwards. 00:48:41.910 --> 00:48:44.870 align:middle line:90% And now let's go to vNE. 00:48:44.870 --> 00:48:46.690 align:middle line:84% We're going to have a similar formula. 00:48:46.690 --> 00:48:51.390 align:middle line:84% What's our formula going to be down here? 00:48:51.390 --> 00:48:54.010 align:middle line:84% Well, I have the same optimal deviation. 00:48:54.010 --> 00:48:59.590 align:middle line:84% If I'm going to deviate, I might as well do the best thing. 00:48:59.590 --> 00:49:02.450 align:middle line:84% So it's a little tricky in this case. 00:49:02.450 --> 00:49:06.090 align:middle line:90% 00:49:06.090 --> 00:49:06.590 align:middle line:90% Yeah. 00:49:06.590 --> 00:49:07.610 align:middle line:90% So maybe-- 00:49:07.610 --> 00:49:15.132 align:middle line:90% 00:49:15.132 --> 00:49:16.590 align:middle line:84% Here, it's actually a little tricky 00:49:16.590 --> 00:49:20.630 align:middle line:84% because my best deviation since we're in a Nash equilibrium 00:49:20.630 --> 00:49:23.830 align:middle line:84% is actually to do what I'm supposed to do. 00:49:23.830 --> 00:49:25.790 align:middle line:84% But it actually doesn't matter here 00:49:25.790 --> 00:49:28.630 align:middle line:84% because we're going to stay in NE anyway. 00:49:28.630 --> 00:49:32.310 align:middle line:90% So let's see. 00:49:32.310 --> 00:49:35.830 align:middle line:84% Maybe this case, we should just-- 00:49:35.830 --> 00:49:37.870 align:middle line:84% yeah, I feel like this equation is misleading. 00:49:37.870 --> 00:49:39.610 align:middle line:90% It's true, but let's just say-- 00:49:39.610 --> 00:49:42.990 align:middle line:90% 00:49:42.990 --> 00:49:48.710 align:middle line:84% I'll just say best deviation, but my claim 00:49:48.710 --> 00:49:58.910 align:middle line:84% is that this is never profitable because if we're in the Nash 00:49:58.910 --> 00:50:02.337 align:middle line:84% equilibrium state today, whatever we do today, 00:50:02.337 --> 00:50:04.670 align:middle line:84% it can't be any better than playing the Nash equilibrium 00:50:04.670 --> 00:50:07.470 align:middle line:84% action, and it has no effect on what happens tomorrow. 00:50:07.470 --> 00:50:08.230 align:middle line:90% Yeah? 00:50:08.230 --> 00:50:09.890 align:middle line:90% STUDENT: I'm a little confused. 00:50:09.890 --> 00:50:11.690 align:middle line:84% You said how you define the value function, 00:50:11.690 --> 00:50:13.150 align:middle line:90% it's just equal to f. 00:50:13.150 --> 00:50:16.410 align:middle line:84% And as I understand, f is just my payoff today. 00:50:16.410 --> 00:50:17.990 align:middle line:90% IAN BALL: Ah. 00:50:17.990 --> 00:50:18.572 align:middle line:90% Great point. 00:50:18.572 --> 00:50:19.530 align:middle line:90% That is very confusing. 00:50:19.530 --> 00:50:20.822 align:middle line:90% Yeah, I see where you're going. 00:50:20.822 --> 00:50:22.640 align:middle line:84% It happens to agree with f, but that's 00:50:22.640 --> 00:50:24.890 align:middle line:84% simply because we're using average discounted payoffs. 00:50:24.890 --> 00:50:40.550 align:middle line:84% So maybe what I should write is 1 minus delta times f of qM-- 00:50:40.550 --> 00:50:42.470 align:middle line:90% let me get at good point-- 00:50:42.470 --> 00:50:47.810 align:middle line:84% plus delta f of qM, plus dot, dot, dot. 00:50:47.810 --> 00:50:50.350 align:middle line:90% That's really what I mean. 00:50:50.350 --> 00:50:51.950 align:middle line:90% Good point. 00:50:51.950 --> 00:50:54.310 align:middle line:84% So if I want to compute what happens, 00:50:54.310 --> 00:50:58.570 align:middle line:84% I'm starting in M. I'm going to get f of qM today. 00:50:58.570 --> 00:51:00.790 align:middle line:84% I'm going to get f of qM tomorrow and so on. 00:51:00.790 --> 00:51:03.990 align:middle line:84% I'm going to discount, and I'm going to take the average. 00:51:03.990 --> 00:51:10.050 align:middle line:84% It turns out that equals f of qM by the way we've defined it. 00:51:10.050 --> 00:51:10.787 align:middle line:90% So you're right. 00:51:10.787 --> 00:51:13.370 align:middle line:84% It's tricky in this case because it happens to reduce to this. 00:51:13.370 --> 00:51:15.848 align:middle line:84% But this is the interpretation of where that's coming from. 00:51:15.848 --> 00:51:16.890 align:middle line:90% And that reduces to this. 00:51:16.890 --> 00:51:18.090 align:middle line:90% Is that clear? 00:51:18.090 --> 00:51:18.850 align:middle line:90% Yeah, great. 00:51:18.850 --> 00:51:19.350 align:middle line:90% Thanks. 00:51:19.350 --> 00:51:21.392 align:middle line:84% I should have written it that way the first time. 00:51:21.392 --> 00:51:23.250 align:middle line:84% And of course, the same applies here. 00:51:23.250 --> 00:51:23.750 align:middle line:90% Yes? 00:51:23.750 --> 00:51:25.550 align:middle line:84% STUDENT: For the Nash equilibrium, 00:51:25.550 --> 00:51:28.650 align:middle line:84% you just said vNE is [INAUDIBLE] [? never ?] profitable. 00:51:28.650 --> 00:51:33.578 align:middle line:84% Is that just by definition that we are in a Nash equilibrium? 00:51:33.578 --> 00:51:35.370 align:middle line:84% IAN BALL: It's a little trickier than that. 00:51:35.370 --> 00:51:37.967 align:middle line:90% So I'm hesitant to write. 00:51:37.967 --> 00:51:40.050 align:middle line:84% Maybe I should go back and write it the other way. 00:51:40.050 --> 00:51:51.030 align:middle line:84% What's tricky is it's the fact that we're in a Nash equilibrium 00:51:51.030 --> 00:51:54.673 align:middle line:84% and that tomorrow we're going to be 00:51:54.673 --> 00:51:56.090 align:middle line:84% in the same state, no matter what. 00:51:56.090 --> 00:51:58.690 align:middle line:84% So maybe let me write it out more formally. 00:51:58.690 --> 00:52:00.130 align:middle line:90% So what we need to show-- 00:52:00.130 --> 00:52:03.632 align:middle line:90% 00:52:03.632 --> 00:52:05.590 align:middle line:84% so in the first case, I can use this function g 00:52:05.590 --> 00:52:06.490 align:middle line:90% to simplify my life. 00:52:06.490 --> 00:52:07.698 align:middle line:90% Here, it's a little trickier. 00:52:07.698 --> 00:52:14.490 align:middle line:84% So really what I need is it's greater than pi qi prime. 00:52:14.490 --> 00:52:30.198 align:middle line:90% 00:52:30.198 --> 00:52:31.490 align:middle line:90% Yeah, let me write it this way. 00:52:31.490 --> 00:52:32.890 align:middle line:84% And actually, the first one we could write this way, 00:52:32.890 --> 00:52:33.650 align:middle line:90% but it reduces. 00:52:33.650 --> 00:52:36.090 align:middle line:90% So I'm firm i. 00:52:36.090 --> 00:52:39.750 align:middle line:84% What I could do instead of following my equilibrium 00:52:39.750 --> 00:52:43.670 align:middle line:84% strategy is I could instead produce qi prime today. 00:52:43.670 --> 00:52:46.110 align:middle line:84% If I produce qi prime today, and everyone else 00:52:46.110 --> 00:52:48.370 align:middle line:84% is producing the Nash equilibrium amount, 00:52:48.370 --> 00:52:50.270 align:middle line:84% then this is my flow payoff today. 00:52:50.270 --> 00:52:53.030 align:middle line:84% And we multiply that by 1 minus delta. 00:52:53.030 --> 00:52:56.390 align:middle line:84% Whatever I do today, we're going to stay in the state Nash 00:52:56.390 --> 00:52:57.330 align:middle line:90% equilibrium. 00:52:57.330 --> 00:53:00.910 align:middle line:84% Because if we look at the arrows here, whatever I do, 00:53:00.910 --> 00:53:03.370 align:middle line:84% the arrow brings us back to the Nash equilibrium state. 00:53:03.370 --> 00:53:06.050 align:middle line:84% So we're going to get delta and Nash equilibrium tomorrow. 00:53:06.050 --> 00:53:09.590 align:middle line:90% 00:53:09.590 --> 00:53:12.670 align:middle line:84% And how do we see that this is less than or equal to this? 00:53:12.670 --> 00:53:16.368 align:middle line:84% Well, whatever qi prime might choose, 00:53:16.368 --> 00:53:17.910 align:middle line:84% it can't give me a higher flow payoff 00:53:17.910 --> 00:53:19.470 align:middle line:84% than choosing the Nash equilibrium 00:53:19.470 --> 00:53:22.710 align:middle line:84% quantity because that's the definition of a best response. 00:53:22.710 --> 00:53:30.350 align:middle line:84% So we see that this is less than or equal to 1 minus delta f 00:53:30.350 --> 00:53:35.550 align:middle line:90% of qNE plus delta vNE. 00:53:35.550 --> 00:53:39.430 align:middle line:90% And that's exactly vNE. 00:53:39.430 --> 00:53:41.870 align:middle line:84% So how do we see that this right-hand side 00:53:41.870 --> 00:53:43.330 align:middle line:90% is smaller than what we want? 00:53:43.330 --> 00:53:46.390 align:middle line:84% Well, all I've done is I've compared the flow payoff 00:53:46.390 --> 00:53:47.950 align:middle line:90% to the Nash equilibrium payoff. 00:53:47.950 --> 00:53:50.570 align:middle line:84% Whatever quantity I choose, it can't be as good. 00:53:50.570 --> 00:53:53.390 align:middle line:84% It's no better than playing my best response, which is exactly 00:53:53.390 --> 00:53:57.630 align:middle line:84% the Nash equilibrium quantity, and that gives me these payoffs. 00:53:57.630 --> 00:54:01.830 align:middle line:84% And then I substitute in this over here, 00:54:01.830 --> 00:54:03.990 align:middle line:90% and we're going to get this. 00:54:03.990 --> 00:54:07.290 align:middle line:90% So any questions on that? 00:54:07.290 --> 00:54:10.550 align:middle line:90% 00:54:10.550 --> 00:54:12.210 align:middle line:84% So the action is going to be here. 00:54:12.210 --> 00:54:15.790 align:middle line:90% 00:54:15.790 --> 00:54:18.790 align:middle line:84% We want to check that star is satisfied, 00:54:18.790 --> 00:54:22.970 align:middle line:84% and let's plug things in and see what we get. 00:54:22.970 --> 00:54:51.190 align:middle line:90% 00:54:51.190 --> 00:54:53.710 align:middle line:90% So let's look at star. 00:54:53.710 --> 00:54:55.910 align:middle line:84% Well, we have our inequality, star. 00:54:55.910 --> 00:54:58.575 align:middle line:84% And we have our values for the v's on the left side. 00:54:58.575 --> 00:54:59.950 align:middle line:84% So we're just going to substitute 00:54:59.950 --> 00:55:01.950 align:middle line:84% these values in to our inequality over here 00:55:01.950 --> 00:55:02.890 align:middle line:90% and see what we get. 00:55:02.890 --> 00:55:05.370 align:middle line:90% So vM becomes f of qM. 00:55:05.370 --> 00:55:08.590 align:middle line:90% 00:55:08.590 --> 00:55:12.750 align:middle line:84% And that needs to be greater than or equal to 1 00:55:12.750 --> 00:55:20.910 align:middle line:84% minus delta g of qM plus delta times vNE, which we look over 00:55:20.910 --> 00:55:22.250 align:middle line:90% here, that's f of qM. 00:55:22.250 --> 00:55:28.005 align:middle line:90% 00:55:28.005 --> 00:55:29.630 align:middle line:84% And here, I think we can really capture 00:55:29.630 --> 00:55:32.510 align:middle line:84% exactly the idea of the temptation of the deviation. 00:55:32.510 --> 00:55:36.350 align:middle line:84% If I don't deviate, I get f of qM forever after. 00:55:36.350 --> 00:55:40.910 align:middle line:84% If I do deviate, I get maybe a higher number today because I 00:55:40.910 --> 00:55:42.990 align:middle line:90% increase my flow payoff today. 00:55:42.990 --> 00:55:45.510 align:middle line:84% But then forever after, I get a lower number. 00:55:45.510 --> 00:55:53.950 align:middle line:84% So let's note that g of qM is strictly greater than f 00:55:53.950 --> 00:55:58.570 align:middle line:84% of qM, which is, in turn, strictly greater than f of qNE. 00:55:58.570 --> 00:56:01.550 align:middle line:90% 00:56:01.550 --> 00:56:03.190 align:middle line:84% So this is what I'm supposed to get. 00:56:03.190 --> 00:56:06.830 align:middle line:84% If I deviate, I bump up my payoff to this today, 00:56:06.830 --> 00:56:09.310 align:middle line:84% but then I drop down to this forever after. 00:56:09.310 --> 00:56:11.310 align:middle line:84% And we want to make sure that I'm patient enough 00:56:11.310 --> 00:56:14.390 align:middle line:84% that the dropdown forever after offsets, or sufficiently 00:56:14.390 --> 00:56:16.210 align:middle line:84% offsets the potential game today. 00:56:16.210 --> 00:56:18.790 align:middle line:90% 00:56:18.790 --> 00:56:20.110 align:middle line:90% Any questions? 00:56:20.110 --> 00:56:29.670 align:middle line:90% 00:56:29.670 --> 00:56:34.950 align:middle line:84% So now let's just do some algebra. 00:56:34.950 --> 00:56:42.170 align:middle line:84% Let's bring-- we want to bring everything to this side I think. 00:56:42.170 --> 00:56:45.410 align:middle line:84% So what we're going to have is delta-- 00:56:45.410 --> 00:56:48.480 align:middle line:90% 00:56:48.480 --> 00:56:50.230 align:middle line:84% I want to bring everything involving delta 00:56:50.230 --> 00:56:52.397 align:middle line:84% to the left side, and everything not involving delta 00:56:52.397 --> 00:56:53.390 align:middle line:90% to the right side. 00:56:53.390 --> 00:56:55.790 align:middle line:90% So I have minus delta g of qM. 00:56:55.790 --> 00:56:58.510 align:middle line:84% I'll bring that to the left side, and that becomes plus. 00:56:58.510 --> 00:57:00.990 align:middle line:90% Then I have plus delta f of qNE. 00:57:00.990 --> 00:57:02.890 align:middle line:84% I'll subtract that and bring it to this side. 00:57:02.890 --> 00:57:09.090 align:middle line:84% So I get minus f of qNE greater than or equal to-- 00:57:09.090 --> 00:57:20.710 align:middle line:90% 00:57:20.710 --> 00:57:25.230 align:middle line:84% So now let me just divide, and I just get delta greater than 00:57:25.230 --> 00:57:27.470 align:middle line:84% or equal to this threshold, maybe 00:57:27.470 --> 00:57:33.110 align:middle line:84% delta upper bar, which is defined to equal g of qM 00:57:33.110 --> 00:57:34.650 align:middle line:90% minus f of qM. 00:57:34.650 --> 00:57:50.070 align:middle line:90% 00:57:50.070 --> 00:57:55.110 align:middle line:84% And I sometimes think it's easier to decompose it this way. 00:57:55.110 --> 00:57:56.330 align:middle line:90% This is g of-- 00:57:56.330 --> 00:58:31.310 align:middle line:90% 00:58:31.310 --> 00:58:34.590 align:middle line:84% So as we expected, this is going to be a subgame perfect Nash 00:58:34.590 --> 00:58:36.670 align:middle line:84% equilibrium if delta is large enough, 00:58:36.670 --> 00:58:38.550 align:middle line:84% if the players are patient enough. 00:58:38.550 --> 00:58:44.350 align:middle line:84% And let's try to interpret the payoffs over here. 00:58:44.350 --> 00:58:46.590 align:middle line:84% Well, what I want to think is I want 00:58:46.590 --> 00:58:50.830 align:middle line:84% to think that this gap represents my one-shot gain 00:58:50.830 --> 00:58:51.870 align:middle line:90% from deviating. 00:58:51.870 --> 00:58:56.870 align:middle line:90% 00:58:56.870 --> 00:59:02.530 align:middle line:84% But this gap-- this is my gain today from deviating, 00:59:02.530 --> 00:59:04.850 align:middle line:84% And this is my loss forever after. 00:59:04.850 --> 00:59:07.950 align:middle line:90% 00:59:07.950 --> 00:59:11.790 align:middle line:84% Maybe we'll call this g equals gain, and this loss equals l. 00:59:11.790 --> 00:59:14.950 align:middle line:84% So we're supposed to be playing the monopoly 00:59:14.950 --> 00:59:16.250 align:middle line:90% quantity every period. 00:59:16.250 --> 00:59:18.630 align:middle line:90% If I deviate, I can overproduce. 00:59:18.630 --> 00:59:22.110 align:middle line:84% I do strictly better today, I gain g for myself. 00:59:22.110 --> 00:59:25.590 align:middle line:84% But forever after, instead of getting the monopoly quantity, 00:59:25.590 --> 00:59:27.490 align:middle line:84% we all produce the Nash equilibrium quantity, 00:59:27.490 --> 00:59:28.490 align:middle line:90% which is strictly worse. 00:59:28.490 --> 00:59:32.470 align:middle line:90% So we lose this amount l. 00:59:32.470 --> 00:59:34.350 align:middle line:84% And it turns out, if you just look 00:59:34.350 --> 00:59:37.630 align:middle line:84% at the algebra I've done here, g of qM minus f 00:59:37.630 --> 00:59:39.810 align:middle line:90% of qM, that's exactly the gain. 00:59:39.810 --> 00:59:42.110 align:middle line:90% So we get g over here. 00:59:42.110 --> 00:59:45.610 align:middle line:84% And at the bottom, we exactly have the gain plus the loss. 00:59:45.610 --> 00:59:49.390 align:middle line:90% 00:59:49.390 --> 00:59:52.830 align:middle line:84% And this is kind of a general formula that says, 00:59:52.830 --> 00:59:54.750 align:middle line:90% what's the requirement on delta? 00:59:54.750 --> 00:59:56.950 align:middle line:84% It has to be large enough, roughly, 00:59:56.950 --> 01:00:01.470 align:middle line:84% that the potential one period gain I get 01:00:01.470 --> 01:00:04.830 align:middle line:84% is offset by the eventual loss I experience. 01:00:04.830 --> 01:00:08.898 align:middle line:84% So let's make sure we understand how this varies with g and l. 01:00:08.898 --> 01:00:09.690 align:middle line:90% So let me ask this. 01:00:09.690 --> 01:00:13.170 align:middle line:84% So if g gets bigger, what happens to delta bar? 01:00:13.170 --> 01:00:16.820 align:middle line:90% 01:00:16.820 --> 01:00:18.570 align:middle line:84% Let's start with L. That's the easier one. 01:00:18.570 --> 01:00:21.950 align:middle line:90% If L gets bigger what happens? 01:00:21.950 --> 01:00:22.657 align:middle line:90% Yeah? 01:00:22.657 --> 01:00:23.990 align:middle line:90% STUDENT: I'm going to be honest. 01:00:23.990 --> 01:00:25.710 align:middle line:84% I'm having trouble working out the math. 01:00:25.710 --> 01:00:27.550 align:middle line:90% IAN BALL: But yeah. 01:00:27.550 --> 01:00:28.718 align:middle line:90% So this is just algebra. 01:00:28.718 --> 01:00:30.010 align:middle line:90% You can check that on your own. 01:00:30.010 --> 01:00:34.030 align:middle line:84% Let's just look at this and this formula over here, just g over g 01:00:34.030 --> 01:00:34.910 align:middle line:90% plus l. 01:00:34.910 --> 01:00:36.470 align:middle line:84% STUDENT: As the loss gets bigger, 01:00:36.470 --> 01:00:38.035 align:middle line:84% delta is going to have to be smaller. 01:00:38.035 --> 01:00:39.410 align:middle line:84% IAN BALL: Delta bar gets smaller. 01:00:39.410 --> 01:00:41.108 align:middle line:90% And what is that? 01:00:41.108 --> 01:00:42.150 align:middle line:90% So can we interpret that? 01:00:42.150 --> 01:00:44.410 align:middle line:84% As the loss gets bigger, delta bar gets smaller. 01:00:44.410 --> 01:00:45.302 align:middle line:90% So what's the-- 01:00:45.302 --> 01:00:47.510 align:middle line:84% STUDENT: That just means you have to put more value-- 01:00:47.510 --> 01:00:50.790 align:middle line:84% or put less value on the future if you're going to be moving 01:00:50.790 --> 01:00:51.723 align:middle line:90% more in the future. 01:00:51.723 --> 01:00:52.390 align:middle line:90% IAN BALL: Right. 01:00:52.390 --> 01:00:54.050 align:middle line:84% So it's easier to sustain equilibrium. 01:00:54.050 --> 01:00:56.550 align:middle line:84% If the loss gets larger, that means the punishment 01:00:56.550 --> 01:00:57.990 align:middle line:90% is very powerful. 01:00:57.990 --> 01:01:00.230 align:middle line:84% If I get caught for deviating, tomorrow I 01:01:00.230 --> 01:01:01.650 align:middle line:90% experience this huge loss. 01:01:01.650 --> 01:01:03.630 align:middle line:84% If I experience a huge loss for deviating, 01:01:03.630 --> 01:01:05.490 align:middle line:84% it's not very attractive for me to deviate. 01:01:05.490 --> 01:01:07.750 align:middle line:84% Therefore, it's easier to sustain the equilibrium. 01:01:07.750 --> 01:01:09.870 align:middle line:84% Because if delta bar is smaller, that 01:01:09.870 --> 01:01:11.990 align:middle line:84% means I don't have to be as patient for this 01:01:11.990 --> 01:01:13.430 align:middle line:90% to be an equilibrium. 01:01:13.430 --> 01:01:16.150 align:middle line:84% If l is really, really big, and the punishment is really, really 01:01:16.150 --> 01:01:18.048 align:middle line:84% bad, even if I'm pretty impatient, 01:01:18.048 --> 01:01:19.590 align:middle line:84% this still constitutes an equilibrium 01:01:19.590 --> 01:01:22.670 align:middle line:84% because the punishment is so severe. 01:01:22.670 --> 01:01:25.830 align:middle line:90% So more delta satisfy this-- 01:01:25.830 --> 01:01:28.230 align:middle line:84% even smaller deltas, more impatient deltas, 01:01:28.230 --> 01:01:30.190 align:middle line:90% when l is very large. 01:01:30.190 --> 01:01:30.930 align:middle line:90% What about g? 01:01:30.930 --> 01:01:32.252 align:middle line:90% Can we also reason through g? 01:01:32.252 --> 01:01:33.710 align:middle line:84% This is a little trickier because g 01:01:33.710 --> 01:01:37.250 align:middle line:84% is on the top and the bottom, but it is increasing in g. 01:01:37.250 --> 01:01:41.190 align:middle line:84% Maybe it's not quite so easy to see, but it is. 01:01:41.190 --> 01:01:44.950 align:middle line:90% So why is this increasing in g? 01:01:44.950 --> 01:01:45.590 align:middle line:90% Yeah? 01:01:45.590 --> 01:01:47.132 align:middle line:84% STUDENT: Well, if you're able to gain 01:01:47.132 --> 01:01:49.270 align:middle line:84% more than the threshold for deciding 01:01:49.270 --> 01:01:51.703 align:middle line:84% that you shouldn't deviate would need to be higher. 01:01:51.703 --> 01:01:52.370 align:middle line:90% IAN BALL: Right. 01:01:52.370 --> 01:01:55.750 align:middle line:84% If I can gain a lot today, it's going to be really, really 01:01:55.750 --> 01:01:57.230 align:middle line:90% tempting for me to deviate. 01:01:57.230 --> 01:01:59.710 align:middle line:84% So it's going to be harder to sustain the equilibrium. 01:01:59.710 --> 01:02:01.810 align:middle line:84% And that's only going to work if the players are really, really, 01:02:01.810 --> 01:02:02.750 align:middle line:90% really patient. 01:02:02.750 --> 01:02:06.510 align:middle line:84% So this threshold gets higher, meaning fewer deltas satisfy it 01:02:06.510 --> 01:02:10.870 align:middle line:84% because the temptation to deviate gets really, really big. 01:02:10.870 --> 01:02:13.150 align:middle line:90% And you can check that-- 01:02:13.150 --> 01:02:14.870 align:middle line:84% everything's in terms of f and g. 01:02:14.870 --> 01:02:18.057 align:middle line:84% But remember, at the end of the day, f and g involve n. 01:02:18.057 --> 01:02:20.390 align:middle line:84% So let's just see if we can get some intuition for this. 01:02:20.390 --> 01:02:21.890 align:middle line:84% Well, maybe this is too hard to see, 01:02:21.890 --> 01:02:24.990 align:middle line:84% but it turns out that delta bar, you can check, 01:02:24.990 --> 01:02:28.430 align:middle line:90% is increasing in n. 01:02:28.430 --> 01:02:30.290 align:middle line:84% So can we give an economic story? 01:02:30.290 --> 01:02:34.110 align:middle line:90% What does this tell us? 01:02:34.110 --> 01:02:36.990 align:middle line:84% Or what might we expect about which cartels are successful 01:02:36.990 --> 01:02:40.428 align:middle line:84% and which cartels are not successful based on this fact? 01:02:40.428 --> 01:02:41.470 align:middle line:90% STUDENT: Smaller cartels. 01:02:41.470 --> 01:02:43.338 align:middle line:84% IAN BALL: Smaller cartels meaning what? 01:02:43.338 --> 01:02:44.130 align:middle line:90% Yeah, you're right. 01:02:44.130 --> 01:02:44.590 align:middle line:90% But-- 01:02:44.590 --> 01:02:46.790 align:middle line:84% STUDENT: Like the ones that have less players involved. 01:02:46.790 --> 01:02:47.790 align:middle line:90% IAN BALL: Fewer firms, right? 01:02:47.790 --> 01:02:48.890 align:middle line:90% The players themselves are big. 01:02:48.890 --> 01:02:50.290 align:middle line:84% So we have to be careful about what small means. 01:02:50.290 --> 01:02:53.250 align:middle line:84% The players are bigger, but the number of players is smaller. 01:02:53.250 --> 01:02:57.110 align:middle line:84% So as n gets bigger, we have to be really, really patient 01:02:57.110 --> 01:02:59.350 align:middle line:90% to sustain cartel behavior. 01:02:59.350 --> 01:03:02.630 align:middle line:84% And given that people aren't necessarily that patient, 01:03:02.630 --> 01:03:03.990 align:middle line:90% that may not work out. 01:03:03.990 --> 01:03:05.750 align:middle line:84% So we would expect that cartels might 01:03:05.750 --> 01:03:09.490 align:middle line:84% be more successful and last longer when n is small 01:03:09.490 --> 01:03:11.490 align:middle line:84% and we just have a few big players interacting. 01:03:11.490 --> 01:03:12.990 align:middle line:84% And indeed, that's pretty consistent 01:03:12.990 --> 01:03:14.830 align:middle line:84% with the empirical evidence, that it's 01:03:14.830 --> 01:03:18.710 align:middle line:84% hard for many, many firms to coordinate and work together 01:03:18.710 --> 01:03:20.310 align:middle line:90% and sustain cartel behavior. 01:03:20.310 --> 01:03:22.430 align:middle line:84% Even when there's few firms, things do break down. 01:03:22.430 --> 01:03:27.042 align:middle line:84% So this may seem kind of just a math story. 01:03:27.042 --> 01:03:28.250 align:middle line:90% We have the Nash equilibrium. 01:03:28.250 --> 01:03:29.042 align:middle line:90% We have a monopoly. 01:03:29.042 --> 01:03:32.670 align:middle line:84% But this is actually exactly what happened in the 1980s. 01:03:32.670 --> 01:03:35.390 align:middle line:84% Saudi Arabia got frustrated that everyone else was overproducing 01:03:35.390 --> 01:03:37.710 align:middle line:84% a little bit, and they flooded the market, and the price of oil 01:03:37.710 --> 01:03:38.690 align:middle line:90% dropped dramatically. 01:03:38.690 --> 01:03:40.770 align:middle line:84% So we see exactly this kind of punishment. 01:03:40.770 --> 01:03:43.365 align:middle line:84% Saudi Arabia said, look, I'm tired of these other countries 01:03:43.365 --> 01:03:45.490 align:middle line:84% overproducing relative to what they're supposed to. 01:03:45.490 --> 01:03:47.573 align:middle line:84% And they punished everyone by flooding the market, 01:03:47.573 --> 01:03:48.730 align:middle line:90% and the price dropped. 01:03:48.730 --> 01:03:52.870 align:middle line:84% And this had huge effects globally. 01:03:52.870 --> 01:03:56.150 align:middle line:84% Let's do one more equilibrium construction 01:03:56.150 --> 01:03:57.650 align:middle line:90% that I think just kind of. 01:03:57.650 --> 01:04:00.710 align:middle line:90% 01:04:00.710 --> 01:04:02.030 align:middle line:90% Illustrates things. 01:04:02.030 --> 01:04:04.530 align:middle line:90% So let's do a final equilibrium. 01:04:04.530 --> 01:04:07.130 align:middle line:84% So before, we did a Nash reversion equilibrium. 01:04:07.130 --> 01:04:09.470 align:middle line:84% Now we're going to do what's called a carrot 01:04:09.470 --> 01:04:10.730 align:middle line:90% and stick equilibrium. 01:04:10.730 --> 01:04:14.070 align:middle line:90% 01:04:14.070 --> 01:04:17.350 align:middle line:84% And what was distinctive about this equilibrium 01:04:17.350 --> 01:04:20.510 align:middle line:84% is once someone deviated, we reverted to Nash equilibrium 01:04:20.510 --> 01:04:21.605 align:middle line:90% forever after. 01:04:21.605 --> 01:04:23.730 align:middle line:84% But if you look at what happened with Saudi Arabia, 01:04:23.730 --> 01:04:26.790 align:middle line:84% they didn't just produce a high amount of oil forever. 01:04:26.790 --> 01:04:28.608 align:middle line:84% They flooded the market for a few years, 01:04:28.608 --> 01:04:31.150 align:middle line:84% and then they eventually brought their production down again, 01:04:31.150 --> 01:04:32.410 align:middle line:90% and prices went up again. 01:04:32.410 --> 01:04:34.150 align:middle line:84% So what we often see in these cartels 01:04:34.150 --> 01:04:36.650 align:middle line:84% is punishments of limited duration. 01:04:36.650 --> 01:04:37.525 align:middle line:90% Someone overproduces. 01:04:37.525 --> 01:04:39.692 align:middle line:84% We punish them for a year or two, or a month or two, 01:04:39.692 --> 01:04:40.330 align:middle line:90% or whatever. 01:04:40.330 --> 01:04:42.370 align:middle line:84% And then we get back to the good thing that we are doing. 01:04:42.370 --> 01:04:44.030 align:middle line:84% Because ultimately, that's what we want to do. 01:04:44.030 --> 01:04:45.863 align:middle line:84% So the question is, can we have these carrot 01:04:45.863 --> 01:04:46.950 align:middle line:90% and stick strategies? 01:04:46.950 --> 01:04:48.350 align:middle line:84% And the way this is going to work 01:04:48.350 --> 01:04:51.190 align:middle line:84% is we're going to have two production levels-- qc 01:04:51.190 --> 01:04:53.230 align:middle line:90% less than qs. 01:04:53.230 --> 01:04:55.590 align:middle line:84% So carrot and stick are generic terms for carrot 01:04:55.590 --> 01:04:57.850 align:middle line:84% is a reward and stick is a punishment. 01:04:57.850 --> 01:04:59.610 align:middle line:84% This is like, I guess, animals or pets. 01:04:59.610 --> 01:05:00.830 align:middle line:90% You give it a carrot or you-- 01:05:00.830 --> 01:05:02.070 align:middle line:90% I mean, it's kind of violent. 01:05:02.070 --> 01:05:04.630 align:middle line:90% But anyway, that's what it is. 01:05:04.630 --> 01:05:05.290 align:middle line:90% So OK. 01:05:05.290 --> 01:05:07.582 align:middle line:84% I'll use this-- maybe I shouldn't use this terminology, 01:05:07.582 --> 01:05:09.370 align:middle line:84% but I've inherited this terminology. 01:05:09.370 --> 01:05:11.510 align:middle line:90% I'll use it. 01:05:11.510 --> 01:05:16.410 align:middle line:84% So the carrot is the low production level. 01:05:16.410 --> 01:05:18.390 align:middle line:84% This is the good thing that we like. 01:05:18.390 --> 01:05:20.158 align:middle line:84% The stick is the high production level. 01:05:20.158 --> 01:05:22.450 align:middle line:84% So this could be the Nash equilibrium and the monopoly, 01:05:22.450 --> 01:05:23.170 align:middle line:90% but it doesn't have to be. 01:05:23.170 --> 01:05:24.710 align:middle line:84% It's just general production levels. 01:05:24.710 --> 01:05:28.030 align:middle line:84% And the idea is we're going to start with the carrot. 01:05:28.030 --> 01:05:29.750 align:middle line:84% And if someone deviates, we're all 01:05:29.750 --> 01:05:32.270 align:middle line:84% going to produce the stick level tomorrow, but only 01:05:32.270 --> 01:05:32.952 align:middle line:90% for one period. 01:05:32.952 --> 01:05:34.910 align:middle line:84% Then if everyone does what they're supposed to, 01:05:34.910 --> 01:05:36.910 align:middle line:84% we come back to the carrot level. 01:05:36.910 --> 01:05:41.270 align:middle line:84% So let's try to represent this as an automaton. 01:05:41.270 --> 01:05:43.610 align:middle line:84% So let me erase this automaton and get a new one. 01:05:43.610 --> 01:05:48.350 align:middle line:90% 01:05:48.350 --> 01:05:50.310 align:middle line:84% So now our automaton is going to have 01:05:50.310 --> 01:05:57.150 align:middle line:84% two states-- the carrot state and the stick state, so c and s. 01:05:57.150 --> 01:05:59.495 align:middle line:84% And we're going to start in the carrot state. 01:05:59.495 --> 01:06:00.870 align:middle line:84% And in the carrot state, everyone 01:06:00.870 --> 01:06:04.670 align:middle line:84% produces this low-level c, which is good. 01:06:04.670 --> 01:06:06.790 align:middle line:84% And in the stick state everyone produces 01:06:06.790 --> 01:06:13.390 align:middle line:84% this high-level qs, which is not so good for everyone. 01:06:13.390 --> 01:06:14.890 align:middle line:84% And now we have to say what happens. 01:06:14.890 --> 01:06:16.150 align:middle line:84% I think in this case, the transitions 01:06:16.150 --> 01:06:17.590 align:middle line:90% are a little more complicated. 01:06:17.590 --> 01:06:19.550 align:middle line:84% So in the carrot state, we're going 01:06:19.550 --> 01:06:22.470 align:middle line:84% to stay in the carrot state if everyone produces the carrot 01:06:22.470 --> 01:06:24.350 align:middle line:90% level. 01:06:24.350 --> 01:06:26.270 align:middle line:84% So we have an arrow here if what? 01:06:26.270 --> 01:06:31.170 align:middle line:90% If qi equals qc for all i. 01:06:31.170 --> 01:06:34.710 align:middle line:84% If everyone does what they're supposed to, we stay here. 01:06:34.710 --> 01:06:37.330 align:middle line:84% Otherwise, we go to the sixth state. 01:06:37.330 --> 01:06:44.270 align:middle line:90% 01:06:44.270 --> 01:06:47.970 align:middle line:84% But now this is where it gets a little tricky. 01:06:47.970 --> 01:06:49.030 align:middle line:90% Let's see. 01:06:49.030 --> 01:06:51.642 align:middle line:84% Can we figure out what happens in the stick state? 01:06:51.642 --> 01:06:53.350 align:middle line:84% What happens if we're in the stick state, 01:06:53.350 --> 01:06:55.250 align:middle line:84% and everyone produces the stick quantity? 01:06:55.250 --> 01:06:56.490 align:middle line:90% What state do we move to? 01:06:56.490 --> 01:06:59.743 align:middle line:90% Do we stay, or do we move? 01:06:59.743 --> 01:07:00.910 align:middle line:90% STUDENT: Move to the carrot. 01:07:00.910 --> 01:07:01.470 align:middle line:90% IAN BALL: We move back. 01:07:01.470 --> 01:07:03.890 align:middle line:84% Because now if everyone does what they're supposed to-- 01:07:03.890 --> 01:07:05.730 align:middle line:90% OK, we did our punishment. 01:07:05.730 --> 01:07:08.950 align:middle line:84% We're now ready to get back to where we were before. 01:07:08.950 --> 01:07:17.010 align:middle line:84% So now we're going to have this arrow if qi equals qs for all i. 01:07:17.010 --> 01:07:20.350 align:middle line:90% 01:07:20.350 --> 01:07:24.710 align:middle line:84% And then what happens if someone doesn't play the stick? 01:07:24.710 --> 01:07:27.488 align:middle line:90% What do we do? 01:07:27.488 --> 01:07:29.030 align:middle line:84% Well, then we stay in the stick state 01:07:29.030 --> 01:07:30.390 align:middle line:90% and punish them again tomorrow. 01:07:30.390 --> 01:07:35.390 align:middle line:90% 01:07:35.390 --> 01:07:37.710 align:middle line:84% So at first this may look confusing. 01:07:37.710 --> 01:07:40.900 align:middle line:84% But I think the way to interpret it is in either state, 01:07:40.900 --> 01:07:42.650 align:middle line:84% if everyone does what they're supposed to, 01:07:42.650 --> 01:07:44.870 align:middle line:84% we move to the carrot state tomorrow. 01:07:44.870 --> 01:07:47.270 align:middle line:84% That's this arrow and this arrow. 01:07:47.270 --> 01:07:49.190 align:middle line:84% If anyone does what they're not supposed to, 01:07:49.190 --> 01:07:51.150 align:middle line:84% we go to the stick state tomorrow. 01:07:51.150 --> 01:07:52.708 align:middle line:84% That's this arrow and this arrow. 01:07:52.708 --> 01:07:54.750 align:middle line:84% And that's true whether we're in the carrot state 01:07:54.750 --> 01:07:56.290 align:middle line:90% today or the stick state today. 01:07:56.290 --> 01:07:59.267 align:middle line:84% Either way, we go to the stick state tomorrow. 01:07:59.267 --> 01:08:00.850 align:middle line:84% I should say when I write "otherwise," 01:08:00.850 --> 01:08:02.350 align:middle line:90% it can be a bit ambiguous. 01:08:02.350 --> 01:08:05.510 align:middle line:84% Otherwise is referring to the source of the node. 01:08:05.510 --> 01:08:08.610 align:middle line:84% So in the c state think of, we're in this state. 01:08:08.610 --> 01:08:09.870 align:middle line:90% Look at all the arrows. 01:08:09.870 --> 01:08:12.288 align:middle line:84% We could either do this, or otherwise we do this. 01:08:12.288 --> 01:08:14.830 align:middle line:84% And then similarly, if I'm in the stick state, if we do this, 01:08:14.830 --> 01:08:15.410 align:middle line:90% we go here. 01:08:15.410 --> 01:08:16.550 align:middle line:90% And otherwise, we go here. 01:08:16.550 --> 01:08:18.270 align:middle line:84% But when you just see all the otherwises 01:08:18.270 --> 01:08:19.729 align:middle line:84% you might think, what are they referring to? 01:08:19.729 --> 01:08:21.396 align:middle line:84% They're referring to the state we're in, 01:08:21.396 --> 01:08:23.750 align:middle line:90% the source of all the arrows. 01:08:23.750 --> 01:08:26.990 align:middle line:84% So now, I don't want to do tons of algebra, 01:08:26.990 --> 01:08:29.149 align:middle line:84% but I just want to see if we can set up 01:08:29.149 --> 01:08:31.710 align:middle line:84% the value function and the equilibrium conditions 01:08:31.710 --> 01:08:32.651 align:middle line:90% for this equilibrium. 01:08:32.651 --> 01:08:34.109 align:middle line:84% And then on the problem set, you'll 01:08:34.109 --> 01:08:35.569 align:middle line:84% get to actually do all the algebra. 01:08:35.569 --> 01:08:37.790 align:middle line:84% But I think here, we'll see the benefits of this value function 01:08:37.790 --> 01:08:38.590 align:middle line:90% approach. 01:08:38.590 --> 01:08:53.434 align:middle line:90% 01:08:53.434 --> 01:08:56.149 align:middle line:90% So there's two steps always. 01:08:56.149 --> 01:08:58.229 align:middle line:84% On the left side, we compute the value functions, 01:08:58.229 --> 01:09:01.630 align:middle line:90% as we did above. 01:09:01.630 --> 01:09:04.170 align:middle line:84% And the right side, we write out the equilibrium conditions. 01:09:04.170 --> 01:09:15.149 align:middle line:90% 01:09:15.149 --> 01:09:17.189 align:middle line:84% If one of these is the Nash equilibrium level, 01:09:17.189 --> 01:09:18.090 align:middle line:90% it's a little tricky. 01:09:18.090 --> 01:09:20.430 align:middle line:84% So maybe just to simplify my life, 01:09:20.430 --> 01:09:25.090 align:middle line:84% let's assume that these are not equal to qNE. 01:09:25.090 --> 01:09:27.590 align:middle line:84% So the carrot and the stick are both different from the Nash 01:09:27.590 --> 01:09:30.910 align:middle line:84% equilibrium level, just it avoids this kind of issue 01:09:30.910 --> 01:09:33.529 align:middle line:84% we got before, where the best deviation is not a deviation. 01:09:33.529 --> 01:09:36.529 align:middle line:84% And let's just not worry about that. 01:09:36.529 --> 01:09:39.149 align:middle line:90% 01:09:39.149 --> 01:09:41.710 align:middle line:84% Now, I want us to conceptually understand 01:09:41.710 --> 01:09:43.430 align:middle line:90% what's happening here. 01:09:43.430 --> 01:09:46.069 align:middle line:84% The left-hand side is nothing to do with equilibrium. 01:09:46.069 --> 01:09:47.990 align:middle line:84% The left-hand side is just saying, 01:09:47.990 --> 01:09:50.830 align:middle line:84% if this is the strategy we were to play, what would everyone's 01:09:50.830 --> 01:09:53.990 align:middle line:90% payoffs be in each state? 01:09:53.990 --> 01:09:55.572 align:middle line:84% Then using that information, we go 01:09:55.572 --> 01:09:57.030 align:middle line:84% to the right side, which is saying, 01:09:57.030 --> 01:09:59.630 align:middle line:84% is it actually an equilibrium for people to play this way? 01:09:59.630 --> 01:10:02.790 align:middle line:84% But notice, we can ask, what would my payoff be in each state 01:10:02.790 --> 01:10:04.470 align:middle line:90% if this is the strategy we play? 01:10:04.470 --> 01:10:07.190 align:middle line:84% I can ask that question whether or not that strategy is actually 01:10:07.190 --> 01:10:08.670 align:middle line:90% in equilibrium. 01:10:08.670 --> 01:10:11.430 align:middle line:84% So let's try to write down our value function formulas here. 01:10:11.430 --> 01:10:18.710 align:middle line:84% So we have Vc and Vs. So we have a carrot state and a stick 01:10:18.710 --> 01:10:19.590 align:middle line:90% state. 01:10:19.590 --> 01:10:20.610 align:middle line:90% And we want to compute. 01:10:20.610 --> 01:10:23.910 align:middle line:84% Remember, the interpretation is the average discounted payoff 01:10:23.910 --> 01:10:26.990 align:middle line:84% from today onward if we're in this state. 01:10:26.990 --> 01:10:29.230 align:middle line:84% So let's say we're in the c state. 01:10:29.230 --> 01:10:30.850 align:middle line:84% What is our payoff going to be today. 01:10:30.850 --> 01:10:32.453 align:middle line:90% If we all follow the c state? 01:10:32.453 --> 01:10:34.370 align:middle line:84% We can use the same notation that was up here? 01:10:34.370 --> 01:10:39.670 align:middle line:90% 01:10:39.670 --> 01:10:41.850 align:middle line:90% Well, it's f of qc. 01:10:41.850 --> 01:10:48.830 align:middle line:90% 01:10:48.830 --> 01:10:51.950 align:middle line:84% And we'll multiply that by 1 minus delta. 01:10:51.950 --> 01:10:57.170 align:middle line:84% And then if we follow f of qc, what happens tomorrow? 01:10:57.170 --> 01:10:59.510 align:middle line:90% What state are we in? 01:10:59.510 --> 01:11:01.350 align:middle line:90% We're still in the carrot state. 01:11:01.350 --> 01:11:05.130 align:middle line:90% So now we get delta v of c. 01:11:05.130 --> 01:11:08.310 align:middle line:90% 01:11:08.310 --> 01:11:11.590 align:middle line:84% Now, this is the one that's a bit trickier. 01:11:11.590 --> 01:11:14.150 align:middle line:84% If we're in the stick state, today we 01:11:14.150 --> 01:11:20.350 align:middle line:90% get 1 minus delta f of qs. 01:11:20.350 --> 01:11:23.090 align:middle line:84% But then what do we get on the right-hand side here? 01:11:23.090 --> 01:11:28.150 align:middle line:90% 01:11:28.150 --> 01:11:29.090 align:middle line:90% Yeah? 01:11:29.090 --> 01:11:31.470 align:middle line:90% STUDENT: Is it delta qc? 01:11:31.470 --> 01:11:32.330 align:middle line:90% IAN BALL: Exactly. 01:11:32.330 --> 01:11:34.372 align:middle line:84% Because if we do what we're supposed to do today, 01:11:34.372 --> 01:11:36.890 align:middle line:84% we end up in the carrot state the following period. 01:11:36.890 --> 01:11:40.790 align:middle line:90% 01:11:40.790 --> 01:11:43.190 align:middle line:84% This is saying, what is my average discounted payout 01:11:43.190 --> 01:11:44.850 align:middle line:84% from today onward in the stick state? 01:11:44.850 --> 01:11:46.550 align:middle line:84% Well, it's the payoff I get today 01:11:46.550 --> 01:11:49.270 align:middle line:84% plus delta-- we've got a discount-- 01:11:49.270 --> 01:11:52.570 align:middle line:84% times my average discounted payoff from tomorrow onwards. 01:11:52.570 --> 01:11:53.110 align:middle line:90% Yes? 01:11:53.110 --> 01:11:55.270 align:middle line:84% STUDENT: So value function is assuming that you 01:11:55.270 --> 01:11:56.720 align:middle line:90% follow the strategy profile. 01:11:56.720 --> 01:11:57.470 align:middle line:90% IAN BALL: Exactly. 01:11:57.470 --> 01:11:58.610 align:middle line:84% So that's the key thing over here. 01:11:58.610 --> 01:11:59.690 align:middle line:84% We're not saying about equilibrium. 01:11:59.690 --> 01:12:00.730 align:middle line:84% We're not talking about deviations. 01:12:00.730 --> 01:12:02.990 align:middle line:84% We're just saying given the strategy profile, 01:12:02.990 --> 01:12:06.050 align:middle line:84% let's compute the value functions in each state. 01:12:06.050 --> 01:12:09.910 align:middle line:90% 01:12:09.910 --> 01:12:11.790 align:middle line:84% Now in general, you can see we're 01:12:11.790 --> 01:12:15.313 align:middle line:84% going to have a system of equations with however 01:12:15.313 --> 01:12:16.230 align:middle line:90% many states there are. 01:12:16.230 --> 01:12:17.110 align:middle line:90% Call it n. 01:12:17.110 --> 01:12:18.830 align:middle line:84% No, that's the number of players. 01:12:18.830 --> 01:12:19.950 align:middle line:90% t-- k. 01:12:19.950 --> 01:12:22.330 align:middle line:84% If there's k states-- so I'll write this down here. 01:12:22.330 --> 01:12:26.550 align:middle line:84% If there's k states, in general, we have k equations and k 01:12:26.550 --> 01:12:30.590 align:middle line:84% unknowns because we have to specify the value in each state, 01:12:30.590 --> 01:12:32.490 align:middle line:84% and we have a formula for each of these. 01:12:32.490 --> 01:12:38.990 align:middle line:84% So we get k states and k unknowns. 01:12:38.990 --> 01:12:41.955 align:middle line:84% And, as you know, this is going to be a linear system. 01:12:41.955 --> 01:12:44.330 align:middle line:84% And it turns out it will actually always have a solution. 01:12:44.330 --> 01:12:46.455 align:middle line:84% That's something that's a math result you can show. 01:12:46.455 --> 01:12:49.230 align:middle line:84% So this is always a way that we can compute what the values are 01:12:49.230 --> 01:12:50.892 align:middle line:90% going to be in each state. 01:12:50.892 --> 01:12:53.350 align:middle line:84% Notice that this is kind of-- there's something very clever 01:12:53.350 --> 01:12:54.150 align:middle line:90% going on. 01:12:54.150 --> 01:12:56.950 align:middle line:84% Because if I just started in one state-- 01:12:56.950 --> 01:12:58.690 align:middle line:84% let's suppose I didn't use this approach, 01:12:58.690 --> 01:13:00.590 align:middle line:84% and I just wanted to say-- let's say 01:13:00.590 --> 01:13:03.587 align:middle line:84% we had a really complicated-- let's say we had 10 states. 01:13:03.587 --> 01:13:06.170 align:middle line:84% And I wanted to say, what is my payoff starting in this state? 01:13:06.170 --> 01:13:07.290 align:middle line:84% Well, I'd have to keep going through it. 01:13:07.290 --> 01:13:09.450 align:middle line:84% I'd say, well, I do this today, then I go to this state. 01:13:09.450 --> 01:13:10.650 align:middle line:84% Then I do this tomorrow, I go to this state. 01:13:10.650 --> 01:13:11.210 align:middle line:90% I go to this state. 01:13:11.210 --> 01:13:12.130 align:middle line:90% I go to this state. 01:13:12.130 --> 01:13:14.380 align:middle line:84% You might think I'd have to compute the whole infinite 01:13:14.380 --> 01:13:17.430 align:middle line:84% horizon of all the different states I could pass through. 01:13:17.430 --> 01:13:19.430 align:middle line:84% What's kind of magical about this approach 01:13:19.430 --> 01:13:21.710 align:middle line:84% is we've just reduced all these computations 01:13:21.710 --> 01:13:23.750 align:middle line:84% to a single system of linear equations, which 01:13:23.750 --> 01:13:24.950 align:middle line:90% is much easier to solve. 01:13:24.950 --> 01:13:27.910 align:middle line:84% I don't have to explicitly compute my entire path 01:13:27.910 --> 01:13:29.055 align:middle line:90% through the automaton. 01:13:29.055 --> 01:13:30.930 align:middle line:84% I just have to solve the system of equations. 01:13:30.930 --> 01:13:32.850 align:middle line:84% And this is related to some kind of nice math. 01:13:32.850 --> 01:13:33.390 align:middle line:90% Yeah? 01:13:33.390 --> 01:13:35.950 align:middle line:84% STUDENT: Each unknown corresponds [INAUDIBLE] 01:13:35.950 --> 01:13:37.680 align:middle line:84% one-to-one correspondence between states. 01:13:37.680 --> 01:13:38.430 align:middle line:90% IAN BALL: Exactly. 01:13:38.430 --> 01:13:40.190 align:middle line:84% So formally, you can think of v as a value 01:13:40.190 --> 01:13:43.730 align:middle line:84% function that's defined on the domain of the set of states. 01:13:43.730 --> 01:13:46.650 align:middle line:84% So a function needs to specify a number for every state, 01:13:46.650 --> 01:13:50.588 align:middle line:84% and that's precisely the value we get starting in that state. 01:13:50.588 --> 01:13:52.630 align:middle line:84% And if we solve this-- we can pretty easily solve 01:13:52.630 --> 01:13:54.922 align:middle line:84% this first equation just to make sure this makes sense. 01:13:54.922 --> 01:13:58.710 align:middle line:84% If we subtract delta Vc from the right side, 01:13:58.710 --> 01:14:04.230 align:middle line:84% then we get 1 minus delta Vc equals 1 minus delta f of qc. 01:14:04.230 --> 01:14:07.190 align:middle line:84% So that tells us that Vc equals f of qc. 01:14:07.190 --> 01:14:09.530 align:middle line:90% Does that seem right? 01:14:09.530 --> 01:14:10.030 align:middle line:90% Yeah. 01:14:10.030 --> 01:14:12.670 align:middle line:84% Because if we're in state c, we never leave. 01:14:12.670 --> 01:14:15.000 align:middle line:84% So every period we get f of qc, and that's 01:14:15.000 --> 01:14:16.250 align:middle line:90% our average discounted payoff. 01:14:16.250 --> 01:14:16.630 align:middle line:90% Question? 01:14:16.630 --> 01:14:17.190 align:middle line:90% Yeah? 01:14:17.190 --> 01:14:19.790 align:middle line:84% STUDENT: I mean, because we know we stayed there-- or for Vc, 01:14:19.790 --> 01:14:23.813 align:middle line:84% we know we stay in state c if we do it the first time. 01:14:23.813 --> 01:14:25.230 align:middle line:84% Would there have been [INAUDIBLE]. 01:14:25.230 --> 01:14:27.670 align:middle line:84% Like, mathematically, you would get the same answer [INAUDIBLE] 01:14:27.670 --> 01:14:29.110 align:middle line:90% instead of writing [INAUDIBLE]. 01:14:29.110 --> 01:14:29.830 align:middle line:90% IAN BALL: You're right. 01:14:29.830 --> 01:14:30.990 align:middle line:84% You could have just jumped-- you're 01:14:30.990 --> 01:14:31.950 align:middle line:84% saying you can basically see what 01:14:31.950 --> 01:14:33.430 align:middle line:90% the payoff is going to be here. 01:14:33.430 --> 01:14:34.013 align:middle line:90% STUDENT: Yeah. 01:14:34.013 --> 01:14:37.030 align:middle line:90% But delta times f of Vc. 01:14:37.030 --> 01:14:38.790 align:middle line:90% IAN BALL: You could have, yeah. 01:14:38.790 --> 01:14:41.090 align:middle line:84% But it's down here where the benefit comes in. 01:14:41.090 --> 01:14:44.350 align:middle line:84% Because here, and more generally, what happens tomorrow 01:14:44.350 --> 01:14:45.790 align:middle line:90% depends on a third state. 01:14:45.790 --> 01:14:47.873 align:middle line:84% So this is the general approach that always works. 01:14:47.873 --> 01:14:48.582 align:middle line:90% But you're right. 01:14:48.582 --> 01:14:51.290 align:middle line:84% In this case, if we're in what's called an absorbing state, 01:14:51.290 --> 01:14:52.990 align:middle line:84% once we're there, we never leave, it's pretty easy 01:14:52.990 --> 01:14:54.130 align:middle line:90% to calculate what happens. 01:14:54.130 --> 01:14:56.130 align:middle line:84% We don't need to worry about what happens in any other state. 01:14:56.130 --> 01:14:58.130 align:middle line:84% But when we have transitions between the states, 01:14:58.130 --> 01:14:59.290 align:middle line:90% then it becomes trickier. 01:14:59.290 --> 01:15:00.910 align:middle line:90% But yeah, exactly right. 01:15:00.910 --> 01:15:03.030 align:middle line:84% But this will never give you the wrong answer. 01:15:03.030 --> 01:15:03.910 align:middle line:90% Yeah. 01:15:03.910 --> 01:15:04.450 align:middle line:90% OK. 01:15:04.450 --> 01:15:06.630 align:middle line:84% So now let's write down our equilibrium conditions. 01:15:06.630 --> 01:15:10.987 align:middle line:84% Once again, we have a condition for each state. 01:15:10.987 --> 01:15:13.070 align:middle line:84% But remember, we're actually capturing a lot here. 01:15:13.070 --> 01:15:14.653 align:middle line:84% Because in each state, there are a lot 01:15:14.653 --> 01:15:17.050 align:middle line:84% of different one-shot deviations we could use. 01:15:17.050 --> 01:15:19.750 align:middle line:84% So when we say equilibrium condition, what we really mean 01:15:19.750 --> 01:15:21.430 align:middle line:84% is we're saying in each state we want 01:15:21.430 --> 01:15:25.562 align:middle line:84% to check that no one-shot deviation is profitable. 01:15:25.562 --> 01:15:27.270 align:middle line:84% And the way we can check that is to check 01:15:27.270 --> 01:15:29.830 align:middle line:84% that the best one-shot deviation is not profitable. 01:15:29.830 --> 01:15:32.020 align:middle line:84% Because if the best deviation is not profitable, 01:15:32.020 --> 01:15:33.770 align:middle line:84% then none of the deviations is profitable. 01:15:33.770 --> 01:15:35.210 align:middle line:84% There may be some really bad deviations. 01:15:35.210 --> 01:15:36.250 align:middle line:90% We don't care about those. 01:15:36.250 --> 01:15:38.290 align:middle line:84% We only care about the most attractive deviations. 01:15:38.290 --> 01:15:40.415 align:middle line:84% And that's exactly what we're going to put in here. 01:15:40.415 --> 01:15:42.640 align:middle line:84% So in the carrot state, we need Vc greater than 01:15:42.640 --> 01:15:44.390 align:middle line:84% or equal to something, and Vs greater than 01:15:44.390 --> 01:15:45.310 align:middle line:90% or equal to something. 01:15:45.310 --> 01:15:46.935 align:middle line:84% Eventually, we'll substitute things in. 01:15:46.935 --> 01:15:49.150 align:middle line:84% But I want to just do it conceptually this way. 01:15:49.150 --> 01:15:53.990 align:middle line:84% So in the carrot state, once again, if we deviate 01:15:53.990 --> 01:15:55.390 align:middle line:90% we're going to get punished. 01:15:55.390 --> 01:15:57.670 align:middle line:84% And the punishment we experience doesn't depend 01:15:57.670 --> 01:15:59.250 align:middle line:90% on the way in which we deviate. 01:15:59.250 --> 01:16:01.870 align:middle line:84% So we might as well choose the deviation that maximizes 01:16:01.870 --> 01:16:03.455 align:middle line:90% our flow payoff today. 01:16:03.455 --> 01:16:04.830 align:middle line:84% And what is our flow payoff going 01:16:04.830 --> 01:16:07.990 align:middle line:84% to be if we choose the deviation that maximizes our flow payoff? 01:16:07.990 --> 01:16:09.790 align:middle line:90% We have a function for this. 01:16:09.790 --> 01:16:11.770 align:middle line:90% It's g of qc. 01:16:11.770 --> 01:16:17.870 align:middle line:90% 01:16:17.870 --> 01:16:22.390 align:middle line:84% So this says, if I choose the best deviation today, 01:16:22.390 --> 01:16:23.630 align:middle line:90% this is what I get today. 01:16:23.630 --> 01:16:26.730 align:middle line:84% If I deviate, what state are we going to be in tomorrow? 01:16:26.730 --> 01:16:30.210 align:middle line:90% 01:16:30.210 --> 01:16:30.710 align:middle line:90% Yeah? 01:16:30.710 --> 01:16:31.530 align:middle line:90% STUDENT: Vc. 01:16:31.530 --> 01:16:32.530 align:middle line:90% IAN BALL: Yeah, exactly. 01:16:32.530 --> 01:16:35.423 align:middle line:84% So now we're going to get the s over here. 01:16:35.423 --> 01:16:37.090 align:middle line:84% And now I think this is the hardest one, 01:16:37.090 --> 01:16:39.030 align:middle line:90% and we'll stop here. 01:16:39.030 --> 01:16:41.030 align:middle line:90% Now we're in the stick state. 01:16:41.030 --> 01:16:43.690 align:middle line:84% We're going to choose our best deviation in the stick state. 01:16:43.690 --> 01:16:48.630 align:middle line:90% 01:16:48.630 --> 01:16:50.790 align:middle line:84% Again, I'm only allowed to write it this way 01:16:50.790 --> 01:16:53.110 align:middle line:84% because the punishment I experience doesn't 01:16:53.110 --> 01:16:56.230 align:middle line:84% depend on exactly which quantity I deviate to. 01:16:56.230 --> 01:16:59.250 align:middle line:84% If the deviation I chose affected the punishment, 01:16:59.250 --> 01:17:00.165 align:middle line:90% then we'd have to-- 01:17:00.165 --> 01:17:01.790 align:middle line:84% things would be a lot more complicated. 01:17:01.790 --> 01:17:04.610 align:middle line:84% But all we have to ask is, am I deviating or not? 01:17:04.610 --> 01:17:07.365 align:middle line:84% However I deviate, it's treated the same way in the future. 01:17:07.365 --> 01:17:08.990 align:middle line:84% So I might as well choose the deviation 01:17:08.990 --> 01:17:10.530 align:middle line:90% that maximizes my flow payoff. 01:17:10.530 --> 01:17:13.790 align:middle line:90% 01:17:13.790 --> 01:17:22.310 align:middle line:84% And then what do we get on the right-hand side? 01:17:22.310 --> 01:17:23.390 align:middle line:90% Delta times what? 01:17:23.390 --> 01:17:27.470 align:middle line:90% What happens if I deviate? 01:17:27.470 --> 01:17:28.130 align:middle line:90% Vs again. 01:17:28.130 --> 01:17:32.372 align:middle line:90% 01:17:32.372 --> 01:17:35.090 align:middle line:84% So notice when we were defining these, 01:17:35.090 --> 01:17:37.097 align:middle line:84% we always had Vc here because we were 01:17:37.097 --> 01:17:39.430 align:middle line:84% looking at what happens if I do what I'm supposed to do. 01:17:39.430 --> 01:17:41.170 align:middle line:84% And therefore, we go to the carrot state. 01:17:41.170 --> 01:17:43.170 align:middle line:84% When we're looking at the equilibrium condition, 01:17:43.170 --> 01:17:45.510 align:middle line:84% we're considering deviations on the right-hand side. 01:17:45.510 --> 01:17:48.630 align:middle line:84% Whenever I deviate, regardless of which state I'm in, 01:17:48.630 --> 01:17:50.830 align:middle line:90% I go to the stick state. 01:17:50.830 --> 01:18:02.550 align:middle line:90% 01:18:02.550 --> 01:18:06.750 align:middle line:84% Now, let's just understand what I think is kind of tricky here, 01:18:06.750 --> 01:18:10.430 align:middle line:84% is this is actually capturing a lot. 01:18:10.430 --> 01:18:13.430 align:middle line:90% If I deviate today-- 01:18:13.430 --> 01:18:15.310 align:middle line:84% so let's just understand the anatomy 01:18:15.310 --> 01:18:19.430 align:middle line:90% of my one-shot deviation. 01:18:19.430 --> 01:18:28.050 align:middle line:84% So we have today, tomorrow, and then thereafter. 01:18:28.050 --> 01:18:31.270 align:middle line:90% 01:18:31.270 --> 01:18:36.390 align:middle line:84% If I deviate today to, say, qi prime in the stick state, 01:18:36.390 --> 01:18:38.743 align:middle line:84% we had a really simple formula on the right-hand side. 01:18:38.743 --> 01:18:40.910 align:middle line:84% But what's actually happening is pretty complicated. 01:18:40.910 --> 01:18:44.070 align:middle line:84% If I deviate today to qi prime and then 01:18:44.070 --> 01:18:46.350 align:middle line:84% return to my equilibrium strategy, 01:18:46.350 --> 01:18:48.090 align:middle line:84% what am I going to produce tomorrow? 01:18:48.090 --> 01:18:56.870 align:middle line:90% 01:18:56.870 --> 01:18:58.150 align:middle line:90% [INAUDIBLE] want to guess? 01:18:58.150 --> 01:18:59.430 align:middle line:90% You see it? 01:18:59.430 --> 01:19:02.030 align:middle line:84% Well, tomorrow I go to the stick state. 01:19:02.030 --> 01:19:05.030 align:middle line:84% So I'm going to produce qs tomorrow. 01:19:05.030 --> 01:19:06.950 align:middle line:84% And then the day after that, I'm actually 01:19:06.950 --> 01:19:12.473 align:middle line:84% going to produce thereafter, I'm going to produce qc. 01:19:12.473 --> 01:19:14.890 align:middle line:84% So this one-shot deviation is actually really complicated. 01:19:14.890 --> 01:19:16.150 align:middle line:90% I'm in the stick state. 01:19:16.150 --> 01:19:17.190 align:middle line:90% I deviate. 01:19:17.190 --> 01:19:19.070 align:middle line:84% That means I stay in the stick state. 01:19:19.070 --> 01:19:21.030 align:middle line:84% Tomorrow, I'm following my equilibrium strategy 01:19:21.030 --> 01:19:23.390 align:middle line:84% in the stick state, so I choose qs. 01:19:23.390 --> 01:19:25.650 align:middle line:84% But if I choose qs and I'm in the stick state, 01:19:25.650 --> 01:19:27.850 align:middle line:84% then we transition to the carrot state the next day. 01:19:27.850 --> 01:19:30.070 align:middle line:84% And then I choose qc the next day. 01:19:30.070 --> 01:19:32.910 align:middle line:84% So the path of play following this deviation-- 01:19:32.910 --> 01:19:36.390 align:middle line:84% and then it's going to be qc forever after-- 01:19:36.390 --> 01:19:38.310 align:middle line:90% is actually really complicated. 01:19:38.310 --> 01:19:42.230 align:middle line:84% When we wrote our formula here, things looked much simpler. 01:19:42.230 --> 01:19:45.150 align:middle line:84% How is this formula reflecting this complicated path 01:19:45.150 --> 01:19:46.010 align:middle line:90% moving forward? 01:19:46.010 --> 01:19:48.410 align:middle line:90% 01:19:48.410 --> 01:19:48.910 align:middle line:90% Yeah? 01:19:48.910 --> 01:19:50.243 align:middle line:90% STUDENT: I just have a question. 01:19:50.243 --> 01:19:52.830 align:middle line:84% Why are sometimes s And c subscripts, other times 01:19:52.830 --> 01:19:56.030 align:middle line:90% superscripts? 01:19:56.030 --> 01:19:57.710 align:middle line:90% Good point. 01:19:57.710 --> 01:20:03.170 align:middle line:84% I think I want these to be superscripts. 01:20:03.170 --> 01:20:06.868 align:middle line:90% 01:20:06.868 --> 01:20:08.410 align:middle line:84% IAN BALL: There's not a great reason. 01:20:08.410 --> 01:20:11.993 align:middle line:84% I often had q sub i here, so I made this a superscript. 01:20:11.993 --> 01:20:14.410 align:middle line:84% But then I never have v sub i, so I made this a subscript. 01:20:14.410 --> 01:20:15.730 align:middle line:90% But everything's the same. 01:20:15.730 --> 01:20:16.230 align:middle line:90% Yeah? 01:20:16.230 --> 01:20:17.870 align:middle line:84% STUDENT: Also, was qi prime-- that 01:20:17.870 --> 01:20:20.790 align:middle line:84% could be an either stick or a carrot state, right? 01:20:20.790 --> 01:20:22.090 align:middle line:90% IAN BALL: I'm looking at this-- 01:20:22.090 --> 01:20:23.673 align:middle line:84% I'm looking specifically at this case. 01:20:23.673 --> 01:20:25.230 align:middle line:90% I'm in the stick state. 01:20:25.230 --> 01:20:27.270 align:middle line:90% I deviate. 01:20:27.270 --> 01:20:29.410 align:middle line:84% Because I deviate, I'm in the stick state tomorrow. 01:20:29.410 --> 01:20:31.910 align:middle line:84% So then I follow my strategy since it's a one-shot deviation 01:20:31.910 --> 01:20:33.490 align:middle line:90% and choose the stick quantity. 01:20:33.490 --> 01:20:35.750 align:middle line:84% But then the next period, I'm in the carrot state. 01:20:35.750 --> 01:20:36.930 align:middle line:90% And then I choose qc. 01:20:36.930 --> 01:20:39.230 align:middle line:84% So my point here is that a one-shot deviation 01:20:39.230 --> 01:20:42.510 align:middle line:84% can lead to a really complicated path of play. 01:20:42.510 --> 01:20:45.070 align:middle line:84% The reason I don't have to worry about it is Vs 01:20:45.070 --> 01:20:47.390 align:middle line:84% is already taking into account this future path. 01:20:47.390 --> 01:20:52.150 align:middle line:84% So this path here of SCCC is already 01:20:52.150 --> 01:20:54.870 align:middle line:84% reflected in the value function Vs. 01:20:54.870 --> 01:20:57.390 align:middle line:84% In fact, we can see over here Vs is precisely 01:20:57.390 --> 01:20:59.310 align:middle line:84% reflecting that I get the stick today and then 01:20:59.310 --> 01:21:00.365 align:middle line:90% the carrot forever after. 01:21:00.365 --> 01:21:02.990 align:middle line:84% So this is showing you the power of the value functions, that I 01:21:02.990 --> 01:21:05.673 align:middle line:84% don't really have to keep track of the total sequence of play. 01:21:05.673 --> 01:21:08.090 align:middle line:84% I just have to keep track of what state I get to tomorrow, 01:21:08.090 --> 01:21:09.465 align:middle line:84% and then it takes care of itself. 01:21:09.465 --> 01:21:10.290 align:middle line:90% Let me stop there. 01:21:10.290 --> 01:21:11.540 align:middle line:90% Sorry for going slightly over. 01:21:11.540 --> 01:21:14.500 align:middle line:84% And good luck on the midterm on Tuesday. 01:21:14.500 --> 01:21:17.000 align:middle line:90%